the remainder of this chapter, however, we will restrict our attention to the region

r > 2GM/c2 . We will often use the shorthand ≡ GM/c2 when writing down

this metric.

9.3 Birkhoff™s theorem

If we do not demand that our original metric is static (or stationary) but only that

it is isotropic, then we would substitute the more general form (9.3),

ds2 = A t r dt2 ’ B t r dr 2 ’ r 2 d + sin2 d

2 2

into Einstein™s empty-space field equations R = 0 in order to determine the

functions A t r and B t r . On repeating our earlier analysis, we would find

some additional non-zero connection coefficients and components of the Ricci

tensor. However, on solving this new set of equations, one discovers that the

resulting metric must still be the Schwarzschild metric (9.12). Thus, we obtain

Birkhoff™s theorem, which states that the spacetime geometry outside a general

spherically symmetric matter distribution is the Schwarzschild geometry.

This is an unexpected result because in Newtonian theory spherical symmetry

has nothing to do with time dependence. This highlights the special character of

the empty-space Einstein equations and of the solutions they admit. In particular,

Birkhoff™s theorem implies that if a spherically symmetric star undergoes strictly

radial pulsations then it cannot propagate any disturbance into the surrounding

space. Looking ahead to Chapter 18, this means that a radially pulsating star

cannot emit gravitational waves.

One can show that the converse of Birkhoff™s theorem is not true, i.e. a matter

distribution that gives rise to the Schwarzschild geometry outside it need not be

spherically symmetric. Indeed, some specific counter-examples are known.

9.4 Gravitational redshift for a fixed emitter and receiver

We begin our discussion of the physics in the vicinity of a spherical mass M by

considering the phenomenon of gravitational redshift. In particular, we consider

the specific example of an emitter, at fixed spatial coordinates rE E E ,

which emits a photon that is received by an observer at fixed spatial coordinates

rR R R . If tE is the coordinate time of emission and tR the coordinate time

203

9.4 Gravitational redshift for a fixed emitter and receiver

D (tR + ∆tR, rR, θR, φR)

ic

es

od

ge

l

ul

N

C (tR , rR, θR, φR)

sic

B (tE + ∆tE, rE, θE, φE)

de

eo

lg

ul

N

A (tE , rE, θE, φE)

Emitter Receiver

at fixed at fixed

(rE, θE, φE) (rR, θR, φR)

Figure 9.1 Schematic illustration of the emission and reception of two light signals.

of reception then the photon travels from the event tE rE E E to the event

tR rR R R along a null geodesic in the Schwarzschild spacetime. This is

illustrated schematically in Figure 9.1, which also shows a second photon, emitted

at a later coordinate time tE + tE and received at tR + tR .

In Appendix 9A we present an approach for calculating gravitational redshifts

in more general situations. Nevertheless, in this simple case, it is instructive to

derive the result in a more elementary manner: we need only use the fact that the

photon geodesic is a null curve.3 Thus ds2 = 0 at all points along it, and from the

Schwarzschild metric (9.12), we find that

’1

2 2

1’ dt = 1 ’ dr 2 + r 2 d + r 2 sin2 d

2 2 2 2

c

r r

where we have written ≡ GM/c2 . Let us consider the first signal. Thus, if is

some affine parameter along the null geodesic then we have

’1/2 1/2

dxi dxj

1 2

dt

= 1’ ’gij

d c r dd

3

This approach is based on that presented in J. Foster & J. D. Nightingale, A Short Course in General Relativity,

Springer-Verlag, 1995.

204 The Schwarzschild geometry

where as before we use the notation x = t r , the g are the components

of the Schwarzschild metric and Latin indices run from 1 to 3. On integrating,

we obtain

’1/2 1/2

dxi dxj

1 2

R

tR ’ tE = 1’ ’gij d

c r dd

E

where E is the value of at emission and R the value at reception. The

important thing to notice about this expression is that the integral on the right-

hand side depends only on the path through space. Thus, for a spatially fixed

emitter and receiver, tR ’ tE is the same for all signals sent. Thus the coordinate

time difference tE separating events A and B is equal to the coordinate time

tR between events C and D,

tR = tE

Now let us consider the proper time intervals along the worldlines of the emitter

and receiver between each pair of events. Along both the emitter™s and receiver™s

worldlines, dr = d = d = 0. Thus, from the Schwarzschild line element (9.12),

in both cases

2

c2 d 2 ≡ ds2 = c2 1 ’ dt2

r

Moreover, in each case r is constant along the worldline, so we can immediately

integrate this equation to obtain

1/2 1/2

2 2

= 1’ = 1’

and

tE tR

E R

rE rR

tR = tE , we find that

Thus, since

1/2

1 ’ 2 /rR

R

=

1 ’ 2 /rE

E

which forms the basis of the formula for gravitational redshift. If we think of the

two light signals as, for example, the two wavecrests of an electromagnetic wave,

then it is clear that this ratio must also be the ratio of the period of the wave

as observed by the receiver and emitter respectively. Thus the frequencies of the

photon as measured by each observer are related by

1/2

1 ’ 2GM/ rE c2

R

= (9.13)

1 ’ 2GM/ rR c2

E

which shows that if rR > rE . The photon redshift z is defined by

<

R E

’1

1+z = R/ E

205

9.5 Geodesics in the Schwarzschild geometry

There is an important point to notice about this derivation. It can be generalised

very easily to any spacetime in which we can choose coordinates such that the

spacetime is stationary 0 g = 0 and g0i x = 0. In this case,

ds2 = g00 x dt2 + gij x dxi dxj

where, as indicated, all the metric components are independent of t. By repeating

the above derivation for an emitter and observer at fixed spatial coordinates in

this more general spacetime, we easily find that

1/2

gx

= 00 E

R

(9.14)

g00 xR

E

The derivations presented here depend crucially upon the fact that the emitter

and receiver are spatially fixed. However, this is not often physically realistic.

