Bimetric theory
Encyclopedia
Bimetric theory refers to a class of modified theories of gravity in which two metric tensor
s 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 space-time a Euclidean metric tensor
in addition to the Riemannian metric tensor 
. Thus at each point of space-time there are two metrics:
The first metric tensor
describes the geometry of space-time and thus 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. The quantities
are defined such that
Now there arise two kinds of covariant differentiation:
differentiation based on 
(denoted by semicolon ), 3- differentiation based on 
(denoted by a slash), ordinary partial derivatives are denoted by comma ( ). 
and 
be the curvature tensors calculated from 
and 
respectively. In the above approach as 
is the flat space-time metric, the curvature tensor 
is zero.
From (1) one finds that though {:} and
are not tensors, but 
is a tensor having the same form as {:} except that the ordinary partial derivative is replaced by 3-covariant 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
, ordinary differentiation by 3-covariant differentiation, 
by 
, in the integration 
by 
, where 
, 
and 
. It is necessary to point out that having once introduced 
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
can be regarded as describing the inertial field because it vanishes by a suitable coordinate transformation.
The quantity
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
It is found that the theories BR and GR differ in the following cases:
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
s are used instead of one. Often the second metric is introduced at high energies, with the implication that the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
may be energy dependent.
In general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, it is assumed that the distance between two points in spacetime is given by the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
. 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 space-time a Euclidean metric tensor
in addition to the Riemannian metric tensor . Thus at each point of space-time there are two metrics:The first metric tensor
describes the geometry of space-time and thus 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. The quantities are defined such thatNow there arise two kinds of covariant differentiation:
differentiation based on (denoted by semicolon ), 3- differentiation based on (denoted by a slash), ordinary partial derivatives are denoted by comma ( ). and be the curvature tensors calculated from and respectively. In the above approach as is the flat space-time metric, the curvature tensor is zero.From (1) one finds that though {:} and
are not tensors, but is a tensor having the same form as {:} except that the ordinary partial derivative is replaced by 3-covariant derivative. The straightforward calculations yieldEach 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
, ordinary differentiation by 3-covariant differentiation, by , in the integration by , where , and . It is necessary to point out that having once introduced 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
can be regarded as describing the inertial field because it vanishes by a suitable coordinate transformation.The quantity
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
It is found that the theories BR and GR differ in the following cases:- propagation of electromagnetic waves
- an external field of high density star
- the behaviour of the intense gravitational waves propagation through strong static gravitational field