Bimetric gravity

Bimetric gravity or bigravity refers to two different classes of theories. The first class of theories relies on modified mathematical theories of gravity (or gravitation) in which two metric tensors are used instead of one.^[1][2] The second metric may be introduced at high energies, with the implication that the speed of light could be energy-dependent, enabling models with a variable speed of light.

If the two metrics are dynamical and interact, a first possibility implies two

On the contrary, the second class of bimetric gravity theories does not rely on massive gravitons and does not modify

Rosen's bigravity (1940 to 1989)

In

Einstein's field equation

is then used to calculate the form of the metric based on the distribution of energy and momentum.

In 1940, Rosen[1]^[2] proposed that at each point of space-time, there is a Euclidean metric tensor ${\displaystyle \gamma _{ij}}$ in addition to the Riemannian metric tensor ${\displaystyle g_{ij}}$. Thus at each point of space-time there are two metrics:

1. ${\displaystyle ds^{2}=g_{ij}dx^{i}dx^{j}}$
2. ${\displaystyle d\sigma ^{2}=\gamma _{ij}dx^{i}dx^{j}}$

The first metric tensor, ${\displaystyle g_{ij}}$, describes the geometry of space-time and thus the gravitational field. The second metric tensor, ${\displaystyle \gamma _{ij}}$, refers to the flat space-time and describes the inertial forces. The Christoffel symbols formed from ${\displaystyle g_{ij}}$ and ${\displaystyle \gamma _{ij}}$ are denoted by ${\displaystyle \{_{jk}^{i}\}}$ and ${\displaystyle \Gamma _{jk}^{i}}$ respectively.

Since the difference of two connections is a tensor, one can define the tensor field ${\displaystyle \Delta _{jk}^{i}}$ given by:

${\displaystyle \Delta _{jk}^{i}=\{_{jk}^{i}\}-\Gamma _{jk}^{i}}$

1

Two kinds of covariant differentiation then arise: ${\displaystyle g}$-differentiation based on ${\displaystyle g_{ij}}$ (denoted by a semicolon, e.g. ${\displaystyle X_{;a}}$), and covariant differentiation based on ${\displaystyle \gamma _{ij}}$ (denoted by a slash, e.g. ${\displaystyle X_{/a}}$). Ordinary partial derivatives are represented by a comma (e.g. ${\displaystyle X_{,a}}$). Let ${\displaystyle R_{ijk}^{h}}$ and ${\displaystyle P_{ijk}^{h}}$ be the Riemann curvature tensors calculated from ${\displaystyle g_{ij}}$ and ${\displaystyle \gamma _{ij}}$, respectively. In the above approach the curvature tensor ${\displaystyle P_{ijk}^{h}}$ is zero, since ${\displaystyle \gamma _{ij}}$ is the flat space-time metric.

A straightforward calculation yields the Riemann curvature tensor

${\displaystyle {\begin{aligned}R_{ijk}^{h}&=P_{ijk}^{h}-\Delta _{ij/k}^{h}+\Delta _{ik/j}^{h}+\Delta _{mj}^{h}\Delta _{ik}^{m}-\Delta _{mk}^{h}\Delta _{ij}^{m}\\&=-\Delta _{ij/k}^{h}+\Delta _{ik/j}^{h}+\Delta _{mj}^{h}\Delta _{ik}^{m}-\Delta _{mk}^{h}\Delta _{ij}^{m}\end{aligned}}}$

Each term on the right hand side is a tensor. It is seen that from GR one can go to the new formulation just by replacing {:} by ${\displaystyle \Delta }$ and ordinary differentiation by covariant ${\displaystyle \gamma }$-differentiation, ${\displaystyle {\sqrt {-g}}}$ by ${\displaystyle {\sqrt {\tfrac {g}{\gamma }}}}$, integration measure ${\displaystyle d^{4}x}$ by ${\displaystyle {\sqrt {-\gamma }}\,d^{4}x}$, where ${\displaystyle g=\det(g_{ij})}$, ${\displaystyle \gamma =\det(\gamma _{ij})}$ and ${\displaystyle d^{4}x=dx^{1}dx^{2}dx^{3}dx^{4}}$. Having once introduced ${\displaystyle \gamma _{ij}}$ into the theory, one has a great number of new tensors and scalars at one's disposal. One can set up other field equations other than Einstein's. It is possible that some of these will be more satisfactory for the description of nature.

