Suppose that a photon moving in the equatorial plane passes an observer at rest at

a coordinate radius in the range 2 < r < 3 . Show that the observer measures the

radial and azimuthmal components of the photon™s velocity to be

1/2

vr = ±c 1 ’ b2 U r v = cb U r 1/2

and

If the observer emits a photon that makes an angle with the outward radial

direction, show that the photon will escape to infinity provided that

√

sin < 3 3 U r 1/2

Find the values of at r = 2 and r = 3 .

9.25 Alice and Bob are astronauts in a space capsule, with no engine, in a circular orbit

at r = R (where R > 3 ) in the equatorial plane of the Schwarzschild geometry. At

some point in the orbit, Bob leaves the capsule, uses his rocket-pack to maintain

a hovering position at that fixed point in space and then rejoins the capsule after

it has completed one orbit. Show that the proper time interval measured by Alice

while Bob is out of the capsule is

1/2

R2

=2 R’3

A

c2

If B is the corresponding proper time interval measured by Bob between these

two events, show that

1/2

R’2

=

B

R’3

A

Briefly compare this result with the ˜twin paradox™ result in special relativity. If

Bob chooses not to rejoin the capsule but instead observes it fly past him, show

that he will measure the capsule™s speed as

1/2

c2

v=

R’2

9.26 A particle A and its antiparticle B are travelling in opposite senses in free circular

orbits in the equatorial plane of the Schwarzschild geometry, one at coordinate

228 The Schwarzschild geometry

radius rA and the other at rB > rA . At some instant, A emits a photon of frequency

A that travels radially and is received by B with frequency B . Show that

1/2

1 ’ 3 /rA

=

B

1 ’ 3 /rB

A

Suppose now that rA = rB = r, so that the particles collide and annihilate each other.

Show that the total radiated energy as measured by an observer at rest at the point

of collision is given by

1/2

1 ’ 2 /r

E = 2m0 c 2

1 ’ 3 /r

where m0 is the rest mass of each particle.

9.27 If the cosmological constant is non-zero, show that the line element outside a

static spherically symmetric matter distribution is given by

’1

r2 r2

2 2

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

2 2 2 2 2

3 3

r r

Hence show that, in the weak-field Newtonian limit, a spherically symmetric mass

M produces a gravitational field strength g given by

GM c2 r ˆ

g= ’ 2 + r

3

r

Show that the shapes of massive particle orbits in the above geometry differ

from those in the Schwarzschild geometry, but that the shapes of photon orbits

do not.

9.28 Consider a static axisymmetric spacetime that is invariant under translations and

reflections along the axis of symmetry. Show that, in general, the line element for

such a spacetime can be written in the form

ds2 = A dt2 ’ d ’B ’C

2 2

dz2

d

for arbitrary functions A, B and C. Show that the non-zero connection coefficients

for this line element are given by

A B C

= =’ =’

0 1 1

01 22 33

2A 2 2

A B C

00 = = 31 =

1 2 3

21

2 2B 2C

229

Exercises

where the primes denote d/d . Hence show that the non-zero components of the

Ricci tensor are given by

A A A B C

R00 = ’ + ’ ’

2 2 2A 2B 2C

A2 B B2 C C2

A

R11 = ’ + ’ + ’

4A2 4B2 4C 2

2A 2B 2C

B BB A C

= ’ ’ ’

R22

2 2 2B 2A 2C

C C C A B

R33 = ’ ’ ’

2 2 2C 2A 2B

9.29 Consider a static, infinitely long, cylindrically symmetric matter distribution of

constant radius that is invariant to Lorentz boosts along the symmetry axis (a

˜cosmic string™). Show that the line element outside the body can be written as

ds2 = c2 dt2 ’ d ’ + ’ dz2

2 2 2

d

where and are constants. For the case = 0, consider the spacelike surfaces

defined by t = constant and z = constant and calculate the circumference of a circle

of constant coordinate radius in such a surface. Hence show that, for < 1, the

geometry on the spacelike surface is that of a two-dimensional cone embedded in

three-dimensional Euclidean space.

10

Experimental tests of general relativity

Most of the experimental tests of general relativity are based on the Schwarzschild

geometry in the region r > 2GM/c2 . Some are based on the trajectories of massive

particles and others on photon trajectories. Most of the ˜classic™ tests are in the

weak-field limit, but more recent observations have begun to probe the more

discriminative strong-field regime. We will now discuss both these ˜classic™ exper-

imental tests and some of the more recent findings and proposals. Some later

tests are in fact more closely linked to the Kerr geometry (see Chapter 13), which

describes spacetime outside a rotating massive body, but the basic principles can

still be understood in terms of the simpler Schwarzschild geometry.

10.1 Precession of planetary orbits

For a general non-circular orbit in Newtonian theory the equation of motion is

d2 u GM

+u = 2

d2 h

where u ≡ 1/r and h is the angular momentum per unit mass of the orbiting

particle. For a bound orbit, the equation has the solution

GM

u= 1 + e cos (10.1)

h2

which describes an ellipse; the parameter e measures the ellipticity of the orbit.

Thus, for example, we can draw the orbit of a planet around the Sun as in

Figure 10.1. We can write the distance of closest approach (perihelion) as r1 =

a 1 ’ e and the distance of furthest approach (aphelion) as r2 = a 1 + e . The

equation of motion then requires that the semi-major axis is given by

h2

a= (10.2)

GM 1 ’ e2

230

231

10.1 Precession of planetary orbits

Planet

φ

Aphelion Perihelion

Sun

a (1 “ e)

Figure 10.1 The elliptical orbit of a planet around the Sun; e is the ellipticity

of the orbit.

