Bimetric gravity

Bimetric gravity or bigravity refers to two different classes 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. 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. Several bimetric theories with massive gravitons exist, such as those attributed to Nathan Rosen (1909–1995) or Mordehai Milgrom with relativistic extensions of 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. Bimetric gravity is also at odds with measurements of gravitational waves emitted by the neutron-star merger GW170817.

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.

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.

In 1940, Rosen 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:

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:

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