On The Many Principles of Equivalence


The definition of the Equivalence Principle has undergone several changes over the years. The original principle, usually attributed to Galileo, asserts that all material objects in free fall undergo the same acceleration in a gravitational field, regardless of their mass and composition. In Newtonian terms this can be expressed by saying that inertial mass is strictly proportional to gravitational mass (a fact which Newton himself verified by means of pendulum experiments on a variety of substances). This proposition is often now called the Weak Equivalence Principle.


While Einstein was searching (in 1907) for a way of generalizing the theory of relativity to include the effects of acceleration he was impressed by the fact that a person in free fall doesn’t feel their own weight. This of course can be seen as a simple consequence of the Weak Equivalence Principle, i.e., the proportionality of inertial and gravitational mass, but Einstein suspected that it was a manifestation of a stronger principle. Just as Galileo had argued (300 years earlier) that in the interior of a uniformly moving ship it was as if the motion did not exist, so Einstein proposed that for someone in an enclosed capsule in free-fall in a gravitational field it is as if the gravitational field does not exist. In other words, all physical processes (not just the trajectories of material objects) would take place within that capsule in just the same way that they would if the gravitational field was absent and the capsule was in uniform motion. (Naturally we must add suitable caveats, such as the requirement for the capsule to be very small and for the duration of the processes to be very short, to avoid “tidal” complications due to variations in the gravitational field.) To distinguish this from Galileo’s version, we will refer to this stronger statement as Einstein’s Equivalence Principle. In crude terms it asserts a fundamental equivalence between gravitation and acceleration.


Before describing some additional forms of the Equivalence Principle, a brief digression will be useful. One of the earliest applications of general relativity was in cosmology. This was a natural development, since the field equations are differential equations, which can be solved only by specifying suitable boundary or initial conditions. Hence the theory is (in a sense) incomplete without some means of constraining those conditions. This led Einstein in 1917 to consider global solutions of the field equations, especially finite (spherical) solutions not requiring any boundary conditions at infinity. He hoped to account for the total inertial/gravitational field purely in terms of the distribution of matter (or perhaps mass-energy), along the lines suggested by Ernst Mach. Later he recognized that this is problematic at best, because in general relativity even gravity itself possesses energy, albeit not usually localizable. This was probably on his mind when he wrote to Lorentz in November of 1919, conceding that the spacetime of general relativity, viz, the inertial/gravitational field, could be called an “ether” (provided we do not assign to it any state of motion), and then going on to say


I see another thing, too. My view that the state of the ether (i.e., the gmn’s) must be determined by the matter alone has nothing compelling about it. That is why one cannot argue for the closure of the world with as much certainty as I have done.


Nevertheless, Einstein’s early cosmological work created an entire academic discipline, and almost all of it has been based on the assumption that the field equations, and all other physical laws, including the constants appearing in those laws, are uniformly applicable throughout the universe and for all time. For example, the gravitational constant G and the speed of light c were both assumed to be universal constants, as was the so-called “cosmological constant” l. Of course, there is a good theoretical reason for l being literally constant, because otherwise the covariant derivative of the “left side” of the field equations would not vanish. We could also argue that the speed of light is unity more or less by definition (although this may entail variations in other parameters such as the fine structure constant). On the other hand, the gravitational constant could conceivably have a different numerical value in different regions of the universe and/or at different epochs.


The simplest assumption is obviously that all the “constants” of nature really are constant, i.e., the same everywhere and for all time. We’re naturally inclined to assume this unless we’re forced by observation to some other conclusion. We might even be tempted to elevate this assumption to the status of a “principle”, and in fact it is closely related to what some cosmologists have called The Cosmological Principle. This consists of the idea that we are not located in any “special” position in the universe, and furthermore that the universe “looks the same” when viewed from any spatial location. Often when people talk about the Cosmological Principle they think in terms of contingent macroscopic attributes such as the distribution of stars and galaxies, but it applies just as well to things like the gravitational constant and all the other phenomena of physics. Some cosmologists have gone even further, and proposed that the universe “looks the same” not only at every location throughout the universe, but also at all times. This has been called the Perfect Cosmological Principle, which led Bondi to the idea of a steady-state cosmology. This cosmology was later abandoned based on empirical evidence that the universe is evolving and looked very different in the past (as indicated by, for example, the cosmic microwave background radiation). Hence the “principles” of cosmologists are not invincible. (As an aside, the cantankerous British astronomer Herbert Dingle ridiculed cosmologists like Milne and Bondi for their practice of enshrining unsupported assumptions as “principles”. Dingle said he had nothing against making assumptions when necessary, but he believed we should “call a spade a spade, not a Perfect Agricultural Principle”.)


Returning now to the Equivalence Principle, it has become customary (at least in some circles) to express the principle by asserting that the laws of physics in the context of special relativity (including the values of the physical constants in those laws) hold good in any sufficiently small free-falling region of spacetime, anywhere in the universe, and at any time, and at any velocity. (Note that the condition of “free fall” allows for different velocities.) Two distinct versions of this principle are discussed. The semi-strong (also called the medium-strong) principle exempts gravitation from “the laws of physics” that are held to be invariant, whereas the strong principle includes gravitation.