The general-relativistic equation of motion is

d2 u GM 3GM

+ u = 2 + 2 u2 (10.3)

d2 h c

If the gravitational field is weak, as it is for planetary orbits around the Sun, then

we expect Newtonian gravity to provide an excellent approximation to the motion

in general relativity. We can therefore treat the Newtonian solution (10.1) as the

zeroth-order solution to the general-relativistic equation of motion. Thus let us

write the general-relativistic solution as

GM

u= 1 + e cos +u

h2

where u is a perturbation. Substituting this expression into the general-relativistic

equation (10.3) we find that, to first-order in u,

d2 u

+ u = A 1 + e2 cos2 + 2e cos

2

d

where the constant A = 3 GM 3 / c2 h4 is very small. A particular integral of

this equation is easily found to be

u = A 1 + e2 ’ 1 cos 2 + e sin

1

(10.4)

2 6

which can be checked by direct differentiation.

Since the constant A is very small, the first two terms on the right-hand side

of (10.4) are tiny, and of no use in testing the theory. However, the last term,

Ae sin , might be tiny at first but will gradually grow with time, since the factor

232 Experimental tests of general relativity

means that it is cumulative. We must therefore retain it, and so our approximate

solution reads

GM

u= 1 + e cos + sin (10.5)

h2

= 3 GM 2 / h2 c2

where 1. Using the relation

1’ = cos cos + sin sin

cos

≈ cos + sin for 1 (10.6)

we can therefore write

GM

u≈ 1 + e cos 1’ (10.7)

h2

From this expression, we see that the orbit is periodic, but with a period

2 / 1 ’ , i.e. the r-values repeat on a cycle that is larger than 2 . The result

is that the orbit cannot ˜close™, and so the ellipse precesses (see Figure 10.2). In

one revolution, the ellipse will rotate about the focus by an amount

2

2 6 GM

= ’2 ≈ 2 =

1’ h2 c2

Substituting for h from (10.2), we finally obtain

6 GM

= (10.8)

a 1 ’ e2 c 2

∆φ

Figure 10.2 Precession of an elliptical orbit (greatly exaggerated).

233

10.2 The bending of light

Let us apply equation (10.8) to the orbit of Mercury, which has the following

parameters: period = 88 days, a = 5 8 — 1010 m e = 0 2. Using M = 2 — 1030 kg,

we find

= 43 per century

In fact, the measured precession is

5599 7 ± 0 4 per century

but almost all of this is caused by perturbations from other planets. The residual,

after taking perturbations into account, is in remarkable agreement with general

relativity. The residuals for a number of planets (and Icarus, which is a large

asteroid with a perihelion that lies within the orbit of Mercury) may also be

calculated (in arcseconds per century):

Observed residual Predicted residual

43 1 ± 0 5

Mercury 43.03

8±5

Venus 8.6

5±1

Earth 3.8

10 ± 1

Icarus 10.3

In each case, the results are in excellent agreement with the predictions of general

relativity. Einstein included this calculation regarding Mercury in his 1915 paper

on general relativity. He had solved one of the major problems of celestial

mechanics in the very first application of his complicated theory to an empirically

testable problem. As you can imagine, this gave him tremendous confidence in

his new theory.

10.2 The bending of light

We have already noted that a massive object can have a significant effect on

the propagation of photons. For example, photons can travel in a circular orbit

at r = 3GM/c2 . We do not, however, expect to observe this effect directly, but

a more modest bending of light can be observed. For investigating the slight

deflection of light by, for example, the Sun, it is easiest to follow an approximation

technique analogous to that used in predicting the perihelion shift of Mercury.

As we showed in Chapter 9, the ˜shape™ equation for a photon trajectory in the

equatorial plane of the Schwarzschild geometry is

d2 u 3GM

+ u = 2 u2 (10.9)

d2 c

234 Experimental tests of general relativity

∆φ / 2

r

b

φ

Figure 10.3 Angles and coordinates in the deflection of light by a spherical mass.

where u ≡ 1/r. In the absence of matter, the right-hand side vanishes and we may

write the solution as

sin

u= (10.10)

b

which represents a straight-line path with impact parameter b (see Figure 10.3).

We again treat (10.10) as the zeroth-order solution to the equation of motion.

Thus we write the general-relativistic solution as

sin

u= +u

b

where u is a perturbation. Substituting this expression into (10.9), we find that,

to first order in u,

d2 u 3GM

+ u = 2 2 sin2

d2 cb

This is satisfied by the particular integral

3GM

u= 1 + 1 cos 2 (10.11)

3

2 b2

2c

and adding (10.11) into the original solution yields

sin 3GM

u= + 1 + 1 cos 2 (10.12)

3

2c2 b2

b

Now consider the limit r ’ , i.e. u ’ 0. Clearly, for a slight deflection we

can take sin ≈ and cos 2 ≈ 1 at infinity, to obtain = ’2GM/ c2 b . Thus

the total deflection (see Figure 10.3) is

4GM

= (10.13)

c2 b

This is the famous gravitational deflection formula (which incidentally is twice

what had previously been worked out using a Newtonian approach). For light

235

10.2 The bending of light

= 1 75. The 1919 eclipse expedition led by Eddington

grazing the Sun it yields

gave two sets of results:

= 1 98 ± 0 16

= 1 61 ± 0 4

both consistent with the theory. This provided the first experimental verification

of a prediction of Einstein™s theory (the ˜anomalous™ perihelion shift of Mercury

had been known for many years) and turned Einstein into a scientific superstar.1

Some historians have argued that Eddington ˜fiddled™ the results to agree with the

theory. If Eddington did indeed massage the results, then he gambled correctly.

