Vector-tensor nature of Bekenstein's relativistic theory of modified gravity

There is a distinct possibility that Newtonian gravity, and its relativistic generalization, General Relativity, breaks down in regions of small acceleration. Modified Newtonian Dynamics (MOND) proposes a fix to the law of gravity in the non-relativistic regime which fares well in explaining the dynamics of galaxies. In the past few years, a relativistic generalization of MOND has been proposed by Bekenstein with interesting consequences on cosmological scales. One might expect that a modified theory of gravity must tamper with the way the gravitational field (or metric), couples or responds to sources. Modifications of gravity typically involve modifying the Einstein-Hilbert action or introducing extra degrees of freedom (or fields) that distort the way the metric enters the action for all forms of matter. A well known example is Jordan-BransDicke theory where an extra scalar field can be interpreted as a time-varying Newton’s “constant”. Such a theory can be rewritten (or transformed) with a redefinition of the metric in such a way that Newton’s “constant” becomes constant but the matter action picks up couplings to the scalar field. More generally one can think of such theories as having two metrics. One metric satisfies the Einstein-Hilbert action while the other defines the stress-energy tensory and the geodesic equations. A rule must then be posited that links the two metrics which typically involves new dynamical fields with their own actions. Bekenstein’s theory falls in this class of theories.