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

[PeterDonis]

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

Yes, in the trivial sense that it's always the same--the tangent space at any given point is always a Minkowski space and always has topology $R^4$. (Note that tangent spaces do not get "carried along curves"; the best you can do is construct a map, using the connection, between the tangent space at one point and the tangent space at another point. But in general in a curved spacetime, this map will not be unique.)

 

[stevendaryl]

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.

On the other hand, how is topology derived in standard GR? It seems to me that it isn't. The field equations only describe how the metric tensor varies locally, while the topology is a global property. So how, in general, do you get the topology?

 

[PeterDonis]

how is topology derived in standard GR? It seems to me that it isn't.

I guess it depends on what you mean by "derived". Take the Schwarzschild solution as an example. If I just look at this solution locally, a small patch of it looks like a small patch of Minkowski spacetime, and I can think of the patch as having topology $R^4$. But if I try to find its maximal global extension, I find that that maximal extension has topology $R^2 \times S^2$, not $R^4$. To me that counts as a "derivation" of the global topology.

 

[Haelfix]

I think that its exactly the same thing in the spin 2 case as well, the only difference is that if you truncate the series, you will have a mismatch on the junction conditions in the overlap and you won't necessarily be able to derive the correct global topology. However if you sum the infinite series and bootstrap the resulting field equations then you have the powerful theorems by Deser that guarantees that the overlap conditions generates the correct conditions and thus the correct topology.

 

[PeterDonis]

you have the powerful theorems by Deser that guarantees that the overlap conditions generates the correct conditions and thus the correct topology

But when the original spin-2 field theory is set up, it's on flat Minkowski spacetime, which already assumes a global topology. If you end up deriving a solution with a different global topology, then you've derived a contradiction; you've shown that the spin-2 field model can't consistently generate that solution, because it has a topology that's inconsistent with the original assumption on which the solution is based.

 

[Haelfix]

Yes but they are two different manifolds with different physical interpretations.. The flat Minkowski spacetime is an unobservable artifact that was utilized as a crutch to set up coordinates in the initial stages of the calculation but it is effectively replaced by the new effective perturbatively generated manifold that the formalism spits out. I don't think there is a contradiction there at least at the level of the mathematics. For instance if you ask the physical question, can light rays return to the same spot in a spatially closed FRW solution then the answer in both formalisms is unambiguous.

 

[PeterDonis]

The flat Minkowski spacetime is an unobservable artifact that was utilized as a crutch to set up coordinates in the initial stages of the calculation but it is effectively replaced by the new effective perturbatively generated manifold that the formalism spits out. I don't think there is a contradiction there at least at the level of the mathematics.

I don't understand how there can't be a contradiction mathematically since the coordinates in which the spin-2 field is initially defined are on a manifold with one topology, but the coordinates in which the final solution is expressed are on a manifold with different topology. You can't even construct a map between the two charts, so how can you possibly show that they are consistent with each other?

 

[Ben Niehoff]

I don't think it's a "contradiction" if you are allowed to glue together several Minkowski spaces. So I guess it depends on how you define your original field theory. But I should think that if you've built in the notion that you can glue together several copies of the coordinate space via certain kinds of overlap conditions on your spin-2 field, then you've already anticipated that this spin-2 field actually describes geometry.

If we take the naive view that physical space is flat and fields propagate in it, then one can never build proper GR out of a spin-2 field.

 

[martinbn]

That's very interesting but also a bit confusing. Is there an explicit example?

 

[haushofer]

I don't have an answer to this topology issue, but I guess you encounter the same issue when claiming that GR is derivable from String Theory (apart from the problem of having time dependent background from a worldsheet point of view ) and viewing a solution to the graviton eoms as a coherent state of gravitonic oscillations.

 

[stevendaryl]

I'm still a little puzzled about using topology as a proof that GR is inequivalent to spin-2 field theory. The argument is that spin-2 field theory on top of flat Minkowsky spacetime will never produce a solution such as the Schwarzschild solution, which has a different topology. I think that's true, but I'm not sure what's supposed to follow from that. What seems to me to be the case is that there is a connection between field theory and topology, regardless of whether that field theory is about the metric or something else.

For instance, let's turn from GR to seemingly simpler electromagnetism.

In Minkowsky spacetime, you can have an arbitrary charge distribution at a given time. But if we change the topology, to make space into a hypersphere, instead of euclidean 3D space, we can still have a theory of electromagnetism, and locally it will look the same as the usual electromagnetism. But there will be one difference: the total charge on a hypersphere must be zero. You can prove that by Gauss' law: the electric flux through any closed surface is equal to the charge enclosed. But on a hypersphere, any closed surface splits the universe into two parts, and the surface can equally well be considered to enclose either one. So the flux through the surface can equally well be considered to be the flux due to the enclosed charge, and also the flux due to the rest of the universe (with the opposite sign). So any collection of charges has to be equal and opposite from the charges of the rest of the universe. So topology imposes constraints on electromagnetism.

If we go from electromagnetism, with an associated spin-1 field, to spin-2 field theory, topology would still impose constraints on the spin-2 field theory. In the same way that only certain charge distributions are consistent with certain topologies, only certain mass/energy distributions are consistent with certain topologies. (But interestingly, the constraint is the opposite--nontrivial topologies may force the mass/energy density to be nonzero.

 

[martinbn]

Can anyone give references for the spin-2 field theory, which have examples? To me it is unclear how exactly it works. By which I don't mean to just get the Hilbert action but to actually do some calculations. As already asked, how is the Schwarzschild solution described? In what sense is it singular and how is the event horizon described, how are the usual questions answered ect.?

 

[Mentz114]

I started this topic because I could not see how the Lagrangian in question could be shown to be invariant under the transformation and hoped there was an easy way. It turns out fairly simply. There are several terms like this

$2\partial_\lambda \phi_{\mu\nu} \partial^\lambda \partial^\mu X^\nu + 2\partial_\lambda \phi_{\mu\nu} \partial^\lambda \partial^\nu X^\mu$

which can be factored into ( I worked this out from a big clue in one of the papers I was reading)

$2\partial^\lambda \partial_\lambda \phi_{\mu\nu} \left( \partial^\mu X^\nu + \partial^\nu X^\mu \right)=2\delta\phi^{\mu\nu}\ \square \phi_{\mu\nu}$

This looks like the harmonic gauge condition. The terms purely in $X$ (the gauge parameter) give rise to terms like

$2\partial_\lambda \partial_\mu X_\nu \partial^\lambda \partial^\mu X^\nu + 2\partial_\lambda \partial_\mu X_\nu \partial^\lambda \partial^\nu X^\mu$

which go to

$2\delta\phi^{\mu\nu}\ \square \left( \partial_\mu X\nu \right) = 2\delta\phi^{\mu\nu}\ Tr(\partial_\mu X_\nu)$

which could be the traceless gauge condition. $\square$ is the Laplace–Beltrami operator ( or generalised Laplacian).

Combining all the terms cancels these conditions leaving just the diffeomorphism invariance $\delta\phi_{\mu\nu}=\partial_\mu X_\nu + \partial_\nu X_\mu$.

Various combinations of the Lagrangian terms can give Lagrangians that are free of either condition, but it takes all four terms for the strike out.

I have a question about this phrase 'The first two terms are all that is needed for gravitons to propagate.'
I thought it is the field that propagates and the vector bosons do the transfer of whatever is transferred in quanta. Maybe the phrase refers to the weak field radiation mode ?