ñòð. 11 |

Ï„=0

0

r/Âµ

0 2 4 6 8

Figure 11.2 Trajectory of a radially infalling particle released from rest at

infinity. The dots correspond to unit intervals of c / , where is the particleâ€™s

proper time and we have taken = t = 0 at r0 = 8 .

254 Schwarzschild black holes

The corresponding curve is shown in Figure 11.2, which is a more quantitative

version of Figure 9.6; we have taken = t = 0 at r0 = 8 . Also plotted are dots

showing unit intervals of c / , together with the light-cone structure at particular

points on the trajectory. We see from the plot that the particle worldline has a

singularity at r = 2 and that it takes an infinite coordinate time t for the particle

to travel from r = 8 to r = 2 . Since t is the time experienced by a stationary

observer at large radius, to such an observer it thus takes an infinite time for the

particle to reach r = 2 . However, the proper time taken by the particle to reach

r = 2 is finite ( = 9 33 /c). Moreover, we see that for later values of the

particle worldline lies in the region r < 2 , which was not plotted in Figure 9.6.

In this region the coordinates t and r swap character, as indicated by the fact

that the light-cone is flipped by 90 . For r < 2 , we also note that, although

continues to increase until r = 0 is reached ( = 10 67 /c), the coordinate time

t decreases along the particle worldline.

Clearly, although the coordinate t is useful and physically meaningful as r â†’ ,

it is inappropriate for describing particle motion at r â‰¤ 2 . Therefore, in the

following section we introduce a new time coordinate that is adapted to describing

radial infall, and in the process we shall remove the coordinate singularity at

r =2 .

11.5 Eddingtonâ€“Finkelstein coordinates

The spacetime diagrams in Figures 11.1 and 11.2 show that the worldlines of

both radially moving photons and massive particles cross r = 2 only at t = Â± .

This suggests that the â€˜lineâ€™ r = 2 , âˆ’ < t < might really not be a line at all,

but a single point. That is, our coordinates may go bad owing to the expansion

of a single event into the whole line r = 2 . One technique for circumventing

the problem of unsatisfactory coordinates is to â€˜probeâ€™ spacetime with geodesics,

which after all are coordinate independent and will not be affected in any way by

the boundaries of coordinate validity. Of the many possibilities, we will use as

probes the null worldlines of radially moving photons.1

Advanced Eddingtonâ€“Finkelstein coordinates

Since in particular, we wish, to develop a better description of infalling particles,

let us begin by constructing a new coordinate system based on radially infalling

1

It is also possible to use the timelike worldlines of freely falling radially moving massive particles as probes of

the spacetime geometry. The traditional approach leads to useful new coordinates, called Novikov coordinates,

but they are related to Schwarzschild coordinates by transformations that are algebraically very complicated.

A more physically meaningful set of new coordinates that are also based on radially moving massive particle

geodesics is discussed in Exercise 11.9.

255

11.5 Eddingtonâ€“Finkelstein coordinates

photons. Recall that the worldline of a radially ingoing photon is given by

r

ct = âˆ’r âˆ’ 2 ln âˆ’ 1 + constant

2

The trick is to use the integration constant as the new coordinate, which we

denote by p. Thus, we make the coordinate transformation

r

p = ct + r + 2 ln âˆ’1 (11.5)

2

where p, for historical reasons, is known as the advanced time parameter and is

clearly a null coordinate (see Section 11.1). Since p is constant along the entire

worldline of the radially ingoing photon, it will be a â€˜goodâ€™ coordinate wherever

that worldline penetrates.

Differentiating (11.5), we obtain

r

dp = c dt + dr

r âˆ’2

and, on substituting for dt in the Schwarzschild line element, we find that in

terms of the parameter p the line element takes the simple form

2

ds2 = 1 âˆ’ dp2 âˆ’ 2 dp dr âˆ’ r 2 d + sin2 d

2 2

(11.6)

r

We see immediately from (11.6) that ds2 is now regular at r = 2 ; indeed it is

regular for the whole range 0 < r < , which is the range of r-values probed by

an infalling photon geodesic. Thus, in some sense, the transformation (11.5) has

extended the coordinate range of the solution in a way reminiscent of the analytic

continuation of a complex function.

One might object that the coordinate transformation (11.5) cannot be used at

r = 2 because it becomes singular. This must happen, however, if one is to

remove the coordinate singularity there. In any case, this transformation takes the

standard form (11.1) for the Schwarzschild line element to the form (11.6). Given

these two solutions, we can simply ask, what is the largest range of coordinates

for which each solution is regular? For the standard form this is 2 < r < ,

whereas for the new form (11.6) it is 0 < r < . In the overlap region 2 < r <

the two solutions are related by (11.5), and hence they must represent the same

solution in this region.

256 Schwarzschild black holes

As one might expect, the metric (11.6) is especially convenient for calculating

the paths of null geodesics. In particular, we see that radial null geodesics (for

which ds = d = d = 0) are given by

2

2 dp dp

1âˆ’ âˆ’2 =0

r dr dr

which has the two solutions

dp

=0 â‡’ p = constant

dr

âˆ’1

2

dp r

= 2 1âˆ’ â‡’ p = 2r + 4 ln âˆ’ 1 + constant (11.7)

2

dr r

which correspond to incoming and outgoing radial null geodesics respectively

(the former being valid by construction).

Since p is a null coordinate, which might be intuitively unfamiliar, it is common

practice to work instead with the related timelike coordinate t , defined by

r

ct â‰¡ p âˆ’ r = ct + 2 ln âˆ’1 (11.8)

2

The line element then takes the form

2 4c 2

ds2 = c2 1 âˆ’ dt 2 âˆ’ dt dr âˆ’ 1 + dr 2 âˆ’ r 2 d 2 + sin2 2

d

r r r

(11.9)

which is again regular for the whole range 0 < r < . The coordinates t r

are called advanced Eddingtonâ€“Finkelstein coordinates. We note that the line

element (11.9) is not invariant with respect to the transformation t â†’ âˆ’t , under

which the second term on the right-hand side changes sign. From (11.7), we see

that incoming and outgoing photon worldlines are given by

ct = âˆ’r + constant (11.10)

r

ct = r + 4 ln âˆ’ 1 + constant (11.11)

2

The first equation, for ingoing photons, corresponds to a straight line making

an angle of 45 with the r-axis and is valid for 0 < r < . Thus the photon

geodesics are continuous straight lines across r = 2 . The spacetime diagram

257

11.5 Eddingtonâ€“Finkelstein coordinates

t'

Radially

infalling

particle

Singularity

p = constant

I

II

r = 2Âµ

r=0 r

Figure 11.3 Lightcone structure in advanced Eddingtonâ€“Finkelstein coordinates.

of the Schwarzschild geometry in advanced Eddingtonâ€“Finkelstein coordinates is

shown in Figure 11.3.

