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before the Correspondence, and the arguments of the Scholium were not written with Leibniz in mind, but Clarke himself suggests (CIV.13) that those arguments — specifically those concerning the bucket — are telling against Leibniz. That argument is indeed devastating to a general principle of relativity — the parity of... |
is highly questionable whether Leibniz's equivalence of hypotheses amount to such a view. That said, his statements in the first four letters of the Correspondence could understandably mislead Clarke on this point — it is in reply to Clarke's challenge that Leibniz explicitly denies the parity of relative motions. But ... |
argument — despite some involvement of Newton in writing the replies. Instead of the argument from the uniqueness of the rate of rotation, he argues that systems with different velocities must be different because the effects observed if they were brought to rest would be different. This argument is of course utterly q... |
there is no privileged standard of rest! As we discuss in Section 8, Mach attributed to Newton the fallacious argument that because the surface of the water curved even when it was not in motion relative to the bucket, it must be rotating relative to absolute space. Our discussion of Newton showed how misleading such a... |
first place he also argues that there must be some privileged sense of rotation, and hence not all relative motions are equal. Second, the argument is ad hominem against Descartes, in which context a disjunctive syllogism — motion is either proper or ordinary or relative to absolute space — is argumentatively legitimat... |
that Newton's argument in the Scholium leaves open the logical possibility that the privileged, true sense of rotation (and acceleration more generally) is some species of relative motion; if not motion properly speaking, then relative to the fixed stars perhaps. (In fact Newton rejects this possibility in De Gravitati... |
at a distance; an ironic position given his theory of universal gravity.) However the kind of folk-reading of Newton that underlies much of the contemporary literature replaces Mach's interpretation with a more charitable one. According to this reading, Newton's point is that his mechanics — unlike Descartes' — could e... |
his explanation involves a privileged sense of rotation, and that absent an alternative hypothesis about its relative nature, we should accept absolute space. But our discussion of Newton's argument showed that it simply does not have an ‘abductive’, ‘best explanation’ form, but shows deductively, from Cartesian premis... |
that Newton had no understanding of how such effects would be explained in his mechanics. For instance, in Corollaries 5 and 6 to the Definitions of the Principles he states in general terms the conditions under which different states of motion are not — and so by implication are — discernible according to his laws of ... |
to say that Newton's contemporaries weren't seriously concerned with explaining inertial effects. Leibniz, for instance, analyzed a rotating body (in the Specimen). In short, parts of a rotating system collide with the surrounding matter and are continuously deflected, into a series of linear motions that form a curved... |
plenum of elastic particles of matter — is far too complex for him to offer any quantitative model based on this qualitative picture. (In the context of the proposed ‘abductive’ reading of Newton, note that this point is telling against a rejection of intrinsic rigidity or forces acting at a distance, not narrow relati... |
in a plenum that stymies analysis. And since Leibniz's collision theory requires a standard of inertial motion, even if he had explained inertial effects, he would not have thereby shown that all motions are relative, much less that all are equal.) Although the argument is then not Newton's, it is still an important re... |
by the folk-Leibniz, especially when it is extended by bringing in a further example from Newton's Scholium. Newton considered a pair of identical spheres, connected by a cord, too far from any bodies to observe any relative motions; he pointed out that their rate and direction of rotation could still be experimentally... |
and by pushing on opposite faces of the two globes to see whether the tension increased or decreased. He intended this simple example to demonstrate that the project he intended in the Principia, of determining the absolute accelerations and hence gravitational forces on the planets from their relative motions, was pos... |
cord are rigid and that they are the only things in their universe, then the example can be used to point out that there are infinitely many different rates of rotation all of which agree on the relations between bodies. Since there are no differences in the relations between bodies in the different situations, it foll... |
between the states of rotation cannot be explained in terms of the relations between bodies. Therefore, a theory of the kind attributed to the folk's Leibniz cannot explain all the phenomena of Newtonian mechanics, and again we can argue abductively for absolute space. (Of course, the argument works by showing that, gr... |
of rotation that cannot merely be relative rotations of any kind; for the differences cannot be traced to any relational differences. That is, granted the assumptions of the argument, rotation is not true relative motion of any kind.) This argument (neither the premises nor conclusion) is not Newton's, and must not be ... |
is not to say that the argument is fallacious, and indeed many have found it attractive, particularly as a defense not of Newton's absolute space, but of Galilean spacetime. That is, Newtonian mechanics with Galilean spacetime can explain the phenomena associated with rotation, while theories of the kind proposed by Ma... |
mechanics, but these explanations rely on the geometric structure of Galilean spacetime — particularly its connection, to interpret acceleration. And thus — the argument goes — those explanations commit us to the reality of spacetime — a manifold of points — whose properties include the appropriate geometric ones. This... |
or regions, distinct from matter, with geometric properties, is what we earlier identified as ‘substantivalism’. There are two points to make about this line of argument. First, the relationist could reply that he need not explain all situations which are possible according to Newtonian mechanics, because that theory i... |
