Radiation paradoxes 7: Accelerating frames and the general theory of relativity

(Previous posts in this series: Part 1, Part 2, Part 3, Part 4, Part 5, Part 6)

At this point, we need to take a slight detour and examine more closely the role of accelerating frames because that is central to resolving the paradox that started this series if posts, of whether an electric charge falling freely under gravity radiates or not. The discussion up to this point has seemed to privilege inertial frames when it came to discussing the laws of physics. This was because we knew how to transform physical quantities between any two inertial frames, using Galilean transformations for Newtonian motion at low speeds and Lorentz transformations when we needed more accurate results or were dealing with speeds comparable to that of light in the regime of special relativity. But transforming between inertial frames and accelerating ones was another story.

Einstein used the insight that any two masses will fall at the same rate in a gravitational field to argue that the distinction between inertial frames (where the laws of physics such as Maxwell’s equations are supposedly valid) and non-inertial frames (where they are not) should not matter and that we should be able to find transformational relations between them.

The experimental fact that the acceleration in free-fall is independent of the material, therefore, is a powerful argument in favor of expanding the postulate of relativity to coordinate systems moving nonuniformly relative to each other.

This paragraph illustrates Einstein’s concern that Newtonian physics ascribed properties to space that required one to give a special status to inertial frames when discussing the laws of physics. Non-inertial (i.e., accelerating frames) had a kind of second class status. He felt that one should be able to describe the laws in any frame of reference. Another problem, as Einstein saw it, was that according to Newton, space seemed to have an effect on matter by determining whether one was in an inertial frame or not but matter had no effect on space. Einstein’s general theory of relativity changed that one-way interaction by proposing that the geometry of space was not fixed. Matter had the effect of changing the geometry of space and the geometry of space affected the motion of matter, thus creating symmetry.

In arriving at his alternative formulation Einstein was following Berkeley, that Newton’s concept of space was meaningless because space was not measurable and that it was the effect of the distant stars that needed to be taken into account. Einstein had a flaw is that he was less than conscientious, cavalier even, in crediting the work of predecessors. Historians have had to work hard to find some of the influences on his work and, for example, there were interminable debates as to whether he was aware of the Michelson-Morley experiments on the aether when he was developing his special theory of relativity, even though those experiments impinged directly on his work and the null results obtained provided evidence in support of his theory.

In this case, he ignored (or was unaware of) Berkeley’s work but fortunately that idea had been taken up again in the late 19th and early 20th century by Ernst Mach (1838-1916) (whom Einstein was aware of) who was more explicit as to the problems with the Newtonian approach and who also said that a passive fixed space had no meaning. Mach defined inertial frames as those that are unaccelerated with respect to the “fixed stars”, by which Einstein said that he meant some suitably defined average of all the matter in the universe. In this approach, matter has inertia only because of the presence of other matter in the universe, not because of the properties of space. Einstein found Mach’s ideas persuasive and gave them the label of Mach’s Principle.

Classical mechanics and the theory of special relativity know of admissible coordinate systems (inertial systems) and nonadmissable coordinate systems. Relative to the first ones, the laws of nature (e.g., the law of inertia and the theorem of the constancy of the speed of light) are supposed to hold; relative to the latter, they do not. In vain one asks for an objective reason for the different quality of systems; and one is forced to explain them as an independent, very strange property of the space-time continuum. Newton was only very reluctantly content with this opinion (of “absolute space”), but he believed that in centrifugal effects he had an objective proof for it in hand.

However, E. Mach was the first to recognize the weakness of this argument. Perhaps it was not a physical quality of space that determined the inertial behavior of bodies; it could also be possible that inertia was not a reaction against a (conceptually empty) acceleration with respect to space, but rather against an acceleration with respect to the rest of gravitating matter in the world. Such a hypothesis appeared more satisfactory to Mach than the old concept of inertia, because it did not attribute to space any mechanically distinct properties but rather, in principle, accepted all coordinate systems as equal. According to this interpretation, inertia also was an interaction between bodies just as is the case in Newtonian gravitation. It is true that this idea did not yet point to a rigorous (quantitative) treatment of the problem, and in essence the natural equality between inertia and gravitation—as laid out earlier (hypothesis of equivalence)—remained hidden to Mach. But he was (after Newton) the first to vividly feel and clearly illuminate the epistemological weakness of classical mechanics.

It should by no means be claimed that the basically unsubstantiated preference of inertial systems over other coordinate systems constitutes an error of classical mechanics. The preference of certain states of motion (namely, of inertial systems) in nature could be a final fact that we have to accept without being able to explain it (or reduce it to some cause). However, a theory in which all states of motion of coordinate systems are—in principle—equal has to be appreciated from an epistemological point of view as being far more satisfying. For the following consideration we want to use this equivalence as a basis under the name of “general principle of relativity.” [Italics in original-MS]

It was his feeling that we needed to treat accelerating frames on an equal footing with inertial frames that stimulated Einstein to develop his general theory of relativity.


  1. says

    In a way he also represented the worst. Here was an articulate, gentle, and caring man whose initial access to the public sphere wad because of religion and none of the apparently better reasons. He did not fall back on religious authoritarianism but it’s where his gravitas came from. For every conman who calls themself “reverend” Tutu is just one example of the few who apparently aren’t completely corrupt -- but he relied on the same social mechanism that they did, to claim attention and respect.

    To me “archbishop” translates into “skilled organizational/political operative” not holy authority or blessed enlightenment. Take away the title and what is he?

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