Does GR imply small Lorentz violations in practice?

[TrickyDicky]

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?

 

[Mentz114]

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.

 

[TrickyDicky]

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.

 

[Mentz114]

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.

 

[TrickyDicky]

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?

 

[muppet]

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.

 

[muppet]

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.

 

[TrickyDicky]

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.

 

[atyy]

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

I am not.

 

[TrickyDicky]

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").

 

[TrickyDicky]

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?

 

[muppet]

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]

atyy, are you talking about a comoving frame here?

Yes, if the worldline is geodesic.

My understanding is that a comoving frame for arbitrary worldlines corresponds to Fermi normal coordinates, and if the wordline is geodesic then we the Christoffel symbols disappear, and the frame is "locally Lorentz". Such a frame is not truly Lorentz, even at the origin, since the derivatives of the Christoffel symbols do not disappear. However, if matter couples "minimally" to the spacetime metric, then there are experiments we can do that give don't probe the derivatives of the Christoffel symbols, and so things will look "locally Lorentz".

 

[PAllen]

I still go with my criterion: If tidal gravity is below the level of detectability for the intended measurements, the region involved can be considered locally Lorentz. It's a question of precision in relation to the gravitational field.

 

[atyy]

I still go with my criterion: If tidal gravity is below the level of detectability for the intended measurements, the region involved can be considered locally Lorentz. It's a question of precision in relation to the gravitational field.

Yes, that's the more complete answer to "SLAC".

 

[TrickyDicky]

Stanford linear accelerator

Close but not quite, the discrepancy is "below the level of detectability" with our current technology, but I am not speaking about "practical" LI, thus not about our capacity to detect a small discrepancy. I'm interested in the theoretical model, is the appeal to the level of detectability compatible with the fact that even an elementary relativistic particle according to GR curves spacetime not infinitesimally (therefore cannot be treated as a example of local flatness) regardless if we can detect it with our actual experiments? after all we cannot rule out that with better instruments we could ever measure a tiny violation of LI, but as I'm saying I'm referring to what GR asserts in principle not to what we can measure.

 

[TrickyDicky]

I still go with my criterion: If tidal gravity is below the level of detectability for the intended measurements, the region involved can be considered locally Lorentz. It's a question of precision in relation to the gravitational field.

Yes,unless the intention is derive physical principles from a lack of precision, or ignore what the theory(GR) says clearly: any amount of mass curves spacetime.

 

[TrickyDicky]

Such a frame is not truly Lorentz, even at the origin, since the derivatives of the Christoffel symbols do not disappear. However, if matter couples "minimally" to the spacetime metric, then there are experiments we can do that give don't probe the derivatives of the Christoffel symbols, and so things will look "locally Lorentz".

As I'm saying, for practical reasons one can choose a experiment where matter "couples minimally" and things will look locally Lorentz for some intended calculation or purpose, but if we are dealing with a question of principle or we want to derive the universality of Lorentz symmetry from it, we have to admit as you do that "such a frame is not truly Lorentz".

 

[Dale]

we have to admit as you do that "such a frame is not truly Lorentz".

I think this is correct. We can say that it is Lorentz to first-order (in distance and time).

 

[TrickyDicky]

I think this is correct. We can say that it is Lorentz to first-order (in distance and time).

Exactly, and would you say one can base a principle in something that is valid only to first order approximation. To give an example, how can one not run into infinities in QED(wich demands renormalization to get reasonable figures), where perfect LI is assumed if it uses an assumption that is only valid to first order?

 

[atyy]

Close but not quite, the discrepancy is "below the level of detectability" with our current technology, but I am not speaking about "practical" LI, thus not about our capacity to detect a small discrepancy. I'm interested in the theoretical model, is the appeal to the level of detectability compatible with the fact that even an elementary relativistic particle according to GR curves spacetime not infinitesimally (therefore cannot be treated as a example of local flatness) regardless if we can detect it with our actual experiments? after all we cannot rule out that with better instruments we could ever measure a tiny violation of LI, but as I'm saying I'm referring to what GR asserts in principle not to what we can measure.

Yes. I did not say otherwise.

 

[atyy]

Exactly, and would you say one can base a principle in something that is valid only to first order approximation. To give an example, how can one not run into infinities in QED(wich demands renormalization to get reasonable figures), where perfect LI is assumed if it uses an assumption that is only valid to first order?

QED has infinities (after renormalization) even if there is global Lorentz invariance.

 

[Dale]

Exactly, and would you say one can base a principle in something that is valid only to first order approximation.

Sure, why not? Tons of science is done to first order. If you try to do too high of an order approximation then you wind up just fitting to the noise. It is very important to use the lowest order you can.

