Chapter 2

Black Holes

2.1 Dark Stars: A Historical Note

It is usual in textbooks to credit John Michell and Pierre-Simon Laplace for the idea
of black holes, in the XVIII Century. The idea of a body so massive that even light
could not escape was put forward by geologist Rev. John Michell in a letter written
to Henry Cavendish in 1783 to the Royal Society:
If the semi-diameter of a sphere of the same density as the Sun were to exceed that of the
Sun in the proportion of 500 to 1, a body falling from an infinite height toward it would
have acquired at its surface greater velocity than that of light, and consequently supposing
light to be attracted by the same force in proportion to its inertia, with other bodies, all
light emitted from such a body would be made to return toward it by its own proper gravity.
(Michell 1784).

In 1796, the mathematician Pierre-Simon Laplace promoted the same idea in the
first and second editions of his book Exposition du système du Monde (it was re-
moved from later editions). Such “dark stars” were largely ignored in the nineteenth
century, since light was then thought to be a massless wave and therefore not influ-
enced by gravity. Unlike the modern concept of black hole, the object behind the
horizon in black stars is assumed to be stable against collapse. Moreover, no equa-
tion of state was adopted neither by Michell nor by Laplace. Hence, their dark stars
were Newtonian objects, infinitely rigid, and they have nothing to do with the nature
of space and time, which were considered by them as absolute concepts. Nonethe-
less, Michel and Laplace could calculate correctly the size of such objects from the
simple device of equating the potential and escape energy from a body of mass M:
1 2 GMm
mv = . (2.1)
2 r2
Just setting v = c and assuming that the gravitational and the inertial mass are the
same, we get

2GM
rdark star = . (2.2)
c2

G.E. Romero, G.S. Vila, Introduction to Black Hole Astrophysics, 31

Lecture Notes in Physics 876, DOI 10.1007/978-3-642-39596-3_2,
© Springer-Verlag Berlin Heidelberg 2014
32 2 Black Holes

Dark stars are objects conceivable only within the framework of the Newtonian
theory of matter and gravitation. In the context of general relativistic theories of
gravitation, collapsed objects have quite different properties. Before exploring par-
ticular situations that can be represented by different solutions of Einstein’s field
equations, it is convenient to introduce a general definition of a collapsed gravita-
tional system in a general space-time framework. This is what we do in the next
section.

2.2 A General Definition of Black Hole

We shall now provide a general definition of a black hole, independently of the co-
ordinate system adopted in the description of space-time, and even of the exact form
of the field equations. First, we shall introduce some preliminary useful definitions
(e.g. Hawking and Ellis 1973; Wald 1984).

Definition A causal curve in a space-time (M, gμν ) is a curve that is non space-like,
that is, piecewise either time-like or null (light-like).

We say that a given space-time (M, gμν ) is time-orientable if we can define over
M a smooth non-vanishing time-like vector field.

Definition If (M, gμν ) is a time-orientable space-time, then ∀p ∈ M, the causal

future of p, denoted J + (p), is defined by:
 
J + (p) ≡ q ∈ M|∃ a future-directed causal curve from p to q . (2.3)

Similarly,

Definition If (M, gμν ) is a time-orientable space-time, then ∀p ∈ M, the causal

past of p, denoted J − (p), is defined by:

J − (p) ≡ {q ∈ M|∃ a past-directed causal curve from p to q}. (2.4)

The causal future and past of any set S ⊂ M are given by:

J + (S) = J + (P ) (2.5)
p∈S

and,

J − (S) = J − (P ). (2.6)
p∈S

A set S is said achronal if no two points of S are time-like related. A Cauchy

surface (Sect. 1.8) is an achronal surface such that every non space-like curve in M
2.3 Schwarzschild Black Holes 33

crosses it once, and only once. A space-time (M, gμν ) is globally hyperbolic if it
admits a space-like hypersurface S ⊂ M which is a Cauchy surface for M.
Causal relations are invariant under conformal transformations of the metric. In
this way, the space-times (M, gμν ) and (M, gμν ), where gμν = Ω 2 gμν , with Ω a
r
non-zero C function, have the same causal structure.
Particle horizons occur whenever a particular system never gets to be influenced
by the whole space-time. If a particle crosses the horizon, it will not exert any further
action upon the system with respect to which the horizon is defined.

Definition For a causal curve γ the associated future (past) particle horizon is de-
fined as the boundary of the region from which the causal curves can reach some
point on γ .

Finding the particle horizons (if one exists at all) requires a knowledge of the
global space-time geometry.
Let us now consider a space-time where all null geodesics that start in a region
J − end at J + . Then, such a space-time, (M, gμν ), is said to contain a black hole if
M is not contained in J − (J + ). In other words, there is a region from where no null
geodesic can reach the asymptotic flat1 future space-time, or, equivalently, there is
a region of M that is causally disconnected from the global future. The black hole
region, BH , of such space-time
 is BH = [M − J − (J + )], and the boundary of BH
− +
in M, H = J (J ) M, is the event horizon.
Notice that a black hole is conceived as a space-time region, i.e. what character-
izes the black hole is its metric and, consequently, its curvature. What is peculiar
of this space-time region is that it is causally disconnected from the rest of the
space-time: no events in this region can make any influence on events outside the
region. Hence the name of the boundary, event horizon: events inside the black hole
are separated from events in the global external future of space-time. The events in
the black hole, nonetheless, as all events, are causally determined by past events.
A black hole does not represent a breakdown of classical causality. As we shall see,
even when closed time-like curves are present, local causality still holds along with
global consistency constrains. And in case of singularities, they do not belong to
space-time, so they are not predicable (i.e. we cannot attach any predicate to them,
nothing can be said about them) in the theory. More on this in Sect. 3.6.

2.3 Schwarzschild Black Holes

The first exact solution of Einstein’s field equations was found by Karl Schwarz-
schild in 1916. This solution describes the geometry of space-time outside a spheri-
cally symmetric matter distribution.

1 Asymptotic flatness is a property of the geometry of space-time which means that in appropriate

coordinates, the limit of the metric at infinity approaches the metric of the flat (Minkowskian)
space-time.
34 2 Black Holes

2.3.1 Schwarzschild Solution

The most general spherically symmetric metric is

ds 2 = α(r, t)dt 2 − β(r, t)dr 2 − γ (r, t)dΩ 2 − δ(r, t)drdt, (2.7)

where dΩ 2 = dθ 2 +sin2 θ dφ 2 . We are using spherical polar coordinates. The metric

(2.7) is invariant under rotations (isotropic).
The invariance group of general relativity is formed by the group of general trans-
formations of coordinates of the form x  μ = f μ (x). This yields 4 degrees of free-
dom, two of which have been used when adopting spherical coordinates (the trans-
formations that do not break the central symmetry are r  = f1 (r, t) and t  = f2 (r, t)).
With the two available degrees of freedom we can freely choose two metric coeffi-
cients, whereas the other two are determined by Einstein’s equations. Some possi-
bilities are:
• Standard gauge.

ds 2 = c2 A(r, t)dt 2 − B(r, t)dr 2 − r 2 dΩ 2 .

• Synchronous gauge.

ds 2 = c2 dt 2 − F 2 (r, t)dr 2 − R 2 (r, t)dΩ 2 .

• Isotropic gauge.

ds 2 = c2 H 2 (r, t)dt 2 − K 2 (r, t) dr 2 + r 2 (r, t)dΩ 2 .

• Co-moving gauge.

ds 2 = c2 W 2 (r, t)dt 2 − U (r, t)dr 2 − V (r, t)dΩ 2 .

Adopting the standard gauge and a static configuration (no dependence of the
metric coefficients on t), we can get equations for the coefficients A and B of the
standard metric:
ds 2 = c2 A(r)dt 2 − B(r)dr 2 − r 2 dΩ 2 . (2.8)
Since we are interested in the solution outside the spherical mass distribution, we
only need to require the Ricci tensor to vanish:

Rμν = 0.

According to the definition of the curvature tensor and the Ricci tensor, we have:

Rμν = ∂ν Γμσ
σ
− ∂σ Γμν
σ
+ Γμσ
ρ σ
Γρν − Γμν
ρ σ
Γρσ = 0. (2.9)

If we remember that the affine connection depends on the metric as

1
σ
Γμν = g ρσ (∂ν gρμ + ∂μ gρν − ∂ρ gμν ),
2
2.3 Schwarzschild Black Holes 35

we see that we have to solve a set of differential equations for the components of
the metric gμν .
The metric coefficients are:

g00 = A(r),
g11 = −B(r),
g22 = −r 2 ,
g33 = −r 2 sin2 θ,
g 00 = 1/A(r),
g 11 = −1/B(r),
g 22 = −1/r 2 ,
g 33 = −1/r 2 sin2 θ.

Then, only nine of the 40 independent connection coefficients are different from
zero. They are:
1
Γ01 = A /(2A),
1
Γ22 = −r/B,
2
Γ33 = − sin θ cos θ,
1
Γ00 = A /(2B),


1
Γ33 = − r sin2 /B ,
3
Γ13 = 1/r,
1
Γ11 = B  /(2B),
2
Γ12 = 1/r,
3
Γ23 = cot θ.

