Bimetric gravity or bigravity refers to two different classes of theories.
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.
[3] Several bimetric theories with massive gravitons exist, such as those attributed to Nathan Rosen (1909–1995)[4][5][6] or Mordehai Milgrom with relativistic extensions of Modified Newtonian Dynamics (MOND).
[8] 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.
[5] Some of these theories lead to cosmic acceleration at late times and are therefore alternatives to dark energy.
[9][10] Bimetric gravity is also at odds with measurements of gravitational waves emitted by the neutron-star merger GW170817.
[11] 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).
[12] 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.
[13][14][15][16][17] 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[1][2] proposed that at each point of space-time, there is a Euclidean metric tensor
, refers to the flat space-time and describes the inertial forces.
into the theory, one has a great number of new tensors and scalars at one's disposal.
can be regarded as describing the inertial field because it vanishes by a suitable coordinate transformation.
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
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
Rosen continued improving his bimetric gravity theory with additional publications in 1978[18] and 1980,[19] in which he made an attempt "to remove singularities arising in general relativity by modifying it so as to take into account the existence of a fundamental rest frame in the universe."
In 1985[20] Rosen tried again to remove singularities and pseudo-tensors from General Relativity.
Twice in 1989 with publications in March[21] and November[22] Rosen further developed his concept of elementary particles in a bimetric field of General Relativity.
It is found that the BR and GR theories differ in the following cases: The predictions of gravitational radiation in Rosen's theory have been shown since 1992 to be in conflict with observations of the Hulse–Taylor binary pulsar.
[5] Since 2010 there has been renewed interest in bigravity after the development by Claudia de Rham, Gregory Gabadadze, and Andrew Tolley (dRGT) of a healthy theory of massive gravity.
[23] Massive gravity is a bimetric theory in the sense that nontrivial interaction terms for the metric
is introduced, and the interaction terms are built out of the matrix square root of
In dRGT massive gravity, the reference metric must be specified by hand.
[25] The dRGT potential is constructed specifically to render this ghost nondynamical, and as long as the kinetic term for the second metric is of the Einstein–Hilbert form, the resulting theory remains ghost-free.
[3] The action for the ghost-free massive bigravity is given by[26] As in standard general relativity, the metric
The interaction potential is built out of the elementary symmetric polynomials