Tensor-vector-scalar gravitation theory

The tensor-vector-scalar-gravitation theory ( TeVeS ) is a theory of gravitation , which presents itself as an alternative to the general theory of relativity for the description of the processes in cosmology .

motivation

Cosmological observations, assuming the validity of the general theory of relativity, have led to the postulation of the existence of dark energy and dark matter . The TeVeS tries to explain these observations without these two phenomena. The theory, first formulated by Jacob Bekenstein in 2004 , emerged from Modified Newtonian Dynamics (MOND) and was adapted to the findings of Einstein's special theory of relativity .

The main difference to the general theory of relativity lies in how the gravitational strength is formulated as a function of the distance to the mass. In TeVeS, this is defined by means of a scalar , a tensor and a vector , while the general theory of relativity represents the spatial geometry by means of a single tensor.

Basics

The TeVeS theory uses a modified metric of shape

${\displaystyle {\tilde {g}}_{\mu \nu }=e^{-2\phi }\left(g_{\mu \nu }+v_{\mu }v_{\nu }\right)-e^{2\phi }v_{\mu }v_{\nu }}$

where the metric corresponds to the general theory of relativity, is a vector field that fulfills the condition , i.e. is time-like and is a scalar. ${\displaystyle g_{\mu \nu }}$ ${\displaystyle v^{\mu }}$ ${\displaystyle g_{\mu \nu }v^{\mu }v^{\nu }=-1}$ ${\displaystyle \phi }$

As in the general theory of relativity, the dynamics of the metric are given by the Einstein-Hilbert effect , while the modified metric is used in the effect for matter . ${\displaystyle S_{m}}$

For the vector field there will be an effect of the shape

${\displaystyle S_{v}=-{\frac {K}{32\pi G}}\int \left(g^{\mu \nu }g^{\alpha \beta }v_{[\mu ,\alpha ]}v_{[\nu ,\beta ]}-2{\frac {\lambda }{K}}\left(g^{\mu \nu }v_{\mu }v_{\nu }+1\right)\right)|\det {g}|^{\frac {1}{2}}d^{4}x}$

accepted. There is a coupling constant and a Lagrange multiplier , which ensures the condition that it is time-like. ${\displaystyle K}$ ${\displaystyle \lambda }$ ${\displaystyle v^{\mu }}$

This effect leads to a set of equations which, in addition to the Einstein equation, determine gravity.

${\displaystyle Kv_{;\nu }^{[\mu ;\nu ]}+\lambda v^{\mu }+8\pi G\sigma ^{2}v^{\nu }\phi _{,\nu }g^{\mu \lambda }\phi _{,\lambda }=8\pi G\left(1-e^{-4\phi }\right)g^{\mu \nu }v^{\lambda }{\tilde {T}}_{\nu \lambda }}$

where is the modified energy-momentum tensor and an auxiliary field that is used in the effect of the scalar field. ${\displaystyle {\tilde {T}}_{\mu \nu }=-2|\det {g}|^{-{\frac {1}{2}}}{\frac {\delta S_{m}}{\delta {\tilde {g}}^{\mu \nu }}}}$ ${\displaystyle \sigma }$

literature

• M. Milgrom : A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis . In: ApJ , 270, 1983, p. 365
• JD Bekenstein: Relativistic gravitation theory for the MOND paradigm . arxiv : astro-ph / 0403694
• JD Bekenstein: The modified Newtonian dynamics - MOND - and its implications for new physics . arxiv : astro-ph / 0701848