newton's thought experiment

The future of the cosmos probably was not on Isaac Newton's mind as he pondered the relation between a falling object on Earth and the Moon in orbit. He did, however, conduct a thought experiment. It went something like this: If he dropped a ball, it would descend toward Earth in free fall until it hit the ground. Launching that ball parallel to the ground from a cannon would propel it some distance, but Earth's gravity would still pull it downward in free fall. Hauling the cannon to a mountaintop would let him shoot the ball still farther. But what if he climbed the highest mountain and fired the ball fast enough--5 miles per second--so that it never touched the ground as Earth's surface curved beneath it? The ball would still fall toward Earth's center, but its fast sideways motion would keep it in a low "orbit." With this intuitive leap, Newton realized that all orbits in the solar system are never-ending free falls. Their motions are determined by the same laws that govern the flights of baseballs, the trajectories of rocks belched from a volcano, and the paths of other freely moving projectiles on Earth.

Newton acknowledged that many predecessors had set the stage for his insights. One was the German astronomer Johannes Kepler, who worked out the correct mathematical details of planetary orbits for the first time in 1609. His painstaking calculations and the observations of his mentor, the Danish astronomer Tycho Brahe, enabled Kepler to derive three key principles. First, planets move around the Sun not in circles, as Copernicus had thought, but in oval paths called ellipses. Second, planets move faster when closer to the Sun and slower when farther away, in such a fashion that their motions sweep out equal areas of their ellipses in equal times. Third, the orbital period of a planet--its year--depends predictably upon its distance from the Sun. We can apply those laws anywhere in the universe where planetary systems revolve around other stars. We can also use them to understand other basic systems, such as two stars that orbit each other closely. But the laws are too simplistic to hold in star clusters, galaxies, and groups of galaxies, where the complexities of gravity's dances require stronger analytical tools.

Newton showed that his laws of motion and gravitation lead to other families of trajectories in the solar system besides ellipses: parabolas, hyperbolas, and, of course, circles. (Gravitational nudges from other bodies in the solar system prevent any object from orbiting the Sun in a perfect circle.) These are "conic sections"-- (continued)

• History & Society
• Science & Tech
• Biographies
• Animals & Nature
• Geography & Travel
• Arts & Culture
• Games & Quizzes
• On This Day
• One Good Fact
• New Articles
• Lifestyles & Social Issues
• Philosophy & Religion
• Politics, Law & Government
• World History
• Health & Medicine
• Browse Biographies
• Birds, Reptiles & Other Vertebrates
• Bugs, Mollusks & Other Invertebrates
• Environment
• Fossils & Geologic Time
• Entertainment & Pop Culture
• Sports & Recreation
• Visual Arts
• Demystified
• Image Galleries
• Infographics
• Top Questions
• Britannica Kids
• Saving Earth
• Space Next 50
• Student Center
• Introduction
• The Newtonian conception of the universe
• The logical structure of Newtonian mechanics
• Relationism and absolutism
• Kant on incongruent counterparts

The question of motion

• The special theory of relativity
• The general theory of relativity
• The problem of the direction of time
• Thermodynamics
• The foundations of statistical mechanics
• The principle of superposition
• The measurement problem
• The theory of Bohm
• The theory of Ghirardi, Rimini, and Weber
• Nonlocality
• Quantum theory and the structure of space-time
• Quantum theory and the foundations of statistical mechanics

• Is mathematics a physical science?
• Why does physics work in SI units?

Our editors will review what you’ve submitted and determine whether to revise the article.

• Table Of Contents

Long before Kant, Newton himself designed a thought experiment to show that relationism must be false. What he hoped to establish was that relationism defeats itself, because there can be no relationist account of those properties of the world that relationism itself seeks to describe.

Consider a universe that consists entirely of two balls attached to opposite ends of a spring. Suppose that the length of the spring, in its relaxed—unstretched and uncompressed—configuration is L . Imagine also that there is some particular moment in the history of this universe at which (1) the length of the spring is greater than L and (2) there are no two material components of this universe whose distance from each other is changing with time—that is, there are no two material components whose relative velocity is anything other than zero. Suppose, finally, that one wishes to know something about the dynamical evolution of this universe in the immediate future: Will the spring oscillate or not?

In the conventional way of understanding Newtonian mechanics , whether the spring will oscillate depends on whether, and to what extent, at the moment in question, it is rotating with respect to absolute space . If the spring is stationary, it will oscillate, but if it is rotating at just the right speed, it will remain stretched. The trouble for the relationist is that relationism cannot accommodate rotation with respect to absolute space. The relationist, who must hold that there is no matter of fact about whether the spring is rotating, cannot predict whether the spring will oscillate or explain why some such springs eventually begin to oscillate and others do not.

The standard relationist response to this argument is to point out that the actual universe contains a great deal more than the hypothetical universe of Newton’s thought experiment. The idea is that there is myriad other stuff that might serve as a concrete material stand-in for absolute space—a concrete material system of reference on which a fully relationist analysis of rotation could be based.

The Austrian physicist Ernst Mach (1838–1916), speaking in absolutist language, pointed out that the universe itself appears not to be rotating (that is, the total angular momentum of the actual universe appears to be zero). As far as the actual universe is concerned, therefore, rotation with respect to absolute space amounts to precisely the same thing as rotation with respect to the universe’s own centre of gravity or to its “bulk mass” or to its “fixed stars” (which were thought, in Mach’s time, to make up the overwhelming majority of the universe’s bulk mass). Mach’s proposal, then, was that rotation simply be defined as rotation with respect to the bulk mass of the universe and that motion in general simply be defined as motion with respect to the bulk mass of the universe. If this proposal were accepted, then a relationist theory of the motions of particles could be formulated as F = ma , where a is understood as acceleration with respect to the bulk mass of the universe.

Note that the cost to relationism in this case, as in the case of the relationist response to the argument from incongruent counterparts, is nonlocality. Whereas the Newtonian law of motion governs particles across the face of an absolute space that is always and everywhere exactly where the particles themselves are, what the Machian laws govern are merely the rates at which spatial relations (distances) between different particles change over time—and these particles may in principle be arbitrarily far apart ( see below Nonlocality ).

There is at least one other way of realizing the relationist’s aspirations in the context of a classical mechanics of the motions of particles. The idea would be not to look for a concrete material stand-in for absolute space but to discard systematically the commitments of Newtonian mechanics regarding absolute space that do not bear directly on the rates at which distances between particles change over time, keeping all and only those that do.

Once the problem is conceived in these terms, its solution is perfectly straightforward. A complete relationist theory of the motions of particles could be formulated as follows:

A given history of changes in the distances between certain particles is physically possible if, and only if, it can be conceived to take place within Newtonian absolute space in such a way as to satisfy F = ma .

This theory, like Mach’s, satisfies all of the standard relationist desiderata: it is exclusively concerned with changes in the distances between particles over time; it makes no assertions about the motion of a single particle alone in the universe or about the motion of the universe’s bulk mass; and it is invariant under all transformations that leave the time-evolutions of interparticle distances invariant.

Unlike Mach’s theory, however, this one reproduces all of the consequences of Newtonian mechanics for the time-evolutions of interparticle distances. It can explain why the spring of Newton’s thought experiment does or does not oscillate , because it need not assume that the total angular momentum of the universe is zero. Although the theory is no less nonlocal than Mach’s, it entails that the law of motion governing isolated subsystems of the universe will make no reference to what is going on in the rest of the universe.

It is clear that the empiricist considerations that have been brought to bear on questions about the nature of space also have implications for the nature of time. Note, first of all, that one’s position within “absolute time” is no more detectable than one’s location within absolute space. Therefore, from an empiricist perspective, there cannot be any matter of fact about what absolute time it currently is. Mach reasoned, moreover, that there can be no direct observational access to the lengths of intervals of time; the most that can be determined is whether a given event occurs before, after, or simultaneously with another event.

In Newtonian mechanics, a “clock” (or a “good clock”) is a physical system with a certain sort of dynamical structure. From a relationist perspective, whether something is a clock (or a good clock) has nothing to do with correlations between the configuration of the clock face and “what time it is” or between changes in the configuration of the clock face and “how much time has passed”—since, for a relationist, there are no facts about what time it is or about how much time a certain process takes. A good clock is simply a physical system with parts whose positions are correlated with the physical properties of the rest of the universe by means of a simple and powerful law. To the extent that time intervals are even intelligible, on this view, they are not measured but rather defined by changes in clock faces.

The technique used above for fashioning a relationist theory of space can be applied more generally to design a relationist theory of both space and time. That is, one proceeds by systematically discarding the commitments of Newtonian mechanics regarding absolute space and absolute time that do not bear directly on sequences of interparticle distances, keeping only those that do.

The resulting theory can be formulated as follows:

A given history of changes in the distances between certain particles is physically possible if, and only if, it can be conceived to take place within Newtonian absolute space-time in such a way as to satisfy F = ma .

Naturally, the concluding points in the preceding section—about the empirical equivalence of the relationist theory to Newtonian mechanics, about locality, and about the applicability of the theory to isolated subsystems of the universe—apply also to the relationist theory of space and time.

• Previous Article
• Next Article

The flight of Newton's cannonball

Electronic mail: [email protected]

• Article contents
• Figures & tables
• Supplementary Data
• Peer Review
• Reprints and Permissions
• Cite Icon Cite
• Search Site

W. Dean Pesnell; The flight of Newton's cannonball. Am. J. Phys. 1 May 2018; 86 (5): 338–343. https://doi.org/10.1119/1.5027489

Download citation file:

• Ris (Zotero)
• Reference Manager

Newton's Cannon is a thought experiment used to motivate orbital motion. Cannonballs were fired from a high mountain at increasing muzzle velocity until they orbit the Earth. We will use the trajectories of these cannonballs to describe the shape of orbital tunnels that allow a cannonball fired from a high mountain to pass through the Earth. A sphere of constant density is used as the model of the Earth to take advantage of the analytic solutions for the interior trajectories that exist for that model. For the example shown, the cannonball trajectories that pass through the Earth intersect near the antipodal point of the cannon.

Citing articles via

Submit your article.

Sign up for alerts

• Online ISSN 1943-2909
• Print ISSN 0002-9505
• For Researchers
• For Librarians
• For Advertisers
• Our Publishing Partners  
• Physics Today
• Conference Proceedings
• Special Topics

pubs.aip.org

• Privacy Policy
• Terms of Use

Connect with AIP Publishing

This feature is available to subscribers only.

Sign In or Create an Account

How Newton Derived the Shape of Earth

To argue for universal gravitation, newton had to become a “geodesist.”.

Isaac Newton’s landmark 1687 work, “Philosophiæ Naturalis Principia Mathematica,” laid the groundwork for classical mechanics, describing the laws of gravitation and predicting astronomical phenomena, like the movement of planets. The work changed physics.

But buried in the Principia is an often-overlooked triumph: Newton’s derivation of Earth’s figure — that is, the calculation of its shape, size, and surface gravity variation, part of a field later known as geodesy — which was crucial to his argument for universal gravitation. Here, I reconstruct Newton’s derivation and its significance 1 .

Newton’s derivation

Newton began his quantitative derivation of Earth’s figure in 1686, after learning about work by the French physicist Jean Richer. In 1671, Richer had traveled to Cayenne, the capital of French Guiana in South America, and experimented with a pendulum clock. Richer found that the clock, calibrated to Parisian astronomical time (48°40’ latitude), lost an average of 2.5 minutes per day in Cayenne (5° latitude). This was surprising, but it could be explained by the theory of centrifugal motion, recently developed by Christian Huygens: The theory suggested that the centrifugal effect is strongest at the equator, so the net effective surface gravity would decrease as you moved from Paris to Cayenne 2 .

Newton accepted Huygens’s theory but realized it meant something strange: If Earth is a sphere and its centrifugal effect is strongest at the equator, gravity would vary across Earth’s surface, and the ocean would bulge up at the equator — a proposition that Newton considered absurd.

To resolve this, he proposed that the solid Earth had behaved like a fluid throughout its formation, gradually bulging up at the equator because of the centrifugal effect. He proposed modeling planets as rotating fluids in equilibrium, where the planet’s shape is stable while the force generated by its rotational motion, and the gravitational attraction between its particles, acts on it 3 .

To derive Earth’s figure based on this theory, Newton first had to calculate the ratio between gravitational acceleration and centrifugal force at the equator (a 1-to-290.8 ratio), based on the period of Earth’s diurnal rotation and estimates of Earth’s equatorial diameter. Newton knew the length of two meridional arcs that he could use for this calculation, measured by surveyors in England and France. He calculated gravitational acceleration at the equator from Richer’s pendulum measurements at 48°50’ latitude, extrapolating the corresponding value at the equator of a homogeneous sphere 4 .

Equipped with the ratio, Newton faced a problem: How could he express mathematically that a rotating fluid, whose constituents attract according to a certain law of gravity, is in a state of equilibrium? To answer this, Newton used an ingenious thought experiment, which he had developed in his 1685 “Liber Secundus” manuscript 5 : A rotating body is in a state of hydrostatic equilibrium if the weight of water in two channels x and y, where x connects the equator to Earth’s center and y connects Earth’s center to one of the poles, is identical. Since x is affected by the centrifugal effect, equilibrium is fulfilled if the overall centrifugal “pull” on the equator is compensated by a change to the figure. In other words, the equatorial regions need to “bulge up” and the poles “flatten” to such an extent that the total weight of x and y (i.e., net gravitational attraction toward the center) is the same. This attempt to define hydrostatic equilibrium was later named the “principle of canals 6 ” (Fig. 1).

Newton then used his theory of gravitational attraction to derive the figure that a rotating body would need to have to balance the net attraction on the two columns — more precisely, the ratio between equatorial diameter and polar axis that would fulfill equilibrium. To calculate this, Newton determined the ratio 7 between polar and equatorial surface gravity for the simpler case of a non-rotating oblate figure, with the axis-diameter ratio of 100-to-101, arriving at 501-to-500.

If this result is multiplied by the length of the two fluid columns (100-to-101), we obtain the ratio of 501-to-505 between the net gravitational forces acting on the polar and equatorial fluid columns. Therefore, the net gravitational force acting on the equatorial fluid column is greater than the corresponding net force acting on the polar fluid column by a magnitude of 4-to-505. So, if a spheroid with the dimensions of 100-to-101 is rotating and in a state of hydrostatic equilibrium, the ratio between equatorial surface gravity and centrifugal force must be 4-to-505.

Since Newton’s premise was that Earth is in a state of hydrostatic equilibrium, he extended this thought experiment to Earth. For his previously determined 1-to-290.1 ratio between equatorial surface gravity and centrifugal force, he calculated a corresponding polar axis and equatorial diameter ratio of 689-to-692. He concluded that Earth, modeled as a homogeneous spheroid that rotates with uniform angular velocity, must have polar and equatorial axes with a length ratio of 689-to-692 to be in a state of hydrostatic equilibrium.

He then calculated the effective surface gravity for this model of Earth — indicated by the length of the seconds-pendulum — to vary as the square of the sine of the latitude 8 . By deriving a general latitudinal variation, he was no longer just concerned with the length ratio between the equatorial diameter and polar axis; instead, he was modeling Earth’s overall figure — an oblate ellipsoid with an ellipticity of 3-to-692 (about 1-to-230.7). Using the pendulum length at 48°40’ astronomical latitude in Paris as a reference point, he predicted that the pendulum has to be shortened by 81/1000 and 89/1000 inches in Gorée and Cayenne, respectively, to preserve its period — close but still inaccurate approximations of the measurements (100/1000 and 125/1000 inches).

Earth’s figure and universal gravitation

Clearly, Newton invested considerable effort in deriving Earth’s figure and latitudinal variation in surface gravity. Besides presenting a novel definition for the hydrostatic equilibrium of rotating bodies, these results presumed Newton’s theory of gravitational attraction. His predictions only hold if all of Earth’s constituent particles mutually attract. Hence, his predictions offered a test for the most fundamental and novel assumption in Newton’s theory of gravitation: that gravity acts universally between all particles of matter. In fact, as George Smith showed, these predictions are the Principia’s only such test 9 .

Newton was aware of this. His editor Roger Cotes kept pushing him to revise the geodetic results in light of new data 10 , and in the second edition of the Principia, Newton revised Earth’s ellipticity from a 689-692 ratio to a 1-to-230 ratio and added a table with detailed predictions of measurements for surface gravity and surface curvature 11 . Newton revised these predictions again for the third edition (Fig. 2).

Were Newton’s geodetic predictions accurate enough to reflect their importance in his argument for universal gravitation? On a naïve reading, the answer is no. When the third and last edition of the Principia was published, Newton had access to one arc measurement of the latitudinal variation in the length of 1° of meridian and five pendulum measurements of the variation of surface gravity with latitude. The arc measurement disagreed with Newton’s predictions, seeming to indicate that Earth is an oblong, rather than oblate, spheroid. Out of existing pendulum measurements, only Jean Richer’s seemed similar, but even that still disagreed with Newton’s prediction. The prediction also does not match current data: Satellite measurements indicate that Earth’s ellipticity has a ratio of 1-to-298.257223563 12 .

However, such a pessimistic view of Newton’s geodetic work misses important nuances of the Principia. As George Smith has argued, the Principia not only proposes theoretical predictions, but a methodology of testing through approximation. Newton accepted that his predictions would likely be inaccurate because he relied on uncertain background hypotheses when deriving them. The success of the universal theory of gravitation, then, should not be measured by the immediate agreement between initial predictions and measurements. Rather, Newton intended that his theory be tested on how well it could guide adjustments to background hypotheses, leading to converging measurements 13 .

In other words, Newton did not aim to establish Earth’s figure once and for all. Rather, he gave approximations, which would allow for adjustments to the assumptions he made in his derivation. With Newton’s early derivation, for example, he assumed the rotating Earth has a homogeneous density. But when his predictions and Richer’s measurements in Cayenne and Gorée disagreed, he modified this assumption in the first edition of the Principia. If Earth is denser at its center, he suggested, the ellipticity of Earth’s equilibrium figure and its surface gravity variation will differ.

With this methodology, Newton passed the torch to future researchers, inviting them to develop hypotheses that would work with these initial measurements, and could then be tested with increasingly precise measurements 14 .

In line with Newton’s methodology, geodesists eventually produced convergent measurements of Earth’s ellipticity based on variation in latitudinal surface gravity and curvature. For about two and a half centuries, they used the theories of gravitation and hydrostatic equilibrium to model Earth’s figure, motion, and constitution, and gradually revised these parameters in light of new measurements. By 1909, all major ellipticity measurements converged within 297.6±0.9, implying that density increased inward 15 . By 1926, Viennese astronomer Samuel Oppenheim concluded that these results offered overwhelming evidence for Newtonian gravity on Earth, vindicating both Newton’s theory of gravitation and his methodology 16 .

For Ohnesorge’s full essay and his reconstruction of Newton’s derivation, visit   go.aps.org/geodesy . Learn more about Ohnesorge’s research at   mohnesorgehps.com .

Other Sources

Subrahmanyan Chandrasekhar, Newton’s Principia for the Common Reader (Clarendon Press, 2003).

Alexis Claude Clairaut, "I. An Inquiry Concerning the Figure of Such Planets as Revolve about an Axis," Philosophical Transactions of the Royal Society of London 40, 449 (1738): 277–306.

John Greenberg, ‘Isaac Newton and the Problem of the Earth’s Shape’. Archive for History of Exact Sciences 49, vol. 4 (1996): 371–91.

Mary Terrall, The Man Who Flattened the Earth: Maupertuis and the Sciences in the Enlightenment (Univ. of Chicago Press, 2002).

