Lorentz invariance and General Coordinate transformations

[kdv]

Sorry to bring up again a question that I asked before but I am still confused about this.

In SR we have Lorentz invariance.

Now we go to GR and one says that the theory is invariant under general coordinate transformations (GCTs). But, as far as I understand, this is simply stating that the equations must be invariant under an arbitrary relabelling of the coordinates (I am not talking about diffeomorphism invariance here, just plain GCT).

My question is: how is Lorentz invariance incorporated in GR? At what point does it enter? I used to think that Lorentz invariance was a subset of GCTs but I am not sure now.

In other words, my question is: how does one show that GR is Lorentz invariant if this does not follow from invariance under GCT?

The issue becomes critical when one incorporates spinors in GR. The theory must be Lorentz invariant but, the experts say, spinors do not form a representation of the general transformations, only a representation of Lorentz transformations.

As George Jones kindly explained to me, the Lorentz transformation of the spinors must be defined only in the tangent space. And this is where one is forced to introduce vielbeins and work in the tangent space etc etc.
But this (naively) seems to indicate that the Lorentz transformations are not a subset of the GCTs.

 

[pmb_phy]

Sorry to bring up again a question that I asked before but I am still confused about this.

In SR we have Lorentz invariance.

Now we go to GR and one says that the theory is invariant under general coordinate transformations (GCTs). But, as far as I understand, this is simply stating that the equations must be invariant under an arbitrary relabelling of the coordinates (I am not talking about diffeomorphism invariance here, just plain GCT).

I too have been wondering about this for sometime. The term general coordinate transformation can give one the impression that it applies to any change in coordinates that your imagination can dream up. Consider the Galilean coordinate transformaion. It would be incorrect to expect invariance when a Galilean coordinate transformation is applied. I believe that the term general coordinate transformation applies to all physically meaningful coordinate transformations and not simply anything that you're mind dreams up.

My question is: how is Lorentz invariance incorporated in GR? At what point does it enter? I used to think that Lorentz invariance was a subset of GCTs but I am not sure now.

The Lorentz transformation is one out of many transformations which are physically meaningful and as such it is contained in the set of all general coordinate transformations, just as you suspected.

In other words, my question is: how does one show that GR is Lorentz invariant if this does not follow from invariance under GCT?

Simply transform from one locally inertial frame to another locally inertial frame.

The issue becomes critical when one incorporates spinors in GR.

I'm not familiar with spinors. They make me dizzy! Seriously though, Can you tell me where spinors come into play in GR? I have a book on spinors and haven't gotten around to reading it yet. I'm wondering where to place it on my list of priorities.

As George Jones kindly explained to me, the Lorentz transformation of the spinors must be defined only in the tangent space.

Sounds right to me.

Pete

 

[kdv]

Thanks for your reply, Pete.

I too have been wondering about this for sometime. The term general coordinate transformation can give one the impression that it applies to any change in coordinates that your imagination can dream up.

However, as far as I can tell, every reference I have read define GCT's as arbitray transformations $$x^\mu \rightarrow x'^\mu(x^\nu)$$ i.e. they never seem to impose any restriction at all on the transformation! Well, beside the fact that they must be " nice" mathematically (continuous, etc). Maybe I misunderstand what I read.

Consider the Galilean coordinate transformaion. It would be incorrect to expect invariance when a Galilean coordinate transformation is applied. I believe that the term general coordinate transformation applies to all physically meaningful coordinate transformations and not simply anything that you're mind dreams up.

You have a very good point there about the fact that Galilean transformations should not be allowed!

But this is what puzzles me. I used to think that the GCTs had to do with physical transformations (between frames of references) but I am not sure now. In SR, a Lorentz transformation corresponds to relabelling events in a way that corresponds to a change of reference frame. So the invariance under Lorentz transformations has some deep physical content.

I used to think that GCT's were the equivalent statement generalized to GR. But I am not sure of this anymore. Your point about Galiean transformations points to a problem. My comment about spinors is a different apparent problem of interpretation.

The Lorentz transformation is one out of many transformations which are physically meaningful and as such it is contained in the set of all general coordinate transformations, just as you suspected.

Simply transform from one locally inertial frame to another locally inertial frame.

This is what I used to think. But then it means that I misinterpret completely textbooks when they present GCT's as if they are arbitrary transformations. They never mention any restriction.

I'm not familiar with spinors. They make me dizzy!

Seriously though, Can you tell me where spinors come into play in GR? I have a book on spinors and haven't gotten around to reading it yet. I'm wondering where to place it on my list of priorities.

I have to deal with them because I am trying to understand the different approaches to quantum gravity. In classical GR I guess they are a non-issue.

 

[Ich]

You have a very good point there about the fact that Galilean transformations should not be allowed!

But this is what puzzles me. I used to think that the GCTs had to do with physical transformations (between frames of references) but I am not sure now. In SR, a Lorentz transformation corresponds to relabelling events in a way that corresponds to a change of reference frame. So the invariance under Lorentz transformations has some deep physical content.

My €0,02:
Coordinates lost their direct physical meaning in GR; there is this additional layer (the metric, especially its components) that translates from coordinates to physics. Therefore, GCT are indeed arbitrary, galileian transformartions are of course allowed - as long as you can write down the metric.
There is a certain subset of coordinates in flat spacetime that is quite easily established physically: cartesian coordinates in intertial frames.
Those are connected by the LT, and you can use the simplest possibe metric (-1,1,1,1) for all of them. If you used GT, the resulting frame would no longer be the standard inertial frame, and you would have a different metric. The LT are unique because they transform orthonormal base vectors to orthonormal base vectors.

Just my thoughts, this might be rubbish. Better wait for more qualified answers.