# Deriving the Schwarzschild metric just by using the equivalence principle

1. Apr 21, 2010

### Huashan

I've read a few papers about derivation of the Schwarzschild metric by using the equivalence principle ( http://cdsweb.cern.ch/record/1000100/files/0611104.pdf" [Broken])... but I couldn't understand them completely

they assume , According to Einstein’s equivalence principle, that the influence of gravitation on phenomena in a local reference frame that is at rest in the field is equivalent to the influence of the accelerated motion of a local reference frame in which phenomena are described in the absence of gravitation . ( till now there is no problem ).

and then for such an accelerated frame they write the metric in the form :

(no problem yet ).

then... they relate the local coordinates x’, t’ in the accelerated frame to the corresponding coordinates in the stationary frame by the usual differential form of the Lorentz transformation for the value of the velocity that has been reached , by the formulae :

here is the problem ... can these frames be related to each other by the above expressions ?? According to the fact that one of them is non-inertial frame??? ... the derivation of the above expressions are based on the fact that the frames are inertial and moving with respect to each other with a constant velocity v .

Can anybody help me?? ...

Last edited by a moderator: May 4, 2017
2. Apr 21, 2010

### atyy

I don't think it can be done, since many metrics (and many theories, not just general relativity) are consistent with some sort of equivalence principle.

3. Apr 22, 2010

### Huashan

yes I know ... my question is whether we can use Lorentz transformations (in the same form as they are locally in SR ) in constant-linear-acceleration frames or nor ?, if yes how ?...

4. Apr 22, 2010

### JustinLevy

Along atyy's comments, I've heard that Nordstrom's theory of gravity satisfies the strong equivalence princple. I searched and found a paper that made this claim in passing, but couldn't find any that had a good discussion on this. Can anyone comment if this is true about Nordstrom's theory?

This sounds a bit over-stating the case, but wikipedia claims:
"All metric theories satisfy the Einstein equivalence principle"
http://en.wikipedia.org/wiki/Brans-Dicke_theory
Is that really true? Then is the only issue whether a metric theory satisfies the strong equivalence princple?

5. Apr 22, 2010

### JustinLevy

You can always locally make the metric match that of flat spacetime. And then yes, the lorentz transformations apply locally at least. I think it is valid only in that sense.

6. Apr 23, 2010

### Ich

Maybe one should mention that the author does not use Lorentz transformations, but timae dilation and length contraction applied to basic vectors, which is wrong.
Lorentz transformations leave all intervals (the metric) unchanged. That's more or less how they are defined.
This paper is cranky, not surprising as the author is a philosopher.

7. Apr 24, 2010

### nutgeb

Huashan, a reliable source you may want to look at for additional perspective on this is Taylor & Wheeler's excellent textbook "Black Holes." In Query 3 on p B-7 and on p B-13 they combine a Lorentz transformation with the Schwarzschild metric to create a metric for an object in radial freefall from infinity toward a black hole, which they refer to as the "rain frame."

This metric demonstrates that in the frame of the falling object, the local space is flat regardless of the spacetime curvature. They credit this metric to Charles Misner, who in turn derived it from Painleve & Gullstrand.

The Painleve-Gullstrand metric for the "river model" described in the Czerniawski paper you cite certainly is a widely accepted and quite interesting variation on the Schwarzschild metric.

Taylor & Wheeler reference a paper providing formal derivations and references:http://arxiv.org/abs/gr-qc/0001069" [Broken]by Martel and Poisson, AJP 69-4, 2001, p. 476-480.

Peacock also combines a relativistic Doppler shift and a gravitational shift to derive a formula for the cosmological redshift in Schwarzschild coordinates, in his http://arxiv.org/abs/0809.4573" [Broken], eq. 16. This equation produces demonstrably correct calculations.

There probably are many other reliable examples to be found where equations using Lorentz transformations are applied in the context of the Schwarzschild metric. Conceptually I see nothing wrong with doing so, as long as one is careful.

Last edited by a moderator: May 4, 2017