Don't try the
when did you stop beating your wife line with me przyk. Instead, stop beating around the bush and respond to
post #158.
I am disappointed that you seem reluctant to show much willingness to consider the points of view of others. One has to wonder why you find this so difficult, since merely acknowledging that someone has a certain opinion is in no way an entrapment for you.
Here are my responses (again) to the three points I summarised in [POST=3067397]post #200[/POST]. Many of them I made, in even more detail, in [THREAD=105796]this[/THREAD] (starting around page 5) and subsequent threads, as well as in a PM discussion about six months ago. If you think they are unfair representations of your end conclusions, or that I have misunderstood something, then now is your opportunity to say so.
1) "There is no motion in spacetime".
Of course, this is a reference to the first few sentences in [POST=3066616]post #158[/POST]:
Spacetime is an abstract mathematical space in which motion does not occur because it models space at all times. You can draw world-lines in it, and you can draw them curved, but that worldline represents the motion of a body through space over time. The body doesn't actually move through spacetime. People tend to talk of "the spacetime around the Earth" and suggest that light moves through it, but that's wrong.
This is the least interesting point to deal with because most physicists would see it as vacuous. To the extent you are right, you are not saying anything that anyone needs to be educated about.
Your argument "spacetime is an abstract mathematical space" is pointless because the same could be said of any mathematical notation or formalism used in physics. There is no testable difference between the "spacetime" view and the "space + time" view, and consequently no meaningful distinction as far as most physicists are concerned. It is simply a question of which notation is the most practical or useful, and in relativistic physics the "spacetime" view and notation just better reflects certain symmetries in relativistic theories. In some cases, the difference is as minor as writing something like $$\phi(x^{\mu})$$ as opposed to $$\phi(\bar{x};\,t)$$.
The next part, "[...] in which motion does not occur" is silly. When a physicist talks about "motion" in the context of spacetime without further qualification, they are almost certainly referring to exactly the same type of "motion" you might in space: changing position over time. You even explain exactly how to describe that in the language of spacetime (with worldlines and so on). Same thing, different notation and language. That's it. A Minkowski diagram of a worldline is essentially the same thing as the distance-time graphs we make kids draw in highschool physics, except with the space and time axes inverted.
The response to the rest of that passage is pretty much the same: when you read someone say something with the word "spacetime" in it, don't simply invent the stupidest interpretation you can come up with and attribute it to that person. That's an instant strawman that serves only to derail the discussion.
Next point:
2) General relativity is, or can be interpreted as, a theory about flat but inhomogeneous space.
As the person claiming this, the burden of proof of course falls on you to adequately support it. Specifically, you have to successfully show that general relativity can be formulated this way and, most importantly,
that it actually works. However, what you offer up in support falls far short of this. Note that merely finding an instance where someone says something is not proof that the idea actually works. That pretty much disqualifies any nontechnical source you might try to use, since the point of nontechnical expositions is just that -- exposition -- and not proofs or derivations.
First, we get the elephant out of the way.
Einstein's 1916 paper was the first complete and general formulation of Einstein's theory that was developed to the point that it could make arbitrary predictions. For some reason you cite it as supporting your case in post #158, despite the fact it is 3+1 dimensional pseudo-Riemannian geometry from beginning to end. The 3+1 dimensional "spacetime" bit is spelled out in the introductory sections that set the stage and define notations for the rest of the paper. For instance, there's a sort of preface on the first page which includes:
The generalization of the theory of relativity has been facilitated considerably by Minkowski, a mathematician who was the first one to recognize the formal equivalence of space coordinates and the time coordinate.
So there already, apparently we're getting from Einstein that Minkowski's spacetime language and formalism is really helpful for understanding and generalising relativity. The very next sentence is:
The mathematical tools that are necessary for general relativity were readily available in the "absolute differential calculus," which is based upon the research on non-Euclidean manifolds by Gauss, Riemann, and Christoffel, and which has been systematized by Ricci and Levi-Civita and has already been applied to problems of theoretical physics
Unsurprisingly, the content of Einstein's paper is pretty much what you would expect from this preface. In fact, the first several section can be read as an abridged recap of special relativity in Minkowski's notation followed by a tutorial on the methods of (pseudo) Riemannian geometry in 3+1 dimensional spacetime. Note the references to "space-time", "four-dimensional", and such language, including definitions of four-vectors, tensors, and so on. Of course, pointing out specific instances like this just amounts to a bit of quote mining, and there is no substitute for actually reading Einstein's paper and understanding why he's doing things that way (which anyone following so far is of course encouraged to do).
