7.8 Curvature contributions to spatial velocity 143
Assuming that we are in vacuum (R
αβ
= 0) and rewriting the frame components
of the Riemann tensor in terms of electric and magnetic parts, the above system
can be written as
δ¨x
ˆα
a
= A
ˆα
a
ˆ
βˆγ
E
ˆ
βˆγ
+ B
ˆα
a
ˆ
βˆγ
H
ˆ
βˆγ
, (7.98)
with a =2, 3, 4andA
ˆα
a
ˆ
βˆγ
and B
ˆα
a
ˆ
βˆγ
being constant matrices. Inverting this
system gives
E
ˆα
ˆ
β
=
¯
A
ˆα
ˆ
βˆγ
d
δ¨x
ˆγ
d
, H
ˆα
ˆ
β
=
¯
B
ˆα
ˆ
βˆγ
d
δ¨x
ˆγ
d
, (7.99)
where again
¯
A
ˆα
a
ˆ
βˆγ
and
¯
B
ˆα
a
ˆ
βˆγ
are constant matrices, the explicit form of which
can be found in Ciufolini (1986) and Ciufolini and Demianski (1986; 1987).
7.8 Curvature contributions to spatial velocity
An observer moving on a curve γ with tangent vector u and proper time s as
parameter can only deduce the spatial velocity of a distant particle relative to
his own local rest frame by exchanging light signals. At the event
A on γ the
observer sends a light signal to the particle, which receives it at the event
P
on γ
.AtP the light signal is reflected back to the observer who receives it at the
event
B on γ. Denote as Υ and Υ
the null geodesics connecting A to P and P to B
respectively. Let A
0
be the event on γ, subsequent to A and antecedent to B,which
is simultaneous with
P with respect to the observer u and such that the space-like
geodesic ζ
P→A
0
joining P to A
0
is extremal with respect to γ.
1
Repeated reading
of the time of emission of light signals at
A and of the time of recording of the
reflected echo at
B allows one to determine the length of the space-like geodesic
segment connecting
P to A
0
, which represents, by definition, the instantaneous
spatial distance of the particle at
P from the observer on γ; see Fig. 7.1.The
relative velocity of the particle with respect to the observer u is then deduced,
differentiating the above spatial distance with respect to the observer’s proper
time.
2
The measurement process involving the events A, P,andB is patently non-
local insofar as the measurement domain is finite. Let us recall, however, that a
standard determination of the particle velocity is based on the measurement of a
frequency shift of the exchanged photons through the application of the Doppler
formula. The velocity so determined, however, is an equivalent velocity because
the frequency shift can also be caused by geometry perturbations which may not
be related to the particle’s motion at all. As already stated, curvature effects are
in general entangled with inertial terms resulting from the choice of the reference
frame, so we shall just term as curvature any possible combination of them. Our
1
We assume that the curve γ
lies in a normal neighborhood of γ, so the geodesic ζ
P→A
0
is
unique.
2
The relative velocity so determined is along the observer’s local line of sight, so it is a
radial velocity, i.e. a velocity either of recession or of approach.