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.
Rosen (1940) has proposed at each point of spacetime a Euclidean metric tensor γ_{ij} in addition to the Riemannian metric tensor g_{ij} . Thus at each point of spacetime there are two metrics:
ds^{2} = g_{ij}dx^{ i}dx^{j}
dσ^{2} = γ_{ij}dx^{i} dx^{j}
The first metric tensor g_{ij} describes the geometry of spacetime and thus the gravitational field. The second metric tensor γ_{ij} refers to the flat spacetime and describes the inertial forces. The Christoffel symbols formed from g_{ij} and γ_{ij} are denoted by and respectively. The quantities
Δ are defined such that
Now there arise two kinds of covariant differentiation:g differentiation based on g_{ij} (is denoted by semicolon (;)), 3 differentiation based on γ_{ij} (denoted by a d), ordinary partial derivatives are denoted by comma ( , ). and be the curvature tensors calculated from g_{ij} and γ_{ij} respectively. In the above approach as γ_{ij} is the flat spacetime metric, the curvature tensor 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 3covariant derivative. The straightforward calculations yield
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 3covariant differentiation, by , in the integration d^{4}x by , where g = det(g_{ij}), 3 = det(γ_{ij}) and d^{4}x = dx^{1}dx^{2} dx^{3}dx^{4}. It is necessary to point out that having once introduced γ_{ij} 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
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
where
or
and = Energy momentum tensor.\\ The variational principle also leads to the relation
Hence from (3)
which implies that in a BR a test particle in a gravitational field moves on a geodesic with respect to g_{ij}. It is found that the theories BR and GR differ in the following cases:

