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Currently, (classical) gravity (General Relativity) is NOT a gauge theory (at least in the sense of a Yang-Mills theory).

Why should "classical" gravity be some (non-trivial or "special" or extended) gauge theory? Should quantum gravity be a gauge theory?

Remark: There are some contradictory claims in the literature to this issue. To what extent gravity is "a gauge" theory?Obviously, GR is not a YM theory. So, why do some people say that gravity "IS" a gauge theory? I found this question relevant, e.g., then we handle GR in the Einstein-Cartan theory or any other theory beyond GR, like teleparallel theories or higher-derivative gravitational theories. So I think it could be useful to discuss the "gauge flavor" of gravity here.

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    $\begingroup$ Why do you say GR is not a gauge theory? I've seen claims both that it is and that it isn't, but the distinction seems somewhat technical to a non-specialist like me. $\endgroup$
    – John Rennie
    Commented Jul 18, 2013 at 10:37
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    $\begingroup$ It depends how rigidly you define gauge theory. Typically gauge theories have compact Lie groups. Gravitational gauge theory, such as it is, is based on the non-compact diffeomorphism group (note I'm really talking about the individual fibres of whatever bundle it is). There is also no analogy to the metric/tetrad in standard gauge theories. Those are the basic differences. It all gets very complicated from there. $\endgroup$
    – Michael
    Commented Jul 18, 2013 at 10:43
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    $\begingroup$ It probably is the KISS principle followed by unification theorists. If unification means all forces become one at very high energies it seems simpler to assume that gravity will wear the same costume as the other three forces. $\endgroup$
    – anna v
    Commented Jul 18, 2013 at 10:57
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    $\begingroup$ As Michael Brown said, the definition of a gauge theory is a key point. I think "gauge theory = a Yang Mills theory" is too restrictive. People also define gauge theories in Hamiltonian terms - phase space/primary constraints/gauge orbits. Not necessary to have the Yang Mills apparatus there. I think some of the arguments in the literature/blogs could be avoided if people gave up front the definition of gauge theory which they were using. $\endgroup$
    – twistor59
    Commented Jul 18, 2013 at 12:37
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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/4359/2451 , physics.stackexchange.com/q/12461/2451 , physics.stackexchange.com/q/46324/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Jul 18, 2013 at 15:18

4 Answers 4

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A theory is usually called a "gauge theory" if all the interactions in that theory are introduced by promoting global symmetries to gauge symmetries. Note that a gauge theory is a gauge invariant theory, but a gauge invariant theory doesn't has to be a gauge theory (for example, the Standard Model is gauge invariant, but it's not a gauge theory since the scalar self-interaction doesn't enlarge the gauge symmetry of the model). Yang-Mills theory is an example of gauge theory, but not all gauge theories are of Yang-Mills type. General Relativity is a gauge theory in three different senses, namely:

  1. Invariance under diffeomorphims. Diffemorphism may be seen as a local (gauged) version of translations $\delta x^{\mu}\rightarrow a^{\mu}(x)$. In order for the theory to be diff. invariant, a covariant derivative $\nabla$ must replace partial derivatives $\partial$ (a general, dynamic metric $g$ tensor must replace Minkowski metric $\eta$ as well). Here, the most similar field to Yang-Mills connection is the Levi-Civita connection $\Gamma$ (note that in Palatini's formulation this field is independent of the metric), which transforms as a tensor plus a term that involves the derivative of $a(x)$, similar to the transformation of a non-abelian field.

  2. Invariance under infinitesimal diff. One can split $g$ in a fixed background and a dynamical perturbation $h$, and the action of an infinitesimal diff. on the perturbation turns out to be $\delta h_{\mu\nu}=\partial_{\mu}a_{\nu}+\partial_{\nu}a_{\mu}$, which is also a gauge symmetry. This is the gauge symmetry connected with the masslessnes of gravitons (much like $SU_c(3)$ is related to the masslessnes of gluons and $U_{em}(1)$ to the masslenes of photons). Here, the most similar field to Yang-Mills connection is $h$, which transforms similarly to the electromagnetic potential, even though $h$ is not a connection in any sense I am aware of.

  3. Invariance under local Lorentz transformations. It turns out that in order for spinors to be coupled to the gravitational field, it is convenient to introduced the tetrad formulation. In this approach, there is a gauge symmetry related to the freedom that one has to choose different basis in different space-time points. One has to introduce a covariant derivative (different from the first one in this answer) that allows us to change basis. This formulation is the closest to Yang-Mills theory (well, Ashtekar variables probably are even closer). The main difference is that in GR, besides a dynamical connection (equivalent to the gauge field in Yang-Mills), there is a tetrad field (due to the fact that the metric is a dynamical field in gravity) that doesn't have a counterpart in Yang-Mills. Here, the closest field to Yang-Mills' is the before-mentioned spin connection, which transforms as a tensor plus a term that involves the derivative of the local Lorentz transformation, very similarly to a Yang-Mills field.

