Tensor–vector–scalar gravity (TeVeS),
[
] developed by
Jacob Bekenstein in 2004, is a relativistic generalization of
Mordehai Milgrom's
Modified Newtonian dynamics (MOND) paradigm.
[
]
The main features of TeVeS can be summarized as follows:
* As it is derived from the
action principle, TeVeS respects
conservation laws;
* In the
weak-field approximation of the spherically symmetric, static solution, TeVeS reproduces the MOND acceleration formula;
* TeVeS avoids the problems of earlier attempts to generalize MOND, such as
superluminal propagation;
* As it is a relativistic theory it can accommodate
gravitational lensing.
The theory is based on the following ingredients:
* A unit
vector field;
* A dynamical
scalar field
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity ...
;
* A nondynamical scalar field;
* A matter
Lagrangian constructed using an alternate
metric;
* An arbitrary dimensionless function.
These components are combined into a relativistic
Lagrangian density
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
, which forms the basis of TeVeS theory.
Details
MOND
[ is a phenomenological modification of the Newtonian acceleration law. In Newtonian gravity theory, the gravitational acceleration in the spherically symmetric, static field of a point mass at distance from the source can be written as
:
where is Newton's constant of gravitation. The corresponding force acting on a test mass is
:
To account for the anomalous rotation curves of spiral galaxies, Milgrom proposed a modification of this force law in the form
:
where is an arbitrary function subject to the following conditions:
:
In this form, MOND is not a complete theory: for instance, it violates the law of momentum conservation.
However, such conservation laws are automatically satisfied for physical theories that are derived using an action principle. This led Bekenstein][ to a first, nonrelativistic generalization of MOND. This theory, called AQUAL (for A QUAdratic Lagrangian) is based on the Lagrangian
:
where is the Newtonian gravitational potential, is the mass density, and is a dimensionless function.
In the case of a spherically symmetric, static gravitational field, this Lagrangian reproduces the MOND acceleration law after the substitutions and are made.
Bekenstein further found that AQUAL can be obtained as the nonrelativistic limit of a relativistic field theory. This theory is written in terms of a Lagrangian that contains, in addition to the ]Einstein–Hilbert action
The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the a ...
for the metric field , terms pertaining to a unit vector field and two scalar fields and , of which only is dynamical. The TeVeS action, therefore, can be written as
:
The terms in this action include the Einstein–Hilbert Lagrangian (using a metric signature