How is general relativity cast as a gauge theory?

[Phrak]

how is general relativity cast as a gauge theory?

 

[atyy]

I've never read this properly, but it's somewhere in here:

Fields
W. Siegel
http://arxiv.org/abs/hep-th/9912205

 

[George Jones]

how is general relativity cast as a gauge theory?

See posts #2 and #3 in

https://www.physicsforums.com/showthread.php?p=1294659#post1294659.

 

[schieghoven]

If your question means 'In what sense does GR possesses gauge invariance?' then, in the classical (Einstein) formulation it has only diffeomorphism invariance; and diffeomorphism transformations are generated by the Lie derivative operator L_v, for arbitrary vector field v.
't Hooft and Veltman, Ann. Inst. Henri Poincare A 20, 69 (1974)
briefly discuss gauge fixing and Feynman rules. (I think this paper is generally available at http://www.numdam.org/numdam-bin/feuilleter?j=AIHPA)

However, in order to treat spinor fields, spinor geometry is a compelling option. In this case the vierbein replaces the metric as the fundamental dynamical variable, and in addition to diffeomorphism invariance it has invariance under local Lorentz transforms. There's an excellent explanation in
Deser and Nieuwenhuizen, Phys. Rev. D 10, 411 (1974)
if you can get it from a library. There's also
Brandt, Lectures on Supergravity
http://xxx.soton.ac.uk/abs/hep-th/0204035
particularly sect 3.3.

BTW, the 't Hooft/Veltman paper and a pair of Deser/Nieuwenhuizen papers are classics from quantum gravity; 't Hooft/Veltman showed (one-loop) renormalizability of source-free GR, but that coupling to a scalar field breaks renormalizability; Deser/Nieuwenhuizen showed that coupling to either electromagnetism or fermions also breaks renormalizability. The lack of renormalizability is a crucial difference between GR and other gauge field theories such as Yang-Mills or QCD. So... where to next?

Dave

 

[Phrak]

If your question means 'In what sense does GR possesses gauge invariance?' then, in the classical (Einstein) formulation it has only diffeomorphism invariance; and diffeomorphism transformations are generated by the Lie derivative operator L_v, for arbitrary vector field v.
't Hooft and Veltman, Ann. Inst. Henri Poincare A 20, 69 (1974)
briefly discuss gauge fixing and Feynman rules. (I think this paper is generally available at http://www.numdam.org/numdam-bin/feuilleter?j=AIHPA)

However, in order to treat spinor fields, spinor geometry is a compelling option. In this case the vierbein replaces the metric as the fundamental dynamical variable, and in addition to diffeomorphism invariance it has invariance under local Lorentz transforms. There's an excellent explanation in
Deser and Nieuwenhuizen, Phys. Rev. D 10, 411 (1974)
if you can get it from a library. There's also
Brandt, Lectures on Supergravity
http://xxx.soton.ac.uk/abs/hep-th/0204035
particularly sect 3.3.

BTW, the 't Hooft/Veltman paper and a pair of Deser/Nieuwenhuizen papers are classics from quantum gravity; 't Hooft/Veltman showed (one-loop) renormalizability of source-free GR, but that coupling to a scalar field breaks renormalizability; Deser/Nieuwenhuizen showed that coupling to either electromagnetism or fermions also breaks renormalizability. The lack of renormalizability is a crucial difference between GR and other gauge field theories such as Yang-Mills or QCD. So... where to next?

Dave

Much appreciated for future reference. They're over my current skills, though.

Is supergravity your main interest in physics?

I was under the impression that general relativity could be caste as a local (and classical) gauge theory where, somehow, some aspect of the connection would manifest as the stress energy tensor. Do you know if this is the case?

 

[Mentz114]

This may be slightly off-topic, but Teleparallel gravity, which is not GR but has the same field equations if one assumes the equivalence bwtween inertial and gravitational mass.

Teleparallel gravity corresponds to a gauge theory of
the translation group. According to this model, to each
point of spacetime there is attached a Minkowski tangent
space, on which the translation (gauge) group acts.

See here and references within -

http://uk.arxiv.org/abs/gr-qc/0312008v1

M

 

[Phrak]

This may be slightly off-topic, but Teleparallel gravity, which is not GR but has the same field equations if one assumes the equivalence bwtween inertial and gravitational mass.

They make an intesting claim. If I read them right, working backwards, the weak equivalence principle is not a direct result of field eqations of general relativity.

There's something large in this that I don't quite understand.

 

[Phrak]

I've never read this properly, but it's somewhere in here:

Fields
W. Siegel
http://arxiv.org/abs/hep-th/9912205

Good grief, that's a 885 page text! Care to summarize it?

 

[atyy]

Good grief, that's a 885 page text! Care to summarize it?

Damn, you didn't fall for my trap! I was hoping you'd summarise it for me. Not sure if this is any more helpful, but it it is shorter (see Peter Woit's response to p falor): http://www.math.columbia.edu/~woit/wordpress/?p=705

 

[atyy]

If I read them right, working backwards, the weak equivalence principle is not a direct result of field eqations of general relativity.

Nordstrom's theory is an alternative that incorporates the weak equivalence principle, so it predicts gravitational redshift. EP alone predicts only half the solar bending of light compared to GR, because of extra bending due to curvature. Nordstrom theory predicts no bending because curvature cancels EP bending.

Another theory that has the equivalence principle is Brans-Dicke theory. It is a modification of GR in which the EP holds except for bodies held together by gravity, a feature discovered by Nordtvedt.

 

[Phrak]

Nordstrom's theory is an alternative that incorporates the weak equivalence principle, so it predicts gravitational redshift. EP alone predicts only half the solar bending of light compared to GR, because of extra bending due to curvature. Nordstrom theory predicts no bending because curvature cancels EP bending.

Another theory that has the equivalence principle is Brans-Dicke theory. It is a modification of GR in which the EP holds except for bodies held together by gravity, a feature discovered by Nordtvedt.

atyy, you've got some hidden talents of which I could not have guessed.

I'd thought there would be a simpler answer to my question where general relativity was cast as a classical, and local gauge theory on the Christoffel connection. Maybe there is. I still don't know. It's assumed I'm asking for the quatized variety, or there's been no classical variety found--thus no comments on null results.

 

[Mentz114]

Phrak:

They make an intesting claim. If I read them right, working backwards, the weak equivalence principle is not a direct result of field eqations of general relativity.

There's something large in this that I don't quite understand.

They do show that without m_inertial=m_grav the GR EOMs are inconsistent and no free-fall can be defined. But I think we knew that already. GR has to assume the weak equivalence principle, otherwise geometry does not fully describe the field and individual particle properties must be taken into account.

M

 

[Phrak]

Damn, you didn't fall for my trap! I was hoping you'd summarise it for me. Not sure if this is any more helpful, but it it is shorter (see Peter Woit's response to p falor): http://www.math.columbia.edu/~woit/wordpress/?p=705

Hey, I've looked ove that encyclopedic tome for the insomniatic, and eternally masochistic, and invented a lot of clever and nasty things say about it! It's certainly no way to learn about any[/] subject, what-so-ever, on this planet or the next. It was undoubtedly composed by a team of grad students in trade for better grades.

After calming down I think it should be an excellent reference, adding depth to something already studied. I've hard-copied 200 pages of mysterious formulae already.