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As weird as it sounds, yes, there are gravity theories with more than one metric tensor. This is called bimetric gravity.

My question to those who have encountered bimetric gravity before:

a) Bimetric gravity was shown to be inconsistent with the data from the Hulse-Taylor pulsar due to contradictory predictions with respect to gravitational waves. Have there been any analyses done on the recent data to this end? I have not been able to find a good resource which tells me exactly what measurable quantity bimetric gravity (BR) can predict and in what way it differs from general relativity (GR).

b) Mathematically, how would one try to understand the construction of a manifold with more than one metric? (and I mean very mathematically - in terms of how to construct the basis for the manifold, in terms of a natural definition of homeomorphisms etc.)

c) What exactly was the physical intuition behind introducing a second metric tensor?

d) Although the wikipedia page suggests that the two metrics have different functionalities, a recent talk I heard at a conference (I don't want to link the talk here because the person also did not seem to understand some of the finer details of BR despite publications, such as a few of the questions asked above.) about BR claimed that the two metrics have roles which are determined by the energy scale i.e. there could be an energy scale at which a normally suppressed metric takes over. Is there any weight to this claim? If so, what is the coupling of this metric? And how does it affect $G_N$?

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(a) Do you have a source in mind for that claim? I think it's a little model dependent exactly what the constraints are. But in ghost-free bigravity (the one you link to on wikipedia), you can set things up where only one metric couples to matter. In that case, insofar as people have been able to compute the answer, the limit from binary pulsars are actually quite weak, because of the Vainshtein screening mechanism.

(b) A nice way to understand this is in terms of Stuckelberg fields. This is described for example in http://arxiv.org/abs/hep-th/0210184. There are different ways of introducing the Stuckelberg fields (so there are different ways of describing the same thing at a mathematical level). Here is one way: imagine you have 2 different manifolds, $M_1$ and $M_2$. Each manifold is endowed with a metric $g_{1,\mu\nu}$ and $g_{2,\mu\nu}$ and with coordinates (say $x_1$ and $x_2$). Then the Stuckelberg fields are (invertible) maps that go from $M_1$ to $M_2$, for example $x_1^\mu = \phi^\mu(x_2)$. Then you can use $\phi$ to map $g_{1,\mu\nu}(x_1)$ from $M_1$ to $M_2$ by \begin{equation} \tilde{g}_{1,\mu\nu}(x_2)=\frac{\partial \phi^\alpha}{\partial x_2^\mu}\frac{\partial \phi^\beta}{\partial x_2^\nu}g_{1,\alpha \beta}(\phi(x_2)) \end{equation} Given this map, you can then write interactions between the two metrics like $g_{2,\mu\nu}(x_2)\tilde{g}^{\mu\nu}_{1}(x_2)$, and now both fields are living on the same coordinates.

There are potential issues with this procedure, for example if $M_1$ and $M_2$ have different topologies, but let's that how things work out in the ideal case.

(c) The physical intuition mostly comes from thinking about the propagating degrees of freedom--perturbatively around a Lorentz invariant background (the simple case is that both metrics are Minkwoski), bimetric theory describes one massless spin-2 and one massive spin-2 particle. Around other backgrounds the notion of spin gets more complicated, but around FRW say you are describing the two normal massless tensor modes of GR, plus additional massive modes (2 tensor, 2 vector, 1 scalar) associated with a massive spin-2.

An important subtlety is that the massive and massless spin-2 degrees of freedom can't be associated with an individual metric--you can't say that metric number 1 carries the massless spin-2 and metric number 2 carries the massive spin-2. You will always see 7 degrees of freedom, but exactly how those degrees of freedom gets distributed among the different metrics depends on the background and on the parameters of the theory.

Geometrically it's a little obscure to think about two metrics, maybe you could say you have two spacetimes that interact with each other through Stuckelberg fields. The original motivation in that Arkhani-Hamed et al paper I linked to is to a procedure called deconstruction--you think of discretizing a continuous dimension into two 'sites', each site has a metric and the sites interact through 'linking fields'. The mass depends on the discretization scale, and the interactions depend on the specific procedure you choose to discretize the derivative along the discretized direction.

(d) I'm not exactly sure what is being asked here so I'll just make a few comments.

In what we might call the 'standard' setup, you have one metric that directly couples to matter and the second metric is (if you like the second metric lives in a 'dark sector'). In that case, the metric that couples to matter serves as the metric that measures distances, etc. The other metric carries the 'dark' degrees of freedom that do not directly couple to matter.

As was said above, the distribution of the massive and massless degrees of freedom between the two metrics can change with time. In particle physics terms, the metrics are basically the 'interaction basis' and the massive/massless modes are the 'mass basis,' and mass eigenstates can be a combination of interaction eigenstates. The particular combination can depend on the background, or on energy scale.

In bigravity there are basically three dimensionful parameters: the planck mass / newton's constant for $g_1$, the planck mass / newton's constant for $g_2$, and a mass parameter that describes the coupling between these two metrics (and the mass of the massive spin-2 dofs). In bi-gravity the actual, physical newton's constant that we measure with, say, solar system experiments is actually a combination of the two newton's constants. The particular combination depends on the way you couple matter. But, if you couple matter to $g_1$ only, then the observed newton's constant would just be the newton's constant for $g_1$.

A special limit of the theory is when you make the planck mass of one of the metrics very large (in particular $g_2$, the metric which does not couple to matter). In this case $g_2$ essentially becomes frozen (making the planck mass for $g_2$ large gives $g_2$ a large inertia) and its fluctuations decouple. In that limit, bigravity reduces to massive gravity, a theory of a single massive spin-2 particle.

Bi-gravity can get very complicated but many of the complications about what the degrees of freedom mean also appear in the standard model, at least in spirit. Physically, bi-gravity is like having two 'generations' of spin-2 particles, one massive and one massless, with mixing: the mass eigenstates and the interaction eigenstates are not the same.

(Hopefully I'm not the person in your last paragraph!)

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