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    $\begingroup$ Wait, with the term "diffeomorphism invariance", doesn't it refer to the symmetry $g_{\mu\nu}\to g_{\mu\nu}-\nabla_{\mu}\xi_{\nu}-\nabla_{\nu}\xi_{\mu}$, i.e., the Lie Derivative along $\xi$ of the metric vanishes? (See section 3.3 of these MIT notes). $\endgroup$
    – Alex Nelson
    Commented Jul 19, 2013 at 0:37
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    $\begingroup$ Hello @AlexNelson . Yes, in the first case, the metric tensor transform as you write. In the second — linearized diff — $h$ transforms with the covariant derivatives replaced by partial derivatives. The metric $g$ is invariant only in the case of a isometry. $\endgroup$
    – Diego Mazón
    Commented Jul 19, 2013 at 2:34
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    $\begingroup$ Drake, the tetrad field can be viewed as a part of the Yang-Mills connection for Poincare or (anti)-de Sitter group. In this case, tetrad and spin-connection are different parts of a single connection. Then gravity looks almost like YM-type gauge theory. There are important differences however. $\endgroup$
    – John
    Commented Jul 19, 2013 at 19:59
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    $\begingroup$ You can look in Blagojevic, "gravitation and gauge symmetries", which is freely downloadable. The matter is quite old. Take connection of the Poincare group $A=\frac12\omega^{a,b}L_{ab}+e^a P_a$, $L_{ab}$ and $P_a$ are Lorentz and translation generators. Then spin-connection and tetrad are the gauge fields corresponding to these generators. Compute the Yang-Mills field strength $F=dA+A^2=R^{a,b}L_{ab}+T^a P_a$. Then $T^a$ is a torsion two-form and $R^{a,b}$ is a Riemann two-form. Einstein-Hilbert action reads $\int R^{a,b} e^c...e^u \epsilon_{abc...u}$ $\endgroup$
    – John
    Commented Jul 19, 2013 at 21:33
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    $\begingroup$ Thanks, @John I don't get your point. Can you write EH action only in terms of a connection (as in Yang-Mills) or you additionally have to write the tetrad field explicitly (which doesn't transform as a connection)? What is the analog of torsion in Yang-Mills? It seems to me you are saying the same thing as me in other variables. $\endgroup$
    – Diego Mazón
    Commented Jul 22, 2013 at 18:01
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Gravity isn't Yang-Mills theory in the narrow sense – well, except for equivalences such as AdS/CFT or Matrix theory that imply that a quantum gravitational theory is fully equivalent to a gauge theory living in a different space (e.g. in AdS/CFT, on the boundary of the AdS space).

However, gravity is a gauge theory in the broader sense because it's conveniently formulated using the diffeomorphism symmetry group. The diffeomorphisms identify physical configurations that are physically equivalent, just like in the Yang-Mills case, so although they are not of the Yang-Mills type, they have to be treated just like Yang-Mills symmetries in Yang-Mills theories.

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    $\begingroup$ Hi, Lubos. Yes, I do knot those known facts. Moreover, I also know that, in the case of gravity, a Lie-group like formalism can be built, but it seems "a formidable mathematical" challenge, since, instead of Lie "structure constants", we are left with Lie "structure functions". I have read and studied this issue, after all, many people have argued and speculated that the issue to "quantize" gravity could be related to the way in which gravity is an extended "gauge gravity". You have also recalled that link, due to the gauge/gravity duality! $\endgroup$
    – riemannium
    Commented Jul 18, 2013 at 16:02
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    $\begingroup$ Indeed, I posted a related question to F.Wilczek twitter account and he (gently) answered me that he believed that GR could be some "kind" of non-linear sigma model. Of course, I have never read a paper about it, but it seems "plausible" (specially due to the string theory connection). Lubos, do you think that Quantum Gravity or gravity itself can be some "deformed" type of gauge theory? $\endgroup$
    – riemannium
    Commented Jul 18, 2013 at 16:05
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    $\begingroup$ Dear Riemannium, the diffeomorphism group is an infinite-dimensional Lie group with its infinite-dimensional Lie algebra that isn't quite the Yang-Mills gauge group but isn't too different. There are various relationships between nonlinear sigma models and gravity but I don't know in what sense they are "the same thing". Not in any sense I am aware of. Gravity is surely a "deformed gauge theory" to the extent that the character of the deformation remains sufficiently vague. ;-) $\endgroup$
    – Luboš Motl
    Commented Jul 20, 2013 at 15:19
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Gravity can be seen as a gauge theory of the Lorentz group (which acts on the tangent space). These was pointed out by Kibble and Sciama during the 50s and 60s.

As John said before, it's better seen in terms of differential forms.

Another reference you might find interesting is the Lecture notes on Chern-Simons gravity by Jorge Zanelli (available in arXiv).

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    $\begingroup$ Well, the distinction between gauge theory and YM theory is important...Spacetime infinitesimal diffemorphisms change the spacetime under the "gauge" variation, a very different behaviour to the YM gauge transformations, that do NOT change the spacetime...Interestingly, the supertranslation group acts (in sugra) more like that... I disagree, ... Gravity is a gauge theory of the Poincaré Group, I presume more than the Lorentz group, because traslations must be included into the game... $\endgroup$
    – riemannium
    Commented Jul 23, 2014 at 19:50
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    $\begingroup$ @riemannium I'm soory... don't undertand whether you are commenting or quoting something. Could you be more explicit? $\endgroup$
    – Dox
    Commented Jul 23, 2014 at 20:14
  • $\begingroup$ @Dox do you have a reference for the work of Kibble and Sciama? $\endgroup$
    – Quillo
    Commented Mar 10, 2021 at 10:20
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    $\begingroup$ @Quillo: T. W. B. Kibble, Lorentz invariance and the gravitational field, J. Math. Phys. 2, 212–221 (1961). The Physical Structure of General Relativity D. W. SCIAMA Rev. Mod. Phys. 36, 463. $\endgroup$
    – Dox
    Commented Mar 11, 2021 at 7:32
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Let's compare a specific instance of gauge theory (electroweak theory) with a specific form of gravity (Lorentz gauge gravity, also know as Einstein-Cartan gravity). Interestingly, in addition to gauge fields, both of them have add-on non-gauge bosonic fields (Higgs in electroweak theory and tetrad/metric in Lorentz gravity) responsible for spontaneous symmetry breaking. The comparison breaks down into two sections with one section for gauge fields, and another section for bosonic fields responsible for spontaneous symmetry breaking:

Gauge fields and gauge symmetries:

Similarities: Both electroweak theory and Lorentz gravity enjoy local gauge symmetries and have corresponding gauge fields as an essential ingredient. The gauge group of electroweak theory is $SU(2)*U_Y(1)$, whereas the gauge group of Lorentz gravity is the local Lorentz group $SO(1,3)$. The gauge fields of former are electroweak gauge fields $W_\mu, A_\mu$, whereas the gauge field of latter is the spin connection field $\omega_\mu$.

Dissimilarities: The Yang-Mills Lagrangian of electroweak theory is quadratic in gauge field curvature ($F = d W + W \wedge W$): $L_{YM} \sim F*F$ , whereas first-order Lagrangian of Lorentz gravity is linear in spin connection gauge field curvature ($R= d \omega + \omega \wedge \omega $): $L_{G} \sim R$.

Add-on bosonic fields (Higgs and tetrad/metric) responsible for spontaneous symmetry breaking:

Similarities: Other than the gauge fields, both electroweak theory and Lorentz gravity have additional bosonic fields that are responsible for spontaneous symmetry breaking. The symmetry-breaking field of electroweak theory is the Higgs field $H$, whereas the symmetry-breaking field of Lorentz gravity is the tetrad/veirbein field $e_\mu$ (which could be understood as the square root of metric $g_{\mu\nu}$). Both the Higgs field $H$ and the tetrad/veirbein field $e_\mu$ break the gauge symmetries upon acquiring a non-zero vacuum expectation value (VEV). The VEV of Higgs field $H$ in electroweak theory breaks the gauge symmetry from $SU(2)*U_Y(1)$ down to $U_{EM}(1)$, whereas the Minkowskian VEV of the tetrad/veirbein field $e_\mu$ in Lorentz gravity breaks the gauge symmetry from the local Lorentz gauge symmetry $SO(1,3)$ down to the global Lorentz symmetry of special relativity.

Dissimilarities: The Higgs field $H$ in electroweak theory is a scalar, whereas the tetrad/veirbein field $e_\mu$ in Lorentz gravity is a vector. Due to the non-trivial diffeomorphism transformation properties of the tetrad/veirbein field $e_\mu$ as a vector, the Minkowskian VEV of the tetrad/veirbein field $e_\mu$ breaks the diffeomorphism symmetry as well, in addition to breaking the local Lorentz symmetry $SO(1,3)$. On the other hand, the VEV of the Higgs field $H$ as a scalar does not break the diffeomorphism symmetry.

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