Coordinate transformations in gr

[cndcnd]

Hi,

My question is the following. In special relativity, the Lorentz transformations correspond to a physical situation in which two frames of reference move with uniform rectilinear motion one with respect to the other. In general relativity, given the physical situation in which one frame of reference moves with uniform curvilinear motion one with respect to another frame of reference (for instance), which are the coordinate transformations between the two systems? Are there closed forms for such transformations in the general case (i.e. motion described by generic equations without any particular regularity) ?

Thanks,

Alex

 

[Mentz114]

This recent thread might be of interest.

https://www.physicsforums.com/showthread.php?t=474662

 

[bcrowell]

Hi, cndcnd,

Welcome to PF!

General relativity doesn't have global frames of reference. Therefore there is nothing like the Lorentz transformation that works in an arbitrarily specified spacetime in GR.

-Ben

 

[cndcnd]

Let me reformulate the question in a more rigorous way. In a flat spacetime (no matter in the whole universe), for two frames moving of uniform rectilinear motion one with respect to the other, the classical transformations are: x'=x+vt, y'=y, z'=z, t'=t and the relativistic transformations are the Lorentz transformations.

For two frames one rotating around the other (in the xy plane), the classical transformations are: x'=rcos(vt)+x, y'=rsin(vt)+y, z'=z,t'=t. Which are the corresponding relativistic transformations?

 

[haushofer]

Hi, cndcnd,

Welcome to PF!

General relativity doesn't have global frames of reference. Therefore there is nothing like the Lorentz transformation that works in an arbitrarily specified spacetime in GR.

-Ben

A globally defined Lorentz transformation ;) Of course, local Lorentz transformations are well-defined, as is clear when you describe GR in terms of vielbeins and spin connections.

 

[bcrowell]

For two frames one rotating around the other (in the xy plane), the classical transformations are: x'=rcos(vt)+x, y'=rsin(vt)+y, z'=z,t'=t. Which are the corresponding relativistic transformations?

I don't understand why you're switching to talking about rotations. Anyway, there is no such thing as a global rotation transformation in GR that is defined on all spacetimes.

 

[cndcnd]

Ok, let’s say I want to perform the following experiment. I have two clocks here on earth, perfectly synchronised. Then I put one clock on a plane which flies around the Earth for 1 week. When the plane lands, I want to calculate what is the discrepancy between the two clocks. Let’s assume for simplicity to ignore the contribution of the space-time curvature induced by the mass of the earth.

Since the Lorentz transformations are only valid for systems moving with rectilinear uniform motion one with respect to the other, how do I calculate the slow-down effect occurring during the phases of acceleration (take of) and deceleration (landing) of the plane and also during the flight (the trajectory is curvilinear) ?

Thanks

 

[Mentz114]

In SR the elapsed time on a clock between two events is the proper length of the worldline.

See for instance http://en.wikipedia.org/wiki/Proper_length