Brans-Dicke theory

Brans-Dicke theory is an extension to Einstein's theory of general relativity. In addition to the metric, $g$ , it introduces a long range scalar field, $φ$ , which acts as the gravitational constant. The difference is that the gravitational "constant" is now a function of spacetime, specifically the totality of mass-energy in the universe. The theory is characterised by a parameter, $ω$ , called the Dicke coupling constant, which describes the coupling between mass-energy and the scalar field. In the basic version of the theory $ω$ is assumed to be a fundamental constant, not changing throughout spacetime. General relativity is recovered in the limit that $ω$ tends to infinity.

Field equations

The single Einstein field equation of general relativity is modified and in addition there is now an equation for the scalar field:

$R_{ab}-\frac{1}{2}g_{ab}R=\frac{8\pi}{\phi}T_{ab}+\frac{\omega}{\phi^2}(\partial_a\phi\partial_b\phi-\frac{1}{2}g_{ab}\partial_c\phi\partial^c\phi) +\frac{1}{\phi}(\nabla_a\nabla_b\phi-g_{ab}\Box\phi)$

$\Box\phi=\frac{8\pi}{3+2\omega}T$

where

$g a b$ is the metric tensor,
$R a b$ is the Ricci tensor constructed out of the metric,
$R$ is the Ricci scalar,
$T a b$ is the matter stress-energy tensor,
$T$ is the trace of $T a b$ , and
$φ$ is the scalar field with $\Box\phi\equiv\nabla_c\nabla^c\phi$ .

Comparison to general relativity

Like general relativity, this theory satisfies the equivalence principle, which implies that the scalar field only affects the geometry of spacetime without having any direct influence on matter. However, unlike general relativity, this theory is considered to more fully satisfy Mach's principle.

General relativity and Brans-Dicke theory predict exactly the same gravitational redshift effect. However, they give different formulas for light deflection and the precession of orbiting bodies, with the Brans-Dicke results depending on the parameter $ω$ . The Brans-Dicke results tend towards those of general relativity in the limit that $ω$ tends towards infinity.

It is in principle possible to measure the value of $ω$ experimentally by, for example, measuring how much light deviates when it passes near the Sun. In previous tests no confirmed results have shown any deviation from general relativity, but this does not mean that Brans-Dicke theory is not correct, merely that $ω$ must be large. The current experimental constraint is that $ω$ must be greater than a couple of hundred.

Derivation from an action principle

The field equations can be derived from the following action:

$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}(\phi R - \omega\frac{\partial_a\phi\partial^a\phi}{\phi} + L_M)$

where

$g$ is the determinant of the metric, and
$L M$ is the matter Lagrangian.

Categories: General relativity

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Brans-Dicke theory

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Brans-Dicke theory

Field equations

Comparison to general relativity

Derivation from an action principle