118
5 Gravitational (Clock) Compass
Fig. 5.5 Symbolical sketch of the explicit solution for the curvature (5.92)–(5.115). In total 21
suitably prepared clocks (hollow circles) are needed to determine all curvature components. The
observer is denoted by the black circle
(5.92)–(5.109), and taking into account the symmetries of the Weyl tensor, we
may use a reduced clock setup to completely determine the gravitational field. All
other components may be obtained from the double self-duality property C abcd =
−
1
4 ε abef ε cdgh C ef gh .
01 : C (2)(3)(2)(3) = −
(1,1) B,
(5.116)
02 : C (0)(3)(2)(3) =
3
4
c
−1
22 c
−1
42 (c 22 − c 42 )
−1
(1,1) Bc
2
22 −
(1,1) Bc
2
42
+
(1,2) Bc
2
42 −
(1,4) Bc
2
22
,
(5.117)
03 : C (3)(0)(3)(0) = 3c
−1
22 c
−1
42 (c 22 − c 42 )
−1
(1,1) Bc 22 −
(1,1) Bc 42
+
(1,2) Bc 42 −
(1,4) Bc 22
,
(5.118)
04 : C (2)(0)(2)(0) =
(2,2) B,
(5.119)
05 : C (3)(2)(2)(0) =
3
4
c
−1
33
(1,3) B + C (2)(3)(2)(3) −
1
3
C (2)(0)(2)(0) c
2
33
, (5.120)