# Riemannian geometry a beginners guide - Morgan F.

ISBN 0-86720-242-4

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7.4. The Schwarzschild metric. The most basic example in general relativity is the effect on the Lorentz metric of a single point mass, such as a sun at the center of a solar system. We will assume that the metric takes the simple form

dr2 = dr2 - r2(d<p2 + sin2 <p dd2) + e*r) dt2, (1)

where A(r) and v{r) are functions to be determined. This metric is spherically symmetric and time-independent. For physical reasons, Einstein further assumed that what is now called the Einstein tensor vanishes:

dr2 = —dr2 - r2 dcp2 — r2 sin2 <p dd2 + dt2.

(4)

C?k = tf/RJk-5R8lk = 0.

(2)

GENERAL RELATIVITY 59

To employ this assumption, we now compute the Einstein tensor for the metric (1). We order the variables r, <p, 6, t. We compute first the metric

gll = - ek, g22 = - Г2, g33 = - r2 sin2 <p, g44 = ev,

others vanish,

gn=-e~k, g22=-r~2, g33 = - r~2sin_2<p,

g44 = e~v, others vanish;

then the Christoffel symbols

Til = §A', ГІ2 = -г<ГА, Г33 = -re~A sin2 <p, (3)

ТІ4 = kv'eV~X, Г22Гіз = г_1, Гзз = -sin <pcos <p,

Г2з = cot <p, Г14 = 2v', others vanish,

where A' denotes dXldr, then some components of the Riemannian curvature tensor

•^121 = *131 = 2.Г XA', *141 = — 2.V" + (2 ^)(2 A9 — 4^' 2,

*212 ~ 2rX' e A, R232 =l—e A, R242 = 2v'{—re A),

*зіз = \rX' sin2 <pe~k, R\23 = sin2 <p(l - e~k),

R343 = hv’(-r)e~ksin2 <p,

R\x4=l2eV-\v" + \v'2 -\v'X'),

R\24 = Rl34 = \r-1v'el'-k-then some components of the Ricci curvature

*n = r_1 X’ -\v" + \v’X’ -\v'2,

R22 = 1 + \re~\X’ - v') - e~k,

R33 = sin2 (p R22,

R44 = \ev-\v" + У2 -\v'X' + 2r"V),

R = —2r~2 + e~\v" - 2r_1A' -\v'X' + \v'2 + 2+ 2r~2)\ and finally some components of the Einstein tensor

Gi = g%k ~ s*5‘*,

G\ = r-2 + e-\-r1i/ -r~2),

Gl = G33 = e~\-\v" + §r_1A' - + b'A' -\v’2),

G44 = r2 + e-\r~lX' - Г2).

Since G$ = 0, e~k = 1 - yr-1 for some constant у (just check that

60 CHAPTER 7

dyldr = 0). Consideration of a test particle with 0 velocity and large r (see Exercise 7.1) leads to the conclusion that y = 2GM, where M is the central mass and G is the gravitational constant. Therefore

e~A = l- 2GMr~\

Since G{ = Gt = 0, A + v is constant. Since the metric should look like the Lorentz metric for r huge, we conclude that A+ v—0. Therefore

ev = e~K=\-2 GMr~\ (4)

Now Gl = G\ = 0 automatically.

We have obtained the famous Schwarzschild metric

dr2 = -(1 - 2GMr~1)~1 dr2 - r2(d<p2 + sin2 <pd62)

+ (1 - 2GMr1) dt2. (5)

Notice that if M = 0, the Schwarzschild metric (5) reduces to the Lorentz metric 7.2(4). Notice too that as r decreases to 1I2GM, dr2 blows up: shrinking the sun to a point mass has created a black hole of radius 1/2GM!

7.5. Relativistic celestial mechanics. Now we are ready to see what differences general relativity predicts for Mercury’s orbit. The physics is embodied in the four equations for geodesics 6.5(2) in the Schwarzschild metric 7.4(5). Four equations should let us solve for r, (p, 6, and t as functions of r. Actually, instead of the first equation for geodesics involving d2r/dT2, we will use the identity dr2 = gij dxl dx!\

To compute the three other geodesic equations, we proceed from 7.4(4) to compute

Л' = -v' = -2GM(r2 - 2GMr)~\

A" =-v" = 2GM(r2 - 2GMr)-\2r - 2GM),

and then, from 7.4(3),

fix = -GM{r2 - 2GMr)~x

GENERAL RELATIVITY 61

T\2 = -r( 1 - 2GMr“1) = 2GM - r

r^3 = (2GM - r) sin2 <p

fi* = 1(1 - 2GMr~1)(2GMr~ 2)

p2 _ p3 _ -1

1 17 ” 1 H ” /

12 — 1 13 -2 _

r33 = —sin <pcos <p I23 = cot cp

T\4 = GM(r2 - 2GMr)-\

Hence the last three geodesic equations [compare to 6.5(2)] are

d2<p . _xdrd<p . (dti\ _ /TTN

—- + 2r sin <p cos <p ( — I =0, (II)

dr2 dr dr \dr]

dd _ _j dr dd _ dipdd _ /TTTN

—- + 2r 1----------+ 2cot<p —— = 0, (III)

dr dr dr dr dr

dh 2 GM dLdL = ()

dr r —2 GMrdrdr

The solution of equations I through IV will give Mercury’s orbit. Assuming that initially dipt dr and cos <p are 0, by (II) <p remains 7t/2. Thus, even relativistically, the orbit remains planar. The other three equations become

-d - r-(f)’ * o - -1. m

+ 2GM(r2 - 2GMr)~1 — — = 0. (IV')

dr2 dr dr

Integrating III' and IV' yields

r2 — = h (a constant), (III")

dr

(1 - 2GMr~l) — = /3 (a constant). (IV")

dr

62 CHAPTER 7

Therefore I' becomes

~r~4{d~d) ~ r ^ ~ 2GMr ^ + P2h~2 = h~2^ ~ 2GMr_1)- (*")

Putting r = u~1 yields

(—) = 2Gm( u3 —u2 + ptu + Po

\dd) \ 2 GM >

for some constants /30, Pi. The maximum and minimum values ulf u2 of u must be roots. Since the roots sum to 1I2GM, the third root is 1I2GM — Mi - u2, and hence

2

(dtf) = 2GM^U ~ Ul^u ~ “2^(M ~~ 2GM + Ul + u2)’

[1 - 2GM(u + Ui + u2)\

\du\ V(mi — u)(u — u2)

_ 1 + GM{u + Ux + u2)

V(«i — u)(u — u2)

To first approximation the orbit is the classical ellipse

u = /-1(1 + ecos 6),

with Mi = l~x{\ + e), u2 = l~l{ 1 — e), and mean distance

/

2 \Mi u2) 1 — e2

For one revolution,

277

f 1 + GMl~\3 + ecos 0) „ J/4

A0« . v ; I esm 6\ dd

J V/ e(l — cos 6)1 e(l + COS 6)

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