Later high-precision tests using radio sources, which can be observed near the

Sun even when there is no lunar eclipse, show there is now no doubt that the

general-relativistic prediction is accurate to a fraction of a percent. Modern radio

experiments using very long baseline interferometry (VLBI) have been performed

to measure the gravitational deflection of the positions of radio quasars as they are

eclipsed by the Sun. Such experiments can be performed to an accuracy of better

than ∼10’4 arcseconds. Figure 10.4 summarizes the results of measurements of

the deflection angle from experiments conducted over the period 1969“75.

The results are in excellent agreement with the predictions of general relativity.

Moreover, as one can see from the figure, the results constrain the parameter

in the Brans“Dicke theory of gravity (see Appendix 8A): we have ≥ 40.

For more dramatic light deflection, our adopted approach of successive approx-

imations is unsuitable. In this case, it is more appropriate to use the exact equation

for d /dr derived in Chapter 9, which reads

’1/2

11 1 2

d

= 2 2 ’ 2 1’

dr rb r r

where b is the impact parameter at infinity. We also showed in Chapter 9 that if

√

b > 3 3 then the photon is not captured by the mass; the resulting general orbit

shape is illustrated in Figure 10.5. From the figure, we see that the deflection

angle is given by

’1/2

11 1 2

=2 ’ 2 1’ (10.14)

dr

r 2 b2 r r

r0

where r0 is the point of closest approach, at which the expression in the square

brackets in (10.14) vanishes.

1

The media had a great story. Remember that this was just after the end of the First World War, and so

the headlines read something like ˜Newton™s theory of gravity overthrown by German physicist, verified by

British scientists.™

236 Experimental tests of general relativity

± 0.88 0.92 0.96 1.00 1.04 1.08

Radio deflection experiments

Muhleman et al. (1970)

1969

Seilestad et al. (1970)

Hill (1971)

1970 Shapiro (quoted in Weinberg 1972)

Stramek (1971)

1971 Stramek (1974)

Riley (1973)

1972 Weiler et al. (1974)

Counselman et al. (1974)

1973 Weiler et al. (1974)

1974 Fomalont and Stramek (1975)

1975 Fomalont and Stramek (1976)

10 40 ∞

5

Value of scalar-tensor ω

Figure 10.4 Results of radio-wave deflection measurements of the positions

of quasars in the period 1969“75 (from C. Will, Theory and Experiment in

Gravitational Physics, Cambridge University Press, 1981). The deflection angle

= 4GM/ R c2 and the error bars are plotted on the parameter .

is

If general relativity is correct, we expect = 1. The abscissa scale gives the

measured values of the parameter in the Brans“Dicke scalar-tensor theory of

gravity, discussed in Appendix 8A.

closest

approach

r0

b

∆φ

Figure 10.5 Angles and coordinates in the deflection of light by a spherical mass.

10.3 Radar echoes

In Chapter 9, we showed that the ˜energy™ equation for a photon orbit in the

Schwarzschild geometry is

h2 2

r + 2 1’

™ = c 2 k2

2

r r

237

10.3 Radar echoes

Using the result

2 2 2

k2

dr dr dt dr

= =

1 ’ 2 /r 2

d dt d dt

we can rewrite the energy equation as

2

h2 c2

1 dr

+ 2 2’ =0 (10.15)

1 ’ 2 /r 1 ’ 2 /r

3 dt kr

Now consider a photon path from Earth to another planet (say Venus), as shown

in Figure 10.6. Evidently the photon path will be deflected by the gravitational

field of the Sun (assuming that the planets are in a configuration like that shown

in the figure, where the photon has to pass close to the Sun in order to reach

Venus). Let r0 be the coordinate distance of closest approach of the photon to the

Sun; then

dr

=0

dt r0

and so from (10.15) we have

h2 c2

=

1 ’ 2 /r0

2

k2 r0

Thus, after rearrangement, we can write (10.15) as

1/2

r0 1 ’ 2 /r

2

dr

= c 1 ’ 2 /r 1’ 2

r 1 ’ 2 /r0

dt

Earth

r

r0

Sun

Venus

Figure 10.6 Photon path from Earth to Venus deflected by the Sun.

238 Experimental tests of general relativity

which can be integrated to give for the time taken to travel between points r0

and r

’1/2

r0 1 ’ 2 /r

2

r 1

t r r0 = 1’ 2 dr

c 1 ’ 2 /r r 1 ’ 2 /r0

r0

The integrand can be expanded to first order in /r to obtain

r 2

r r0

t r r0 = 1+ + dr

r r + r0

c r 2 ’ r0

2 r

r0

which can be evaluated to give

1/2

r 2 ’ r0 r + r 2 ’ r0 r ’ r0

2 2

1/2 1/2

2

= + +

ln (10.16)

t r r0

r + r0

c c r0 c

The first term on the right-hand side is just what we would have expected if

the light had been travelling in a straight line. The second and third terms give us

the extra coordinate time taken for the photon to travel along the curved path to

the point r. So, you can see from Figure 10.6 that if we bounce a radar beam to

Venus and back then the excess coordinate-time delay over a straight-line path is

1/2 1/2

rE ’ r0 rV ’ r0

2 2 2 2

t = 2 t rE r0 + t rV r0 ’ ’

c c

where the factor 2 is included because the photon has to go to Venus and back.

