Space isotropy and weak equivalence principle in a scalar theory of gravity

Abstract

We consider a preferred-frame bimetric theory in which the scalar gravitational field both influences the metric and has direct dynamical effects. A modified version ("v2") is built, by assuming now a locally-isotropic dilation of physically measured distances, as compared with distances evaluated with the Euclidean space metric. The dynamical equations stay unchanged: they are based on a consistent formulation of Newton's second law in a curved space-time. To obtain a local conservation equation for energy with the new metric, the equation for the scalar field is modified: now its l.h.s. is the flat wave operator. Fluid dynamics is formulated and the asymptotic scheme of post-Newtonian approximation is adapted to v2. The latter also explains the gravitational effects on light rays, as did the former version (v1). The violation of the weak equivalence principle found for gravitationally-active bodies at the point-particle limit, which discarded v1, is proved to not exist in v2. Thus that violation was indeed due to the anisotropy of the space metric assumed in v1.

Dirac-Type Equations in a Gravitational Field, with Vector Wave Function

“…the "Einstein-aether theory" of gravitation, endowed with a dynamical preferred reference frame [35]; cf. also this author's scalar theory [36,37]. This physical possibility may turn out to be experimentally relevant to quantum mechanics in a gravitational field, or not.…”

Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation

“…For light rays, it has been checked in Ref. [11] that also for v2 the 1PN predictions of this theory for photons are indistinguishable from the standard 1PN predictions of GR. This means that the gravitational effects on electromagnetic rays: the gravitational redshift, the deflection of light, and the time delay, are predicted by this theory as they are in GR.…”

“…with ψ ≡ −Log β, the flat-spacetime wave operator, and σ ≡ T 00 , the energy component of the energy-momentum tensor [11]. This equation is valid in coordinates adapted to the preferred reference frame, and with x 0 = cT where X ↦ → T is a preferred time map on the spacetime manifold V. The equation for the scalar field will not be used in the present work, however.…”

“…We have developed an asymptotic scheme of post-Newtonian approximation, in accordance with the general principles of asymptotic analysis [23]. Using this scheme, one shows first that v1 has the correct Newtonian limit [23] (and v2 also [11]). At the second approximation (1PN), one has to take into account the motion of the mass center of the gravitating system, assumed isolated -the solar system, saywith respect to the ether frame.…”

“…(26) below [11]. In particular, in the static spherical case, the 1PN approximation of the spacetime metric is the same as the standard 1PN metric of GR { [11], Eq. (88)}.…”