 

[TrickyDicky]

QED has infinities

So far so good

after renormalization

You've got this backwards, renormalization is the procedure to get rid of the infinite terms appearing at intermediate stages of every calculation of the bare masses and charges of electrons in QED.
These infinities arise because of the postulated pointlike nature of elementary particles, implying electrons could transmit arbitrary high momenta, integrating over these arbitrary high momenta makes quantities diverge to infinity.
What I'm arguing is that the pointlike particle assumption of QED is founded in Lorentz invariance for particles and seems like GR contradicts it insofar this particles have mass.

even if there is global Lorentz invariance.

not even, but because of, is what I'm pointing out. the global LI comes from the infinite degrees of freedom of the fields (thus its quantum must be pointlike)

I guess what is needed is a QFT in curved space not in flat Minkowskian space, but can that be achieved holding on to Lorentz invariance even if we know it's just an approximation?

 

[TrickyDicky]

Sure, why not? Tons of science is done to first order. If you try to do too high of an order approximation then you wind up just fitting to the noise. It is very important to use the lowest order you can.

You can do it alright. That's not my point.
You can't pretend that what is an approximation is exact and derive of that assumption physical axioms.

 

[Dale]

I don't know what you are talking about. What physical axioms are you referring to?

Science is about getting accurate models. As long as the approximations are buried in the noise you have an accurate model. Approximations are a good thing and a staple of calculus as well as physics. We cannot experimentally distinguish between a low-order model and a high-order model when the high-order terms are undetectable, so we use the low-order model until the high-order terms become detectable.

 

[TrickyDicky]

I don't know what you are talking about. What physical axioms are you referring to?

QFT Lorentz symmetry.

Science is about getting accurate models. As long as the approximations are buried in the noise you have an accurate model. Approximations are a good thing and a staple of calculus as well as physics. We cannot experimentally distinguish between a low-order model and a high-order model when the high-order terms are undetectable, so we use the low-order model until the high-order terms become detectable.

This is right of course, is not what I'm debating though, don't know why you bring it up.

 

[Dale]

QFT Lorentz symmetry.

The Lorentz symmetry of QFT depends only on the mathematical form of the equations and has nothing to do with whether or not a given laboratory experiment is affected by spacetime curvature.

 

[atyy]

You've got this backwards, renormalization is the procedure to get rid of the infinite terms appearing at intermediate stages of every calculation of the bare masses and charges of electrons in QED.
These infinities arise because of the postulated pointlike nature of elementary particles, implying electrons could transmit arbitrary high momenta, integrating over these arbitrary high momenta makes quantities diverge to infinity.
What I'm arguing is that the pointlike particle assumption of QED is founded in Lorentz invariance for particles and seems like GR contradicts it insofar this particles have mass.

No I did not. The infinities remain after renormalization. QED may even break down before Lorentz violation is detected.

not even, but because of, is what I'm pointing out. the global LI comes from the infinite degrees of freedom of the fields (thus its quantum must be pointlike)

I guess what is needed is a QFT in curved space not in flat Minkowskian space, but can that be achieved holding on to Lorentz invariance even if we know it's just an approximation?

QFT in curved spacetime is an approximation, because it treats the gravitational field classically, while the other fields are treated quantum mechanically. QFT in flat spacetime is also an approximation. We have no theory that is not an approximation.

 

[TrickyDicky]

The Lorentz symmetry of QFT depends only on the mathematical form of the equations and has nothing to do with whether or not a given laboratory experiment is affected by spacetime curvature.

That's what I'm saying. I'm centering on the form of those equations, and where they come from.

 

[PAllen]

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?

Let's go back to the beginning. Yes, there is no global Lorentz invarianced in GR (invariants come from the metric, and the Minkowski metric is not globally valid).

But then, "locally" means limit at infinity?? Nope.

Locally is code word for "limit as region goes to zero size". A sphere is locally Euclidean, but no finite section is exactly Euclidean. It is mathematically precise to say that it is in the limit of zero size. This is not a trivial statement, in that sums of angles of triangles go to pi; ratio of circumference of circle to to radius goes to pi. Thus, such a region, blown up, is still much closer to Euclidean, than a larger section of the orginal sphere.

Similarly, locally lorentz is making an exact limiting statement about GR geometry.

 

[TrickyDicky]

The infinities remain after renormalization. QED may even break down before Lorentz violation is detected.

Please explain, your telegraphic style may lead to misunderstandings.

QFT in curved spacetime is an approximation, because it treats the gravitational field classically, while the other fields are treated quantum mechanically. QFT in flat spacetime is also an approximation. We have no theory that is not an approximation.

Of course in a trivial sense all science is just an approximaton, but there are right approximations and wrong ones. If you are alluding to the fact that we don't have a Quantum gravity theory, I'm just suggesting that an obstacle towards that goal might be using equations in a Lorentz invariant form. That's what I'm trying to get across without much success, probably this is not the right forum to debate it.

 

[TrickyDicky]

Let's go back to the beginning. Yes, there is no global Lorentz invarianced in GR (invariants come from the metric, and the Minkowski metric is not globally valid).

But then, "locally" means limit at infinity?? Nope.

Locally is code word for "limit as region goes to zero size". A sphere is locally Euclidean, but no finite section is exactly Euclidean. It is mathematically precise to say that it is in the limit of zero size. This is not a trivial statement, in that sums of angles of triangles go to pi; ratio of circumference of circle to to radius goes to pi. Thus, such a region, blown up, is still much closer to Euclidean, than a larger section of the orginal sphere.

Similarly, locally lorentz is making an exact limiting statement about GR geometry.

I can perfectly agree with this,I have no problem wth GR being locally Lorentz invariant because I understand what that means in geometrical terms,now this is the "geometrical part", I find issue with the physical interpretation of that mathematical statement, let's look at the right hand side, the matter-energy source, an elementary particle is not "local" in the geometrical sense. That's all I'm saying. But hey if I'm wrong, please enlighten me.

 

[Dale]

That's what I'm saying. I'm centering on the form of those equations, and where they come from.

Then I don't understand why you care that a real lab will have some second or higher order tidal gravity effects. If you are just dealing with the math then the form of the equations can be exactly Lorentz invariant.

As far as where the QFT equations come from, you may be better off asking in the QM folder.

 

[atyy]

Please explain, your telegraphic style may lead to misunderstandings.

The fixed point of the renormalization flow in QED is an infrared fixed point. It means that we can do perturbation theory at low energies, which accounts for the empirical success of QED. However, the theory diverges as we go to higher energies. Look up eg. the Landau pole. So even without empirical evidence, we expect the theory to fail at sufficiently high energies.

QCD's renormalization flow is a UV fixed point, meaning that the theory is consistent at arbitrarily high energies. In this case, it is expected to fail not on mathematical grounds, but on empirical grounds, such as not containing gravity.

Of course in a trivial sense all science is just an approximaton, but there are right approximations and wrong ones. If you are alluding to the fact that we don't have a Quantum gravity theory, I'm just suggesting that an obstacle towards that goal might be using equations in a Lorentz invariant form. That's what I'm trying to get across without much success, probably this is not the right forum to debate it.

No, you are not having much success because no one debates it (ie. what you are saying is not controversial). However, there seems to be a coherent theory of quantum gravity that is exactly globally Lorentz invariant at the boundary. The boundary theory completely specifies the theory in the bulk, where one only has local Lorentz invariance. Look up AdS/CFT or gauge/gravity duality. This theory seems not to get the correct cosmology, or the matter content. There are attempts to formulate QG as a Lorentz violating theory, such as Horava-Lifgarbagez gravity.

The major results in QG are:
1) The spacetime metric field treated as a quantum field is not perturbatively renormalizable.
2) The spacetime metric field cannot arise from more fundamental degrees of freedom that are also quantum fields in flat 4D Minkowski spacetime (Weinberg-Witten theorem).

So the possibilities people are trying are:
1) The spacetime metric field is renormalizable, if one looks non perturbatively (asymptotic safety). If this fails, then we must conclude that the spacetime metric field arises from more fundamental degrees of freedom that are either
2) not 4D quantum fields (string theory)
3) or Lorentz violating (Horava Lifgarbagez)
4) you can think of more possibilities yourself ;)

 

[TrickyDicky]

Thanks everyone.
Communiction is hard but I think I can get something out of all this, hopefully someone else can too.

 

[TrickyDicky]

So the possibilities people are trying are:
1) The spacetime metric field is renormalizable, if one looks non perturbatively (asymptotic safety). If this fails, then we must conclude that the spacetime metric field arises from more fundamental degrees of freedom that are either
2) not 4D quantum fields (string theory)
3) or Lorentz violating (Horava Lifgarbagez)
4) you can think of more possibilities yourself ;)

There seems to be more possibilities, this is from the wikipedia article about Lorentz violating theories:
Quote
"Lorentz violation refers to theories which are approximately relativistic when it comes to experiments that have actually been performed (and there are quite a number of such experimental tests) but yet contain tiny or hidden Lorentz violating corrections."End quote

The second model of such theories that describes is close to what I was suggesting here:

Quote
"Similar to the approximate Lorentz symmetry of phonons in a lattice (where the speed of sound plays the role of the critical speed), the Lorentz symmetry of special relativity (with the speed of light as the critical speed in vacuum) is only a low-energy limit of the laws of Physics, which involve new phenomena at some fundamental scale. Bare conventional "elementary" particles are not point-like field-theoretical objects at very small distance scales, and a nonzero fundamental length must be taken into account. Lorentz symmetry violation is governed by an energy-dependent parameter which tends to zero as momentum decreases. Such patterns require the existence of a privileged local inertial frame (the "vacuum rest frame"). They can be tested, at least partially, by ultra-high energy cosmic ray experiments like the Pierre Auger Observatory."

"Models belonging to this class can be consistent with experiment if Lorentz breaking happens at Planck scale or beyond it, and if Lorentz symmetry violation is governed by a suitable energy-dependent parameter. One then has a class of models which deviate from Poincaré symmetry near the Planck scale but still flows towards an exact Poincaré group at very large length scales."
End quote

does anyone know of any references of people working on this approach?
Or experiments related in the Pierre Auger Observatory?

 

[atyy]

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

Non-relativistic AdS/CFT: http://arxiv.org/abs/0804.4053