Since rE r0 and rV r0 we have

1/2

rE ’ r0

2 2

2 r

ln E +

t rE r0 ’ ≈

c c r0 c

and likewise for tV and rV . Thus, the excess coordinate-time delay is

4GM rr

ln E 2V + 1

t≈

c3 r0

Of course, clocks on earth do not measure coordinate time but the corresponding

proper time. Assuming the Earth to be at fixed coordinates r during the

travel time of the signal, this is given by

1/2

2GM

= 1’ 2 t

c rE

GM/c2 , we can ignore this effect to the accuracy of our

However, since rE

calculation. For Venus, when it is opposite to the Earth on the far side of the Sun,

≈ 220 s

239

10.3 Radar echoes

The idea of the experiment is as follows. Fire an intense radar beam towards

Venus when it is almost opposite to the Earth on the far side of the Sun and

measure the time delay of the radar echo with a sensitive radio telescope. The

excess time delay gives us a test of the principle of equivalence. This sounds

straightforward, but the time delay is very small and depends on the values of

rE rV r0 . How can one determine these parameters to the required precision?

The answer is to fit the measured delays over a long period of time to a curve

chosen by varying rE rV etc. as free parameters (see Figure 10.7). There are a

number of technical problems that limit the accuracy of this method. Firstly, we

must correct for the motion of Venus and the Earth in their orbits and for their

individual gravitational fields. Also, in practice, the radar beam is reflected from

different points on the surface of Venus (mountain peaks, valleys, etc.) and this

introduces a dispersion in the time delay of several hundred s. This problem

can be solved by bouncing the radar beams from a mirror “ as has since been

done using the Viking landers on Mars. Another, more complicated, problem is

correcting for refraction by the Solar corona “ this can be important for photon

paths that graze the surface of the Sun. Nevertheless, Figure 10.7 confirms that the

corrected measurements are in excellent agreement with the general-relativistic

prediction.

200

160 Superior conjunction

25 Jan 1970

Excess delay (µs)

120

80

40

0

“300 “200 “100 0 100 200 300

Time (days)

Figure 10.7 The Earth“Venus time-delay measurement compared with the

general-relativistic prediction.

240 Experimental tests of general relativity

10.4 Accretion discs around compact objects

As we have seen, the orbits of particles and photons are probes of the geometry of

spacetime. Information about the geometry produced by compact massive objects

or black holes can be obtained from observations of the orbits of particles in the

accretion disc that often surrounds them. As we saw in Chapter 9, the radiation

efficiency of the accretion disc around a Schwarzschild black hole is 10 times

greater than the efficiency of the nuclear burning of hydrogen, and such disks are

very strong emitters in X-rays.

Even at the temperatures of ∼ 107 K that characterise an accretion disc, some

heavy nuclei retain bound electrons. The small trace of iron found in the accreting

matter is such a nucleus. Incident radiation from X-ray flares above and below

the disc can lead to fluorescence from the highly ionised atoms in the disc; in

this process an electron in the atom is de-excited from a higher energy level

to a lower one and emits a photon. For iron atoms, this results in photons of

energy 6.4 keV, giving a spectral line roughly in the middle of the X-ray band.

As one might expect, however, the frequency of the emitted photons as measured

by some observer at infinity (i.e. an astronomer on Earth) will differ from the

frequency with which the photons were emitted. Qualitatively, there are two

effects that cause this frequency shift. First, the photons will be gravitationally

redshifted by an amount that depends on the radius from which they were emitted.

Second, they will be Doppler shifted by an amount that depends on the speed and

direction (relative to the distant observer) of the material from which they were

emitted, in particular whether the material was moving towards or away from the

observer.

Unfortunately, given the typical size of accretion disks around compact objects,

and their large distance from us, the angular size of such systems as viewed

from Earth is typically far smaller than the width of the observing beam of any

telescope. Thus when an astronomer measures the spectrum (i.e. the photon flux as

a function of frequency) of such an object, the radiation received at each frequency

comes from various parts of the disc. Nevertheless, the observed spectrum is seen

to consist of a much-broadened iron line, whose shape contains information about

the spacetime geometry around the accreting object. In spite of the integration of

contributions from across the disc, the photons coming from the inner parts of

the accretion disc close to the compact object allow one to use the line profile to

probe the strong-field regime of gravity.

As an illustration, let us calculate in some simple cases the redshift one would

expect if the central object were not rotating, so that the geometry outside it is

given by the Schwarzschild metric. For simplicity, take the disc to be oriented

edge-on to the observer, as shown in Figure 10.8. All orbits are then in the plane

of the observer and the disc, which we take to be the equatorial plane = /2.

241

10.4 Accretion discs around compact objects

uE

p(E)

Photon

path

r

φ

Observer

Figure 10.8 The emission of a photon by matter in an accretion disc around a

compact object. The observer is viewing the disc edge-on.

