A Is gravity simply an interaction in the gauge theory of gravity?

[Dale]

This means that the two theories cannot be equivalent.

Well, there is a large class of physical solutions in common, but there are some solutions in one that are not in the other. The class of common solutions includes all local solutions and some global ones.

 

[vanhees71]

I don't understand, why you think the resulting theory of gravity from the gauge principle is different from GR (it's even more general, because it includes also torsion in the case of matter with spin). You get the very same equations as in GR, only the derivation is different, and the spacetime cannot be Minkowski space, because the curvature is not 0. For me it's obvious that this is the same physical theory as (extended) GR.

It's also explicitly mentioned in Sect. 6 of Kibbles paper:

T. W. B. Kibble, Lorentz Invariance and the Gravitational
Field, Jour. Math. Phys. 2, 212 (1960),
https://doi.org/10.1063/1.1703702

 

[PeterDonis]

I don't understand, why you think the resulting theory of gravity from the gauge principle is different from GR

A very specific concern has been raised in this thread, which you have not addressed. We all agree that the field equation of the theory of gravity derived from the gauge principle is the Einstein Field Equation. That's not an issue. The issue is that the underlying manifold (not the metric but the underlying manifold) that is permitted by the gauge theory construction, as far as we can tell, must be $R^4$, whereas GR considered as a geometric theory of gravity without any gauge theory justification admits other underlying manifolds.

So far your only response to this concern has been "the gauge theory construction allows other underlying manifolds, but it's too complicated to explain how here, look at these papers". As I have already posted, I don't see anything in the papers you referenced that justifies the claim that other underlying manifolds than $R^4$ are permitted.

 

[Dale]

I don't understand, why you think the resulting theory of gravity from the gauge principle is different from GR (it's even more general, because it includes also torsion in the case of matter with spin).

In GR you start with a pseudo-Riemannian manifold which is a topological manifold equipped with a metric (with a -+++ signature). That topological manifold is arbitrary and may have highly nontrivial global topology. However, locally, in a neighborhood around any event in the manifold you have that simple topology.

If you start with an underlying Lorentzian manifold and construct gauge fields on top of that Lorentzian manifold then your global topology is inherited from the underlying manifold. You can only have the trivial global topology by construction.

You get the very same equations as in GR, only the derivation is different

You get the same equations as in GR (neglecting matter with spin), but those equations are local equations. They don’t inform the global topology. That global topology must be constructed in some other manner.

In GR you can use any topological manifold. But in the gauge approach the topology is necessarily trivial by construction.

Frankly, I don’t think this is a particular weakness. I don’t think that we have any evidence that the physical spacetime of the universe has non-trivial topology.

 

[PeterDonis]

In GR you start with a pseudo-Riemannian manifold

I would say that GR, considered simply as a geometric theory of gravity, can be even more general than this: you can start with a solution on a local patch and ask what its maximal analytic extension is, and then see what topological manifold that maximal analytic extension has. For example, you can start by specifying a vacuum, spherically symmetric solution on a local patch, and ask what the maximal analytic extension is, and find out that it's the global Kruskal solution with topology $R^2 \times S^2$.

The issue with the gauge theory version, as you note, is that your model already starts with an assumption about the global topology. In fact, I think the assumption is even more restrictive than that; see below.

I don’t think that we have any evidence that the physical spacetime of the universe has non-trivial topology.

But we do have evidence that the physical spacetime of the universe is not conformally the same as Minkowski spacetime (since any FRW spacetime other than the edge case of the empty Milne universe is not); it doesn't have the same structure at infinity. I think having the same structure at infinity as Minkowski spacetime is required for the gauge theory construction to make sense.

Also, if "non-trivial topology" means "anything other than $R^4$", then I don't think we can say for sure that we have no evidence of non-trivial topology for our actual universe. The topology of de Sitter spacetime, which is what our universe, at least in our best current model, appears to be approaching asymptotically, is $R \times S^3$, not $R^4$.

 

[Dale]

I would say that GR, considered simply as a geometric theory of gravity, can be even more general than this: you can start with a solution on a local patch and ask what its maximal analytic extension is, and then see what topological manifold that maximal analytic extension has.

