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

If the two metrics are dynamical and interact, a first possibility implies twograviton modes, one massive and one massless; such bimetric theories are then closely related tomassive gravity.^[3] Several bimetric theories with massive gravitons exist, such as those attributed toNathan Rosen (1909–1995)^[4][5][6] orMordehai Milgrom with relativistic extensions ofModified Newtonian Dynamics (MOND).^[7] More recently, developments in massive gravity have also led to new consistent theories of bimetric gravity.^[8] Though none has been shown to account for physical observations more accurately or more consistently than the theory ofgeneral relativity, Rosen's theory has been shown to be inconsistent with observations of theHulse–Taylor binary pulsar.^[5] Some of these theories lead tocosmic acceleration at late times and are therefore alternatives todark energy.^[9][10] Bimetric gravity is also at odds with measurements of gravitational waves emitted by the neutron-star mergerGW170817.^[11]

On the contrary, the second class of bimetric gravity theories does not rely on massive gravitons and does not modifyNewton's law, but instead describes the universe as amanifold having two coupledRiemannian metrics, where matter populating the two sectors interact through gravitation (and antigravitation if thetopology and theNewtonian approximation considered introducenegative mass andnegative energy states incosmology as an alternative todark matter and dark energy).^[12] Some of thesecosmological models also use a variable speed of light in the highenergy density state of theradiation-dominated era of the universe, challenging theinflation hypothesis.^{[13][14][15][16][17]}

Ingeneral relativity (GR), it is assumed that the distance between two points inspacetime is given by themetric tensor.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 aEuclidean metrictensor${\displaystyle {\gamma }_{ij}}$ in addition to the Riemannian metric tensor${\displaystyle g_{ij}}$. Thus at each point of space-time there are two metrics:

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

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. TheChristoffel symbols formed from${\displaystyle g_{ij}}$ and${\displaystyle {\gamma }_{ij}}$ are denoted by${\displaystyle {\{}_{jk}^i\}}$ and${\displaystyle {\mathrm{\Gamma }}_{jk}^i}$ respectively.

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

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

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 theRiemann 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 theRiemann curvature tensor

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${\displaystyle \mathrm{\Delta }}$ and ordinary differentiation by covariant${\displaystyle \gamma }$-differentiation,${\displaystyle \sqrt{-g}}$ by${\displaystyle \sqrt{\frac{g}{\gamma }}}$, 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${\displaystyle d^{4}x=d{x}^{1}d{x}^{2}d{x}^{3}d{x}^4}$. 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 inbimetric relativity (BR) takes the form

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

2

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

Being the quantity${\displaystyle \mathrm{\Delta }}$ 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

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 ageodesic 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 theHulse–Taylor binary pulsar.^[5]

Since 2010 there has been renewed interest in bigravity after the development byClaudia de Rham,Gregory Gabadadze, andAndrew 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 acosmological constant. In the dRGT theory, a nondynamical "reference metric"${\displaystyle f_{\mu \nu }}$ is introduced, and the interaction terms are built out of thematrix square root of${\displaystyle g^{-1}f}$.

In dRGT massive gravity, the reference metric must be specified by hand. One can give the reference metric anEinstein–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. Thismassive bigravity was introduced byFawad Hassan andRachel 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 theBoulware–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 thekinetic term for the second metric is of the Einstein–Hilbert form, the resulting theory remains ghost-free.^[3]

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

${\displaystyle S=-\frac{M_g^2}{2}\int d^{4}x\sqrt{-g}R(g)-\frac{M_f^2}{2}\int d^{4}x\sqrt{-f}R(f)+m^2{M}_g^2\int d^{4}x\sqrt{-g}{\displaystyle \sum _{n=0}^4{\beta }_n{e}_n(\mathbb{X})+\int d^{4}x\sqrt{-g}{\mathcal{L}}_{\mathrm{m}}(g,{\mathrm{\Phi }}_i).}}$

As in standard general relativity, the metric${\displaystyle g_{\mu \nu }}$ has an Einstein–Hilbert kinetic term proportional to theRicci scalar${\displaystyle R(g)}$ and a minimal coupling to the matter Lagrangian${\displaystyle {\mathcal{L}}_{\mathrm{m}}}$, with${\displaystyle {\mathrm{\Phi }}_i}$ representing all of the matter fields, such as those of theStandard Model. An Einstein–Hilbert term is also given for${\displaystyle f_{\mu \nu }}$. Each metric has its ownPlanck mass, 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 theelementary symmetric polynomials${\displaystyle e_n}$ 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 thematrix square root 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

where brackets indicate atrace,${\displaystyle [\mathbb{X}]\equiv X^{\mu }{}_{\mu }}$. 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

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