The ratio of the photon™s frequency at reception to that at emission is given by

p R · uR p R uR

R

= = (10.17)

p E · uE p E uE

E

where p E and p R are the photon 4-momenta at emission and reception respec-

tively, uE is the 4-velocity of the material at emission and uR is the 4-velocity

of the observer at reception. Assuming the observer to be fixed at infinity, the

components of his 4-velocity in the t r coordinate system are

uR = 1 0 0 0

Now consider the 4-velocity of the emitting material. Since we are assuming that

this material is moving in a circular orbit it must have a 4-velocity of the form

uE = u0 0 0 u3

E E

Using the fact that

d d dt d0

u3 =

= = u

E

dt E

d dt d

we can write the emitter™s 4-velocity as

u E = u0 1 0 0

E

≡ d /dt = GM/r 3 1/2 , which we derived in

where, for circular motion,

Chapter 9. We can now fix u0 by using the fact that g u u = c2 . If the emitting

E

material is at a coordinate radius r, we have

’1/2 ’1/2

2 3

=c c 1’ ’r = 1’

u0 2 2 2

E

r r

Our general expression (10.17) therefore yields

’1

1/2

3

p0 R p0 R pE

1± 3

R

= = 1’

p0 E u0 + p3 E u3 r p0 E p0 E

E E E

242 Experimental tests of general relativity

where the plus sign corresponds to the emitting matter on the side of the disc

moving towards the observer and the minus sign corresponds to the matter on

the other side. However, the Schwarzschild metric is stationary, i.e. the metric

components g do not depend explicitly on t. Thus p0 is conserved along the

geodesic, and so

’1

1/2

3 pE

1± 3

R

= 1’

r p0 E

E

It therefore remains only to fix the ratio p3 E /p0 E in order to determine the

observed redshift. In general, we must use the fact that the photon worldline is

null, and so g p p = 0. As we are working in the equatorial plane = /2,

this yields

’1

1 2 2 1

1’ p0 2 ’ 1 ’ p1 2 ’ =0

2

(10.18)

p

r2 3

c2 r r

For photons emitted from material at a general position angle , one would now

need to use the geodesic equations for the photon worldline in order to eliminate

p1 . There are, however, two special cases for which this is not necessary.

The simplest case occurs when the photon is emitted from matter moving

transverse to the observer, i.e. when = 0 or = . We then have p3 E = 0,

and so the observed frequency ratio is

1/2

3

R

= 1’ (10.19)

r

E

The other simple cases occur when the matter is moving either directly towards

or away from the observer, i.e. when = ’ /2 or = /2. Then the radial

components of the photon 4-momentum, p1 E , will be zero. From (10.18) we

obtain

p3 E r

=

c 1 ’ 2 /r 1/2

p0 E

= “ /2 is given by

so that the photon frequency shift for

1 ’ 3 /r 1/2

R

= (10.20)

1 ± r/ ’ 2 ’1/2

E

The above discussion has been for a disc viewed edge-on. The other limiting case,

when the disc is viewed face-on, is easier to analyse. Since the motion of the

emitting matter is always transverse to the observer, the frequency shift is given

by (10.19).

243

10.4 Accretion discs around compact objects

Although the observed iron line consists of photons coming from different

radii in the disc, we may still calculate the smallest possible frequency (or largest

redshift) present in the observed spectrum. It is clear that such photons must

be emitted from the smallest possible value of r. As discussed in the previous

chapter, the innermost stable circular orbit for the Schwarzschild metric is at

r = 6 . Thus the smallest frequency represented is therefore given by

√

2/3 = 0 47 for a disc viewed edge-on

√

R/ E =

1/ 2 = 0 71 for a disc viewed face-on

If the central object were rotating (so that the exterior geometry is given by the

Kerr metric “ see Chapter 13), then the smallest frequencies could be even lower.

Figure 10.9 shows the iron spectral line measured in the galaxy MCG-6-30-15.

In general the detailed shape of the line profile depends on the mass and rotation

of the central object, the inclination of the disc to the line of sight and relativistic

beaming effects. It is hoped that, in the future, line profiles can be measured to

1.5—10“4

Line flux (ph cm“2 s“1 keV“1)

10“4

5—10“5

0

4 6 8

Energy (keV)

Figure 10.9 The line profile of the iron 6.4 keV spectral line from MCG-6-30-15

observed by the ASCA satellite (Y. Tanaka et al., Nature 375, 659, 1995). The

emission line is extremely broad, the width indicating velocities of order one-

third the speed of light. There is a marked asymmetry towards energies lower

than the rest energy of the emission line, with a smallest energy of about 4 keV.

The solid line shows a fit to the data assuming a disc around a non-rotating

Schwarzschild black hole, extending between 3 and 10 Schwarzschild radii and

inclined at an angle of 30 to the line of sight. Certain features suggest that the

central object may in fact be rapidly rotating.

244 Experimental tests of general relativity

sufficient accuracy to determine the mass and angular momentum of the central

compact object.

10.5 The geodesic precession of gyroscopes

We have seen how the motion of test bodies can be used to explore the geometry

of a curved spacetime. If the test body has spin then the motion of its spin vector

can also be used to probe the spacetime geometry. Here we discuss the idealised

case of an infinitesimally small test body with spin, such as a small gyroscope.

