lecture ch18

Chapter 18

Black Holes

A spectacular consequences of GR is the prediction that

• gravitational fields can become so strong that theycan effectively trap even light,

• because space becomes so curved that there are nopaths for light from an interior to exterior region.

Such objects are calledblack holes.

We shall discuss 2 solutions to GR that can lead to black holes:

• TheSchwarzschild solution, which can be used to describe spher-ical black holes with no charge or angular momentum

• The Kerr solution, which can be used to describe rotating, un-charged black holes.

We shall also consider how gravity is altered ifquantum mechanicscomes into play, and considerobservational evidencefor black holes.

767

768 CHAPTER 18. BLACK HOLES

18.1 The Schwarzschild Solution

One of the simplest solutions corresponds to a metric thatdescribes the gravitational field exterior to a static, spher-ical, uncharged mass without angular momentum and iso-lated from all other mass(Schwarzschild, 1916).

The Schwarzschild solution is

• A solution to the vacuum Einstein equationsGµν = Rµν = 0.

• Thus valid only in the absence of matter and non-gravitationalfields (Tµν = 0).

• Spherically symmetric.

Thus, the Schwarzschild solution is valid outside spher-ical mass distributions, but the interior of a star will bedescribed by a different metric that must be matched atthe surface to the Schwarzschild one.

18.1. THE SCHWARZSCHILD SOLUTION 769

18.1.1 The Form of the Metric

Work in spherical coordinates(r,θ ,ϕ) and seek a time-independentsolution assuming

• The angular part of the metric will be unchanged from its formin flat space because of the spherical symmetry.

• The parts of the metric describingt and r will be modified byfunctions that depend on the radial coordinater .

Therefore, let us write the 4-D line element as

ds2 = −B(r)dt2+A(r)dr2︸︷︷︸

Modified from flat space

+r2dθ2+ r2sin2θdϕ2︸︷︷︸

Same as flat space

,

whereA(r) andB(r) are unknown functions that may depend onr butnot time. They may be determined by

1. Requiring that this metric be consistent with the Einstein fieldequations forTµν = 0.

2. Imposing physical boundary conditions.


Boundary conditions:Far from the star gravity becomesweak so

Limr→∞

A(r) = Limr→∞

B(r) = 1.

Substitute the metric form in vacuum Einstein and impose these bound-ary conditions (Exercise):

1. With the assumed form of the metric,

gµν =

−B(r) 0 0 0

0 A(r) 0 0

0 0 r2 0

0 0 0 r2sin2θ

.

compute the non-vanishing connection coefficientsΓλµν .

Γσλ µ = 1

2gνσ(

∂gµν

∂xλ +∂gλν∂xµ −

∂gµλ∂xν

)

2. Use the connection coefficients to construct the Ricci tensorRµν .

Rµν = Γλµν ,λ −Γλ

µλ ,ν +ΓλµνΓσ

λσ −Γσµλ Γλ

νσ ,

(Only needRµν , not full Gµν , since we will solvevacuum Ein-stein equations.


3. Solve the coupled set of equations

Rµν = 0

subject to the boundary conditions

Limr→∞

A(r) = Limr→∞

B(r) = 1.

The solution requires some manipulation but is remarkably simple:

B(r) = 1− 2Mr A(r) = B(r)−1,

whereM is the single parameter. The line element is then

ds2 =−(

1− 2Mr

)

dt2 +

(

1− 2Mr

)−1

dr2+ r2dθ2 + r2sin2θdϕ2,

wheredτ2 =−ds2. The corresponding metric tensor is

gµν =

−(

1− 2Mr

)

0 0 0

0

(

1− 2Mr

)−1

0 0

0 0 r2 0

0 0 0 r2sin2θ

.

which is diagonal butnot constant.


