# GR as a gauge theory

## Main Question or Discussion Point

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), wich 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?

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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 xμ → x'μ = xμ + ξμ(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.

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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 xμ → x'μ = xμ + ξμ, 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.

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
2019 Award
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.

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.

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.

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
2019 Award
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?

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.

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
2019 Award
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
2019 Award
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.

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
2019 Award
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.

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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
2019 Award
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.

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.

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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.

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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