Mathematics of Physics and Engineering

114 Theory of Relativity

in (2.4-35) approaches the flat Minkowski metric in spherical coordinates
(dr)^2 +r^2 (sin^2 ip(d9)^2 + (d<p)^2 )-c^2 (dt)^2.
The reader can derive (2.4.35) by following the steps below. The com-
putations are not hard, but Steps 2 and 3 require a lot of care and patience.
Step 1. Argue that spherical symmetry and stationarity imply ds^2 —
F(r)(dr)^2 + r^2 (sin^2 (p{d6)^2 + (dtp)^2 ) + H(r)(dt)^2 for some functions F,H
that are not equal to zero. With x^1 = r, x^2 = 0, x^3 = tp, x^4 = t, we have
the components of the metric tensor gn = F(r), 022 = r^2 sin^2 <p, g 33 = r^2 ,
044 = H(r) and all other g^- are zero. Similarly, for the inverse tensor,
0 ii = l/F(r), g^22 = l/(r^2 sin^2 <p), g^33 = 1/r^2 , g^44 = l/H(r) and all other
gu are zero.
Step 2. Using formulas (2.4.27), verify that, of the 40 values of rjfc, only
nine are non-zero:

,i F'(r) pi rsin^2 <p ril

u~2F(r)' ™~ F(r) ' 133 _

l2 1 p 2 COStp 3 1 3

12 ~ ' l 23 — • i i 13 "~ i L 22 —

r sinw r

r pi H'(r).

F(rY^44 2F(r)'

sin <p cos </>;F r^4 ^'(r) 14 =

(2.4.37)

57ep 5. Using formulas (2.4.26), verify that, out of 10 values of R^, only
four are non-zero:

_ H"(r) (H'(r))^2 F'(r) F'(r)H'(r)

(^11) ~ 2H(r) 4H (^2) (r) rF(r) AF(r)H(r)'
, , 1 rF'(r) rH'ir) . ,
1 rF'(r) rH'(r) R 22
F(r) 2F^2 (r) + F(r)H(r) ~ sin^2 <p]
(2.4.38)
H"(r) F'(r)H'(r) H'(r) (H'(r))^2
2F(r) AF^2 (r) rF(r) 4F(r)H(r)'
Step 4- (a) Multiply by 4rF(r)H^2 (r) the equation Rn = 0, multiply by
4rF^2 (r)H(r) the equation R44 — 0, and take the difference to conclude
that AH(r)(H(r)F(r))' = 0 or F(r)H(r) = A for some real number A. (b)
Multiply by 2F^2 (r)H(r) the equation R 33 = 0 and use F(r)H(r) = A to
conclude that rF'(r) = F(r)(l - F(r)) or F(r) = (1 + lf(Br))-^1 for some
real number B.
Step 5. We have lim^oo F(r) = lim,.-^ H(r) = 1 and, by (2.4.18) on page
104, far away from the gravitating mass the metric should be flat, that is