Replacing in the expression for Rμν :


A A A B  A
R00 = − + + − ,
2B 4B A B rB

A A A B  B
R11 = − + − ,
2A 4A A B rB

1 r A B 
R22 = − 1 + − ,
B 2B A B
R33 = R22 sin2 θ.
36 2 Black Holes

Einstein’s field equations for the region of empty space then become:

R00 = R11 = R22 = 0

(the fourth equation has no additional information). Multiplying the first equation
by B/A and adding the result to the second equation, we get:

A B + AB  = 0,

from which AB = constant. We can write then B = αA−1 . Going to the third equa-
tion and replacing B we obtain: A + rA = α, or:

d(rA)
= α.
dr
The solution of this equation is:

k
A(r) = α 1 + ,
r

with k another integration constant. For B we get:


k −1
B = 1+ .
r

If we now consider the Newtonian limit:

A(r) 2Φ
2
=1+ 2 ,
c c
with Φ = −GM/r the Newtonian gravitational potential, we conclude that

2GM
k=−
c2
and
α = c2 .
Therefore, the Schwarzschild solution for a static mass M can be written in spher-
ical coordinates (t, r, θ, φ) as
 
2GM 2 2 2GM −1 2

ds 2 = 1 − 2
c dt − 1 − 2
dr − r 2 dθ 2 + sin2 θ dφ 2 . (2.10)
rc rc

As mentioned, this solution corresponds to the vacuum region exterior to the

spherical object of mass M. Inside the object, space-time will depend on the pecu-
liarities of the physical object.
2.3 Schwarzschild Black Holes 37

The metric given by Eq. (2.10) has some interesting properties. Let’s assume that
the mass M is concentrated at r = 0. There seems to be two singularities at which
the metric diverges: one at r = 0 and the other at

2GM
rSchw = . (2.11)
c2
The length rSchw is know as the Schwarzschild radius of the object of mass M.
Usually, at normal densities, rSchw is well inside the outer radius of the physical
system, and the solution does not apply in the interior but only to the exterior of
the object. For instance, for the Sun rSchw ∼ 3 km. However, for a point mass, the
Schwarzschild radius is in the vacuum region and space-time has the structure given
by (2.10). In general, we can write

M
rSchw ∼ 3 km,
M

where M = 1.99 × 1033 g is the mass of the Sun.

It is easy to see that strange things occur close to rSchw . For instance, for the
proper time we get:

2GM 1/2
dτ = 1 − dt, (2.12)
rc2
or

2GM −1/2
dt = 1 − dτ. (2.13)
rc2
When r → ∞ both times agree, so t is interpreted as the proper time measured
from an infinite distance. As the system with proper time τ approaches to rSchw , dt
tends to infinity according to Eq. (2.13). The object never reaches the Schwarzschild
surface when seen by an infinitely distant observer. The closer the object is to the
Schwarzschild radius, the slower it moves for the external observer.
A direct consequence of the difference introduced by gravity in the local time
with respect to the time at infinity is that the radiation that escapes from a given
radius r > rSchw will be redshifted when received by a distant and static observer.
Since the frequency (and hence the energy) of the photon depends on the time inter-
val, we can write, from Eq. (2.13):

2GM −1/2
λ∞ = 1 − λ. (2.14)
rc2

Since the redshift is:

λ∞ − λ
z= , (2.15)
λ
38 2 Black Holes

Fig. 2.1 Space-time diagram

in Schwarzschild coordinates
showing the light cones of
events at different distances
from the event horizon.
Adapted form Carroll (2003)

then

2GM −1/2
1+z= 1− , (2.16)
rc2
and we see that when r → rSchw the redshift becomes infinite. This means that a
photon needs infinite energy to escape from inside the region determined by rSchw .
Events that occur at r < rSchw are disconnected from the rest of the universe. Hence,
we call the surface determined by r = rSchw an event horizon. Whatever crosses the
event horizon will never return. This is the origin of the expression “black hole”,
introduced by John A. Wheeler in the mid 1960s. The black hole is the region of
space-time inside the event horizon. We can see in Fig. 2.1 what happens with the
light cones as an event is closer to the horizon of a Schwarzschild black hole. The
shape of the cones can be calculated from the metric (2.10) imposing the null con-
dition ds 2 = 0. Then,

dr 2GM
=± 1− , (2.17)
dt r
where we made c = 1. Notice that when r → ∞, dr/dt → ±1, as in Minkowski
space-time. When r → 2GM, dr/dt → 0, and light moves along the surface r =
2GM, which is consequently a null surface. For r < 2GM, the sign of the derivative
is inverted. The inward region of r = 2GM is time-like for any physical system that
has crossed the boundary surface.
What happens to an object when it crosses the event horizon? According to
Eq. (2.10), there is a singularity at r = rSchw . The metric coefficients, however,
can be made regular by a change of coordinates. For instance we can consider
Eddington-Finkelstein coordinates. Let us define a new radial coordinate r∗ such
that radial null rays satisfy d(ct ± r∗ ) = 0. Using Eq. (2.10) it can be shown that:
 
2GM  r − 2GM/c2 
r∗ = r + 2 log  .
c 2GM/c2 
Then, we introduce:
v = ct + r∗ .
2.3 Schwarzschild Black Holes 39

Fig. 2.2 Space-time diagram in Eddington-Finkelstein coordinates showing the light cones close
to and inside a black hole. Here, r = 2GM/c2 = rSchw is the Schwarzschild radius where the event
horizon is located. Adapted form Townsend (1997)

The new coordinate v can be used as a time coordinate replacing t in Eq. (2.10).
This yields:

2GM
2 2 
ds 2 = 1 − 2
c dt − dr∗2 − r 2 dΩ 2
rc
or

2GM
ds 2 = 1 − dv 2 − 2drdv − r 2 dΩ 2 , (2.18)
rc2
where
dΩ 2 = dθ 2 + sin2 θ dφ 2 .
Notice that in Eq. (2.18) the metric is non-singular at r = 2GM/c2 . The only real
singularity is at r = 0, since there the Riemann tensor diverges. In order to plot the
space-time in a (t, r)-plane, we can introduce a new time coordinate ct∗ = v − r.
From the metric (2.18) or from Fig. 2.2 we see that the line r = rSchw , θ = constant,
and φ = constant is a null ray, and hence, the surface at r = rSchw is a null surface.
This null surface is an event horizon because inside r = rSchw all cones have r = 0
in their future (see Fig. 2.2). The object in r = 0 is the source of the gravitational
field and is called the singularity. We shall say more about it in Sect. 3.6. For the
moment, we only remark that everything that crosses the event horizon will end at
the singularity. This is the inescapable fate for everything inside a Schwarzschild
black hole. There is no way to avoid it: in the future of every event inside the event
40 2 Black Holes

Fig. 2.3 Embedding

space-time diagram in
Eddington-Finkelstein
coordinates showing the light
cones of events at different
distances from a
Schwarzschild black hole.
From http://www.oglethorpe.
edu/faculty/~m_rulison/
ChangingViews/Lecture7.htm

horizon is the singularity. There is no escape, no hope, no freedom, inside the black
hole. There is just the singularity, whatever such a thing might be.
We see now that the name “black hole” is not strictly correct for space-time re-
gions isolated by event horizons. There is no hole to other place. Whatever falls
into the black hole, goes to the singularity. The central object increases its mass and
energy with the accreted bodies and fields, and then the event horizon grows. This
would not happen if what falls into the hole were able to pass through, like through
a hole in a wall. A black hole is more like a space-time precipice, deep, deadly,
and with something unknown at the bottom. A graphic depiction with an embed-
ding diagram of a Schwarzschild black hole is shown in Fig. 2.3. An embedding
is an immersion of a given manifold into a manifold of lower dimensionality that
preserves the metric properties.

2.3.2 Birkhoff’s Theorem

If we consider the isotropic but not static line element,

ds 2 = c2 A(r, t)dt 2 − B(r, t)dr 2 − r 2 dΩ 2 , (2.19)

and substitute it into Einstein’s empty-space field equations Rμν = 0 to obtain the
functions A(r, t) and B(r, t), the result would be exactly the same:

2GM
A(r, t) = A(r) = 1 − ,
rc2

and
 −1
2GM
B(r, t) = B(r) = 1 − .
rc2
This result is general and known as Birkhoff’s theorem:
2.3 Schwarzschild Black Holes 41

The space-time geometry outside a general spherically symmetric matter distribution is the
Schwarzschild geometry.

Birkhoff’s theorem implies that strictly radial motions do not perturb the space-
time metric. In particular, a pulsating star, if the pulsations are strictly radial, does
not produce gravitational waves.
The converse of Birkhoff’s theorem is not true, i.e.,
If the region of space-time is described by the metric given by expression (2.10), then the
matter distribution that is the source of the metric does not need to be spherically symmetric.

