136 Non-local measurements
monitor the strains on the springs. The relative acceleration between two nearby
particles is completely determined by the electric part of the Riemann tensor,
which can be thought of as a symmetric force distribution whose six indepen-
dent components are the strains on the six springs. When the off-diagonal terms
(i.e. the transverse strains) vanish, the springs connecting the test particles to
the observer lie along the principal axes of the tidal force matrix, so that the
apparatus maps out the local gravitational field, acting just as a compass. In
the case of a vacuum space-time, the electric part of the Weyl tensor represents
the only curvature contribution to the deviation between any two neighboring
trajectories introducing shearing forces, due to its property of being symmetric
and trace-free. Actually Szekeres’ gravitational compass only describes an ideal-
ized situation. For any practical use, in fact, it should be replaced by a “gravity
gradiometer,” i.e. a device to perform measurements of the local gradient of the
tidal gravitational force. The theory of a relativistic gravity gradiometer has been
developed by many authors (Mashhoon and Theiss, 1982; Mashhoon, Paik, and
Will, 1989) in view of satellite experiments around the Earth in the framework
of the post-Newtonian approximation. It should also be noted that a modern
observational trend is to use atomic interferometry to build a future generation
of highly precise gravity gradiometers (see Matsko, Yu, and Maleki, 2003, and
references therein).
More recently Chicone and Mashhoon (2002) have obtained a generalized
geodesic deviation equation in Fermi coordinates as well as in arbitrary coor-
dinates as a Taylor expansion in powers of the components of the deviation
vector, retaining terms up to first order, but without any restriction on the rela-
tive spatial velocities. They then investigated in a number of papers (Chicone
and Mashhoon, 2005a; 2005b) the motion of a swarm of free particles (in
both non-relativistic and relativistic regimes) relative to a free reference par-
ticle which is on a radial escape trajectory away from a collapsed object (a
Schwarzschild as well as a Kerr black hole), discussing the astrophysical implica-
tions of the related (observer-dependent) tidal acceleration mechanism. The fur-
ther dependence of the deviation equation on the 4-acceleration of the observer
as well as his 3-velocity has been accounted for very recently by Mullari and
Tammelo (2006).
In what follows we shall develop the general theory of relativistic strains, gener-
alizing Szekeres’ picture as well as the one associated with the relativistic gravity
gradiometry in the case when the acceleration strains are present. We also iden-
tify which frame is the most convenient for measuring either tidal or inertial
forces experienced by an extended body.
The relative acceleration equation
Let us consider a collection of test particles, i.e. a congruence C
U
of time-like world
lines, with unit tangent vector U (U ·U = −1) parameterized by the proper time