For example, we might want to calculate the gravitational redshift of a photon

if the emitter or receiver (or both) are in free fall or moving in some arbitrary

manner. A method for calculating redshifts in such general situations is given in

Appendix 9A. In order to use this formalism, however, we require knowledge

of the paths followed by freely falling particles and photons. Therefore, we now

consider geodesics in the Schwarzschild geometry.

9.5 Geodesics in the Schwarzschild geometry

In deriving the Schwarzschild line element,

’1

2 2

ds = c 1’ dt ’ 1 ’ dr 2 ’ r 2 d ’ r 2 sin2 d

2 2 2 2 2

(9.15)

r r

we also calculated the connection coefficients for this metric. Thus we could

now write down the geodesic equations for the Schwarzschild geometry in the

form

d2 x dx dx

+ =0

d2 dd

where is some affine parameter along the geodesic x . It is more instruc-

tive, however, to obtain the geodesic equations using the very neat ˜Lagrangian™

procedure discussed in Chapter 3.

Thus, let us consider the ˜Lagrangian™ L = g x x , where x ≡ dx /d .

™™ ™

Using (9.15), L is given by

’1

2 2

r 2 ’ r 2 ™ 2 + sin2 ™2

™2

L=c 1’ t ’ 1’ ™

2

(9.16)

r r

206 The Schwarzschild geometry

The geodesic equations are then obtained by substituting this form for L into the

the Euler“Lagrange equations

d L L

’ =0

™

d x x

Performing this calculation, we find that the four resulting geodesic equations (for

= 0 1 2 3) are given by

2

™

1’ t=k (9.17)

r

’1 ’2

c2 2

2 2

r 2 ’ r ™ 2 + sin2 ™2 = 0

™

1’ r + 2 t ’ 1’

¨ ™ (9.18)

2

r r r r

¨ + 2 r ™ ’ sin cos ™ 2 = 0

™ (9.19)

r

r 2 sin2 ™ = h (9.20)

In (9.17) and (9.20) respectively, the quantities k and h are constants. These two

equations are derived immediately since L is not an explicit function of t or .

We see immediately that = /2 satisfies the third geodesic equation (9.19).

Because of the spherical symmetry of the Schwarzschild metric we can therefore,

with no loss of generality, confine our attention to particles moving in the ˜equato-

rial plane™ given by = /2. In this case our set of geodesic equations reduces to

2

™

1’ t=k (9.21)

r

’1 ’2

c2 2

2 2

r2 ’ r ™ 2 = 0

™

1’ r + 2 t ’ 1’

¨ ™ (9.22)

r2

r r r

r2 ™ = h (9.23)

These equations are valid for both null and non-null affinely parameterised

geodesics. In each of these cases, however, it is easier to replace the rather

complicated r-equation (9.22) by a first integral of the geodesics equations. For

a non-null geodesic this first integral is simply

g x x = c2

™™ (9.24)

whereas for a null geodesic it is

g x x =0

™™ (9.25)

Before going on to discuss separately non-null and null geodesics in the equa-

torial plane = /2, it is instructive to discuss the physical interpretation of the

207

9.6 Trajectories of massive particles

constants k and h. One can arrive at equations (9.21) and (9.23) simply by using

the fact that the components p0 and p3 of a particle™s 4-momentum are conserved

along geodesics since L does not depend explicitly on t and (remember that p

is proportional to the tangent vector to the geodesic at each point). For notational

simplicity, for a massive particle, we shall take the particle to have unit rest

mass and choose the affine parameter to be the particle™s proper time , so that

p = x . Similarly, for a massless particle we are free to choose an appropriate

™

affine parameter along the null geodesic, once again such that p = x . Thus, for

™

= /2 we may write

2

™ ™

p0 = g00 t = c2 1 ’ t = kc2 (9.26)

r

p3 = g33 ™ = ’r 2 ™ = ’h (9.27)

where in the last equality on each line we have defined the constants in a manner

that coincides with (9.21) and (9.23). Let us first consider the constant k. If, at some

event, an observer with 4-velocity u encountered a particle with 4-momemtum p

then he would measure the particle™s energy to be

E = p·u = p u

For an observer at rest at infinity we have u = 1 0 0 0 and so E = p0 = kc2

(which is conserved along the particle geodesic). Thus we may take k = E/c2 ,

where E is the total energy of the particle in its orbit. Since for massive particles

we have assumed unit mass, in the general case we have k = E/ m0 c2 , where

m0 is the rest mass of the particle. For the constant h, we can see immediately

from (9.27) that it equals the specific angular momentum of the particle and that

