Brans Dicke Theory

The Brans Dicke theory (sometimes referred to as the Jordan-Brans Dicke theory ) is a classical field theory and one of the simplest extensions of General Relativity (GTR). It was developed in 1961 by Robert Henry Dicke and Carl H. Brans , using earlier work by Pascual Jordan . It is the best known and simplest representative of scalar - tensor theories of gravity in which the space-time curvature of the metric of ART and additional scalar fields is generated.

The theory contains a free parameter via which the scalar fields couple to the curvature. For the Brans-Dicke theory approximates the GTR to the point of indistinguishability, so that in principle it cannot be falsified by experiments . Precision measurements during the Cassini-Huygens mission, however, pushed the allowed range up , which is a big step up from the previous strongest results. ${\displaystyle \omega }$ ${\displaystyle \omega \rightarrow \infty }$ ${\displaystyle \omega >40\,000}$

Brans and Dicke also developed the model in order to have an alternative to the general theory of relativity, in which Mach's principle is implemented (the scalar field is determined by the masses in the universe).

As a metric theory of gravity, it fulfills the principle of equivalence and therefore, like the GTR, predicts a gravitational redshift.

definition

The effect of the Brans-Dicke theory is: ${\displaystyle S}$

${\displaystyle S={\frac {1}{16\,\pi }}\int \mathrm {d} ^{4}x{\sqrt {-g}}\left(\phi \,R-\omega \,\phi ^{-1}\partial _{\mu }\phi \,\partial ^{\mu }\phi \right)+S_{\mathrm {M} }}$

Here is

• g the metric
• R is the trace of the Ricci tensor
• ${\displaystyle \omega }$a dimensionless parameter
• ${\displaystyle \phi }$ a scalar field
• ${\displaystyle S_{\mathrm {M} }}$the effect of the matter fields, which is assumed to be independent of . ${\displaystyle \phi }$

In contrast to ART, the effect of which is given by:

${\displaystyle S_{\text{ART}}={\frac {1}{16\,\pi }}\int \mathrm {d} ^{4}x\,{\sqrt {-g}}\,R+S_{\mathrm {M} }}$

there is an additional scalar field . ${\displaystyle \phi }$

This leads to modified equations of motion :

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

${\displaystyle {\begin{aligned}G_{\mathrm {ab} }&:=R_{\mathrm {ab} }-{\frac {1}{2}}R\,g_{\mathrm {ab} }\\&\;={\frac {8\,\pi }{\phi }}\,T_{\mathrm {ab} }+{\frac {\omega }{\phi ^{2}}}\left(\partial _{a}\phi \,\partial _{b}\phi -{\frac {1}{2}}\,g_{\mathrm {ab} }\partial _{c}\phi \,\partial ^{c}\phi \right)+{\frac {1}{\phi }}\,\left(\nabla _{a}\,\nabla _{b}\,\phi -g_{\mathrm {ab} }\,\Box \phi \right)\end{aligned}}}$

With

• ${\displaystyle G_{ab}}$ is the Einstein tensor, a kind of mean curvature
• ${\displaystyle \Box }$the Laplace – Beltrami operator
• ${\displaystyle R_{ab}}$the Ricci tensor
• ${\displaystyle T_{\mathrm {ab} }}$the energy-momentum tensor
• T his trail.

According to the first equation, T represents a source for the scalar field which, as can be seen in the second equation, contributes to the curvature. This distinguishes the theory from the GTR, whose equations of motion are given by: ${\displaystyle \phi }$

${\displaystyle G_{\mathrm {ab,ART} }=8\,\pi \,T_{\mathrm {ab} }}$

This modification leads to changed predictions for certain gravitational effects, such as B. the deflection of light by massive bodies or the perihelion of the planets . Through experiments it was therefore possible to restrict the permitted values ​​for the coupling constant , which can be chosen as a free parameter and which controls the size of the deviations from the predictions of the ART, in the direction of ever smaller deviations from the ART. ${\displaystyle \omega }$

literature

• Pascual Jordan : Gravity and Space . Vieweg, Braunschweig 1955
• CH Brans: The roots of scalar-tensor theory: an approximate history . arxiv : gr-qc / 0506063
• C. Misner, K. Thorne , JA Wheeler : Gravitation . WH Freeman, San Francisco 1973, ISBN 0-7167-0344-0 (especially box 39.1)

Individual evidence

1. ↑ Carl Brans, Robert H. Dicke: Mach's Principle and a Relativistic Theory of Gravitation . In: Physical Review . tape 124 , p. 925-935 , doi : 10.1103 / PhysRev.124.925 .
2. ↑ Clifford M. Will: The Confrontation between General Relativity and Experiment . In: Living Rev. Relativity , 9, 2006, relativity.livingreviews.org ( Memento of the original from June 13, 2007 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.