The test body moves along a timelike geodesic curve, so its 4-velocity u

is parallel-transported along its worldline. Thus, in some coordinate system, its

components satisfy

du

+ u u =0

d

Suppose that the spin of the test body is described by the 4-vector s along the

geodesic. Since this vector can have no timelike component in the instantaneous

rest frame of the test body, we require that at all points along the geodesic

s·u = g s u = 0 (10.21)

Since the 4-velocity u of the test body is parallel-transported along its geodesic,

to ensure that the inner product is conserved at all points along the worldline

we require that s is also parallel-transported along the geodesic. Hence its

components must satisfy

ds

+ s u =0 (10.22)

d

Let us now suppose that the test body is in a circular orbit of coordinate radius

r in the equatorial plane of the Schwarzschild geometry. Using the expressions we

derived in Chapter 9 for the connection coefficients for the Schwarzschild

coordinates (with = /2), one finds that most of the

metric in t r

are zero. Moreover, for a test body in a circular orbit we have u1 = u2 = 0, and

we find that the equations (10.22) reduce to

ds0

+ u =0

0 10

(10.23)

10 s

d

ds1

+ u+ u =0

1 00 1 33

(10.24)

00 s 33 s

d

ds2

=0 (10.25)

d

ds3

+ u =0

3 13

(10.26)

13 s

d

245

10.5 The geodesic precession of gyroscopes

where the connection coefficients are given by

’1

2 2

= 1’ = 1’

0 1

10 00

r2 r2

r r

2 1

33 = ’r 1 ’ 13 =

1 3

r r

Moreover, from our discussion in the previous section, we can write the test

body™s 4-velocity as

= u0 1 0 0

u

where u0 = dt/d = 1 ’ 3 /r ’1/2 and = d /dt = c2 /r 3 1/2 are both

constants.

Since u1 = u2 = 0 the orthogonality condition (10.21) reduces to

c2 1 ’ 2 /r s0 u0 ’ r 2 s3 u3 = 0

and noting that u3 /u0 = we may express s0 in terms of s3 :

r2

s=2

0

s3

c 1 ’ 2 /r

Using this result it is straightforward to show that equation (10.23) is equivalent

to equation (10.26). Thus the system of equations reduces to

ds2 r 3 ds2 ds3 u0 1

’ 0s =0 =0 + s =0 (10.27)

d u d d r

It is more convenient to convert the -derivatives to t-derivatives using u0 =

dt/d . Then, on using the third equation to eliminate s3 from the first, the system

of equations becomes

2

d2 s 1 ds2 ds3

+0 s =0 =0 + s1 = 0

1

dt2 u dt dt r

Let us take the initial spatial direction s of the spin vector to be radial, so that

s2 0 = s3 0 = 0. The corresponding solution to our system of equations is easily

shown to be

s1 t = s1 0 cos s2 t = 0 s3 t = ’ s1 0 sin (10.28)

t t

r

= /u0 = 1 ’ 3 /r 1/2 . This solution shows that the spatial part

where

s of the spin vector rotates relative to the radial direction with a coordinate

angular speed in the negative -direction. However, the radial direction itself

rotates with coordinate angular speed in the positive -direction, and it is the

difference between these two speeds that gives rise to the geodesic precession

246 Experimental tests of general relativity

Radial

direction

„¦′t

’

Final s

„¦t

± ’

Initial s

Figure 10.10 The geodesic precession effect for a spinning object in a circular

orbit in the equatorial plane of the Schwarzschild geometry. Here the initial

direction t = 0 is radial.

effect. This is illustrated in Figure 10.10. Since one revolution is completed in

a coordinate time t = 2 / , the final direction of s is therefore 2 + , where

=2/ ’ . Thus, after one revolution the spatial spin vector is rotated

in the direction of the orbital motion by an angle

=2 1 ’ 1 ’ 3 /r 1/2

The geodesic precession effect may be observable experimentally by measuring

the spacelike spin vector of a gyroscope in an orbiting spacecraft. Although the

effect is small, it is cumulative. Thus, for a gyroscope in a near-Earth orbit, the

precession rate is about 8 per year, which should be measurable. (In fact there is

an additional very small effect, which may also be measurable, due to the fact that

the Earth is slowly rotating and so the geometry outside it is correctly described

by the Kerr metric). In April 2004, NASA launched the Gravity Probe B (GP-B)

satellite to carry out this experiment and it is currently recording measurements;

the results are eagerly awaited.

Exercises

10.1 Show that the equation of motion for planetary orbits in Newtonian gravity is

d2 u GM

+u = 2

d2 h

where u = 1/r and r is the radial distance from the centre of mass of the central

object.

247

Exercises

10.2 Show that the equation of motion in Exercise 10.1 has the solution

GM

u= 1 + e cos

h2

and that this describes an ellipse. Show further that

h2

a=

GM 1 ’ e2

where r1 = a 1 ’ e and r2 = a 1 + e are the distances of closest and furthest

approach respectively.

10.3 Verify that the general-relativistic equation of motion for planetary orbits (10.3)

has the solution

2

3 GM 11

GM

u = 2 1 + e cos + 1 + e2 ’ cos 2 + e sin

c 2 h4 26

h

to first order in the relativistic perturbation to the Newtonian solution.

10.4 Verify that the general-relativistic equation of motion for a photon trajectory (10.9)

has the solution

sin 3GM 1

u= + 2 2 1 + cos 2

2c b 3

b

to first order in the relativistic perturbation to a straight-line path.

10.5 Show that the gravitational deflection of light by the Sun predicted in the Newto-

nian theory of gravity is exactly half the value predicted in general relativity.

10.6 In the radar-echoes test, a photon travels from radial coordinate r to r0 , which is

the radial coordinate of the closest approach of the photon to the Sun. Verify that,

to first order in /r, the elapsed coordinate time is given by

1/2

r 2 ’ r0 r + r 2 ’ r0 r ’ r0

2 1/2 2 1/2

2

t r r0 = + +

ln

r + r0

c c r0 c

An accretion disc extends from r = 6 to r = 20 in the equatorial plane of the

10.7

Schwarzschild geometry. A photon is emitted radially outwards by a particle on

the inner edge of the disc and is absorbed by a particle on the outer edge of the

disc. Find the ratio of the energy absorbed to that emitted.

