Is GR Considered a Gauge Theory?

[TrickyDicky]

I understand that writing the E-H action in terms of tetrads makes evident GR is a gauge theory. IOW general covariance/diffeomorphism invariance in GR is a form of gauge invariance.
However unlike other gauge theories(for instance EM dependence on Minkowski spacetime), this gauge invariance in GR is accompanied by background independence(no prior global geometry/topology), which makes us see GR's manifold abstractly as a simple differentiable manifold, that only has a casual geometric appearance when for practical requirements of the problem at hand like exploiting its symmetries we make a gauge choice and use some preferred coordinates(i.e. FRW coordinates in cosmology or Schwarzschild coordinates for isolated sources) but if GR is a gauge theory those gauge choices are not physical.
Assuming this is not a wrong depiction of GR's general covariance, background independence and gauge invariance, a questions occurs to me:

If GR is background independent, solutions of the EFE are valid locally like are all the observables derived from this local curvature geometry, but we shouldn't be able to infer any global geometry/topology from them, so I don't see any justification for "maximally extended solutions" like say, Kruskal-Szekeres solution. Being rigorous it seems there is no physical grounds to use it if we take seriously gauge invariance. How is this usually dealt with?

 

[Bill_K]

I understand that writing the E-H action in terms of tetrads makes evident GR is a gauge theory. IOW general covariance/diffeomorphism invariance in GR is a form of gauge invariance.

Be clear that there are two quite different gauge groups you're talking about. The tetrad gauge group involves a Lorentz transformation at each spacetime point. Whereas the general covariance group is a change of coordinates. An infinitesimal coordinate transformation is x^μ → x'^μ = x^μ + ξ^μ(x), i.e. an infinitesimal translation at each point.

I don't see any justification for "maximally extended solutions" like say, Kruskal-Szekeres solution.

Einstein's equations are hyperbolic, which means their solutions are not unique unless you specify Cauchy initial conditions on an entire hypersurface. Kruskal's solution and others like it are maximal analytic extensions, which exclude the presence of "white hole" information.

 

[TrickyDicky]

Be clear that there are two quite different gauge groups you're talking about. The tetrad gauge group involves a Lorentz transformation at each spacetime point. Whereas the general covariance group is a change of coordinates. An infinitesimal coordinate transformation is x^μ → x'^μ = x^μ + ξ^μ, i.e. an infinitesimal translation.

It is true that locally GR has Lorentz invariance(that's basically the equivalence principle) but not globally. So even though tetrads happen to have Lorentz invariance it is not that gauge group I'm referring to, it is rather the diffeomorphism group.
Expressing the Einstein-Hilbert action in terms of local frames however implies IMO that GR has gauge invariance, this would be possible only by virtue of GR's background independence .

 

[Bill_K]

It is true that locally GR has Lorentz invariance(that's basically the equivalence principle) but not globally. So even though tetrads happen to have Lorentz invariance it is not that gauge group I'm referring to, it is rather the diffeomorphism group.

The tetrad group I'm referring to is the (position-dependent) transformation from one tetrad to another. This is not an automatic feature of general relativity, it's an add-on that you only get if and when you express things in terms of tetrads.

Expressing the Einstein-Hilbert action in terms of local frames however implies IMO that GR has gauge invariance, this would be possible only by virtue of GR's background independence.

It does indeed have gauge invariance, namely general coordinate invariance, even without the frames.

 

[TrickyDicky]

Einstein's equations are hyperbolic, which means their solutions are not unique unless you specify Cauchy initial conditions on an entire hypersurface. Kruskal's solution and others like it are maximal analytic extensions, which exclude the presence of "white hole" information.

All this is fine but I don't see the connection with my question.

 

[Bill_K]

solutions of the EFE are valid locally like are all the observables derived from this local curvature geometry, but we shouldn't be able to infer any global geometry/topology from them, so I don't see any justification for "maximally extended solutions" like say, Kruskal-Szekeres solution.

The justification, like I said, is that Kruskal is singled out as an analytic extension. The global geometry/topology is not unique, unless you add the assumption of analyticity.