The spacetime diagram now appears more sensible. It is straightforward to see

that the radial trajectory of an infalling particle or photon is continuous at the

Schwarzschild radius r = 2 . The lightcone structure changes at the Schwarzschild

radius and, as you can see from the diagram, once you have crossed the boundary

r = 2 your future is directed towards the singularity. Similarly, it can be seen

that a photon (or particle) starting at r < 2 cannot escape to the region r > 2 .

The Schwarzschild radius r = 2 defines an event horizon, a boundary of no

return. Once a particle crosses the event horizon it must fall to the singularity

at r = 0. Moreover, from the paths of the â€˜outgoingâ€™ null geodesics, we see

that any photons emitted by the infalling particle at r < 2 will not reach an

observer in region I. Thus to such an observer the particle appears never to cross

the event horizon. A compact object that has an event horizon is called a black

hole.

Retarded Eddingtonâ€“Finkelstein coordinates

One may reasonably ask what occurs if one instead chooses to construct a

new coordinate system based on the worldlines of radially outgoing photons.

258 Schwarzschild black holes

By analogy with our discussion above, this is achieved straightforwardly by intro-

ducing the new null coordinate q defined by

r

q = ct âˆ’ r âˆ’ 2 ln âˆ’1

2

which is known as the retarded time parameter. The line element of the

Schwarzschild geometry then becomes

2

ds2 = 1 âˆ’ dq 2 + 2 dq dr âˆ’ r 2 d + sin2 d

2 2

r

which is again regular for 0 < r < . Similarly, it is common practice to introduce

a new timelike coordinate tâˆ— defined by

r

ctâˆ— â‰¡ q + r = ct âˆ’ 2 ln âˆ’1

2

The coordinates tâˆ— r are called retarded Eddingtonâ€“Finkelstein coordi-

nates, and the corresponding line element in these coordinates is simply the time

reversal of the advanced Eddingtonâ€“Finkelstein line element (11.9).

It is straightforward to draw an spacetime diagram analogous to Figure 11.3 in

retarded Eddingtonâ€“Finkelstein coordinates, and one finds that (by construction)

the outgoing radial null geodesics are continuous straight lines at 45 but the

ingoing null rays are discontinuous, tending to tâˆ— = + at r = 2 . In this case,

the surface r = 2 again acts as a one-way membrane, but this time letting only

outgoing timelike or null geodesics cross from inside to outside. Indeed, particles

must move away from the singularity at r = 0 and are forcibly expelled from the

region r < 2 . Such an object is called a white hole.

This behaviour appears completely at odds with our intuition regarding the

gravitational attraction of a massive body. Moreover, how can the physical

processes that occur be so radically different depending on oneâ€™s choice of coordi-

nates, since we have maintained throughout that coordinates are merely arbitrary

labels of spacetime events? The key to resolving this apparent paradox is to

realise that our original coordinates t r covered only a part of the â€˜fullâ€™

Schwarzschild geometry. This topic is discussed fully in Section 11.9, in which we

introduce Kruskal coordinates, which cover the entire geometry and which show

that it possesses both a black-hole and a white-hole singularity. The advanced

Eddingtonâ€“Finkelstein coordinates â€˜extendâ€™ the solution into the (more familiar)

part of the manifold that constitutes a black hole, whereas the retarded Eddingtonâ€“

Finkelstein coordinates extend the solution into a different part of the manifold,

corresponding to a white hole. As we will discuss in Section 11.9, the existence

of white holes as a physical reality (as opposed to a mathematical curiosity) is

259

11.6 Gravitational collapse and black-hole formation

rather doubtful. Black holes, however, are likely to occur physically, as we now

go on to discuss.

11.6 Gravitational collapse and black-hole formation

Our investigation of the properties of a black hole would be largely academic

unless there were reasons for believing that they might exist in Nature. The

possibility of their existence arises from the idea of gravitational collapse.

A star is held up by a mixture of gas and radiation pressure, the relative

contributions depending on its mass. The energy to provide this pressure support is

derived from the fusion of light nuclei into heavier ones, predominantly hydrogen

into helium, which releases about 26 MeV for each atom of He that is formed.

When all the nuclear fuel is used up, however, the star begins to cool and collapse

under its own gravity. For most stars, the collapse ends in a high-density stellar

remnant known as a white dwarf. In fact, we expect that in around 5 billion years

the Sun will collapse to a form a white dwarf with a radius of about 5000 km and

a spectacularly high mean density of about 109 kg mâˆ’3 .

Astronomers have known about white dwarfs since as long ago as 1915 (the

earliest example being the companion to the bright star Sirius, known as Sirius

B), but nobody at the time knew how to explain them. The physical mechanism

providing the internal pressure to support such a dense object was a mystery. The

answer had to await the development of quantum mechanics and the formulation

of Fermiâ€“Dirac statistics. Fowler realised in 1926 that white dwarfs were held

up by electron degeneracy pressure. The electrons in a white dwarf behave like

the free electrons in a metal, but the electron states are widely spaced in energy

because of the small size of the star in its white-dwarf form. Because of the

Pauli exclusion principle, the electrons completely fill these states up to a high

characteristic Fermi energy. It is these high electron energies that save the star

from collapse.

In 1930, Chandrasekhar realized that the more massive a white dwarf, the denser

it must be and so the stronger the gravitational field. For white dwarfs over a

critical mass of about 1 4 M (now called the Chandrasekhar limit), gravity would

overwhelm the degeneracy pressure and no stable solution would be possible.

Thus, the gravitational collapse of the object must continue. At first it was thought

that the white dwarf must collapse to a point. After the discovery of the neutron,

however, it was realized that at some stage in the collapse the extremely high

densities occurring would cause the electrons to interact with the protons via

inverse -decay to form neutrons (and neutrinos, which simply escape). A new

stable configuration â€“ a neutron star â€“ was therefore possible in which the pressure

support is provided by degenerate neutrons. A neutron star of one solar mass

260 Schwarzschild black holes

would have a radius of only 30 km, with a density of around 1016 kg mâˆ’3 . Since

the matter in a neutron star is at nuclear density, the gravitational forces inside the

star are extremely strong. In fact, it is the first point in the evolution of a stellar

object at which general relativistic effects are expected to be important (we will

discuss relativistic stars in Chapter 12).

Given the extreme densities inside a neutron star, there remain uncertainties

in the equation of state of matter. Nevertheless, it is believed that (as for white

dwarfs), there exists a maximum mass above which no stable neutron-star config-

uration is possible. This maximum mass is believed to be about 3 M (which

is known as the Oppenheimerâ€“Volkoff limit). Thus, we believe that stars more

massive than this limit should collapse to form black holes. Moreover if the

collapse is spherically symmetric then it must produce a Schwarzschild black

hole.

Some theorists were very sceptical about the formation of black holes. The

Schwarzschild solution in particular is very special â€“ it is exactly spherically

symmetric by construction. In reality, a star will not be perfectly symmetric

and so perhaps, as it collapses, the asymmetries will amplify and avoid the

formation of an event horizon. In the early 1960s, however, Penrose applied global

geometrical techniques to prove a famous series of â€˜singularity theoremsâ€™. These

showed that in realistic situations an event horizon (a closed trapped surface)

will be formed and that there must exist a singularity within this surface, i.e. a

point at which the curvature diverges and general relativity ceases to be valid.