2.3.3 Orbits

Orbits around a Schwarzschild black hole can be easily calculated using the metric
and the relevant symmetries (see. e.g. Raine and Thomas 2005; Frolov and Zelnikov
2011). Let us call k μ a vector in the direction of a given symmetry (i.e. k μ is a
Killing vector). A static situation is symmetric in the time direction, hence we can
write k μ = (1, 0, 0, 0). The 4-velocity of a particle with trajectory x μ = x μ (τ ) is
uμ = dx μ /dτ . Then, since u0 = E/c, where E is the energy, we have:
E E
gμν k μ uν = g00 k 0 u0 = g00 u0 = η00 = = constant. (2.20)
c c
If the particle moves along a geodesic in a Schwarzschild space-time, we obtain
from Eq. (2.20):

2GM dt E
c 1− 2 = . (2.21)
c r dτ c
Similarly, for the symmetry in the azimuthal angle φ we have k μ = (0, 0, 0, 1),
in such a way that:

gμν k μ uν = g33 k 3 u3 = g33 u3 = −L = constant. (2.22)

In the Schwarzschild metric we find, then,

dφ
r2 = L = constant. (2.23)
dτ
If we now divide the Schwarzschild interval (2.10) by c2 dτ 2 we get
  2    2
2GM dt −2 2GM −1 dr 2 −2 2 dφ
1= 1− 2 −c 1− 2 −c r , (2.24)
c r dτ c r dτ dτ
and using the conservation equations (2.21) and (2.23) we obtain:
 2  
dr E2 L2 2GM
= 2 − c + 2
2
1− 2 . (2.25)
dτ c r c r
42 2 Black Holes

Fig. 2.4 General relativistic effective potential plotted for several values of angular momentum

Then, expressing the energy in units of mc2 and introducing an effective potential
Veff ,
 2
dr E2
= 2 − Veff2
. (2.26)
dτ c
For circular orbits of a massive particle we have the conditions

dr d 2r
=0 and = 0.
dτ dτ 2
The orbits are possible only at the turning points of the effective potential:
 
L2 2rg
Veff = c + 2
2 1− , (2.27)
r r

where L is the angular momentum in units of mc and rg = GM/c2 is the gravita-

tional radius. Then,

L2 1 L4
r= ± − 12L2 . (2.28)
2crg 2 c2 rg2
The effective potential is shown in Fig. 2.4 for different values of the angular mo-
mentum.
For L2 > 12c2 rg2 there are two solutions. The negative sign corresponds to a max-
imum of the potential and is unstable. The positive sign corresponds to a minimum,
which is, consequently, stable. At L2 = 12c2 rg2 there is a single stable orbit. It is the
innermost marginally stable orbit, and it occurs at r = 6rg = 3rSchw . The specific
2.3 Schwarzschild Black Holes 43

angular momentum of a particle in a circular orbit at r is:

 1/2
rg r
L=c .
1 − 3rg /r

Its energy (units of mc2 ) is:

 
2rg 3rg −1/2
E = 1− 1− .
r r

The proper and observer’s periods are:

 1/2  1/2
2π r3 3rg
τ= 1−
c rg r

and
 1/2
2π r3
T= .
c rg
Notice that when r → 3rg both L and E tend to infinity, so only massless particles
can orbit at such a radius.
The local velocity at r of an object falling from rest to the black hole is (e.g.
Raine and Thomas 2005):
proper distance dr
vloc = = .
proper time (1 − 2GM/c2 r)dt
Hence, using the expression for dr/dt from the metric (2.10)
 
dr 2GM 1/2 2GM
= −c 1 − , (2.29)
dt c2 r c2 r

we have,
 1/2
2rg
vloc = (in units of c). (2.30)
r
Then, the differential acceleration the object will experience along an element dr
is:2
2rg
dg = 3 c2 dr. (2.31)
r
The tidal acceleration on a body of finite size Δr is simply (2rg /r 3 )c2 Δr. This
acceleration and the corresponding force becomes infinite at the singularity. As the
object falls into the black hole, tidal forces act to tear it apart. This painful process is

2 Notice that dvloc /dτ = (dvloc /dr)(dr/dτ ) = (dvloc /dr)vloc = rg c2 /r 2 .

44 2 Black Holes

known as “spaghettification”. The process can last significant long before the object
crosses the event horizon, depending on the mass of the black hole.
The energy of a particle in the innermost stable orbit can be obtained from the
above equation for the energy setting r = 6rg . This yields (units of mc2 ):
 
2rg 3rg −1/2 2 √
E = 1− 1− = 2.
6rg 6rg 3

Since a particle at rest at infinity has E = 1, then

√ the energy that the particle should
release to fall into the black hole is 1 − (2/3) 2 = 0.057. This means 5.7 % of its
rest mass energy, significantly higher than the energy release that can be achieved
through nuclear fusion.
An interesting question is what is the gravitational acceleration at the event hori-
zon as seen by an observer from infinity. The acceleration relative to a hovering
frame system of a freely falling object at rest at r is (Raine and Thomas 2005):
 
GM/c2 2GM/c2 −1/2
gr = −c2 1− .
r2 r
So, the energy spent to move the object a distance dl will be dEr = mgr dl. The
energy expended respect to a frame at infinity is dE∞ = mg∞ dl. Because of the
conservation of energy, both quantities should be related by a redshift factor:

Er gr 2GM/c2 −1/2
= = 1− .
E ∞ g∞ r
Hence, using the expression for gr we get:

GM/c2
g∞ = c 2 . (2.32)
r2
Notice that for an observer at r, gr → ∞ when r → rSchw . From infinity, however,
the required force to hold the object hovering at the horizon is

GmM/c2 mc4
mg∞ = c2 2
= .
rSchw 4GM

This is the surface gravity of the black hole.

2.3.4 Radial Motion of Photons

In the case of photons we have that ds 2 = 0. The radial motion, then, satisfies:
 
2GM 2 2 2GM −1 2
1− c dt − 1 − dr = 0. (2.33)
rc2 rc2
2.3 Schwarzschild Black Holes 45

From here,

dr 2GM
= ±c 1 − . (2.34)
dt rc2
Integrating, we have:
 
2GM  rc2 
ct = r + 2 ln − 1 + constant outgoing photons, (2.35)
c 2GM
 
2GM  rc2 
ct = −r − 2 ln − 1 + constant incoming photons. (2.36)
c 2GM

Notice that in a (ct, r)-diagram the photons have world-lines with slopes ±1 as
r → ∞, indicating that space-time is asymptotically flat. As the events that generate
the photons approach to r = rSchw , the slopes tend to ±∞. This means that the
light cones become thinner and thinner for events close to the event horizon. At
r = rSchw the photons cannot escape and they move along the horizon (see Fig. 2.1).
An observer in the infinity will never detect them.

2.3.5 Circular Motion of Photons

In this case, fixing θ = constant due to the symmetry, we have that photons will
move in a circle of r = constant and ds 2 = 0. Then, from (2.10), we have:

2GM 2 2
1− c dt − r 2 dφ 2 = 0. (2.37)
rc2

This means that


c 2GM
φ̇ = 1− = constant.
r rc2
The circular velocity is:

r φ̇ Ωr
vcirc = √ = . (2.38)
g00 (1 − 2GM/c2 r)1/2

Setting vcirc = c for photons and using Ω = (GM/r 3 )1/2 , we get that the only pos-
sible radius for a circular photon orbit is:

3GM
rph = . (2.39)
c2
For a compact object of 1 M , rph ≈ 4.5 km, in comparison with the Schwarzschild
radius of 3 km. Photons moving at this distance form the “photosphere” of the black
46 2 Black Holes

hole. The orbit, however, is unstable, as it can be seen from the effective potential:

L2ph  2rg

Veff = 1− . (2.40)
r2 r

Notice that the four-acceleration for circular motion is aμ = uμ uν ;ν . The radial

component in the Schwarzschild metric is:

GM/r 2 − Ω 2 r
ar = . (2.41)
1 − 2GM/c2 r − Ω 2 r 2 /c2

The circular motion along a geodesic line corresponds to the case ar = 0 (free mo-
tion). This gives from Eq. (2.41) the usual expression for the Keplerian angular
velocity

GM 1/2
ΩK = ,
r3
already used in deriving rph . The angular velocity, however, can have any value de-
termined by the metric and can be quite different from the corresponding Keplerian
value. In general:
 1/2 
rΩK GM 2GM −1/2
v= = 1− 2 . (2.42)
(1 − 2GM/c2 r)1/2 r c r

From this latter equation and the fact that v ≤ c it can be concluded that pure Keple-
rian motion is only possible for r ≥ 1.5rSchw . At r ≤ 1.5rSchw any massive particle
will find its mass increased by special relativistic effects in such a way that the
gravitational attraction will outweigh any centrifugal force.

2.3.6 Gravitational Capture

A particle coming from infinity is captured if its trajectory ends in the black hole.
The angular momentum of a non-relativistic particle with velocity v∞ at infinity is
L = mv∞ b, where b is an impact parameter. The condition L/mcrSchw = 2 defines
bcr,non-rel = 2rSchw (c/v∞ ). Then, the capture cross section is:
2
c2 rSchw
σnon-rel = πbcr
2
= 4π 2
. (2.43)
v∞
√
For an ultra-relativistic particle, bcr = 3 3rSchw /2, and then

27 2
σrel = πbcr
2
= πr . (2.44)
4 Schw
2.3 Schwarzschild Black Holes 47

2.3.7 Other Coordinate Systems

Other coordinates can be introduced to study additional properties of black holes.

We refer the reader to the books of Frolov and Novikov (1998), Raine and Thomas
(2005), and Frolov and Zelnikov (2011) for further details. Here we shall only intro-
duce the Kruskal-Szekeres coordinates. These coordinates have the advantage that
they cover the entire space-time manifold of the maximally extended Schwarzschild
solution and are well-behaved everywhere outside the physical singularity. They al-
low to remove the non-physical singularity at r = rSchw and provide new insights
on the interior solution, on which we shall return later.
Let us consider the following coordinate transformation:
 1/2 
r r ct
u= −1 e 2rSchw cosh ,
rSchw 2rSchw
 1/2 
r r ct
v= −1 e 2rSchw sinh , (2.45)
rSchw 2rSchw
if r > rSchw ,

and
 1/2 
r r ct
u= 1− e 2rSchw sinh ,
rSchw 2rSchw
 1/2 
r r ct
v= 1− e 2rSchw cosh , (2.46)
rSchw 2rSchw
if r < rSchw .