(as result of the choice of signature for the metric) p3 is equal to minus the

specific angular momentum. Finally, we note that the results (9.17“9.20) can also

be derived using the alternative form (3.56) of the geodesic equations, which may

be written

p= 1

(9.28)

g pp

2

9.6 Trajectories of massive particles

The trajectory of a massive particle is a timelike geodesic. Considering motion in

the equatorial plane, we replace the geodesic equation (9.22) by (9.24), where g

is taken from (9.15) with = /2. Moreover, since we are considering a timelike

geodesic we can choose our affine parameter to be the proper time along the

208 The Schwarzschild geometry

path. Thus we find that the worldline x of a massive particle moving in the

equatorial plane of the Schwarzschild geometry must satisfy the equations

2

™

1’ t=k (9.29)

r

’1

2 2

r 2 ’ r 2 ™ 2 = c2

™2

1’ t ’ 1’ ™

2

(9.30)

c

r r

r2 ™ = h (9.31)

By substituting (9.29) and (9.31) into (9.30), we obtain the combined ˜energy™

equation for the r-coordinate,

h2 2GM 2GM

r + 2 1’ 2

™ ’ = c 2 k2 ’ 1

2

(9.32)

r cr r

where we have written = GM/c2 . We shall use this ˜energy™ equation to discuss

radial free fall and the stability of orbits. Note that the right-hand side is a constant

of the motion. We can verify the physical meaning of the constant k by noting

that E ∝ k. The constant of proportionality is fixed by requiring that, for a particle

at rest at r = , we have E = m0 c2 . Letting r ’ and r = 0 in (9.32), we thus

™

2 = 1. Hence, as previously, we must have k = E/ m c2 , where E is

require k 0

the total energy of the particle in its orbit.

A second useful equation, which enables us to determine the shape of a particle

orbit (i.e. r as a function of ), may be found by using h = r 2 ™ to express r in ™

the energy equation (9.32) as

dr dr d h dr

= =2

d dd rd

We thus obtain

2

h2 2GM 2GMh2

h dr

+ 2 = c k ’1 + + 23

22

r2 d r r cr

If we make the substitution u ≡ 1/r that is usually employed in Newtonian orbit

calculations, we find that

2

c2 2 2GMu 2GMu3

du

+u = 2 k ’1 + +

2

h2 c2

d h

We now differentiate this equation with respect to to obtain finally

d2 u GM 3GM

+ u = 2 + 2 u2 (9.33)

d2 h c

209

9.7 Radial motion of massive particles

In Newtonian gravity, the equations of motion of a particle of mass m in the

equatorial plane = /2 may be determined from the Lagrangian

GMm

L = 2 m r2 + r2 ™ 2 +

™

1

r

From the Euler“Lagrange equations we have

r2 ™ = h

h GM

r=

¨ ’2

r3 r

where the integration constant h is the specific angular momentum of the particle.

If we now substitute u = 1/r and eliminate the time variable, the Newtonian

equation of motion for planetary orbits is obtained:

d2 u GM

+u = 2 (9.34)

d2 h

We must remember, however, that in this equation u = 1/r, where r is the radial

distance from the mass, whereas in (9.33) r is a radial coordinate that is related

to distance through the metric. Nevertheless, the forms of the two equations are

very similar except for the extra term 3GMu2 /c2 in (9.33). We note that this term

correctly tends to zero as c ’ .

Two interesting special cases of massive-particle orbits are worth investigating

in detail, namely radial motion and motion in a circle.

9.7 Radial motion of massive particles

is constant, which implies that h = 0. Thus, (9.32) reduces to

For radial motion

2GM

r 2 = c 2 k2 ’ 1 +

™ (9.35)

r

™

Differentiating this equation with respect to and dividing through by r gives

GM

r =’

¨ (9.36)

r2

which has precisely the same form as the corresponding equation of motion in

Newtonian gravity. This does not imply, however, that general relativity and

Newtonian gravity predict the same physical behaviour. It should be remem-

bered that in (9.36) the coordinate r is not the radial distance, and dots indicate

derivatives with respect to proper time rather than universal time.

210 The Schwarzschild geometry

As a specific example, consider a particle dropped from rest at r = R. From

(9.35) we see immediately that k2 = 1 ’ 2GM/ c2 R , so (9.35) can be written

™

r2 11

= GM ’ (9.37)

2 rR

This has the same form as the Newtonian formula equating the gain in kinetic

energy to the loss in gravitational potential energy for a particle (of unit mass)

falling from rest at r = R. This provides a useful way to remember this equation,

but the different meanings of r and of the dot should again be borne in mind.

We could continue our analysis of this quite general situation, but we can

illustrate the main physical points by considering a particle dropped from rest at

infinity. In this case k = 1 and the algebra is much less complicated. Thus, setting

k = 1 in the geodesic equation (9.29) and in (9.35), we obtain

’1

2

dt

= 1’ (9.38)

d r

1/2

2 c2

dr

=’ (9.39)

d r

where in (9.39) we have taken the negative square root. These equations form the

basis of our discussion of a radially infalling particle dropped from rest at infinity.

From these equations we see immediately that the components of the 4-velocity

of the particle in the t r coordinate system are simply

’1 1/2

2 c2

2

dx

≡ = 1’ ’ 00

u

d r r

Equation (9.39) determines the trajectory r . On integrating (9.39) we imme-

diately obtain

3

r3

2 2

r0

= ’

2 c2 3 2 c2

3

where we have written the integration constant in a form such that = 0 at r = r0 .

Thus is the proper time experienced by the particle in falling from r = r0 to a

coordinate radius r.

