102
5 Gravitational (Clock) Compass
For (n) bodies at locations (n) η a relative to the reference body, moving with
relative (m) velocities (m) u a one ends up with the system
(m,n) A a
| Y
=
4
3
R abcd
(m) u
b (n) η
c (m) u
d .
(5.6)
Here the (m,n) A a denote the measured accelerations relative to the reference point
Y . Physically, these A’s correspond to the springs in the mechanical compass picture
of Szekeres [6].
General Solution
As was shown in [5], the 20 independent components of the curvature tensor can be
explicitly determined in terms of the accelerations (m,n) A a and velocities (m) u a . The
algebraic system (5.6) allows to express the Riemann curvature tensor components
as follows:
01 : R 1010 =
3
4
(1,1) A 1 c
−2
10 ,
(5.7)
02 : R 2010 =
3
4
(1,1) A 2 c
−2
10 ,
(5.8)
03 : R 3010 =
3
4
(1,1) A 3 c
−2
10 ,
(5.9)
04 : R 2020 =
3
4
(1,2) A 2 c
−2
10 ,
(5.10)
05 : R 3020 =
3
4
(1,2) A 3 c
−2
10 ,
(5.11)
06 : R 3030 =
3
4
(1,3) A 3 c
−2
10 ,
(5.12)
07 : R 2110 =
3
4
(2,1) A 2 c
−1
21 c
−1
20 − R 2010 c
−1
21 c 20 ,
(5.13)
08 : R 3110 =
3
4
(2,1) A 3 c
−1
21 c
−1
20 − R 3010 c
−1
21 c 20 ,
(5.14)
09 : R 0212 =
3
4
(3,1) A 0 c
−2
32 + R 2010 c
−1
32 c 30 ,
(5.15)
10 : R 1212 =
3
4
(2,2) A 2 c
−2
21 − R 2020 c
2
20 c
−2
21 − 2R 0212 c
−1
21 c 20 ,
(5.16)
11 : R 3220 =
3
4
(3,2) A 3 c
−1
32 c
−1
30 − R 3020 c
−1
32 c 30 ,
(5.17)