The line element in the Kruskal-Szekeres coordinates is completely regular, ex-

cept at r = 0:
3
4rSchw r

ds 2 = e rSchw dv 2 − du2 − r 2 dΩ 2 . (2.47)
r
The curves at r = constant are hyperbolic and satisfy:
 1/2
r r
u2 − v 2 = −1 e rSchw , (2.48)
rSchw

whereas the curves at t = constant are straight lines that pass through the origin:
u ct
= tanh , r < rSchw ,
v 2rSchw
(2.49)
u ct
= coth , r > rSchw .
v 2rSchw
48 2 Black Holes

Fig. 2.5 The Schwarzschild

metric in Kruskal-Szekeres
coordinates (c = 1)

In Fig. 2.5 we show the Schwarzschild space-time in Kruskal-Szekeres coordi-

nates. Each hyperbola represents a set of events of constant radius in Schwarzschild
coordinates. A radial worldline of a photon in this diagram (ds = 0) is represented
by a straight line forming an angle of ±45◦ with the u axis. A time-like trajectory has
always a slope larger than that of 45◦ ; and a space-like one, a smaller slope. A par-
ticle falling into the black hole crosses the line at 45◦ and reaches the future singu-
larity at r = 0. For an external observer this occurs in an infinite time. The Kruskal-
Szekeres coordinates have the useful feature that outgoing null geodesics are given
by u = constant, whereas ingoing null geodesics are given by v = constant. Further-
more, the (future and past) event horizon(s) are given by the equation uv = 0, and
the curvature singularity is given by the equation uv = 1.
A closely related diagram is the so-called Penrose or Penrose-Carter diagram.
This is a two-dimensional diagram that captures the causal relations between dif-
ferent points in space-time. It is an extension of a Minkowski diagram (light cone)
where the vertical dimension represents time, and the horizontal dimension repre-
sents space, and slanted lines at an angle of 45◦ correspond to light rays. The biggest
difference with a Minkowski diagram is that, locally, the metric on a Penrose dia-
gram is conformally equivalent3 to the actual metric in space-time. The conformal
factor is chosen such that the entire infinite space-time is transformed into a Pen-
rose diagram of finite size. For spherically symmetric space-times, every point in
the diagram corresponds to a 2-sphere. In Fig. 2.6 we show a Penrose diagram of a
Minkowskian space-time.
This type of diagrams can be applied to Schwarzschild black holes. The result
is shown in Fig. 2.7. The trajectory represents a particle that goes from some point

3 We remind that two geometries are conformally equivalent if there exists a conformal transfor-

mation (an angle-preserving transformation) that maps one geometry to the other. More generally,
two (pseudo) Riemannian metrics on a manifold M are conformally equivalent if one is obtained
from the other through multiplication by a function on M.
2.4 Kerr Black Holes 49

Fig. 2.6 Penrose diagram of

a Minkowskian space-time

Fig. 2.7 Penrose diagram of

a Schwarzschild black hole

in our universe into the black hole, ending in the singularity. Notice that there is
a mirror extension, also present in the Kruskal-Szekeres diagram, representing a
white hole and a parallel, but inaccessible universe. A white hole presents a naked
singularity. These type of extensions of solutions of Einstein’s field equations will
be discussed later.
Now, we turn to axially symmetric (rotating) solutions of the field equations.

2.4 Kerr Black Holes

A Schwarzschild black hole does not rotate. The solution of the field equations
(1.36) for a rotating body of mass M and angular momentum per unit mass a was
found by Roy Kerr (1963):

ds 2 = gtt dt 2 + 2gtφ dtdφ − gφφ dφ 2 − ΣΔ−1 dr 2 − Σdθ 2 (2.50)

50 2 Black Holes


gtt = c2 − 2GMrΣ −1 (2.51)
gtφ = 2GMac−2 Σ −1 r sin2 θ (2.52)

2
gφφ = r 2 + a 2 c−2 − a 2 c−2 Δ sin2 θ Σ −1 sin2 θ (2.53)
Σ ≡ r 2 + a 2 c−2 cos2 θ (2.54)
−2 2 −2
Δ ≡ r − 2GMc
2
r +a c . (2.55)

This is the Kerr metric in Boyer-Lindquist coordinates (t, r, θ, φ), which reduces
to Schwarzschild metric for a = 0. In Boyer-Lindquist coordinates the metric is
approximately Lorentzian at infinity (i.e. we have a Minkowski space-time in the
usual coordinates of Special Relativity).
The element gtφ no longer vanishes. Even at infinity this element remains (hence
we wrote approximately Lorentzian above). The Kerr parameter ac−1 has dimen-
sions of length. The larger the ratio of this scale to GMc−2 (the spin parameter
a∗ ≡ ac/GM), the more aspherical the metric. Schwarzschild’s black hole is the
special case of Kerr’s for a = 0. Notice that, with the adopted conventions, the an-
gular momentum J is related to the parameter a by:

J = Ma. (2.56)

Just as the Schwarzschild solution is the unique static vacuum solution of

Eqs. (1.36) (a result called Israel’s theorem), the Kerr metric is the unique stationary
axisymmetric vacuum solution (Carter-Robinson theorem).
The horizon, the surface which cannot be crossed outward, is determined by the
condition grr → ∞ (Δ = 0). It lies at r = rhout where

2 1/2
rhout ≡ GMc−2 + GMc−2 − a 2 c−2 . (2.57)

Indeed, the track r = rhout , θ = constant with dφ/dτ = a(rh2 + a 2 )−1 dt/dτ has
ds = 0 (it represents a photon circling azimuthally on the horizon, as opposed
to hovering at it). Hence the surface r = rhout is tangent to the local light cone.
Because of the square root in Eq. (2.57), the horizon is well defined only for
a∗ = ac/GM ≤ 1. An extreme (i.e. maximally rotating) Kerr black hole has a spin
parameter a∗ = 1. Notice that for (GMc−2 )2 − a 2 c−2 > 0 we have actually two
horizons. The second, the inner horizon, is located at:

2 1/2
rhinn ≡ GMc−2 − GMc−2 − a 2 c−2 . (2.58)

This horizon is not seen by an external observer, but it hides the singularity to any
observer that has already crossed rh and is separated from the rest of the universe.
For a = 0, rhinn = 0 and rhout = rSchw . The case (GMc−2 )2 − a 2 c−2 < 0 corresponds
to no horizons and it is thought to be unphysical.
A study of the orbits around a Kerr black hole is beyond the limits of the present
text (the reader is referred to Frolov and Novikov 1998; Pérez et al. 2013), but
2.4 Kerr Black Holes 51

we shall mention several interesting features. One is that if a particle initially falls
radially with no angular momentum from infinity to the black hole, it gains angular
motion during the infall. The angular velocity as seen from a distant observer is:
dφ (2GM/c2 )ar
Ω(r, θ ) = = . (2.59)
dt (r 2 + a 2 c−2 )2 − a 2 c−2 Δ sin2 θ
The particle will acquire angular velocity in the direction of the spin of the black
hole. As the black hole is approached, the particle will find an increasing tendency
to get carried away in the same sense in which the black hole is rotating. To keep
the particle stationary with respect to the distant stars, it will be necessary to apply
a force against this tendency. The closer the particle will be to the black hole, the
stronger the force. At a point re it becomes impossible to counteract the rotational
sweeping force. The particle is in a kind of space-time maelstrom. The surface deter-
mined by re is the static limit: from there in, you cannot avoid rotating. Space-time
is rotating here in such a way that you cannot do anything in order to not co-rotate
with it. You can still escape from the black hole, since the outer event horizon has not
been crossed, but rotation is inescapable. The region between the static limit and the
event horizon is called the ergosphere. The ergosphere is not spherical but its shape
changes with the latitude θ . It can be determined through the condition gtt = 0.
Consider a stationary particle, r = constant, θ = constant, and φ = constant. Then:
 2
dt
c = gtt
2
. (2.60)
dτ
When gtt ≤ 0 this condition cannot be fulfilled, and hence a massive particle
cannot be stationary inside the surface defined by gtt = 0. For photons, since ds =
cdτ = 0, the condition is satisfied at the surface. Solving gtt = 0 we obtain the shape
of the ergosphere:
GM 1
1/2
re = 2
+ 2 G2 M 2 − a 2 c2 cos2 θ . (2.61)
c c
The static limit lies outside the horizon except at the poles where both surfaces
coincide. The phenomenon of “frame dragging” is common to all axially symmetric
metrics with gtφ = 0.
Roger Penrose (1969) suggested that a projectile thrown from outside into the
ergosphere begins to rotate acquiring more rotational energy than it originally had.
Then the projectile can break up into two pieces, one of which will fall into the
black hole, whereas the other can go out of the ergosphere. The piece coming out
will then have more energy than the original projectile. In this way, we can extract
energy from a rotating black hole. In Fig. 2.8 we illustrate the situation and show
the static limit, the ergosphere and the outer/inner horizons of a Kerr black hole.
The innermost marginally stable circular orbit rms around an extreme rotating
black hole (ac−1 = GM/c2 ) is given by Raine and Thomas (2005):
 2   1/2
rms rms rms
−6 ±8 − 3 = 0. (2.62)
GM/c2 GM/c2 GM/c2
52 2 Black Holes

Fig. 2.8 Left: a rotating black hole and the Penrose process. From Luminet (1998). Right: sketch
of the interior of a Kerr black hole

For the “+” sign this is satisfied by rms = GM/c2 , whereas for the “−‘” sign the
solution is rms = 9GM/c2 . The first case corresponds to a co-rotating particle and
the second one to a counter-rotating
√ particle. The energy of the co-rotating particle
in the innermost orbit is 1/ 3 (units of mc2 ). The binding energy of a particle
in an orbit is the difference between the orbital energy and its energy at infinity.
This means a binding energy of 42 % of the rest energy at infinity! For the counter-
rotating particle, the binding energy is 3.8 %, smaller than for a Schwarzschild black
hole.
An essential singularity occurs when gtt → ∞; this happens if Σ = 0. This con-
dition implies:

r 2 + a 2 c−2 cos2 θ = 0. (2.63)

Such a condition is fulfilled only by r = 0 and θ = π

2. This translates in Cartesian
coordinates to:4

x 2 + y 2 = a 2 c−2 and z = 0. (2.64)

The singularity is a ring of radius ac−1 on the equatorial plane. If a = 0, then

Schwarzschild’s point-like singularity is recovered. If a = 0 the singularity is not
necessarily in the future of all events at r < rhinn : the singularity can be avoided by
some geodesics.

