Schwarzschild's solution, classical (local) GR tests and general covariance

[TrickyDicky]

At the time Schwarzschild derived his solution (1915) he only had a version of the EFE that was not fully coordinate free, he used the equations in unimodular form, and therefore he could only consider the "outside of the star" part of the fully general covariant form we know now.
So does a "local" approximation of GR always imply loss of general covariance?
I guess that as long as we impose the coordinate asymptotic flatness condition to derive the metric (like is done in the usual textbook Schwarzschild metric derivation) the answer is affirmative.

 

[Ben Niehoff]

As far as I can tell, Schwarzschild merely chose a convenient way to fix a gauge (i.e., choose coordinates). Since ultimately, any explicit solution of the EFE must be written in some coordinates, at some point one has to make decisions about coordinates.

That Schwarzschild seemed to think that his choice of coordinates was special and meaningful is merely a misconception of the times. In fact, there were some disagreements about which coordinate system was most suitable for the Schwarzschild geometry; cf. "isotropic" coordinates, where the coordinate speed of light (a meaningless quantity) is equal in all directions. A few physicists in those times thought this was very important to be arguing about.

I don't think it is worth putting too much stake in 90-year-old misconceptions. The EFE are the same now as they were written down then (as far as I know), and clearly they are coordinate-invariant.

Whatever condition you apply to fix your coordinates is irrelevant, because once you find a solution, you can always change coordinates to your heart's desire, and you will still have a solution (and it will still be the same solution).

Edit: Also, what does this have to do with classical tests of GR? You didn't mention anything about that.

 

[Ben Niehoff]

Incidentally, an English translation of Schwarzschild's paper can be found here:

http://arxiv.org/abs/physics/9905030

Hoenstly, I am not convinced that Schwarzschild actually thought there was anything special about his coordinates, but merely used the unit determinant condition as a way to choose one system of coordinates out of many. So it may be that the misconception is entirely on your part rather than his.

At any rate, the point is: Yes, you have to gauge fix in order to write down an explicit solution. And no, this does not break coordinate invariance; you are still free to change coordinates after you have a solution.

 

[atyy]

't Hooft, http://www.staff.science.uu.nl/~hooft101/lectures/gr.html (p49, footnote 9)

"In his original paper, using a slightly different notation, Karl Schwarzschild replaced ... by a new coordinate r that vanishes at the horizon, since he insisted that what he saw as a singularity should be at the origin, claiming that only this way the solution becomes "eindeutig" (unique), so that you can calculate phenomena such as the perihelion movement (see Chapter 12) unambiguously. The substitution had to be of this form as he was using the equation that only holds if g = 1 . He did not know that one may choose the coordinates freely, nor that the singularity is not a true singularity at all. This was 1916. The fact that he was the first to get the analytic form, justifies the name Schwarzschild solution."

 

[atyy]

The EFE are the same now as they were written down then (as far as I know)

Only since 1998, right? :tongue2:

 

[Ben Niehoff]

't Hooft, http://www.staff.science.uu.nl/~hooft101/lectures/gr.html (p49, footnote 9)

"In his original paper, using a slightly different notation, Karl Schwarzschild replaced ... by a new coordinate r that vanishes at the horizon, since he insisted that what he saw as a singularity should be at the origin, claiming that only this way the solution becomes "eindeutig" (unique), so that you can calculate phenomena such as the perihelion movement (see Chapter 12) unambiguously. The substitution had to be of this form as he was using the equation that only holds if g = 1 . He did not know that one may choose the coordinates freely, nor that the singularity is not a true singularity at all. This was 1916. The fact that he was the first to get the analytic form, justifies the name Schwarzschild solution."

Interesting. So yeah, people back then really had not much idea what they were doing. :P

 

[TrickyDicky]

Interesting. So yeah, people back then really had not much idea what they were doing. :P

To be fair and give it some historical context, Schwarzschild received the early November 1915 EFE while at the german front in WWI, and derived his vacuum solution with a not fully generally covariant version of them restricted to unimodular coordinate solutions. And then he unfortunately died (of an uncommon skin condition) within a few months and had no oportunity to work with the final version of the EFE. So he knew what he was doing after all. ;)
About the relation of this with the classical tests of GR, well they use the Schwarzschild solution with the coordinate condition of asymptotical flatness right? I was just remarking that they are local approximations in the sense that we impose unphysical conditions to obtain them (besides the vacuum condition, that is) like flatness at spatial infinity even if we know that is not realistic , because we are abstracting a certain portion of the universe like the solar system and the coordinate asymptotic flatness allows a good enough approximation. But it also means loss of general covariance as long as that gauge is fixed. Of course we know this is part of an extended general covariant solution that has no coordinate dependence. But I was stressing that the coordinate condition seems important to derive the metric form usually employed for the classical tests. Particularly the preccession and deflection of light ones.