The static spacetime relative acceleration for general free fall and its possible experimental test

Abstract

Mishra has recently established, using a generic static metric, the relative local proper-time 3-acceleration of a test-particle in one-dimensional free fall relative to a static reference frame in any static spacetime. In this paper, on the grounds of gravitoelectromagnetism we establish, in a covariant spacetime form, the relative 4-acceleration for the general free fall, indicating its canonical representation with its 3-space cinematical content. Then we obtain the relation between this representation and the very known expression for the relative free fall acceleration in Fermi coordinates. Taking this into account, it is shown that an experiment with relativistic beams in a circular accelerator, modelled by Fermi coordinates, recently proposed by Moliner et al, can test the here established covariant result and, therefore, can also verify Mishra's formula. This possibility of experimental verification, besides its intrinsic importance, can answer a recent inquire by Vigier, related to his recent proposal of derivation of inertial forces.

Kerr metric, static observers and Fermi coordinates

“…Marck [13] showed how to work with the Fermi metric alone along geodesics in Kerr to study this question, recently revisited by Ishii, Shibata and Mino [14]. Chicone and Mashhoon [15,16,17,18] have extended the discussion to more general accelerated world lines (see also [19,20,21,22] and references therein). However, because of the importance of the local observer frame in gravitational calculations, it is satisfying to finally understand how Fermi coordinates may be explicitly constructed for interesting world lines in the complicated geometry of the Kerr spacetime.…”

Ultrarelativistic motion: inertial and tidal effects in Fermi coordinates

“…Inertial accelerations have been discussed by a number of authors [3,4,7,[10][11][12][13][14][15]. In particular, it has been shown [7] that for ω = 0 andȧ = 0, the inertial acceleration experienced by the particle parallel to its motion is given to the lowest order in a by −a ·V(1 − 2V 2 /c 2 ), whereV = V/V is the unit vector tangent to the spatial path of the particle; therefore, there is a sign reversal for V > V c = c/ √ 2 with consequences that are contrary to Newtonian expectations.…”

A Lorentz-Poincaré Type Interpretation of the Weak Equivalence Principle

“…A free-falling observer crossed by some free-falling system with a non-zero relative velocity, must -according to the WEP-have a zero relative acceleration only while meeting at the intersection point of their trajectories. That this is the case for a LIF observer is a priori not intuitive given the free-fall acceleration relative to a static observer [4,9,19]. Damour's stipulation of the Weak Equivalence Principle, or "universality of free-fall", is precise in this sense ( [12], emphasis added):…”

“…In GRT, a static observer will attribute a free-fall acceleration with explicit dependence on the kinematics of a particle -using generic coordinates [3,9,19] (a coordinate-free space-time decomposition of a covariant expression):…”