Analogue of Lorentz transformation in General Relativity.

[arroy_0205]

Can anybody tell me if there is anything analogous to Lorentz transformation in General Relativity (ie, in curved space)? If there is then what are the corresponding group generators and what is the corresponding algebra? I just wondered this question while reading about Lorentz group and I am not very if such analogy will exist in curved space. But may be using the concept of parallel transport such analogies may be made but that will be complicated.

 

[Mentz114]

As you know, the LT connects a special class of frames in SR. It seems the only possible set of special frames in GR are the freely-falling frames. Each FF observer can construct a local SR frame whose spatial axes are local null geodesics (in reality this frame may be limited in size by the local field). If we know the details of the space-time, it ought to be possible to devise a transformation that connects two such frames, but I don't know what it is or even if it exists.

[edit] Of course there's no reason whatever why two different FFFs should be inertial wrt to each other, so whatever transformation connects them will not look much like the LT.

 

[HallsofIvy]

Not an "analogue" but a "generalization". And that is that that the length of the "world line" is the same from any frame of reference.

 

[Mentz114]

HallsofIvy,

Is there a general transformation between freely-falling frame coords ?

$$x_{\mu} \rightarrow x'_{\mu} = f(x_{\mu})$$

I'm trying to work something out in Painleve-Gullstrand co-ords.

M