C Gravitational (Clock) Compass
233
and so the light-like propagation direction calculated on r = 0 is k i = p i + v i . The
vacuum field equations
R (a)(b) = −R (α)(a)(b)(α) + R (4)(a)(b)(4) = 0 ,
(C.91)
and the radiative conditions on the Riemann tensor (that the Riemann tensor be type
N in the Petrov classification with k (a) as degenerate principal null direction)
R (a)(b)(c)(d) k
(d)
= R (a)(b)(c)(3) + R (a)(b)(c)(4) = 0 ,
(C.92)
must be satisfied on r = 0 for substitution into (5.87). As a consequence of (C.91)
and (C.92) there are only two independent non-vanishing components of the vacuum
Riemann tensor calculated on r = 0, namely, R (1)(4)(1)(4) = −R (2)(4)(2)(4) and
R (1)(4)(2)(4) . All remaining non-vanishing curvature components are given in terms
of these by
R (1)(3)(1)(3) = −R (2)(3)(2)(3) = −R (1)(3)(1)(4)
= R (2)(3)(2)(4) = R (1)(4)(1)(4) ,
(C.93)
and
R (1)(3)(2)(3) = −R (1)(4)(2)(3) = −R (2)(4)(1)(3)
= R (1)(4)(2)(4) .
(C.94)
When these are substituted into the Riemann tensor terms in (5.87) we find that
R (α)(4)(β)(4) p
(α) p (β) = R (A)(4)(B)(4) p
(A) p
(B) ,
(C.95)
R (α)(4)(β)(γ ) p
(α) u
(β) p (γ )
= R (A)(4)(B)(4) { u (3) p
(A)
− u
(A) p
(3)
} p
(B) ,
(C.96)
and
R (α)(β)(γ )(σ ) u
(α) p
(β) u
(γ ) p
(σ )
= R (A)(4)(B)(4)
×{u
(A) p
(3)
− u
(3) p
(A)
}{u
(B) p
(3)
− u
(3) p
(B)
},
(C.97)
where capital letters take values 1, 2.