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2

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

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'

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†
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

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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