Sure, as far as finding solutions that is fine. But mathematically the primitive piece is the pseudo-Lorentzian manifold. That is the mathematical foundation on which everything else is built. Of course you can infer the foundation from measurements or start with the foundation. But from a mathematical construction, the manifold is the basis.

we do have evidence that the physical spacetime of the universe is not conformally the same as Minkowski spacetime (since any FRW spacetime other than the edge case of the empty Milne universe is not); it doesn't have the same structure at infinity.

I could easily be wrong, but I think that conformal changes are not topological changes. I.e. the fact that they are not conformally the same does not imply that they are topologically different.

Of course, curvature singularities make topological defects, but we expect GR to break down there, so I wouldn’t count the existence of singularities in GR as evidence of non-trivial physical topology, even where GR is known to accurately describe the physical spacetime far from the singularity.

 

[vanhees71]

A very specific concern has been raised in this thread, which you have not addressed. We all agree that the field equation of the theory of gravity derived from the gauge principle is the Einstein Field Equation. That's not an issue. The issue is that the underlying manifold (not the metric but the underlying manifold) that is permitted by the gauge theory construction, as far as we can tell, must be $R^4$, whereas GR considered as a geometric theory of gravity without any gauge theory justification admits other underlying manifolds.

So far your only response to this concern has been "the gauge theory construction allows other underlying manifolds, but it's too complicated to explain how here, look at these papers". As I have already posted, I don't see anything in the papers you referenced that justifies the claim that other underlying manifolds than $R^4$ are permitted.

I don't know, how I should explain this in other words than I've tried to do it again. One last try:

The only difference is the derivation of Einstein's theory of the gravitational interaction. The resulting theory is GR. The point is that you derive the possibility of reinterpreting the gravitational interaction as a (pseudo-)Riemannian spacetime manifold. In the standard textbook approach you simply start from this assumption arguing with the "equivalence principle" as Einstein historically did. The resulting description of the gravitational interaction is the same in both approaches. In the standard approach, of course you can go the other way around and show that GR in fact IS a gauge theory with Poincare symmetry made local. The only thing you have to do is to introduce tetrad fields and express everything in terms of them.

Also, I don't understand, why I should write up the entire theory in forum postings after having given 2 (by the way pretty famous) papers and a reference to a also highly established textbook (Ramon, Quantum Field Theory, a modern primer, 2nd edition).

 

[PeterDonis]

conformal changes are not topological changes

Agreed.

 

[PeterDonis]

I don't know, how I should explain this in other words than I've tried to do it again.

You can either explain how the specific issue I raised is addressed, which you haven't (you just keep repeating that the field equation that gets derived is the same--not in so many words, but that's what your statements amount to--which, as I've already said, is not the issue, we all agree that the field equation that gets derived is the same), or point out where specifically in the papers you referenced the specific issue I raised is addressed, since, as I've already said, I can't find anything in those papers that addresses that specific issue. If you are unable to do that, then I guess this discussion is done.

I don't understand, why I should write up the entire theory in forum postings after having given 2 (by the way pretty famous) papers

Which, as I have already said, do not appear to address the specific issue I raised.

 

[Dale]

The only difference is the derivation of Einstein's theory of the gravitational interaction. The resulting theory is GR.

Locally that is true, but the question is: how can the gauge approach lead to a non-trivial topology.

For instance, in GR you could have a flat vacuum solution which differs from Minkowski spacetime because of topology. For example a “hole” or a “torus” topology. As far as I understand, the gauge approach would not allow such solutions. If it is flat vacuum then it is Minkowski topologically.

Do you understand the flat “hole” or “torus” as being valid solutions for the gauge approach. If so, how are they constructed given the starting point for the approach which assumes an underlying background Minkowski spacetime.

 

[vanhees71]

Obviously I don't understand the issue. If I reinterpret the gauge fields in a geometrical way as said in Kibbles paper (Sect. 6) then I've an Einstein-Cartan manifold as spacetime (at least as far its local properties are concerned) and not a global affine Minkowski space. BTW, you can find all this in the more elaborate review by Hehl et al:

F. Hehl, P. Von Der Heyde, G. Kerlick and J. Nester, General
Relativity with Spin and Torsion: Foundations and Prospects,
Rev. Mod. Phys. 48, 393 (1976),
https://doi.org/10.1103/RevModPhys.48.393

There it's explicitly stated that the final spacetime model is an Einstein-Cartan manifold and not an affine Minkowski space.