and time relations between bodies, but which approximates to Newton's if matter is distributed suitably. Such a relationist would be following Mach's proposal, which we will discuss next. Such a position would be satisfactory only to the extent that a suitable concrete replacement theory to Newton's theory is developed... |
been made. Second, one must be careful in understanding just how the argument works, for it is tempting to gloss it by saying that in Newtonian mechanics the connection is a crucial part of the explanation of the surface of the water in the bucket, and if the spacetime which carries the connection is denied, then the e... |
But this gloss tacitly assumes that Newtonian mechanics can only be understood in a substantial Galilean spacetime; if an interpretation of Newtonian mechanics that does not assume substantivalism can be constructed, then all Newtonian explanations can be given without a literal connection. Both Sklar (1974) and van Fr... |
as a primitive quantity not defined in terms of motion relative to anything, be it absolute space, a connection or other bodies. (Notice the family resemblance between this proposal and Leibniz's view of force and speed.) Van Fraassen proposes formulating mechanics as ‘Newton's Laws hold in some frame’, so that the for... |
picks out a standard of inertial motion, not absolute space or a connection, or any instantaneous relations. These proposals aim to keep the full explanatory resources of Newtonian mechanics, and hence admit ‘true acceleration’, but deny any relations between bodies and spacetime itself. Like the actual Leibniz, they a... |
are nothing but the relations between bodies. Of course, such views raise the question of how a motion can be not relative to anything at all, and how we are to understand the privileging of frames; Huggett (2006) contains a proposal for addressing these problems. (Note that Sklar and van Fraassen are committed to the ... |
Newton's laws are capable of explaining all the phenomena without recourse to spacetime geometry; that the connection and the metrical properties are explanatorily redundant. A similar view is defended in the context of relativity in Brown 2005.) Between the time of Newton and Leibniz and the 20th century, Newton's mec... |
period of dominance, 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 provok... |
Newtonian space, Neumann (1870) and Lange (1885) developed more concrete definitions of the reference frames in which Newton's laws hold. In these and a few other works, the concept of the set of inertial frames was first clearly expressed, though it was implicit in both remarks and procedures to be found in the Princi... |
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 methodologi... |
us concerning motion and acceleration. Mach at least partly misinterpreted Newton's aims in the Scholium, and inaugurated a reading of the bucket argument (and by extension the globes argument) that has largely persisted in the literature since. Mach viewed the argument as directed against a ‘strict’ or ‘general-relati... |
absolute space. 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 can... |
less the existence of absolute space. (The reader will recall that Newton's actual aim was simply to show that Descartes' two kinds of motion are not adequate to accounting for rotational phenomena.) Although Mach does not mention the globes thought experiment specifically, it is easy to read an implicit response to it... |
competent to say what would happen, or what would be possible, in a universe devoid of matter other than two globes. So 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 vi... |
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 alr... |
its place we need only substitute the 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 (w... |
his official positivist account of scientific laws), then Mach suggests that we replace the 1st law with an empirically equivalent mathematical rival: Mach's Equation (1960, 287) The sums in this equation are to be taken over all massive bodies in the universe. Since the top sum is weighted by distance, distant masses ... |
world with a (reasonably) static distribution of heavy distant bodies, such as we appear to live in, the equation entails local conservation of linear momentum in ‘inertial’ frames. The upshot of this equation is that the frame of the fixed stars plays exactly the role of absolute space in the statement of the 1st law.... |
Newton's first law, is not vectorial.) 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 19... |
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 to... |
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 Einstei... |
relative distances and (perhaps) their 1st and 2nd time derivatives, and thus ‘generally relativistic’ in the sense sometimes read into Leibniz's remarks about motion. Mach wished to eliminate absolute time from physics too, so he would have wanted a proper relationist reduction of these derivatives also. The Barbour-B... |
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... |
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... |
outward force (before any rotation is communicated to the water), if the sides of the bucket were massive enough. 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 ... |
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 recently finished G... |
due to the Earth's rotation.) Its specific form must fall off with distance much more slowly than 1/r2, if it 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 vin... |
of absolute quantities of motion from mechanics. Direct assault on the problem of satisfying Mach-heavy in a classical framework proved unsuccessful, despite the efforts of others besides Mach (e.g., Friedländer 1896, Föpl 1904, Reissner 1914, 1915), until the work of Barbour and Bertotti in the 1970s and 80s. (Between... |
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.) Rather than formulating a revised law of gravity/inertia ... |
replacing the elements of the action that involve absolute quantities of motion with new terms invoking only relative distances, velocities etc. Their first (1977) theory uses a very simple and elegant action, and satisfies everything one could wish for from a Mach-heavy theory: it is relationally pure (even with respe... |
