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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.

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.

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