Instead of parameterising the worldline in terms of the proper time , we can

alternatively describe the path as r t , thereby mapping out the trajectory of the

particle in the t r coordinate plane. This is easily achieved by writing

1/2

2 c2 2

dr dr d

= =’ 1’ (9.40)

dt d dt r r

211

9.7 Radial motion of massive particles

On integrating, we find

⎛ ⎞

3

r3 ⎠ 4

2⎝ r0 r0 r

t= ’ + ’

2 c2 2 c2

3 2 2

c

+1 ’1

r/ 2 r0 / 2

2

+ ln

’1 +1

c r/ 2 r0 / 2

where the choice of the integration constant gives t = 0 at r = r0 .

In particular we note that

3

2 r0

’ as r ’ 0

2 c2

3

t’ as r ’ 2

Evidently, the particle takes a finite proper time to reach r = 0. When the worldline

is expressed in the form r t , however, we see that r asymptotically approaches 2

as t ’ . Since the coordinate time t corresponds to the proper time experienced

by a stationary observer at large radius, we must therefore conclude that, to such

an observer, it takes an infinite time for the particle to reach r = 2 . We return

to this point later.

It is interesting to ask what velocity a stationary observer at r measures for

the infalling particle as it passes. From the Schwarzschild metric (9.15) we see

that, for a stationary observer at coordinate radius r, a coordinate time interval dt

corresponds to a proper time interval

1/2

2

dt = 1 ’ dt

r

Similarly, a radial coordinate separation dr corresponds to a proper radial distance

measured by the observer equal to

’1/2

2

dr = 1 ’ dr

r

Thus the velocity of the radially infalling particle, as measured by a stationary

observer at r, is given by

’1 1/2

2 c2

2

dr dr

= 1’ =’ (9.41)

dt r dt r

Thus we find the rather surprising result that, as the particle approaches r = 2 ,

a stationary observer at that radius observes that the particle™s velocity tends to c.

We note that the equation (9.41) is only physically valid for r > 2 since, as we

shall see, it is impossible to have a stationary observer at r ¤ 2 .

212 The Schwarzschild geometry

9.8 Circular motion of massive particles

For circular motion in the equatorial plane we have r = constant, and so r = r = 0.

™¨

Setting u = 1/r = constant in the ˜shape™ equation (9.33) we have

GM 3GM 2

u= + 2u

h2 c

from which we find that

c2 r 2

h=2

r ’3

Putting r = 0 in the energy equation (9.32) and substituting the above expression

™

2 allows us to identify the constant k:

for h

1 ’ 2 /r

k= (9.42)

1 ’ 3 /r 1/2

The energy of a particle of rest mass m0 in a circular of radius r is then given by

E = km0 c2 . We can use this result to determine which circular orbits are bound.

For this we require E < m0 c2 , so the limits on r for the orbit to be bound are

given by k = 1. This yields

1 ’ 2 /r = 1 ’ 3 /r

2

which is satisfied when r = 4 or r = . Thus, over the range 4 < r < ,

circular orbits are bound. A plot of E/ m0 c2 as a function of r/ is shown in

Figure 9.2.

1.1

1.05

c2)

E / (m0

1

0.95

4 6 8 10 12 14 16 18 20

r/µ

Figure 9.2 The variation of k = E/ m0 c2 as a function of r/ for a circular

orbit of a massive particle in the Schwarzschild geometry.

213

9.9 Stability of massive particle orbits

We can obtain another useful result by substituting our expression for h2 into

the geodesic equation r 2 ™ = h; then we can write

2

c2

d

=2

r r ’3

d

The significance of this equation is that it cannot be satisfied for circular orbits

with r < 3 . Such orbits cannot be geodesics (since they do not satisfy the

geodesic equations) and so cannot be followed by freely falling particles. Thus,

according to general relativity a free massive particle cannot maintain a circular

orbit with r < 3 around a spherical massive body, no matter how large the

angular momentum of the particle. This is very different from Newtonian theory.

It is also useful to calculate the expression for d /dt, which is given by

2 2 2

1 ’ 2 /r 2 c2 GM

d dd d

= = = 3= 3

k2

dt d dt d r r

This expression is exactly the same as the Newtonian expression for the period

of a circular orbit of radius r. Although we cannot say that r is the radius of

the orbit in the relativistic case, we see that the spatial distance travelled in one

complete revolution is 2 r, just as in the Newtonian case.

9.9 Stability of massive particle orbits

The above analysis appears to suggest that the closest bound circular orbit around

a massive spherical body is at r = 4 . However, we have not yet determined

whether this orbit is stable.

In Newtonian dynamics the equation of motion of a particle in a central potential

can be written

1 dr 2

+ Veff r = E

2 dt

where Veff r is the effective potential and E is the total energy of the particle

per unit mass. For an orbit around a spherical mass M, the effective potential is

h2

GM

Veff r = ’ +2 (9.43)

2r

r

where h is the specific angular momentum of the particle. This effective potential

is shown in Figure 9.3. It can be seen that bound orbits have two turning points

and that a circular orbit corresponds to the special case where the particle sits at

the minimum of the effective potential. Furthermore one sees that, in Newtonian

dynamics, a finite angular momentum provides an angular momentum barrier

preventing a particle reaching r = 0. This is not true in general relativity.

214 The Schwarzschild geometry

V(r)

effective potential

unbound orbit

r

elliptical orbit

circular orbit

“GM

r

Figure 9.3 The Newtonian effective potential for h = 0, showing how an angular

momentum barrier prevents particles reaching r = 0.