The singularity theorems were important in convincing people that black holes

must form in Nature. In Appendices 11A and 11B, we discuss some of the

observational evidence for the existence of black holes. As we will see, there is

compelling evidence that black holes do indeed exist. Furthermore, as mentioned

in Section 10.4, it should become possible within the next few years not only to

measure the masses of black holes but also to measure their angular momenta,

using powerful X-ray telescopes! Direct experimental probes of the strong-gravity

regime are now possible.

11.7 Spherically symmetric collapse of dust

Let us consider the spherically symmetric collapse of a massive star to form a

Schwarzschild black hole and also the view this process seen by a stationary

observer at large radius. For simplicity, we consider the case in which the star has

a uniform density and the internal pressure is assumed to be zero. In the absence

of pressure gradients to deflect their motion, the particles on the outer surface of

this â€˜ball of dustâ€™ will simply follow radial geodesics. In order to simplify our

analysis still further, we will assume that initially the surface of the â€˜starâ€™ is at

261

11.7 Spherically symmetric collapse of dust

rest at infinity.2 In this case, the particles on the surface will follow the radial

geodesics we discussed earlier.

Consider two observers participating in the gravitational collapse of the spher-

ical star. One observer rides the surface of the star down to r = 0, and the other

observer remains fixed at a large radius. Moreover, suppose that the infalling

observer carries a clock and communicates with the distant one by sending out

radial light signals at equal intervals according to this clock. Figure 11.4 shows

the relevant spacetime diagram in advanced Eddingtonâ€“Finkelstein coordinates

ct r , with and suppressed. The dots denote unit intervals of ct/ and we

have chosen = t = 0 at r = 8 . This diagram is easily constructed from the

results that were used to obtain Figure 11.2.

For a distant observer at fixed r, we know that the standard Schwarzschild

coordinate time t measures proper time. From (11.8), however, we see that if r is

fixed then dt = dt. Thus, a unit interval of t corresponds to a unit interval of

ct'/Âµ

Ï„ = 10.67Âµ /c

Observer

14

Ï„ = 10Âµ /c

12

on

ot

Ï„ = 8Âµ /c

10

ph

8

Ï„ = 6Âµ /c

6

Ï„ = 4Âµ /c

4

Ï„ = 2Âµ /c

2

Ï„=0

0

r/Âµ

0 2 4 6 8

Figure 11.4 Collapse of the surface of a pressureless star to form a black hole

in advanced Eddingtonâ€“Finkelstein coordinates. The starâ€™s surface started at rest

at infinity, and we have chosen = t = 0 at r = 8 .

This is equivalent to the collapse commencing with the starâ€™s surface at some finite radius r = r0 with some

2

finite inwards velocity.

262 Schwarzschild black holes

proper time for a distant fixed observer. From the diagram, we see that the light

pulses are not received at equal intervals of t . Rather, the proper time interval

measured by the distant observer between each pulse steadily increases. Indeed,

the last light pulse to reach this observer is the one emitted just before the surface

of the star crosses r = 2 . The worldine of this photon is simply the vertical line

r = 2 , and so this pulse would only â€˜reachâ€™ the distant observer at t = . Pulses

emitted after the surface of the star has crossed the event horizon do not progress

to larger r but instead progress to smaller r and end up at the singularity at r = 0.

Thus, the distant observer never sees the starâ€™s surface cross the radius r = 2 .

Furthermore, the pulses emitted at equal intervals by the falling observerâ€™s clock

arrive at the distant observer at increasingly longer intervals. Correspondingly, the

photons received by the distant observer are increasingly redshifted, the redshift

tending to infinity as the starâ€™s surface approaches r = 2 . Both these effects

mean that the distant observer sees the luminosity of the star fall to zero. To

summarise, the distant observer sees the collapse slow down and the starâ€™s state

approach that of a quasi-equilibrium object with radius r = 2 , which eventually

becomes totally dark. Thus, the distant observer sees the formation of a black hole.

Let us quantify further what the observer sees as the star collapses to form

a black hole. Since we are interested in measurements made by a distant

fixed observer, we may use either advanced Eddingtonâ€“Finkelstein coordinates

or traditional Scharwzschild coordinates t r , as both correspond

tr

to physical quantities at large r. We shall use the latter simply because we

have already obtained the equations for a massive radially infalling particle in

Schwarzschild coordinates. Suppose that a particle on the surface of the star emits

a radially outgoing pulse of light at coordinates tE rE , which is received by the

distant fixed observer at tR rR . Since the photon follows a radially outgoing

null geodesic, we can write

rE r

âˆ’ 1 = ctR âˆ’ rR âˆ’ 2 ln R âˆ’ 1

ctE âˆ’ rE âˆ’ 2 ln (11.12)

2 2

The radial coordinate â€˜seenâ€™ by the distant observer at time tR is the function

rE tR obtained by solving (11.12). Using the fact that the coordinates tE and rE of

the freely falling emitter are related by (11.4), we find that, if r is very close to 2 ,

ctR

rE tR â‰ˆ 2 + a exp âˆ’ (11.13)

4

where a is an unimportant constant depending on and rR . The important

consequence of this result is that the radius r = 2 is approached exponentially,

as seen by the distant observer, with a characteristic time 4 /c. Since

GM M

= 5 Ã— 10âˆ’6

= seconds

c3

c M

263

11.7 Spherically symmetric collapse of dust

the time scale for stellar-size objects is very small by the usual astrophysical

standards. Thus for any collapse even approximately like the free-fall collapse

described here, the approach to a black hole is extremely rapid.

Let us work out the redshift seen by the distant observer as a function of time t.

The ratio of the frequencies of a photon at emission and reception is

uR p R

R

= (11.14)

uE p E

E

where uE and uR are the 4-velocities of the emitter and receiver respectively and

p is the photon 4-momentum. The 4-velocity of our emitter riding on the starâ€™s

surface is

âˆ’1

uE = 1 âˆ’ 2 /r âˆ’ 2 c2 /r 1/2

00

whereas the 4-velocity of the stationary observer at infinity is

uR = 1 0 0 0

Hence (11.14) reduces to

âˆ’1

p0 R pE 1

= u0 + 1

R

=0 u

E

p0 E E

uE p0 E + u1 p1 E

E E

where we have used that fact that the Schwarzschild metric is stationary and so

p0 is conserved along the photon geodesic. Moreover, since p is null we require

g p p = 0, which in our case reduces to

âˆ’1

1 2 2

1âˆ’ p0 2 âˆ’ 1 âˆ’ =0

2

p1

c2 r r

âˆ’1 p /c

So, for a radially outgoing photon, p1 = âˆ’ 1 âˆ’ 2 /r and we find that

0

1/2 âˆ’1 1/2

2 2 2

R

= 1âˆ’ 1+ = 1âˆ’ (11.15)

r r r

E

As r â†’ 2 we see that R â†’ 0, so the redshift is infinite. By Taylor-expanding

(11.15) about r = 2 , we find that for r close to 2 we can write

r âˆ’2

R

â‰ˆ

4

E

however, near the event horizon the time of reception is given by (11.13). Hence

ct

R

âˆ¼ exp âˆ’

4

E

264 Schwarzschild black holes

so that the redshift goes exponentially to infinity with a characteristic time 4 /c.