The Mathematical Works of Isaac Newton , ed. Derek Thomas Whiteside, vol. 6 (Cambridge University Press, 1974).

"Closing the Loop: Testing Newtonian Gravity, Then and Now," in Newton and Empiricism , ed. Zvi Biener and Eric Schliesser, vol. 262–353 (Oxford Univ. Press, 2014).

Miguel Ohnesorge

Miguel Ohnesorge is a doctoral student at the University of Cambridge and visiting Fellow at Boston University’s Philosophy of Geoscience Lab.

The only published discussions of the work’s methodological importance are in Schliesser and Smith 2000 and Smith 2014, who do not reconstruct Newton’s derivation. Todhunter 1873 and Greenberg 1996 discuss the derivation in plain English, without methodological contextualization. Chandrasekhar 2003 also reconstructs the derivation in modern algebra, which contains some minor but confusing mistakes and ambiguities.

Christian Huygens, "Discours de La Cause de La Pesanteur," in Traité de Lumierè Avec Discours de La Cause de La Pesanteur, ed. Christian Huygens (Leiden: Pierre Vander, 1690).

The Principia: Mathematical Principles of Natural Philosophy , trans. Bernhard Cohen and Ann Whitman (Univ. of California Press, 1999).

The Principia

Isaac Newton, De moto Corporum Liber Secundus (Cambridge Univ. Library, MS Add. 3990, 1685).

Isaac Todhunter, A History of the Mathematical Theories of Attraction and the Figure of the Earth, from the Time of Newton to That of Laplace , vol. 1 (London: Macmillan, 1873).

Newton’s derivation for the attraction of perfectly spherical and spheroidal compound bodies is given in Book 1, Prop. 91, coroll. 1 and 2 and reconstructed in its original notation at go.aps.org/geodesy .

Eric Schliesser and George E. Smith, "Huygens’s 1688 Report to the Directors of the Dutch East India Company on the Measurement of Longitude at Sea and the Evidence It Offered Against Universal Gravity," preprint (2000).

Cotes to Newton, The Correspondence of Isaac Newton , vol. 5 (1712): 232-236.

World Geodetic System (WGS) 84 Model.

George E. Smith, "Essay Review: Chandrasekhar’s Principia: Newton’s Principia for the Common Reader," Journal for the History of Astronomy 27, vol. 4 (1996): 353–62.

Miguel Ohnesorge, "Pluralizing Measurement: Physical Geodesy’s Measurement Problem and Its Resolution, 1880-1924," Studies in History and Philosophy of Science Part A (forthcoming).

A. Sommerfeld and Samuel Oppenheim, Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen: Fünfter Band: Physik (Vieweg+Teubner Verlag, 1926).

Join your Society

If you embrace scientific discovery, truth and integrity, partnership, inclusion, and lifelong curiosity, this is your professional home.

• Table of Contents
• Random Entry
• Chronological
• Editorial Information
• About the SEP
• Editorial Board
• How to Cite the SEP
• Special Characters
• Advanced Tools
• Support the SEP
• PDFs for SEP Friends
• Make a Donation
• SEPIA for Libraries
• Entry Contents

Bibliography

Academic tools.

• Friends PDF Preview
• Author and Citation Info
• Back to Top

Absolute and Relational Space and Motion: Post-Newtonian Theories

What is the nature of motion in physical theories and theorising, and is there any significance to the distinction between ‘absolute’ and ‘relative’ motion? In the companion article, on absolute and relational space and motion: classical theories , we discussed how such questions were addressed in the history of physics from Aristotle through to Newton and Leibniz. In this article, we explore the ways in which the selfsame issues have been taken up by contemporary authors, beginning with Mach, moving on to Einstein, and concluding with a discussion of two highly relevant modern research programmes: shape dynamics and the so-called ‘dynamical approach’ to spacetime. Readers interested in following up either the historical or the current debates about the natures of space, time and motion will find ample links and references scattered through the discussion and in the Other Internet Resources section below.

The reader should note at the outset that this article presupposes familiarity with some of the basic concepts of relativity theory; in addition, section 3 presupposes familiarity with some relativity standard machinery from theoretical physics (e.g., Lagrangian mechanics). It would not be appropriate, in this philosophical article, to explain all of the background details here from the ground up. In lieu of doing so, we have (a) provided extensive references to literature in which the relevant concepts are explained further, (b) highlighted more technical subsections of this article which can be skipped on first reading, and (c) provided throughout non-technical summaries of the relevant conceptual points.

1.1 Two Interpretations of Mach on Inertia

1.2 implementing mach-heavy, 1.3 mach-lite versus mach-heavy, 2.1 relations determine state of motion, 2.2 the relationist roots of str and gtr, 2.3 from special relativity to general relativity, 2.4 general relativity and relativity of motion, 3.1. configuration space, 3.2. emergent temporality, 3.3. best matching, 3.4. relativistic best matching, 3.5. conceptual matters, 4.1 the dynamical approach and regularity relationism, 4.2 space-time and explanation on the dynamical approach, 4.3 the dynamical approach and general relativity, 5. conclusion, works cited in text, notable philosophical discussions of the absolute-relative debates, other internet resources, related entries.

Between the time of Newton and Leibniz and the 20th century, Newton’s mechanics and gravitation theory reigned essentially unchallenged, and with that long period of dominance, Newton’s absolute space came to be widely accepted. At least, no natural philosopher or physicist offered a serious challenge to Newton’s absolute space, in the sense of offering a rival theory that dispenses with it. But like the action at a distance in Newtonian gravity, absolute space continued to provoke philosophical unease. Seeking a replacement for the unobservable Newtonian space, Neumann (1870) and Lange (1885) developed more concrete definitions of the reference frames in which Newton’s laws hold. [ 1 ] In these and a few other works, the concept of the set of inertial frames (those in which material bodies obey Newton’s three laws of motion) was first clearly expressed, though it was implicit in both remarks and procedures found in Newton’s Principia . (See the entries on space and time: inertial frames and Newton’s views on space, time, and motion ) The most sustained, comprehensive, and influential attack on absolute space was made by Ernst Mach in his Science of Mechanics (1883).

In a lengthy discussion of Newton’s Scholium on absolute space, Mach accuses Newton of violating his own methodological precepts by going well beyond what the observational facts teach us concerning motion and acceleration. Mach at least partly misinterpreted Newton’s aims in the Scholium , and inaugurated a reading of Newton’s bucket argument (and by extension the globes argument) that has largely persisted in the literature since. (See absolute and relational space and motion: classical theories , section 4, for discussion of Newton’s bucket argument.) Mach viewed the argument as directed against a ‘strict’ or ‘general-relativity’ form of relationism, and as an attempt to establish the existence of absolute space. (Strict relationism denies that there is any such thing as an absolute motion; all motion is merely relative, i.e., is nothing more than changes of location relative to some arbitrarily chosen reference frame.) Mach points out the obvious gap in the argument when so construed: the experiment only establishes that acceleration (rotation) of the water with respect to the Earth, or the frame of the fixed stars , produces the tendency to recede from the center; it does not prove that a strict relationist theory cannot account for the bucket phenomena, much less the existence of absolute space.

The reader of the entry on absolute and relational space and motion: classical theories will recall that Newton’s actual aim was simply to show that Descartes’ two kinds of motion are not adequate to account for rotational phenomena. Newton’s bucket argument showed that the effects of rotational motion could not be accounted for by means of the motion of the water relative to its immediate surroundings (the bucket walls); Newton’s thought experiment with two globes connected by a cord was meant to show that one can determine whether they are rotating about their common center (and if so, in which direction) without needing any reference to anything external. By pushing on opposite faces of the two globes and checking for an increase or decrease in the tension in the cord, one can determine in which sense the spheres are in rotation, if they are rotating at all.

Although Mach does not mention the globes thought experiment specifically, it is easy to read an implicit response to it in the things he does say: nobody is competent to say what would happen, or what would be possible, in a universe devoid of matter other than two globes. In other words, Mach would question Newton’s starting premise that the cord connecting the two globes in an otherwise empty universe might be under tension, and indeed under a wide range of different quantities of tension. So, for Mach, neither the bucket nor the globes can establish the existence of absolute space.

Both in Mach’s interpretations of Newton’s arguments and in his replies, one can already see two anti-absolute space viewpoints emerge, though Mach himself never fully kept them apart. The first strain, which we may call ‘Mach-lite’, criticizes Newton’s postulation of absolute space as a metaphysical leap that is neither justified by actual experiments, nor methodologically sound. The remedy offered by Mach-lite is simple: we should retain Newton’s mechanics and use it just as we already do, but eliminate the unnecessary posit of absolute space. In its place we need only substitute the reference frame of the fixed stars, as is the practice in astronomy in any case. If we find the incorporation of a reference to contingent circumstances (the existence of a single reference frame in which the stars are more or less stationary) in the fundamental laws of nature problematic (which Mach need not, given his official positivist account of scientific laws), then Mach suggests that we replace the 1st law with an empirically equivalent mathematical rival, such as this one:

Mach’s Equation (1960, 287) \[ \frac{d^2 (\frac{\Sigma mr}{\Sigma m})}{dt^2} = 0 \]

In this equation the sums are to be taken over all massive bodies in the universe. Since the top sum is weighted by distance, distant masses count much more than near ones. In a world with a (reasonably) static distribution of heavy distant bodies, such as we appear to live in, the equation entails that the velocity of a free body will be constant (to an extremely good approximation) in precisely those frames that we already consider to be ‘inertial’ frames. The upshot of this equation is that the frame of the fixed stars plays the role of absolute space in the statement of the 1st law. This proposal does not, by itself, offer an alternative to Newtonian mechanics, and as Mach himself pointed out, the law is not well-behaved in an infinite universe filled with stars; but the same can perhaps be said of Newton’s law of gravitation (see Malament 1995, and Norton 1993). But Mach did not offer this equation as a proposed law valid in any circumstances; he avers, “it is impossible to say whether the new expression would still represent the true condition of things if the stars were to perform rapid movements among one another.” (p. 289)

It is not clear whether Mach offered this revised first law as a first step toward a theory that would replace Newton’s mechanics, deriving inertial effects from only relative motions, as Leibniz desired. But many other remarks made by Mach in his chapter criticizing absolute space point in this direction, and they have given birth to the Mach-heavy view, later to be christened “Mach’s Principle” by Albert Einstein. [ 2 ] The Mach-heavy viewpoint calls for a new mechanics that invokes only relative distances and (perhaps) their 1st and 2nd time derivatives, and thus is ‘generally relativistic’ in the sense sometimes read into Leibniz’s remarks about motion (see absolute and relational space and motion: classical theories , section 6). Mach wished to eliminate absolute time from physics too, so he would have wanted a proper relationist reduction of these derivatives also. The Barbour-Bertotti theories, discussed below, provide this.

Mach-heavy apparently involves the prediction of novel effects due to ‘merely’ relative accelerations. Mach hints at such effects in his criticism of Newton’s bucket:

Newton’s experiment with the rotating vessel of water simply informs us that the relative rotation of the water with respect to the sides of the vessel produces no noticeable centrifugal forces, but that such forces are produced by its relative rotation with respect to the mass of the earth and the other celestial bodies. No one is competent to say how the experiment would turn out if the sides of the vessel [were] increased until they were ultimately several leagues thick. (1883, 284)

The suggestion here seems to be that the relative rotation in stage (i) of the experiment might immediately generate an outward force (before any rotation is communicated to the water), if the sides of the bucket were massive enough. (Note that this response could not have been made by Leibniz – even if he had wanted to defend Machian relationism – because it involves action at a distance between the water and the parts of the bucket, something he regarded as a metaphysical absurdity.)

Mach’s thick-sided bucket

More generally, Mach-heavy involves the view that all inertial effects should be derived from the motions of the body in question relative to all other massive bodies in the universe. The water in Newton’s bucket feels an outward pull due (mainly) to the relative rotation of all the fixed stars around it. Mach-heavy is a speculation that an effect something like electromagnetic induction should be built into gravity theory. (Such an effect does exist according to the General Theory of Relativity, and is called ‘gravitomagnetic induction’. The Gravity Probe B mission was designed to measure a gravitomagnetic induction effect on orbiting gyroscopes due to the Earth’s rotation.) Its specific form must fall off with distance much more slowly than \(1/r^2\), if the theory is to be empirically similar to Newtonian physics; but it will certainly predict experimentally testable novel behaviors. A theory that satisfies all the goals of Mach-heavy would appear to be ideal for the vindication of strict relationism and the elimination of absolute quantities of motion from mechanics.

Direct assault on the problem of satisfying Mach-heavy in a classical framework proved unsuccessful for a long time, despite the efforts of others besides Mach – for example, Friedländer (1896), Föpl (1904), and Reissner (1914, 1915). (Between the late 19th century and the 1970s, there was of course one extremely important attempt to satisfy Mach-heavy: the work of Einstein that led to the General Theory of Relativity. Since Einstein’s efforts took place in a non-classical (Lorentz/Einstein/Minkowski) spacetime setting, we discuss them in the next section.) One very influential approach to implementing Mach-heavy was promulgated in the work of Barbour and Bertotti (1977); this has since developed into the research programme of ‘shape dynamics’, and will be discussed in more detail in section 3 below.

Mach-lite, like the relational interpretations of Newtonian physics reviewed in the entry on absolute and relational space and motion: classical theories , section 5, offers us a way of understanding Newtonian physics without accepting absolute position, velocity or acceleration. But it does so in a way that lacks theoretical clarity and elegance, since it does not delimit a clear set of cosmological models. We know that Mach-lite makes the same predictions as Newtonian physics for worlds in which there is a static frame associated with the stars and galaxies; but if asked about how things will behave in a world with no frame of fixed stars, or in which the stars are far from ‘fixed’, it shrugs and refuses to answer. (Recall that Mach-lite simply says: “Newton’s laws hold in the frame of reference of the fixed stars.”) This is perfectly acceptable according to Mach’s philosophy of science, since the job of mechanics is simply to summarize observable facts in an economical way. But it is unsatisfying to those with stronger realist intuitions about laws of nature.

If there is, in fact, a distinguishable privileged frame of reference in which the laws of mechanics take on a specially simple form, without that frame being determined in any way by relation to the matter distribution, a realist will find it hard to resist the temptation to view motions described in that frame as the ‘true’ or ‘absolute’ motions. If there is a family of such frames, disagreeing about velocity but all agreeing about acceleration, then the realist will feel a temptation to think of at least acceleration as ‘true’ or ‘absolute’. If such a realist believes motion to be by nature a relation rather than a property (and not all philosophers accept this; see the entry on absolute and relational space and motion: classical theories , section 1) then they will feel obliged to accord some sort of existence or reality to the structure – e.g., the structure of Galilean spacetime – in relation to which these motions are defined. For philosophers with such realist inclinations, the ideal relational account of motion would therefore be some version of Mach-heavy.

2. Einstein

Einstein’s Special Theory of Relativity (STR) is notionally based on a principle of relativity of motion; but that principle is ‘special’ – meaning, restricted. The relativity principle built into STR is in fact nothing other than the Galilean principle of relativity, which is built into Newtonian physics. [ 3 ] In other words, while there is no privileged standard of velocity, there is nevertheless a determinate fact of the matter about whether a body has accelerated or non-accelerated (i.e., inertial) motion. In this regard, the spacetime of STR is exactly like Galilean spacetime (discussed in the entry on absolute and relational space and motion: classical theories , section 5). In terms of the question of whether all motion can be considered purely relative, one could argue that there is nothing new brought to the table by the introduction of Einstein’s STR – at least, as far as mechanics is concerned. (See the entry on space and time: inertial frames for a more detailed discussion.)

In this subsection we will discuss an interesting sense in which, in STR, the letter (if not the spirit) of classical relationism can be considered vindicated: the spatio-temporal relations between material things are, on their own, sufficient to fully determine the state of motion of a body. The discussion here presupposes acquaintance with STR and its basic mathematics, and will be hard to follow for readers lacking that background; such readers should feel free to skip this subsection, which is not necessary for following the material in the rest of section 2.

As Dorling (1978) first pointed out, there is a sense in which the standard absolutist arguments against ‘strict’ relationism using rotating objects (buckets or globes) fail in the context of STR. Maudlin (1993) used the same considerations to show that there is a way of recasting relationism in STR that appears to be successful. STR incorporates certain novelties concerning the nature of time and space, and how they mesh together; perhaps the best-known examples are the phenomena of ‘length contraction’, ‘time dilation’, and the ‘relativity of simultaneity.’ [ 4 ] In STR both spatial distances and time intervals between events – when measured in the standard ways – are frame-relative (observers in different states of motion, i.e. at rest in different reference frames, will ‘disagree’ about their sizes). The standard classical relationist starting point – the configuration of relative distances between the existing bodies at a moment of time – does not exist, at least not as an objective, observer- or frame-independent set of facts. Because of this, when considering what spatial or temporal relations a relationist should postulate as fundamental, it is arguably most natural to restrict oneself to the frame-invariant spatiotemporal ‘distance’ between events in spacetime. This is given by the interval between two points: \([\Delta x^2 + \Delta y^2 + \Delta z^2 - \Delta t^2]\) – the four-dimensional analog of the Pythagorean theorem, for spacetime distances. If one regards the spacetime interval relations between point-masses-at-times as one’s basis, on which spacetime is built up as an ideal entity (analogously to how Leibniz thought of 3-d space as an ideal entity abstracted from spatial distance relations), then with only mild caveats relationism works: the spacetime interval relations suffice to uniquely fix how the material systems can be embedded (up to isomorphism) in the ‘Minkowski’ spacetime of STR. The modern variants of Newton’s bucket and globes arguments no longer stymie the relationist because (for example) the spacetime interval relations among bits of matter in Newton’s bucket at rest are quite different from the spacetime interval relations found among those same bits of matter after the bucket is rotating. For example, the spacetime interval relation between a bit of water near the side of the bucket, at one time, and itself (say) a second later is smaller than the interval relation between a center-bucket bit of water and itself one second later (times referred to inertial-frame clocks). The upshot is that, unlike the situation in classical physics, a non-rotating body cannot have all the same spatiotemporal relations among its parts as a similar body in rotation. We cannot put a body or system into a state of rotation (or other acceleration) without thereby changing the spacetime interval relations between the various bits of matter at different moments of time, compared to what they would have been if the body had remained non-accelerated or non-rotated. The facts about rotation and acceleration, thus, supervene on spacetime interval relations. [ 5 ]

It is worth pausing to consider to what extent this victory for (some form of) relationism satisfies the classical ‘strict’ relationism traditionally ascribed to Mach and Leibniz. The spatiotemporal relations that save the day against the bucket and globes are, so to speak, mixed spatial and temporal distances. They are thus quite different from the spatial-distances-at-a-time presupposed by classical relationists; moreover they do not correspond to relative velocities (-at-a-time) either. Their oddity is forcefully captured by noticing that if we choose appropriate bits of matter at ‘times’ eight minutes apart, I-now am at zero distance from the surface of the sun (of eight minutes ‘past’, since it took 8 minutes for light from the sun to reach me-now). So we are by no means dealing here with an innocuous, ‘natural’ translation of classical relationist quantities into the STR setting. On the other hand, in light of the relativity of simultaneity (see footnote 5 ), it can be argued that the absolute simultaneity presupposed by classical relationists and absolutists alike was, in fact, something that relationists should always have regarded with misgivings. From this perspective, instantaneous relational configurations – precisely what one starts with in the theories of Barbour and Bertotti discussed below – would be the things that should be treated with suspicion.