 

[WannabeNewton]

See chapter 10 of Wald and/or Chapter 7 of Hawking and Ellis for a discussion of what Bill has already said regarding the Cauchy initial conditions and the global topology (the chapter in Wald is quite heavy in math, more so than any other chapter I would say; Hawking and Ellis is heavy on math throughout so that goes without saying).

 

[PAllen]

I'll make an analogy with a much simpler case.

Consider Riemannian (regular, not pseudo) 2 manifolds, and ask which is a solution of the differential equation:

R=k, k constant > 0

Typically, one answers "2-sphere". But really this is only unique if you implicitly specify maximal solution (such that it is not a subset of any other solution). Otherwise, any connected open subset of a 2-sphere satisfies the equation at every point.

Specifically, one might solve using some funny coordinates and end up with a coordinate description of a 2-hemisphere with equator and pole removed. This satisfies the equation everywhere. Then you might follow geodesics and find that you can complete and join geodesics by adding a point at the pole; and you can complete geodesics the other way by mirroring; then find there are no more extensions you can make consistent with R=k.

This is quite analogous to the relation between SC exterior solution and K-S solution: the extension past the equator is like adding the WH component; filling the pole is like extending geodesics through the horizon.

Why prefer maximal solutions? Mathematically, it leads to unique solutions much more often. Also, there is little point in talking about a solution that is a subset of another as a distinct solution. Physically, you may find this criterion not compelling.

IMO, the compelling argument for maximal solutions in GR as a physical theory, is a physical argument: modern formulations of the equivalence principle that include the statement "local physics is everywhere consistent with SR, and everwhere and everywhen, follows the same laws". If this is the case, and you have a timelike geodesic that ends, when it could be extended, you have a violation of this principle. Two identical clocks, identical physical situation set up. For one, a local hour after 1 pm comes 2 pm. For the other, 2 pm never arrives for no reason that can be locally specified.

 

[TrickyDicky]

The justification, like I said, is that Kruskal is singled out as an analytic extension. The global geometry/topology is not unique, unless you add the assumption of analyticity.

Ok, I see what you mean, but this seems reinforce my point, it is not a warranted assumption for a differentiable real manifold, it is a rather strict constraint.
I guess what I wanted to highlight when I brought up tetrads was the idea of locality by their association to normal coordinates, but in a (no prior geometry) differentiable manifold it would be more correct to refer to something like harmonic coordinates, and precisely using these coordinates is one of the requirements for the EFE to count as a well posed initial value formulation.

 

[TrickyDicky]

IMO, the compelling argument for maximal solutions in GR as a physical theory, is a physical argument: modern formulations of the equivalence principle that include the statement "local physics is everywhere consistent with SR, and everwhere and everywhen, follows the same laws". If this is the case, and you have a timelike geodesic that ends, when it could be extended, you have a violation of this principle. Two identical clocks, identical physical situation set up. For one, a local hour after 1 pm comes 2 pm. For the other, 2 pm never arrives for no reason that can be locally specified.

The problem with this argument is that if you really follow the statement "local physics is everywhere consistent with SR, and everwhere and everywhen, follows the same laws" strictly, you cannot have a timelike geodesic that ends in the first place, it will locally continue as long as the statement is true.

 

[TrickyDicky]

See chapter 10 of Wald and/or Chapter 7 of Hawking and Ellis for a discussion of what Bill has already said regarding the Cauchy initial conditions and the global topology (the chapter in Wald is quite heavy in math, more so than any other chapter I would say; Hawking and Ellis is heavy on math throughout so that goes without saying).

Thanks a lot for those references, very well suited.

 

[PAllen]

The problem with this argument is that if you really follow the statement "local physics is everywhere consistent with SR, and everwhere and everywhen, follows the same laws" strictly, you cannot have a timelike geodesic that ends in the first place, it will locally continue as long as the statement is true.

