Understanding Gauge Symmetry in Massive Gravity: Analysis of Fierz-Pauli Action

[Mentz114]

Refering to this paper "Theoretical Aspects of Massive Gravity" (http://arxiv.org/abs/1105.3735) about the spin-2 boson field and GR.

The author uses the Fierz-Pauli action ( I quote the massless part)

$-\frac{1}{2}\partial_\lambda h_{\mu\nu}\partial^\lambda h^{\mu\nu} + \partial_\mu h_{\nu\lambda}\partial^\nu h^{\mu\lambda}$and states that these terms have the gauge symmetry

$\delta h_{\mu\nu} = \partial_\mu \xi_\nu + \partial_\nu \xi_\mu$

for a spacetime dependent gauge parameter $\xi_\mu(x)$.

No problem there but I wonder if someone could show this explicitly ?

If I understand correctly this symmetry is diffeomorphism invariance, since Killings equations arise as the analog of conserved charges and currents.

 

[Matterwave]

You should plug in $h'_{\mu\nu}=h_{\mu\nu}+\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$ into your equation and hopefully see that all the terms with $\xi$ cancel out so that you get the original equation back again.

 

[Mentz114]

Thanks, I will try that.

Something that bothers me is that $h'_{\mu\nu}$ should be traceless, so shouldn't the gauge transformation be something like,

$h'_{\mu\nu}=h_{\mu\nu}+\partial_{(\mu}\xi_{\nu)}-\eta_{\mu\nu}\partial_\rho\xi^\rho$

 

[haushofer]

Your h' "should not be traceless"; that's a specific gauge condition you can impose (see e.g. Zwiebach's stringtheory book). This gaugechoice is not preserved by a general gauge transformation as is often the case, but imposes a condition on the gauge parameter (vector field): it should be divergenceless, as you have noticed.

What you did is to introduce a compensating transformation, which pulls you back into your gauge choice again. You should then check if the Fierz Pauli action is actually invariant under such a transformation.

You may want to check similar cases, like the Donder gauge in treating grav. waves, or choosing the Minkowski metric for the worldsheet in string theory.

 

[haushofer]

So, to be clear:

1) You have an action
2) This action is invariant under gct's (linearized)
3) To analyze e.g. the degrees of freedom of the field h you choose a specific gauge, e.g. one in which h is traceless
4) This gaugechoice breaks the gct's to a subgroup of transformations

 

[Mentz114]

So, to be clear:

1) You have an action
2) This action is invariant under gct's (linearized)
3) To analyze e.g. the degrees of freedom of the field h you choose a specific gauge, e.g. one in which h is traceless
4) This gauge choice breaks the gct's to a subgroup of transformations

[edit]

I just deleted a line because

2) is true because all tensorial contractions are invariant under gct's

So I suppose we add a condition (gauge choice) to get some physics.

Thank you for the response.

 

[haushofer]

No, that's not true. E.g., the divergence of a vector field using the partial derivative instead of the covariant one is not covariant wrt gct's. You can check this explicitly.

But yes, gauge degrees of freedom are just redundancies, so to extract physics you often have to gaugefix

 

[Mentz114]

OK. So should I have said

2) is true because all covariant tensorial contractions are invariant under gct's.

Is the Fierz-Pauli action covariant ?

I'm having trouble relating this to all the material I'm reading on 'invariance under gauge transformations' which always use the EM field and the transformation of the 4-potential. This process is making a global symmetry (invariance) into a local one.

$\vec{A} \rightarrow \vec{A} + \nabla \Lambda$
$\phi \rightarrow \phi + \partial_t \Lambda$

If the transformed potential is plugged back into the EOMs they are unchanged.

I don't see how $h'_{\mu\nu}=h_{\mu\nu}+\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$ is making a global symmetry into a local one.

 

[Ben Niehoff]

A gauge transformation in EM acts on the 4-vector potential like this:

$$A'_\mu = A_\mu + \partial_\mu \zeta.$$
Surely you can see the resemblance to

$$h'_{\mu\nu} = h_{\mu\nu} + \partial_\mu \xi_\nu + \partial_\nu \xi_\mu.$$
Although in this case you may have to think of a gauge symmetry as "a local redundancy in the degrees of freedom that simplifies the expression of the equations of motion" rather than "a global symmetry made into a local symmetry."

 

[Mentz114]

A gauge transformation in EM acts on the 4-vector potential like this:

Right.