If we now return to our questions about motions – about the nature of velocities and accelerations – we find, as noted above, that matters in the interval-relational interpretation of STR are much the same as in Newtonian mechanics in Galilean spacetime. There are no well-defined absolute velocities, but there are indeed well-defined absolute accelerations and rotations. In fact, the difference between an accelerating body (e.g., a rocket) and an inertially moving body is codified directly in the cross-temporal interval relations of the body with itself . So we are very far from being able to conclude that all motion is relative motion of a body with respect to other bodies. It is true that the absolute motions are in 1–1 correlation with patterns of spacetime interval relations, but it is not at all correct to say that they are, for that reason, eliminable in favor of merely relative motions. Rather we should simply say that no absolute acceleration can fail to have an effect on the material body or bodies accelerated. But this was already true in classical physics if matter is modeled realistically: the cord connecting the globes does not merely tense, but also stretches; and so does the bucket, even if imperceptibly, i.e., the spatial relations change.

Maudlin does not claim this version of relationism to be victorious over an absolutist or substantivalist conception of Minkowski spacetime, when it comes time to make judgments about the theory’s ontology. There may be more to vindicating relationism than merely establishing a 1–1 correlation between absolute motions and patterns of spatiotemporal relations.

The simple comparison made above between STR and Newtonian physics in Galilean spacetime is somewhat deceptive. For one thing, Galilean space time is a mathematical innovation posterior to Einstein’s 1905 theory; before then, Galilean spacetime had not been conceived, and full acceptance of Newtonian mechanics implied accepting absolute velocities and, arguably, absolute positions, just as laid down in the Scholium . So Einstein’s elimination of absolute velocity was a genuine conceptual advance. Moreover, the Scholium was not the only reason for supposing that there existed a privileged reference frame of ‘rest’: the working assumption of almost all physicists in the latter half of the 19th century was that, in order to understand the wave theory of light, one had to postulate an aetherial medium filling all space, wave-like disturbances in which constituted electromagnetic radiation. It was assumed that the aether rest frame would be an inertial reference frame; and physicists felt some temptation to equate its frame with the absolute rest frame, though this was not necessary. Regardless of this equation of the aether with absolute space, it was assumed by all 19th century physicists that the equations of electrodynamic theory would have to look different in a reference frame moving with respect to the aether than they did in the aether’s rest frame (where they presumably take their canonical form, i.e., Maxwell’s equations and the Lorentz force law). So while theoreticians labored to find plausible transformation rules for the electrodynamics of moving bodies, experimentalists tried to detect the Earth’s motion in the aether. Experiment and theory played collaborative roles, with experimental results ruling out certain theoretical moves and suggesting new ones, while theoretical advances called for new experimental tests for their confirmation or – as it happened – disconfirmation.

As is well known, attempts to detect the Earth’s velocity in the aether were unsuccessful. On the theory side, attempts to formulate the transformation laws for electrodynamics in moving frames – in such a way as to be compatible with experimental results – were complicated and inelegant. [ 6 ] A simplified way of seeing how Einstein swept away a host of problems at a stroke is this: he proposed that the Galilean principle of relativity holds for Maxwell’s theory, not just for mechanics. The canonical (‘rest-frame’) form of Maxwell’s equations should be their form in any inertial reference frame. Since the Maxwell equations dictate the velocity c of electromagnetic radiation (light), this entails that any inertial observer, no matter how fast she is moving, will measure the velocity of a light ray as c – no matter what the relative velocity of its emitter may be. Einstein worked out logically the consequences of this application of the special relativity principle, and discovered that space and time must be rather different from how Newton described them. STR undermined Newton’s absolute time just as decisively as it undermined his absolute space.

Einstein’s STR was the first clear and empirically successful physical theory to overtly eliminate the concepts of absolute rest and absolute velocity while recovering most of the successes of classical mechanics and 19th century electrodynamics. It therefore deserves to be considered the first highly successful theory to explicitly relativize motion, albeit only partially. But STR only recovered most of the successes of classical physics: crucially, it left out gravity. And there was certainly reason to be concerned that Newtonian gravity and STR would prove incompatible: classical gravity acted instantaneously at a distance, while STR eliminated the privileged absolute simultaneity that this instantaneous action presupposes.

Several ways of modifying Newtonian gravity to make it compatible with the spacetime structure of STR suggested themselves to physicists in the years 1905–1912, and a number of interesting Lorentz-covariant theories were proposed (i.e., theories compatible with the spacetime of STR, which is called ‘Minkowski spacetime’ because Hermann Minkowski first revealed the spacetime structure that Einstein’s postulates in STR entail). Einstein rejected these proposed theories one and all, for violating either empirical facts or theoretical desiderata. But Einstein’s chief reason for not pursuing the reconciliation of gravitation with STR’s spacetime appears to have been his desire, beginning in 1907, to replace STR with a theory in which not only velocity could be considered merely relative, but also acceleration. That is to say, Einstein wanted if possible to completely eliminate all absolute quantities of motion from physics, thus realizing a theory that satisfies at least one kind of ‘strict’ relationism. (Regarding Einstein’s rejection of Lorentz-covariant gravity theories, see Norton 1992; regarding Einstein’s quest to fully relativize motion, see Hoefer 1994.)

Einstein began to see this complete relativization as possible in 1907, thanks to his discovery of the Equivalence Principle (cf. Lehmkuhl forthcoming ). Imagine we are far out in space, in a rocket ship accelerating at a constant rate \(g = 9.81 m/s^2.\) Things will feel just like they do on the surface of the Earth; we will feel a clear up-down direction, bodies will fall to the floor when released, etc. Indeed, due to the well-known empirical fact that gravity affects all bodies by imparting a force proportional to their matter (and energy) content, independent of their internal constitution, we know that any experiment performed on this rocket will give the same results that the same experiment would give if performed on the Earth. Now, Newtonian theory teaches us to consider the apparent downward, gravity-like forces in the rocket ship as ‘pseudo-forces’ or ‘inertial forces’, and insists that they are to be explained by the fact that the ship is accelerating in absolute space. But Einstein asked whether there is any way for the person in the rocket to regard him/herself as being ‘at rest’ rather than in absolute (accelerated) motion? And the answer he gave is: Yes. The rocket traveler may regard him/herself as being ‘at rest’ in a homogeneous and uniform gravitational field. Such a field would entail an accelerative force “downward” on every body that is equal in magnitude and direction everywhere in space. This is unlike the Earth’s gravitational field, which varies depending on distance from the Earth’s center and points in different directions at different locations. Positing the existence of such a field will explain all the observational facts just as well as the supposition that he/she is accelerating relative to absolute space (or, absolutely accelerating in Minkowski spacetime). But is it not clear that the latter is the truth, while the former is a fiction? By no means; if there were a uniform gravitational field filling all space, then it would affect all the other bodies in the world – the Earth, the stars, etc, – imparting to them a downward acceleration away from the rocket; and that is exactly what the traveler observes.

In 1907 Einstein published his first gravitation theory (Einstein 1907), treating the gravitational field as a scalar field that also represented the (now variable and frame-dependent) speed of light. Einstein viewed the theory as only a first step on the road to eliminating absolute motion. In the 1907 theory, the theory’s equations take the same form in any inertial or uniformly accelerating frame of reference. One might say that this theory reduces the class of absolute motions, leaving only rotation and other non-uniform accelerations as absolute. But, Einstein reasoned, if uniform acceleration can be regarded as equivalent to being at rest in a constant gravitational field, why should it not be possible also to regard inertial effects from these other, non-uniform motions as similarly equivalent to “being at rest in a (variable) gravitational field”? Thus Einstein set himself the goal of expanding the principle of equivalence to embrace all forms of ‘accelerated’ motion.

Einstein thought that the key to achieving this aim lay in further expanding the range of reference frames in which the laws of physics take their canonical form, to include frames adapted to any arbitrary motions. More specifically, since the class of all continuous and differentiable coordinate systems includes as a proper subclass the coordinate systems adapted to any such frame of reference, if he could achieve a theory of gravitation, electromagnetism and mechanics that was generally covariant – its equations taking the same form in any coordinate system from this general class – then the complete relativity of motion would be achieved. If there are no special frames of reference in which the laws take on a simpler canonical form, there is no physical reason to consider any particular state or states of motion as privileged, nor deviations from those as representing ‘absolute motion’. (Here we are just laying out Einstein’s train of thought; later we will see reasons to question the last step.) And in 1915, Einstein achieved his aim in the General Theory of Relativity (GTR).

There is one key element left out of this success story, however, and it is crucial to understanding why most physicists reject Einstein’s claim to have eliminated absolute states of motion in GTR. Going back to our accelerating rocket, we accepted Einstein’s claim that we could regard the ship as hovering at rest in a universe-filling gravitational field. But a gravitational field, we usually suppose, is generated by matter. How is this universe-filling field linked to generating matter? The answer may be supplied by Mach-heavy. Regarding the ‘accelerating’ rocket which we decide to regard as ‘at rest’ in a gravitational field, the Machian says: all those stars and galaxies, etc., jointly accelerating downward (relative to the rocket), ‘produce’ that gravitational field. The mathematical specifics of how this field is generated will have to be different from Newton’s law of gravity, of course; but it should give essentially the same results when applied to low-mass, slow-moving problems such as the orbits of the planets, so as to capture the empirical successes of Newtonian gravity. Einstein thought, in 1916 at least, that the field equations of GTR are precisely this mathematical replacement for Newton’s law of gravity, and that they fully satisfied the desiderata of Mach-heavy relationism. But it was not so. (See the entry on early philosophical interpretations of general relativity .)

In GTR, spacetime is locally very much like STR’s flat Minkowski spacetime. There is no absolute velocity locally, but there are clear local standards of accelerated vs non-accelerated motion, i.e., local inertial frames. In these ‘freely falling’ frames bodies obey the usual rules for non-gravitational physics familiar from STR, albeit only approximately (this is sometimes called the ‘strong equivalence principle’, and is discussed further in section 4 below). But overall spacetime is curved, and local inertial frames may tip, bend and twist as we move from one region to another. The structure of curved spacetime is encoded in the metric field tensor g ab , with the curvature encoding gravity at the same time: gravitational forces are so to speak ‘built into’ the metric field, geometrized away. Since the spacetime structure encodes gravity and inertia, and in a Mach-heavy theory these phenomena should be completely determined by the relational distribution of matter (and relative motions), Einstein wished to see the metric as entirely determined by the distribution of matter and energy. But what the GTR field equations entail is, in general, only a partial-determination relation.

We cannot go into the mathematical details necessary for a full discussion of the successes and failures of Mach-heavy in the GTR context. But one can see why the Machian interpretation Einstein hoped he could give to the curved spacetimes of his theory fails to be plausible, by considering a few simple ‘worlds’ permitted by GTR. In the first place, for our hovering rocket ship, if we are to attribute the gravity field it feels to matter, there has got to be all this other matter in the universe. But if we regard the rocket as a mere ‘test body’ (not itself substantially affecting the gravity present or absent in the universe), then we can note that according to GTR, if we remove all the stars, galaxies, planets etc. from the world, the gravitational field does not disappear. On the contrary, it stays basically the same locally, and globally, in the simplest solution of the field equations, it takes the form of empty Minkowski spacetime – precisely the quasi-absolute structure Einstein was hoping to eliminate. Solutions of the GTR field equations for arbitrary realistic configurations of matter (e.g., a rocket ship ejecting a stream of particles to push itself forward) are hard to come by, and in fact a realistic two-body exact solution has yet to be discovered. But numerical methods can be applied for many purposes, and physicists do not doubt that something like our accelerating rocket – in otherwise empty space – is possible according to the theory. [ 7 ] We see clearly, then, that GTR fails to satisfy Einstein’s own understanding of Mach’s Principle, according to which, in the absence of matter, space itself should not be able to exist.

A second example: GTR allows us to model a single rotating object in an otherwise empty universe (e.g., a neutron star). Relationism of the Machian variety says that such rotation is impossible, since it can only be understood as rotation relative to some sort of absolute space. In the case of GTR, this is indeed the natural way to understand such a model: the rotation is best understood as rotation relative to a ‘background’ spacetime that is identical to the Minkowski spacetime of STR, only ‘curved’ by the presence of matter in the region of the star.

On the other hand, there is one charge of failure-to-relativize-motion sometimes leveled at GTR that is unfair. It is sometimes asserted that the simple fact that the metric field (or the connection it determines) distinguishes, at every location, motions that are ‘absolutely’ accelerated and/or ‘absolutely rotating’ from those that are not, by itself entails that GTR fails to embody a folk-Leibniz style general relativity of motion (e.g. Earman (1989), ch. 5). We think this is incorrect, and leads to unfairly harsh judgments about confusion on Einstein’s part. The local inertial structure encoded in the metric would not be ‘absolute’ in any meaningful sense, if that structure were in some clear sense fully determined by the relationally specified matter-energy distribution. Einstein was not simply confused when he named his gravity theory. (Just what is to be understood by “the relationally specified matter-energy distribution” is a further, thorny issue, which we cannot enter into here.)

GTR does not fulfill all the goals of Mach-heavy, at least as understood by Einstein, and he recognized this fact by 1918 (Einstein 1918). And yet … GTR comes tantalizingly close to achieving those goals, in certain striking ways (cf. Hoefer 2014). For one thing, GTR does predict Mach-heavy effects, known as ‘frame-dragging’: if we could model Mach’s thick-walled bucket in GTR, it seems clear that it would pull the water slightly outward, and give it a slight tendency to begin rotating in the same sense as the bucket (even if the big bucket’s walls were not actually touching the water). While GTR does permit us to model a lone rotating object, if we model the object as a shell of mass (instead of a solid sphere) and let the size of the shell increase (to model the ‘sphere of the fixed stars’ we see around us), then as Brill & Cohen (1966) showed, the frame-dragging becomes complete inside the shell. In other words: our original Minkowski background structure effectively disappears, and inertia becomes wholly determined by the shell of matter, just as Mach posited was the case. This complete determination of inertia by the global matter distribution appears to be a feature of other models, including the Friedman-Lemâitre-Robertson-Walker Big Bang models that best match observations of our universe.

Finally, it is important to recognize that GTR is generally covariant in a very special sense: unlike all other prior theories (and unlike many subsequent quantum theories), it postulates no fixed ‘prior’ or ‘background’ spacetime structure. As mathematicians and physicists realized early on, other theories, e.g., Newtonian mechanics and STR, can be put into a generally covariant form. But when this is done, there are inevitably mathematical objects postulated as part of the formalism, whose role is to represent absolute elements of spacetime structure (see Friedman 1983, Pooley 2017). What is unique about GTR is that it was the first, and is still the only ‘core’ physical theory, to have no such absolute elements in its covariant equations. (Whether these claims are exactly correct is a matter of ongoing debate, relating to the question of the ‘background independence’ of GTR: for discussion, see e.g. Belot (2011), Pitts (2006), Read (2016), and Pooley (2017).) The spacetime structure in GTR, represented by the metric field, is at least partly ‘shaped’ by the distribution of matter and energy. And in certain models of the theory, such as the Big Bang cosmological models, some authors have claimed that the local standards of inertial motion – the local ‘gravitational field’ of Einstein’s equivalence principle – are entirely fixed by the matter distribution throughout space and time, just as Mach-heavy requires (see, for example, Wheeler and Cuifollini 1995).

Absolutists and relationists are thus left in a frustrating and perplexing quandary by GTR. Considering its anti-Machian models, we are inclined to say that motions such as rotation and acceleration remain absolute, or nearly-totally-absolute, according to the theory. On the other hand, considering its most Mach-friendly models, which include all the models taken to be good candidates for representing the actual universe, we may be inclined to say: motion in our world is entirely relative; the inertial effects normally used to argue for absolute motion are all understandable as effects of rotations and accelerations relative to the cosmic matter, just as Mach hoped. But even if we agree that motions in our world are in fact all relative in this sense, this does not automatically settle the traditional relationist/absolutist debate, much less the relationist/substantivalist debate. Many philosophers (including, we suspect, Nerlich 1994 and Earman 1989) would be happy to acknowledge the Mach-friendly status of our spacetime, and argue nevertheless that we should understand that spacetime as a real thing, more like a substance than a mere ideal construct of the mind as Leibniz insisted. By contrast, other philosophers (e.g., Rynasiewicz 1995) argue that due to the conceptual and mathematical novelties introduced in GTR, the traditional absolute vs. relational motion debate simply fails to make sense any more (on this question, see also Hoefer 1998).

3. Shape Dynamics

We turn now to a modern-day attempt to implement Mach-heavy known as ‘shape dynamics’. (In fact, shape dynamics is just one theory within this tradition, as we will see below.) This approach was initiated – albeit not under that name – by Barbour and Bertotti (1977, 1982). In tackling the problem of implementing Mach-heavy, rather than formulating a revised law of gravity/inertia using relative quantities, Barbour and Bertotti used the framework of Lagrangian mechanics, replacing elements of the mathematics referring to absolute quantities of motion with new terms invoking only relative distances, velocities, etc. In this section, we presuppose a basic familiarity with the Lagrangian framework. For a non-technical introduction to shape dynamics, see Barbour (1999); for an up-to-date review of recent work in the field, see Mercati (2018).

In this section, we survey the results and motivations of the shape dynamics research program, focussing first on the above-mentioned theory of Barbour and Bertotti (which recovers a subsection of the solution space of Newtonian particle theory), before turning to the Machian alternative to general relativity developed by Barbour and collaborators: it is this latter theory which is shape dynamics ‘proper’. Readers uninterested in the technical details of this work can skip to section 3.5, in which its conceptual upshots are discussed.

For a given physical system, define its ‘configuration space’ to be the space of possible instantaneous states of that system. (For example, the space of possible distributions of N particles in Euclidean space, according to a Cartesian coordinate system laid down on that space.) As the system evolves, the point in configuration space representing the system’s instantaneous state traces out a continuous curve. On this picture, metaphysically possible worlds are represented by (rising) curves in the product space formed from configuration space and a one-dimensional space representing time. Nomologically possible worlds are represented by those curves that are allowed by the dynamics. For example, in the Lagrangian formalism, the nomologically possible worlds are represented by those curves which extremize the action: a particular functional of such curves.

Consider now, for the sake of concreteness, two Newtonian worlds which differ by either a static or a kinematic Leibniz shift: that is, constant translations or velocity boosts of the material content of the universe (see the companion entry on absolute and relational space and motion: classical theories , for further discussion of such shifts). These two worlds will be represented by distinct curves in configuration space. However, given a configuration space, one can construct a ‘reduced’ configuration space, in which certain such histories are mathematically identified, or ‘quotiented’, such that they are mapped to the same unique history in reduced configuration space. Specifically, proponents of this approach define two such reduced configuration spaces:

• ‘Relative configuration space’, which is configuration space quotiented by rigid translations and rotations. (I.e., curves in configuration space which represent worlds which differ by rigid translations or rotations are mathematically identified.)
• ‘Shape space’, which its configuration space quotiented by rigid translations, rotations, and dilatations. (I.e., curves in configuration space which represent worlds which differ by rigid translations, rotations, or dilatations are mathematically identified.)

(Two points here. First, recall that a ‘dilatation’ is a scale transformation. Second, in what follows we will refer to the group which consists of the union of translations, rotations and dilatations as the ‘similarity group’.) If these Machian theorists are able to formulate a dynamics on shape space (i.e., a dynamics which identifies the curves in shape space which represent nomologically possible worlds), then that dynamics will, in light of the above reduction, not bring with it a meaningful notion of absolute position, or absolute velocity, or absolute scales. Barbour and collaborators take such a dynamics to realize Mach-heavy: the undetectable spacetime structure associated with such quantities has been expunged. Below, we will see how this can be done in the concrete contexts of Newtonian particle dynamics and general relativity.

