# Lorentz invariance and GR

A. Neumaier
How is Lorentz invariance handled in GR? [...] does GR imply small Lorentz violations in practice?

No. Lorentz invariance is independent of general covariance = diffeomorphism invariance. The latter only replaces the translation invariance of flat field theories.

Lorentz invariance is seen explicitly in general relativity coupled to a spinor field, where gravity is represented by a frame field and local Lorentz invariance is imposed on the frames.

PAllen
No. Lorentz invariance is independent of general covariance = diffeomorphism invariance. The latter only replaces the translation invariance of flat field theories.

Lorentz invariance is seen explicitly in general relativity coupled to a spinor field, where gravity is represented by a frame field and local Lorentz invariance is imposed on the frames.

This is way above my math level (conventional treatments of GR). Physically, the way I see Lorentz invariance being violated in GR is:

1) There is no coordinate system where the Minkowski metric can be used to compute path invariants.

2) If two inertial observers set up overlapping Fermi-normal coordinates, the Lorentz transform will not work to translate observations from one to the other.

People have gotten special relativity as a low energy limit, even Maxwell's equations. Gravity, not yet. There are non-relativistic versions of AdS/CFT, but they are probably not UV complete.

2+1D: http://www.als.lbl.gov/als/science/sci_archive/144dirac_fermions.html
3+1D: http://arxiv.org/abs/hep-th/0507118
Long list of references: http://arxiv.org/abs/1102.0789

Thanks for the refs, looks like interesting stuff.

A. Neumaier
This is way above my math level (conventional treatments of GR). Physically, the way I see Lorentz invariance being violated in GR is:

1) There is no coordinate system where the Minkowski metric can be used to compute path invariants.

2) If two inertial observers set up overlapping Fermi-normal coordinates, the Lorentz transform will not work to translate observations from one to the other.

Local Lorentz invariance is a property of what one finds in the (flat!) tangent hyperplane at each point, which reflects faithfully a neighborhood of that point in terms of Riemann normal coordinates. You can see it if you consider gravity formulated in terms of local orthonormal frames.

PAllen
Local Lorentz invariance is a property of what one finds in the (flat!) tangent hyperplane at each point, which reflects faithfully a neighborhood of that point in terms of Riemann normal coordinates. You can see it if you consider gravity formulated in terms of local orthonormal frames.

Then I don't think we're disagreeing. If you consider any region of finite size in GR, you will not find exact Lorentz invariance.

If you consider any region of finite size in GR, you will not find exact Lorentz invariance.

Exactly. At least we seem to agree on that.

So if in GR, fermionic particles can't define an SR inertial frame due its having a finite size in the 4-manifold, we must conclude only the vacuum and massless bosons can be real inertial frames as defined in SR, and wouldn't be exactly exchangeable with the fermion frames.
In other words all fermionic matter undergoes some degree of acceleration-is affected by curvature. Is this logic correct?