Not true. That is exactly what you have for an infall geodesic in exterior SC solution. If you set up local normal coordinates for the infaller near the horizon, its geodesic ends for no local physical reason. That was the one of the very first clues about what was going on, by Roberson of Robertson-Walker fame, in the 1930s. Other people around this time got hints from coordinate transforms (e.g. Lemaitre), but Robertson is credited with first noticing what happens if you set up a local frame for the infaller near the horizon. Specifically, the transform from SC to a local free fall frame maps SC t=∞ to t'=<finite value>. The frame is then chopped at finite t' for no local physical reason.

 

[WannabeNewton]

Specifically, the transform from SC to a local free fall frame maps SC t=∞ to t'=<finite value>. The frame is then chopped at finite t' for no local physical reason.

This is very cool! Know of any text or what have you where I could see the calculations?

 

[Passionflower]

Not true. That is exactly what you have for an infall geodesic in exterior SC solution. If you set up local normal coordinates for the infaller near the horizon, its geodesic ends for no local physical reason. That was the one of the very first clues about what was going on, by Roberson of Robertson-Walker fame, in the 1930s. Other people around this time got hints from coordinate transforms (e.g. Lemaitre), but Robertson is credited with first noticing what happens if you set up a local frame for the infaller near the horizon. Specifically, the transform from SC to a local free fall frame maps SC t=∞ to t'=<finite value>. The frame is then chopped at finite t' for no local physical reason.

Not sure what your point is.
Worldlines end at the singularity not at the event horizon.

 

[TrickyDicky]

Not true. That is exactly what you have for an infall geodesic in exterior SC solution. If you set up local normal coordinates for the infaller near the horizon, its geodesic ends for no local physical reason. That was the one of the very first clues about what was going on, by Roberson of Robertson-Walker fame, in the 1930s. Other people around this time got hints from coordinate transforms (e.g. Lemaitre), but Robertson is credited with first noticing what happens if you set up a local frame for the infaller near the horizon.

I'm not sure I follow you, I just applied logic to your argument. You argue from the point of view that the analytical extension is alrady there. I have no problem with that, this thread considers the requirements for the analytical extension to be made and their physical justification.
It's good you remark "no local physical reason", the reason is mathematical of course and applies once the analytical extension is performed. But your argument was supposedly physical.

 

[PAllen]

Not sure what your point is.
Worldlines end at the singularity not at the event horizon.

The singularity is where they can't be extended, and 'really' end. However, if you consider the manifold defined by SC exterior coordinates, the horizon is an (excluded) boundary of the manifold. The manifold and geodesics can then be continued through the horizon consistent with the requirement of vacuum (vanishing Einstein tensor) and spherical symmetry.

 

[PAllen]

I'm not sure I follow you, I just applied logic to your argument. You argue from the point of view that the analytical extension is alrady there. I have no problem with that, this thread considers the requirements for the analytical extension to be made and their physical justification.
It's good you remark "no local physical reason", the reason is mathematical of course and applies once the analytical extension is performed. But your argument was supposedly physical.

I don't follow how you don't follow. I do not assume the extension is already there. My argument is purely physical. Unfortunately, I don't see any way to explain it better or differently. Perhaps someone else can.

 

[TrickyDicky]

A far from suspect routinely used text on GR, d'Inverno's says this:
"The intriguing question of whether or not the mathematical procedure for extending [the SC solution...] has any physical significance is still an open one. Although Einstein's equations fix the local geometry of spacetime, they do not fix its global geometry or its topology".
This is what basically triggers my OP.
Bill correctly confirmed that analiticity is the basic mathematical requirement for the extension, now I'm exploring what could physically justify such mathematical constraint on GR's manifold.

 

[PAllen]

A far from suspect routinely used text on GR, d'Inverno's says this:
"The intriguing question of whether or not the mathematical procedure for extending [the SC solution...] has any physical significance is still an open one. Although Einstein's equations fix the local geometry of spacetime, they do not fix its global geometry or its topology".
This is what basically triggers my OP.
Bill correctly confirmed that analiticity is the basic mathematical requirement for the extension, now I'm exploring what could physically justify such mathematical constraint on GR's manifold.

And I'm answering that, without extension, you could set up two clocks in identical local physical circumstances, and they behave differently. The extension then removes the difference and restores the EEP. This argument does not apply near the singularity, because the singularity is a local feature than cannot be replicated elsewhere in the solution. What d'Inverno would think of this argument, I do not know. I'm giving my reason for thinking maximal extension has a physical motivation in GR, and that motivation is the EEP.