10.8 For a gyroscope in a circular orbit in the equatorial plane of the Schwarzschild

geometry, show that the components s of its spin 4-vector satisfy equa-

tions (10.23“10.26).

10.9 Show that the system of equations (10.27) has the solution (10.28).

10.10 A gyroscope in a circular orbit of radius r in the equatorial plane of the

Schwarzschild geometry has its spatial spin vector s also lying in the equatorial

plane. Show that, after one complete orbit, the angle between the initial and final

directions of the spatial spin vector is given by

=2 1 ’ 1 ’ 3 /r 1/2

irrespective of the initial direction of the spin vector. Does this still hold if the

original spatial spin vector does not lie in the plane of the orbit?

11

Schwarzschild black holes

In our discussion of the Schwarzschild geometry, we have thus far used the

coordinates t r to label events in the spacetime. In this context, t r

are called the Schwarzschild coordinates. Moreover, until now we have been

concerned only with the exterior region r > 2 . We now turn to the discussion

of the Schwarzschild geometry in the interior region r < 2 , and the significance

of the hypersurface r = 2 . We shall see that, in order to understand the entire

Schwarzschild geometry, we must relabel the events in spacetime using different

sets of coordinates.

11.1 The characterisation of coordinates

Before discussing the Schwarzschild geometry in detail, let us briefly consider the

characterisation of coordinates. In general, if we wish to write down a solution

of Einstein™s field equations then we need to do so in some particular coordinate

system. But what, if any, is the significance of any such system? For example,

suppose we take the Schwarzschild solution and apply some complicated coor-

dinate transformation x ’ x . The resulting metric will still be a solution of

the empty-space field equations, of course, but there is likely to be little or no

physical or geometrical significance attached to the new coordinates x .

One thing we can do, however, is to establish whether at some event P a

coordinate x is timelike, null or spacelike. This corresponds directly to the nature

of the tangent vector e to the coordinate curve at P. The easiest way to determine

this property of the coordinate is to fix the other coordinates at their values at P

and consider an infinitesimal variation dx in the coordinate of interest. If the

corresponding change in the interval ds2 is positive, zero or negative then x is

timelike, null or spacelike respectively. This, in turn, corresponds simply to the

sign of the relevant diagonal element g (no sum) of the metric.

248

249

11.2 Singularities in the Schwarzschild metric

11.2 Singularities in the Schwarzschild metric

With these ideas in mind, let us look at the Schwarzschild metric in the traditional

coordinate system. We have

tr

’1

2GM 2GM

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

2 2 2 2 2

(11.1)

cr cr

Inspection of this line element shows immediately that the metric is singular at

r = 0 and r = 2GM/c2 . The latter value is known as the Schwarzschild radius

and is often denoted rS , so that

2GM

rS =

c2

We must remember, however, that we derived the Schwarzschild solution by

solving the vacuum field equations R = 0, and so the metric given by (11.1) is

only valid down to the surface of the spherical matter distribution. For example,

the Schwarzschild radius for the Sun is

2GM

rS = = 2 95 km

c2

which is much smaller than the radius of the Sun R = 7 — 105 km . Similarly,

the Schwarzschild radius for a proton is

2GMp

= 10’52 m

rS = 2

c

again much smaller than the characteristic radius of a proton (Rp = 10’15 m). In

fact, for most real objects the Schwarzschild radius lies deep within the object,

where the vacuum field equations do not apply. But what if there exist objects so

compact that they lie well within the Schwarzschild radius? For such an object, the

Schwarzschild solution looks very odd. Ignoring for the moment the singularity

in the metric at r = rS , let us denote the region r > rS as region I, and r < rS as

region II.

From the Schwarzschild metric (11.1) we see that, in region I, the metric

coefficient g00 is positive and the gii (for i = 1 2 3) are negative. It therefore

follows that for r > rS the coordinate t is timelike and the coordinates r

are spacelike. Indeed, in region I we may attach simple physical meanings to the

coordinates. For example, t is the proper time measured by an observer at rest at

infinity. Similarly, r is a radial coordinate with the property that the surface area

of a 2-sphere t = constant, r = constant is 4 r 2 . In region II, however, the metric

250 Schwarzschild black holes

coefficients g00 and g11 change sign. Hence, for r < rS , t is a spacelike coordinate

and r is timelike. Thus ˜time™ and ˜radial™ coordinates swap character on either

side of r = rS . It is natural to ask what this means, and, indeed, whether it is

physically meaningful.

Let us therefore consider in more detail the singularities in the metric at

r = 0 and r = rS . We must remember that coordinates are simply a way of

labelling events in spacetime. The physically meaningful geometric quantites are

the 4-tensors defined at any point on the spacetime manifold. Spacetime curvature

is described covariantly by the components of the curvature tensor R (and its

contractions), which we may easily calculate for the Schwarzschild metric (11.1).

For example, the curvature scalar at any point is given by

48 2

=6 (11.2)

R R

r

which we see is finite at r = rS . Moreover, since it is a scalar, its value remains

the same in all coordinate systems. Thus the spacetime curvature at r = rS is

perfectly well behaved, and so we see that r = rS is a coordinate singularity. By

the same token, (11.2) is singular at r = 0 and so this point is a true intrinsic

singularity of the Schwarzschild geometry.