So with $F_{\mu\nu} = \partial_\mu A_\nu-\partial_\nu A_\mu$ if we put in $A'_\mu$ the $\zeta$ terms cancel out.

What is the equivalent calculation in the F-P case ? I'll try the substitution again.

I hope I'm not looking argumentative - I need guidance.

[later]
I may have made a bad mistake. In the paper the phrase '.. surviving two derivative terms ...' is used in connection with the action. Later the author refers to 'two-derivative terms'. So I could have dropped 2 terms from the action if I mis-interpreted the 'two derivative' bit.

I think in view of this ghastly ambiguity I'm dropping this paper in protest.

I need to go to the original paper ( or maybe Carroll has something I should read).

Thanks to all respondees.

 

[Mentz114]

This paper gives much more detail and will probably set me straight

http://arxiv.org/pdf/hep-th/0606019v2.pdf

 

[dextercioby]

Refering to this paper "Theoretical Aspects of Massive Gravity" (http://arxiv.org/abs/1105.3735) about the spin-2 boson field and GR.

The author uses the Fierz-Pauli action ( I quote the massless part)

$-\frac{1}{2}\partial_\lambda h_{\mu\nu}\partial^\lambda h^{\mu\nu} + \partial_\mu h_{\nu\lambda}\partial^\nu h^{\mu\lambda}$
[...]

If you leave aside 2 terms of the Lagrangian action, you can't show the gauge symmetry anymore.

 

[Mentz114]

If you leave aside 2 terms of the Lagrangian action, you can't show the gauge symmetry anymore.

Yes, I dropped two terms. But, I was right in that the two terms I used are invariant under the 'traceless gauge' I proposed. See equations (5), (6), (7) and (8) in this

http://arxiv.org/pdf/hep-th/0606019v2.pdf

However I can see now that the full Lagrangian is invariant under the transformation $h'_{\mu\nu} = h_{\mu\nu} + \partial_\mu \xi_\nu + \partial_\nu \xi_\mu$ which is diffeomorphism.

So all my original doubts are settled. Progress at last.

Thanks again.

 

[dextercioby]

Yes, the reduction from 4 to 2 terms in the presence of a subsequent tracing condition reminds one of the reduction of the standard E-m Lagrangian density to the so-called "Fermi form', if the Lorenz gauge is imposed from the start.

 

[Mentz114]

This is from the first paper I cited

The real underlying principle of GR has nothing to do with coordinate invariance
or equivalence principles or geometry, rather it is the statement: general relativity is the
theory of a non-trivially interacting massless helicity 2 particle. The other properties are
consequences of this statement, and the implication cannot be reversed.
(my bold emphasis)

Is this graviton-graviton interaction ?

If this is the case, are there interaction terms in the Lagrangian ?

If so, I take that to mean an interchange of something between gravitons, or possibly gravitons changing into other gravitons.

To get sourced gravitons do we need a source term in the Lagrangian ?

 

[dextercioby]

Yes, GR is perturbatively an infinite sum of graviton-graviton interaction terms. Yes, gravitons can self-couple, but there is no Yang-Mills version of GR, as shown by Henneaux et al. here: N. Boulanger, T. Damour, L. Gualtieri, M. Henneaux, Inconsistency of
interacting, multi-graviton theories, Nucl. Phys. B597 (2001) 127–171

 

[haushofer]

OK. So should I have said

2) is true because all covariant tensorial contractions are invariant under gct's.

Is the Fierz-Pauli action covariant ?

If you see a tensor, you should always ask the question "tensor under WHAT transformations?". In this case you have linearized gct's; linearized, because there is only a partial derivative in the transformation of the metric perturbation h, and not a covariant one. In this case: Yes, the FP-action is covariant wrt these linearized gct's, but you should check that explicitly by plugging the transformation into the action and see that you get at most total derivatives in the Lagrangian.

Another example: the Newtonian potential is often called a scalar, but it is only a scalar under the Galilei group. Most certainly NOT under gct's! Otherwise we wouldn't have a Newtonian form of the equivalence principle. A related example is the connection, which is not a tensor under gct's, but it is a tensor under the Poincare group.

 

[Mentz114]

Yes, GR is perturbatively an infinite sum of graviton-graviton interaction terms. Yes, gravitons can self-couple, but there is no Yang-Mills version of GR, as shown by Henneaux et al. here: N. Boulanger, T. Damour, L. Gualtieri, M. Henneaux, Inconsistency of
interacting, multi-graviton theories, Nucl. Phys. B597 (2001) 127–171

Interesting.