The geodesic equation in bimetric relativity (BR) takes the form

${\displaystyle {\frac {d^{2}x^{i}}{ds^{2}}}+\Gamma _{jk}^{i}{\frac {dx^{j}}{ds}}{\frac {dx^{k}}{ds}}+\Delta _{jk}^{i}{\frac {dx^{j}}{ds}}{\frac {dx^{k}}{ds}}=0}$

2

It is seen from equations (1) and (2) that ${\displaystyle \Gamma }$ can be regarded as describing the inertial field because it vanishes by a suitable coordinate transformation.

Being the quantity ${\displaystyle \Delta }$ a tensor, it is independent of any coordinate system and hence may be regarded as describing the permanent gravitational field.

Rosen (1973) has found BR satisfying the covariance and equivalence principle. In 1966, Rosen showed that the introduction of the space metric into the framework of general relativity not only enables one to get the energy momentum density tensor of the gravitational field, but also enables one to obtain this tensor from a variational principle. The field equations of BR derived from the variational principle are

${\displaystyle K_{j}^{i}=N_{j}^{i}-{\frac {1}{2}}\delta _{j}^{i}N=-8\pi \kappa T_{j}^{i}}$

3

where

${\displaystyle N_{j}^{i}={\frac {1}{2}}\gamma ^{\alpha \beta }(g^{hi}g_{hj/\alpha })_{/\beta }}$

or

${\displaystyle {\begin{aligned}N_{j}^{i}&={\frac {1}{2}}\gamma ^{\alpha \beta }\left\{\left(g^{hi}g_{hj,\alpha }\right)_{,\beta }-\left(g^{hi}g_{mj}\Gamma _{h\alpha }^{m}\right)_{,\beta }-\gamma ^{\alpha \beta }\left(\Gamma _{j\alpha }^{i}\right)_{,\beta }+\Gamma _{\lambda \beta }^{i}\left[g^{h\lambda }g_{hj,\alpha }-g^{h\lambda }g_{mj}\Gamma _{h\alpha }^{m}-\Gamma _{j\alpha }^{\lambda }\right]-\right.\\&\qquad \Gamma _{j\beta }^{\lambda }\left[g^{hi}g_{h\lambda ,\alpha }-g^{hi}g_{m\lambda }\Gamma _{h\alpha }^{m}-\Gamma _{\lambda \alpha }^{i}\right]+\Gamma _{\alpha \beta }^{\lambda }\left.\left[g^{hi}g_{hj,\lambda }-g^{hi}g_{mj}\Gamma _{h\lambda }^{m}-\Gamma _{j\lambda }^{i}\right]\right\}\end{aligned}}}$

with

${\displaystyle N=g^{ij}N_{ij}}$, ${\displaystyle \kappa ={\sqrt {\frac {g}{\gamma }}}}$

and ${\displaystyle T_{j}^{i}}$ is the energy-momentum tensor.

The variational principle also leads to the relation

${\displaystyle T_{j;i}^{i}=0}$.

Hence from (3)

${\displaystyle K_{j;i}^{i}=0}$,

which implies that in a BR, a test particle in a gravitational field moves on a geodesic with respect to ${\displaystyle g_{ij}.}$

Rosen continued improving his bimetric gravity theory with additional publications in 1978^[18] and 1980,^[19] in which he made an attempt "to remove singularities arising in general relativity by modifying it so as to take into account the existence of a fundamental rest frame in the universe." In 1985^[20] Rosen tried again to remove singularities and pseudo-tensors from General Relativity. Twice in 1989 with publications in March^[21] and November^[22] Rosen further developed his concept of elementary particles in a bimetric field of General Relativity.

It is found that the BR and GR theories differ in the following cases:

• propagation of electromagnetic waves
• the external field of a high density star
• the behaviour of intense gravitational waves propagating through a strong static gravitational field.

The predictions of gravitational radiation in Rosen's theory have been shown since 1992 to be in conflict with observations of the

Massive bigravity

Since 2010 there has been renewed interest in bigravity after the development by Claudia de Rham, Gregory Gabadadze, and Andrew Tolley (dRGT) of a healthy theory of massive gravity.^[23] Massive gravity is a bimetric theory in the sense that nontrivial interaction terms for the metric ${\displaystyle g_{\mu \nu }}$ can only be written down with the help of a second metric, as the only nonderivative term that can be written using one metric is a cosmological constant. In the dRGT theory, a nondynamical "reference metric" ${\displaystyle f_{\mu \nu }}$ is introduced, and the interaction terms are built out of the

of ${\displaystyle g^{-1}f}$.