In general relativity, the ˜energy™ equation (9.32) for the motion of a particle

around a central mass can be written

2

h2 c2 c2 2

1 2

dr

+ 2 1’ ’ = k ’1

2 2r 2

d r r

where we recall that the constant k = E/ m0 c2 . Thus in general relativity we

identify the effective potential per unit mass as

c2 h2 h2

r =’ + 2’ 3 (9.44)

Veff

2r

r r

which has an additional term proportional to 1/r 3 as compared with the Newtonian

case (9.43). Remembering that = GM/c2 , we see that (9.44) reduces to the

form (9.43) in the non-relativistic limit c ’ .

Figure 9.4 shows the general relativistic effective potential for several values of

¯

h ≡ h/ c . The dots indicate the locations of stable circular orbits, which occur

at the local minimum of the potential. The local maxima in the potential curves

¯

are the locations of unstable circular orbits. For any given value of h, circular

orbits occur where dVeff /dr = 0. Differentiating (9.44) gives

c2 h2 3 h2

dVeff

= 2 ’ 3+ 4

dr r r r

and so the extrema of the effective potential are located at the solutions of the

quadratic equation

c2 r 2 ’ h2 r + 3 h2 = 0

215

9.9 Stability of massive particle orbits

0.1

0.05

4.5

0

4.0

Veff /c 2

3.75

“ 0.05

2*(3)1/2

“ 0.1 h=0

“ 0.15

5 10 15 20 25

r /µ

Figure 9.4 The general relativistic effective potential plotted for several values

¯

of the angular momentum parameter h.

which occur at

h

r= h ± h2 ’ 12 2 c2

2

2c

√ √

We note, in particular, that if h = 12 c = 2 3 c then there is only one

extremum, and there are no turning points in the orbit for lower values of h. The

significance of this result is that the innermost stable circular orbit has

6GM

rmin = 6 =

c2

√

This orbit, with r = 6 and h/ c = 2 3, is unique in satisfying both

dVeff /dr = 0 and d2 Veff /dr 2 = 0, the latter being the condition for marginal

stability of the orbit.

The existence of an innermost stable orbit has some interesting astrophysical

consequences. Gas in an accretion disc around a massive compact central body

settles into circular orbits around the compact object. However, the gas slowly

loses angular momentum because of turbulent viscosity (the turbulence is thought

to be generated by magnetohydrodynamic instabilities). As the gas loses angular

momentum it moves slowly inwards, losing gravitational potential energy and

heating up. Eventually it has lost enough angular momentum that it can no longer

follow a stable circular orbit, and so it spirals rapidly inwards onto the central

object.

We can make an estimate of the efficiency of energy radiation in an accretion

disc. The maximum efficiency is of the order of the ˜gravitational binding energy™

216 The Schwarzschild geometry

at the innermost stable circular orbit (i.e. the energy E lost as the particle moves

from infinity to the innermost orbit) divided by the rest mass energy of the particle.

Setting r = 6 in (9.42) and remembering that k = E/ m0 c2 , we find that

√

22

E

= ≈ 0 943

m0 c2 3

Thus the maximum radiation efficiency of the accretion disc is

≈ 1 ’ 0 943 = 5 7%

acc

Thus, an accretion disc around a highly compact astrophysical object can convert

perhaps a few percent of the rest mass energy of the gas into radiation; this may be

compared with the efficiency of nuclear burning of hydrogen to helium (26 MeV

per He nucleus),

∼ 0 7%

nuclear

Accretion discs are therefore capable of converting rest mass energy into radiation

with an efficiency that is about 10 times greater than the efficiency of the nuclear

burning of hydrogen. The ˜accretion power™ of highly compact objects (such as

black holes) cause some of the most energetic phenomena known in the universe.

A physically intuitive picture of a non-circular orbit and the capture of a particle

with non-zero angular momentum h may be obtained by differentiating the energy

12

10

8

6

4

2

0

“5 0 5 10 15 20

“2

“4

Figure 9.5 Orbit for a particle projected azimuthally from r = 20GM/c2 with

√

h = 3 5GM/c. A circular orbit would require h = 20/ 17 GM/c. The points

are plotted at equal intervals of the particle™s proper time.

217

9.10 Trajectories of photons

equation (9.32) for massive particle orbits with respect to proper time . Using

the original equation to remove the first derivative dr/d , we find that

d2 r GM h2 3h2 GM

=’ 2 + 3 ’ 2 4

d2 r r cr

As we might expect, the first two terms on the right-hand side are very like

the Newtonian expressions corresponding to an inward gravitational force and a

repulsive ˜centrifugal force™ proportional to h2 . The third term is new, however,

and is also proportional to h2 but this time acts inwards. This shows that close to a

highly compact object, specifically within the radius r = 3GM/c2 , the centrifugal

force ˜changes sign™ and is directed inwards, thus hastening the demise of any

particle that strays too close to the object. This leads to spiral orbits of the type

shown in Figure 9.5.