The computation of the luminosity is more complicated since it involves non-radial

photon geodesics also. Nevertheless, using the above analysis we see that the time

intervals between successive photons will also decrease as âˆ¼ exp âˆ’ct/ 4 and

so we expect the luminosity to decay exponentially as âˆ¼ exp âˆ’ct/ 2 .

11.8 Tidal forces near a black hole

As discussed in Section 7.14, in Newtonian gravity a distribution of non-

interacting particles freely falling towards the Earth will be elongated in the

direction of motion and compressed in the transverse directions, as a result of

gravitational tidal forces. The same effect occurs in a body falling towards a

spherical object in general relativity, but if the object is a black hole then the

effect becomes infinite at r = 0.

We may calculate the tidal forces in the Schwarzschild geometry, working in

traditional Schwarzschild coordinates t r . At any particular point in space,

the tidal forces have the same form for any (close) pair of particles that are in

free fall. Thus, it is easiest to calculate the tidal forces at some coordinate radius

r for the case in which the two particles are released from rest at r. In this case, a

frame of orthonormal basis vectors defining the inertial instantaneous rest frame

of one of the particles may be taken as

âˆ’1/2 1/2

1 1 2 2

Ë† Ë†

=u= 1âˆ’ = 1âˆ’

e0 e1

0 1

c c r r

1 1

Ë† Ë†

= =

e2 e3

2 3

r sin

r

Substituting these expressions into (7.28), together with the appropriate expres-

sions for the components of the Riemann tensor in Schwarzschild coordinates,

from (7.27) we obtain (after some algebra) that the spatial components of the

orthogonal connecting vector between the two particles satisfy

Ë† Ë†

Ë†

d2 2 c2 d2 c2 d2 c2

r

Ë† Ë†

Ë†

=+ 3 =âˆ’ 3 =âˆ’ 3

r

2 2 2

d r d r d r

Ë†

The positive sign in the r -equation indicates a tension or stretching in the radial

Ë† Ë†

direction and the negative signs in the - and - equations indicate a pressure or

compression in the transverse directions. Note the 1/r 3 radial dependence in each

case, which is characteristic of tidal gravitational forces. Moreover, the equations

reveal that the tidal forces do not undergo any â€˜transitionâ€™ at r = 2 but become

infinite at r = 0.

265

11.8 Tidal forces near a black hole

Let us consider an intrepid astronaut falling feet first into a black hole. The

equations derived above will not hold exactly, since there will exist forces between

the particles (atoms) that comprise the astronaut. Nevertheless, when the tidal

gravitational forces become strong we can neglect the interatomic forces, and

the equations derived above will be valid to an excellent approximation. Thus

the unfortunate astronaut would be stretched out like a piece of spaghetti (!), as

illustrated in Figure 11.5. In fact, not only do the tidal forces tear the astronaut

to pieces, but the very atoms of which the astronaut is composed must ultimately

suffer the same fate! Assuming that the limit of tolerance to stretching or compres-

sion of a human body is an acceleration gradient of âˆ¼ 400 m sâˆ’2 per metre, for

a human to survive the tidal forces at the Schwarzschild radius requires a very

massive black hole with

105 M

M

If you fell towards a supermassive black hole, with say M âˆ¼ 109 M (such black

holes are believed to lie at the centres of some galaxies; see Appendix 11B) you

would cross the event horizon without feeling a thing. However, your fate will

have been sealed â€“ you will end up shredded by the tidal forces of the black

hole as you approach the singularity, from which there is no escape. If you fell

towards a â€˜smallâ€™ black hole, of mass say 10 M , you would be shredded by the

tidal forces of the hole well before you reached the event horizon.

Figure 11.5 An astronaut stretched by the tidal forces of a black hole. For a

human to survive this stretching at the Schwarzschild radius requires a very

massive black hole, with M 105 M

266 Schwarzschild black holes

11.9 Kruskal coordinates

In our discussion of advanced and retarded Eddingtonâ€“Finkelstein coordinates, we

found that neither coordinate system was completely satisfactory. In the advanced

coordinates the outgoing null rays are discontinuous, and in the retarded coor-

dinates the ingoing null rays are discontinuous. It is natural to ask whether it is

possible to find a system of coordinates in which both the incoming and outgoing

radial photon geodesics are continuous straight lines. Such a coordinate system

was indeed discovered in 1961 by Martin Kruskal, and it serves also to clarify

the structure of the complete Schwarzschild geometry.

An obvious way to begin is to introduce both the advanced null coordinate p

and the retarded null coordinate q that we met during our discussion of Eddingtonâ€“

Finkelstein coordinates. In the coordinates p q the Schwarzschild metric

becomes

2

ds2 = 1 âˆ’ dp dq âˆ’ r 2 d 2 + sin2 d 2 (11.16)

r

where r is considered as a function of p and q, defined implicitly by

r

p âˆ’ q = r + 2 ln âˆ’1

1

2 2

Our new system of coordinates has some appealing properties. Most impor-

tantly, the 2-space defined by = constant, = constant has the simple metric

2

ds2 = 1 âˆ’ (11.17)

dp dq

r

Transforming from the null coordinates p and q to the new coordinates

ct = p+q

1

(11.18)

2

r

r=

Ëœ p âˆ’ q = r + 2 ln âˆ’1

1

(11.19)

2 2

Ëœ

where t is the standard Schwarzschild timelike coordinate and r is a radial space-

like coordinate (sometimes called the tortoise coordinate!), the 2-space metric

then becomes

2

ds2 = 1 âˆ’ c2 dt2 âˆ’ dËœ 2 (11.20)

r

r

= 2

(11.21)

x dx dx

where x0 = ct and x1 = r . This line element has the same form as that of a

Ëœ

Minkowski 2-space (which is spatially flat) but it is multiplied by what mathemati-

cians call a conformal scaling factor, 2 x , which is a function of position. The

267

11.9 Kruskal coordinates

2-space itself is curved, because the derivatives of the function x enter into the

components of the curvature tensor, but the line element (11.21) of the 2-space

is manifestly conformally flat. In fact, any two-dimensional (pseudo-)Riemannian

manifold is conformally flat (see Appendix 11C), in that a coordinate system

always exists in which the line element takes the form (11.21). We have thus

succeeded in finding such a coordinate system for the 2-space (11.17).

The form of the line element (11.21) has an important consequence for studying

the paths of radially moving photons (for which d = d = 0). Since the conformal

factor 2 x is just a scaling, it does not change the lightcone structure and so

the latter should just look like that in Minkowski space. Thus, in a spacetime

Ëœ

diagram in ct r coordinates, both ingoing and outgoing radial null geodesics

are straight lines with slope Â±1, as is easily seen by setting ds2 = 0 in (11.20).