By comparing

g00 =−(

1− 2GMrc2

)

︸︷︷︸

Weak gravity (earlier)

←→ g00 =−(

1− 2GMrc2

)

︸︷︷︸

Schwarzschild (G & c restored)

we see that the parameterM (mathematically the single free parameterof the solution) may be identified with the total mass that is the sourceof the gravitational curvature. It has 3 contributions:

• Rest mass

• Contributions from mass–energy densities and pressure

• Energy from spacetime curvature

From the structure of the metric

ds2 =−B(r)dt2+A(r)dr2+ r2dθ2 + r2sin2θdϕ2,

• θ andϕ have similar interpretations as for flat space.

• r generallycannot be interpreted as a radiusbecauseA(r) 6= 1.

• The coordinate timet generallycannot be interpreted as a clocktimebecauseB(r) 6= 1.

The quantityrS≡ 2M

is called theSchwarzschild radius.It plays a central rolein the description of the Schwarzschild spacetime.


r/Mgµν

r =

2M

g11

g11

g00

+

−

Figure 18.1:The componentsg00 andg11 in the Schwarzschild metric.

The line element (metric)

ds2 =−(

1− 2Mr

)

dt2+

(

1− 2Mr

)−1

dr2+ r2dθ2+ r2sin2θdϕ2

appears to contain two singularities

1. A singularity atr = 0 (anessential singularity).

2. A singularity atr = rS = 2M (acoordinate singularity).

Coordinate Singularity:Place where a chosen set of coor-dinates does not describe the geometry properly.

Example:At North Pole the azimuthal angleϕ takes a continuum of values0–2π , so allthose values correspond to a single point. Butthis hasno physical significance.

Coordinate singularities not essential and can be removedby adifferent choice of coordinate system.


18.1.2 Measuring Distance and Time

What is the physical meaning of the coordinates(t, r,θ ,ϕ)?

• We may assign a practical definition to the radial coordinate r by

1. Enclosing the origin of our Schwarzschild spacetime in aseries of concentric spheres,

2. Measuring for each sphere a surface area (conceptually bylaying measuring rods end to end),

3. Assigning a radial coordinater to that sphere usingArea =4πr2.

• Then we can use distances and trigonometry to define the angu-lar coordinate variablesθ andϕ.

• Finally we can define coordinate timet in terms of clocks at-tached to the concentric spheres.

For Newtonian theory with its implicit assumption thatevents occur on a passive background of euclidean spaceand constantly flowing time, that’s the whole story.


But in curved Schwarzschild spacetime

• The coordinates(t, r,θ ,ϕ) provide a global reference frame foran observer making measurements an infinite distance from thegravitational source of the Schwarzschild spacetime.

• However, physical quantities measured by arbitrary observersare not specified directly by these coordinates but rather mustbe computed from the metric.

Proper and Coordinate Distances

Consider distance measured in the radial direction. Setdt = dθ =

dϕ = 0 in the line element to obtain an interval of radial distance

ds2 =−(

1− 2Mr

)

dt2︸︷︷︸

=0

+

(

1− 2Mr

)−1

dr2+ r2 dθ2︸︷︷︸

=0

+r2sin2θ dϕ2︸︷︷︸

=0

−→ ds2 =

(

1− 2Mr

)−1

dr2 −→ ds=dr

√

1− 2GMrc2

,

• In this expression we termds the proper distanceand dr thecoordinate distance.

• The physical interval in the radial direction measured by alocalobserver is given by the proper distanceds, not bydr.

• GM/rc2 is a measure of the strength of gravity so the properdistance and coordinate distance are equivalent only if gravity isnegligibly weak, either because

1. The sourceM is small, or

2. we are a very large coordinate distancer from the source.


Asymptotically

flat space

(dr ~ ds)

Asymptotically

flat space

(dr ~ ds)

ds

dr

C1

C2

C3C4

(1-2M/r)-1/2

Curved space

(dr < ds )

Flat space

(dr = ds)

Figure 18.2:Relationship between radial coordinate distancedr and proper dis-tanceds in Schwarzschild spacetime.

The relationship between the coordinate distance intervaldr and theproper distance intervalds is illustrated further in Fig. 18.2.

• The circlesC1 andC3 represent spheres having radiusr in eu-clidean space.

• The circlesC2 andC4 represent spheres having an infinitesimallylarger radiusr +dr in euclidean space.

