Analogy of counterterms of gauge in GR

[bluecap]

In gauge theories, the counterterms in the equation to balance the gauge freedom (like the phase in electrodynamics) produce the forces of nature. In GR.. what is the equivalent of the counterterms and what is the equivalent of the phase or isospins freedom in electroweak)?

 

[vanhees71]

It is not clear to me what you mean with "counterterm" here. Usually you use this notion in renormalization of relativistic quantum field theory to subtract the divergences in higher-order contributions (Feynman diagrams with loops).

Perhaps you mean the covariant derivative, introduced to make a global symmetry local? Then the answer is that to get general relativity in terms of a gauge principle, you gauge the Lorentz symmetry, i.e., you make this a local symmetry. This means that at any point of the general spacetime you can introduce local coordinates such that for local (!) observables you have a Minkowski space, and the laws of physics are valid as in special relativity in the so defined local reference frame. This is the mathematical abstract formulation of the equivalence principle, and in this way you introduce tetrades and the connection. It's too long to explain this formalism in a forums posting. A very clear exposition of this (for me most natural approach to GR from the physicist's point of view, i.e., describing gravity as an interaction as all the other fundamental interactions with the specific property to obey the equivalence principle) can be found in

P. Ramond, Field Theory - A Modern Primer, 2nd edition, Addison Wesley 1989

 

[bluecap]

It is not clear to me what you mean with "counterterm" here. Usually you use this notion in renormalization of relativistic quantum field theory to subtract the divergences in higher-order contributions (Feynman diagrams with loops).

Perhaps you mean the covariant derivative, introduced to make a global symmetry local? Then the answer is that to get general relativity in terms of a gauge principle, you gauge the Lorentz symmetry, i.e., you make this a local symmetry. This means that at any point of the general spacetime you can introduce local coordinates such that for local (!) observables you have a Minkowski space, and the laws of physics are valid as in special relativity in the so defined local reference frame. This is the mathematical abstract formulation of the equivalence principle, and in this way you introduce tetrades and the connection. It's too long to explain this formalism in a forums posting. A very clear exposition of this (for me most natural approach to GR from the physicist's point of view, i.e., describing gravity as an interaction as all the other fundamental interactions with the specific property to obey the equivalence principle) can be found in

P. Ramond, Field Theory - A Modern Primer, 2nd edition, Addison Wesley 1989

in U(1), the covariant derivative for phase becomes the electromagnetic field
in Su(2), the 3 covariant derivative for isospin becomes the 3 weak forces bosons
in Su(3), the 8 covariant derivative for colors becomes the 8 strong color forces.
In General relativity.. the ____ (blank) covariant derivative for ____ (blank) becomes the gravitational force. Please supply the blanks. Ty.

 

[vanhees71]

Of course in general relativity the connection is the connection of the pseudo-Riemannian space-time manifold, and the 24 Christoffel symbols are the corresponding "components of the gravitational field". As I said, what's "gauged", i.e., made from a global to a local symmetry is the symmetry under Lorentz transformations (to be more specific: the proper orthochronous Lorentz transformations).

 

[haushofer]

If one gauges the Poincaré algebra, then one ends up with local translations, local Lorentz transformations and gct's. However, putting the torsion to zero (the field strength of local translations) enables one to rewrite the local translations on both gauge fields as a linear combination of LT's and GCT's. After that one can introduce covariant derivatives of both these transformations.

 

[vanhees71]

What are GCT's?

The space an have torsion if you have fields with spin involved. For spin 1/2 that's shown in the above cited book by Ramond.

 

[pervect]

What are GCT's?

The space an have torsion if you have fields with spin involved. For spin 1/2 that's shown in the above cited book by Ramond.

I'm unfortunately not following the bulk of the discussion (what's a GCT? :-) ), but I do think it's worth mentioning that in GR, the torsion is zero. This does lead to problems with spin 1/2 fields, but as GR is a purely classical theory, so it doesn't need to handle spin 1/2 fields. Of course, a theory of quantum gravity would have to, but we don't really have one yet.

While we don't have an agreed on theory of quantum gravity, Einstein-Cartan theory (which does have torsion) does have the ability to handle the spin 1/2 fields, and it does have torsion. But I'm concerned that some confusion might be generated by talking about Einstein-Cartan theory (or other theories that have torsion) as if they were GR - when GR doesn't have torsion.

It's not terribly clear to me what a "force" is in the context of purely classical GR, and I suspect that may be important to the OP's question at a fundamental level. What GR has is Christoffel symbols, i.e. a connection. Under the right circumstances, a subset of these Christoffel symbols act a lot like forces - but they're not really forces in the ordinary sense. For instance, it's fairly naturall to interpret $\Gamma^x{}_{tt}$ as a "force", but how should we interpret, say $\Gamma^t{}_{xt}$?

