Can an Arbitrary Co-ordinate System Be Used for Observers in GR?

[blue_sky]

The Galileo transformation of co-ordinate let me move from a co-ordinate system to an other in the classic physics like the Lorenz transformation in SR.
What are the co-ordinate transformation in GR which have a physical meaning in the sense that can be associated with a real mass?

 

[pmb_phy]

The Galileo transformation of co-ordinate let me move from a co-ordinate system to an other in the classic physics like the Lorenz transformation in SR.
What are the co-ordinate transformation in GR which have a physical meaning in the sense that can be associated with a real mass?

In GR all coordinate transformations from one set of spacetime coordinates to another are valid so long as they satisfy certain properties, i.e. the Jacobian of the transformation is finite and does not vanish. Hence the name "General" relativity.

Pete

 

[blue_sky]

In GR all coordinate transformations from one set of spacetime coordinates to another are valid so long as they satisfy certain properties, i.e. the Jacobian of the transformation is finite and does not vanish. Hence the name "General" relativity.

Pete

Yup, I agree with that but I'm interested in finding the co-ordinate transformations in GR that can be associated with real bodies. As far I understood, while I can consider a generic co-ordinate transformations, not all of them can be associated to real objects. Am I wrong?

 

[pmb_phy]

Yup, I agree with that but I'm interested in finding the co-ordinate transformations in GR that can be associated with real bodies. As far I understood, while I can consider a generic co-ordinate transformations, not all of them can be associated to real objects. Am I wrong?

I don't understand what you mean by a coordinate transformation associated with real bodies. Coordinate transformations in relativity pertain to events in spacetime and nothing else. You must be thinking of soemthing else.

Pete

 

[blue_sky]

Pete, this is unclear to me.
Let me try to say it in different way; I’m looking to a spaceship with a generic velocity wi with respect to me. If xi are the co-ordinate system associated with me and yj the ones associated with the spaceship, what are the fj such that yj=fj(xi) ? Make this sense?

 

[pmb_phy]

Pete, this is unclear to me.
Let me try to say it in different way; I’m looking to a spaceship with a generic velocity wi with respect to me. If xi are the co-ordinate system associated with me and yj the ones associated with the spaceship, what are the fj such that yj=fj(xi) ? Make this sense?

That is a combination of coordinate transformations and velocity transformations.

They are given here
http://www.geocities.com/physics_world/sr/lorentz_trans.htm
http://www.geocities.com/physics_world/sr/velocity_trans.htm

Pete

 

[blue_sky]

That is a combination of coordinate transformations and velocity transformations.

They are given here
http://www.geocities.com/physics_world/sr/lorentz_trans.htm
http://www.geocities.com/physics_world/sr/velocity_trans.htm

Pete

Thanks Pete.
Are the above transformations valid also in presence of gravitational fields? If not, do you know what are the valid ones in that case?

 

[pmb_phy]

Thanks Pete.
Are the above transformations valid also in presence of gravitational fields?

The Lorentz transformation is in the weal field limit. When done so its called a background Lorentz transformation. See

http://www.geocities.com/physics_world/gr/back_lorentz_trans.htm

Pete

 

[DW]

The Galileo transformation of co-ordinate let me move from a co-ordinate system to an other in the classic physics like the Lorenz transformation in SR.
What are the co-ordinate transformation in GR which have a physical meaning in the sense that can be associated with a real mass?

Since your question is for GR, what you are being told about aplication of Lorentz transformation in accelerated states and all is complete nonsense in that context. This is what you do if you want to relate an arbitrary observer's coordinates to your own in general relativity.
First you choose your own coordinates which in GR is arbitrary, but given what you want to do is most sensible to choose them so that locally the metric reduces to $$g_{\mu }_{\nu }|(x = y = z = 0) = \eta _{\mu } _{\nu }$$. Choose coordinates for the arbitrary observer so that the metric at his location reduces also to $$g'_{\mu }_{\nu }|(x' = y' = z' = 0) = \eta _{\mu } _{\nu }$$. In doing that you will have descided on a transformation between the two frames. Your choice of coordinates for the two frames with their transformation will not be unique, but GR yeilds no motivation whatsoever that they should be. This method will always reduce to Lorentz class transformations in the case that the observers are near each other and will always reduce to Lorentz class transformations when the metrics for the two are globally $$\eta _{\mu } _{\nu }$$.

 

[blue_sky]

Thanks DW.
Where I can read more on this subject?

 

[DW]

Thanks DW.
Where I can read more on this subject?

Since global choice of frame is so arbitrary in GR, texts just don't discuss this particular subject in much more detail than I just gave you. Particular transformations are usually discussed in detail that take the expression for the invariant interval for a particular case of spacetime from a common set of coordinates some other set in order to make a point that isn't obvious otherwise for example Schwarzschild to Kruskal-Szekeres, but other than that sort of thing it just doesn't matter what coordinates you use. Some coordinate choices are more natural to use or yield simpler expressions for the metric especially in special relativity where the expression for the metric is invariant in Lorentz transformations between inertial frames. Where it comes to "arbitrarily" time dependent accelerations in arbitrary directions in flat spacetime the most natural coordinates to use are related by a Lorentz like transformation and the only place you will even find that is
http://www.geocities.com/zcphysicsms/chap5.htm#BM65
Besides that and the natural choice of a coordinates where the metric locally reduces to that of SR, your choice of coordinates is arbitrary so don't worry. You can't go wrong.

 

[blue_sky]

Since global choice of frame is so arbitrary in GR, texts just don't discuss this particular subject in much more detail than I just gave you. Particular transformations are usually discussed in detail that take the expression for the invariant interval for a particular case of spacetime from a common set of coordinates some other set in order to make a point that isn't obvious otherwise for example Schwarzschild to Kruskal-Szekeres, but other than that sort of thing it just doesn't matter what coordinates you use. Some coordinate choices are more natural to use or yield simpler expressions for the metric especially in special relativity where the expression for the metric is invariant in Lorentz transformations between inertial frames. Where it comes to "arbitrarily" time dependent accelerations in arbitrary directions in flat spacetime the most natural coordinates to use are related by a Lorentz like transformation and the only place you will even find that is
http://www.geocities.com/zcphysicsms/chap5.htm#BM65
Besides that and the natural choice of a coordinates where the metric locally reduces to that of SR, your choice of coordinates is arbitrary so don't worry. You can't go wrong.

Thanks DW.
1 more question.
If I choose an arbitrary co-ordinate system, can I be on observer solidal to that co-ordinate system or, in order to be possible that, the co-ordinate system must respect some restrictions?

 

[DW]

Thanks DW.
1 more question.
If I choose an arbitrary co-ordinate system, can I be on observer solidal to that co-ordinate system or, in order to be possible that, the co-ordinate system must respect some restrictions?

Solidal? The only restriction you need in order so that it can be considered rectilinear local to you as an observer is that the metric expressed in your coordinates reduces to $$\eta _{\mu }_{\nu }$$ at your location.