That's not to say that Einstein uses the Minkowski "spacetime" language exclusively throughout the whole paper and never uses the "space + time" language, but the general pattern that emerges is that Einstein uses the "spacetime" language when he's speaking in generalities (i.e. developing the general theoretical framework), and reserves the "space + time" language for far more specific and restricted problems in the latter part of the paper, e.g. low velocity and weak field approximations that make the correspondence with Newtonian physics.
Note that the general "equation of motion" that Einstein (re)derives in this paper, which you refer to in post #158, is the geodesic equation in 3+1 dimensional spacetime given as equation (20d) in section 9 on page 168. (In Minkowski notation, Greek indices run over the four spacetime coordinates. This is more or less explicitly stated toward the end of page 155 in section 4). In [POST=2727436]this post[/POST], I also showed that the time component part of the geodesic equation was vital for the recovery of Newtonian gravity in the weak field approximation. As I also pointed out back then, a more detailed discussion of this approximation is given by Einstein in section 21 around pages 194--195.
Your statement that "
metric is to do with measurement" is also false. The metric components are basically the coefficients that appear in a generalised differential version of Pythagoras' theorem, and are defined and discussed on page 155. They are not defined as being measurable. It is clear from the definition in equation (3) and surrounding discussion that they are coordinate-dependent: pick an arbitrary coordinate system, then $$\mathrm{d}s^{2}$$ is an invariant and the $$g_{\mu\nu}$$s are just whatever they need to be in that coordinate system such that $$\mathrm{d}s^{2} \,=\, g_{\mu\nu} \mathrm{d}x^{\mu} \mathrm{d}x^{\nu}$$. In particular, this implies that the metric components transform according to
$$g'_{\alpha\beta} \,=\, \frac{\partial x^{\mu}}{\partial x'^{\alpha}} \, \frac{\partial x^{\nu}}{\partial x'^{\beta}} \, g_{\mu\nu} \,.$$
This is a special case of the (covariant) rank 2 tensor component transformation rule given in equation (11) on page 159.
So then what? Honestly as far as your case goes, that's pretty much it, really. Of the various sources you cited, Einstein's 1916 paper is
by far the most important for two reasons: 1) it is the only one that defines general relativity formally and precisely enough that predictions can be derived from it (which he does for certain specific circumstances in all the gory details), and 2) being a technical paper it is addressed primarily at physicists who needed to fully understand his work. This is how Einstein wanted his
scientific peers to understand his theory.
Your case is undermined by the
absence of any paper by Einstein or anyone else establishing a formal equivalence between the 3+1 dimensional Riemannian geometry and an alternative formulation that is based around some idea of inhomogeneous space. The best evidence you have is that some Chinese researchers
had a shot at it in 2008. (!) I thought I already made this point as clearly as possible at the conclusion of [POST=3051334]this post[/POST]:
If there is an alternative version of GR, why isn't there a complete treatise on it in the literature, say something analogous to Einstein's 1916 paper? Where is this alternative version of GR actually formally developed and advanced? Why is it that the best you seem able to do is come up with scattered quotes mined from different places and a 2008 article in Chinese Physics Letters that is hardly contemporaneous with Einstein?
(I [POST=2707463]previously responded[/POST] with regard to the
Chinese Physics Letters paper too, by the way.)
Of course, you're thinking about the Leyden address, which finally brings us to the last point:
3) Albert Einstein was personally a proponent of point #2.
As explained above, and many times to you before, you won't find anything to back this up in Einstein's 1916 paper. Instead, you offer what seems to be one of your all-time favourite Einstein quotes, from the 1920 Leyden address:
According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that "empty space" in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν), has, I think, finally disposed of the view that space is physically empty.
from which you somehow jump to "Space is inhomogeneous, not curved." But the part you bolded doesn't support that. First, it merely states that "empty space" is inhomogeneous, which is not controversial (e.g. the geometry or curvature of spacetime within the solar system is inhomogeneous). You don't explain where you get this dichotomy between inhomogeneity and curvature -- that's not in your quote. And since we're happily quote mining, did you notice the bits I highlighted in red for you?
The whole passage basically amounts to arguing that spacetime should be viewed as having properties. That is entirely consistent with Einstein's 1916 paper, and there is no indication that Einstein is recanting anything from his original formulation of the theory.
Once again, I point out that there is no article by Einstein formally establishing a version or interpretation of general relativity specifically based around inhomogeneity of space. All you have are a few isolated quotes mined from various places.