• In euclidean space the distance that would be measured betweenthe spheres isdr

• But in the curved space the measured distance between the spheresis ds, which is larger thandr, by virtue of

ds=dr

√

1− 2GMrc2

,

• Notice however that at large distances from the source of thegravitational field the Schwarzschild spacetime becomes flat andthendr ∼ ds.


Proper and Coordinate Times

Likewise, to measure a time interval for a stationary clock at r setdr = dθ = dϕ = 0 in the line element and useds2 =−dτ2c2 to obtain

dτ =

√

1− 2GMrc2 dt.

• In this expressiondτ is termed theproper timeanddt is termedthecoordinate time.

• The physical time interval measured by a local observer is givenby the proper timedτ, not by the coordinate timedt.

• dt anddτ coincide only if the gravitational field is weak.

Thus we see that for the gravitational field outside a spher-ical mass distribution

• The coordinatesr andt correspond directly tophysi-cal distance and time in Newtonian gravity.

• In general relativity the physical (proper) distancesand times must becomputed from the metricand arenot given directly by the coordinates.

• Only in regions of spacetime where gravity is veryweak do we recover the Newtonian interpretation.

This is as it should be:The goal of relativity is to make the lawsof physics independent of the coordinate system in which they areformulated.


The coordinates in a physical theory are likestreet num-bers.

• They provide alabeling that locates points in aspace,but knowing the street numbers is not suffi-cient to determine distances.

• We can’t answer the question of whether the distancebetween 36th Street and 37th Street is the same as thedistance between 40th Street and 41st Street until weknow whether the streets are equally spaced.

• We must compute distances from ametric that givesa distance-measuring prescription.

– Streets that are always equally spaced corre-spond to a “flat” space.

– Streets with irregular spacing correspond toa position-dependent metric and thus to a“curved” space.

For the flat space the difference in street number cor-responds directly (up to a scale) to a physical dis-tance, but in the more general (curved) case it doesnot.


18.1.3 Embedding Diagrams

It is sometimes useful to form a mental image of the struc-ture for a curved space by embedding the space or a subsetof its dimensions in 3-D euclidean space.

Such embedding diagrams can be misleading, as illustrated well bythe case of a cylinder embedded in 3-D euclidean space, whichsug-gests that a cylinder is curved. But it isn’t:

• The cylinder is intrinsically a flat 2-D surface:

– Cut it and roll it out into a plane, or

– calculate its vanishing gaussian curvature.

• The cylinder hasno intrinsic curvature;the appearance of cur-vature derives entirely from the embedding in 3-D space and istermedextrinsic curvature.

Nevertheless, the image of the cylinder embedded in 3D euclideanspace is a useful representation of many properties associated with acylinder.


We can embed only 2 dimensions of Schwarzschild spacetime in3Deuclidean space.

• Illustrate by choosingθ = π/2 andt = 0, to give a 2-D metric

dℓ2 =

(

1− 2Mr

)−1

dr2+ r2dϕ2.

• The metric of the 3-D embedding space is conveniently repre-sented in cylindrical coordinates as

dℓ2 = dz2+dr2+ r2dϕ2

• This can be written onz= z(r) as

dℓ2 =

(dzdr

)2

dr2+dr2+ r2dϕ2 =

[

1+

(dzdr

)2]

dr2+ r2dϕ2

• Comparing

dℓ2 =

[

1+

(dzdr

)2]

dr2+r2dϕ2 ↔ dℓ2 =

(

1− 2Mr

)−1

dr2+r2dϕ2

implies that

z(r) = 2√

2M(r−2M),

which defines anembedding surfacez(r) having a geometry thatis the same as the Schwarzschild metric in the(r−ϕ) plane.


Figure 18.3:An embedding diagram for the Schwarzschild(r−ϕ) plane.