 

[TrickyDicky]

And I'm answering that, without extension, you could set up two clocks in identical local physical circumstances, and they behave differently. The extension then removes the difference and restores the EEP. This argument does not apply near the singularity, because the singularity is a local feature than cannot be replicated elsewhere in the solution. What d'Inverno would think of this argument, I do not know. I'm giving my reason for thinking maximal extension has a physical motivation in GR, and that motivation is the EEP.

That difference would exist with or without extensión, in both cases there is no geodesic completeness.

 

[PAllen]

That difference would exist with or without extensión, in both cases there is no geodesic completeness.

Geodesic incompleteness due to a singularity has a local physical cause - infinite physical quantities. This does not violate EEP because there is no nonsingular local region that is physically equivalent (alternatively, you can say there is no problem for the EEP because all singularities involve geodesic incompleteness, so all similar regions follow the same laws). Geodesic incompleteness that is removable by maximal extension has no local physical cause. You have physical history that is locally the same as any other non-singular region - up to the incompleteness; yet the (removable) geodesic incompleteness gives a different history than equivalent regions elsewhere. To me, this violates EEP. Performing the extension removes the violation.

 

[TrickyDicky]

PAllen, when you say: "Geodesic incompleteness that is removable by maximal extensión... " you are already making a physical assumption, namely that is removable. I've tried to explain that the issue in this thread is set in a previous step to making that assumption. For instance what could be a physical reason for a differentiable real manifold to be analytic.

 

[TrickyDicky]

By the way singularities are not usually considered physical, just like Infinite physical quantities like density. And they are equally unphysical in the Extended case as in a putative non-extendable case.

 

[TrickyDicky]

Let's put this another way since I only used the SC case as an example.
If GR is to be considered a gauge theory, and so far nobody has questioned it, any gauge dependent preferred coordinate choice shouldn't have any physical content over any other gauge choice.
In that sense only local observables that can be described by local coordinates such as harmonic or normal coordinates would be physical observables (Dirac observables defined as those that are invariant under gauge transformations of the theory) .

 

[PAllen]

PAllen, when you say: "Geodesic incompleteness that is removable by maximal extensión... " you are already making a physical assumption, namely that is removable. I've tried to explain that the issue in this thread is set in a previous step to making that assumption. For instance what could be a physical reason for a differentiable real manifold to be analytic.

The ability to extend is not a physical assumption, it is a mathematical fact within the mathematical formulation of the theory. The question (for physicists) is physical significance of the extension. I have given a physical argument about why not extending is physically implausible compared to extending, in the context of comparing the mathematical model to known classical physical behavior. It is a plausibility argument, not a proof. Take it or leave it, I am not going to say more on this because it nothing new is being said.

Note, that quantum behavior may lead to completely different conclusions about a certain case. For example, classical EM + local Lorentz invariance, insists that if an atom is a classical object it must radiate and decay. QM changes this, in part, by adding to the description of a physical system the number of available quantum states. Thus, though governed by the same fundamental law, this new law explains the in-equivalence between charged metal spheres and electrons in an atom. My argument applies only to GR as a classical theory of fields.

 

[PAllen]

Let's put this another way since I only used the SC case as an example.
If GR is to be considered a gauge theory, and so far nobody has questioned it, any gauge dependent preferred coordinate choice shouldn't have any physical content over any other gauge choice.
In that sense only local observables that can be described by local coordinates such as harmonic or normal coordinates would be physical observables (Dirac observables defined as those that are invariant under gauge transformations of the theory) .

In GR, since it is routine not to be able to cover a manifold with a single coordinate chart, you have to allow for a gauge to be a collection of coordinate patches.

 

[TrickyDicky]

In GR, since it is routine not to be able to cover a manifold with a single coordinate chart, you have to allow for a gauge to be a collection of coordinate patches.