Unfortunately, however, the coordinates ct r are pathological when r = 2 ,

Ëœ

as is easily seen from (11.19). This suggests that, instead of using the parameters

p and q directly, we should look for a coordinate transformation that preserves

the manifest conformal nature of the 2-space defined by (11.17) but removes the

offending factor 1 âˆ’ 2 /r, which is the cause of the pathological behaviour. It

Ëœ Ëœ

is straightforward to see that a transformation of the form p p and q q will

achieve this goal, since, in this case, the metric becomes

2 dp dq

ds2 = 1 âˆ’ ËœËœ

dp dq

ËœËœ

r dp dq

which has the same general form as (11.17). An appropriate choice of the functions

p p and q q that removes the factor 1 âˆ’ 2 /r in the line element is (as

Ëœ Ëœ

suggested by Kruskal)

p q

p = exp

Ëœ q = âˆ’ exp âˆ’

Ëœ

4 4

for which we find that

3

32 r

ds = exp âˆ’ ËœËœ

2

dp dq

2

r

The usual form of the metric is then obtained by defining a timelike variable v

and a spacelike variable u by

v= p+q

ËœËœ u= pâˆ’q

ËœËœ

1 1

2 2

Thus, the full line element for the Schwarzschild geometry in Kruskal coordinates

is given by

vu

3

32 r

ds = exp âˆ’ dv2 âˆ’ du2 âˆ’ r 2 d + sin2 d

2 2 2

(11.22)

2

r

268 Schwarzschild black holes

where r is considered as a function of v and u that is defined implicitly by

r r

u2 âˆ’ v2 = âˆ’ 1 exp (11.23)

2 2

It is straightforward to show that the coordinates v and u are related to the

original Schwarzschild coordinates t and r by the following transformations. For

r > 2 we have

1/2

r r ct

v= âˆ’1 exp sinh

2 4 4

1/2

r r ct

u= âˆ’1 exp cosh

2 4 4

whereas, for r < 2 ,

1/2

r r ct

v = 1âˆ’ exp cosh

2 4 4

1/2

r r ct

u = 1âˆ’ exp sinh

2 4 4

Considerable insight into the nature of the Schwarzschild geometry can be

obtained by plotting its spacetime diagram in Kruskal coordinates. The causal

structure defined by radial light rays is (by construction) particularly easy to

analyse in Kruskal coordinates. From the metric (11.22), we see that for ds =

d = d = 0 we have

v = Â±u + constant

which represents straight lines at Â±45 to the axes. This is a direct consequence of

the fact that the 2-space with d = d = 0 is manifestly conformally flat in v u

coordinates. Thus, the lightcone structure should look like that in Minkowski

space. Also, we note that a massive particle worldline must always lie within the

future light-cone at each point.

It is also instructive to plot lines of constant t and r. From (11.23) we see that

lines of constant r are curves of constant u2 âˆ’ v2 and are hence hyperbolae. In

particular, the value r = 2 correpsonds to either of the straight lines u = Â±v,

constant-r hyperbolae, and the value r = 0

which are the asymptotes to the set of âˆš

corresponds to the hyperbolae v = Â± u2 + 1. Thus the â€˜pointâ€™ in space r = 0

is mapped into two lines. However, not too much can be made of this since it

is a singularity of the geometry. We should not glibly speak of it as a part of

269

11.9 Kruskal coordinates

spacetime with a well-defined dimensionality. Similarly, lines of constant t may

be mapped out. It is straightforward to show that

for r > 2

v/u

=

tanh ct/ 4

for r < 2

u/v

so fixed values of t correspond to lines of constant u/v, i.e. straight lines through

the origin. The value t = âˆ’ corresponds to u = âˆ’v, while t = corresponds

to u = v. The value t = 0 for r > 2 corresponds to the line v = 0, whereas for

r < 2 it is the line u = 0.

We note that the entire region covered by the Schwarzschild coordinates âˆ’ <

t < , 0 < r < is mapped onto the regions I and II in Figure 11.6. Thus, we

would require two Schwarzschild coordinate patches (I, II) and (I II ) to cover the

entire Schwarzschild geometry, but a single Kruskal coordinate system suffices.

The diagonal lines r = 2 , t = and r = 2 , t = âˆ’ define event horizons

separating the regions of spacetime II and II from the other regions, I and I .

The Kruskal diagram has some curious features. There are two â€˜Minkowskiâ€™

regions, I and I , so apparently there are two universes. We can identify region I

as the spacetime region outside a Schwarzschild black hole and region II as the

interior of the black-hole event horizon. Any particle that travels from region I to

v

r=

4Âµ

ct = 2Âµ

r=

r= Âµ

3

Radially

2Âµ

,t

infalling

II

=â€“

ct = Âµ particle

âˆž

I' I ct = 0 u

ct = â€“Âµ

âˆž II'

=

,t

2Âµ ct = â€“2Âµ

r=

r = 4Âµ

r = 3Âµ

r= Âµ

Figure 11.6 Spacetime diagram of the Schwarzschild geometry in Kruskal coor-

dinates. The lower and upper wavy lines at the boundaries of the shaded regions

are respectively the past singularity and the future singularity at r = 0. The

broken-line arrows show escaping signals.

270 Schwarzschild black holes

region II can never return and, moreover, must eventually reach the singularity

r = 0. Regions I and II are completely inaccessible from regions I or II. Region

II is similar to region II but in reverse: it is a part of spacetime from which

a particle can escape (into regions I and I ) but not enter. Moreover, there is

a singularity in the past â€“ a white hole â€“ from which particles can emanate.

Indeed, we may now understand more clearly our discussion of the advanced

and retarded Eddingtonâ€“Finkelstein coordinates in Section 11.5: the advanced

coordinates describe the Schwarzschild geometry in regions I and II, whereas the

retarded coordinates cover the regions I and II . The two universes I and I are

actually connected by a wormhole at the origin, which we discuss in more detail

in the next section, but, as we will show, no particle can travel between regions

I and I .

It is worth asking what has happened here. How can a few simple coordinate

transformations lead to what is apparently new physics? What we have done

amounts to mathematically extending the Schwarzschild solution. Mathematicians

would call this a maximal extension of the Schwarzschild solution because all

geodesics either extend to infinite values of their affine parameter or end at a

past or future singularity. Thus Kruskal coordinates probe all the Schwarzschild

geometry. Hence, we find that the complete Schwarzschild geometry consists of

a black hole and white hole and two universes connected at their horizons by a

wormhole.