9.10 Trajectories of photons

The trajectory of a photon (and of any other particle having zero rest mass) is a

null geodesic. We cannot use the proper time as a parameter, so instead we use

some affine parameter along the geodesic. Considering motion in the equatorial

plane, the equations of motion are given by the geodesic equations (9.21) and

(9.23), and we replace the r-equation (9.22) by the condition g x x = 0. Thus

™™

we have

2

™

1’ t=k (9.45)

r

’1

2 2

r2 ’ r2 ™ 2 = 0

™2

1’ t ’ 1’ ™

2

(9.46)

c

r r

r2 ™ = h (9.47)

For photon trajectories, an analogue of the energy equation (9.32) can again be

obtained by substituting (9.45) and (9.47) into (9.46), which gives

h2 2

r + 2 1’

™ = c 2 k2

2

(9.48)

r r

Similarly, the analogue for photons of the shape equation (9.33) is obtained by

substituting h = r 2 ™ into (9.46) and using the fact that

dr dr d h dr

= =

r2 d

d dd

218 The Schwarzschild geometry

Making the usual substitution u ≡ 1/r and differentiating with respect to we

find

d2 u 3GM

+ u = 2 u2 (9.49)

d2 c

It is again worth mentioning the two special cases of radial motion and motion in

a circle.

9.11 Radial motion of photons

For radial motion ™ = 0 and (9.46) reduces to

’1

2 2

™2

1’ t ’ 1’ r2 = 0

™

2

c

r r

from which we obtain

2

dr

= ±c 1 ’ (9.50)

dt r

On integrating, we have

r

ct = r + 2 ln ’ 1 + constant outgoing photon

2

r

ct = ’r ’ 2 ln ’ 1 + constant incoming photon

2

Notice that under the transformation t ’ ’t, incoming and outgoing photon paths

are interchanged, as we would expect. In fact for the moment the differential

equation (9.50) is more useful. In a ct r -diagram, we see that the photon

worldlines will have slopes ±1 as r ’ (forming the standard special-relativistic

lightcone), but their slopes approach ± as r ’ 2 . This means that they become

more vertical; the cone ˜closes up™.

Our knowledge of the lightcone structure allows us to construct the ˜picture™

behind our earlier algebraic result that a particle takes infinite coordinate time to

reach the horizon; this is illustrated in Figure 9.6. The curved solid line is the

worldline of a massive particle dropped from rest by an observer fixed at r = R.

Since massive particle worldlines are confined within the forward lightcone in any

event, the closing up of the lightcones forces the worldlines of massive particles

to become more vertical as r ’ 2 . Thus, the particle ˜reaches™ r = 2 only

at t = . Further, suppose that at some point along its trajectory the particle

emits a radially outgoing photon in the direction of the observer. The tangent to

219

9.12 Circular motion of photons

Particle Observer

ct

n

oto

Ph

r = 2µ r=R

Figure 9.6 A radially infalling particle emitting a radially outgoing photon. The

wavy line indicates the singularity at r = 0.

the resulting photon worldline must, at any event, lie along the outward-pointing

forward lightcone at that point. This is illustrated by the broken line in Figure 9.6.

Thus, in the limit where the particle approaches r = 2 , the initial direction of

the photon worldline approaches the vertical and so the photon will be received

by the observer only at t = . Thus to an external observer the particle appears

to take an infinite time to reach the horizon.

As discussed earlier, however, the proper time experienced by a massive

particle in falling to r = 2 is finite. Moreover, dr/d does not tend to zero at this

point, so the particle has not ˜run out of steam™ and presumably passes beyond

this threshold. Thus, our present coordinate system is inadequate for discussing

what happens at and within r = 2 , and our ct r -diagram is in some respects

misleading in these regions. We discuss this further in Chapter 11.

9.12 Circular motion of photons

For motion in a circle we have r = constant. Thus, from the shape equation (9.49),

we see that the only possible radius for a circular photon orbit is

3GM

r=

c2

220 The Schwarzschild geometry

Therefore a massive object can have a considerable effect on the path of a photon.

There is no such orbit around the Sun, for example, since the solar radius is much

larger than 3GM /c2 ≈ 4 5 km, but outside a black hole there can be such an

orbit. As we shall see below, however, the orbit is not stable.

9.13 Stability of photon orbits

We can rewrite the ˜energy™ equation (9.48) for photon orbits as

™

r2 1

+ Veff r = 2 (9.51)

h2 b

where we have defined the quantity b = h/ ck and the effective potental

1 2

Veff r = 1’

r2 r

In fact, by rescaling the affine parameter along the photon geodesic in such a way

that ’ h the explicit h-dependence in (9.51) may be removed.

The effective potential is plotted in Figure 9.7, from which we see that Veff r

has a single maximum at r = 3 , where the value of the potential is 1/ 27 2 .

Thus the circular orbit at r = 3 is unstable. We conclude that there are no stable

circular photon orbits in the Schwarzschild geometry.

The character of general photon orbits is determined by the value of the

constant b. To find the physical meaning of b, we begin by using the geodesic

equation (9.45) and the energy equation (9.47) to write

™ ’1/2

11 1 2

d

= = 2 2 ’ 2 1’

™

dr r rb r r

Veff

1

27µ2

r

r = 2µ r = 3µ

Figure 9.7 The effective potential for photon orbits.

221

Appendix 9A: General approach to gravitational redshifts

closest

approach

r

b

φ

√

Figure 9.8 The shape of a photon orbit passing a spherical mass if b > 3 3 .

Thus, for a photon orbit, as r ’ we have

d

= ±b

r2

dr

’ 0 as r ’

Assuming that , the solution to this equation is

b

r =±

sin

which gives the equations of two straight lines with impact parameter b passing

on either side of the origin.

The nature of the orbits depends very much on the value of the impact param-

eter b. Let us first consider inward-moving photons, i.e. photons for which r is

initially decreasing. From (9.51) and Figure 9.7 we see that if 1/b2 < 1/ 27 2 ,

√

so that b > 3 3 , then the orbit will have a single turning point of closest

approach and escape again to infinity. This situation is illustrated in Figure 9.8.

√

If b < 3 3 , however, then the light ray will be captured by the massive body

and spiral in towards the origin. √

Similar considerations hold for trajectories that start at small radii. If b > 3 3

then the photon will escape, and at infinity its straight-line path will have an impact

√

parameter b. If b < 3 3 then the photon path will have a turning point, and it

will fall back towards the origin. In this case the particle does not reach infinity,

so b cannot be interpreted simply as an impact parameter. It is straightfoward to

show that if a photon is emitted from within the region r = 2 to r = 3 then the

opening angle from the radial direction for the photon to escape varies from

= 0 at r = 2 to = /2 at r = 3 .

Appendix 9A: General approach to gravitational redshifts

Consider a general spacetime with metric g in some arbitrary coordinate system

x , where x0 is a timelike coordinate and the xi are spacelike. Suppose that an

222 The Schwarzschild geometry

uR(B)

µ

p(B)

B

uE (A)

p(A)

A

Figure 9.9 Schematic illustration of the emission and reception of a photon.

emitter and a receiver have worldlines xE E and xR R respectively, where

E and R are the proper times of each observer. At some event A, emits a

photon with 4-momentum p A that is received by at an event B. Furthermore,

let us assume that at event A the emitter has 4-velocity uE A and that at event

B the receiver has 4-velocity uR B . This is illustrated schematically in Figure 9.9.

The energies of the photon as observed by the emitter at A and by the receiver

at B are respectively given by

E A = p A · uE A = p A uE A

E B = p B · uR B = p B uR B

Since in both cases E = h , the ratio of the photon frequencies is given by the

general result

p B uR B

R

= (9.52)

p A uE A

E

If we know the components of the 4-momentum p A at emission then we can

calculate the components p B at reception, using the fact that the photon travels

along a null geodesic. Since the photon 4-momentum p at any point is tangent

to this geodesic, it is parallel-transported along the path. Thus, if the photon

223

Appendix 9A: General approach to gravitational redshifts

geodesic x is described in terms of some affine parameter then

dp dx

’ =0

p

d d

Moreover, since p is tangent to the geodesics, we can choose the affine parameter

so that p = dx /d , in which case

dp

= pp

d

It is also worth remembering that a first integral of the equation for a photon

geodesic, which can prove very useful, is

p p =0 (9.53)

Let us now examine some special cases of the general formula (9.52). We

begin by considering the case in which both the emitter and the receiver

have fixed spatial coordinates. Thus, for i = 1 2 3 the spatial components of

their 4-velocities are

i i

dxE dxR

≡ =0 ≡ =0

ui ui

and

E R

dE dR

Moreover, in each case, the squared length of the 4-velocity is g u u = c2 . In

our situation, this reduces to g00 u0 2 = c2 , so we find that

c

u0 = 1/2

g00

Hence the formula (9.52) reduces to

1/2

pB g00 A

=0

R

(9.54)

p0 A g00 B

E

Let us now make the additional assumption that the metric is stationary in our

chosen cooordinate system, i.e.

=0

0g

Thus, the metric components g cannot depend explicitly on the coordinate x0 .

As shown in Section 3.19, this means that the zeroth covariant component of the

tangent vector is constant along an affinely parameterised geodesic. Since the

photon 4-momentum is simply proportional to the tangent vector, this means that

224 The Schwarzschild geometry

p0 is constant along the photon™s geodesic. Thus, in this case, (9.54) reduces

further to

1/2

gA

= 00

R

(9.55)

g00 B

E

and we have recovered the result (9.14) derived earlier.

Exercises

9.1 Show that surfaces of constant t and r in the general isotropic metric (9.3) have

surface area 4 r 2 .

9.2 For the general static isotropic metric (9.4), show that the off-diagonal components of

the Ricci tensor R are zero and that the diagonal components are given by (9.8“9.11).

9.3 The Schwarzschild line element is

’1

2 2

ds = c 1’ dt ’ 1 ’ dr 2 ’ r 2 d ’ r 2 sin2 d

2 2 2 2 2

r r

By considering the ˜Lagrangian™ L = g x x , where the dots denote differentiation

™™

with respect to an affine parameter , calculate the connection coefficients .

Hence verify that the geodesic equations are given by (9.17“9.20).

9.4 Derive the results (9.17“9.20) using the alternative form (9.28) of the geodesic

equations.

9.5 Calculate the connection coefficients and the Ricci tensor for the general isotropic

metric (9.3). Hence prove Birkoff™s theorem.

9.6 Use Birkhoff™s theorem to show that a particle inside a spherical shell of matter

experiences no gravitational force.

9.7 Show that the ˜Lemaitre™ line element

2/3 2/3

4 9 9

ds = c dw ’ dz ’ z ’ cw

2 2 2 2 2 2

d

9 2 z ’ cw 2

where d 2 = d 2 + sin2 d 2 , describes the Schwarzschild geometry. Show that

observers with fixed spatial coordinates z are in free fall and had zero velocity

at infinity, and that the proper time of such observers is w.