It seems the summing of a series is not required, as Deser says

We now derive the full Einstein equations, on the basis of the same self-coupling requirement,
but with the advantages that the full theory emerges in closed form with just one added (cubic)
term, rather than as an infinite series,

I don't know if this is important but it makes it easier for me to understand the theory.

 

[PeterDonis]

GR is perturbatively an infinite sum of graviton-graviton interaction terms.

There's another wrinkle here, too, though. Suppose I start with a spin-2 field theory on flat Minkowski spacetime, and compute this perturbative sum. As mentioned here, Deser showed that this infinite sum actually converges to the Einstein-Hilbert action. But many spacetimes described by this action do not have the same topology as flat Minkowski spacetime; for example, Schwarzschild spacetime has topology $R^2 \times S^2$, not $R^4$. So viewing GR this way requires one to believe that summing the infinite perturbation series can change the topology of the underlying spacetime in which the fields are defined, which doesn't seem right; or, alternatively, one must believe that this view of GR is only valid for solutions that have the same topology as the underlying spacetime, which leaves out many important solutions.

 

[dextercioby]

But perturbative methods are not simple approximations. They neglect the full topology, they are only 1st order variations of the flat spacetime metric. That way the full nonlinear theory becomes linear. The result of Henneaux et al. merely states that the only way a spin 2 field can self-couple is the perturbative Hilbert-Einstein action, plus the important fact that there's only 1 type of gravitons.

 

[PeterDonis]

perturbative methods are not simple approximations. They neglect the full topology, they are only 1st order variations of the flat spacetime metric.

No, that's not correct. Full perturbation theory can be carried to any order you like; there is no restriction to only first-order terms in the perturbative series. You can even, as Deser did, show that an infinite sum to all orders in a particular perturbation series can in fact be expressed in closed form.

 

[stevendaryl]

There's another wrinkler here, too, though. Suppose I start with a spin-2 field theory on flat Minkowski spacetime, and compute this perturbative sum. As mentioned here, Deser showed that this infinite sum actually converges to the Einstein-Hilbert action. But many spacetimes described by this action do not have the same topology as flat Minkowski spacetime; for example, Schwarzschild spacetime has topology $R^2 \times S^2$, not $R^4$. So viewing GR this way requires one to believe that summing the infinite perturbation series can change the topology of the underlying spacetime in which the fields are defined, which doesn't seem right; or, alternatively, one must believe that this view of GR is only valid for solutions that have the same topology as the underlying spacetime, which leaves out many important solutions.

Just thinking out loud here: Given an arbitrary topology, you can break things up into little overlapping patches that are topologically little open sets of $R^4$. Then I think that solving GR for the original topology is equivalent to solving the spin-2 field equations in each patch, together with the constraint that the solutions must agree (up to a coordinate transformation) on the overlap.

 

[Ben Niehoff]

Just thinking out loud here: Given an arbitrary topology, you can break things up into little overlapping patches that are topologically little open sets of $R^4$. Then I think that solving GR for the original topology is equivalent to solving the spin-2 field equations in each patch, together with the constraint that the solutions must agree (up to a coordinate transformation) on the overlap.

Yes, but that process does not allow you to think of gravity as a spin 2 field propagating on a background (global) Minkowski space.

Also, even locally you will have problems. In regions of high curvature, you will certainly notice something wonky happens with rulers, clocks, and neighboring parallel geodesics. You could argue that the interactions with the spin 2 graviton bring about those effects, sure. But if it looks like geometry, it's certainly easier to describe it as geometry.

 

[martinbn]

Just one more thought. If you look locally and try to patch things up, there might be a problem if sequences of patches become disjoint and then meet again. Similarly as having more than one path between two diferent points. The topology may cause trouble.

My question is, which might be off topic, in what sense are the usuall formulation of GR and the spin-2 field equivalent?

 

[Ben Niehoff]

Just one more thought. If you look locally and try to patch things up, there might be a problem if sequences of patches become disjoint and then meet again. Similarly as having more than one path between two diferent points. The topology may cause trouble.

My question is, which might be off topic, in what sense are the usuall formulation of GR and the spin-2 field equivalent?

Depends which spin-2 theory you mean. The truncated, linearized one? Obviously not. But if, as someone mentioned earlier, one can reproduce the Einstein-Hilbert action by summing this infinite series of graviton interactions, then they are the same theory, in patches.

