120
5 Gravitational (Clock) Compass
Fig. 5.6 Symbolical sketch of the explicit vacuum solution for the curvature (5.116)–(5.130). In
total 11 suitably prepared clocks (hollow circles) are needed to determine all curvature components.
The observer is denoted by the black circle
the remaining three curvature components read
10 : C (1)(0)(2)(3) =
1
3
(K 3 − K 1 ) ,
(5.128)
11 : C (2)(0)(1)(3) =
1
3
(K 2 − K 1 ) ,
(5.129)
12 : C (3)(0)(2)(1) =
1
3
(K 3 − K 2 ) .
(5.130)
A symbolical sketch of the solution is given in Fig. 5.6.
Special Spacetimes
In the following we consider setups of clocks which allow for a determination of
the field in special space-times. As in the previous section on general and vacuum
spacetimes we look for arrangements of n clocks, at positions (n) p α , with velocities
(m) u α w.r.t. the reference world line of the observer. While possible in principle, and
in particular covered by our general formalism, we are again not going to allow for
situations with additional acceleration or rotations. The explicit derivation of the
frequency ratios is given in Appendices C.2–C.4.