Is there a relation between GCT and Lorentz invariance?

[kdv]

I am not sure where to post this question since it involves GR and particle physics but here it goes.

I am reading in a book that when coupling a spinor to gravity, one replaces [itex] \partial_\mu \psi [/tex] by a covariant derivative [itex] D_\mu \psi [/tex] which must transform like a spinor under local lorentz transformations but as a vector under general coordinate transformations. (and does that mean that [itex] \gamma^\mu [/itex] must transform as a vector under GCTs ?)

Can someone explain to me the logic involved here? I know that the D contains an index "mu" which indicates that this should be a vector under GCT's, but I don't really understand the rationale. I don't really understand the physical distinction between the LLT's and the GCT's. I thought that there was only the GCT which included the LLT as special case but this doesn't seem to be the case.



And (and I know this is a different issue), how is it possible to do a GCT in the first place? Imean, the curvature had a physical impact on the geodesics of particles so if I do a GCT that turns a flat region into a curved one, the physics is changed, obviously. So how can the theory be invariant under GCTs?? I know that this is something that bothered Einstein but I never understood the resolution of this issue. Invariance under GCT does not make sense to me since physics seems to be changed.


What does it mean to say that [itex] D_\mu \psi [/itex] must transform as a vector under a GCT? It's not possible to define spinors under GCT?

 

[George Jones]

I am not sure where to post this question since it involves GR and particle physics but here it goes.

I am reading in a book that when coupling a spinor to gravity, one replaces [itex] \partial_\mu \psi [/tex] by a covariant derivative [itex] D_\mu \psi [/tex] which must transform like a spinor under local lorentz transformations but as a vector under general coordinate transformations. (and does that mean that [itex] \gamma^\mu [/itex] must transform as a vector under GCTs ?)

Which book?

From the context, it looks like [itex]\psi[/itex] is Dirac spinor, not a 2-component spinor. Not that I can say anything useful but I just want to confirm this.

And (and I know this is a different issue), how is it possible to do a GCT in the first place? Imean, the curvature had a physical impact on the geodesics of particles so if I do a GCT that turns a flat region into a curved one, the physics is changed, obviously. So how can the theory be invariant under GCTs?? I know that this is something that bothered Einstein but I never understood the resolution of this issue. Invariance under GCT does not make sense to me since physics seems to be changed.

Moving from coordinate system to coordinate system does not change whether curvature is zero or non-zero, i.e., if (non) zero in one coordinate system, then (non) zero in all.

General coordinate systems are allowed in special relativity and even in Newtonian mechanics. For example, consider a free particle moving along a straight line in a plane, with the line expressed in Cartesian coordinates. By using the standard transformations, this line also can be expressed in polar coordinates. Or one could start in polar coordinates, solve the geodesic equation, and arrive at the same thing.

Coordinates are just labels (for the same points), so changing coordinates changes the labels, but doesn't change the physics.

What does it mean to say that [itex] D_\mu \psi [/itex] must transform as a vector under a GCT? It's not possible to define spinors under GCT?

There's something a little deeper going on, and if I find some time, I'll try and have a look at it, but, for now, I'll make a guess.

Forget spinors for a few moments.

Consider an event P on the worldline of observer A. To make measurements at P, A sets up an orthonormal frame of 4-vectors (tetrad) at P. Suppose the worldline of observer B intersects A's worldline at P. B also sets up an orthonormal frame at P. Even in general relativity, the two frames are related by a Lorentz transformation. Now, P could be contained in different coordinate patches, with a general coordinate transformation relating the two coordinate systems.

Sometimes each (4-vector) member of a tetrad has two indices. One index denotes which members of the tetrad, i.e, which (complete) 4-vector, and one index denotes the *components* of the tetrad members with respect to a general coordinate system. Lorentz transformations act the first index, i.e., transform from physical frame to physical frame, while general coordinate transformations act on the second index. Sometimes one of the indices is suppressed.

I gave an example of Lorentz transformation between frames in Schwarzschild geometry in https://www.physicsforums.com/showpost.php?p=848684&postcount=4". I also expressed t each member of each frame has components with respect to the coordinate basis (which isn't a frame)

[tex]\left \{ \frac{\partial}{\partial x^{\mu}} \right\}.[/tex]

These components transform under general coordinate transformations.

I think something similar happens for spinors. Instead of orthonormal frames and Lorentz transformations, there are spin frames and (appropriate representations) of SL(2,C) transformations, and there still are are general coordinate systems that label the events.

Not sure of the details.

 

[kdv]

Which book?

Particle Physics and Cosmology by P.D.B. Collins, A.D. Martin and E.J.Squires, John Wiley and Sons, 1989.

From the context, it looks like [itex]\psi[/itex] is Dirac spinor, not a 2-component spinor. Not that I can say anything useful but I just want to confirm this.

Thanks for the reply George.

Well, it's a four-component spinor but I guess it could be Dirac or Majorana.

Moving from coordinate system to coordinate system does not change whether curvature is zero or non-zero, i.e., if (non) zero in one coordinate system, then (non) zero in all.

Oh yes, of course. Sorry for my lapse there. So if the Riemann tensor is non-zero in one coordinate system, it will be non-zero after any GCT.

General coordinate systems are allowed in special relativity and even in Newtonian mechanics. For example, consider a free particle moving along a straight line in a plane, with the line expressed in Cartesian coordinates. By using the standard transformations, this line also can be expressed in polar coordinates. Or one could start in polar coordinates, solve the geodesic equation, and arrive at the same thing.

Coordinates are just labels (for the same points), so changing coordinates changes the labels, but doesn't change the physics.




There's something a little deeper going on, and if I find some time, I'll try and have a look at it, but, for now, I'll make a guess.

Forget spinors for a few moments.

Consider an event P on the worldline of observer A. To make measurements at P, A sets up an orthonormal frame of 4-vectors (tetrad) at P. Suppose the worldline of observer B intersects A's worldline at P. B also sets up an orthonormal frame at P. Even in general relativity, the two frames are related by a Lorentz transformation. Now, P could be contained in different coordinate patches, with a general coordinate transformation relating the two coordinate systems.

Good, that's a useful situation to focus on before even talking about spinors. I guess I am confused about the relation between GCT and the Lorentz transformations. I always thought that teh GCT were an extension of teh LT. I thought that special relativity said "physics is invariant under LT" and that GR said "well, we can be even more general and say that physics is invariant under arbitrary coordinate transformations". But then I thought that this meant that the Lorentz transformations were a special case of the GCT.

So now I am confused about having to think separately about LT and GCT. if I understand correctly now, the GCT have no physical content? So any theory describing spacetime should be invariant under GCT for a purely mathematical reason: different labelling of points should not matter. And this does not imply at all Lorentz invariance??
I though the two were related. Can you elaborate a bit more on this?

Thanks a lot.