The extended Schwarzschild metric is a solution of Einsteinâ€™s theory and hence

is allowed by classical general relativity. Thus, for example, classical general

relativity allows the existence of white holes. Photons or particles could, in

principle, emanate from a past singularity. But, as you can see from the Kruskal

spacetime diagram, you cannot â€˜fall intoâ€™ a white hole since it can only exist in

your past. Can a white hole really exist? The answer is that we donâ€™t know for

sure. Classical GR must break down at singularities. We would expect quantum

effects to become important at ultra-short distances and ultra-high energies. In

fact, from the three fundamental constants G, and c we can form the following

energy, mass, time, length and density scales:

1/2

EP = = 1 22 Ã— 1019 GeV

c5 /G

Planck energy

= 2 18 Ã— 10âˆ’5 g

1/2

mP =

Planck mass c/G

1/2

= 5 39 Ã— 10âˆ’44 s

tP = G/c5

Planck time

1/2

= 1 62 Ã— 10âˆ’33 cm

lP = G/c3

Plancklength

= 5 16 Ã— 1093 g cmâˆ’3

= c5 / G2

Planck density P

271

11.10 Wormholes and the Einsteinâ€“Rosen bridge

These Planck scales define the characteristic energies, lengths, times, etc. at which

we expect quantum gravitational effects to become important. To put it into some

kind of perspective, an elementary particle with the Planck mass would weigh

about the same as a small bacterium.

Nobody really expects the centres of black holes to harbour true singularities.

Instead, it is expected that, close to the classical singularity, quantum gravitational

effects will occur that will prevent the divergences of classical general relativity.

We do not yet have a complete theory of quantum gravity, though many people

hope that M-theory (formerly known as superstring theory) might one day provide

such a theory. Theorists have developed semi-classical theories, however, which

might (or might not) contain some of the features of a complete theory of quantum

gravity. Such calculations suggest that white holes would be unstable and could

not exist for more than about a Planck time. It is interesting that within a few

pages we have pushed Einsteinâ€™s theory of gravity to the edge of known physics.

11.10 Wormholes and the Einsteinâ€“Rosen bridge

Although it is not obvious from Figure 11.6, the two universes I and I are actually

connected by a wormhole at the origin. To understand the structure at the origin,

you must realize that the coordinates and have been suppressed in this figure;

each point in Figure 11.6 actually represents a 2-sphere.

We can gain some intuitive insight into wormholes by considering the geometry

of the spacelike hypersurface v = 0, which extends from u = + to u = âˆ’ .

The line element for this hypersurface is

3

32 r

ds = âˆ’ exp âˆ’ du2 âˆ’ r 2 d + sin2 d

2 2 2

2

r

We can draw a cross-section of this hypersurface corresponding to the equatorial

plane = /2, in which the line element reduces further to

3

32 r

ds = âˆ’ exp âˆ’ du2 âˆ’ r 2 d

2 2

(11.24)

2

r

To interpret this, we may consider a two-dimensional surface possessing a

line element d 2 given by minus (11.24) and embed it in a three-dimensional

Euclidean space.

This embedding is most easily performed by re-expressing d 2 in terms of the

coordinates r and , which is easily shown to yield the familiar form

âˆ’1

2

= 1âˆ’ dr 2 + r 2 d

2 2

(11.25)

d

r

272 Schwarzschild black holes

However, we must remember that, in the spacelike hypersurface v = 0, as we

move along the u-axis from + to âˆ’ the value of r decreases to a minimum

value r = 2 (at u = 0) and then increases again. In general, in Euclidean space, a

2-surface parameterised by arbitrary coordinates can be specified by giving

a = 1 2 3 , where the xa define some coordinate system

three functions xa

in the three-dimensional Euclidean space. In our particular case, it will be useful

to use cylindrical polar coordinates z , in which case the line element of

the three-dimensional space is

=d + + dz2

2 2 2 2

(11.26)

d d

Moreover, since the 2-surface we wish to embed (which is parameterised by the

coordinates r and ) is clearly axisymmetric, we may take the three functions

specifying this surface to have the form

= = z=z r

r

Substituting these forms into (11.26), we may thus write the line element on the

embedded 2-surface as

2 2

d dz

= + dr 2 +

2 2 2

(11.27)

d d

dr dr

For the geometry of the embedded 2-surface to be identical to the geometry of

the 2-space of interest, we require the line elements (11.25) and (11.27) to be

identical, and so we require r = r and thus

âˆ’1

2

2

dz

1+ = 1âˆ’

dr r

The solution to this differential equation is easily found to be

zr = r âˆ’2 + constant

8

and substituting r = gives us the equation of the cross-section of the embedded

2-surface in the z -plane of the Euclidean 3-space. Taking the constant of

integration to be zero, and remembering that r (and hence ) is never less than

2 , we find that the surface has the form shown in Figure 11.7. Thus, the

geometry of the spacelike hypersurface at v = 0 can be thought of as two distinct,

but identical, asymptotically flat Schwarzschild manifolds joined at the â€˜throatâ€™

r = 2 by an Einsteinâ€“Rosen bridge. If one so wishes, one can also connect the

two asymptotically flat regions together in a region distant from the throat. In this

case, the wormhole connects two distant regions of a single universe.

In either case, the structure of the wormhole is dynamic. One is used to thinking

of the Schwarzschild geometry as â€˜staticâ€™. However, working for the moment in

273

11.10 Wormholes and the Einsteinâ€“Rosen bridge

Ï† = constant u = constant

Figure 11.7 The structure of the Einsteinâ€“Rosen bridge.

terms of the traditional Schwarzschild coordinate, it is only in regions I and I that

t is timelike and the fact that the metric coefficients are independent of t means

that spacetime is static. In regions II and II , the t-coordinate is spacelike and

the r-coordinate is timelike. Since the metric coefficients do depend explicitly on

r, the spacetime in these regions is no longer static but evolves with respect to

this timelike coordinate. Returning to Kruskal coordinates, consider the spacelike

hypersurface v = 0. As this surface is pushed forwards in time (in the +v direction

in the Kruskal diagram), part of it enters region II and begins to evolve.

As v increases, the picture of the geometry of the hypersurface is qualitatively

the same as that illustrated in Figure 11.7, but the bridge narrows, the universes

now joining at r < 2 . At v = 1, the bridge pinches off completely and the two

universes simply touch at the singularity r = 0. For larger values of v the two

universes, each containing a singularity at r = 0, are completely separate. Since

the Kruskal solution is symmetric in v, the same things happen for negative values

of v. The full time evolution is shown schematically in Figure 11.8. Thus, the

two universes are disconnected at the beginning, each containing a singularity of

infinite curvature r = 0 . As they evolve in time, their singularities join each

other and form a non-singular bridge. The bridge enlarges until at v = 0 it reaches

a maximum radius at the throat equal to r = 2 . It then contracts and pinches off,

v < â€“1 v = â€“1 â€“1 < v < 0 v=0 0<v <1 v=1 v>1

Figure 11.8 Time evolution of the Einsteinâ€“Rosen bridge.

274 Schwarzschild black holes

leaving the two universes disconnected and containing singularities r = 0 once

again.

Sadly, it is impossible for a traveller to pass through the wormhole from one

universe into the other, since the formation, expansion and collapse of the bridge

occur too rapidly. By examining the paths of light rays in the Kruskal diagram, we

can deduce that no particle or photon can pass across the bridge from the faraway

region of one universe to the faraway region of the other without getting caught

and crushed in the throat as it pinches off. Nevertheless, after falling through

the horizon of the black hole, a traveller could see light signals from the other

universe through the throat of the wormhole. Unfortunately, the penalty for seeing

the other universe is death at the singularity.