The Machian ambitions of Barbour and collaborators do not end there, for these authors also seek to excise primitive temporal structure. Initially, one might distinguish histories that correspond to a single curve in configuration space being traced out at different rates with respect to the primitive temporal parameter. Those working in this tradition, however, view each curve in configuration space as corresponding to exactly one possible history. They therefore elect to dispose of the auxiliary one-dimensional space representing a primitive absolute time which was introduced above. Instead, they seek to construct an ‘emergent’ notion of temporality from dynamics defined on configuration space alone. By way of a procedure known as ‘Jacobi’s principle’, the Machian relationist selects a unique temporal parameter which maximally simplifies this dynamics defined on configuration space. For the details of Jacobi’s principle, see Pooley (2013).

It is all well and good speaking of a dynamics defined on relative configuration space, or shape space. However, it remains incumbent on our Machian theorists to construct explicit dynamics appropriate for these spaces: i.e., dynamics which do not recognise solutions related by the action of the similarity group (viz., translations, rotations, and dilatations) as being distinct. Given a dynamics on configuration space, one can indeed achieve this task. The procedure which implements this is known as ‘best matching’, and was developed in the seminal work of Barbour and Bertotti (1982), in which a version of Newtonian particle mechanics with dynamics formulated on relative configuration space was first constructed. The extension to shape space was undertaken in (Barbour 2003).

Informally, the goal of best matching is to use the similarity group to minimize the intrinsic difference between successive points along a history in configuration space. To take a simple example drawn from Barbour (1999), consider the history of a particular triangle: the triangle may, along that history, rotate, dilate, alter its internal angles, and so on. However, at each point best matching allows one to act on the triangle with similarity transformations; thereby, triangles which at successive points along a history differ merely by a translation, rotation or dilatation will be regarded as being identical after best matching. In this way, a ‘best matched’ history is selected, in which the intrinsic differences between successive states of the system under consideration (in the above example, the triangle) are minimised. While a metric on configuration space will in general assign a different length to histories differing by the action of the similarity group, the length of the best matched history, constructed via the above procedure, will induce a unique length of paths, and therefore metric, on shape space.

A little more formally, the best matching procedure works as follows. Consider a class of paths in configuration space, all corresponding to the same path in shape space (i.e., consider a class of paths in configuration space related by the action of the similarity group). As mentioned above, a given metric on configuration space will in general assign to each path in that space a different length; as a result, the length of the associated path in shape space will be underdetermined. However, starting from any given point p in configuration space, one can use the action of the similarity group on configuration space to define a unique curve, by shifting the points of any curve through p along the corresponding orbits of the similarity group (think of these ‘orbits’ as contour lines in configuration space, relating points which differ only by the action of the similarity group) so as to extremize the length assigned to the curve (relative to the metric under consideration). It is this extremized length which is assigned to the unique curve in shape space. With each curve in shape space assigned a unique length, one can then, as usual, specify a principle which selects some such curves as representing nomologically possible worlds, based upon their lengths. (Recall again, for example, that in Lagrangian mechanics it is those curves which extremize an action which are regarded as being dynamically possible.)

The best matching prescription can be applied not only to Newtonian particle theories, but also to other spacetime theories, including GTR. (There is no reason why best matching cannot be applied to Newtonian field theories, or to special relativistic particle dynamics, but these steps are usually skipped by Machian relationists following in the tradition of Barbour and Bertotti, who proceed at this stage straight to GTR.)

To see how best matching works in the case of GTR, first note that a certain subclass of solutions of that theory (namely, those which are globally hyperbolic) can be formulated in terms of the ‘3+1 formalism’, according to which the state of the universe at a particular time is represented by a determinate 3-manifold with associated Riemannian metric; dynamical equations then determine how such 3-geometries evolve in time. (For a summary of the 3+1 formalism, see e.g. Gourgoulhon (2012).) The Machian relationists working in the shape dynamics research program take this 3+1 approach to GTR as their starting point. They thus assume that instantaneous spaces which are the points in configuration space have the determinate topology of some closed 3-manifold without boundary. Configuration space is the space of Riemannian 3-metrics on that 3-manifold. The natural analogue of relative configuration space is, then, this space of Riemannian 3-metrics quotiented by diffeomorphisms, which are the generalisations of Leibniz shifts appropriate to GTR (see the entry on the hole argument ). The analogue of shape space in this case is the space of Riemannian 3-metrics, but quotiented in addition by local dilatations (by ‘local’, we mean here a transformation which can vary from point to point).

Having constructed shape space in the relativistic case, one may then best match in order to construct one’s relational theory implementing Mach-heavy (the metric on configuration space is defined from the 3+1 dynamics of GTR): conceptually, the approach here is the same as that presented in the previous section. Moreover, one can again apply Jacobi’s principle, in order to eliminate a commitment to primitive temporal structure. In this case, the resulting theory is known as ‘shape dynamics’, which involves a commitment only to primitive conformal structure (i.e., facts about angles between objects) on the 3-geometries: all other absolute quantities, the claim goes, have been excised. One way to understand the relationship between GTR and shape dynamics is that one trades the relativity of simultaneity but absoluteness of scales in the former theory, for absolute simultaneity but the relativity of scales in the latter.

There are important differences between the relationship between ‘standard’ Newtonian particle mechanics and its best-matched alternative on the one hand, and the relationship between GTR and shape dynamics on the other. In the former case, the class of solutions of the best-matched theory is a proper subset of the solutions of Newtonian mechanics: for example, it includes only the sector of the solution space of Newtonian mechanics which ascribes zero angular momentum to the entire universe. Sometimes, this is marketed as an advantage of the latter theory: the best-matched theory predicts what was, in the Newtonian case, an unexplained coincidence. (For discussion, see Pooley & Brown 2002.) In the latter case, by contrast, it has been discovered that one can ‘glue’ solutions of shape dynamics to construct new solutions, which are not associated with any particular solution of GTR (in the sense that they are not the best matched equivalents of any solution of GTR): see (Mercati 2018). Thus, the solution spaces of GTR and shape dynamics overlap, but the latter is not a proper subset of the former. Given this, it is no longer clear that shape dynamics can be presented as a ‘more predictive’ alternative to GTR.

A second conceptual point to make regarding the Machian relationism of Barbour and collaborators pertains to its motivations. Barbour claims, as we have already seen above, that only spatial angles – and not spatial scales, or a temporal timescale, or absolute velocities or positions – are directly empirically observable. Thus, the thought goes that an empiricist of good standing should favour (say) shape dynamics over GTR, for the former theory, unlike the latter, renders only such ‘directly observable’ quantities meaningful; it does not commit to any absolute quantities which are not ‘directly observable’. There are, however, two central points at which this reasoning could be questioned. First: one could repudiate Barbour’s empiricist motivations. Second: one could deny that only angles are directly observable, or, indeed, that this structure is directly observable at all (see Pooley 2013, p. 47). As Pooley points out, these are not the strongest grounds on which to motivate Barbour’s project. Rather, a better motivation is this: best-matched theories have the merit of ontological parsimony, as compared with the theories such as Newtonian particle mechanics or general relativity, to which the best-matching procedure is applied. A second motivation has to do with the potential of this research programme to present new avenues for exploration in the quest for a quantum theory of gravity.

Our third and final point is this. Although it is possible to couple shape dynamics to matter (see e.g. (Gomes 2012)), in this theory, just as in GTR as discussed in the previous section, one also has vacuum solutions, with primitive conformal structure on the 3-geometries. Given the existence of these vacuum solutions, as with GTR, it is far from clear that the theory makes good on the ambitions of Mach and the early Einstein to construct a theory in which all spatiotemporal notions are reduced to facts about matter. That said, it is worth noting that, unlike in GTR, in shape dynamics one cannot have a solution consisting of a single rotating body: the overall angular momentum of the universe must vanish.

4. The Dynamical Approach

Since 2000, a new ‘dynamical’ approach to spacetime structure has emerged in the works of Robert DiSalle (2006) and especially Oliver Pooley and Harvey Brown (2001, 2006). This approach is to be situated against an opposing, supposedly orthodox ‘geometrical’ approach to spacetime structure, as encapsulated in the works of e.g. Janssen (2009) and Maudlin (2012). (This is not to say that either the dynamical view or the opposing geometrical view is a unified edifice, as we will see below.) The dynamical-geometrical debate has many facets, but one can take the central bone of contention to pertain to the arrow of explanation: is it the case that the geometrical structures of spacetime explain why material bodies behave as they do (as the geometrical view would have it), or is it rather the case that the geometrical structure of spacetime is explained by facts about the behaviour of material bodies (as the dynamical view would have it)? Although this debate connects with historical debates between substantivalists and relationists, it should be regarded as a distinct dispute, for reasons to which we will come.

While it is important to keep in mind the above disagreement regarding the arrow of explanation when one is considering the dynamical-geometrical debate, it will be helpful in this article to hone in on two more specific claims of the dynamical approach, as presented by Brown (2005), consistent with the above claim that it is facts about the dynamics of material bodies which explain facts about spatiotemporal structure, rather than vice versa. These two claims are the following (Read 2020a):

• Fixed background space-time structures, such as the Minkowski space-time of STR, or Newton’s absolute space, are to be ontologically reduced to the symmetries of the dynamical equations governing matter fields.
• No piece of space-time structure, whether fixed or dynamical (in the latter case, as in GTR) is necessarily surveyed by physical bodies; rather, in order to ascertain whether this is so, one must attend carefully to the details of the dynamics governing the particular matter fields which constitute physical bodies.

On the first of these two points: proponents of the dynamical approach maintain that the spacetime structure of our world is what it is because of the dynamical laws of nature and their symmetries . That is, the dynamical laws are (at least, relative to spacetime) fundamental, and spacetime structure is derivative; in this sense, the view is (at least in some cases) a modern-day form of relationism (Pooley 2013, §6.3.2) – albeit of a very different kind from the relationist approaches considered up to this point. (Note, though, that this relationism is a corollary of the above explanatory contention of the dynamical approach; moreover, it is one which is applicable only to theories which fixed spacetime structure such as Newtonian mechanics or STR – and therefore not to theories with dynamical spacetime structure, such as GTR. For this reason, as already indicated above, proponents of the dynamical view are not to be identified naïvely with relationists.)

On the second of these two points: the idea – what Butterfield (2007) calls ‘Brown’s moral’ – is that one cannot simply posit a piece of geometrical structure in one’s theory, e.g. a Minkowski metric field in STR, and know ab initio that material bodies (in particular rods and clocks) will read off intervals of that structure; rather, whether this is the case or not will depend upon the constitution of, and dynamics governing, those material bodies. We will see below specific theories in which any such assumption seems to fail. Note that this second point is again consistent with the explanatory contention taken above to be characteristic of the dynamical approach: a given piece of structure inherits its operational significance as spacetime by dint of the behaviour of material bodies.

Before addressing the second of these two points, we should consider the first in greater detail. The claim that fixed spatiotemporal structure is to be ontologically reduced to facts about material bodies invites many questions, chief among which is perhaps the following: to what could this ontological reduction possibly amount? In the following section, we will see one particular metaphysical programme which promises to make good on this claim.

There is arguably a tight relationship between the geometrical symmetries of a spacetime and the symmetries of a theory that describes the physics of matter (in a broad sense, including fields) in it. (Theories such as GTR, in which space-time has its own dynamics are more complicated, and will be discussed later; for further discussion of symmetries in physics, see the entry on symmetry and symmetry breaking .) Each symmetry is a set of transformations, with a rule of composition: formally a ‘group’. For instance, the group of rotations in the plane has a distinct element for every angle in the range 0–360 degrees; the composition of two rotations is the single rotation through the sum of their angles. Spacetime symmetries are those transformations which leave invariant a piece of spacetime structure (e.g., the symmetries of Minkowski spacetime are translations, spatial rotations and Lorentz boosts: together, the so-called Poincaré transformations); dynamical symmetries are those transformations which leave invariant a set of dynamical equations (e.g., the symmetries of Maxwell’s equations of electromagnetism are again the Poincaré transformations). There are good reasons to hold that the symmetry groups of theory and spacetime must agree. First, since the theory describes matter, and hence (arguably) what is measurable, any theoretical symmetries not reflected in the postulated spacetime structure indicate unmeasurable geometry: for instance, if an absolute present were postulated in relativistic physics. While in the other direction, if there were extra spacetime symmetries beyond those found in the dynamics, then per impossible one could measure nonexistent geometric quantities: for instance, a theory that depends on absolute velocities cannot be formulated in Galilean spacetime (see the entry on absolute and relational space and motion: classical theories for further discussion of these Newtonian spacetime structures). Famously, Earman (1989, ch. 3) declares that the matching of space-time and dynamical symmetries is, thus, an ‘adequacy condition’ on a physical theory.

Given this ‘adequacy condition’, a given geometry for spacetime formally constrains the allowable theories to those with the just the right symmetries: not too many, and not too few. It was an assumption of many substantivalists (the views of whom are discussed below) that this constraint was not merely formal, but ontological: that the geometry is more fundamental than the laws, or that geometry offers a ‘real’ explanation of the form of the laws – such authors would, by the above categorization, qualify as proponents of a geometrical view. However, that the symmetries should agree does not specify any direction of dependence, and it could be reversed, so that the geometric symmetries are ontologically determined by those of the laws of the theory: hence the geometry itself is an expression of the (symmetry properties of the) dynamics of matter – transparently, this is consistent with the first of the two specific commitments of the dynamical view discussed above. In the words of Brown and Pooley (2006) (making these points about STR): “… space-time’s Minkowskian structure cannot be taken to explain the Lorentz covariance of the dynamical laws. From our perspective … the direction of explanation goes the other way around. It is the Lorentz covariance of the laws that underwrites the fact that the geometry of space-time is Minkowskian.”

Of the opposing geometrical approach to spacetime, Brown and Pooley (2006, p. 84) question the mechanism by which autonomous spacetime structure is supposed to explain or constrain the behaviour of material bodies. Although we will keep our attention focussed on the dynamical view in this subsection, rather than upon its opponents (see the following subsection for more on the explanatory capacities of spacetime), one might, however, ask at this point: does the dynamical view really do better in this regard? How is it that dynamical symmetries are supposed to explain, or account for, spacetime structure? In the context of theories with fixed spacetime structure, this question is answered by proponents of the dynamical view via an ontological reduction of spatiotemporal structure to symmetries of dynamical equations governing matter fields, as indicated in (1) above. (In fact, this ‘reduction’ is better described as a form of elimination , as we will see.) But this, in turn, invites yet more questions: how, metaphysically, is this ontological reduction operating? Can one in fact state dynamical laws, or understand them as “holding” or “governing”, without presupposing facts about spacetime structure?

Take, for example, Newton’s laws of motion. The 1st law asserts that bodies not acted upon by an external force will move with constant velocity; similarly for the 2nd law and acceleration. These laws seem to presuppose that these are meaningful terms, but in spacetime terms their meaning is given by geometric structures: for instance, constant velocity in Galilean spacetime means having a straight spacetime trajectory. And the problem is not restricted to Newtonian physics; the same point can be made regarding theories that presuppose the Minkowski background spacetime structure, e.g., the quantum field theories of the Standard Model.

It is increasingly well-appreciated that one suitable metaphysical program to which the dynamical approach can appeal here is Huggett’s (2006) regularity relationism: see (Huggett 2009; Pooley 2013; Stevens 2020). The idea is to consider the dynamical laws as regularities that systematize and describe the patterns of events concerning an underlying ontology/ideology that involves or presupposes only very limited spatiotemporal features. To illustrate how this approach might go, consider Pooley’s (2013, §6.3) proposal that the dynamical approach to STR might postulate only R 4 topological spatiotemporal structure, which could be (for example) attributed to a massive scalar field. Suppose we are given a full 4D field description of such a field, in terms of some arbitrary coordinate system. This would describe a simple ‘Humean mosaic’, to use David Lewis’ term for one’s basic spatiotemporal and material commitments (see article David Lewis for further discussion). Now, smooth coordinate changes applied to such a description will generate distinct mathematical representations of that Humean mosaic, given using distinct coordinatizations of the field-stuff. It might happen that, among all such representations, there is a subclass of coordinate systems which are such that (i) when the scalar field is described using a member of the class, it turns out that its values at spacetime points satisfy some simple/elegant mathematical equation; and moreover, (ii) the members of the class are related by a nicely-specifiable symmetry group. If this is so, then the simple/elegant equation can be taken as expressing a dynamical law for the world of this mosaic (understood as a statement: “There are frames in which …”), and the symmetry group of the law can be seen as capturing the derivative , not intrinsic, spacetime structure of the world. If the symmetry group is the Poincaré group, for example, then the field behaves ‘as if’ it were embedded in a spacetime with Minkowski geometry. But all this means is that the dynamics is representationally equivalent to a theory with an autonomous Minkowski geometry. From the point of view of the dynamical approach, such a theory is merely an interesting, and perhaps useful, representation of the real facts: and it’s a mistake to take every feature of a representation to correspond to something in reality (Brown & Read 2020, §5).

Even granting that this regularity relationist understanding of the dynamical approach goes through, three outstanding issues for the dynamical approach deserve to be mentioned. First: given that the proponent of the view seeks to excise metrical (more generally: geometrical) spacetime structure, one might ask: why stop there? Is there not something unnatural about excising fixed metric structure, while taking topological structure to be primitive? Such a concern was raised by Norton (2008), to which two responses have been offered in the literature: (I) In direct response to Norton, Pooley points out that “the project was to reduce chronogeometric facts to symmetries, not to recover the entire spatiotemporal nature of the world from no spatiotemporal assumptions whatsoever” (2013, p. 57). (II) Menon (2019) has argued that the machinery of ‘algebraic fields’ can be deployed in order to reduce topological structure to facts about matter, thereby, if successful, meeting Norton’s challenge head-on. Second: how is one to extend the dynamical approach, understood as a form of Huggett’s regularity relationism, to theories of dynamical space-time such as GTR? Here, the lack of spacetime symmetries in the theory has posed problems for the successful implementation of any such account (Stevens 2014), although arguably initial progress in this regard has been made by Vassallo and Esfeld (2016). Third: to which symmetries of the laws is the dynamical approach supposed to be sensitive? In the philosophy of physics, it is common to draw a distinction between ‘internal’ and ‘external’ symmetries: examples of the former include U(1) gauge transformations in electromagnetism; examples of the latter are coordinate transformations, such as Galilean boosts in Newtonian mechanics. But there are many questions here, such as: (i) how, precisely, is the distinction between internal and external symmetries to be drawn? (ii) why should the proponent of the dynamical approach stop at external symmetries? For discussion of these questions, see (Dewar 2020).

We have already seen how the dynamical approach, qua programme of ontological reduction, is supposed to play out in the context of theories with fixed spacetime structure, including both Newtonian theories and STR. We have also witnessed Brown and Pooley’s concerns about the ability of a substantival spacetime to explain facts about the behavior of matter. These concerns are motivated by apparent problem cases, in which the symmetries of a substantival spacetime seem to come apart from those of the dynamical laws governing matter. Such cases include: (i) Newtonian mechanics set in Newtonian spacetime (Read 2020a); (ii) the Jacobson-Mattingly theory (Jacobson & Mattingly 2001), in which dynamical symmetries are a subset of spacetime symmetries, as a result of the presence of an additional (dynamical) symmetry-breaking vector field (Read, Brown & Lehmkuhl 2018).

