116
5 Gravitational (Clock) Compass
13 : R (0)(3)(1)(3) =
3
4
c
−1
11
(3,1) B − R (3)(0)(3)(0) +
1
3
R (1)(3)(1)(3) c
2
11
,
(5.104)
14 : R (0)(3)(2)(3) =
3
4
c
−1
22
(3,2) B − R (3)(0)(3)(0) +
1
3
R (2)(3)(2)(3) c
2
22
,
(5.105)
15 : R (3)(1)(3)(2) =
3
2
c
−1
41 c
−1
42
−
(3,4) B + R (3)(0)(3)(0) +
4
3
R (0)(3)(1)(3) c 41
+
4
3
R (0)(3)(2)(3) c 42 −
1
3
R (1)(3)(1)(3) c
2
41 −
1
3
R (2)(3)(2)(3) c
2
42
,
(5.106)
16 : R (2)(0)(1)(0) =
1
2
(4,1) B − R (1)(0)(1)(0) − R (2)(0)(2)(0) −
4
3
R (0)(2)(1)(2) c 11
−
4
3
R (2)(1)(1)(0) c 11 +
1
3
R (1)(2)(1)(2) c
2
11
,
(5.107)
17 : R (3)(0)(2)(0) =
1
2
(5,2) B − R (2)(0)(2)(0) − R (3)(0)(3)(0) −
4
3
R (0)(3)(2)(3) c 22
−
4
3
R (3)(2)(2)(0) c 22 +
1
3
R (2)(3)(2)(3) c
2
22
,
(5.108)
18 : R (3)(0)(1)(0) =
1
2
(6,1) B − R (1)(0)(1)(0) − R (3)(0)(3)(0) −
4
3
R (0)(3)(1)(3) c 11
−
4
3
R (3)(1)(1)(0) c 11 +
1
3
R (1)(3)(1)(3) c
2
11
.
(5.109)
Introducing abbreviations
K 1 :=
3
4
c
−1
33
−
(4,3) B + R (1)(0)(1)(0) + 2R (2)(0)(1)(0) + R (2)(0)(2)(0)
−
1
3
(R (1)(3)(1)(3) + 2R (3)(1)(3)(2) + R (2)(3)(2)(3) )c
2
33
−
4
3
(R (3)(1)(1)(0) + R (3)(2)(2)(0) )c 33
,
(5.110)