 

[martinbn]

Yes, in patches. But what about globally?

 

[PeterDonis]

what about globally?

They are the same theory globally only in spacetimes that have the same topology as Minkowski spacetime. As I noted in a previous post, this leaves out many important solutions.

 

[martinbn]

They are the same theory globally only in spacetimes that have the same topology as Minkowski spacetime. As I noted in a previous post, this leaves out many important solutions.

Which means that strictly speaking they are not equivalent.

 

[Mentz114]

Which means that strictly speaking they are not equivalent.

Good point.

 

[stevendaryl]

Which means that strictly speaking they are not equivalent.

Definitely, spin-2 field theory on top of Minkowsky spacetime can't produce GR on some manifold with a different topology. However, it seems possible to me that you could do spin-2 field theory on top of some other manifold, other than Minkowsky spacetime.

But now that I think about it, I'm not sure how that would work. It would be fine for topologies that are flat (for instance, a torus). But if the topology doesn't admit a metric with zero curvature, then it seems that you would have to do something like this:

1. Start with any old metric that is compatible with the topology.
2. Do spin-2 field theory as a perturbation of that metric (instead of as a perturbation of the minkowsky metric).

This would be kind of a weird hodge-podge, though, because some part of the curvature would be "background curvature" due to the background metric, and part of the curvature would be due to the spin-2 field theory, even though only the sum of the two would have any physical significance.

 

[PeterDonis]

it seems possible to me that you could do spin-2 field theory on top of some other manifold, other than Minkowsky spacetime.

Yes, but as you point out, this would be a sort of hodgepodge. This type of theory could not give a complete explanation, for example, of how the Schwarzschild geometry is produced by a spin-2 field, because the only way to construct a theory of a spin-2 field that produces Schwarzschild geometry would be to put in, at a minimum, the $R^2 \times S^2$ topology by hand, and do the spin-2 field theory on some "background" manifold with that topology.

 

[Mentz114]

Yes, but as you point out, this would be a sort of hodgepodge. This type of theory could not give a complete explanation, for example, of how the Schwarzschild geometry is produced by a spin-2 field, because the only way to construct a theory of a spin-2 field that produces Schwarzschild geometry would be to put in, at a minimum, the $R^2 \times S^2$ topology by hand, and do the spin-2 field theory on some "background" manifold with that topology.

Is this what happens between equations (3.20) and (3.22) in http://arxiv.org/abs/1105.3735 ?

This is the solution for the massless graviton field with a (point ?) source.

 

[Haelfix]

I don't see why it would be a 'hodgepodge' and is done all the time in the literature. If you couldn't do spin 2 theory in the Schwarzschild geometry, people would have abandoned the approach a long time ago. Weinberg wrote an entire textbook dealing with the subject nearly 50 years ago.

The real problem is when you work in an initial value formulation where one works in the standard field theory linearized approximation, and then you ask what the physics is like in a situation where you might have topology changing physics (say by bringing in very large amounts of stress energy inwards from infinity). There you are stuck in a mess that really requires a step by step iteration where each patch is regulated and checked for self consistency. I have never seen such a calculation done in the context of pure GR.

 

[PeterDonis]

Weinberg wrote an entire textbook dealing with the subject nearly 50 years ago.

As I understand it, Weinberg's textbook, like all of the treatments of this subject in the 1960s and 1970s, deals with a spin-2 field theory on a flat spacetime background, not a Schwarzschild spacetime background. Yes, they derive results showing that you can model a portion of Schwarzschild spacetime using this approach, but only a portion. You cannot possibly model the entire Schwarzschild spacetime this way because, as noted, the topology is different; and AFAIK Weinberg's text does not claim you can.

 

[Mentz114]

I don't see why it would be a 'hodgepodge' and is done all the time in the literature. If you couldn't do spin 2 theory in the Schwarzschild geometry, people would have abandoned the approach a long time ago. Weinberg wrote an entire textbook dealing with the subject nearly 50 years ago.

The real problem is when you work in an initial value formulation where one works in the standard field theory linearized approximation, and then you ask what the physics is like in a situation where you might have topology changing physics (say by bringing in very large amounts of stress energy inwards from infinity). There you are stuck in a mess that really requires a step by step iteration where each patch is regulated and checked for self consistency. I have never seen such a calculation done in the context of pure GR.

Is it meaningful to talk about the topology of the local frame carried along a curve ( tangent space ) ?