It is not obvious that these critiques are fair to proponents of a geometrical view. One might take their position not to be that a certain piece of geometrical structure (e.g., the Minkowski metric of STR) invariably constrains matter, whenever it is present in a theory, to manifest its symmetries (a claim which seems to be false, in light of the above cases). Instead, one might take their claim to be conditional: if one has matter which couples to this piece of geometrical structure in such-and-such a way, then that geometrical structure can explain why the laws have the such-and-such symmetries. In (Read, 2020a), the (arguably) straw man version of a geometrical view critiqued by Brown and Pooley is dubbed the ‘unqualified geometrical approach’, in contrast with this more nuanced and defensible version of the view, which is dubbed the ‘qualified geometrical approach’. (Brown might still reject the qualified geometrical approach on the grounds that it makes explanatory appeal to objects which violate the ‘action-reaction principle’, which states that every entity physical should both act on, and react to, other physical entities (Brown 2005, p. 140). If so, that this is the real reason for the rejection deserves to be flagged; moreover, it remains open whether the objection succeeds against the non-substantivalist versions of the geometrical view which are discussed below.)

Focussing on the qualified geometrical approach, there are also questions regarding the particular sense in which spacetime structure can be said to be explanatory of dynamical symmetries. One notion of explanation discussed in this literature is that of a ‘constructive explanation’.This is derivative on Einstein’s distinction between ‘principle theories’ and ‘constructive theories’ (Einstein 1919): for detailed discussion, see (Brown 2005, §5.2). In brief, a constructive explanation is one in which phenomenological effects are explained by reference to real (but possibly unobservable) physical bodies. (For further discussion of how to understand constructive theories and explanations, see (Frisch 2011).) With the idea of a constructive explanation in mind, one can say this: if a proponent of a geometrical view hypostatizes spacetime, then they can give constructive explanations of certain physical effects by appeal to that spacetime structure; otherwise, they cannot. That said, even if one does not hypostatise spacetime, and so concedes that spacetime cannot offer constructive explanations of the behaviour of matter, it is not obvious that spacetime cannot still facilitate other kinds of explanation. For discussions of these issues, see (Acuña 2016; Dorato & Felline 2010; Frisch 2011; Read 2020b).

As we have already seen in section 2, spacetime in GTR is dynamical. This leads Brown to maintain that there is no substantial conceptual distinction between the metric field of GTR and matter fields: “Gravity is different from the other interactions, but this doesn’t mean that it is categorically distinct from, say, the electromagnetic field” (Brown 2005, p. 159). In this sense, Brown is a relationist about GTR, and counts authors such as (Rovelli, 1997) as allies. However, much caution is needed concerning this use of the term ‘relationism’. In particular, in the context of GTR – and in significant contrast with his approach to theories such as STR – Brown makes no claim that the metric field should be ontologically reduced to properties of (the laws governing) matter fields; rather, in light of its dynamical status, the metric field of GTR “cries out for reification” (Brown, personal communication). Indeed, even if Brown did not maintain this, we have already registered above that there are technical problems with attempting to apply the dynamical approach, understood as a version of regularity relationism, to theories such as GTR.

In light of these issues, when considering GTR, Brown (2005, ch. 9) focuses entirely on thesis (2), presented in the introduction to this section: no piece of geometrical structure has its ‘chronogeometric significance’ of necessity – that is, no piece of geometrical structure is necessarily surveyed by physical bodies; rather, in order to ascertain whether such is the case, one must pay detailed attention to the dynamics of the matter fields constituting those physical bodies. This, indeed, should already be evident in light of the examples discussed in the previous subsection, such as the Jacobson-Mattingly theory, in which matter does not ‘advert’ to the designated piece of spacetime structure.

This thesis (2) should be uncontroversial. There are, however, concerns that the thesis is so uncontroversial that any distinction between the dynamical approach and its opponents in the context of theories such as GTR (and, in particular, without the regularity relationist approach to ontological reduction applied in the case of theories with fixed spacetime structure) has been effaced (Pooley 2013; Read 2020a). Even setting this aside, there are also disagreements regarding how exactly a piece of structure in a given theory is to acquire its ‘chronogeometric significance’ – that is, for the intervals which it determines to be accessible operationally to physical bodies and measuring devices. Brown’s preferred answer to this question (Brown 2005, ch. 9) makes appeal to the ‘strong equivalence principle’. There are a great many subtleties and technical difficulties which need to be overcome in order to attain a clear understanding of this principle (Read, Brown & Lehmkuhl 2018; Weatherall 2020), but, roughly speaking, it states that, in local regions in GTR, matter fields can be understood to obey Lorentz covariant dynamical equations, just as in STR (we have already seen something of this in section 2 above). Absent further details, pace Brown, it is not clear why this is sufficient to secure the ‘chronogeometric significance’ of the metric field in GTR. Even setting this aside, there are questions regarding whether the strong equivalence principle is necessary for chronogeometric significance. For example, an alternative approach might make appeal to the results of (Ehlers, Pirani & Schild, 1972), in which the authors demonstrate that the trajectories of massive and massless bodies are sufficient to reconstruct the metric field in GTR (cf. (Malament 2012, §2.1)). These issues are raised in (Read 2020a), but much work remains to be done in uncovering the full range of ways in which a given piece of structure might come to have chronogeometric significance.

This entry, and its companion on classical theories , have been concerned with tracing the history and philosophy of ‘absolute’ and ‘relative’ theories of space and motion. Along the way we have been at pains to introduce some clear terminology for various different concepts (e.g., ‘true’ motion, ‘substantivalism’, ‘absolute space’), but what we have not really done is say what the difference between absolute and relative space and motion is: just what is at stake? Rynasiewicz (2000) argued that there simply are no constant issues running through the history from antiquity through general relativity theory; that there is no stable meaning for either ‘absolute motion’ or ‘relative motion’ (or ‘substantival space’ vs ‘relational space’). While we agree to a certain extent, we think that nevertheless there are a series of issues that have motivated thinkers again and again. Rynasiewicz is probably right that the issues cannot be expressed in formally precise terms, but that does not mean that there are no looser philosophical affinities that shed useful light on the history and on current theorizing.

Our discussion has revealed several different issues, of which we will highlight three as components of the ‘absolute-relative debate’. (i) There is the question of whether all motions and all possible descriptions of motions are equal, or whether some are ‘real’ – what we have called, in Seventeenth Century parlance, ‘true’. There is a natural temptation for those who hold that there is ‘nothing but the relative positions and motions between bodies’ to add ‘and all such motions are equal’, thus denying the existence of true motion. However, arguably – perhaps surprisingly – no one we have discussed has unreservedly held this view (at least not consistently): Descartes considered motion ‘properly speaking’ to be privileged, Leibniz introduced ‘active force’ to ground motion (arguably in his mechanics as well as metaphysically), and Mach’s view seems to be that the distribution of matter in the universe determines a preferred standard of inertial motion. In general relativity there is a well-defined distinction between inertial and accelerated motion, given by the spacetime metric, but Einstein initially hoped that the metric itself would be determined in turn by relative locations and motions of the matter distribution in spacetime.

That is, relationists can allow ‘true’ motions if they offer an analysis of them in terms of the relations between bodies. Given this logical point, we are led to the second question: (ii) is true motion definable in terms of relations or not? (And if one hopes to give an affirmative answer, what kinds of relations are acceptable to use in the reductive definition?) It seems reasonable to call this the issue of whether motion is absolute or relative. Descartes and Mach were relationists about motion in this sense, while Newton was an absolutist. In the case of Einstein and GTR we linked relational motion to the satisfaction of Mach’s Principle, just as Einstein did in the early years of the theory. Despite some promising features displayed by GTR, and certain of its models, we saw that Mach’s Principle is certainly not fully satisfied in GTR as a whole. We also noted that in the absence of absolute simultaneity, it becomes an open question what relations are to be permitted in the definition (or supervience base) – spacetime interval relations? Instantaneous spatial distances and velocities on a 3-d hypersurface? The shape dynamics program comes at this question from a new perspective, starting with momentary slices of space (with or without matter contents) which are given a strongly relational – as opposed to absolute – interpretation. However, we argued that it ultimately remains unclear whether this approach vindicates Mach’s Principle.

The final issue we have discussed in this article is that of (iii) whether spacetime structures are substantial entities in their own right, metaphysically speaking not grounded on facts about dynamical laws, or whether instead it is best to think of the reality of spacetime structures as dependent upon, and explained by, facts about the world’s dynamical laws, as advocates of the dynamical approach maintain. The debate here is not the same as that between classical relationism and substantivalism, although there are clear affinities between the dynamical approach and classical relationism. We explored how this issue takes quite different forms in the context of special relativistic (Lorentz covariant) physical theories and in the context of general relativistic theories.

• Acuña, P. 2016, “Minkowski space-time and Lorentz Invariance: The Cart and the Horse or Two Sides of a Single Coin?”, Studies in History and Philosophy of Modern Physics , 55: 1–12.
• Barbour, J., 2012, Shape Dynamics. An Introduction , in F. Finster, et al ., Quantum Field Theory and Gravity: Conceptual and Mathematical Advances in the Search for a Unified Framework , Basel: Springer. doi:10.1007/978-3-0348-0043-3_13
• –––, 1977, “Gravity and Inertia in a Machian Framework,” Nuovo Cimento , 38B: 1–27.
• ––– and Bertotti, B., 1982, “Mach’s Principle and the Structure of Dynamical Theories,” Proceedings of the Royal Society (London), 382: 295–306.
• Belot, G., 2011, “Background-Independence”, General Relativity and Gravitation , 43: 2865–2884.
• Brill, D. R. and Cohen, J., 1966, “Rotating Masses and their effects on inertial frames,” Physical Review , 143: 1011–1015.
• Brown, H. R., 2005, Physical Relativity: Space-Time Structure from a Dynamical Perspective , Oxford: Oxford University Press.
• Brown, H. R. and Pooley, O., 2001, “The Origins of the space-time Metric: Bell’s Lorentzian Pedagogy and its Significance in General Relativity”, in C. Callender and N. Huggett (eds.), Physics Meets Philosophy at the Plank Scale, Cambridge: Cambridge University Press.
• –––, 2006, “Minkowski space-time: A glorious non-entity”, in Dennis Dieks (ed.), The Ontology of Spacetime , Amsterdam: Elsevier, 67–89.
• Brown, H. R. and Read, J., 2021, “The Dynamical Approach to space-time Theories”, in E. Knox and A. Wilson (eds.), The Routledge Companion to Philosophy of Physics , London: Routledge.
• Butterfield, J., 2007, “Reconsidering Relativistic Causality”, International Studies in the Philosophy of Science , 21(3): 295–328.
• Dewar, N., 2020, “General-Relativistic Covariance”, Foundations of Physics , 50: 294–318.
• Disalle, R., 2006, Understanding Space-time: The Philosophical Development of Physics from Newton to Einstein , Cambridge: Cambridge University Press.
• Dorato, M. and Felline, L., 2010, “Structural Explanations in Minkowski space-time: Which Account of Models?”, in V. Petkov (ed.), Space, Time, and space-time: Physical and Philosophical Implications of Minkowski’s Unification of Space and Time , Berlin: Springer.
• Dorling, J., 1978, “Did Einstein need General Relativity to solve the Problem of Space? Or had the Problem already been solved by Special Relativity?,” British Journal for the Philosophy of Science , 29: 311–323.
• Earman, J., 1970, “Who’s Afraid of Absolute Space?,” Australasian Journal of Philosophy , 48: 287–319.
• –––, 1989, World Enough and Spacetime: Absolute and Relational Theories of Motion , Cambridge, MA: MIT Press.
• Ehlers, J., Pirani, F. A. E., and Schild, A., 1972, “The Geometry of Free Fall and Light Propagation”, in L. O’Reifeartaigh (ed.), General Relativity: Papers in Honour of J. L. Synge , Oxford: Clarendon Press, pp. 63–84.
• Einstein, A., 1907, “Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen,” Jahrbuch der Radioaktivität und Electronik , 4: 411–462.
• –––, 1918, “Prinzipielles zur allgemeinen Relativitätstheorie,” Annalen der Physik , 51: 639–642.
• –––, 1919, “Time, Space, and Gravitation”, in The Times , 28th November 1919; reprinted as Document 26, in J. Stachel, et al. (eds.), The Collected Papers of Albert Einstein (Volume 7), Princeton, NJ: Princeton University Press, 1987–2006.
• Einstein, A., Lorentz, H. A., Minkowski, H. and Weyl, H., 1952, The Principle of Relativity , W. Perrett and G.B. Jeffery, (trans.), New York: Dover Books.
• Föppl, A., 1904, “Über absolute und relative Bewegung,” Sitzungsberichte der Mathematisch-Naturwissenschaftlichen Klasse der Bayerischen Akademie der Wissenschaften zu München , 34: 383–395.
• Friedländer, B. and J., 1896, Absolute oder relative Bewegung , Berlin: Leonhard Simion.
• Friedman, M., 1983, Foundations of Space-Time Theories: Relativistic Physics and Philosophy of Science , Princeton: Princeton University Press.
• Frisch, M., 2011, “Principle or constructive relativity”, Studies in History and Philosophy of Science (Part B), 42(3): 176–183.
• Gomes, H., 2012, “The Coupling of Shape Dynamics to Matter”, Journal of Physics: Conference Series , 360: 012058.
• Gourgoulhon, E., 2012, 3+1 Formalism in General Relativity: Bases of Numerical Relativity , Berlin/Heidelberg: Springer Press.
• Hoefer, C., 1994, “Einstein’s Struggle for a Machian Gravitation Theory,” Studies in History and Philosophy of Science , 25: 287–336.
• –––, 2014, “Mach’s principle as action-at-a-distance in GR:The causality question”, Studies in History and Philosophy of Modern Physics , 48: 128–136.
• Huggett, N., 2006, “The Regularity Account of Relational Spacetime,” Mind , 115: 41–74.
• –––, 2009, “Essay Review: Physical Relativity and Understanding Space‐Time,” (joint review of Harvey R. Brown, Physical Relativity: Space‐Time Structure from a Dynamical Perspective , Oxford: Oxford University Press, and and Robert DiSalle, Understanding Space‐Time: The Philosophical Developments of Physics from Newton to Einstein , Cambridge: Cambridge University Press), Philosophy of Science , 76(3): 404–422.
• Jacobson, T. and Mattingly, D., 2001, “Gravity with a Dynamical Preferred Frame”, Physical Review D , 64: 024028.
• Janssen, M., 2009, “Drawing the Line Between Kinematics and Dynamics in Special Relativity”, Studies in History and Philosophy of Modern Physics , 40: 26–52.
• Lange, L., 1885, “Ueber das Beharrungsgesetz,” Berichte der Königlichen Sachsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-physische Classe , 37: 333–51.
• Lehmkuhl, D., 2021,“The Equivalence Principle” in E. Knox and A. Wilson (eds.), The Routledge Companion to Philosophy of Physics , London: Routledge.
• Mach, E., 1883, Die Mechanik in ihrer Entwickelung, historisch-kritisch dargestellt , 2nd edition, Leipzig: Brockhaus; English translation, The Science of Mechanics , La Salle, Illinois: Open Court Press, 1960, 6th edition.
• Malament, D., 1995, “Is Newtonian Cosmology Really Inconsistent?,” Philosophy of Science , 62(4): 489–510.
• –––, 2012, Topics in the Foundations of General Relativity and Newtonian Gravitation Theory , Chicago, IL: University of Chicago Press.
• Maudlin, T., 1993, “Buckets of Water and Waves of Space: Why Space-Time is Probably a Substance,” Philosophy of Science , 60: 183–203.
• –––, 2012, Philosophy of Physics: Space and Time , Princeton, NJ: Princeton University Press.
• Menon, T., 2019, “Algebraic Fields and the Dynamical Approach to Physical Geometry”, Philosophy of Science 86: 1273–83.
• Mercati, Flavio, 2018, Shape Dynamics: Relativity and Relationalism , Oxford: Oxford University Press.
• Minkowski, H., 1908. “Space and time,” in Einstein, et al ., 1952, pp. 75–91.
• Nerlich, Graham, 1994, The Shape of Space , 2nd edition, Cambridge: Cambridge University Press.
• Neumann, C., 1870, Ueber die Principien der Galilei-Newton’schen Theorie , Leipzig: B. G. Teubner.
• Newton, I., 2004, Newton: Philosophical Writings , A. Janiak (ed.), Cambridge: Cambridge University Press.
• Newton, I. and I. B. Cohen, 1999, The Principia: Mathematical Principles of Natural Philosophy , I. B. Cohen and A. M. Whitman (trans.), Berkeley: University of California Press.
• Norton, J., 1992, “Einstein, Nordström and the Early Demise of Scalar, Lorentz-Covariant Theories of Gravitation,” Archive for History of Exact Sciences , 45: 17–94.
• –––, 1993, “A Paradox in Newtonian Cosmology,” in M. Forbes , D. Hull and K. Okruhlik (eds.), PSA 1992: Proceedings of the 1992 Biennial Meeting of the Philosophy of Science Association (Volume 2), East Lansing, MI: Philosophy of Science Association, pp. 412–20.
• –––, 2008, “Why constructive relativity fails”, British Journal for Philosophy of Science , 59(4): 821–834.
• Pitts, J. B., 2006, “Absolute Objects and Counterexamples: Jones-Geroch Dust, Torretti Constant Curvature, Tetrad-Spinor, and Scalar Density”, Studies in History and Philosophy of Modern Physics , 37(2): 347–371.
• Pooley, O., 2002, The Reality of Spacetime , D. Phil thesis, Oxford: Oxford University.
• –––, 2013, “Substantivalist and Relationalist Approaches to Spacetime”, in Robert Batterman (ed.), The Oxford Handbook of Philosophy of Physics , Oxford: Oxford University Press.
• –––, 2017, “Background Independence, Diffeomorphism Invariance, and the Meaning of Coordinates,” In Dennis Lehmkuhl (ed.), Towards a Theory of Spacetime Theories . Birkhäuser.
• Read, J., 2016, “Background Independence in Classical and Quantum Gravity”, B.Phil. thesis, University of Oxford.
• –––, 2020a, “Explanation, Geometry, and Conspiracy in Relativity Theory”, in C. Beisbart, T. Sauer and C. Wüthrich (eds.), Thinking About Space and Time: 100 Years of Applying and Interpreting General Relativity, Einstein Studies series, vol. 15 , Basel: Birkhäuser.
• –––, 2020b, “Geometrical Constructivism and Modal Relationalism: Further Aspects of the Dynamical/Geometrical Debate”, International Studies in the Philosophy of Science , 33(1): 23–41.
• Read, J., Brown, H. R., and Lehmkuhl, D., 2018, “Two Miracles of General Relativity”, Studies in History and Philosophy of Modern Physics , 64: 14–25.
• Read, J., and Menon, T., 2019, “The limitations of inertial frame spacetime functionalism”, Synthese , first online 29 June 2019. doi:10.1007/s11229-019-02299-2
• Rovelli, C., 1997, “Halfway Through the Woods: Contemporary Research on Space and Time”, in J. Earman and J. D. Norton, (eds.), The Cosmos of Science , Pittsburgh, PA: University of Pittsburgh Press, pp. 180–223.
• Roberts, J. T., 2003, “Leibniz on Force and Absolute Motion,” Philosophy of Science , 70: 553–573.
• Rynasiewicz, R., 1995, “By their Properties, Causes, and Effects: Newton’s Scholium on Time, Space, Place, and Motion – I. The Text,” Studies in History and Philosophy of Science , 26: 133–153.
• –––, 2000, “On the Distinction between Absolute and Relative Motion,” Philosophy of Science , 67: 70–93.
• Sklar, L., 1974, Space, Time and Spacetime , Berkeley: University of California Press.
• Stevens, S., 2014, The Dynamical Approach to Relativity as a Form of Regularity Relationalism , D.Phil. thesis, University of Oxford.
• –––, 2020, “Regularity Relationalism and the Constructivist Project”, British Journal for the Philosophy of Science , 71: 353–72.
• Vassallo, A. and M. Esfeld, 2016, “Leibnizian relationism for general relativistic physics,” Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics , 55: 101–107.
• Weatherall, J. O., 2020, “Two dogmas of dynamicism,” Synthese , first online 16 November 2020. doi:10.1007/s11229-020-02880-0
• Wheeler, J.A. and Ciufolini, I., 1995, Gravitation and Inertia , Princeton, N.J.: Princeton U. Press.
• Barbour, J. B., 1982, “Relational Concepts of Space and Time,” British Journal for the Philosophy of Science , 33: 251–274.
• Belot, G., 2000, “Geometry and Motion,” British Journal for the Philosophy of Science , 51: 561–595.
• Butterfield, J., 1984, “Relationism and Possible Worlds,” British Journal for the Philosophy of Science , 35: 101–112.
• Callender, C., 2002, “Philosophy of Space-Time Physics,” in The Blackwell Guide to the Philosophy of Science , P. Machamer (ed.), Cambridge: Blackwell, pp. 173–198.
• Carrier, M., 1992, “Kant’s Relational Theory of Absolute Space,” Kant Studien , 83: 399–416.
• Dasgupta, S., 2015, “Substantivalism vs Relationalism About Space in Classical Physics”, Philosophy Compass , 10: 601–624.
• Dieks, D., 2001, “Space-Time Relationism in Newtonian and Relativistic Physics,” International Studies in the Philosophy of Science , 15: 5–17.
• Disalle, R., 1995, “Spacetime Theory as Physical Geometry,” Erkenntnis , 42: 317–337.
• –––, 2006, Understanding Space-time: The Philosophical Development of Physics from Newton to Einstein , Cambridge: Cambridge University Press.
• –––, 1986, “Why Space is Not a Substance (at Least Not to First Degree),” Pacific Philosophical Quarterly , 67: 225–244.
• Earman, J. and J. Norton, 1987, “What Price Spacetime Substantivalism: The Hole Story,” British Journal for the Philosophy of Science , 38: 515–525.
• Hoefer, C., 2000, “Kant’s Hands and Earman’s Pions: Chirality Arguments for Substantival Space,” International Studies in the Philosophy of Science , 14: 237–256.
• –––, 1998, “Absolute Versus Relational Spacetime: For Better Or Worse, the Debate Goes on,” British Journal for the Philosophy of Science , 49: 451–467.
• –––, 1996, “The Metaphysics of Space-Time Substantialism,” Journal of Philosophy , 93: 5–27.
• Huggett, N., 2000, “Reflections on Parity Nonconservation,” Philosophy of Science , 67: 219–241.
• Le Poidevin, R., 2004, “Space, Supervenience and Substantivalism,” Analysis , 64: 191–198.
• Malament, D., 1985, “Discussion: A Modest Remark about Reichenbach, Rotation, and General Relativity,” Philosophy of Science , 52: 615–620.
• Maudlin, T., 1990, “Substances and Space-Time: What Aristotle would have Said to Einstein,” Studies in History and Philosophy of Science , 21(4): 531–561.
• –––, 1993, “Buckets of Water and Waves of Space: Why Space-Time is Probably a Substance,” Philosophy of Science , 60: 183–203.
• Mundy, B., 1983, “Relational Theories of Euclidean Space and Minkowski Space-Time,” Philosophy of Science , 50: 205–226.
• –––, 1992, “Space-Time and Isomorphism,” Proceedings of the Biennial Meetings of the Philosophy of Science Association , 1: 515–527.
• Nerlich, G., 1973, “Hands, Knees, and Absolute Space,” Journal of Philosophy , 70: 337–351.
• –––, 1994, What Spacetime Explains: Metaphysical Essays on Space and Time , New York: Cambridge University Press.
• –––, 1996, “What Spacetime Explains,” Philosophical Quarterly , 46: 127–131.
• –––, 2003, “Space-Time Substantivalism,” in The Oxford Handbook of Metaphysics , M. J. Loux (ed.), Oxford: Oxford Univ Press, 281–314.
• Norton, J., 1995, “Mach’s Principle before Einstein,” in J. Barbour and H. Pfister (eds.), Mach’s Principle: From Newton’s Bucket to Quantum Gravity: Einstein Studies (Volume 6), Boston: Birkhäuser, pp. 9–57.
• –––, 1996, “Absolute Versus Relational Space-Time: An Outmoded Debate?,” Journal of Philosophy , 93: 279–306.
• Pooley, O., 2013, “Substantivalist and Relationalist Approaches to Spacetime”, in Robert Batterman (ed.), The Oxford Handbook of Philosophy of Physics , Oxford: Oxford University Press.
• Stein, H., 1967, “Newtonian Space-Time,” Texas Quarterly , 10: 174–200.
• –––, 1977, “Some Philosophical Prehistory of General Relativity,” in Minnesota Studies in the Philosophy of Science 8: Foundations of Space-Time Theories , J. Earman, C. Glymour and J. Stachel (eds.), Minneapolis: University of Minnesota Press.
• Teller, P., 1991, “Substance, Relations, and Arguments about the Nature of Space-Time,” Philosophical Review , 100(3): 363–397.
• Torretti, R., 2000, “Spacetime Models for the World,” Studies in History and Philosophy of Modern Physics (Part B), 31(2): 171–186.

