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Bimetric theory refers to a class of modified theories of gravity in which two metric tensors are used instead of one. Often the second metric is introduced at high energies, with the implication that the speed of light may be energy dependent.

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

See also

Rosen (1940) has proposed at each point of space-time a Euclidean metric tensor γij in addition to the Riemannian metric tensor gij . Thus at each point of space-time there are two metrics:

ds2 = gijdx idxj

dσ2 = γijdxi dxj

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

Δ are defined such that

 \Delta^{i}_{jk}=\{^{i}_{jk}\}-\Gamma^{i}_{jk}~~~~~~~~~~~~~~(1)

Now there arise two kinds of covariant differentiation:g differentiation based on gij (is denoted by semicolon (;)), 3- differentiation based on γij (denoted by a d), ordinary partial derivatives are denoted by comma ( , ). R^{\lambda}_{ij \sigma} and P^{\lambda}_{ij \sigma} be the curvature tensors calculated from gij and γij respectively. In the above approach as γij is the flat space-time metric, the curvature tensor P^{\lambda}_{ij \sigma} is zero.

From (1) one finds that though {:} and Γ are not tensors, but Δ is a tensor having the same form as {:} except that the ordinary partial derivative is replaced by 3-covariant derivative. The straightforward calculations yield

 R^{h}_{ijk}=-\Delta^{h}_{ij/k}+\Delta^{h}_{ik/j}+\Delta^{h}_{mj}\Delta^{m}_{ik}-\Delta^{h}_{mk}\Delta^{m}_{ij}

Each term on right hand side of (1.6.4) is a tensor. It is seen that from general relativity (GR) one can go to new formulation just by replacing {:} by Δ , ordinary differentiation by 3-covariant differentiation, \sqrt {-g} by \sqrt{\frac{g}{\gamma}}, in the integration d4x by \sqrt {-\gamma}d^{4}x , where g = det(gij), 3 = detij) and d4x = dx1dx2 dx3dx4. It is necessary to point out that having once introduced γij into the theory, one has a great number of new tensors and scalars at one's disposal. One can set up field equations other than Einstein's field equations. 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

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

It is seen from equation (1) and (2) that Γ can be regarded as describing the inertial field because it vanishes by a suitable coordinate transformation.

The quantity Δ being 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 has shown 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 variational principle are

 K^{i}_{j}= N^{i}_{j}-\frac{1}{2}\delta^{i}_{j}N = -8 \pi \kappa T^{i}_{j}~~~~~~~~~~~~~~(3)

where

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

or

 N^{i}_{j}= \gamma^{\alpha \beta}\left\{(g^{hi}g_{hj, \alpha}),\beta - (g^{hi}g_{mj}\Gamma^{m}_{h\alpha}),\beta\right\} -\gamma^{\alpha \beta}(\Gamma^{i}_{j\alpha}),\beta+ \Gamma^{i}_{\lambda \beta}[g^{h\lambda}g_{hj},\alpha - g^{h\lambda}g_{mj}\Gamma^{m}_{h\alpha} -\Gamma^{\lambda}_{j\alpha}]-\Gamma^{\lambda}_{j\beta}[g^{hi}g_{h\lambda},\alpha - g^{hi}g_{m\lambda}\Gamma^{m}_{h\alpha} -\Gamma^{i}_{\lambda\alpha}]

 + \Gamma^{\lambda}_{\alpha \beta}[g^{hi}g_{hj},\lambda - g^{hi}g_{mj}\Gamma^{m}_{h\lambda} -\Gamma^{i}_{j\lambda}]

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

and T^{i}_{j} = Energy momentum tensor.\\ The variational principle also leads to the relation

 T^{i}_{j;i}=0.

Hence from (3)

 K^{i}_{j;i}=0,

which implies that in a BR a test particle in a gravitational field moves on a geodesic with respect to gij. It is found that the theories BR and GR differ in the following cases:

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

References

External links

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