√
4 Therelation with Boyer-Lindquist coordinates is z = r cos θ , x = r 2 + a 2 c−2 sin θ cos φ, y =
√
r 2 + a 2 c−2 sin θ sin φ.
2.5 Reissner-Nordström Black Holes 53

2.4.1 Pseudo-Newtonian Potentials for Black Holes

The full effective general relativistic potential for particle orbits around a Kerr black
hole is quite complex. Instead, pseudo-Newtonian potentials can be used. The first of
such potentials, derived by Bohdam Paczyński and used by first time by Paczyński
and Wiita (1980), for a non-rotating black hole with mass M, is:
GM
Φ =− , (2.65)
r − 2rg

where as before rg = GM/c2 is the gravitational radius. With this potential one can
use Newtonian theory and obtain the same behavior of the Keplerian circular orbits
of free particles as in the exact theory: orbits with r < 9rg are unstable, and orbits
with r < 6rg are unbound. However, velocities of material particles obtained with
the potential (2.65) are not accurate, since special relativistic effects are not included
(Abramowicz et al. 1996). The velocity vp−N calculated with the pseudo-Newtonian
potential should be replaced by the corrected velocity vp−N corr such that

1
vp−N = vp−N
corr corr
γp−N , corr
γp−N = . (2.66)
corr 

vp−N 2
1− c

This re-scaling works amazingly well (see Abramowicz et al. 1996) compared with
the actual velocities. The agreement with General Relativity is better than 5 %.
For the Kerr black hole, a pseudo-Newtonian potential was found by Semerák
and Karas (1999). It can be found in the expression (19) of their paper. However,
the use of this potential is almost as complicated as dealing with the full effective
potential of the Kerr metric in General Relativity.

2.5 Reissner-Nordström Black Holes

The Reissner-Nordström metric is a spherically symmetric solution of Eqs. (1.36).
However, it is not a vacuum solution, since the source has an electric charge Q, and
hence there is an electromagnetic field. The energy-momentum tensor of this field
is:

−1 1
Tμν = −μ0 Fμρ Fν − gμν Fρσ F
ρ ρσ
, (2.67)
4
where Fμν = ∂μ Aν − ∂ν Aμ is the electromagnetic field strength tensor and Aμ is
the electromagnetic 4-potential. Outside the charged object the 4-current j μ is zero,
so Maxwell’s equations are:
μν
F;μ = 0, (2.68)
Fμν;σ + Fσ μ;ν + Fνσ ;μ = 0. (2.69)
54 2 Black Holes

Einstein’s and Maxwell’s equations are coupled since F μν enters into the gravi-
tational field equations through the energy-momentum tensor and the metric gμν
enters into the electromagnetic equations through the covariant derivative. Because
of the symmetry constraints we can write:

 μ ϕ(r)
A = , a(r), 0, 0 , (2.70)
c2
where ϕ(r) is the electrostatic potential, and a(r) is the radial component of the
3-vector potential as r → ∞.
The solution for the metric is given by
ds 2 = Δc2 dt 2 − Δ−1 dr 2 − r 2 dΩ 2 , (2.71)
where
2GM/c2 q 2
Δ=1− + 2. (2.72)
r r
In this expression, M is once again interpreted as the mass of the hole and
GQ2
q= (2.73)
4π0 c4
is related to the total electric charge Q.
The metric has a coordinate singularity at Δ = 0, in such a way that:

1/2
r± = rg ± rg2 − q 2 . (2.74)

Here, rg = GM/c2 is the gravitational radius. For rg = q, we have an extreme

Reissner-Nordström black hole with a unique horizon at r = rg . Notice that a
Reissner-Nordström black hole can be more compact than a Schwarzschild black
hole of the same mass. For the case rg2 > q 2 , both r± are real and there are two hori-
zons as in the Kerr solution. Finally, in the case rg2 < q 2 both r± are imaginary there
is no coordinate singularities, no horizon hides the intrinsic singularity at r = 0. It
is thought, however, that naked singularities do not exist in Nature (see Sect. 3.6
below).

2.6 Kerr-Newman Black Holes

The Kerr-Newman metric of a charged spinning black hole is the most general black
hole solution. It was found by Ezra “Ted” Newman in 1965 (Newman et al. 1965).
This metric can be obtained from the Kerr metric (2.50) in Boyer-Lindquist coordi-
nates with the replacement:
2GM 2GM
2
r −→ 2 r − q 2 ,
c c
where q is related to the charge Q by Eq. (2.73).
2.6 Kerr-Newman Black Holes 55

The full expression reads:

ds 2 = gtt dt 2 + 2gtφ dtdφ − gφφ dφ 2 − ΣΔ−1 dr 2 − Σdθ 2 (2.75)



gtt = c2 1 − 2GMrc−2 − q 2 Σ −1 (2.76)


gtφ = a sin2 θ Σ −1 2GMrc−2 − q 2 (2.77)

2
gφφ = r 2 + a 2 c−2 − a 2 c−2 Δ sin2 θ Σ −1 sin2 θ (2.78)
Σ ≡ r 2 + a 2 c−2 cos2 θ (2.79)



Δ ≡ r 2 − 2GMc−2 r + a 2 c−2 + q 2 ≡ r − rhout r − rhinn , (2.80)

where all symbols have the same meaning as in the Kerr metric and the outer horizon
is located at

2 1/2
rhout = GMc−2 + GMc−2 − a 2 c−2 − q 2 . (2.81)
The inner horizon is located at:

2 1/2
rhinn = GMc−2 − GMc−2 − a 2 c−2 − q 2 . (2.82)

The Kerr-Newman solution is a non-vacuum solution either. It shares with the

Kerr and Reissner-Nordström solutions the existence of two horizons, and as the
Kerr solution it presents an ergosphere. At a latitude θ , the radial coordinate for the
ergosphere is:

2 1/2
re = GMc−2 + GMc−2 − a 2 c−2 cos2 θ − q 2 . (2.83)

Like the Kerr metric for an uncharged rotating mass, the Kerr-Newman interior
solution exists mathematically but is probably not representative of the actual met-
ric of a physically realistic rotating black hole due to stability problems (see next
chapter). The surface area of the horizon is:


A = 4π rhout2 + a 2 c−2 . (2.84)

The Kerr-Newman metric represents the simplest stationary, axisymmetric,

asymptotically flat solution of Einstein’s equations in the presence of an electro-
magnetic field in four dimensions. It is sometimes referred to as an “electrovac-
cuum” solution of Einstein’s equations. Any Kerr-Newman source has its rotation
axis aligned with its magnetic axis (Punsly 1998a). Thus, a Kerr-Newman source is
different from commonly observed astronomical bodies, for which there might be a
substantial angle between the rotation axis and the magnetic moment.
Since the electric field cannot remain static in the ergosphere, a magnetic field
is generated as seen by an observer outside the static limit. This is illustrated in
Fig. 2.9. You can find the expressions for the components of the fields in Punsly
(2001).
Pekeris and Frankowski (1987) have calculated the interior electromagnetic field
of the Kerr-Newman source, i.e., the ring singularity. The electric and magnetic
56 2 Black Holes

Fig. 2.9 The electric (solid)

and magnetic (dashed) field
lines of a Kerr-Newman black
hole. The rotation axis of the
hole is indicated

Fig. 2.10 Magnetic field of a

Kerr-Newman source. See
text for units. Reprinted figure
with permission from Pekeris
and Frankowski (1987).
Copyright (1987) by the
American Physical Society

fields are shown in Figs. 2.10 and 2.11, in a (λ, z)-plane, with λ = (x 2 + y 2 )1/2 . The
general features of the magnetic field are that at distances much larger than ac−1 it
resembles closely a dipole field, with a dipolar magnetic moment μd = Qac−1 . On
the disc of radius ac−1 the z-component of the field vanishes, in contrast with the
interior of Minkowskian ring-current models. The electric field for a positive charge
distribution is attractive for positive charges toward the interior disc. At the ring
there is a charge singularity and at large distances the field corresponds to that of a
point-like charge Q.
2.6 Kerr-Newman Black Holes 57