How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• St. Andrews School of Mathematics and Statistics Index of Biographies
• The Pittsburgh Phil-Sci Archive of pre-publication articles in philosophy of science
• Ned Wright’s Special Relativity tutorial
• Andrew Hamilton’s Special Relativity pages
• The Collected Papers of Albert Einstein (online, Princeton University Press)

general relativity: early philosophical interpretations of | Mach, Ernst | Newton, Isaac: views on space, time, and motion | space and time: absolute and relational space and motion, classical theories | space and time: the hole argument

Acknowledgments

Carl Hoefer’s research for this entry was supported by his employer, ICREA (Pg. Lluís Companys 23, 08010 Barcelona, Spain), and by Spanish MICINN grant FFI2016-76799-P.

Copyright © 2021 by Nick Huggett < huggett @ uic . edu > Carl Hoefer < carl . hoefer @ ub . edu > James Read < james . read @ philosophy . ox . ac . uk >

• Accessibility

Support SEP

Mirror sites.

View this site from another server:

• Info about mirror sites

The Stanford Encyclopedia of Philosophy is copyright © 2024 by The Metaphysics Research Lab , Department of Philosophy, Stanford University

Library of Congress Catalog Data: ISSN 1095-5054

Newton’s Cannon

This simulation is based on Isaac Newton’s famous thought experiment and illustration of firing a projectile from a high mountaintop at various speeds, as he described in A Treatise of the System of the World . The calculation behind it uses Newton's Second Law and the Law of Gravitation to predict step by step the progress of the projectile. Each time step lasts only 5 seconds. The projectile starts from a mountain that is 0.165 times Earth's radius above the mean surface to match Newton's original drawing scale. MIT License

To revisit this article, visit My Profile, then View saved stories .

• The Big Story
• Newsletters
• Steven Levy's Plaintext Column
• WIRED Classics from the Archive
• WIRED Insider
• WIRED Consulting

What Would It Take to Shoot a Cannonball Into Orbit?

Gravity is pretty complicated if you think about it. The motion of a ball falling on the surface of the Earth is caused by the same interaction as the moon orbiting the Earth. That's crazy. It's even crazier to realize that humans figured out that these two motions (falling ball and moon) are from the same gravitational force. It sure doesn't look the same.

Now imagine that you are around during the time of Isaac Newton (let's say early 1700s). How do you make this model of universal gravity? I don't know how he did it, but Newton finally made the connection between the motion of planets (and moons) and the motion of objects on the surface of the Earth. He explains this connection with his famous thought experiment of a cannon firing a ball from a tall mountain. Here is his diagram from A Treatise of the System of the World .

The diagram shows that an object moving on the surface of the Earth could eventually become an object orbiting the Earth. He does this by imagining a cannonball—a super fast cannonball fired from a super high mountain. The actual range of this ball would be farther than a normal cannon ball because you would have take into the curvature of the Earth. Oh, right—you have to ignore air resistance. In fact, if you shoot the ball fast enough it will "miss" the Earth entirely and enter a low Earth orbit.

There you have it. Gravity on the surface of the Earth is the same as the gravity between the moon and Earth. Like I said, this is a big deal.

I don't know who drew this diagram. Maybe it's been modified over time. However, I'm going to start with a version of it and then check it for accuracy. In particular, I want to know the height of the mountain and I want to check the trajectory of the cannon ball paths. It's just what I do.

Although it's just an image, it is still useful to video analysis software to analyze this image. Of course I'm going to use the free (and awesome) Tracker Video Analysis . From the image, I want to find the height of the "mountain". That's fairly straightforward. I can just set the radius of the Earth diagram to the radius of the Earth and then measure the height. This puts the top of the mountain at 1.198 times the radius of the Earth from the center of the Earth (it's easier to deal with this in terms of the Earth's radius). Oh, that makes the height of the mountain 19.8 percent the radius of the Earth—see how nice that is?

Just for comparison, Mount Everest has a height of 8848 meters . In terms of the Earth, this is 0.139 percent of the Earth's radius. Or to put this another way, Newton's mountain is seriously ginormous. I guess there are a bunch of other cool things we could consider regarding a mountain as high as Mt. Newton (that's what I'm calling it now)—but I will just leave those questions alone for now.

What about the trajectory of these cannonballs launched from Mt. Newton? Let's consider the trajectories of the first three cannonballs (the three lowest speeds). I don't know anything about the time for these motions, but I can get the x and y positions. Since this doesn't really fit with your normal projectile motion plots (y vs. x), I'm going to instead plot r vs. θ where r is the distance of the ball from the center of the Earth and θ is the angular position from the center of the Earth. Yes, this is polar coordinates.

Since the angle is measured from the horizontal x-axis, these cannonball trajectories start from the left and move right. I hope that's not too confusing. But the real question: are these trajectories real? I could show that an object interacting gravitationally with the Earth should have an elliptical trajectory, but I'm not going to do that. Nope. Instead I am going to make a numerical model and adjust the starting velocity until I get something close to one of these. It's going to be fun.

Here's the plan (it's the same plan as most numerical calculations ). But here are the basic steps.

• Break the problem into very small steps of time (about 1 second in this case).
• Calculate the gravitational force vector on the cannon ball based on the position of the ball with respect to the Earth.
• Use this force to update the momentum of the ball.
• Use the momentum (and thus velocity) to update the position of the ball.
• Repeat until you want to stop.

That's it. Here's what you get. Yes, this is an actual and real numerical calculation. You can see the code by clicking the "pencil" icon. In the code view, you can change the starting velocity. If you don't change that velocity, you are only cheating yourself. Seriously. Try changing the velocity.

OK, that's starting to look like the Newton picture. But can I get a trajectory that is JUST like the picture? Let me adjust the starting velocity to see if I get data that is very similar to the above three shots (from the picture). Here's what I get (as a plot of radial distance vs. θ).

This is for the first three cannon shots and you can see the trajectories don't quite match up. With my best estimates, these are cannonball speeds of 2800 m/s, 4200 m/s, and 6200 m/s. Oh, just for fun the orbital velocity at the height of Mount Newton would be 7252 m/s. Go ahead and use that velocity in the code example above. It should make a nice orbit.

So, let me summarize what we have. The diagram of Newton's cannonball seems nice, but it's just a sketch. If only Newton had python.

• What's the fastest 100 meter dash a human can run ?
• Amazon wants you to code the AI brain for this little car
• Spotify's year-end ads highlight the weird and wonderful
• You can pry my air fryer out of my cold, greasy hands
• Airports cracked Uber and Lyft— cities should take note
• Looking for more? Sign up for our daily newsletter and never miss our latest and greatest stories

Advertisement

The impossible barber and other bizarre thought experiments

How inventing riddles has revealed the flaws in our grasp of reality

By Stephen Battersby

11 May 2016

Dmytro Zinkevych/Alamy Stock Photo

If you imagined that thought experiments were mere mental gymnastics meant to bamboozle the uninitiated, think again. Take Schrödinger’s cat, perhaps the most famous example, which involves a cat that is simultaneously alive and dead. It seems bizarre – and that’s the point. It was designed as a slap on the wrist for quantum theorists, to show that a theory that predicts such nonsense must be missing something. Current thinking is that perhaps nothing is missing, and quantum theory really is as weird as it seems.

But other thought experiments have forced us to reformulate the laws that describe nature. Take Maxwell’s demon, which appears to break the laws of thermodynamics. It showed us that thermodynamics really was missing something (see “ Matter, energy… knowledge: How to harness physics’ demonic power “).

Here are seven classic thought experiments that might make you think…

The impossible barber

A certain barber is very particular about his work. He shaves every person who does not shave themselves, and no one who does shave themselves. So: does the barber shave himself? It doesn’t take long to see the contradiction: If he does, he can’t; if he doesn’t, he must. Such a barber can’t exist.

This barber is often used to illustrate a more abstract puzzle known as Russell’s paradox. In 1901, mathematician and philosopher Bertrand Russell was investigating set theory, a formal way of defining and dealing with collections of anything. At the time, one of its central ideas was that for every property you can define, there must be a set. There’s the set of all green things, and the set of all whole numbers except 4. You can also define sets of sets: say, the set of all sets that contain exactly two elements. The problem comes when pondering the possibility of a set of all sets that do not contain themselves — like the barber, this seems to be impossible.

The paradox exposed contradictions in much of the mathematics of the time , forcing Russell and others to try to devise more intricate logical footings for mathematics. Russell’s approach was to say that mathematical objects fall into a hierarchy of different “types”, each one built only from objects of lower type. Type theory has been used to design computer programming languages that reduce the chance of creating bugs. But it’s not the definitive solution – more than a century later, mathematicians are still arguing over the answer to Russell’s paradox.

Galileo’s balls

Galileo may never have dropped balls from the top of the leaning tower of Pisa, as the legend goes. But he did devise a simple thought experiment that told us something profound about gravity. Take two weights, one light, one heavy. If heavier objects fall faster than light ones, as Aristotle said, then the lighter weight will lag behind. That implies that when the two are tied together, they will fall more slowly than the heavy weight alone. But together, they weigh more than the heavy alone, so they should fall faster. Wait, so is it faster or slower?

As Galileo realised, acceleration due to gravity doesn’t depend on the mass of an object. This was a crucial result for the emerging science of physics, and Isaac Newton’s ideas of motion and universal gravitation. It even holds a germ of Einstein’s subtle theory of gravity. His general theory of relativity is rooted in the equivalence principle , the idea that gravity and acceleration are essentially the same thing— as Galileo glimpsed back in the 17th century.

Newton’s cannon

Take one gigantic cannon, put it on top of a mountain so high it reaches above the atmosphere, and fire horizontally. Irresponsible, perhaps, but instructive. If the cannonball is fired at a low speed, gravity will soon drag it down to the ground along a tightly curving arc. If you add more gunpowder, the ball will go faster and its arc will be more gradual, taking it further around the curve of the Earth. Fire it fast enough and the cannonball’s path will not meet the ground at all – it will fly all the way around and hit you in the back of the head. Here, have a go .

This thought experiment helped Newton show that gravity is a universal force: the force we see pulling cannonballs and apples to Earth can also explain the orbit of the moon around Earth, and Earth around the sun.

We are used to the idea of universal forces now. We know that nuclear reactions fuel the distant stars, and that exoplanets can be magnetic. But before Newton there was no expectation that the celestial realm should have the same rules as Earth. His cannonball blew a big hole in such heavenly pretentions.

Achilles and the tortoise

Two-and-a-half millennia ago, the Greek philosopher Zeno of Elea apparently proved that motion is an illusion. One of his paradoxes sets fleet-of-foot Achilles to chase a tortoise that has a small head start. Achilles can never catch the tortoise, argued Zeno, because first he must reach the point where the tortoise started, but by then the tortoise has moved on to a new position. So then Achilles must run there – by which time the tortoise has moved on again. The “dichotomy paradox” is more general: to cover any distance, you must first cover half that distance, then half of what’s left, then half of what’s left, and so on for ever. It seems that you can never get there, no matter what the original distance or how fast you move.

Since then, mathematicians have pointed out that although these arguments take an infinite amount of time to pan out, real motion doesn’t have to. We know for instance that that an infinite series of terms can add up to something finite. If you add an infinite series of fractions starting with ½ and halving in value with each new term (½ + ¼ +1/8…), the infinite sum is equal to 1. You can use maths like this to represent the distance travelled or the time taken in Zeno’s paradox, so — phew – motion is possible after all. That said, Zeno’s paradox may manifest itself for real in the quantum world .

The Chinese room

Can a computer be conscious? In an attempt to disprove this idea of “strong artificial intelligence ”, John Searle , a philosopher at the University of California, Berkeley, imagined himself inside a room of dictionaries and rule books that hold instructions for translating Chinese to English and vice versa. Someone posts a question through the door written in Chinese, and using his rule books Searle works out an appropriate answer. To the questioner it would seem there is a mind in the room that understands Chinese, even though there isn’t. Searle claims that a hypothetical rule-bound computer designed to speak Chinese would be the same — a mere machine with no understanding.

There are many objections to this thought experiment. Some argue that although Searle does not understand Chinese, he is part of a larger system, including the rule books, that does. You might baulk at the idea that a mind could be made from a person plus some books, but it’s only a very dim mind, taking perhaps years or millennia to respond to one question.

Another interpretation is that Searle’s idea merely highlights the mystery of “other minds”: that you can’t know whether a computer, a penguin or the person next door is conscious in the same way as you are . If the Chinese room doesn’t disprove strong AI, thinking about it could help us to find out what’s missing from our understanding of consciousness.

Ride a light beam

In his Autobiographical Notes , Albert Einstein tells us how as a 16-year-old he imagined riding along with a light beam. If you could keep pace with it, the light must appear stationary, he imagined. Its oscillating electric and magnetic fields would be frozen. But that seems impossible. The equations developed by James Clerk Maxwell that describe the oscillations of electromagnetic fields forbid it, and we’ve certainly never seen such a thing as frozen light.

“One sees in this paradox the germ of the special relativity theory is already contained,” he wrote in 1947. As Einstein came to realise, the motion of light is the same no matter how fast you are moving. Even if you were travelling at almost the speed of light, the ray would still zip away from you at the same constant speed. This idea eventually led Einstein to an entirely new way of seeing the universe through the equations of special relativity, with their extraordinary predictions that time is elastic and that inert matter holds vast quantities of energy.

Laplace’s demon

Imagine a being that knows the place and motion of every particle in the universe. It also knows physics, and its mind works so fast that it can calculate how these particles will exert forces on one another, changing their motions. Can this intellect, described by Pierre Laplace in 1814, see the future of everything?

“Laplace’s demon”, as it became known, probes the idea of determinism. In a purely classical world, the demon seems to work. Chaos theory means that the future is ultra-sensitive to the past, but if the demon’s knowledge is infinitely precise, it could still know the fate of all.

Quantum mechanics may slay the demon. In mainstream quantum theory, events do not always have causes: radioactive decay and other things can happen spontaneously. But not all interpretations of quantum mechanics include this indeterminacy.