9.8 For a general stationary spacetime with line element

ds2 = g00 x dt2 + gij x dxi dxj

show that, for a fixed emitter and receiver, the ratio of the received photon frequency

to the emitted frequency is

1/2

gx

= 00 E

R

g00 xR

E

where xE and xR are the fixed spatial coordinates of the emitter and receiver

respectively.

225

Exercises

9.9 An isolated thin rigid spherical shell has mass M and radius R. Suppose that a

small hole is drilled through the shell, so that an observer O at the shell™s centre

can observe the outside universe. Show that a photon emitted by a fixed observer

E at r = rE (where rE > R) and received by O is blueshifted by the amount

1/2

1 ’ 2 /rE

=

O

1 ’ 2 /R

E

9.10 Show that the quantity

p2

L =p +

2 2

sin2

where p is the 4-momentum of a particle, is a constant of motion along any

geodesic in the Schwarzschild geometry. Hence show that the particle orbits in a

Schwarzschild geometry are stably planar.

9.11 For a particle dropped from rest at infinity in the Schwarzschild geometry, find

expressions for t r and r , where t is the coordinate time and is the proper

time of the particle.

9.12 A particle is dropped from rest at a coordinate radius r = R in the Schwarzschild

geometry. Obtain an expression for the 4-velocity of the particle in t r

coordinates when it passes coordinate radius r.

9.13 A particle at infinity in the Schwarzschild geometry is moving radially inwards with

coordinate speed u0 . Show that at any coordinate radius r the coordinate velocity

is given by

2 2

2GM 1 2GM

dr

= 1’ 2 c2 1 ’ 1’

2

c2 r

dt cr 0

’1/2

where 0 = 1 ’ u2 /c2 . Determine the velocity relative to a stationary observer

0

at r, and show that this velocity tends to c as r tends to 2GM/c2 , irrespective of

the value of u0 .

9.14 Suppose that the particle in Exercise 9.13 has rest mass m0 and that it stopped at

r = r1 . If its excess energy was converted to radiation that is observed at infinity,

show that the energy released as seen by a stationary observer at r1 is

E = m0 c 2 ’1

0

1 ’ 2GM/ c2 r1

What is the energy released as observed at infinity? Show that this tends to 0 m0 c2

as r1 tends to 2GM/c2 .

9.15 For a particle in a circular orbit of radius r in the Schwarzschild geometry, use the

alternative form (9.28) of the geodesic equations to show that

d GM

=3

dt r

9.16 In the Schwarzschild geometry, a photon is emitted from a coordinate radius r = r2

and travels radially inwards until it is reflected by a fixed mirror at r = r1 , so

226 The Schwarzschild geometry

that it travels radially outwards back to r = r2 . How long does the round trip take

according to a stationary observer in infinity?

9.17 A photon moves in a circular orbit at r = 3 in the Schwarzschild geometry. Show

that the period of the orbit as measured by a stationary observer at this radius is

= 6 /c. What is the period of the orbit as measured by a stationary observer

infinity?

9.18 Show that a massive particle moving in the innermost stable circular orbit in the

Schwarzschild geometry has speed c/2 as measured by a stationary observer at this

radius. Hence calculate the period of the orbit as measured by the local observer.

What is the period of the orbit as measured by a stationary observer infinity?

9.19 Alice is situated at a fixed position on the equator of the Earth (which is assumed to

be spherical). In Schwarzschild coordinates t r , her worldline is described

in terms of a parameter by

t= r =R = /2 =

where and are constants and R is the coordinate radius of the Earth™s surface.

Bob is a distant stationary observer in space. Show that he will measure the orbital

speed of Alice to be v = R / . By considering the magnitude of Alice™s 4-velocity,

show that

’1/2

v2 GM

= 1’ 2 + 2

c cr

where M is the mass of the Earth. Interpret this result physically.

9.20 All massive objects look larger than they really are. Show that a light ray grazing

the surface of a massive sphere of coordinate radius r > 3GM/c2 will arrive at

infinity with impact parameter

1/2

r

b=r

r ’ 2GM/c2

Hence show that the apparent diameter of the Sun M = 2 — 1030 kg R =

7 — 108 m exceeds the coordinate diameter by nearly 3 km.

9.21 The Hipparcos satellite can measure the positions of stars to an accuracy of 0.001

arcseconds. If it is measuring the position of a star in a direction perpendicular

to the plane of the Earth™s orbit, do Hipparcos observers need to account for the

gravitational bending of light by the Sun?

9.22 A massive particle is moving in the equatorial plane of the Schwarzschild geometry.

Show that at infinity the particle moves in a straight line with impact parameter

√

b = h/ c k2 ’ 1 .

9.23 An observer at rest at coordinate radius r = R in the Schwarzschild geometry

drops a massive particle which free-falls radially inwards. When the particle is at a

coordinate radius r = rE it emits a photon radially outwards. Find an expression for

the redshift z of the photon when it is received by the observer. Show that z ’

as rE ’ 2GM/c2 .

227

Exercises

9.24 Show that the geodesic equations for photon motion in the equatorial plane = /2

of the Schwarzschild geometry can be written in the form

’1

1 2 ™=1 1

™

t= 1’ r2 =

™ ’U r

r2 b2

bc r

where b is a constant, the dots correspond to differentiation with respect to some

affine parameter and

1 2

Ur = 1’