We may illustrate the idea of coordinate singularities with a simple example.

As discussed in Chapter 2, one may write the line element for the surface of a

2-sphere as

a2 d 2

ds = 2 +

2 2 2

d

a’ 2

This line element has a singularity at = a. Embedding this manifold, for the

moment, in three-dimensional Euclidean space, we know that = a corresponds

simply to the equator of the sphere (relative to the origin of the coordinate system)

and it is clear why the coordinates cover the surface of the sphere uniquely

only up to this point. There is nothing pathological occurring in the intrinsic geom-

etry of the 2-sphere at the equator, i.e. there is no ˜real™ (or intrinsic) singularity

in the metric. As shown in Appendix 7A, the Gaussian curvature of a 2-sphere

is simply K = 1/a2 , which does not ˜blow up™ anywhere. Thus, = a is only a

coordinate singularity, which has resulted simply from choosing coordinates with

a restricted domain of validity. In an analogous way, the coordinate singularity

of the Schwarzschild metric is simply a result of the coordinate system that we

have chosen to use. We can remove it by making appropriate transformations

of coordinates, which we will discuss later. For the time being, however, let us

continue our investigation of the Schwarzschild geometry using the Schwarzschild

coordinates t r .

251

11.3 Radial photon worldlines in Schwarzschild coordinates

11.3 Radial photon worldlines in Schwarzschild coordinates

Let us investigate the spacetime diagram of the Schwarzschild solution in

coordinates. The metric reads

tr

’1

2 2

ds = c 1’ dt ’ 1 ’ dr 2 ’ r 2 d

2 2 2 2

r r

where d is an element of solid angle. We have written it in this form because

we shall usually ignore the angular coordinates in drawing spacetime diagrams,

i.e. these diagrams will show the r ct -plane for fixed values of and .

We begin by determining the lightcone structure in the diagram, by considering

the paths of radially incoming and outgoing photons; these were discussed briefly

in Section 9.11. From the metric, for a radially moving photon we have

’1

1 2

dt

=± 1’

dr c r

where the plus sign corresponds to a photon that is outgoing (in that dr/dt is

positive in the region r > 2 ) and the minus sign corresponds to a photon that

is incoming (in that dr/dt is negative in the region r > 2 ). On integrating, we

obtain

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 the incoming and outgoing photon

paths are interchanged, as we would expect. We can now plot these curves in the

r ct -plane, as shown in Figure 11.1. The diagram is drawn for fixed and .

Since the diagram will be the same for all other and , we should think of each

point r ct in the diagram as representing a 2-sphere of area 4 r 2 .

Figure 11.1 requires some words of explanation. At large radii in region I the

gravitational field becomes weak and the metric tends to the Minkowski metric

of special relativity. Thus, as expected, the lightcone structure becomes that of

Minkowski spacetime, where incoming and outgoing light rays define straight

lines of slope ±1 in the diagram. As we approach the Schwarzschild radius, the

ingoing light rays tend to the ordinate t ’ + and outgoing light rays tend to

t ’ ’ . This seems to suggest that it takes an infinite time for an incoming signal

to cross the Schwarzschild radius, but in this respect the diagram is misleading,

as we shall see shortly (we discussed this point briefly in Section 9.11).

252 Schwarzschild black holes

Ingoing null

congruence

ct

Singularity

II I

Outgoing null

congruence

r=0 r = 2µ

Figure 11.1 Lightcone structure of the Schwarzschild solution.

In region II the lightcones flip their orientation by 90 , since the coordinates

t and r reverse their character. We see that all photons in this region must end

up at r = 0. At this point there is real singularity, where the curvature of the

Schwarzschild solution diverges. Moreover, any massive particle in region II

must also end up at the singularity, since a timelike worldline must lie within

the forward light-cone at each point. Thus we conclude that once within the

Schwarzschild radius you necessarily end up at a spacelike singularity at r = 0.

To escape would require a violation of causality.

11.4 Radial particle worldlines in Schwarzschild coordinates

The causal structure in Figure 11.1 is determined by radially moving photons.

It is also of interest to determine the worldlines of radially moving massive

particles in Schwarzschild coordinates. For simplicity let us consider an infalling

particle released from rest at infinity, which we investigated in detail in Chapter 9.

Parameterising the particle worldline in terms of the proper time , we found that

the trajectory r could be written implicitly as

3

r3

2 2

r0

= ’ (11.3)

2 c2 3 2 c2

3

253

11.4 Radial particle worldlines in Schwarzschild coordinates

taking = 0 at r = r0 . Alternatively, if the trajectory is described as r t , where

t is the coordinate time, we found that

⎛ ⎞

3

r3 ⎠ 4

2⎝ r0 r0 r

t= ’ + ’

2 c2 2 c2

3 2 2

c

+1 ’1

r/ 2 r0 / 2

2

+ ln (11.4)

’1 +1

c r/ 2 r0 / 2

where t = 0 at r = r0 . Using equations (11.3) and (11.4), we can associate a given

value of the particle™s proper time with a point in a r ct -diagram. Thus, as

increases, we can plot out the particle trajectory in the r ct -plane.

ct/µ

18

„ = 10µ /c

„ = 10.67µ /c

14

„ = 8µ /c

12

10

„ = 6µ /c

8

6

„ = 4µ /c

4

„ = 2µ /c