# Bimetric gravity

(Redirected from Bimetric theory)

Bimetric gravity or bigravity refers to two different class 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] 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 graviton modes, one massive and one massless; such bimetric theories are then closely related to massive gravity.[2] Several bimetric theories with massive gravitons exist, such as those attributed to Nathan Rosen (1909–1995)[3][4][5] or Mordehai Milgrom with Modified Newtonian Dynamics (MOND). More recently, developments in massive gravity have also led to new consistent theories of bimetric gravity.[6] Though none has been shown to account for physical observations more accurately or more consistently than the theory of general relativity, Rosen's theory has been shown to be inconsistent with observations of the Hulse–Taylor binary pulsar.[4] Some of these theories lead to cosmic acceleration at late times and are therefore alternatives to dark energy.[7][8]

On the contrary, the second class of bimetric gravity theories does not rely on massive gravitons and does not modify Newton's law, but instead describes the universe as a manifold having two coupled Riemannian metrics, where matter populating the two sectors interact through gravitation (and antigravitation if the topology and the Newtonian approximation considered introduce negative mass and negative energy states in cosmology as an alternative to dark matter and dark energy). Some of these cosmological models also use a variable speed of light in the high energy density state of the radiation-dominated era of the universe, challenging the inflation hypothesis.[9][10][11][12]. The Janus cosmological model introduced by Jean-Pierre Petit belongs to this class.

## Rosen's bigravity (1940)

In general relativity (GR), it is assumed that the distance between two points in spacetime is given by the metric tensor. Einstein's field equation is then used to calculate the form of the metric based on the distribution of energy and momentum.

Rosen (1940)[13][14] has 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:

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

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:

${\displaystyle \Delta _{jk}^{i}=\{_{jk}^{i}\}-\Gamma _{jk}^{i}\qquad (1)}$

Two kinds of covariant differentiation then arise: ${\displaystyle g}$-differentiation based on ${\displaystyle g_{ij}}$ (denoted by a semicolon), and covariant differentiation based on ${\displaystyle \gamma _{ij}}$ (denoted by a slash). Ordinary partial derivatives are represented by a comma. 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

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

Each term on 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 ${\displaystyle g=\det(g_{ij}),\gamma =det(\gamma _{ij})}$ 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

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

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.

The quantity ${\displaystyle \Delta ,}$ being a tensor, 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

${\displaystyle K_{j}^{i}=N_{j}^{i}-{\frac {1}{2}}\delta _{j}^{i}N=-8\pi \kappa T_{j}^{i}\qquad (3)}$

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

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 ${\displaystyle g_{ij}.}$

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 to be in conflict with observations of the Hulse–Taylor binary pulsar.[4]

## Janus cosmological model

From 1977, Petit starts to build an atypical bimetric theory of gravity called the Janus cosmological model in reference to the two-faced god who "looks simultaneously to the future and to the past".[15] Petit produces science comics and videos to popularize the various aspects of this cosmological model.[16][17][18][19]

Previously known as the twin universe theory, it would explain various observational facts that the standard model cannot answer, the gravitational interaction of positive and negative masses being an alternative candidate for the explanation of dark matter, dark energy, cosmic inflation and the accelerating expansion of the universe.[20] Despite being peer reviewed, this non-standard cosmological model has not triggered much interest in the scientific community throughout the years, except with mathematicians and geometers who seem more interested than cosmologists in its topological subtleties.[21][22][23][24]

However, in particle physics, the theory shares similarities with the mirror matter of hidden sectors addressing CP violation.[25][26][27] In general relativity, later independent work about bimetric gravity with positive and negative masses lead to the same conclusions regarding the laws of gravitation.[9][10][11]

2D didactic image of Sakharov's twin universe model.

The Janus model has the same foundation as a model previously published by Andrei Sakharov ten years before.[28] In 1967, Sakharov addressed the baryon asymmetry of the universe considering for the first time events in CPT symmetry occurring before the Big Bang:

Sakharov was the first scientist to introduce twin universes he called "sheets". He achieved a complete CPT symmetry since the second sheet is populated by invisible "shadow matter" which is antimatter (C-symmetry) because of an opposite CP-violation there, and the two sheets are mirror of each other both in space (P-symmetry) and time (T-symmetry) through the same initial gravitational singularity. He continued developing this idea for twenty years.[30][31][32][33][34][35][36]

2D didactic image of Janus model.

Ignoring the prior existence of this work translated in a book only fifteen years after its Russian publication,[29] Petit publishes his first paper about two enantiomorphic universes with opposite arrows of time in 1977.[37][38] Unlike Sakharov, he makes the two parallel universes interacting through gravity straightforward. In this first non-relativistic Newtonian dynamics model, galaxies are imbedded in repellent invisible negative mass, so they can be modeled as an exact solution of two Vlasov equations, coupled by Poisson's equation.

In 1994, the model is developed as a bimetric description of the universe.[39] However this bimetry is not similar to independent work done in the field of classical bimetric gravity where the second metric refers to gravitons with nonzero mass. In the janus model, the bigravity is an extension of general relativity describing the universe as a Riemannian manifold associated to two conjugated metrics generating their own geodesics, solutions of two coupled Einstein field equations:[40]

${\displaystyle R_{\mu \nu }^{(+)}-{\tfrac {1}{2}}\,R^{(+)}g_{\mu \nu }^{(+)}=\chi \left(T_{\mu \nu }^{(+)}+{\sqrt {\frac {g^{(-)}}{g^{(+)}}}}T_{\mu \nu }^{(-)}\right)}$
${\displaystyle R_{\mu \nu }^{(-)}-{\tfrac {1}{2}}\,R^{(-)}g_{\mu \nu }^{(-)}=-\chi \left({\sqrt {\frac {g^{(+)}}{g^{(-)}}}}T_{\mu \nu }^{(+)}+T_{\mu \nu }^{(-)}\right)}$

Petit's system of two coupled field equations reduces to Einstein's field equations in the case of a portion of spacetime where positive mass matter dominates and no negative mass is present, like in the Solar System. Similarly to this Einsteinian approximation, the Newtonian approximation allows to recover Newton's law of universal gravitation and formula for gravitational potentials from the field equations in the limit of weak fields and low velocities with respect to the speed of light.