In dRGT massive gravity, the reference metric must be specified by hand. One can give the reference metric an Einstein–Hilbert term, in which case ${\displaystyle f_{\mu \nu }}$ is not chosen but instead evolves dynamically in response to ${\displaystyle g_{\mu \nu }}$ and possibly matter. This massive bigravity was introduced by Fawad Hassan and Rachel Rosen as an extension of dRGT massive gravity.^[3][24]

The dRGT theory is crucial to developing a theory with two dynamical metrics because general bimetric theories are plagued by the Boulware–Deser ghost, a possible sixth polarization for a massive graviton.^[25] The dRGT potential is constructed specifically to render this ghost nondynamical, and as long as the kinetic term for the second metric is of the Einstein–Hilbert form, the resulting theory remains ghost-free.^[3]

The action for the ghost-free massive bigravity is given by^[26]

${\displaystyle S=-{\frac {M_{g}^{2}}{2}}\int d^{4}x{\sqrt {-g}}R(g)-{\frac {M_{f}^{2}}{2}}\int d^{4}x{\sqrt {-f}}R(f)+m^{2}M_{g}^{2}\int d^{4}x{\sqrt {-g}}\displaystyle \sum _{n=0}^{4}\beta _{n}e_{n}(\mathbb {X} )+\int d^{4}x{\sqrt {-g}}{\mathcal {L}}_{\mathrm {m} }(g,\Phi _{i}).}$

As in standard general relativity, the metric ${\displaystyle g_{\mu \nu }}$ has an Einstein–Hilbert kinetic term proportional to the

${\displaystyle R(g)}$ and a minimal coupling to the matter Lagrangian ${\displaystyle {\mathcal {L}}_{\mathrm {m} }}$, with ${\displaystyle \Phi _{i}}$ representing all of the matter fields, such as those of the Standard Model. An Einstein–Hilbert term is also given for ${\displaystyle f_{\mu \nu }}$. Each metric has its own , denoted ${\displaystyle M_{g}}$ and ${\displaystyle M_{f}}$ respectively. The interaction potential is the same as in dRGT massive gravity. The ${\displaystyle \beta _{i}}$ are dimensionless coupling constants and ${\displaystyle m}$ (or specifically ${\displaystyle \beta _{i}^{1/2}m}$) is related to the mass of the massive graviton. This theory propagates seven degrees of freedom, corresponding to a massless graviton and a massive graviton (although the massive and massless states do not align with either of the metrics).

The interaction potential is built out of the

${\displaystyle e_{n}}$ of the eigenvalues of the matrices ${\displaystyle \mathbb {K} =\mathbb {I} -{\sqrt {g^{-1}f}}}$ or ${\displaystyle \mathbb {X} ={\sqrt {g^{-1}f}}}$, parametrized by dimensionless coupling constants ${\displaystyle \alpha _{i}}$ or ${\displaystyle \beta _{i}}$, respectively. Here ${\displaystyle {\sqrt {g^{-1}f}}}$ is the of the matrix ${\displaystyle g^{-1}f}$. Written in index notation, ${\displaystyle \mathbb {X} }$ is defined by the relation

${\displaystyle X^{\mu }{}_{\alpha }X^{\alpha }{}_{\nu }=g^{\mu \alpha }f_{\nu \alpha }.}$

The ${\displaystyle e_{n}}$ can be written directly in terms of ${\displaystyle \mathbb {X} }$ as

${\displaystyle {\begin{aligned}e_{0}(\mathbb {X} )&=1,\\e_{1}(\mathbb {X} )&=[\mathbb {X} ],\\e_{2}(\mathbb {X} )&={\frac {1}{2}}\left([\mathbb {X} ]^{2}-[\mathbb {X} ^{2}]\right),\\e_{3}(\mathbb {X} )&={\frac {1}{6}}\left([\mathbb {X} ]^{3}-3[\mathbb {X} ][\mathbb {X} ^{2}]+2[\mathbb {X} ^{3}]\right),\\e_{4}(\mathbb {X} )&=\operatorname {det} \mathbb {X} ,\end{aligned}}}$

where brackets indicate a trace, ${\displaystyle [\mathbb {X} ]\equiv X^{\mu }{}_{\mu }}$. It is the particular antisymmetric combination of terms in each of the ${\displaystyle e_{n}}$ which is responsible for rendering the Boulware–Deser ghost nondynamical.

• Alternatives to general relativity
• DGP model
• Scalar–tensor theory