# Is General Relativity a Gauge Theory?

Many seem to argue it is but Steven Weinstein argues in his paper http://philsci-archive.pitt.edu/archive/00000834/00/gr_gauge.pdf" [Broken] it is not.

He argues that the diffeomorphism invariance of GR is more restrictive than gauge invariance since in the case of diffeomorphism invariance the space-time points (by themselves) are meaningless because the physics is in the description of the correlations, while in the case of gauge invariance the space-time points remain during a gauge transformation and are accompanied by a change in the gauge field.

Any opinions as to the validity of the argument?

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It depends upon how you define a gauge theory. Personally speaking, I define a gauge theory as one which can be derived from a singular Lagrangian. If the Lagrangian for a theory is singular, then it's (almost) guaranteed to be a gauge theory. To be more precise, consider the following.

All physical theories of which I am aware can be derived from an action principle. Let's take the simple case of a discrete theory as an example. We assume that we have some system which is described by generalised coordinates $q^i$. More specifically, take a manifold $B$ and its tangent bundle $TB$ with projection $\pi:TB\to B$ such that $\pi^{-1}(b)=T_bB$. We define a map $L:TB\mapsto\mathbb{F}$ where $\mathbb{F}$ is some target value field (typically taken to be $\mathbb{R}$ or $\mathbb{C}$ in physics). In coordinates, we say that the classical trajectories of a system with $N$ degrees of freedom are those for which the action

$$S=\int_{t_1}^{t_2}dt\,L(q^i,\dot{q}^i),\qquad i=1,\ldots,N$$

is stationary with respect to arbitrary variations $\delta q(t)$ which vanish at the endpoint. (There are one or two technical considerations involving the argument of the Lagrangian which are surprisingly never discussed in elementary treatments, but I digress.) So far so bog standard, right? Everyone knows that a necessary and sufficient condition for the action to be stationary are that the Euler-Lagrange equations are satisfied:

$$L_i = -\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}^i}\right) + \frac{\partial L}{\partial q^i} = 0$$

What is rarely mentioned, however, is that a much more enlightening way to write the Euler-Lagrange equations is

$$L_i = -W_{ij}(q,\dot{q})\ddot{q}^j + V_i = 0,$$

where

$$W_{ij}(q,\dot{q}) = -\frac{\partial^2L}{\partial\dot{q}^i\partial\dot{q}^j}$$
$$V_i = -\frac{\partial^2L}{\partial\dot{q}^i\partial q^j}$$

From this you can see that the $\ddot{q}^i$s at a given time are uniquely determined by $(q,\dot{q})$ at that time if and only if $W_{ij}$ is invertible. If $\det W_{ij}\ne0$ the system is called regular and we can do this. However, if $\det W_{ij}=0$ then the accelerations and time evolution cannot be uniquely determined. Such systems are called singular and one gets different time evolutions for the same initial conditions. For these types of systems one also has constraints, i.e., some relations $\Psi^A(q,\dot{q})=0$ that must be solved before one goes to treat the initial value problem of the theory (we're all familiar with these constraints from, say, electromagnetism).

One can now look through the generalised Bianchi identities corresponding to this idea and conclude that gauge theories are necessarily singular systems, i.e., all gauge theories can (and must) be derivable from a Lagrangian for which $W_{ij}$ is non-invertible.

How does this answer your original question? Well, every time I read one of these claims that general relativity is not a gauge theory, it is inevitably as a result of a misunderstanding of the difference between gauge theories which have internal symmetries (i.e., those for which the gauge group acts on some internal degrees of freedom - these necessarily involve gauge groups acting on some fibre) and gauge theories which are generally covariant.

For the first type of theory, all the constraints are linear in the momenta and the Hamiltonian doesn't vanish. The local symmetries are then generated by first class constraints (a constraint is first class if it's Poisson bracket with all other constraints vanishes).

For the second type, at least one constraint must be quadratic in the momenta. Furthermore, there is always a choice of canonical variables for which the Hamiltonian is itself a constraint (MTW call this a super-Hamiltonian in chapter 21). This leads to all sorts of interesting questions regarding the interpretation of time and, indeed, is the very basis of the celebrated "problem of time" in quantum gravity. (I'd go one further and say that this is actually the key to understanding quantum gravity, but I may be alone on this.) The answer to your question is that in regarding the diffeomorphism group $Diff(M)$ as a gauge group, the people who claim that GR is not a gauge theory get lost in all sorts of tortuous metaphysical ramblings about the nature of observables and so on. My counter argument is that, at a basic technical level, there is no controversy. GR is a gauge theory since it fits into the above classification. It's the interpretation of GR within this framework which is hard.

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john baez
Gold Member
What's a 'gauge theory'? Presumably a theory with gauge transformations as symmetries.