Even if the demon can live on in a universe governed by quantum mechanics, however, it probably doesn’t live around here. There is a mathematical argument that shows “the entire physical Universe cannot be fully understood by any single inference system that exists within it”. You might conceive of the demon as some kind of outside observer, but that opens another philosophical can of worms: is it meaningful to say that something can know all about our universe without having any physical effect on us?

• general relativity /
• mathematics /
• consciousness

Sign up to our weekly newsletter

Receive a weekly dose of discovery in your inbox! We'll also keep you up to date with New Scientist events and special offers.

More from New Scientist

Explore the latest news, articles and features

Mathematics

The mathematical theory that made the internet possible.

Subscriber-only

The surprising connections between maths and poetry

DeepMind AI gets silver medal at International Mathematical Olympiad

A skilful primer makes sense of the mathematics beneath AI's hood

Popular articles.

Trending New Scientist articles

• Skip to Page Content
• Skip to Site Navigation
• Skip to Search
• Skip to Footer

OLogy Cards > thought experiment

thought experiment

A thought experiment is an experiment that can be described but is not actually performed. Instead, it is carried out by the mind, using reasoning and logic. Thought experiments have been used by great thinkers since ancient times, and they are still used today to explain concepts such as quantum physics and relativity. These "virtual experiments" help us explore the world.

What it is : a virtual experiment carried out by the mind Famous "thought experimenters" : Galileo, Newton, Einstein When it's used : often in astronomy and physics to understand something about the Universe How it's done : 1) Ask a question; 2) visualize the situation; 3) make a prediction based on logic; 4) develop a theory.

Famous Thought Experiments Since ancient times, great thinkers have used thought experiments to understand our Universe. In the first century B.C.E., the philosopher and poet Lucretius used a thought experiment to show that space is infinite: if the Universe had a boundary, that would mean you could throw a spear at it. If it pierced the boundary, there is something on the other side. If it bounced back, there must be some kind of wall existing in further space. Either way, there could be no edge of the Universe. In the 17th century, the physicist and mathematician Isaac Newton wondered, if he shot a cannonball hard enough, would it circle the planet without touching the ground? Later, this thought experiment would be carried out by satellites. In orbit, things are "falling" towards the Earth, but at speeds fast enough so that they never reach the ground.

Einstein predicted the twin who traveled into space would be ??? than his brother.

the same age

The "twin paradox" is a famous thought experiment: one twin stays on Earth while the other takes a trip into space. The twin who traveled to space would be younger than his brother, since for him, time moved more slowly.

In one of Einstein's most famous thought experiments, he wondered what it would be like to ride a:

beam of light

At the age of 16, Einstein wondered: "If I could ride a beam of light, how would the world look?" This question inspired one of the greatest thought experiments of all time, and helped Einstein figure out his Special Theory of Relativity.

A thought experiment is impossible to carry out for physical, technical, or practical reasons.

Some thought experiments can be done in reality but many are conducted using scientific reasoning and a little imagination.

You might also like...

Train of thought.

Take your imagination on a wonderful, mind-bending trip with these "thought experiments"!

Views from Windows

Take a personal look at how the height and velocity of your vantage point can affect the view.

Space and Time

How do you describe your place in the 4th dimension?

Sean P. Robinson

Lecturer, m.i.t. physics department, search form, you are here, newton's cannon, newton's cannon.

This simulation is based on Isaac Newton's thought experiment and illustration of firing a projectile from a high mountaintop at various speeds, as he described in A Treatise of the System of the World .

The past and present of thought experiments’ research at Glancy: bibliometric review and analysis

• Open access
• Published: 07 September 2024
• Volume 3 , article number  142 , ( 2024 )

Cite this article

You have full access to this open access article

• Hartono Bancong 1  

35 Accesses

Explore all metrics

In the development of physical theories, thought experiments play a crucial role. Research on this topic began in 1976 and has continued to the present. This study aims to provide a more complete picture of the progress of thought experiments over the past two decades. To achieve this, this study employs bibliometric mapping methods. A total of 679 published papers were analyzed, including articles (504), conference papers (92), and book chapters (83). This data was retrieved from the Scopus database. The study's findings reveal that research and publications on thought experiments are highly valued and have received significant attention over the past eight years. According to the findings, 90% of the top 20 source titles contributing to thought experiments are from journals in the first and second quartiles (Q1 and Q2). This quartile ranking shows the quality and significant influence of a journal. The geographical distribution indicates that the United States contributes the most to thought experiments research, with 213 documents, 2592 citations, and 47 links. We also identified several prospective keywords that could be the focus of future research, including artificial intelligence, physics education, fiction, God, theology, productive imagination, technology, speculative design, and critical design. Therefore, this study provides a thorough picture of thought experiment research trends and future directions of potential topics that can be the focus of future researchers.

Similar content being viewed by others

Design Science Research: Progression, Schools of Thought and Research Themes

The Origins of Design Principles: Where do… they all come from?

Thought Experiments in Science and in Science Education

Explore related subjects.

• Artificial Intelligence

Avoid common mistakes on your manuscript.

1 Introduction

Thought experiments (TEs) have a long history in science. Since Ernst Mach, the term TEs, a direct translation of the phrase Gedankenexperimente , has been widely discussed in the philosophy of science [ 18 ]. Thought and experiments are two components of TEs [ 4 , 18 , 29 ]. The thought element involves visualizing an imaginary world based on theory and experience, whereas the experimental aspect entails practical tasks in a physical laboratory, such as manipulating items and related variables. While some authors consider TEs to be mere arguments [ 24 ], others believe TEs are a form of fiction since their function is comparable to literary fiction in that both have a narrative framework by creating scenarios of occurrences from beginning to end [ 13 , 22 ]. However, unlike fiction, which frequently provides contradictory discourses, we believe that TEs should be logically and conceptually cohesive. TEs are structured imaginative actions based on the theory and experience of thought experimenters to achieve certain goals.

The contributions of TEs to the growth of scientific theories, particularly in physics, are essential. Physicists have employed TEs several times throughout history to either come up with new hypotheses or disprove previous ones. As the most representative examples, Newton used the TEs of cannonballs to support his hypothesis that the force of gravity is universal and the principal force of planetary motion, or Galileo used the TEs of free-falling bodies to disprove Aristotle's theory of gravity, which stated that the speed of falling objects is proportional to their weight. Galileo’s falling body, Newton's bucket and cannon, Maxwell's demon, and Schrodinger’s cat are just a few of the well-known TEs in physics [ 4 ]. These are only a few examples of the significant role TEs played in the development of scientific theories.

In the past 10 years, several works have studied TEs from the perspectives of history and philosophy of science [ 7 , 8 , 10 , 30 , 33 ]. Because most existing historical work on TEs focuses on individual TEs or individual accounts of TEs, reassessing the history of the philosophical debate on TEs becomes essential [ 33 ]. In the philosophy of science, historical debates regarding interactions between various philosophers or philosophical explanations across time in developed TEs are sometimes disregarded. Several studies have also used TEs as an imaginative tool in the classroom to teach science subjects. Velentzas and Halkia [ 37 ], for example, used TEs from Newton's Cannon to teach satellite physics. They then assert that TEs, as a teaching tool, can assist students in strengthening their syllogistic abilities and help them conceive scenarios beyond their everyday experience [ 37 ]. El Skaf and Palacios [ 12 ] have also systematically analyzed the epistemic role of TEs from Wheeler's demon and Geroch's engine, which gave rise to black hole thermodynamics. Recently, Bancong et al. [ 2 ] reported that physics teachers in Indonesia have a high awareness of the importance of TEs in learning physics, especially atomic theory and relativity, even though they lack skills in the pedagogic aspects of TEs. Therefore, Indonesian physics teachers also suggest using technology such as virtual reality to help visualize an imaginary world when performing TEs.

Although a number of studies on TEs from various perspectives have been conducted, no study has yet completely examined this field to look at the trend of this topic in recent years. Therefore, it becomes essential to conduct a bibliometric study of TEs over time based on authoritative databases like Scopus. Because of Scopus's comprehensive coverage of scholarly articles in the field of education [ 23 , 27 , 34 ], it was chosen as the database for this study. Scopus is also a popular resource for bibliometric research [ 23 , 28 ]. For this reason, we use data sources from the Scopus database to carry out the bibliometric method. Our study covers journal articles, conference papers, and book chapters from the last 20 years to provide a more complete view.

To highlight the significance of TEs research, we compare its growth to other scientific topics. While many scientific fields have seen growth over the past two decades, TEs research has also shown a unique and sustained increase in interest and publications. This trend contrasts sharply with the decline in research focus on traditional physics experiments [ 41 ]. Similarly, other topics in physics education, such as methodological issues, textbook analysis, and pre-service physics teachers, are also experiencing reduced research interest [ 25 ]. Additionally, the integration of TEs with emerging technologies, such as artificial intelligence, underscores their evolving relevance and potential for future research [ 21 ].

Therefore, this study aims to provide an up-to-date overview of trends in TEs research. The research questions in this study are as follows:

How is the growth of research output on the topic of TEs over the last 20 years?

Which source titles have contributed the most to the publication of papers on TEs in the last 20 years?

Who are the most prominent authors on the topic of TEs in the last 20 years?

Which countries have published the most articles on TEs over the past 20 years?

What are the most relevant keywords that can be found in the studies of TEs over the last 20 years?

2.1 Research design

This study aims to analyze the trends in TEs research over the past 20 years by using a bibliometric mapping method. To ensure a thorough analysis of recent trends and developments, this study focused on studies published between 2003 and 2022. This period was chosen because of significant advancements in research methodologies and bibliometric analysis tools in the early 2000s, as well as the consistent growth and comprehensive coverage of the Scopus database since that time. Bibliometric analysis is a well-known statistical method for examining and analyzing a large amount of scientific data on a certain topic [ 26 , 39 ]. Metrics studied in bibliometric research include annual publications, source titles, authors, institutions, nations, and keywords, covering data from primary, secondary, and tertiary journals over a specific time period. It should be noted that no ethical approval was required for this study as it did not involve humans or animals.

2.2 Data collection

In this study, data were gathered from the Scopus database ( https://www.scopus.com ). Scopus was chosen because it covers a wider range of documents than any other scientific database [ 23 , 28 , 35 ]. Scopus is the world's largest abstracting and indexing database, with 84 million records containing over 18.0 million open access items, including gold, hybrid gold, green, and bronze, as well as 10.9 million conference papers, 25.8 thousand active peer-reviewed journals, and over 7000 publishers [ 14 ]. In addition, Scopus covers a wider range of educational disciplines than other databases, such as the Web of Science (WoS) [ 23 , 27 , 34 ]. As a result, using the Scopus database enables researchers to shed light on areas that may not be covered in WoS.

Electronic data search and retrieval were conducted on February 25, 2023. Keyword search was set to include title, abstract, and keywords. The keyword search was set to include the title, abstract, and keywords. The combination of search strings, operators, and filters used in this study was TITLE-ABS-KEY ("Thought-experiments" AND "Science" OR "Physics"). Quotation marks were used to focus on documents containing this exact phrase, thus ensuring high relevance to the study's scope. The Scopus database retrieved 898 documents related to these keywords with full bibliographical information, including articles (67.04%), paper proceedings (10.13%), book chapters (10.02%), and other types of documents (12.81%). By using the Scopus filter, other types of publications (12.81%), including review articles, were excluded from the list of documents. The exclusion of review articles was intentional to focus on original research contributions that advance the field of TEs directly. Including reviews could confound the analysis as they often summarize existing research rather than introduce new findings. Therefore, concentrating on the three most prevalent types of documents—articles, conference papers, and book chapters—allowed for a clearer interpretation of trends and patterns in original research outputs over the specified period. Additionally, we limited the year of publication to studies published within the last 20 years (2003–2022) to ensure the relevance and currency of our analysis. After using a filtering process to eliminate papers that did not meet the inclusion and exclusion criteria, a total of 679 articles were identified for bibliometric analysis. These articles included 504 articles, 92 book chapters, and 83 conference papers.

2.3 Data analysis

The data analysis process began with acquiring the necessary raw data by downloading it from the Scopus database in either comma-separated value (CSV) or research information system (RIS) format. For data analysis and visualization, we used VOSviewer and Microsoft Excel. VOSviewer, a sophisticated mapping tool, was employed to create collaborative networks for various variables and keywords, while Microsoft Excel was used for descriptive analysis, such as determining the number of articles published each year and identifying the most prolific source titles.

The network graphs in this study were generated using VOSviewer, based on co-authorship, co-occurrence, and citation data from Scopus. The analysis type focused on the co-occurrence of keywords and co-authorship, with a full counting method. Keywords with a minimum of four occurrences were included. The visualization settings in VOSviewer were mainly default, with the attraction parameter set to 2 and the repulsion parameter set to 0. These settings ensured that the most relevant and frequently occurring terms were highlighted, providing a clear overview of research trends and collaborations in the field of TEs over the past 20 years.

In this study, we explored the most productive publishers, the most referenced articles, the most productive authors, the most productive nations, and author keyword occurrences across time. An analysis of co-authorship and co-occurrence was performed at this stage. The analysis of co-authorship provides insights into the interactions between authors. This methodology was also used for metrics related to countries. For country attribution, we included all the countries of all authors involved in each publication, not just the corresponding author. This method ensures that all co-authors' contributions are acknowledged and provides a comprehensive representation of the global distribution of research. Co-occurrence analysis was employed as a means of investigating current keywords and their interrelationships with other phrases associated with TEs. Within this particular framework, the term “node size” refers to the frequency at which a certain keyword appears in comparison to other words. Additionally, interconnected nodes are visually represented by lines known as connections. The link establishes a connection between two nodes, while the width of the link signifies the intensity or potency of the connection between the aforementioned nodes [ 36 , 39 ].

In the context of network map visualization, nodes that exhibit a high degree of association are categorized into clusters. The clustering of items was performed using the Louvain algorithm, a popular method for community detection in large networks due to its efficiency and accuracy in handling large datasets [ 36 , 39 ]. This algorithm was chosen for its ability to uncover modular structures within large networks, which is particularly useful for identifying distinct research themes and collaboration groups in bibliometric data. Subsequently, a distinct color code was assigned to each cluster, wherein nodes within the same cluster exhibit a high degree of homogeneity. Therefore, this bibliometric mapping approach enabled researchers to discern patterns and emerging areas of interest throughout the timeframe spanning from 2003 to 2022. Figure 1 shows the stages in the process of collecting and analyzing data in this study.

The steps in collecting and analyzing the data

3.1 Statistics analysis

In this analysis, we use statistical data to observe differences in the number of articles published each year. The goal is to determine whether the quantity of publications on the topic of TEs has increased or decreased annually. Figure 2 illustrates the number of papers published over the last 20 years (2003–2022). As we can see, there has been an increase in the interest and attention of researchers, scholars, and experts in researching TEs. The growth started in 2004 and continued until 2006. The number of papers published then fluctuated between 2006 and 2015. The increase started again in 2015 and continued until 2021. The number of publications increased significantly in 2021, with 69 articles published. This growth demonstrates that research and publications on TEs are in high demand and have garnered significant attention globally in the last eight years despite a reduction in 2022. Although studies in this area are still ongoing, these findings indicate an annual growth in the writing and publication of TEs on Scopus.

Number of articles published each year

Statistical data are also used to see the number of source titles that have made the greatest contributions to TEs during the last 20 years. A total of 679 papers have been published from various sources with different types of documents in the form of articles (504), conference papers (92), and book chapters (83). According to statistical data in the Scopus database, publication in journals is very significant in publishing research on the topic of TEs, while publication in proceedings and book chapters with the main scope of TEs is not very significant. Therefore, researchers, academics, and experts are advised to submit their articles focused on TEs to journals rather than proceedings and chapter books. Table 1 lists the top 20 sources of scientific research publications covering the topic of TEs from 2003 to 2022.

As seen in Table 1 , 90% of the source titles contributing to the TEs topic are journals, with only one publishing conference proceedings. Philosophical studies ranks first, with 17 documents published in the last 20 years. This is followed by the AIP Conference Proceedings with 15 documents. The American Journal of Physics, Science and Education, and Studies in History and Philosophy of Science Part A have published 11 documents each. Other source titles, such as Synthese (10), Foundations of Science (9), Physics Teacher (9), Journal for General Philosophy of Science (8), and Philosophy of Science (8), also contributed to publishing TEs topics. Minds and Machines and Physics Education each published seven documents. Erkenntnis, European Journal of Physics, Physics Essays, and Religions each published six documents, Acta Analytica published five documents, while Axiomathes, Boston Studies in the Philosophy of and History of Science, and European Journal for Philosophy of Science each published four documents.

3.2 Bibliometric analysis

3.2.1 contributions of authors.

Table 2 shows the 10 most prolific authors based on the total number of published articles from 2003 to 2022. As shown in this list, Stuart is the most significant author with 7 papers (51 citations), followed by Bancong from Universitas Muhammadiyah Makassar, Indonesia, with 5 papers (15 citations). Following Bancong, Fehige from the University of Toronto, Canada, has also published 5 articles. The majority of Fehige’s research focuses on TEs in the context of religion. In contrast to Fehige, Brown, also from the University of Toronto in Canada, has studied TEs through the lens of history and philosophy of science in several of his works (4 documents, 52 citations). Similarly, Buzzoni (3 documents, 15 citations) and El Skaf (3 documents, 29 citations) from Italy, discuss TEs from historical and philosophical perspectives of science. Meanwhile, Halkia and Velentzas from the University of Athens, Greece, have analyzed TEs thoroughly from an educational standpoint, with the number of documents being 4 and 86 citations.

3.2.2 Contributions of country

In the context of the leading countries, authors from 64 different countries/territories published a total of 679 documents. Table 3 lists the top 20 countries in terms of TE contributions based on the number of papers published. As shown, the United States contributes the most to TEs research, with 213 documents, 2592 citations, and 47 links. The number of papers is about three times that of the United Kingdom, which comes second (75 documents, 1016 citations, and 31 links). European countries continue to hold third to sixth place, with Germany publishing 50 documents with 634 citations, followed by Canada (43 documents, 410 citations, and 17 links), Italy (33 documents, 96 citations, 6 links), and the Netherlands (28 documents, 342 citations, and 12 links). This suggests that countries in America and Europe contribute the most to TEs. The Asian country that has contributed the most to TEs is China, with 18 documents, 286 citations, and 11 links, followed by India (14 documents), Japan (12 documents), and South Korea (12 documents), with 97, 111, and 27 citations, respectively. The three countries below these are European countries, with Austria having issued 10 documents related to TEs with a total of 135 citations, followed by Finland (9 documents, 31 citations) and Spain (9 documents, 47 citations).

3.2.3 Keywords

The results of a keyword analysis can be used in further investigation of the topic at hand. This study employs a minimum threshold of two occurrences of keywords in all research articles that were examined using VOSviewer. Figure 3 displays the 253 authors' keywords detected from 1990, which may be categorized into six distinct clusters. Cluster 1 is characterized by a red color, Cluster 2 by a green color, while Cluster 3 is distinguished by a blue color. In addition, Cluster 4 is characterized by a yellow color, Cluster 5 has a purple hue, and Cluster 6 is distinguished by a light blue shade. Each cluster is comprised of interconnected keywords that are visually represented by the same colors. It is important to note that the size and shape of the node are indicative of the frequency of its occurrences [ 36 , 39 ]. In other words, there is a positive correlation between the size of the node and the frequency of occurrences of these terms. Clustering is employed as a means to gain insights or a comprehensive understanding of bibliometric groupings, whereas image mapping serves the purpose of obtaining a holistic depiction of a bibliometric network.