If it's about the global topology of the universe, I don't think that this can be uniquely stated from GR or Einstein-Cartan theory. I'm not even sure whether it can be inferred from empirical evidence, because all we have are pretty local observations in our neighborhood. The current spacetime model of the large-scale structure is a flat FLRW solution of GR, but also this is inferred from the cosmological data (CMBR fluctuations, redshift-distance relation of standard candles) via the cosmological model based on the cosmological principle (i.e., the ansatz that the large-scale coarse-grained spacetime is an FLRW solution). So I don't think that the status of this question is much different between standard GR and the Einstein-Cartan theory following from the gauge approach.

Further FAPP (i.e., for the astronomical observables we have to check our model of gravity) the gauge-theoretical Einstein-Cartan theory and standard GR are indistinguishable since it's anyway only different in matter, and there the spin contribution is negligible (and thus the torsion is negligible too).

Overall I don't understand the hostility against the gauge approach to gravity. I think it's at least as convincing as the standard textbook approach to GR via Einstein's original argument via the various forms of the equivalence principle(s). It's clear that he had no idea about spin in 1907-1915. Nevertheless interestingly Cartan had the idea with torsion already in 1922, long before the notion of spin was established (1926 non-relativisticall, 1928 relativistically).

 

[martinbn]

@vanhees71 My guess is that, when you think of GR, you are focused on the metric and nothing else. The metric is a tensor on some manifold, which is important but often not explicitly mentioned. Let me ask you these questions. I assume that the answer is "yes", but I'll ask them just to try to bring your attention to the issue you seem to miss. Do you know what a manifold is? Do you understand that there are four dimensional manifolds that are different from $\mathbb R^4$? Just to be clear neither of the questions is concerned with a metric, it just about the manifolds. Now the issue is that a theory that uses only $\mathbb R^4$ cannot be the same as a theory that uses any manifold, including different from $\mathbb R^4$.

 

[martinbn]

By the way I don't think the question of what the acctual space-time is, is relevent. A gravity theory should be able to handle all kinds of situations. For example if you want to model an isolated system the space-time will be assymptotically flat and vacuum near infinity. One cannot say "Ah, but the universe we live in is homogenous." It is true but irrelevent.

 

[vanhees71]

@vanhees71 My guess is that, when you think of GR, you are focused on the metric and nothing else. The metric is a tensor on some manifold, which is important but often not explicitly mentioned. Let me ask you these questions. I assume that the answer is "yes", but I'll ask them just to try to bring your attention to the issue you seem to miss. Do you know what a manifold is? Do you understand that there are four dimensional manifolds that are different from $\mathbb R^4$? Just to be clear neither of the questions is concerned with a metric, it just about the manifolds. Now the issue is that a theory that uses only $\mathbb R^4$ cannot be the same as a theory that uses any manifold, including different from $\mathbb R^4$.

Yes, of course. In GR you start with a pseudo-Riemannian manifold, i.e., a differentiable manifold with a fundamental form of signature (1,3) (or (3,1) depending on your preferred sign convention) with an affine connection uniquely determined by the demand that it must be metric compatible and torsion free.

The gauge theory of gravitation does NOT only use $\mathbb{R}^4$ but at the end you end up with a general Einstein-Cartan manifold, which is a more general concept than the pseudo-Riemannian spacetime manifold of GR. It's a manifold with a fundamental form of signature (1,3) (or (3,1)) and a metric-compatible connection, which however is not necessarily torsion free. The torsion is related to spin and obeys a separate equation of motion.

Which manifold you concretely get depends in both GR and the gauge theory of gravity on the "matter content" and the concrete physical situation you consider.

 

[vanhees71]

By the way I don't think the question of what the acctual space-time is, is relevent. A gravity theory should be able to handle all kinds of situations. For example if you want to model an isolated system the space-time will be assymptotically flat and vacuum near infinity. One cannot say "Ah, but the universe we live in is homogenous." It is true but irrelevent.

I fully agree with that too, and the theory following from the gauge approach also doesn't say anything different.

 

[Dale]

If it's about the global topology of the universe, I don't think that this can be uniquely stated from GR or Einstein-Cartan theory.

Yes, it is about the global topology. I know how you could get a torus or hole topology in GR. How could you do that in the Einstein Cartan theory?

Overall I don't understand the hostility against the gauge approach to gravity.

