104
5 Gravitational (Clock) Compass
Fig. 5.2 Symbolical sketch of the explicit compass solution in (5.7)–(5.26). In total 13 suitably
prepared test bodies (hollow circles) are needed to determine all curvature components. The
reference body is denoted by the black circle
This implies that 13 test bodies are needed to measure the gravitational field
completely. In Fig. 5.2 a graphical form of the solution in (5.7)-(5.26) is given.
Vacuum Solution
In vacuum the number of independent components of the curvature is reduced to
the 10 components of the Weyl tensor C abcd . Replacing R abcd in the compass
solution (5.7)-(5.26), and taking into account the symmetries of Weyl (in particular
the double-self-duality property C abcd = −
1
4 abef cdgh C ef gh , where abcd is the
totally antisymmetric Levi-Civita tensor with 0123 = 1), the gravitational field may
be explicitly expressed as follows:
01 : C 1010 =
3
4
(1,1) A 1 c
−2
10 ,
(5.27)
02 : C 2010 =
3
4
(1,1) A 2 c
−2
10 ,
(5.28)
03 : C 3010 =
3
4
(1,1) A 3 c
−2
10 ,
(5.29)