Fig. 2.11 Electric field of a

Kerr-Newman source. See
text for units. Reprinted figure
with permission from Pekeris
and Frankowski (1987).
Copyright (1987) by the
American Physical Society

Fig. 2.12 Charged and

rotating black hole
magnetosphere. The black
hole has charge +Q whereas
the current ring circulating
around it has opposite charge.
The figure shows (units
G = c = 1) the region of
closed lines determined by
the light cylinder, the open
lines that drive a
magnetohydrodynamical
wind, and the vacuum region
in between. Adapted from
Punsly (1998a). Reproduced
by permission of the AAS

Charged black holes might be a natural result from charge separation during the
gravitational collapse of a star. It is thought that an astrophysical charged object
would discharge quickly by accretion of charges of opposite sign. There remains
the possibility, however, that the charge separation could lead to a configuration
where the black hole has a charge and a superconducting ring around it would have
the same but opposite charge, in such a way the whole system seen from infinity is
neutral. In such a case a Kerr-Newman black hole might survive for some time, de-
pending on the environment. For further details, the reader is referred to the highly
technical book by Brian Punsly (2001) and related articles (Punsly 1998a, 1998b,
and Punsly et al. 2000). In Figs. 2.12 and 2.13 the magnetic field around a Kerr-
Newman black hole surrounded by a charged current ring is shown. The opposite
charged black hole and ring are the minimum energy configuration for the system
black hole plus magnetosphere. Since the system is neutral from the infinity, it dis-
58 2 Black Holes

Fig. 2.13 Three different scales of the Kerr-Newman black hole model developed by Brian Punsly.
From Punsly (1998a). Reproduced by permission of the AAS

charges slowly and can survive for a few thousand years. During this period, the
source can be active through the capture of free electrons from the environment
and the production of gamma rays by inverse Compton up-scattering of synchrotron
photons produced by electrons accelerated in the polar gap of the hole. In Fig. 2.14
we show the corresponding spectral energy distribution obtained by Punsly et al.
(2000) for such a configuration of Kerr-Newman black hole magnetosphere.

2.6.1 Einstein-Maxwell Equations

In order to determine the gravitational and electromagnetic fields over a region of a

space-time we have to solve the Einstein-Maxwell equations:
2.7 Other Black Holes 59

Fig. 2.14 The spectral

energy distribution resulting
from a Kerr-Newman black
hole slowly accreting from
the interstellar medium. From
Punsly et al. (2000),
reproduced with permission
©ESO

1 8πG
Rμν − Rgμν + Λgμν = − 4 (Tμν + Eμν ), (2.85)
2 c
4π μ 1
E = −F μρ Fρν + δνμ F σ λ Fσ λ , (2.86)
c ν 4
Fμν = Aμ;ν − Aν;μ , (2.87)
4π
Fμν;ν = Jμ . (2.88)
c
Here Tμν and Eμν are the energy-momentum tensors of matter and electromag-
netic fields, Fμν and Jμ are the electromagnetic field and current density, Aμ is the
4-dimensional potential, and Λ is the cosmological constant.
The solution of this system of equations is non-trivial since they are coupled.
The electromagnetic field is a source of the gravitational field and this field enters
into the electromagnetic equations through the covariant derivatives indicated by
the semi-colons. For an exact and relevant solution of the problem see Manko and
Sibgatullin (1992).

2.7 Other Black Holes

2.7.1 Born-Infeld Black Holes

Born and Infeld (1934) to avoid the singularities associated with charged point par-
ticles in Maxwell theory. Almost immediately, Hoffmann (1935) coupled General
60 2 Black Holes

Relativity with Born-Infeld electrodynamics to obtain a spherically symmetric solu-

tion representing the gravitational field of a charged object. This solution, forgotten
during decades, can represent a charged black hole in nonlinear electrodynamics. In
Born-Infeld electrodynamics the trajectories of photons in curved space-times are
not null geodesics of the background metric. Instead, they follow null geodesics of
an effective geometry determined by the nonlinearities of the electromagnetic field.
The action of Einstein gravity coupled to Born-Infeld electrodynamics has the
form (in this section we adopt, for simplicity, c = G = 4π0 = (4π)−1 μ0 = 1):
 
√ R
S = dx 4 −g + LBI , (2.89)
16π
with
 
1 1 1
LBI = 1 − 1 + Fσ ν F b − F̃σ ν F b ,
σ ν 2 σ ν 4 (2.90)
4πb2 2 4
where g is the determinant of the metric tensor, R is the√scalar of curvature,
Fσ ν = ∂σ Aν − ∂ν Aσ is the electromagnetic tensor, F̃σ ν = 12 −gεαβσ ν F αβ is the
dual of Fσ ν (with εαβσ ν the Levi-Civita symbol), and b is a parameter that indi-
cates how much Born-Infeld and Maxwell electrodynamics differ. For b → 0 the
Einstein-Maxwell action is recovered. The maximal possible value of the electric
field in this theory is b, and the self-energy of point charges is finite. The field equa-
tions can be obtained by varying the action with respect to the metric gσ ν and the
electromagnetic potential Aν .
We can write LBI in terms of the electric and magnetic fields:

 
b2 B 2 − E 2 (E · B)2
LBI = 1− 1− − . (2.91)
4π b2 b4
The Lagrangian depends non-linearly of the electromagnetic invariants:
1 1

F = Fαβ F αβ = B 2 − E 2 , (2.92)
4 2
1
G̃ = Fαβ F̃ αβ = −B · E. (2.93)
4
Introducing the Hamiltonian formalism,
∂L ∂L αβ ∂L αβ
P αβ = 2 = F + F̃ , (2.94)
∂Fαβ ∂F ∂ G̃
1

H = P αβ Fαβ − L F, G̃2 , (2.95)
2
and adopting the notation
1
P = Pαβ P αβ , (2.96)
4
2.7 Other Black Holes 61

1
Q̃ = Pαβ P̃ αβ , (2.97)
4

we can express F αβ as a function of P αβ , P , and Q̃:

∂H ∂H αβ ∂H αβ
F αβ = 2 = P + P̃ . (2.98)
∂Pαβ ∂P ∂ Q̃

The Hamiltonian equations in the P and Q̃ formalism can be written as:


∂H αβ ∂H αβ
P̃ + P = 0. (2.99)
∂P ∂ Q̃ ,β

The coupled Einstein-Born-Infeld equations are:


∂H ∂H ∂H
4πTμν = Pμα Pν − gμν 2P
α
+ Q̃ −H , (2.100)
∂P ∂P ∂ Q̃

∂H ∂H
R=8 P + Q̃ −H . (2.101)
∂P ∂ Q̃

The field equations have spherically symmetric black hole solutions given by

ds 2 = ψ(r)dt 2 − ψ(r)−1 dr 2 − r 2 dΩ 2 , (2.102)

with
 ∞

2M 2 
ψ(r) = 1 − + 2 x 4 + b2 Q2 − x 2 dx, (2.103)
r b r r
QE
D(r) = , (2.104)
r2
B(r) = QM sin θ, (2.105)

where M is the mass, Q2 = Q2E + Q2M is the sum of the squares of the electric QE
and magnetic QM charges, B(r) and D(r) are the magnetic and the electric induc-
tions in the local orthonormal frame. In the limit b → 0, the Reissner-Nordström
metric is obtained. The metric (2.102) is also asymptotically Reissner-Nordström
for large values of r. With the units adopted above, M, Q and b have dimensions of
length. The metric function ψ(r) can be expressed in the form
 
2M 2
ψ(r) = 1 − + 2 r 2 − r 4 + b2 Q2
r 3b
   2 √ 
|bQ|3 r − |bQ| 2
+ F arccos 2 , , (2.106)
r r + |bQ| 2
62 2 Black Holes

where F (γ , k) is the elliptic integral of the first kind.5 As in Schwarzschild and

Reissner-Nordström cases, the metric (2.102) has a singularity at r = 0.
The zeros of ψ(r) determine the position of the horizons, which have to be ob-
tained numerically. For a given value of b, when the charge is small, 0 ≤ |Q|/M ≤
ν1 , the function ψ(r) has one zero and there is a regular event horizon. For inter-
mediate values of charge, ν1 < |Q|/M < ν2 , ψ(r) has two zeros, so there are, as
in the Reissner-Nordström geometry, an inner horizon and an outer regular event
horizon. When |Q|/M = ν2 , there is one degenerate horizon. Finally, if the val-
ues of charge are large, |Q|/M > ν2 , the function ψ(r) has no zeros and a naked
√ values of |Q|/M where the number of horizons change,
singularity is obtained. The
ν1 = (9|b|/M)1/3 [F (π, 2/2)]−2/3 and ν2 , which should be calculated numerically
from the condition ψ(rh ) = ψ  (rh ) = 0, are increasing functions of |b|/M. In the
Reissner-Nordström limit (b → 0) it is easy to see that ν1 = 0 and ν2 = 1.
The paths of photons in nonlinear electrodynamics are not null geodesics of the
background geometry. Instead, they follow null geodesics of an effective metric
generated by the self-interaction of the electromagnetic field, which depends on the
particular nonlinear theory considered. In Einstein gravity coupled to Born-Infeld
electrodynamics the effective geometry for photons is given by Bretón (2002):
2
dseff = ω(r)1/2 ψ(r)dt 2 − ω(r)1/2 ψ(r)−1 dr 2 − ω(r)−1/2 r 2 dΩ 2 , (2.107)

where
Q2 b2
ω(r) = 1 + . (2.108)
r4
Then, to calculate the deflection angle for photons passing near the black holes, it
is necessary to use the effective metric (2.107) instead of the background metric
(2.102). The horizon structure of the effective metric is the same as that of metric
(2.102), but the trajectories of photons are different.

