JohnNemo said:
OK, returning to the question of does the Sun go round the Earth or does the Earth go round the Sun, obviously if we take the Sun as our reference frame, the Earth is in orbit, whereas if we take the Earth as our reference frame the Sun is in orbit. However if we take a reference frame which is non-rotating relative to the distant stars, it looks very much like the Earth is orbiting the Sun and not the other way around. But is there something special about a reference frame which is non-rotating relative to the distant stars?
In
The Evolution of Physics (1938) - available at
https://archive.org/details/evolutionofphysi033254mbp - Einstein wrote
"Can we formulate physical laws so that they are valid for all CS, not only those moving uniformly, but also those moving quite arbitrarily, relative to each other? If this can be done, our difficulties will be over. We shall then be able to apply the laws of nature to any CS. The struggle, so violent in the early days of science, between the views of Ptolemy and Copernicus would then be quite meaningless. Either CS could be used with equal justification. The two sentences, 'the sun is at rest and the Earth moves', or 'the sun moves and the Earth is at rest', would simply mean two different conventions concerning two different CS. Could we build a real relativistic physics valid in all CS; a physics in which there would be no place for absolute, but only for relative, motion? This is indeed possible!" (page 224)
which seems to indicate that he thought that there was nothing special at all about a reference frame which was non-rotating relative to the distant stars ("The struggle... between the views of Ptolemy and Copernicus would then be quite meaningless") but, when he wrote this, was he hoping that GR would be more Machian than it turned out to be?
I'll give the super short answer that I would up writing at the conclusion to this rather long post first, in the hopes it will avoid the too long, didn't read issue. Then comes the bulk of the post, which seems to have grown quite a bit over my original intent.
The short version: When we talk about objects never moving faster than "c", we are not using tensor language. When we talk about objects having time-like worldlines, we are using tensor language. The intent is basically the same, only the semantics are different. However, the tensor language statements won't necessarily be recongizable to people who are not familiar with tensors.
The longer disucssion, in a good-news, bad-news format.
Good news: Using tensor methods, we can indeed express the laws of physics in a coordinate system (CS) where the sun orbits the Earth - but there are some potential misunderstandings and limitatoins here, see below.
Bad news #1. One doesn't generally learn tensor methods until one is in graduate school. The methods one learns in high school physics will NOT allow one to think of the Sun as orbiting the Earth. Assuming that the tensor methods needed work in the ways that one is (presumably) familiar with from high school leads to misunderstandings.
Bad news #2. The coordinate systems in which the Sun orbit the Earth do not necessarily cover all of space-time. There are limits on the size of accelerating frames, for instance. There is a bit more below.
Bad news #3. The relationship between the coordinates and physically measurable quantities becomes considerably less straightforwards in general coordinates.
It's helpful to consider a specific example, which we will take to be a rotating frame of reference (such as the rotating Earth) using tensor methods. For convenience we will omit gravity, and just talk about a rotating frame of reference in a space-time without gravity. This relates to the title question of this thread as well, though the omission of gravity makes it not quite the same.
At some distance, the object at rest in these coordinates has what we call a null worldline. Having a null-worldline is a coordinate independent tensor-language statement that's roughly equivalent to the coordinate dependent statement "moving at the speed of light".
In tensor language, we would talk about the Born coordinate chart,
<<link>>, and we'd concisely specify the coordinates by giving the metric tensor in the form of its line element:
$$ds^2 = -\left( 1- \frac{\omega^2 r^2}{c^2} \right) \mathrm dt^2 + 2 \omega r^2 \mathrm dt \, \mathrm d\phi + \mathrm dz^2 + \mathrm dr^2 + r^2 \mathrm d\phi^2$$
Physicisits familar with the methods generally regard the specification of such a metric as a complete description of a coordinate system, because they know how to calculate anything they need to calculate about the physics just from being given this mathematical expression.
The issues with the size of the coordinate system, by the way, shows up in the above line element because ##\left( 1- \omega^2 r^2 / c^2 \right)## vanishes when ##\omega r## = c, making the tensor singular at this point. This is called a coordinate singularity. So we can see some differences between the rotating coordinates and the non-rotating coordinates, the former has a coordinate singularity, and the later doesn't.
Basically, the physics doesn't change, just the language changes, and people who haven't learned the tensor methods generally don't understand the tensor language. So we use language that is hopefully familiar to them instead.
If we take a physical experiment like that documented in "The Ultimate Speed",
<<link>> we don't get any different results. Electrons (in this particular experiment) still have a limiting speed slower than c no matter how much energy we give them. We just use slightly different language to describe the results.
Rather than talking about the steps needed to make velocities (not tensors) into four-velocities (true tensors), I'll take a different approach to coordinate independence for this experiment. Regardless of coordinates, if we compare a light pulse and a pulse of relativistic electrons, the light pulse will move faster, than the electrons. For instance, if we send both pulses out at the same time (and make sure the electron beam is not deflected by any stray fields), the light pulse will arrive at the agreed-on destination first, the electrons will arrive later. This is true regardless of how much energy we give the electrons. We could perform a similar experiment on a rotating platform if we really wanted to. We might notice the electron beam taking a different path than the light beam in this case unless we could raise the energy of the electron beam high enough to make the differences experiemntally unmeasurable. But we'd never see the electron beam beating the light beam to the destination. The best we could do is make the difference experimentally so tiny that we can't measure it reliably.
I haven't really covered the issue of the physical significance of the coordinates, but it's somewhat important, so I'll try to give it a brief exposition. Basically, the "t" coordinate in the above rotating coordinate system doesn't have any direct relationship to what clocks read. In particular, as clocks approach the critical radius ##\omega r -> c##, the clocks slow down more and more in terms of the time coordinate t. In the limit, the clocks stop. This isn't the fault of the clocks actually stopping, it's just due to our choice of coordinates. We can figure this out by noting that the clocks don't stop in the inertial coordinates, while they do stop in the rotating coordinates. . Basically, the rotating coordinates are poorly behaved, they have coordinate singularities. The mathematical issue of ##g_{00}## disappearing is the same issue as the clocks stopping in my less formal exposition.
Using tensors, there isn't any problem with using generalized coordinates as long as they are well behaved. Guaranteeing that coordinates are well behaved and interpreting the physical significance of the coordinates is not as trivial as one might assume without some experience and practice actually using generalized coordinates.