So I think that the whole idea of gravity as a "force" in the sense that the OP is asking about needs to at least be questioned. Is GR really a "force of nature" in the same sense that the other given examples (electromagnetism, isospin, etc) are?

At the risk of confusion, I'll give a specific example. GR includes effects such as time dilation. E&M doesn't. If we try to view gravity as "a force of nature", how do we include "gravitational time dilation" into the picture?

 

[pervect]

Let me add a bit on the other part of the question. It's unclear to me how to regard gravity "as a force of nature", but we can talk abut what the gauge symmetries in GR are, i.e. their physical interpretation, at least in the context of weak-field, linearized gravity, without worrying about the question of whether gravity is a "force" or not.

My discussion is based on MTW's gravitation $18.1, box 8.2, but most any GR textbook will talk about the issue. Imagine one has some coordinate system that creates maps from points, P, to coordinates $x^\mu$, i.e. one has P -> $x^\mu$ or $x^\mu(P)$.

Then one can consider infinitesimal coordinate transformations of the form $x^\mu$ -> $x^\nu$.

$$x^{\nu}(P) = x^\mu + \xi^\mu (P)$$

where the $\xi^\mu$ are four arbitrary infinitesimal functions.

GR is a diffeomorphism invariant theory, and these small infinitesimal coordinate transformations represent the gauge degree of freedom in linearized gravity. The end effect of these transformation is to transform $h_{\mu\nu}$ to $h_{\mu\nu} - \xi_{\mu,\nu} - \xi_{\nu,\mu}$, where $h_{\mu\nu}$ is the pertubation to the flat-space metric $g_{\mu\nu}$.

I believe that this is basically equivalent to haushoffer's more terse remarks. These infinitesimal coordinate transformations would be what haushoffer called "the Poincare algebra", I think. I start to get lost much beyond this point, alas.

 

[vanhees71]

I agree that standard GR is using a pseudo-Riemannian manifold and as such is by definition torsion free, and indeed as long as you stay on the classical level there's no need to extend it to a more general manifold.

Concerning the question whether gravity is an interaction as the other interactions or rather spacetime geometry (or better "geometrodynamics") is a matter of taste. I think both approaches have their own merit, and it's well worth to follow both approaches. For me the gauge-theoretical approach was a revelation to understand GR and how the "geometrization" of this interaction is quite unavoidable. Another more pedagogical treatment can be found in "Feynamn lectures on gravitation".

 

[pervect]

I agree that standard GR is using a pseudo-Riemannian manifold and as such is by definition torsion free, and indeed as long as you stay on the classical level there's no need to extend it to a more general manifold.

Concerning the question whether gravity is an interaction as the other interactions or rather spacetime geometry (or better "geometrodynamics") is a matter of taste. I think both approaches have their own merit, and it's well worth to follow both approaches. For me the gauge-theoretical approach was a revelation to understand GR and how the "geometrization" of this interaction is quite unavoidable. Another more pedagogical treatment can be found in "Feynamn lectures on gravitation".

Would Straumann's "Reflections on Gravity", https://arxiv.org/abs/astro-ph/0006423, be an example of what you're calling the gauge-theoretical approach?

The thing that bother's me about Straumann's approach is that I think it has problems explaining black holes, or other non-trivial topologies. That's only my personal conclusion, the literature seems very quiet on the topic. I view this as an unfortunate limitation that would make using Straumann's approach the sole approach that one uses to learn GR too limiting. Other than that, I think it does have a lot of appeal as a simple way to describe GR in those cases where it's applicable. I suspect that it would work well as a popularization too, except for the inevitable backlash when people who learned it that way tried to apply it to black holes.

 

[PeterDonis]

The thing that bother's me about Straumann's approach is that I think it has problems explaining black holes, or other non-trivial topologies.

One possible way of dealing with this is to believe that there are no nontrivial topologies in actual fact--i.e., that whatever quantum gravity theory we eventually come up with will tell us that quantum corrections prevent any topology other than $R^4$ from ever actually existing. On this view, what we think of as "black holes" are actually "temporary" apparent horizons (I put "temporary" in quotes because they can last for $10^{67}$ years or more) that eventually disappear due to Hawking radiation and never have singularities or nontrivial topologies inside them--all events in the spacetime can eventually send light signals to future null infinity and there are no actual event horizons. A model of this form could in principle be described as a field on Minkowski spacetime, just like the other interactions. But of course it's still an open question whether the correct theory of quantum gravity will look like this when we finally find it.