Fig. 18.3 illustrates the embedding function

z(r) = 2√

2M(r−2M)

Fig. 18.3 is not what a black hole “looks” like, but it isa striking and useful visualization of the Schwarzschildgeometry. Thus such embedding diagrams are a standardrepresentation of black holes in popular-level discussion.


r

t

Light ray

So

urc

e

Sch

wa

rzsch

ild r

ad

ius

Dis

tan

t o

bse

rve

r

r = 2M0 r = R1

Emission of light

with frequency ω0

Detection of light

with frequency ω

r ~

Figure 18.4:A spacetime diagram for gravitational redshift in the Schwarzschildmetric.

18.1.4 The Gravitational Redshift

Emission of light from a radiusR1 that is then detected bya stationary observer at a radiusr >> R1 (Fig. 18.4).

For an observer with 4-velocityu, the energy measured for a photonwith 4-momentump is

E = h̄ω =−p·u,

Observers stationary in space but not time so

ui(r) = 0 u0 6= 0


Thus the 4-velocity normalization gives

u·u = gµν(x)dxµ

dτdxν

dτ= g00(x)u

0(r)u0(r) =−1.︸︷︷︸

Solve foru0(r)

and we obtain

u0(r) =

√

−1g00

=

(

1− 2Mr

)−1/2

.

Symmetry: Schwarzschild metric independent of time,which implies the existence of aKilling vector

ξ µ = (t, r,θ ,ϕ) = (1,0,0,0)

associated with symmetry under time displacement.

Thus, for a stationary observer at a distancer ,

uµ(r) =

((

1− 2Mr

)−1/2

, 0, 0, 0

)

=

(

1− 2Mr

)−1/2

ξ µ .

The energy of the photon measured atr by stationary observer is

h̄ω(r) =−p·u =−(

1− 2Mr

)−1/2

(ξ ·p)r


But ξ ·p is conservedalong the photon geodesic (ξ is aKilling vector), soξ ·p is in fact independent ofr .

Therefore,

h̄ω0≡ h̄ω(r = R1) =−(

1− 2MR1

)−1/2

(ξ ·p)

h̄ω∞≡ h̄ω(r → ∞) =−(ξ ·p),

and fromh̄ω∞/h̄ω0 we obtain immediately a gravitational redshift

ω∞ = ω0

(

1− 2MR1

)1/2

.

We have made no weak-field assumptions so this resultshould bevalid for weak and strong fields.

For weak fields2M/R1 is small, the square root can be expanded, andtheG andc factors restored to give

ω∞ ≃ ω0

(

1− GMR1c2

)

(valid for weak fields)

which is the result derived earlier using the equivalence principle.

By viewing ω as defining clock ticks, the redshift mayalso be interpreted as agravitational time dilation.


18.1.5 Particle Orbits in the Schwarzschild Metric

Symmetries of the Schwarzschild metric:

1. Time independence→ Killing vector ξt = (1,0,0,0)

2. No dependence onϕ → Killing vector ξϕ = (0,0,0,1)

3. Additional Killing vectors associated with full rotational sym-metry (won’t need in following).

Conserved quantities associated with these Killing vectors:

ε ≡−ξt ·u =

(

1− 2Mr

)dtdτ

ℓ≡ ξϕ ·u = r2sin2θdϕdτ

.

Physical interpretation:

• At low velocitiesℓ∼ (orbital angular momentum / unit rest mass)

• SinceE = p0 = mu0 = mdt/dτ ,

Limr→∞

ε =dtdτ

=Em

andε ∼ energy / (unit rest mass)at large distance.

Also we have the velocity normalization constraint

u·u = gµνuµuν =−1.


The Schwarzschild line element is

ds2 =−(

1− 2Mr

)

dt2+

(

1− 2Mr

)−1

dr2+ r2dθ2+ r2sin2θdϕ2

Conservation of angular momentum confines the particlemotion to a plane, which we conveniently take to be theequatorial plane withθ = π

2 implying thatdθ = 0, so

• u2≡ uθ = dθ/dτ = 0.

• sin2θ = 1

Then writing the velocity constraint

gµνuµuν =−1.

out explicitly for the Schwarzschild metric gives

−(

1− 2Mr

)

(u0)2 +

(

1− 2Mr

)−1

(u1)2 + r2(u3)2 =−1.

which we may rewrite using

uµ =

(dx0

dτ,dx1

dτ,dx2

dτ,dx3

dτ

)

ε =

(

1− 2Mr

)dtdτ

ℓ = r2sin2θdϕdτ

.

in the form

ε2−12

=12

(drdτ

)2

+12

[(

1− 2Mr

)(ℓ2

r2 +1

)

−1

]

.


