Gauge vector–tensor gravity

Details

The physical geometry, as seen by particles, represents the Finsler geometry–Randers type:

d s = − g μ ν d x μ d x ν + ( B μ + B ~ μ ) d x μ

This implies that the orbit of a particle with mass m can be derived from the following effective action:

S = m ∫ d τ ( 1 2 x ˙ μ x ˙ ν g μ ν + ( B μ + B ~ μ ) x ˙ μ ) .

The geometrical quantities are Riemannian. GVT, thus, is a bi-geometric gravity.

Action

The metric's action coincides to that of the Einstein–Hilbert gravity:

S Grav = 1 16 π G ∫ d 4 x − g R

where R is the Ricci scalar constructed out from the metric. The action of the gauge fields follow:

S B = − 1 16 π G κ l 2 ∫ d 4 x − g L ( l 2 4 B μ ν B μ ν )

and

S B ~ = − 1 16 π G κ ~ l ~ 2 ∫ d 4 x − g L ( l ~ 2 4 B ~ μ ν B ~ μ ν )

where L has the following MOND asymptotic behaviors

L ( x ) = { x , for   x ≫ 1 2 3 | x | 3 2 , for   x ≤ 1   ,

and κ , κ ~ represent the coupling constants of the theory while l , l ~ are the parameters of the theory and

l < l ~ .

Coupling to the matter

Metric couples to the energy-momentum tensor. The matter current is the source field of both gauge fields. The matter current is

J μ = ρ u μ

where ρ is the density and u μ represents the four velocity.

Regimes of the GVT theory

GVT accommodates the Newtonian and MOND regime of gravity; but it admits the post-MONDian regime.

Strong and Newtonian regimes

The strong and Newtonian regime of the theory is defined to be where holds:

L ( l 2 4 B μ ν B μ ν ) = l 2 4 B μ ν B μ ν , L ( l ~ 2 4 B ~ μ ν B ~ μ ν ) = l ~ 2 4 B ~ μ ν B ~ μ ν .

The consistency between the gravitoelectromagnetism approximation to the GVT theory and that predicted and measured by the Einstein–Hilbert gravity demands that

κ + κ ~ = 0

which results to B μ + B ~ μ = 0 . So the theory coincides to the Einstein–Hilbert gravity in its Newtonian and strong regimes.

MOND regime

The MOND regime of the theory is defined to be

L ( l 2 4 B μ ν B μ ν ) = | l 2 4 B μ ν B μ ν | 3 2 , L ( l ~ 2 4 B ~ μ ν B ~ μ ν ) = l ~ 2 4 B ~ μ ν B ~ μ ν .

So the action for the B μ field becomes aquadratic. For the static mass distribution, the theory then converts to the AQUAL model of gravity with the critical acceleration of

a 0 = 4 2 κ c 2 l

So the GVT theory is capable of reproducing the flat rotational velocity curves of galaxies. The current observations do not fix κ which is supposedly of order one.

Post-MONDian regime

The post-MONDian regime of the theory is defined where both of the actions of the B μ , B ~ μ are aquadratic. The MOND type behavior is suppressed in this regime due to the contribution of the second gauge field.