Can wormholes exist in Nature? Can they connect different universes, or differ-

ent parts of the same universe? Again, nobody knows for sure. Many theorists

would argue that we need to understand quantum gravity to understand worm-

holes. Wormholes are probably unstable, but â€˜virtualâ€™ wormholes are a feature of

some formulations of quantum gravity.

11.11 The Hawking effect

So far our discussion of black holes has been purely classical. Indeed, we have

found that classically nothing can escape from the within the event horizon

of a black hole; that is why they are called black holes! However, in 1974,

Stephen Hawking applied the principles of quantum mechanics to electromagnetic

fields near a black hole and found the amazing result that black holes radiate

continuously as a blackbody with a temperature inversely proportional to their

mass! Hawkingâ€™s original calculation uses the techniques of quantum field theory,

but we can derive the main results very simply from elementary arguments.

According to quantum theory, even the vacuum of empty space exhibits quan-

tum fluctuations, in which particleâ€“antiparticle pairs are created at one event

only to annihilate one another at some other event. Pair creation violates the

conservation of energy and so is classically forbidden. In quantum mechanics,

however, one form of Heisenbergâ€™s uncertainty principle is t E = , where E

is the minimum uncertainty in the energy of a particle that resides in a quantum

mechanical state for a time t. Thus, provided the pair annihilates in a time less

than t = / E, where E is the amount of energy violation, no physical law

has been broken.

Let us now consider such a process occurring just outside the event horizon of a

black hole. For simplicity, let us consider a Schwarzschild black hole in t r

coordinates. Suppose that a particleâ€“antiparticle pair is produced from the vacuum

Â¯

and that the constituents of the pair have 4-momenta p and p respectively. Since

275

11.11 The Hawking effect

Â¯

the spacetime is stationary 0 g = 0 , the quantities p0 = e0 Â· p and p0 = e0 Â· p are

Â¯

conserved along the particle worldlines; here e0 is the t-coordinate basis vector.

Thus, for a fluctuation from the vacuum, classical conservation requires

Â¯

e0 Â· p + e0 Â· p = 0 (11.28)

The squared â€˜lengthâ€™ of the coordinate basis vector e0 is given by

e0 Â· e0 = g00 = c2 1 âˆ’ 2 /r (11.29)

Â¯

Thus, outside the horizon r > 2 , e0 is timelike. The components e0 Â· p and e0 Â· p

are therefore proportional to the particle energies as measured by an observer

whose 4-velocity is along the e0 -direction. Hence both must be positive, so the

conservation condition (11.28) cannot be satisfied.

However, if the fluctuation occurs near the event horizon then the inward-

moving particle may travel to the region r < 2 . Inside the event horizon e0

is spacelike, as shown by (11.29). Thus e0 Â· p is a component of the spatial

momentum for some observer and so may be negative. Hence the conservation

condition (11.28) can be satisfied if the antiparticle (say) crosses the horizon with

Â¯

negative e0 Â· p and the particle escapes to infinity with positive e0 Â· p. As seen by

an observer at infinity, the black hole has emitted a particle of energy e0 Â· p and

Â¯

the black holeâ€™s mass has decreased by e0 Â· p c2 as a consequence of the particle

falling into it. This is the Hawking effect. Of course, the argument is equally valid

if it is the particle that falls into the black hole and the antiparticle that escapes

to infinity. The black hole emits particles and antiparticles in equal numbers.

For a fluctuation near the horizon, the inward-travelling particle needs to endure

in a prohibited negative e0 Â·p condition only for a short proper time, as measured by

some locally free-falling observer, before reaching the inside of the horizon where

negative e0 Â·p is allowed. The particle has, in fact, tunnelled quantum mechanically

through a region outside the horizon, where negative e0 Â· p is classically forbidden,

to a region inside the event horizon where it is classically allowed. The process

works best where the proper time in the forbidden region is smallest, i.e. close to

the horizon.

The distant observer sees a steady flux of particles and antiparticles. The flux

must be steady, since the geometry is independent of t and so the rate of particle

emission must also be independent of t. Let us calculate the typical energy of

such a particle as measured by the distant observer. Suppose that the particleâ€“

antiparticle pair is created at some event P with coordinate radius R = 2 + . Let

us consider this event as viewed by a freely falling observer, starting from rest at

this point. Since the observer is in free fall, the rules of special relativity apply in

his frame. A typical measure of the proper time elapsed before the observer

276 Schwarzschild black holes

reaches the horizon may be obtained by considering a radially free-falling particle

that starts from rest at r = R. In this case,

1 âˆ’ 2 /R 1/2

Ë™

t=

1 âˆ’ 2 /r

1/2

11

r=âˆ’ 2 c

Ë™ âˆ’

2

rR

Thus the required proper time interval is

âˆ’1/2

2 c2 2 c2 1/2

2 22

=âˆ’ âˆ’ dr â‰ˆ

2+

r c

2+

where the final result is quoted to first order in . From the uncertainty principle,

the typical energy of the particle, as measured by a freely falling observer, is

given by

c

= = âˆ’1/2

22

However, this can also be written as

= p Â· u â‰ˆ p0 u0

where u is the observerâ€™s 4-velocity and the approximation holds since u1 u0 .

Ë™

Now, u0 = t â‰ˆ 2 / 1/2 to first order in . Moreover, p0 is conserved along

the particleâ€™s worldline and is equal to the energy E of the particle as measured

by the distant observer, whose 4-velocity is simply u = 1 0 0 0 . Thus, we

finally obtain

1/2

c3

E= = (11.30)

2 4GM

Remarkably, this result does not depend on ; the particle always emerges with

this characteristic energy.

The full quantum field theory calculation shows that the particles are in

fact received with a blackbody energy spectrum characterised by the Hawking

temperature

c3

T=

8 kB GM

The typical particle energy is thus E = kB T = c3 / 8 GM , which is only a

factor 2 smaller than our crude estimate (11.30). Putting in numbers, we find that

âˆ’1

M

âˆ’8

T = 6 Ã— 10 K

M

277

Appendix 11A: Compact binary systems

Thus the radiation from a solar-mass black hole, such as might be formed by the

gravitational collapse of a massive star, is negligibly small.

It is straightforward to calculate the rate dM/dt at which the black hole loses

mass, as determined by a stationary distant observer whose proper time is t. Since

the black-hole event horizon emits radiation as a blackbody of temperature T , the

black-hole mass must decrease at a rate

T 4A

dM

=âˆ’ 2

dt c

where = 2 kB / 60 3 c2 is the Stefanâ€“Boltzmann constant and A is the proper

4

area of the event horizon. From the Schwarzschild metric we find that A = 16 2 ,

and so we obtain

dM

=âˆ’ 2 (11.31)

dt M

where the dimensionless constant = c4 / 15 360 G2 = 3 76 Ã— 1049 . The solu-

tion M t to (11.31) is easily calculated. For a black hole whose evaporation is

complete at time t0 , we find that

Mt = 3 t0 âˆ’ t 1/3

(11.32)

This result shows that a burst of energy is emitted right at the end of a black

holeâ€™s life. For example, in the final second it should emit âˆ¼ 1022 J of energy,

primarily as -rays. No such events have yet been identified.