 

[vanhees71]

Straumann's paper is more in the spirit of the "Feynman Letures on Gravitation". What I had in mind is described in Ramond's book and is much simpler. It's just gauging the Lorentz (or Poincare) symmetry of special relativity, which leads to GR (with the possible extension to include spinor fields leading to non-symmetric Christoffel symbols, i.e., a spacetime manifold with torsion). It's also worked out in

T. W. B. Kibble, Lorentz invariance and the gravitational field, J. Math. Phys. 2, 212 (1961)
http://dx.doi.org/10.1063/1.1703702

 

[haushofer]

What are GCT's?

The space an have torsion if you have fields with spin involved. For spin 1/2 that's shown in the above cited book by Ramond.

General coördinate transfo.

Indeed, with fermions things change. In supergravity the field strength of translations gets an additional term quadratic in the gravitini due to the {Q,Q }~P commutator. Hence torsion.

 

[haushofer]

For the gauging procedure see https://arxiv.org/abs/1011.1145

I was thinking about writing an insight about this concerning GR whenever I can find the time, if people are interested.

 

[pervect]

One possible way of dealing with this is to believe that there are no nontrivial topologies in actual fact--i.e., that whatever quantum gravity theory we eventually come up with will tell us that quantum corrections prevent any topology other than $R^4$ from ever actually existing. On this view, what we think of as "black holes" are actually "temporary" apparent horizons (I put "temporary" in quotes because they can last for $10^{67}$ years or more) that eventually disappear due to Hawking radiation and never have singularities or nontrivial topologies inside them--all events in the spacetime can eventually send light signals to future null infinity and there are no actual event horizons. A model of this form could in principle be described as a field on Minkowski spacetime, just like the other interactions. But of course it's still an open question whether the correct theory of quantum gravity will look like this when we finally find it.

Are there any papers on this idea? I'd be particularly interested in a discussion of how it could be tested experimentally, either in principle or in practice.

 

[dextercioby]

in U(1), the covariant derivative for phase becomes the electromagnetic field
in Su(2), the 3 covariant derivative for isospin becomes the 3 weak forces bosons
in Su(3), the 8 covariant derivative for colors becomes the 8 strong color forces.
In General relativity.. the ____ (blank) covariant derivative for ____ (blank) becomes the gravitational force. Please supply the blanks. Ty.

I don't know why this thread is marked as "basic", it shouln't be. Gravity was formulated by Ivanenko and Fock as a gauge theory as far back as 1929 (here I omit the conceptually incorrect work by H. Weyl from 1918 which has the historical merit of introducing the word "Eich" = "Gauge"), but the 2 Russians lacked the mathematical theory of fiber bundles which became available only after 1940. It was much later the work by the Polish school (especially Andrzey Trautman) to put the GRT on the same footing with the rest of the fictive field theories in the Standard Model (electromagnetic, weak and strong). The complicated answer to your question in line 4 you can pick either from Trautman's work, or from the nice review by D. Ivanenko and G. Sardanashvili in "Physics Reports", Vol. 94, pp. 1-45, 1983.

 

[PeterDonis]

I don't know why this thread is marked as "basic", it shouln't be.

Good point. I've changed it to advanced.

 

[PeterDonis]

Are there any papers on this idea?

I don't know if there are on the idea exactly as I've expressed it. But it's an obvious implication of a class of models that have been proposed as a solution to the black hole information problem. The key feature that is stressed in those models is that no singularity ever forms and there is no actual event horizon, so confirming that the model preserves quantum unitarity is trivial. But there is also an assumption that the topology of the spacetime must be $R^4$.

An example of such a model is in this paper by Haggard & Rovelli:

https://arxiv.org/pdf/1407.0989v2.pdf

Here the requirement for $R^4$ topology appears as the (stronger) requirement that the causal structure of the spacetime is Minkowski spacetime (p. 5, item vi, bottom right). But this property is not the focus of the paper; the focus is how the model presented solves the black hole information problem by creating a "bounce" due to quantum effects.

I'd be particularly interested in a discussion of how it could be tested experimentally, either in principle or in practice.

In principle the model described in the paper linked to above could be tested--just wait long enough after an apparent black hole forms and see if you see an apparent white hole, i.e., a bunch of stuff exploding back out again, carrying all the information that was carried by the stuff that originally collapsed. But this test is not practical, at least not for a collapsing object of stellar mass or larger, since the time you would have to wait is orders of magnitude longer than the current age of the universe.