Sagittarius A-Star said:
this book about relativity seems to be a bad source
I'm afraid I have to agree. I don't think Gardner meant to say quite what the OP interpreted him as saying, but I can see how what Gardner wrote could have been misunderstood that way, and even after that misunderstanding is corrected there are plenty of issues that remain with what Gardner wrote.
The quote on p. 114 of my copy that I was referring to in post #22 is this:
Martin Gardner said:
The stay-at-home does not move relative to the universe.
Emphasis in original.
Gardner never explains what he means by "relative to the universe", which is the first issue. But from other things he says, it is clear, at least to me, that what he means by "relative to the universe" is something like "relative to the average of all matter in the universe". If we wanted to sharpen this up to be more technical (which Gardner really should have done), we could say "relative to a comoving observer in the same spacetime region as the twins", and we could test that the stay-at-home twin was at rest relative to a comoving observer by confirming that, for example, the stay-at-home twin sees the CMBR as isotropic and the traveling twin does not.
It certainly seems on the surface like this is saying that "the universe" defines a preferred frame. But we have to be careful here. The
laws of physics do not define any preferred frame. But
particular solutions of the laws of physics can have symmetry properties that
do pick out a particular frame as being "preferred" in the sense that the solution looks a lot simpler when described using that frame. For example, in FRW spacetime, standard "comoving" coordinates make the solution look simpler than any other frame, because those coordinates match up with the homogeneity and isotropy of the spacetime geometry.
So all Gardner is actually saying is that, in his version of the scenario (about which more below), the stay-at-home twin is the one who is "at rest" with respect to the symmetry properties of the particular spacetime geometry that describes the universe, i.e., of the particular solution of the laws of physics that describes the universe. He is not saying that the
laws themselves define a preferred frame, as of course they don't.
However, this still leaves a number of issues that are worth pointing out.
Gardner's claimed explanation of the difference in aging, in the traveling twin's rest frame, as being due to a "gravitational field" caused by the traveling twin accelerating "relative to the universe", relies on the claim that accelerating relative to the average of all matter in the universe--or, more technically, relative to a comoving observer in your vicinity--brings into being a "gravitational field". However, that claim, as it stands, is problematic for two reasons.
The first reason, about which more below, is that we can't just assert without proof that accelerating relative to a comoving observer brings into being a "gravitational field"--or, taking account of the second objection I'm going to raise in a moment, that accelerating relative to a comoving observer causes differential aging between twins who separate and then come back together. We have to actually prove it. And proving it requires specifying what spacetime geometry you are using. Gardner is never actually clear about this, but of course the standard formulation of the twin paradox is in flat Minkowski spacetime--and in flat Minkowski spacetime,
there is no matter in the universe. There is no such thing as "relative to the rest of the universe" or 'accelerating relative to the rest of the universe" or "accelerating relative to comoving observers" in Minkowski spacetime. So by bringing in "the rest of the universe" as though it actually plays a role in the scenario, Gardner is
changing the scenario. He is not really talking about the standard twin paradox any more, but about some ad hoc, not well specified variation of it that takes place in some not well specified spacetime geometry where "the rest of the universe" is significant.
(Gardner does eventually, on p. 116, raise the question of what would happen in a hypothetical universe that contained nothing but the two twins and their spaceships--which is to say, in the actual standard twin paradox in its standard formulation. More on that below. But he certainly does nothing to inform the reader that this hypothetical version he talks about almost as an afterthought is actually the standard version that everyone else discusses.)
The second reason the "gravitational field" claim is problematic as it stands is that "gravitational fields" are frame-dependent, and one of the key tenets of relativity is that no actual physical observable--which the difference in aging of the twins is--can be explained by something that is frame-dependent. That is not to say that we can't construct a non-inertial frame in which the traveling twin is at rest, or that in that frame, there will not be something that can be termed a "gravitational field", or that calculating the stay-at-home twin's elapsed time in this frame will bear some useful similarities to calculations done in an ordinary gravitational field, the kind generated by a planet like the Earth. All of those things are true. But that doesn't change the fact that the "gravitational field" present in such a non-inertial frame is a frame-dependent thing, and will not really bear the weight of explanation that Gardner wants to put on it.
As noted above, on p. 116 (of my copy), Gardner discusses the case (which, as noted, he treats as an afterthought, but which is actually the standard twin paradox in flat Minkowski spacetime) of a "universe" that contains nothing but the two spaceships, and asks what the result would be in such a case. Here is what he says:
Martin Gardner said:
The answer [to the question of whether there would be differential aging between the twins] depends on whether you adopt Eddington's view on inertia or the Machian view of Dennis Sciama. In Eddington's view the answer is yes. Ship A accelerates with respect to the metric spacetime structure of the cosmos; ship B does not. The situation remains unsymmetrical and the usual difference in aging results. From Sciama's point of view the answer is no. Acceleration is meaningless except with respect to other material bodies. In this case, the only material bodies are the two spaceships. The situation is perfectly symmetrical. In fact, there are no inertial frames to speak of because there is no inertia (except an extremely feeble, negligible inertia resulting from the presence of the two ships). In a cosmos without inertia it is hard to predict what would happen if a ship turned on its rocket motors! As Sciama says, with British understatement, "Life would be quite different in such a universe".
As
@Sagittarius A-Star notes, Gardner fails to mention that, while the Eddington viewpoint he refers to is in fact just standard relativity theory using flat Minkowski spacetime, which is a perfectly valid solution to the Einstein Field Equation, the Sciama viewpoint he described was purely speculative, based on a purely hypothetical future theory that never actually got developed at all. (Sciama published one paper on it but AFAIK never followed up with anything more, and the one paper, while interesting, contains nothing that could be described as a complete theory, even a simple one, and AFAIK no one else has followed it up either.) The underlying contradiction is evident in Gardner's quote: he notes that "it is hard to predict what would happen" in such a model, but only after giving a definite prediction as to what would happen: there would be no difference in aging between the twins.
The "Machian" issue that Sciama was trying to come to grips with is one that a number of physicists have commented on. The simplest way to describe it is to note that the Einstein Field Equation of GR allows solutions--a number of which are heavily used in physics--in which "inertia", i.e., the spacetime geometry, is
not completely determined by the distribution of matter and energy. One obvious example is that there are multiple
vacuum solutions to the EFE, i.e., specifying that there is
no matter or energy anywhere does not completely specify the spacetime geometry! To know
which vacuum solution you are dealing with, you have to add other specifications. This means GR contains solutions which are not "Machian", if you think "Machian" means that the spacetime geometry
should be completely determined by the distribution of matter and energy.
