5.3 Field Determination by Means of Clocks
107
Fig. 5.4 Sketch of the
operational procedure to
measure the curvature of
spacetime by means of
clocks. An observer with a
clock moving along a world
line Y compares his clock
readings C to a set of suitably
prepared clocks in the vicinity
of Y . The number of clocks
required for the determination
of all curvature components
depends on the type of the
underlying spacetime
Frequency Ratio in Flat Space-Time
Here we review the frequency ratio between a clock carried by the observer and a
clock in the vicinity of his world line is given in a flat background. Readers should
also consult Appendix C.1, in which a suitable form of the flat space-time metric
in the neighbourhood of a time-like world line, in a reference frame carried by a
general observer, is constructed.
With X i = (X, Y, Z, T ) and x i = (x, y, r, u) the parametric equations of an
arbitrary time-like world line, with arc length s as parameter along it, are
X
i
= X
i (s) ⇔ x
i
= x
i (s) ,
(5.37)
with
η ij
dX i
ds
dX j
ds
= +1 .
(5.38)
Using (C.27) and writing
dx i
ds
=
dx i
du
du
ds
,
(5.39)