Non-metric Compatible Connections: Physically Plausible?

[Ibix]

TL;DR Summary

What does a connection that isn't metric compatible mean?

- The connection should be metric compatible.

If this is opening a can of worms then please say so and I'll start a separate thread.

This constraint on the covariant derivative means that transporting the metric is, in fact, path independent. Is this actually a requirement for any physically plausible theory? If you don't impose it, doesn't it mean that a free-falling observer can measure the metric near some event (getting a local Minkowski frame) then find that the theory says that the "natural" transport of that is not a local Minkowski frame at some later event? That is, such a theory wouldn't respect the equivalence principle?

 

[vanhees71]

It opens a can of worms, but in my opinion a very interesting one.

I think it's pretty likely that, given the fact that there are particles/matter with spin, GR should in fact be extended to Einstein-Cartan theory, i.e., the connection is not torsion-free anymore, and then even when the connection is metric compatible, it's not unique anymore. AFAIK you need a model to determine the torsion in addition to the metric, but for sure the GR experts can they something more definite about such extensions of GR.

 

[Orodruin]

It opens a can of worms, but in my opinion a very interesting one.

I think it's pretty likely that, given the fact that there are particles/matter with spin, GR should in fact be extended to Einstein-Cartan theory, i.e., the connection is not torsion-free anymore, and then even when the connection is metric compatible, it's not unique anymore. AFAIK you need a model to determine the torsion in addition to the metric, but for sure the GR experts can they something more definite about such extensions of GR.

I think it is indeed a can of juicy worms. If I do not misremember, if you vary the metric and connection independently, you actually regain the Levi-Civita connection from the Einstein-Hilbert action, but this indeed changes once you introduce matter fields with spin.

Perhaps we should move this to a separate thread though.

 

[Ibix]

In this thread I asked:
Requiring metric compatibility of the covariant derivative means that transporting the metric is, in fact, path independent. Is this actually a requirement for any physically plausible theory? If you don't impose it, doesn't it mean that a free-falling observer can measure the metric near some event (getting a local Minkowski frame) then find that the theory says that the "natural" transport of that is not a local Minkowski frame at some later event? That is, such a theory wouldn't respect the equivalence principle?
@Orodruin and @vanhees71 replied:

I think it's pretty likely that, given the fact that there are particles/matter with spin, GR should in fact be extended to Einstein-Cartan theory, i.e., the connection is not torsion-free anymore, and then even when the connection is metric compatible, it's not unique anymore. AFAIK you need a model to determine the torsion in addition to the metric, but for sure the GR experts can they something more definite about such extensions of GR.

If I do not misremember, if you vary the metric and connection independently, you actually regain the Levi-Civita connection from the Einstein-Hilbert action, but this indeed changes once you introduce matter fields with spin.

I must admit I asked off the cuff, so haven't done any reading on connections other than the Levi-Civita connection, and I'm not sure where to start. But it does seem that my initial reaction ("they wouldn't make physical sense") was wrong. I'd be interested in pointers to appropriate reading material, as well as any commentary.

 

[Orodruin]

Summary:: What does a connection that isn't metric compatible mean?

In this thread I asked:
Requiring metric compatibility of the covariant derivative means that transporting the metric is, in fact, path independent. Is this actually a requirement for any physically plausible theory? If you don't impose it, doesn't it mean that a free-falling observer can measure the metric near some event (getting a local Minkowski frame) then find that the theory says that the "natural" transport of that is not a local Minkowski frame at some later event? That is, such a theory wouldn't respect the equivalence principle?
@Orodruin and @vanhees71 replied:I must admit I asked off the cuff, so haven't done any reading on connections other than the Levi-Civita connection, and I'm not sure where to start. But it does seem that my initial reaction ("they wouldn't make physical sense") was wrong. I'd be interested in pointers to appropriate reading material, as well as any commentary.

It depends on what you mean by ”make physical sense”. Many manifolds that are used in physics do not even have a natural metric that represents something physical (just take phase space of Hamiltonian mechanics or state space of thermodynamics as examples).

 

[vanhees71]

A very condensed review about the idea that one needs the extension of the pseudo-Riemannian spacetime of GR to an Einstein-Cartan manifold can be found in Ramond's QFT textbook, where he treats it from the point of view of gauge theories, i.e., using the Lorentz group of Minkowski spacetime as the local gauge group. It also turns out that one necessarily needs the extension when considering particles (media) with spin:

P. Ramond, Field Theory: A Modern Primer, Addison-Wesley, Redwood City, Calif., 2 ed. (1989).

The classical review paper is by Hehl et al:

https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.48.393

 

[pervect]

A very condensed review about the idea that one needs the extension of the pseudo-Riemannian spacetime of GR to an Einstein-Cartan manifold can be found in Ramond's QFT textbook, where he treats it from the point of view of gauge theories, i.e., using the Lorentz group of Minkowski spacetime as the local gauge group. It also turns out that one necessarily needs the extension when considering particles (media) with spin:

P. Ramond, Field Theory: A Modern Primer, Addison-Wesley, Redwood City, Calif., 2 ed. (1989).

The classical review paper is by Hehl et al:

https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.48.393

A couple of questions. Is this related to what Wiki calls "Einstein-Cartan theory"?

https://en.wikipedia.org/w/index.php?title=Einstein–Cartan_theory&oldid=934447151

Secondly, would it be fair to say that ECKS theory still has a metric compatible connection, but drops the requirment that there be no torsion? Or is that wrong?

 

[vanhees71]

I think it's Einstein-Cartan theory, and indeed the connection is still metric compatible but with torsion and thus not unique. That's why you need additional equations for the torsion with the spin tensor as sources.

 

[haushofer]

Not sure if it adds anything, but if you describe e.g. N=1 supergravity as the gauge theory of the super poincare algebra, the torsion follows from the constraint that the field strength of translations is set to zero; the torsion then depends on the gravitino. See e.g.

https://www.physicsforums.com/insights/general-relativity-gauge-theory/

 

[robphy]

Possibly useful reading (I haven't read them):

https://en.wikipedia.org/wiki/Metric_connection
https://en.wikipedia.org/wiki/Nonmetricity_tensor
https://projecteuclid.org/euclid.cmp/1103858479 (Comm. Math. Phys., Volume 29, Number 1 (1973), 55-59. "Conditions on a connection to be a metric connection", B. G. Schmidt)
https://mathoverflow.net/questions/...annian-metric-for-which-it-is-the-levi-civita
https://inis.iaea.org/collection/NCLCollectionStore/_Public/18/010/18010695.pdf?r=1&r=1
("Nonmetricity and torsion: Facts and fancies in gauge approaches to gravity", Baekler, P.; Hehl, F.W.; Mielke, E.W.)

https://www.mdpi.com/2218-1997/5/7/173 ( Universe 2019, 5(7), 173; https://doi.org/10.3390/universe5070173
"The Geometrical Trinity of Gravity" JB Jiménez, L Heisenberg, and TS Koivisto .
I'm not familiar with this journal but... https://www.mdpi.com/journal/universe/editors and this special issue https://www.mdpi.com/journal/universe/special_issues/feature_papers_2019 has some names I recognize. )