the gravitational field. The second metric tensor, , refers to the flat space-time and describes the inertial forces. The Christoffel symbols formed from and are denoted by and respectively. Since the difference of two connections is a tensor, one can define the tensor field given by: Two kinds of covariant differentiation then arise: -differentiation based on (denoted by a semicolon, e.g. ), and covariant differentiation based on (denoted by a slash, e.g. ). Ordinary partial derivatives are represented by a comma (e.g. ). Let and be the Riemann curvature tensors calculated from and , respectively. In the above approach the curvature tensor is zero, since is the flat space-time metric. A straightforward calculation yields the Riemann 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 and ordinary differentiation by covariant -differentiation, by , integration measure by , where , and . Having once introduced 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 It is seen from equations () and () that can be regarded as describing the inertial field because it vanishes by a suitable coordinate transformation. Being the quantity 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 where or with , and is the energy-momentum tensor. The variational principle also leads to the relation . Hence from () , which implies that in a BR, a test particle in a gravitational field moves on a geodesic with respect to Rosen continued improving his bimetric gravity theory with additional publications in 1978 and 1980, in which he made an attempt "to remove singularities arising in general relativity