It is not a routine, it's simply impossible.
I've never seen a gauge defined as a collection of coordinate patches. You probably mean that by choosing preferred coordinates for whatever practical reasons that suit a specific problem one makes a gauge choice, which is true, I'm stressing this. So what is your point?

 

[TrickyDicky]

The ability to extend is not a physical assumption, it is a mathematical fact

Exactly, as long as you make the assumption of analyticity

within the mathematical formulation of the theory.

The assumption of analyticity is not included in the formulation of GR, as I've said several times GR as a gauge theory only assumes a differentiable manifold, and that is GR considered as a classical theory of fields.

The question (for physicists) is physical significance of the extension.

Right, and its physical justification as quoted in the d'Inverno textbook.

I have given a physical argument about why not extending is physically implausible compared to extending, in the context of comparing the mathematical model to known classical physical behavior. It is a plausibility argument, not a proof.

But it is not even well posed as I have made clear already.

My argument applies only to GR as a classical theory of fields.

Not really, see above.

 

[Bill_K]

I've never seen a gauge defined as a collection of coordinate patches.

Now you have.

In gravity, as in some other gauge theories, it may happen that a single gauge does not suffice to cover the entire space. This situation arises, for example, in electromagnetism, in which the vector potential for the Dirac monopole is singular along one axis, the so-called Dirac string. Quoting a typical source,

Our strategy is to split the sphere surrounding the magnetic monopole into two pieces along the equator. For the northern hemisphere we take the field configuration A and simply throw away the Dirac string running along the south pole. In the southern hemisphere we take instead the field configuration A. Neither A nor A are singular. We piece together these two distinct patches in order to cover the sphere, by making a gauge transformation between the two field configurations along the equator.

This situation is generally referred to as a gauge bundle. See its description in Wikipedia. Also here.

 

[TrickyDicky]

Now you have.

In gravity, as in some other gauge theories, it may happen that a single gauge does not suffice to cover the entire space. This situation arises, for example, in electromagnetism, in which the vector potential for the Dirac monopole is singular along one axis, the so-called Dirac string. Quoting a typical source,


This situation is generally referred to as a gauge bundle. See its description in Wikipedia. Also here.

Thanks for the info and links, it is basically what I figured.
Too bad it doesn't help me answer the question in the OP. Background independence, which is specific to gravity gauge theory, still makes necessary to justify physically the analyticity of the manifold in GR in order to make the analytical extension.
For instance I'm really skeptical that a singular spacetime could be analytic, but It'd be great to find some proof to confirm it or reject it.

 

[Bill_K]

For sure, we often lose sight of the fact that the Kruskal extension of Schwarzschild is representative, but not the only possible extension. It's said to represent a black hole which is "eternal", i.e. somehow formed from quantum gravity during the big bang.

[See this diagram for the notation: interior regions II and IV, exterior regions I and III] But for a more realistic black hole that formed from a collapsed star, regions III and IV would be absent, having been replaced by a nonvacuum solution. Furthermore, region IV would likely contain matter/radiation which came either from the star or from somewhere in region III. And so an observer falling into the hole would not necessarily run into the vacuum Kruskal extension he was expecting to find but something else.

 

[PAllen]

For sure, we often lose sight of the fact that the Kruskal extension of Schwarzschild is representative, but not the only possible extension. It's said to represent a black hole which is "eternal", i.e. somehow formed from quantum gravity during the big bang.

It is the unique maximal extension if you require vacuum (vanishing Einstein tensor) + spherical symmetry throughout the solution.

 

[TrickyDicky]

It is the unique maximal extension if you require vacuum (vanishing Einstein tensor) + spherical symmetry throughout the solution.

+analyticity, remember?

 

[Bill_K]

It is the unique maximal extension if you require vacuum (vanishing Einstein tensor) + spherical symmetry throughout the solution.

I'm just saying that these assumptions are fine mathematically, but there is no physical justification for making them. Regions III and IV are terra incognita, and could plausibly be the source of matter and/or gravitational waves. Especially IV, where we're assuming that an exploding singularity produces nothing but vacuum.

 

[PAllen]

+analyticity, remember?

No additional assumption is needed, for uniqueness in the sense I gave to follow.