But what's a 'gauge transformation'?

It depends on how you define 'gauge transformation'. Lots of people define a gauge transformation as a symmetry of a principal bundle - a 'bundle automorphism', in math-speak. But, many other people define a gauge transformation as a symmetry of a Lagrangian whose generator vanishes 'on shell' - that is, when the equations of motion hold. This is a very useful viewpoint in physics, as coalquay404 so clearly explained.

While everything in Steve Weinstein's paper seems to be correct, it would be nice if he'd said more about the second notion of 'gauge theory', which in some sense is the more physical of the two.

Perhaps Weinstein expects a bit too much from General Relativity, afterall it is not a completely relational theory. Or is relationalism irrelevant here?

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Perhaps Weinstein expects a bit too much from General Relativity, afterall it is not a completely relational theory. Or is relationalism irrelevant here?

Define what you mean by "relational".

Define what you mean by "relational".
In the interest of brevity let's just assume Smolin's definitions in http://arxiv.org/PS_cache/hep-th/pdf/0507/0507235v1.pdf" [Broken].

By the way I do not fully agree with Smolin's points as to why GR is not a completely relational theory, I think he left out one additional argument which does not make the theory relational at all. But that is beyond the scope of this topic,

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There have been some serious attempts of which I know to cast GR as a relational theory. However, depending on one's point of view, it's unfortunate that all of these lead to manifest breaking of general covariance not just at the Hamiltonian level but also at the Lagrangian level. This is usually due to the fact that one's ability to "slice" the spacetime is no longer a gauge freedom but actually gets promoted to the level of a constraint. I'm not an expert on these (in fact, I'm not an expert on anything) but that's my understanding of them.

On another note, Smolin's paper is in my list of papers that I really should have read by this stage but which I'm embarrassed to say I have not. I'm going to try to read it over the next couple of days and get back to you.

Also, Chris Hillman sent me a PM to suggest that my comment above about the nature of gauge theories really does deserve a couple of examples to illustrate things (and also to correct a minor typo). I'll try to do this when I have some free time.

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Interesting alternative for Euler-Lagrange equation, Coalquay404! How do you get it from EL-eq, please? ... and shouldn't $$V_i = \partial L/ \partial q_i$$ be the "force", like in Newton's equation? (the coeff. of q'' looks like a mass matrix - interesting!)

tom.stoer
There have been some serious attempts of which I know to cast GR as a relational theory. However, depending on one's point of view, it's unfortunate that all of these lead to manifest breaking of general covariance not just at the Hamiltonian level but also at the Lagrangian level. This is usually due to the fact that one's ability to "slice" the spacetime is no longer a gauge freedom but actually gets promoted to the level of a constraint.
The reformulation of gravity as a gauge theory and the spacetime foliation have nothing to do with each other. It is true that in some research programs both gauge and canonical formulism are used together, but logically and mathematically they are independent. One can formulate gravity as a manifest covariant gauge theory. In addition it is wrong that the canonical formalism leads to a breaking of general covariance. Covariance is no longer explicitly visible, but it is represented and guarantueed by the constraints (this may become problematic when quantizing gravity, but at the classical level Lagrangian and Hamiltonian / canonical formalism are strictly equivalent).

There have been some serious attempts of which I know to cast GR as a relational theory. However, depending on one's point of view, it's unfortunate that all of these lead to manifest breaking of general covariance not just at the Hamiltonian level but also at the Lagrangian level. This is usually due to the fact that one's ability to "slice" the spacetime is no longer a gauge freedom but actually gets promoted to the level of a constraint. I'm not an expert on these (in fact, I'm not an expert on anything) but that's my understanding of them.

It is true that in some research programs both gauge and canonical formulism are used together, but logically and mathematically they are independent. One can formulate gravity as a manifest covariant gauge theory. In addition it is wrong that the canonical formalism leads to a breaking of general covariance. Covariance is no longer explicitly visible, but it is represented and guarantueed by the constraints (this may become problematic when quantizing gravity, but at the classical level Lagrangian and Hamiltonian / canonical formalism are strictly equivalent).

Tom, aren't you conflating 2 different things here? Coalquay in the quote is talking about casting GR as a relational theory, wich is not exactly the same as formulating gravity as a manifest covariant gauge theory which he agreed in post #2 that is perfectly alright.
Also I think you are expressing basically the same thing he did in the rest of his post but with different words. where he says:"one's ability to "slice" the spacetime is no longer a gauge freedom but actually gets promoted to the level of a constraint" , you say:"Covariance is no longer explicitly visible, but it is represented and guarantueed by the constraints".

tom.stoer
As this thread started with "gravity as a gauge theory" and later mutated into "gravity as a relational theory" I wanted to clarify the differences. Yes, I think we agree and therefore perhaps my post was redundant.