It’s worth noting that these modern statements of the equivalence principle actually combine several conceptually distinct principles, because they entail not just the equivalence of gravitational acceleration and coordinate acceleration, they also assert a uniformity in the laws (and constants) of physics throughout the universe, and for all time, and at all velocities. This posited uniformity does not represent an assertion of equivalence between gravitational and coordinate acceleration, but rather of equivalence of different spatial and temporal regions of the universe, and of different frame of reference (i.e., velocities). Arguably the uniformity over space and time would more properly be associated with the perfect cosmological principle, because it asserts that (in the absence of any evidence to the contrary) we should assume the universe “looks” the same everywhere and at all times. Granted this principle seems to be falsified with respect to the contingent structure of the universe over time, but this wouldn’t necessarily preclude a “weak” version of the principle, applicable to just the underlying physical laws.


The inclusion, within the modern statement of the strong equivalence principle, of the assertion that the laws of physics are the same for all frames of reference (i.e., independent of velocity) is also conceptually quite distinct from the original meaning of Einstein’s equivalence principle. It actually represents the principle of Lorentz covariance, which is to say, the principle of special relativity. There may be some justification for combining these two principles, consistent with Einstein’s view that general relativity was a generalization of special relativity. He did in fact regard the equivalence principle as a generalization of the principle of Lorentz covariance, and of course the term “special relativity” is contained in Einstein’s version of the equivalence principle, but this implies that the additional phrase “at any velocity” is superfluous.


It’s easy to see why the definition of the equivalence principle has tended to evolves and subsume other principles. At first it simply denotes the idea that the effects of gravitation are physically equivalent to the effects of acceleration of the reference frame, without implying what these effects may be, or on what they may operate, or on whether these things may differ at different times and places. But then we re-phrase the principle by saying that the laws of physics are the same in any freely-falling reference frame. Unfortunately this fails to specify what “sameness” is being asserted. The same as what? It could mean the same as each other, but this modifies the original meaning of the principle, which was to assert a “sameness” not between different times and places, but between the effects of two different physics processes at any given time and place. The seemingly innocent linguistic re-phrasing has actually introduced a very significant change in the meaning. By equating the physics in free-fall in a gravitational field with the physics in free-fall in the absence of a gravitational field, we do not thereby claim that physics is the same for all times and places in the universe.


More fundamentally, Einstein’s didn’t regard flat spacetime as devoid of a gravitational field. He considered Minkowski spacetime as just a special case of a gravitational field. From this point of view, the equivalence principle asserts the intrinsic “sameness” of (sufficiently small regions of) curved and flat spacetime – but this doesn’t imply that the physics in curved and flat spacetime in one region of the universe are equivalent to those in all other regions of the universe. The latter hypothesis is more properly regarded as a version of the cosmological principle.


For the sake of clarity, it would be better to keep distinct the different kinds of “sameness”. Hypotheses asserting that separate places and times in the universe are (in various senses) “the same” ought to be recognized as versions of the cosmological principle; the hypothesis of Lorentz covariance ought to be seen as an expression of the principle of special relativity; and the term “equivalence principle” should be reserved for assertions about the “sameness” of the effects of gravitation and extrinsic acceleration. The truth or falsity of these different principles clearly have at least some degree of independence, and it seems of questionable value to refer to each of them as various aspects or versions of “the equivalence principle”. We might as well re-name every principle as some version of the equivalence principle. It’s arguably true that nearly every physical principle asserts some kind of equivalence or symmetry, so we could nominally justify this convention, but it would reduce rather than enhance conceptual clarity.


It also strikes me as historically inaccurate to claim that Einstein’s principle of equivalence entailed some kind of cosmological principle. We’ve seen this recently in popular press reports about claims to have discovered evidence of variations in one or more of the fundamental constants (e.g., the fine structure constant) in distant regions or epochs of the universe. These are reported excitedly as “disproofs of one of Einstein’s basic principles”, namely, the equivalence principle, or even the principle of Lorentz covariance. For example, in 2002 the following appeared in New Scientist:


One of Einstein's most dearly held concepts - that the speed of light is constant - is looking a little fragile. Physicists in Australia claim there is good reason to think the speed of light has slowed over time. "Einstein would have absolutely hated this," said Paul Davies of Macquarie University in Sydney. "His entire theory of relativity was founded on the notion that the speed of light is an absolute fixed universal number."


Again we see the idea of a “universal number”, conveying an aspect of the cosmological principle, being confused with principles whose meaning does not entail any assertion about uniformity of properties on a cosmological scale. The above story would more accurately be described as evidence against some form of the cosmological principle. Admittedly Einstein assumed something like the cosmological principle in his work on cosmology, in the absence of any evidence to the contrary, but surely he recognized it as distinct from the hypothesis of the equivalence of the effects of gravity and acceleration. It is not true that the “entire theory of relativity was founded on the notion that the speed of light is an absolute fixed universal number”. Neither the principle of Lorentz covariance nor Einstein’s equivalence principle necessarily entails uniformity of physical laws and parameters throughout the universe.


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