In yellow, the "preposterous" runaway motion of a positive and negative masses described by Bondi and Bonnor.
In green, gravitational movements in the Janus model which differ from those elaborated by Bondi and Bonnor, solving the runaway paradox.

The theory describes two parallel universes in CPT symmetry interacting through gravity, both originating from the same initial singularity. In the model, four types of matter coexist:

• positive mass matter (baryonic matter)
• positive mass antimatter (C-symmetry, the antimatter according to Dirac)
• negative mass matter (CPT symmetry)
• negative mass antimatter (C × CPT symmetry = PT-symmetry, the antimatter according to Feynman)[41]

As positive mass matter emits positive energy photons travelling along null geodesics of the metric ${\displaystyle g_{\mu \nu }^{(+)}}$, and negative mass matter emits negative energy photons travelling along null geodesics of the metric ${\displaystyle g_{\mu \nu }^{(-)}}$, the exotic matter cannot be detected with optical instruments, besides its gravitational interaction with normal matter.

The Newtonian approximation of the system of two coupled field equations provides the following gravitational interactions:

• particles of same energy attract each other according to Newton's law (positive mass attracts positive mass and negative mass attracts negative mass)
• particles of opposite energy repel each other according to "anti" Newton's law (positive mass and negative mass repel each other)

Those laws are different to the laws spelled out by Hermann Bondi and William Bonnor,[42][43] and solve the runaway paradox,[40] that usually makes scientists think negative mass can not physically exist:

Due to topological considerations, matter populating each fold appears to the other as having an opposite mass and an opposite arrow of time, although the proper time remains positive for both species.[16]

In 1995, Petit combines his bimetric model with his VSL theory into the first paper summarizing the twin universes cosmology.[44]

The main hypotheses stating that negative energy particles exist and result from time reversal, that two particles of opposite mass repel each other, and that physical constants can vary, are in opposition with the standard models of particle physics and cosmology. In quantum field theory, the T operator acting on Hilbert spaces is complex, and can be either linear and unitary, or antilinear and antiunitary; but is arbitrarily chosen antilinear and antiunitary in order to prevent inversion of energy, as the vacuum state of the Zero-point energy must have the lowest possible ground state and can not have negative values.[45] But when this axiom was formulated, the accelerating expansion of the universe, which implies a negative pressure, was not known yet. As a pressure is a volumetric energy density, Petit thinks this problem should be reconsidered.

However, in group theory, the T operator is real and can reverse the energy. Dynamics of relativistic elementary particles is described by the Poincaré group. Currently physics uses the restricted Poincaré group, with only forward in time ("orthochronous") motions. As demonstrated by Jean-Marie Souriau using the complete Poincaré group, including backward in time ("antichronous") motions, arrow of time reversal equals mass inversion of a particle.[46]

In the 2000s, Petit integrates Souriau's mathematical physics and fully geometrize his model with group theory.[47][48][22][49]

In 2014 and 2015 he publishes a set of four papers detailing the most recent developments of the Janus model. The first paper produces an exact solution to the coupled field equations referring to the matter-dominated era which resolves the runaway paradox of negative mass and challenges dark energy to account for the accelerating expansion of the universe.[40] In a second paper this is extended to two metrics with their own speed of light,[20] followed by the Lagrangian derivation of the model.[50] A fourth paper is devoted to the cancellation of the central singularity in the Schwarzschild solution, questioning the classical black hole model.[51]

A comparison of the Janus model in agreement with latest observational data has been published in 2018.[52] The same year, a paper discussed the arbitrary decision of preventing negative energy states in quantum field theory, as such negative energy is compatible with the Dirac equation when considering a unitary time-reversal operator, provided that one considers that negative energy goes with negative mass according to ${\displaystyle -m=-E/c^{2}}$, so the Klein–Gordon probability density ${\displaystyle E/m}$ in relativistic quantum mechanics remains positive.[53]

The model finally considers the possibility of apparent faster-than-light interstellar travel with limited energy. The mechanism would involve an artificial version of the black hole natural inversion mass process.[51] The transferred vehicle would cruise along geodesics of the metric ${\displaystyle g_{\mu \nu }^{(-)}}$ where the speed of light is greater, and the distances shorter. The inverted particles of the ship and its passengers would have to appear at a relativistic speed in the new frame of reference through Lorentz contraction, in order for the energy to be conserved, with no acceleration. After mass inversion, a craft would go so fast that it could not slow down, but arriving at its destination, a new mass inversion would give back its former kinetic parameters, with no deceleration.[20]

## 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.[54] 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 matrix 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 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[2] as an extension of dRGT massive gravity.

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.[55] 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.[2]

The action for the ghost-free massive bigravity is given by[56]

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

As in standard general relativity, the metric ${\displaystyle g_{\mu \nu }}$ has an Einstein-Hilbert kinetic term proportional to the Ricci scalar ${\displaystyle R(g)}$ 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 Planck mass, ${\displaystyle M_{g}}$ and ${\displaystyle M_{f}}$. 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 elementary 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 the matrix 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

{\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.

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