# Lorentz invariance and GR

How is Lorentz invariance handled in GR? I know that there is no global Lorentz invariance in GR, instead it only holds locally, meaning that it is obeyed in the limit at infinity:when r goes to infinity by considering infinite distance or infinitely small point mathematical objects.
But when considering finite distances, does GR imply small Lorentz violations in practice?

If you think of a spacetime as a collection of smooth, non-intersecting geodesic worldlines, representing flow-lines of a perfect fluid, or a set of observers, then in a small enough volume, Lorentz invariance is preserved between the observers in the volume. So, in that sense it's an approximation whose accuracy depends on the spacetime curvature in the region and the size of the volume.

If you think of a spacetime as a collection of smooth, non-intersecting geodesic worldlines, representing flow-lines of a perfect fluid, or a set of observers, then in a small enough volume, Lorentz invariance is preserved between the observers in the volume. So, in that sense it's an approximation whose accuracy depends on the spacetime curvature in the region and the size of the volume.

Right, so I interpret that Lorentz invariance is only "perfectly" realized in a kind of abstract spacetime such as like you say "in a small enough volume" that to be accurate it would have to be infinitely small, so are tiny Lorent violations allowed in GR?

PAllen
Right, so I interpret that Lorentz invariance is only "perfectly" realized in a kind of abstract spacetime such as like you say "in a small enough volume" that to be accurate it would have to be infinitely small, so are tiny Lorent violations allowed in GR?

I think observationally it boils down to tidal gravity. An inertial 'lab' can't erase tidal gravity, so once large enough for this to be significant, it is no longer Lorentz in behavior.

My first post is not correct, strictly speaking. I know of at least one local spacetime that isn't Lorentzian.

I guess any deviations from spacetime flatness will break Lorentz invariance. There isn't a global transformation that connects all IRFs in GR as there is in SR.

I see.
Just to check and avoid confusions, and maybe this question would belong more in the Quantum Physics forum, when they talk about the requirement in QFT of strict Lorentz invariance for the theory to be coherent (together with CPT symmetry implicit in CP violation and T-asymmetry) they obviously refer always to elementary particles obeying strict Lorent symmetry?
and that would be one reason particles in QFT are required to be point-like, without length dimensions?

PAllen
I see.
Just to check and avoid confusions, and maybe this question would belong more in the Quantum Physics forum, when they talk about the requirement in QFT of strict Lorentz invariance for the theory to be coherent (together with CPT symmetry implicit in CP violation and T-asymmetry) they obviously refer always to elementary particles obeying strict Lorent symmetry?
and that would be one reason particles in QFT are required to be point-like, without length dimensions?

With the possible exception of distant entanglement, the scale of quantum phenomena puts them well withing 'locally lorentz' to any measurable precision (so it would seem to me).

The question of the influence of significant tidal gravity on entanglement is one I would like to hear others who know something comment on. Is there an opportunity here to explore the quantum / gravity interface short of the Planck scale?

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Can I clear up: what is actually meant by "Lorentz invariance" in this context? Or by the OP?
Usually, I'd take Lorentz invariance to mean that under Lorentz transformations of a global inertial frame the laws of physics retain their form. That's certainly true in GR; under an arbitrary diffeomorphism of spacetime they retain their form in GR- right?

PAllen
Can I clear up: what is actually meant by "Lorentz invariance" in this context? Or by the OP?
Usually, I'd take Lorentz invariance to mean that under Lorentz transformations of a global inertial frame the laws of physics retain their form. That's certainly true in GR; under an arbitrary diffeomorphism of spacetime they retain their form in GR- right?

That is one way of looking at things, but another more common way is to say that GR is only locally Lorentz invariant in that that the Lorentz transform takes you from one inertial frame to another only locally, not globally. Further, things like being able to unambiguously talk about 'the interval' between events apply only locally in GR.

Can't you define the interval between events via a curve connecting them? It's the construction by which a Riemannian manifold is endowed with the structure of a metric space, although I can see that with pseudo-Riemannian metric, the idea of an extremal length might not carry through in a completely straightforward way...

PAllen
Can't you define the interval between events via a curve connecting them? It's the construction by which a Riemannian manifold is endowed with the structure of a metric space, although I can see that with pseudo-Riemannian metric, the idea of an extremal length might not carry through in a completely straightforward way...