derived from the field equations); it is nearly empirically equivalent to Newton's theory in a world such as ours (with a large-scale uniform, near-stationary matter distribution); yet it does predict novel effects such as the ones Mach posited with his thick bucket. Among these is an ‘anisotropy of inertia’ effect — a... |
more force than accelerating it perpendicular to the galactic plane — large enough to be ruled out empirically. Barbour and Bertotti's second attempt (1982) at a relational Lagrangian mechanics was arguably less Machian, but more empirically adequate. In it, solutions are sought beginning with two temporally-nearby, in... |
‘intrinsic difference’ parameter that measures how different the two configurations are. In the solutions of the theory, this intrinsic difference quantity gets minimized, as well as the ordinary action, and in this way full solutions are derived despite not starting from a privileged inertial-frame description. The th... |
Newtonian theory: the set of models of Newtonian mechanics and gravitation in which there is zero net angular momentum. This result makes perfect sense in terms of strict relationist aims. In a Newtonian world in which there is a nonzero net angular momentum (e.g., a lone rotating island galaxy), this fact reveals itse... |
the center”. Since a strict relationist demands that bodies obey the same mechanical laws even in ‘rotating’ coordinate systems, there cannot be any such tendency to recede from the center (other than in a local subsystem), in any of the relational theory's models. Since cosmological observations, even today, reveal no... |
Bertotti theory can lay claim to exactly the same empirical successes (and problems) that Newtonian physics had. The second theory does not predict the (empirically falsified) anisotropy of inertia derivable from the first; but neither does it allow a derivation of the precession of the orbit of Mercury, which the firs... |
the relational interpretations of Newtonian physics reviewed in 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. ... |
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 referenc... |
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 fr... |
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 agr... |
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 as we saw in the introduction, not all philosophers accept this) then she will feel obliged to accord some sort of existence or reality to the structure —... |
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. The Special Theory of Relativity (STR) is notionally based on a principle of relativity of motion; but ... |
into STR is in fact nothing other than the Galilean principle of relativity, which is built into Newtonian physics. 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. I... |
Galilean spacetime (defined in section 5 above). 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. As Dorling (1978) first pointed out, howev... |
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 very successful. STR incorporates certain novelties conce... |
fix how the material systems are embeddable (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 fro... |
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 cloc... |
situation in classical physics, a body at rest cannot have all the same spatial 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 differe... |
supervene on spacetime interval relations. 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, mi... |
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 su... |
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 note), it can be argued that the absolute simultaneity presupposed by classical relationists and abso... |
have regarded with misgivings. From this perspective, instantaneous relational configurations — precisely what one starts with in the theories of Barbour and Bertotti — 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 acceler... |
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 ... |
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 do... |
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 STR and Newtonian physics in Galilean spacetime is somewhat deceptive. For one thing, Galilean spacetime 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,... |
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... |
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 absol... |
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 transfo... |
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, ... |
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. A simplified way of seeing how Einstein swept away a host of problems at a stroke is this: he proposed that the Galilean princip... |
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 velo... |
relative velocity of its emitter. 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 spa... |
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 S... |
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 comp... |
a number of interesting Lorentz-covariant theories were proposed (set in the Minkowski spacetime of STR). Einstein rejected these efforts 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 appea... |
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 Lor... |
see Hoefer 1994.) Einstein began to see this complete relativization as possible in 1907, thanks to his discovery of the Equivalence Principle. Imagine we are far out in space, in a rocket ship accelerating at a constant rate g = 9.98 m/s2. Things will feel just like they do on the surface of the Earth; we will feel a ... |
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 ... |
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... |
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 abso... |
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 constan... |
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... |
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 subclass the coordinate systems adapted to any such frame of reference, if he could achieve a theory of gravitation, ele... |
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 d... |
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 underst... |
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 ... |
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... |
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 flat Minkowski spacetime. There is no absolute velocity locally, but there are clear local standards of accelerated vs non-accelerated motion, i.e., local ... |
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