Appendix 11A: Compact binary systems

One of the best ways of finding candidate black holes is to search for luminous

compact X-ray sources. The reason is that if a black hole has a stellar companion

then the intense tidal field can pull gas from the companion, producing an accretion

disc around the black hole. A schematic picture is shown in Figure 11.9. As we

showed in Chapter 10, accretion discs can radiate very efficiently and we would

expect to observe high-energy (X-ray) photons emitted from a small region of

space.

Table 11.1 summarizes the two common classes of compact binaries. The

compact object can be a white dwarf, neutron star or black hole. If you find a

compact binary system then you can set limits on the mass of the compact object

from the dynamics of the binary orbit. If you find evidence for a compact object

that is more massive than the Chandrasekhar limit then you have good evidence

that the object might be a black hole.

278 Schwarzschild black holes

Table 11.1 Compact accreting binary systems

Compact object

Companion star White dwarf Neutron star Black hole

Early type, massive None known Massive X-ray binaries Cyg X-I

A0620 âˆ’ 00

Late type, low mass Cataclysmic variables Low mass X-ray binaries

(e.g., dwarf novae)

Figure 11.9 Schematic picture of a compact binary system.

In fact it is not so straightforward. What observers actually measure is the mass

function

PK 3

fM =

2G

where P is the orbital period, and K is the radial velocity amplitude. For example,

for the low-mass X-ray binary A0620-00 the period is P = 7 7 hours and K =

457 km sâˆ’1 . From Keplerâ€™s laws we can show that the mass function is related

to the masses M1 and M2 of the compact object and the companion star and the

279

Appendix 11B: Supermassive black holes

Table 11.2 Derived parameters and dynamical mass measurements of SXTs

g cmâˆ’3 q = M1 /M2

Source fM M i M1 M M2 M

6 08 Â± 0 06 17 Â± 1 55 Â± 4 12 Â± 2

V404 Cyg 0 005 0.6

G2000 + 25 5 01 Â± 0 12 24 Â± 10 56 Â± 15 10 Â± 4

16 0.5

4 86 Â± 0 13 60 Â± 10 6Â±2

N Oph 77 07 >19 0.3

+20

6+5

3 01 Â± 0 15 8Â±2

N Mus 91 10 54âˆ’15 0.8

âˆ’2

A0620 âˆ’ 00 2 91 Â± 0 08 15 Â± 1 37 Â± 5 10 Â± 5

18 0.6

J0422 + 32 1 21 Â± 0 06 20 âˆ’ 40 10 Â± 5

42 >12 0.3

J1655 âˆ’ 40 3 24 Â± 0 14 3 6Â±0 9 67 Â± 3 6 9Â±1

0 03 2.1

4U1543 âˆ’ 47 0 22 Â± 0 02 20 âˆ’ 40 5 0Â±2 5

02 â€” 2.5

0 21 Â± 0 08 5Â±1 43 Â± 11 1 3Â±0 6

Cen X-4 05 0.4

inclination angle i of the orbit to the plane of the sky by

M1 sin3 i

3

fM =

M1 + M2 2

You can see from this equation that the mass function is a strict lower limit on the

mass M1 of the compact object. It is equal to the latter, f = M1 , only if M2 = 0

and the orbit is viewed edge on (so that sin i = 1). For example, for A0620 âˆ’ 00

the lower limit on the mass of the compact object is 2 9M , and this makes it

a very good black hole candidate because this mass limit is very close to the

theoretical upper limit for the mass of a neutron star. In fact, it is possible to

make reasonable estimates3 for M2 and sin i in this system, leading to a probable

mass of â‰ˆ 10M for the compact object â€“ well into the black-hole regime.

Table 11.2 summarises the dynamical mass limits on some good black-hole

candidates (so-called short X-ray transients). As you can see, in several systems,

such as V404 Cyg, G2000 + 25 and N Oph 77, the minimum mass inferred from

the mass function is well above the theoretical maximum mass limit for a neutron

star. As we understand things at present there can be no other explanation than

that the compact objects are black holes.

Appendix 11B: Supermassive black holes

The first quasar4 (3C273) was discovered in 1963 by Maarten Schmidt. He

measured a cosmological redshift of z = 0 15 for this object, which was

3

An estimate of the mass M2 can be made by measuring the spectral type and luminosity of the companion

star. The inclination angle can be estimated from the shape of the starâ€™s light curve by searching for evidence

of eclipsing by the compact object.

4

Quasi-stellar radio source. We now know that the majority of quasars are radio quiet, and so they are often

called QSOs for quasi-stellar object.

280 Schwarzschild black holes

unprecedently high at the time (quasars have since been discovered with redshifts

as high as z = 5 8). Quasars are very luminous, typically 100â€“1000 times brighter

than a large galaxy. However, they are compact, so compact, in fact, that quasars

look like stars in photographs. In fact, from variability and other studies one can

infer that the size of the continuum-emitting region of a quasar is of order a few

parsecs or less. How can we explain such a phenomenon? Imagine an object

radiating many times the luminosity of an entire galaxy from a region smaller

than the Solar System. Donald Lynden-Bell was one of the first to suggest that

the quasar phenomenon is caused by accretion of gas onto a supermassive black

hole residing at the centre of a galaxy. The black-hole masses required to explain

the high luminosities of quasars are truly spectacular â€“ we require black holes

with masses a few million to a few billion times the mass of the Sun.

Do such supermassive black holes exist? The evidence in recent years has

become extremely strong. Using the Hubble Space Telescope it is possible to

probe the velocity dispersions of stars in the central regions of galaxies. According

to Newtonian dynamics, we would expect the characteristic velocities to vary as

GM

v2 âˆ¼

r

If the central mass is dominated by a supermassive black hole then we expect the

typical velocities of stars to increase as we go closer to the centre. This is indeed

what is found in a number of galaxies. From the rate of increase of the velocities

with radius, we can estimate the mass of the central object, which seems to be

correlated with the mass of the bulge component of the galaxy:

Mbh â‰ˆ 0 006Mbulge

It seems as though, at the time of galaxy formation, about half a percent of the

mass of the bulge material collapses right to the very centre of a galaxy to form a

supermassive black hole. During this phase the infalling gas radiates efficiently,

producing a quasar. When the gas supply is used up, the quasar quickly fades

away leaving a dormant massive black hole that is starved of fuel. Nobody has

yet developed a convincing theory of how this happens, or of what determines

the mass of the central black hole.

A sceptic might argue that these observations merely prove that a dense compact

object exists at the centre of a galaxy that is not necessarily a black hole. But there

are two beautiful observational results that probe compact objects on parsecond

scales â€“ making it almost certain that the central objects are black holes. In our

own Milky Way Galaxy it is possible to measure the proper motions of stars in

the Galactic centre (using infrared wavelengths to penetrate through the dense

dust that obscures optical light). This has allowed astronomers to see the stars

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