In GR, there is no unique geodesic path (in general), and it does *not* have as clear extremal properties as SR - again, except very locally.

JesseM
Can I clear up: what is actually meant by "Lorentz invariance" in this context? Or by the OP?
Usually, I'd take Lorentz invariance to mean that under Lorentz transformations of a global inertial frame the laws of physics retain their form. That's certainly true in GR; under an arbitrary diffeomorphism of spacetime they retain their form in GR- right?
My understanding is that any set of physical laws (including, say, Newtonian gravity) can be cast in a diffeomorphism-invariant form by expressing the laws in terms of a metric, so unlike Lorentz-invariance, diffeomorphism-invariance isn't really seen as a symmetry of the laws of physics, it's just an inevitable byproduct of expressing physical laws in a particular mathematical way.

atyy
The availability of locally lorentz coordinates are due to the signature of the metric.

The equivalence principle is preserved by the "minimal coupling" of other fields to the metric.

PAllen
The availability of locally lorentz coordinates are due to the signature of the metric.

The equivalence principle is preserved by the "minimal coupling" of other fields to the metric.

Mathematically, the signature guarantees the existence of coordinates where the Minkowski metric occurs 'locally'. However, isn't it a physical statement that such coordinates match local measurements of an inertial observer? In that sense, can one say there is physical content to the assertion of locally Lorentz character of GR?

(Just asking; nor formal degree here).

atyy
Mathematically, the signature guarantees the existence of coordinates where the Minkowski metric occurs 'locally'. However, isn't it a physical statement that such coordinates match local measurements of an inertial observer? In that sense, can one say there is physical content to the assertion of locally Lorentz character of GR?

(Just asking; nor formal degree here).

I put in the second thing about minimal coupling because the locally Lorentz thing doesn't hold if one looks at second derivatives of the metric. So to see things locally Lorentzian in the same way at every point in spacetime, we also need the laws governing matter not to couple to curvature.

PAllen
I put in the second thing about minimal coupling because the locally Lorentz thing doesn't hold if one looks at second derivatives of the metric. So to see things locally Lorentzian in the same way at every point in spacetime, we also need the laws governing matter not to couple to curvature.

Ah, thanks. I missed that - the minimal coupling statement gives the physical meaning I was looking for.

I put in the second thing about minimal coupling because the locally Lorentz thing doesn't hold if one looks at second derivatives of the metric. So to see things locally Lorentzian in the same way at every point in spacetime, we also need the laws governing matter not to couple to curvature.

But how does this have any physical content? you are describing a flat spacetime.

atyy
Can I clear up: what is actually meant by "Lorentz invariance" in this context? Or by the OP?
Usually, I'd take Lorentz invariance to mean that under Lorentz transformations of a global inertial frame the laws of physics retain their form. That's certainly true in GR; under an arbitrary diffeomorphism of spacetime they retain their form in GR- right?

Yes. So "LI" means we write the laws not in generally covariant form, but only in Lorentz covariant form ("SR" form, no non-zero Christoffel symbols allowed). Then see if they remain the same under a Lorentz transformation.

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atyy
But how does this have any physical content? you are describing a flat spacetime.

I am not.

I am not.
Well in the absence of intrinsic curvature (no non-zero Christoffel symbols allowed) you get flat spacetime don't you? Please explain if not the case.

atyy
Well in the absence of intrinsic curvature (no non-zero Christoffel symbols allowed) you get flat spacetime don't you? Please explain if not the case.

You use Riemann normal coordinates, and the Christoffel symbols disappear at the origin ("locally").

You use Riemann normal coordinates, and the Christoffel symbols disappear at the origin ("locally").

Yes, that's trivial. You are talking about a property of Riemannian geometry that can be used to explain the Equivalence principle mathematically. A Riemannian manifold is "locally" flat.
But I'm asking you to translate to a physical picture, what physical object behaves like an infinitesimal patch of spacetime?

atyy, are you talking about a comoving frame here?

atyy
yes, that's trivial. You are talking about a property of riemannian geometry that can be used to explain the equivalence principle mathematically. A riemannian manifold is "locally" flat.
But i'm asking you to translate to a physical picture, what physical object behaves like an infinitesimal patch of spacetime?

Stanford linear accelerator

atyy