2.7.2 Regular Black Holes

Solutions of Einstein’s field equations representing black holes where the metric
is always regular (i.e. free of intrinsic singularities where R μνρσ Rμνρσ diverges)
can be found for some choices of the equation of state. For instance, Mbonye and
Kazanas (2005) have suggested the following equation:
  m  1/n
ρ ρ
pr (ρ) = α − (α + 1) ρ. (2.109)
ρmax ρmax

γ  sin γ
5 F (γ , k) =
0 (1 − k 2 sin2 φ)−1/2 dφ = 0 [(1 − z2 )(1 − k 2 z2 )]−1/2 dz.
2.7 Other Black Holes 63

The maximum limiting density ρmax is concentrated in a region of radius


1
r0 = . (2.110)
Gρmax

At low densities pr ∝ ρ 1+1/n and the equation reduces to that of a polytrope gas. At
high densities close to ρmax the equation becomes pr = −ρ and the system behaves
as a gravitational field dominated by a cosmological term in the field equations. The
exact values of m, n, and α determine the sound speed in the system. Imposing that
the maximum sound speed cs = (dp/dρ)1/2 be finite everywhere, it is possible to
constrain the free parameters. Adopting m = 2 and n = 1 Eq. (2.109) becomes:
  2 
ρ ρ
pr (ρ) = α − (α + 1) ρ. (2.111)
ρmax ρmax
The model introduced by Mbonye and Kazanas represents a regular static black
hole, with a matter source that smoothly goes from a de Sitter behavior near the
origin to Schwarzschild’s spacetime outside the object. A space-time metric well-
adapted to examine the properties of this system is (Mbonye et al. 2011):

2m(r) −1 2

ds = −B(r)dt + 1 −
2 2
dr + r 2 dθ 2 + sin2 θ dφ 2 , (2.112)
r
where

2 

r
B(r) = exp  2
m r + 4πr  3 psf r 
r0 r
 
1
× ) dr  , (2.113)
(1 − 2m(r r )

and
 r

m(r) = 4π ρ r  r  2 dr  . (2.114)
0
Outside the body ρ → 0, and Eq. (2.112) becomes Schwarzschild solution for
Rμν = 0. When r → 0, ρ = ρmax and the metric becomes of de Sitter type:
 
r2 2 2 r 2 −1 2

ds = 1 − 2 c dt − 1 − 2
2
dr − r 2 dθ 2 + sin2 θ dφ 2 , (2.115)
r0 r0

with

3
r0 = . (2.116)
8πGρmax
There is no singularity at r = 0 and the black hole is regular. For 0 ≤ r < 1 it has
constant positive density ρmax and negative pressure pr = −ρmax and space-time
64 2 Black Holes

becomes asymptotically de Sitter in the innermost region. It might be speculated

that the transition in the equation of state occurs because at very high densities the
matter field couples with a scalar field that provides the negative pressure.
Other assumptions for the equation of state can lead to different (but still regu-
lar) behavior, like a bouncing close to r = 0 and the development of an expanding
closed universe inside the black hole (Frolov et al. 1990). Unstable behavior, both
dynamic and thermodynamic, seems to be a characteristic of these type of black
hole solutions (Pérez et al. 2011).
Regular black holes might also be found in f (R) gravity for some suitable func-
tion of the curvature scalar.

2.7.3 f (R) Black Holes

2.7.3.1 Rewriting the Field Equations of f (R) Gravity

In order to study the possible black hole solutions obtained from any f (R) theory,
we rewrite the action (1.122) in the form

S = S g + Sm , (2.117)

where Sg is the gravitational action, with the scalar function now written as R +
f (R)
 

1 
Sg = d 4 x |g| R + f (R) . (2.118)
16πG
From the matter term Sm , we define the energy momentum tensor as
2 δSm
T μν = − √ . (2.119)
|g| δgμν

By performing variations of (2.117) with respect to the metric tensor, we obtain the
field equations in metric formalism in a more convenient way:

 1

Rμν 1 + f  (R) − gμν R + f (R)
2
+ (∇μ ∇ν − gμν )f  (R) + 8πGTμν = 0, (2.120)

with  = ∇β ∇ β as before and f  (R) = df (R)/dR. Clearly, for f (R) = 0 standard

General Relativity is recovered. Taking the trace of this equation yields:



R 1 + f  (R) − 2 R + f (R) − 3f  (R) + 8πGT = 0, (2.121)
μ
where T = T μ . Unlike the case of General Relativity, vacuum solutions (T = 0)
do not necessarily imply a null curvature R = 0. From Eq. (2.120) we obtain the
2.7 Other Black Holes 65

condition for vacuum constant scalar curvature R = R0 solutions:

 1

Rμν 1 + f  (R0 ) − gμν R0 + f (R0 ) = 0. (2.122)
2
and the Ricci tensor becomes proportional to the metric,
R0 + f (R0 )
Rμν = gμν , (2.123)
2(1 + f  (R0 ))

with 1 + f  (R0 ) = 0. Taking the trace on the previous equation:



R0 1 + f  (R0 ) − 2 R0 + f (R0 ) = 0, (2.124)

and therefore
2f (R0 )
R0 = . (2.125)
f  (R0 ) − 1
If we want to find viable black hole solutions of Eq. (2.120), some conditions
must be imposed in order to make f (R) theories consistent with known gravitational
and cosmological facts. These conditions are (Cembranos et al. 2011):
1. f  (R) ≥ 0 for R  f  (R). This is the stability requirement for a high-curvature
classical regime and that of the existence of a matter dominated era in cosmo-
logical evolution. A simple physical interpretation can be given to this condition:
if an effective gravitational constant Geff ≡ G/(1 + f  (R)) is defined, then the
sign of its variation with respect to R, dGeff /dR, is uniquely determined by the
sign of f  (R), so in case f  (R) < 0, Geff would grow as R does, because R
generates more and more curvature. This mechanism would destabilize the met-
ric field, since it would not have a fundamental state because any small curvature
would grow to infinity. Instead, if f  (R) ≥ 0, a counter reaction mechanism op-
erates to compensate this R growth and stabilize the system.
2. 1 + f  (R) > 0. This conditions ensures that the effective gravitational constant
is positive, as it can be checked from the previous definition of Geff .
3. f  (R) < 0. Keeping in mind the strong restrictions of Big Bang nucleosynthe-
sis and cosmic microwave background, this condition ensures that the expected
behavior be recovered at early times, that is, f (R)/R → 0 and f  (R) → 0 as
R → ∞. Conditions 1 and 2 together demand f (R) to be a monotonously in-
creasing function between the values −1 < f  (R) < 0.
4. f  (R) must be small in recent epochs. This condition is mandatory in order to
satisfy imposed restrictions by local (solar and galactic) gravity tests.
In looking for constant curvature R0 vacuum solutions for fields generated by
massive charged objects we follow Cembranos et al. (2011). The action (in units of
G = c =  = k = 1) is:
 

1 
S= d 4 x |g| R + f (R) − Fμν F μν , (2.126)
16π
66 2 Black Holes

where Fμν = ∂μ Aν − ∂ν Aμ and Aμ the electromagnetic potential. This action leads

to the field equations:

 1

Rμν 1 + f  (R0 ) − gμν R0 + f (R0 )
2

1
− 2 Fμα F αν − gμν Fαβ F αβ = 0. (2.127)
4

If we take the trace of the previous equation, then (2.124) is recovered due to the
μ
fact that F μ = 0.
The axisymmetric, stationary and constant curvature R0 solution that describes a
black hole with mass, electric charge, and angular momentum was originally found
by Carter (1973). In Boyer-Lindquist coordinates, the metric describing with no
coordinate singularities the spacetime exterior to the black hole and interior to the
cosmological horizon (provided it exists), is:
 2
ρ 2 2 ρ 2 2 Δθ sin2 θ dt
2 
2 dφ
ds =
2
dr + dθ + a − r +a
Δr Δθ ρ2 Ξ Ξ
 2
Δr dt dφ
2
− a sin2 θ , (2.128)
ρ Ξ Ξ

with


 R0 2 q2
aΔr ≡ r 2 + a 2 1 − r − 2Mr + ,
12 (1 + f  (R0 ))
ρ 2 ≡ r 2 + a 2 cos2 θ,
(2.129)
R0 2
Δθ ≡ 1 + a cos2 θ,
12
R0 2
Ξ ≡1+ a ,
12
where M, a and q, as before, denote the mass, spin, and electric charge parameters,
respectively.
The potential vector and electromagnetic field tensor in Eq. (2.127) for metric
(2.128) are:

qr dt 2 dφ
A=− 2 − a sin θ ,
ρ Ξ Ξ

q(r 2 − a 2 cos2 θ ) dt 2 dφ
F =− − a sin θ ∧ dr (2.130)
ρ4 Ξ Ξ
 
2qra cos θ sin θ dt
2 
2 dφ
− dθ ∧ a − r + a .
ρ4 Ξ Ξ
2.7 Other Black Holes 67

We adopt Q2 ≡ q 2 /(1 + f  (R0 )) in what follows to refer to the electric charge

parameter of the black hole.
If we take the limits M → 0, Q → 0, a → 0 to the metric, we obtain a constant
curvature spacetime metric:

R0 r 2 1
ds 2 = − 1 − dt 2 + dr 2 + r 2 dΩ 2 , (2.131)
12 R0 r 2
(1 − 12 )

that corresponds to either a de Sitter or an anti-de Sitter space-time depending on the

sign of R0 . Clearly, when a → 0 and Q → 0, Schwarzschild black hole is recovered.
Calculating R μνσρ Rμνσρ , only ρ = 0 happens to be an intrinsic singularity, and
considering the definition of ρ in (2.129), such singularity is given by:

r = 0 and θ = π/2. (2.132)

Keeping in mind that we are working with Boyer-Lindquist coordinates, the set of
points given by r = 0 and θ = π/2 represent a ring in the equatorial plane of radius
a centered on the rotation axis of the black hole, just as in Kerr black holes.
The horizons are found from g rr = 0, i.e. their location is given by the roots of
the equation Δr = 0:

12 2 24M 12
2 
r 4 + a2 − r + r− a + Q2 = 0. (2.133)
R0 R0 R0

This is a fourth order equation that can be rewritten as:

(r − r− )(r − rint )(r − rext )(r − rcosm ) = 0, (2.134)

where r− is always a negative solution with no physical meaning, rint and rext are
the interior and exterior horizons respectively, and rcosm represents the cosmological
event horizon for observers between rext and rcosm . This horizon divides the region
that the observer could see from the region she/he could never see if she/he waited
long enough. The existence of real solutions for this equation is given by a factor h,
called horizon parameter (Cembranos et al. 2011):
  
4 R0 2 2
 3
h≡ 1− a − 4 a 2 + Q2
R0 12
  
4 R0 2 4 R0 2 2
+ 1− a 1− a
R0 12 R0 12
 2

2 
+ 12 a + Q 2
− 18M 2
. (2.135)

For a negative scalar curvature R0 , three options may be considered: (i) h > 0:
there are only two real solutions, rint and rext , lacking this configuration a cosmo-
logical horizon, as it is expected for an anti-de Sitter like Universe. (ii) h = 0: there
68 2 Black Holes

is only a degenerated root, particular case of an extremal black hole, whose interior
and exterior horizons have merged into one single horizon with a null surface grav-
ity. (iii) h < 0: there is no real solution to (2.135), which means absence of horizons
and then a naked singularity.
For a positive curvature R0 , there are also several configurations depending on
the value of h: (i) h < 0: both rint , rext , and rcosm are positive and real, thus the black
hole possesses a well-defined horizon structure in an Universe with a cosmological
horizon. (ii) h = 0: two different cases may be described, either rint and rext become
degenerated solutions, or rext and rcosm do so. The first case represents an extremal
black hole. The second case can be understood as the cosmological limit for which
a black hole preserves its exterior horizon without being “torn apart” by the rela-
tive recession speed between two radially separated points induced by the cosmic
expansion in an Universe described by a constant positive curvature. (iii) h > 0:
there is only one positive root, that may be either rint or rcosm . In the first case, the
mass of the black hole has exceeded the limit imposed by the cosmology (h = 0),
and there are neither exterior nor cosmological horizon. This situation just leaves
the interior horizon to cover the singularity (marginal naked singularity case). If the
root corresponds to rcosm , there is a naked singularity with a cosmological horizon.
From a certain positive value of the curvature R0crit onward, the h factor goes to zero
for two values of a, i.e., apart from the usual amax for which the black hole turns
extremal, there is now a spin lower bound amin , below which the black hole turns
into a marginally extremal black hole. Therefore


h amax , M, |R0 | ≥ 0, Q = 0


⇒ amax ≡ amax M, |R0 | ≥ 0, Q , (2.136)


h amin , M, R0 ≥ R0 > 0, Q = 0
crit


⇒ amin ≡ amin M, R0 ≥ R0crit > 0, Q . (2.137)

Another interesting feature of Kerr-Newman black holes is the presence of an er-

gosphere bounded by a Stationary Limit Surface (SLS), given by gtt = 0. In Boyer-
Lindquist coordinates:

Δθ sin2 θ a 2 Δr
− 2 2 = 0, (2.138)
ρ2Ξ 2 ρ Ξ
that leads to the fourth order equation
 
12 2 24M 12 2 2
r + a −
4 2
r + r − a cos θ +
2 2
a sin θ
R0 R0 R0
12
2 
− a + Q2 = 0, (2.139)
R0
which can be rewritten as:

(r − rS− )(r − rS int )(r − rS ext )(r − rS cosm ) = 0. (2.140)

2.7 Other Black Holes 69

Fig. 2.15 Left: Diagram of a Kerr-Newman black hole structure with negative curvature solution
R0 = −0.4 < 0, M = 1, a = 0.85 and Q = 0.35 (h > 0). Right: Black hole structure with positive
curvature solution R0 = 0.4 > 0, M = 1, a = 0.9 and Q = 0.4 (h < 0). Dotted surfaces represent
the static limit surfaces (SLS) whereas horizons are shown with continuous lines. The rotation
axis of the black hole is indicated by the vertical arrow. In both types of black hole, the region
between the exterior SLS rS ext and its associated exterior horizon rext is known as ergoregion.
From Cembranos et al. (2011). Reproduced by permission of the authors

From this equation it follows that each horizon has an “associated” SLS. Both
hypersurfaces coincide at θ = 0, π as seen when comparing (2.139) with Eq.(2.133).
A scheme of black hole horizons and the corresponding ergospheres is shown in
Fig. 2.15 for both signs of R0 .
For some general properties of static and spherically symmetric black holes in
f (R)-theories see Perez-Bergliaffa and Chifarelli de Oliveira Nunes (2011). For
Kerr-f (R) black holes and disks around them see Pérez et al. (2013).

2.7.4 Mini Black Holes

In principle, a black hole can have any mass above the Planck mass. A black hole
is formed always that there is an energy density large enough as to curve the space-
time forming a null closed surface. A lower possible mass for a black hole is im-
posed by the Compton wavelength, λC = h/Mc, which represents a limit on the
minimum size of the region in which a mass M at rest can be localized. For a suf-
ficiently small M, the reduced Compton wavelength (λ̄C = /Mc) exceeds half the
70 2 Black Holes

Schwarzschild radius, and no black hole description exists. The smallest mass for a
black hole is thus approximately the Planck mass.
Very low-mass or “mini” black holes evaporate very quickly by emission of
Hawking’s radiation (see Sect. 3.3). The absence of notable excesses of particles in
cosmic radiation—especially in the form of anti-protons—compared with the fluxes
expected in a “standard” astrophysical context allows to impose strict constraints on
the number density of black holes evaporating in today’s Universe. In particular, it
can be deduced that their contribution to the total mass of the universe is today no
higher than one ten millionth. As these small black holes are likely to have been
produced in the early cosmos by the fluctuations in energy density at that time—and
with masses that were very low—it is possible to obtain vital information about the
universe’s degree of inhomogeneity shortly after the period of inflation by imposing
upper limits from observations of secondary particles to the possible number of mini
black holes.
In addition to these astrophysical and cosmological aspects, there is another
route of investigation that is particularly promising for microscopic black holes,
namely at particle accelerators. If the center-of-mass energy of two elementary par-
ticles is higher than the Planck scale, and their impact parameter is lower than the
Schwarzschild radius, a black hole must be produced. If the Planck scale is thus
in the TeV range, the 14 TeV center-of-mass energy of the Large Hadron Collider
might allow it to become a black-hole factory with a production rate as high as about
one per second.
The possible presence of extra compact dimensions (Sect. 1.12) would be ben-
eficial for the production of black holes. The key point is that it allows the Planck
scale to be reduced to accessible values, but it also allows the Schwarzschild radius
to be significantly increased, thus making the condition for the impact parameter
to be smaller than the Schwarzschild radius easier to satisfy. It is important to note
that the resulting mini black holes have radii that are much smaller than the size of
extra dimensions, and that they can therefore be considered as totally immersed in
a D-dimensional space, which has, to a good approximation, a time dimension and
D − 1 non-compact space dimensions. The black hole thus acts like a quasi-selective
source of S waves and sees our brane in the same way as the “bulk” associated with
the extra dimensions. As the particles residing in the brane greatly outnumber those
living in the bulk (essentially gravitons), the black hole evaporates into particles
of the Standard Model. Its lifetime is very short (of the order of 10−26 s) and its
temperature (typically about 100 GeV here) is much lower than it would be with
the same mass in a four-dimensional space. The black hole nevertheless retains its
characteristic spectrum in the form of a quasi-thermal law peaked around its temper-
ature. From the point of view of detection, it is not too difficult to find a signature for
such events: they have a high multiplicity, a large transverse energy, a “democratic”
coupling to all particles, and a rapid increase in the production cross-section with
energy.
First of all the reconstruction of temperature (determined by the energy spectrum
of the particles emitted when the black hole evaporates) as a function of mass (de-
termined by the total energy deposited) allows information to be gained about the
References 71

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number of extra dimensions could thus be revealed quite easily by the characteris-
tics of the emitted particles. One can go even further. In particular, quantum gravity
effects could be revealed, as behavior during evaporation in the Planck region is sen-
sitive to the details of the gravitational theory used (for more on mini black holes
see Frolov and Zelnikov 2011, and references therein).

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