We can put this in the form

E =12

(drdτ

)2

+Veff(r),

where we define a fictitious “energy”

E ≡ ε2−12

and an effective potential

Veff(r) =12

[(

1− 2Mr

)(ℓ2

r2 +1

)

−1

]

=−Mr

+ℓ2

2r2︸︷︷︸

Newtonian

− Mℓ2

r3︸︷︷︸

correction

This is analogous to the energy integral of Newtonian me-chanics with an effective potentialVeff having

• A Newtonian gravity plus centrifugal piece,

• an additional term porportional tor−3 that modifiesNewtonian gravity.


Veff

r/M

Newtonian

Schwarzschild

Figure 18.5:Effective potentials for finiteℓ in the Schwarzschild geometry and inNewtonian approximation.

Figure 18.5 compares the Schwarzschild effective potential with aneffective Newtonian potential.

The Schwarzschild potential generally has one maximumand one minimum ifℓ/M >

√12 .

Note the very different behavior of Schwarzschild andNewtonian mechanics at the origin because of the correc-tion term in

Veff(r) =−Mr

+ℓ2

2r2︸︷︷︸

Newtonian

− Mℓ2

r3︸︷︷︸

correction


Veff

Veff

0

0

Stable Equilibrium

Unstable Equilibrium

Veff

0

Veff

0

Turning Points

r/M

Circular

Orbits

Bound

Precessing

Orbits

Scattering

Orbits

Plunging

Orbits

Unstable

Stable

Figure 18.6:Orbits in a Schwarzschild spacetime. Effective potential on left andcorresponding classes of orbits on right.


18.1.6 Innermost Stable Circular Orbit

The radial coordinate of the inner turning point for bound precessingorbits in the Schwarzschild metric is given by (Exercise)

r− =ℓ2

2M

1+

√

1−12

(Mℓ

)2

Thusr− has a minimum possible value when

Mℓ

=1√12

.

The corresponding radius for theinnermost stable circularorbit RISCO is then

RISCO = 6M.

The innermost stable circular orbit is impor-tant in determining how much gravitationalenergy can be extracted from matter accret-ing onto neutron stars and black holes.


r+

δφ

r-

r+

r-

r+

r-

r+

r-

Figure 18.7:Precessing orbits in a Schwarzschild metric

18.1.7 Precession of Orbits

An orbit closes if the angleϕ sweeps out exactly2π in thepassage between two successive inner or two successiveouter radial turning points.

In Newtonian gravity the1/r central potential→ closedelliptical orbits. In Schwarzschild metric the effective po-tential deviates from1/r and orbitsprecess:ϕ changes bymore than2π between successive radial turning points.


To investigate this precession quantitatively we require an expressionfor dϕ/dr. From the energy equation

drdτ

=±√

2(E−Veff(r)),

and from the conservation equation forℓ,

dϕdτ

=ℓ

r2sin2θ.

Combine, recalling that we are choosing an orbital planeθ = π2 ,

dϕdr

=dϕ/dτdr/dτ

=± ℓ

r2√

2(E−Veff(r))

=± ℓ

r2

[

2E−(

1− 2Mr

)(

1+ℓ2

r2

)

+1

]−1/2

=± ℓ

r2

[

ε2−(

1− 2Mr

)(

1+ℓ2

r2

)]−1/2

,

where we have usedE = 12(ε

2−1).


