Bimetric gravity

Bimetric gravity or bigravity refers to a class of modified mathematical theories of gravity (or gravitation) in which two metric tensors are used instead of one. The second metric may be introduced at high energies, with the implication that the speed of light could be energy-dependent. If the two metrics are dynamical and interact then there will be two graviton modes, one massive and one massless, and thus bimetric theories are closely related to massive gravity.

There are several different bimetric theories, such as those attributed to Nathan Rosen (1909–1995) or Mordehai Milgrom with Modified Newtonian Dynamics (MOND). More recently, developments in massive gravity have also led to new consistent theories of bimetric gravity. 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. Some of these theories lead to cosmic acceleration at late times and are therefore alternatives to dark energy.

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) 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:

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