While we were on the topic of whether or not GR is relational, I think you guys should take a look at "Shape Dynamics" which is a renormalizable theory of gravity in which GR emerges. It could help give a new approach to Quantum Gravity since "GR" wouldn't produce nonrenormalizable operators and it gives a relational view of GR with background independence. 4d diffeomorphism invariance is replaced with 3d diffeomorphism invariance and 3d conformal invariance.

@coalquay404: Shape Dynamics doesn't break any covariance in the Lagrangian.

PAllen
While we were on the topic of whether or not GR is relational, I think you guys should take a look at "Shape Dynamics" which is a renormalizable theory of gravity in which GR emerges. It could help give a new approach to Quantum Gravity since "GR" wouldn't produce nonrenormalizable operators and it gives a relational view of GR with background independence. 4d diffeomorphism invariance is replaced with 3d diffeomorphism invariance and 3d conformal invariance.

@coalquay404: Shape Dynamics doesn't break any covariance in the Lagrangian.

Hey, go vote for that in the poll of significant recent papers in the 'beyond the standard model' forum. Last I checked, I was the only vote for this paper.

While we were on the topic of whether or not GR is relational, I think you guys should take a look at "Shape Dynamics" which is a renormalizable theory of gravity in which GR emerges. It could help give a new approach to Quantum Gravity since "GR" wouldn't produce nonrenormalizable operators and it gives a relational view of GR with background independence. 4d diffeomorphism invariance is replaced with 3d diffeomorphism invariance and 3d conformal invariance.

@coalquay404: Shape Dynamics doesn't break any covariance in the Lagrangian.

Hey, go vote for that in the poll of significant recent papers in the 'beyond the standard model' forum. Last I checked, I was the only vote for this paper.

After reading the paper a bit, I have to admit I don't quite understand what is the physical justification or the advantages of imposing a a frame and introducing absoulute time, wasn't this considered anathema by mainstream physics?

Many seem to argue it is but Steven Weinstein argues in his paper http://philsci-archive.pitt.edu/archive/00000834/00/gr_gauge.pdf" [Broken] it is not.

He argues that the diffeomorphism invariance of GR is more restrictive than gauge invariance since in the case of diffeomorphism invariance the space-time points (by themselves) are meaningless because the physics is in the description of the correlations, while in the case of gauge invariance the space-time points remain during a gauge transformation and are accompanied by a change in the gauge field.

Any opinions as to the validity of the argument?

GR is not a gauge theory because it is not a spin-2 field over a flat background.

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tom.stoer
GR is not a gauge theory because it is not a spin-2 field over a flat background.
But as many papers show there is a gauge symmetry - and that's all what counts :-)

I don't think that you need a flat background in order to define a gauge theory.

The most famous application is LQG (for the gauge aspects we need no quantization, the classical theory is sufficient). If you rewrite GR using Ashtekar's variables you end up with a canonical formalism with three constraints which I denote

Gi(x) ~ 0
Va(x) ~ 0
H ~ 0

The spatial diffeomorphism constraint V and the Hamiltonian constraint H are not known from ordinary gauge theories like QCD. The details of the dynamics, e.g. the kinetic energy of the fields differ, of course.

But the first constraint

Gi(x) = (DE)i(x)

and its algebra with the gauge field A and the field strength E is structurally identical to quark-less SU(2) QCD. This so-called Gauss law generates gauge transformations in A and E. In GR (Ashtekar's formulation) that means that gauge transformations are nothing else but local rotations of tangent space frames (E is related to the triad field).

There are other formalisms which end up with different notations but I would say all of them agree that GR can be rewritten as a theory gauging certain aspects of underlying spacetime symmetries.

You may want to have a look at

http://arxiv.org/abs/arXiv:gr-qc/9602013
On the Gauge Aspects of Gravity
F. Gronwald, F.W. Hehl
(Submitted on 8 Feb 1996)
Abstract: We give a short outline, in Sec.\ 2, of the historical development of the gauge idea as applied to internal ($U(1),\, SU(2),\dots$) and external ($R^4,\,SO(1,3),\dots$) symmetries and stress the fundamental importance of the corresponding conserved currents. In Sec.\ 3, experimental results with neutron interferometers in the gravitational field of the earth, as inter- preted by means of the equivalence principle, can be predicted by means of the Dirac equation in an accelerated and rotating reference frame. Using the Dirac equation in such a non-inertial frame, we describe how in a gauge- theoretical approach (see Table 1) the Einstein-Cartan theory, residing in a Riemann-Cartan spacetime encompassing torsion and curvature, arises as the simplest gravitational theory. This is set in contrast to the Einsteinian approach yielding general relativity in a Riemannian spacetime. In Secs.\ 4 and 5 we consider the conserved energy-momentum current of matter and gauge the associated translation subgroup. The Einsteinian teleparallelism theory which emerges is shown to be equivalent, for spinless matter and for electromagnetism, to general relativity. Having successfully gauged the translations, it is straightforward to gauge the four-dimensional affine group $R^4 \semidirect GL(4,R)$ or its Poincar\'e subgroup $R^4\semidirect SO(1,3)$. We briefly report on these results in Sec.\ 6 (metric-affine geometry) and in Sec.\ 7 (metric-affine field equations (\ref{zeroth}, \ref{first}, \ref{second})). Finally, in Sec.\ 8, we collect some models, currently under discussion, which bring life into the metric-affine gauge framework developed.