Network visualization of TEs

Figure 3 shows Cluster 1 (red) with 68 items such as thought experiments, intuition, Science, Kant, Aristotle, Galileo, Platonism, personal identity, theology, fiction narrative, moral motivation, and neuroscience. Cluster 2 (green) consists of 57 categories, such as science fiction, philosophy of science, philosophy of physics, philosophical thought, epistemology, knowledge, scientific reasoning, experiments, models, and realism. Cluster 3 (blue) contains 41 items, such as consciousness, Maxwell's demon, Schrodinger's cat, quantum theory, entropy, uncertainty principle, quantum entanglement, quantum information, quantum physics, and Newton's bucket. Furthermore, cluster 4 (yellow) consists of 30 items: physics education, science education, visualization, special theory of relativity, history of physics, problem-solving, exploration, Einstein, relativity, and falsification. Cluster 5 (purple) consists of 29 items: imagination, ontology, physics, truth time, algorithm of discovery, artificial intelligence, ethics, nanotechnology, fiction, philosophy, and technology. Finally, cluster 6 (light blue) contains 16 categories, including popular science, fictionality, narrative, construction, sensation, a priori, story, Mach, memory, productive imagination, and schema.

Keywords in clusters 1 and 2 have a high number of occurrences and a high total link strength. The term thought experiment ranks first with 85 occurrences and a total link strength of 91. This is followed by the term thought experiment with 60 occurrences, a total link strength of 98, and several other keywords. The high number of occurrences and high total link strength indicate that scientific research publications on the topic of TEs in the 2003–2022 range indexed by Scopus have a strong and direct relationship with these keywords. Table 4 displays the ten keywords with the highest occurrence and overall link strength in the last 20 years on the topic of TEs.

VOSviewer, on the other hand, is also used to visualize the progress of keywords over a certain period. Figure 4 illustrates the overlay visualization of the TEs topic in the time range 2003 to 2022.

Overlay visualization of TEs

Figure 4 depicts the annual distribution of the number of articles containing keywords. The various colors represent the publication dates of the related papers where these keywords appear together. The data in Fig. 4 indicate that the most frequently used topics related to TEs from 2010 to 2014 were quantum theory, ethical naturalism, ethical naturalism, quantum mechanics, scientific discovery, and mental models. Then, from 2014 to 2018, keywords such as scientific reasoning, intuition, science education, computer simulation, history of science, and science fiction began to appear in the TEs topic. The hottest topics in TEs research are shown in yellow color, including fiction, artificial intelligence, God, theology, speculative design, critical design, and methods of case. These findings indicate that these keywords have gained popularity in recent years. It can be concluded that scholars have increasingly turned to research on the mentioned topics in recent years.

4 Discussion

The goal of this study is to use the bibliometric mapping method to examine the trend of studies on TEs during the last 20 years (2003–2022). According to the findings of the study, there has been an increase in the interest and attention of researchers, scholars, and professionals in studying TEs. Although research in this area is ongoing, these findings indicate an annual growth in the writing and publication of TEs on Scopus. This growth demonstrates that research and publications on TEs are in high demand and receive significant global attention.

Interestingly, 90% of the top 20 source titles contributing to TEs research are journals in the first quartile (Q1) and second quartile (Q2). Among these, 10 journals are in the highest quartile, Q1, and 8 journals are in Q2. The quartile level indicates that these journals have the highest quality and the greatest influence [ 39 , 40 ]. Furthermore, 7 source titles (Philosophical Studies, Synthese, Foundations of Science, Minds and Machines, Erkenntnis, Acta Analytica, and Axiomathes) that publish TEs topics focus on the field of philosophy. When studying TEs from a philosophical standpoint, researchers, scholars, and professionals have the option of submitting their articles to these journals. Alternatively, if TEs are studied from a historical perspective, journals such as Science and Education, Studies in History and Philosophy of Science Part A, Journal for General Philosophy of Science, Philosophy of Science, Boston Studies in the Philosophy and History of Science, and European Journal for Philosophy of Science are appropriate. Meanwhile, if TEs are studied from an educational perspective, Physics Teacher, Science and Education, Physics Education, American Journal of Physics, and European Journal of Physics are ideal choices for publishing articles. These journals regularly publish articles in physics education studies.

If we look at the authors who have made the greatest contributions to the topic of TEs in the previous 20 years (2003–2022), Stuart is the most significant author with 7 articles (51 citations). Stuart’s work focuses on the history and philosophy of TEs [ 31 , 32 , 33 ], with the first publication in 2014 in the journal Perspectives of Science. In contrast to Stuart, Bancong's work, which ranks second, investigates various TEs from an educational standpoint. His first work, published in 2018, examined TEs in high school physics textbooks [ 3 ], followed by an investigation of how students construct TEs collaboratively [ 4 ], and an identification of factors influencing TEs during problem-solving activities [ 5 ]. Following Bancong, Fehige from the University of Toronto, Canada, has also published 5 articles. Most of his work examines TEs in religious contexts, such as thought experiments, Christianity and science in novalis [ 15 ], thought experiments and theology [ 16 ], and the book of job as a thought experiment: on science, religion, and literature [ 17 ] which was published in the journal Religions in 2019. Brown examines TEs in several of his works in light of the history and philosophy of science [ 6 , 7 ], as do Buzzoni and El Skaf from Italy, who mostly discuss TEs in light of the history and philosophy of science [ 8 , 12 ]. Meanwhile, Halkia and Velentzas from the University of Athens, Greece, have discussed TEs from an educational perspective, such as using TEs from Newton's Cannon for teaching satellite physics [ 37 ] and using TEs from the theory of relativity for teaching relativity theories [ 38 ].

Over the past two decades, authors have examined TEs from diverse perspectives, including history, philosophy, education, and religion. This variety highlights a significant shift in the disciplinary landscape of TE research, which is historically rooted in the philosophy of science [ 18 , 24 ]. The true strength of TEs lies in their adaptability across disciplines, rather than in resolving philosophical disputes. Although TEs were traditionally centered on history and philosophy of science (HPS), recent trends show a growing application in education and technology, particularly in artificial intelligence and speculative design. This shift indicates that TEs have not lost their significance but have instead found new areas of relevance. In HPS, the focus has moved toward understanding the methodological and epistemological implications of TEs, confirming their essential role in scientific reasoning [ 7 , 30 ]. Additionally, in fields such as physics education, TEs are increasingly utilized to explore complex theoretical concepts and enhance educational methodologies [ 2 , 12 ].

Based on the most commonly used keywords in the last 20 years, research on TEs has mostly focused on understanding TEs from a philosophical perspective in the first five years (2003–2007). Thought experiments rethought and reperceived [ 19 ], on thought experiments: is there more to the argument? [ 24 ] and thought experiments [ 9 ] are a few examples. Then, over the next five years (2008–2012), many studies looked at how TEs contributed to physical theories, including the special theory of relativity and quantum theory. The keywords that emerged frequently during this period were quantum theory, scientific discovery, methodology, quantum mechanics, twin earths, falling bodies, and others. In the last ten years, TEs have been studied from various perspectives. For example, in 2013, Velentzas and Halkia [ 38 ] also used TEs as a didactic tool in teaching physics to upper-secondary students. Fehige, on the other hand, began to connect TEs to theology, with a specific focus on the interaction between Christianity and science [ 15 , 16 ]. There are also researchers who continue to study the existence of TEs from a philosophical point of view and claim that TEs are science fiction [ 1 , 20 ]. In recent years, TEs have become increasingly popular in education and have been linked to artificial intelligence. Artificial intelligence, physics education, productive imagination, technology, and speculative design are some of the keywords that appear frequently. This is not surprising because TEs, as experimental activities using mental models, are not easy for students to perform on their own [ 4 , 5 ]. Therefore, technology that can assist students in creating an imaginative world for constructing TEs is required.

Since no studies have charted the trends in TEs research so far, it is difficult to compare the research results obtained with those of others. Nevertheless, several studies that examine trends in physics education reveal that although research on experiments is declining in physics education, TEs are still important to physics teaching and learning [ 41 ]. Hallswoth et al. [ 21 ] have also used artificial intelligence technologies to support TEs in the field of wet biology research, which is dominated by experiments on microbial growth and survival. The use of artificial intelligence in learning is based on the growing interest in artificial intelligence methods in science, technology, and education [ 11 ]. Overall, our study contributes to a more comprehensive understanding of TEs research trends during the last 20 years. In addition, this research also contributes to providing an overview of several potential topics that can be the focus of future researchers, such as the use of artificial intelligence in TEs. By situating our findings within the broader context of previous studies, we provide a clearer picture of how TE research has evolved and where it is heading.

5 Conclusions

This study aims to present a more comprehensive understanding of the trend of studies on TEs during the last 20 years (2003–2022). Research on this topic began in 1976, and its progress has continued to the present. A total of 679 published papers from various sources, including articles (504), conference papers (92), and book chapters (83), were analyzed. The results of the study show that research and publications on TEs are of interest and have received a lot of attention during the last eight years. A significant increase occurred in 2021, with 69 published articles. According to the findings, 95% of the top 20 source titles contributing to TEs are from journals in the first and second quartiles (Q1 and Q2). This quartile ranking shows the quality and significant influence of a journal. The geographical distribution reveals that the United States contributes the most to TEs research, with 213 documents, 2592 citations, and 47 links. We also identified several prospective keywords that could be the focus of future research, including artificial intelligence, physics education, fiction, God, theology, productive imagination, technology, speculative design, and critical design. Therefore, this study contributes to providing a thorough picture of thought experiment research trends and future directions of potential topics that can be the focus of future researchers.

This research has several limitations. The exclusive source of publication data utilized in this study is the Scopus database, which is recognized as one of the most extensive databases in the field. However, it is worth noting that future research endeavors may consider including publication data from other prominent sources such as WoS and Google Scholar. Furthermore, the utilization of the search function in the TITLE-ABS-KEY field, specifically employing the terms "Thought-experiments" AND "Science" OR "Physics," was used for the purpose of data retrieval. However, it is important to acknowledge that this approach is not infallible, as there is a potential for some papers to be overlooked, making the process less than completely accurate. Despite its limitations, this research is often regarded as a pioneering contribution to the field of bibliometric studies on the subject of TEs during the past two decades.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Asprem E. How Schrödinger’s cat became a zombie. Method Theory Study Relig. 2016;28(2):113–40. https://doi.org/10.1163/15700682-12341373 .

Article   Google Scholar  

Bancong H. High school physics teachers’ perceptions and attitudes towards thought experiments in Indonesia. Phys Educ. 2023;58(4):045011. https://doi.org/10.1088/1361-6552/acdb37 .

Bancong H, Song J. Do physics textbooks present the ideas of thought experiments?: A case in Indonesia. J Pendidik IPA Indones. 2018;7(1):25–33. https://doi.org/10.15294/jpii.v7i1.12257 .

Bancong H, Song J. Exploring how students construct collaborative thought experiments during physics problem-solving activities. Sci Educ. 2020;29(3):617–45. https://doi.org/10.1007/s11191-020-00129-3 .

Bancong H, Song J. Factors triggering thought experiments in small group physics problem-solving activities. New Phys Sae Mulli. 2020;70(5):466–80. https://doi.org/10.3938/NPSM.70.466 .

Brown JR. The promise and perils of thought experiments. Interchange. 2006;37:63–75. https://doi.org/10.1007/s10780-006-8400-6 .

Brown JR. Natural science and supernatural thought experiments. Religions. 2019;10(6):389. https://doi.org/10.3390/rel10060389 .

Buzzoni M. Thought experiments in philosophy: a neo-Kantian and experimentalist point of view. Topoi. 2019;38(4):771–9. https://doi.org/10.1007/s11245-016-9436-6 .

Cooper R. Thought experiments. Metaphilosophy. 2005;36(3):328–47. https://doi.org/10.1111/j.1467-9973.2005.00372.x .

Dohrn D. Thought experiments without possible worlds. Philos Stud. 2018;175(2):363–84. https://doi.org/10.1007/s11098-017-0871-z .

Dushkin RV, Stepankov VY. Semantic supervised training for general artificial cognitive agents. In: Dushkin RV, Stepankov VY, editors. Frontiers in artificial intelligence and applications. Amsterdam: IOS Press BV; 2021. p. 422–30.

Google Scholar  

El Skaf R, Palacios P. What can we learn (and not learn) from thought experiments in black hole thermodynamics? Synthese. 2022;200:434. https://doi.org/10.1007/s11229-022-03927-0 .

Elgin CZ. Fiction as thought experiment. Perspect Sci. 2014;22(2):221–41. https://doi.org/10.1162/POSC_a_00128 .

Elsevier. Scopus: your brilliance, connected. 2022. www.elsevier.com/solutions/scopus/how-scopus-works/content/content-policy-and-selection . Accessed 15 Jan 2023.

Fehige Y. Poems of productive imagination: thought experiments, Christianity and science in Novalis. Neue Z für Syst Theol Relig. 2013;55(1):54–83. https://doi.org/10.1515/nzsth-2013-0004 .

Fehige Y. Theology and thought experiments. In: Stuart MT, Fehige Y, Brown JR, editors. The Routledge companion to thought experiments. London: Routledge; 2017. p. 183–94.

Chapter   Google Scholar  

Fehige Y. The book of job as a thought experiment: On science, religion, and literature. Religions. 2019;10(2):77. https://doi.org/10.3390/rel10020077 .

Galili I. Thought experiments: determining their meaning. Sci Educ. 2009;18(1):1–23. https://doi.org/10.1007/s11191-007-9124-4 .

Gendler TS. Thought experiments rethought—and reperceived. Philos Sci. 2004;71(5):1152–63. https://doi.org/10.1086/425239 .

Gendron C, Ivanaj S, Girard B, Arpin ML. Science-fiction literature as inspiration for social theorizing within sustainability research. J Clean Prod. 2017;164:1553–62. https://doi.org/10.1016/j.jclepro.2017.07.044 .

Hallsworth JE, Udaondo Z, Pedrós-Alió C, Höfer J, Benison KC, Lloyd KG, Cordero RJB, de Campos CBL, Yakimov MM, Amils R. Scientific novelty beyond the experiment. Microb Biotechnol. 2023;16(6):1131–73. https://doi.org/10.1111/1751-7915.14222 .

Ichikawa J, Jarvis B. Thought-experiment intuitions and truth in fiction. Philos Stud. 2009;142(2):221–46. https://doi.org/10.1007/s11098-007-9184-y .

Irwanto I, Saputro AD, Widiyanti W, Laksana SD. Global trends on mobile learning in higher education: a bibliometric analysis (2002–2022). Int J Inform Educ Technol. 2023;13(2):373–83. https://doi.org/10.18178/ijiet.2023.13.2.1816 .

Norton JD. On thought experiments: Is there more to the argument? Philos Sci. 2004;71(5):1139–51. https://doi.org/10.1086/425238 .

Nurazmi N, Bancong H. Exploring physics education research: Popular topics in prestigious international journals in the period of 2009–2019. In: Nurazmi N, Bancong H, editors. AIP conference proceedings. New York: AIP Publishing; 2024. p. 020126.

Pan X, Yan E, Cui M, Hua W. Examining the usage, citation, and diffusion patterns of bibliometric mapping software: a comparative study of three tools. J Informet. 2018;12(2):481–93. https://doi.org/10.1016/j.joi.2018.03.005 .

Prahanı BK, Saphıra HV, Wıbowo FC, Sulaeman NF. Trend and visualization of virtual reality and augmented reality in physics learning From 2002–2021. J Turk Sci Educ. 2022;19(4):1096–118. https://doi.org/10.36681/tused.2022.164 .

Pranckutė R. Web of science (WoS) and scopus: the titans of bibliographic information in today’s academic world. Publications. 2021;9(1):12. https://doi.org/10.3390/publications9010012 .

Reiner M, Gilbert J. When an image turns into knowledge: the role of visualization in thought experimentation. In: Gilbert JK, Reiner M, Nakhleh M, editors. Visualization: theory and practice in science education. Dordrecht: Springer; 2008. p. 295–309.

Schindler S, Saint-Germier P. Are thought experiments “disturbing”? The case of armchair physics. Philos Stud. 2020;177(9):2671–95. https://doi.org/10.1007/s11098-019-01333-w .

Stuart MT. Cognitive science and thought experiments: a refutation of Paul Thagard’s skepticism. Perspect Sci. 2014;22(2):264–87. https://doi.org/10.1162/POSC_a_00130 .

Stuart MT. Telling stories in science: Feyerabend and thought rxperiments. HOPOS J Int Soc Hist Phil Sci. 2021;11:262–81. https://doi.org/10.1086/712946 .

Stuart MT, Fehige Y. Special issue thought experiments in the history of philosophy of science motivating the history of the philosophy of thought experiments. HOPOS J Int Soc Hist Phil Sci. 2021;11:212–21. https://doi.org/10.1086/712940 .

Supriadi U, Supriyadi T, Abdussalam A, Rahman AA. A decade of value education model: a bibliometric study of scopus database in 2011–2020. Eur J Educ Res. 2022;11(1):557–71. https://doi.org/10.12973/EU-JER.11.1.557 .

Thu HLT, Tran T, Phuong TTT, Tuyet TLT, Le HH, Thi TV. Two decades of stem education research in middle school: a bibliometrics analysis in scopus database (2000–2020). Educ Sci. 2021;11(7):353. https://doi.org/10.3390/educsci11070353 .

van Eck NJ, Waltman L. Citation-based clustering of publications using CitNetExplorer and VOSviewer. Scientometrics. 2017;111:1053–70. https://doi.org/10.1007/s11192-017-2300-7 .

Velentzas A, Halkia K. From earth to heaven: using “Newton’s cannon” thought experiment for teaching satellite physics. Sci Educ. 2013;22(10):2621–40. https://doi.org/10.1007/s11191-013-9611-8 .

Velentzas A, Halkia K. The use of thought experiments in teaching physics to upper secondary-level students: two examples from the theory of relativity. Int J Sci Educ. 2013;35(18):3026–49. https://doi.org/10.1080/09500693.2012.682182 .

Visser M, van Eck NJ, Waltman L. Large-scale comparison of bibliographic data sources: Scopus, web of science, dimensions, crossref, and microsoft academic. Quant Sci Stud. 2021;2:20–41. https://doi.org/10.1162/qss_a_00112 .

Yu YC, Chang SH, Yu LC. An academic trend in STEM education from bibliometric and co-citation method. Int J Inf Educ Technol. 2016;6(2):113–6. https://doi.org/10.7763/IJIET.2016.V6.668 .

Yun E. Review of trends in physics education research using topic modeling. J Balt Sci Educ. 2020;19(3):388–400. https://doi.org/10.33225/jbse/20.19.388 .

Download references

Author information

Authors and affiliations.

Department of Physics Education, Universitas Muhammadiyah Makassar, Makassar, 90221, Indonesia

Hartono Bancong

You can also search for this author in PubMed   Google Scholar

Contributions

The author contributed to the conception and design of the study, data collection and analysis as well as the preparation of the manuscript.

Corresponding author

Correspondence to Hartono Bancong .

Ethics declarations

Competing interests.

The authors declare no competing interests.

Additional information

Publisher's note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ .

Reprints and permissions

About this article

Bancong, H. The past and present of thought experiments’ research at Glancy: bibliometric review and analysis. Discov Educ 3 , 142 (2024). https://doi.org/10.1007/s44217-024-00246-z

Download citation

Received : 06 February 2024

Accepted : 29 August 2024

Published : 07 September 2024

DOI : https://doi.org/10.1007/s44217-024-00246-z

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Bibliometrics
• Thought experiments
• Find a journal
• Publish with us
• Track your research