I, for one, have no hostility towards it. In fact, I would be very interested in some experimental results (in matter of course) that could distinguish them. I think there is a good chance that the eventual classical limit of a correct quantum gravity theory will be the Einstein Cartan theory rather than GR.

However, I don’t see any way to get a hole or torus topology in the gauge theory. I don’t think that is a problem because we don’t have any experimental confirmation of non-trivial topologies. It is merely a difference with GR. It is not the most important difference, but it is a legitimate difference as far as I can tell.

 

[martinbn]

Yes, of course. In GR you start with a pseudo-Riemannian manifold, i.e., a differentiable manifold with a fundamental form of signature (1,3) (or (3,1) depending on your preferred sign convention) with an affine connection uniquely determined by the demand that it must be metric compatible and torsion free.

The gauge theory of gravitation does NOT only use $\mathbb{R}^4$ but at the end you end up with a general Einstein-Cartan manifold, which is a more general concept than the pseudo-Riemannian spacetime manifold of GR. It's a manifold with a fundamental form of signature (1,3) (or (3,1)) and a metric-compatible connection, which however is not necessarily torsion free. The torsion is related to spin and obeys a separate equation of motion.

Which manifold you concretely get depends in both GR and the gauge theory of gravity on the "matter content" and the concrete physical situation you consider.

Ok, but then what is the formulation of the theory? The references that you gave seem to consider only $\mathbb{R}^4$.

 

[martinbn]

Yes, it is about the global topology. I know how you could get a torus or hole topology in GR. How could you do that in the Einstein Cartan theory?

For Einstein Cartan it is the same as GR.

 

[Dale]

For Einstein Cartan it is the same as GR.

How? That is what I don’t get. The topology is inherited from the underlying flat manifold.

 

[martinbn]

How? That is what I don’t get. The topology is inherited from the underlying flat manifold.

Not for Einstein Cartan. It is just like GR. You have a four dimenssional manifold with a metric and a connection which is metric but can have torsion. The field equations are the same as in GR plus an equation that relates the torsion and the spin tensor.

 

[vanhees71]

Yes, it is about the global topology. I know how you could get a torus or hole topology in GR. How could you do that in the Einstein Cartan theory?

I, for one, have no hostility towards it. In fact, I would be very interested in some experimental results (in matter of course) that could distinguish them. I think there is a good chance that the eventual classical limit of a correct quantum gravity theory will be the Einstein Cartan theory rather than GR.

However, I don’t see any way to get a hole or torus topology in the gauge theory. I don’t think that is a problem because we don’t have any experimental confirmation of non-trivial topologies. It is merely a difference with GR. It is not the most important difference, but it is a legitimate difference as far as I can tell.

I must admit that I don't know enough about the global topology in GR nor in Einstein Cartan theory. Do you have a reference concerning the torus or hole in GR?

From an experimental point of view, of course, almost everything concerning GR is in the astronomical context, i.e., dealing with macroscopic matter, partially "under extreme conditions" and all observations are "local", and that's why for sure it is hard to imagine that we'll be ever able to empirically learn about the global topology of the universe. Of course, the cosmological "concordance model", i.e., the $\Lambda \text{CDM}$ model is pretty convincing and amazingly successful in the recent years particularly with the more and more precise measurements of the CMBR fluctuations, including recently again the polarization and the redshift-distance relationship/"Hubble Law" via the establishment of "standard candles" (like Supernovae of Type Ia). A caveat is that more recently there are again some inconsistencies between different determinations of the Hubble constant and its evolution. Other tests of GR vs. alternative theories made also a lot of progress, e.g., pulsar timing and the analysis of black-hole mergers and, even more interesting, neutron-star mergers ("kilonovae"), etc.

I think it's very difficult to check for torsion as predicted by the gauge approach leading to an Einstein-Cartan manifold in relation to spin since according to this theory it's only observable in the medium, and the macroscopic bodies we can test the theories of gravity are "spin saturated" many-body systems like neutron stars. FAPP they are well described on the classical level with hydrodynamics (or magneto hydrodynamics as the just yesterday published paper by my colleagues in Frankfurt on the jet of the black hole in M87 in Nature Astronomy [1]), and there you expect the usual torsion free connection. According to the theory, torsion is also not observable outside of the matter, i.e., it cannot be observed by the motion of objects under influence of their mutual gravitational interaction, i.e., a high-precision method like pulsar timing is not expected to observe torsion.