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But as many papers show there is a gauge symmetry - and that's all what counts :-)

I don't think that you need a flat background in order to define a gauge theory.

The most famous application is LQG (for the gauge aspects we need no quantization, the classical theory is sufficient). If you rewrite GR using Ashtekar's variables you end up with a canonical formalism with three constraints which I denote

Gi(x) ~ 0
Va(x) ~ 0
H ~ 0

The spatial diffeomorphism constraint V and the Hamiltonian constraint H are not known from ordinary gauge theories like QCD. The details of the dynamics, e.g. the kinetic energy of the fields differ, of course.

But the first constraint

Gi(x) = (DE)i(x)

and its algebra with the gauge field A and the field strength E is structurally identical to quark-less SU(2) QCD. This so-called Gauss law generates gauge transformations in A and E. In GR (Ashtekar's formulation) that means that gauge transformations are nothing else but local rotations of tangent space frames (E is related to the triad field).

There are other formalisms which end up with different notations but I would say all of them agree that GR can be rewritten as a theory gauging certain aspects of underlying spacetime symmetries.

You may want to have a look at

http://arxiv.org/abs/arXiv:gr-qc/9602013
On the Gauge Aspects of Gravity
F. Gronwald, F.W. Hehl
(Submitted on 8 Feb 1996)
Abstract: We give a short outline, in Sec.\ 2, of the historical development of the gauge idea as applied to internal ($U(1),\, SU(2),\dots$) and external ($R^4,\,SO(1,3),\dots$) symmetries and stress the fundamental importance of the corresponding conserved currents. In Sec.\ 3, experimental results with neutron interferometers in the gravitational field of the earth, as inter- preted by means of the equivalence principle, can be predicted by means of the Dirac equation in an accelerated and rotating reference frame. Using the Dirac equation in such a non-inertial frame, we describe how in a gauge- theoretical approach (see Table 1) the Einstein-Cartan theory, residing in a Riemann-Cartan spacetime encompassing torsion and curvature, arises as the simplest gravitational theory. This is set in contrast to the Einsteinian approach yielding general relativity in a Riemannian spacetime. In Secs.\ 4 and 5 we consider the conserved energy-momentum current of matter and gauge the associated translation subgroup. The Einsteinian teleparallelism theory which emerges is shown to be equivalent, for spinless matter and for electromagnetism, to general relativity. Having successfully gauged the translations, it is straightforward to gauge the four-dimensional affine group $R^4 \semidirect GL(4,R)$ or its Poincar\'e subgroup $R^4\semidirect SO(1,3)$. We briefly report on these results in Sec.\ 6 (metric-affine geometry) and in Sec.\ 7 (metric-affine field equations (\ref{zeroth}, \ref{first}, \ref{second})). Finally, in Sec.\ 8, we collect some models, currently under discussion, which bring life into the metric-affine gauge framework developed.

Everyone can call apples to oranges, but that is not a very smart thing to do...

The reason for the which «the Hamiltonian constraint H are not known from ordinary gauge theories like QCD» is because gauge theories are well-formulated, with the H_QCD being correctly obtained from the associated Lagrangian. Your above vanishing H is not the Hamiltonian for gravity (and it gives the well-known problem of the absence of time).

The paper by F. Gronwald, F.W. Hehl highlights the misunderstanding that I referred to in https://www.physicsforums.com/showpost.php?p=3546480&postcount=15. They start from Gauge theory, i.e., they apply the gauge principle to «a field theory in Minkowski spacetime», but then they confound the resulting gauge theory with GR. Among other mistakes they confound the effective metric obtained from the «gauge potential» with the dynamical metric of GR. Both metric are different by a term which is roughly of the order of the gauge potential, although they treat both as if were the same mathematical objects, without any rigor.

GR is not a gauge theory, although anyone can say the contrary by changing the meaning of "gauge theory" to his/her favorite choice.

tom.stoer