The change inϕ per orbit,∆ϕ, can be obtained by integrating overone orbit,

∆ϕ =

∫ r+

r−

dϕdr

dr +

∫ r−

r+

dϕdr

dr = 2∫ r+

r−

dϕdr

dr

= 2ℓ∫ r+

r−

drr2

[

ε2−(

1− 2Mr

)(

1+ℓ2

r2

)]−1/2

= 2ℓ∫ r+

r−

drr2

c2(ε2−1)+2GM

r− ℓ2

r2︸︷︷︸

Newtonian

+2GMℓ2

c2r3︸︷︷︸

correction

−1/2

where in the last stepG andc have been reinserted through the sub-stitutions

M→ GMc2 ℓ→ ℓ

c,

Evaluation of the integral requires some care becausetheintegrand tends to∞ at the integration limits:From one ofour earlier expressions

drdτ

=±[

ε2−(

1− 2Mr

)(

1+l2

r2

)]1/2

,

which is the denominator of our integrand. But the limitsare turning points of the radial motion anddr/dτ = 0 atr+ or r−.


In the Solar System and most other applications the valuesof ∆ϕ are very small, so it is sufficient to keep only termsof order1/c2 beyond the Newtonian approximation.

Expanding the integrand and evaluating the integral with due care(Exercise) yields

Precession angle= δϕ ≡ ∆ϕ−2π

≃ 6π(

GMcℓ

)2

rad/orbit.

This may be expressed in more familiar classical orbital parameters:

• In Newtonian mechanicsL = mr2ω , whereL is the angular mo-mentum andω the angular frequency.

• For Kepler orbits

ℓ2 =

(Lm

)2

=

(

r2dϕdτ

)2

≃(

r2dϕdt

)2

= GMa(1−e2),

wheree is the eccentricity anda is the semimajor axis.

This permits us to write

δϕ =6πGM

ac2(1−e2)

= 1.861×10−7(

MM⊙

)(AUa

)1

1−e2 rad/orbit,


The form of

δϕ =6πGM

ac2(1−e2)

shows explicitly that the amount of relativistic precession is enhancedby

• largeM for the central mass,

• tight orbits (small values ofa),

• large eccentricitiese.

The precession observed for most objects is small.

• Precession of Mercury’s orbit in the Sun’s gravitational field be-cause of general relativistic effects is observed to be43 arcsec-onds per century.

• The orbit of the Binary Pulsar precesses by about4.2 degreesper year.

The precise agreement of both of these observations with thepredic-tions of general relativity is a strong test of the theory.


18.1.8 Escape Velocity in the Schwarzschild Metric

Consider a stationary observer at a Schwarzschild radialcoordinateR who launches a projectile radially with avelocity v such that the projectile reaches infinity withzero velocity. This defines the escape velocity in theSchwarzschild metric.

• The projectile follows a radial geodesic since thereare no forces acting on it

• The energy per unit rest mass isε and it is conserved(time invariance of metric).

• At infinity ε = 1, since then the particle is at rest andthe entire energy is rest mass energy. Thusε = 1 atall times.

If uobs is the 4-velocity of the stationary observer, the energy measuredby the observer is

E =−p·uobs =−mu·uobs

=−mgµνuµuνobs

=−mg00u0u0

obs (Observer stationary),

wherep = mu, with p the 4-momentum andm the rest mass, and thelast step follows because the observer is stationary. But

g00 =−(1− 2M

r

)

︸︷︷︸

From metric

u0obs=

(1− 2M

R

)−1/2

︸︷︷︸

Stationary observer

u0 =(1− 2M

r

)−1

︸︷︷︸

Fromε = (1− 2Mr )u0 = 1


Therefore,

E =−mg00u0u0

obs

= m(1− 2M

r

)(1− 2M

r

)−1(1− 2M

r

)−1/2

= m(1− 2M

R

)−1/2

But in the observer’s rest frame the energy and velocity are related by

E = mγ = m(1−v2)−1/2

so comparison of the two expressions forE yields

m(1− 2M

R

)−1/2= m(1−v2)−1/2 −→ v2 =

2MR

and thus

v≡ vesc=

√

2MR

Notice that

• This, coincidentally, is the same result as for Newtonian theory.

• At the Schwarzschild radiusR= rS = 2M, the escape velocity isequal toc.

This is the first hint of anevent horizon in theSchwarzschild spacetime.


18.1.9 Radial Fall of a Test Particle in Schwarzschild Geometry

It will be instructive for later discussion to consider theparticular case of a radial plunge orbit that starts from in-finity with zero kinetic energy (ε = 1) and zero angularmomentum (ℓ = 0).

First, let us find an expression for theproper timeas a function of thecoordinater . From earlier expressions

E =ε2−1

2=

12

(drdτ

)2

−Mr

+ℓ2

2r2−Mℓ2

r3 ,

which implies forℓ = 0 andε = 1,

0 =12

(drdτ

)2

−Mr−→ dr

dτ=±

(2Mr

)1/2

.

Choosing the negative sign (infalling orbit) and integrating with initialconditionτ(r = 0) = 0 gives

τ2M

=−23

r3/2

(2M)3/2

for the proper timeτ to reach the origin as a function of the initialSchwarzschild coordinater .


To find an expression for thecoordinate timet as a function ofr , wenote thatε = 1 and is conserved. Then from the previous expressions

ε = 1 =

(

1− 2Mr

)dtdτ

drdτ

=±(

2Mr

)1/2

we have that

dtdr

=dt/dτdr/dτ

=−(

1− 2Mr

)−1(2Mr

)−1/2

,

which may be integrated to give for the coordinate timet

t = 2M

(

−23

( r2M

)3/2+2( r

2M

)1/2+ ln

∣∣∣∣∣

(r/2M)1/2+1

(r/2M)1/2−1

∣∣∣∣∣

)

.


r /M

Proper

time τ

Schwarzschild

coordinate time t

-Time/M

rS = 2M

Figure 18.8:Comparison of proper time and Schwarzschild coordinate time for aparticle falling radially in the Schwarzschild geometry.

The behavior oft andτ are plotted in Fig. 18.8. Notice that

• The proper timeτ to fall to the origin is finite.

• For the same trajectory an infinite amount of coordinate time telapses to reach the Schwarzschild radius.

• The smooth trajectory of the proper time throughrS suggests thatthe apparent singularity of the metric there is not physical.

Later in this chapter we shall introduce alternative coordi-nates that explicitly remove the singularity atr = 2M (butnot atr = 0).


18.1.10 Light Ray Orbits

Calculation of light ray orbits in the Schwarzschild metriclargely parallels that of particle orbits, except that

u·u = gµνdxµ

dλdxν

dλ= 0,

(not u·u =−1!) whereλ is an affine parameter.

For motion in the equatorial plane(θ = π2), this becomes explicitly

−(

1− 2Mr

)(dtdλ

)2

+

(

1− 2Mr

)−1( drdλ

)2

+ r2(

dϕdλ

)2

= 0.

By analogy with the corresponding arguments for particle motion

ε ≡−ξt ·u=

(

1− 2Mr

)dtdλ

,

ℓ = ξϕ ·u= r2sin2θdϕdλ

,

are conserved along the orbits of light rays.

With a proper choice of the normalization of the affineparameterλ , the conserved quantitityε may be interpretedas the photon energy and the conserved quantityℓ as itsangular momentum at infinity.


Veff 0

+

−

Veff 0

+

−

Veff 0

+

−

r orbits

Circular

orbit

Scattering

orbit

Plunging

orbit

Figure 18.9:The effective potential for photons and some light ray orbits in aSchwarzschild metric. The dotted lines on the left side givethe value of 1/b2 foreach orbit.

By following steps analogous to the derivation of particle orbits

1b2 =

1ℓ2

(drdλ

)2

+Veff(r)

Veff(r)≡1r2

(

1− 2Mr

)

b2≡ ℓ2

ε2.

The effective potential for photons and some classes of light ray orbitsin the Schwarzschild geometry are illustrated in Fig. 18.9.


∆φφ

δφ

rr1

b

Figure 18.10:Deflection of light by an angleδϕ in a Schwarzschild metric.

18.1.11 Deflection of Light in a Gravitational Field

Proceeding in a manner similar to that for the calculationof the precession angle for orbits of massive objects, wemay calculate the deflectiondϕ/dr for a light ray in theSchwarzschild metric.

The result is

δϕ =4GMc2b

= 2.970×10−28(

Mg

)(cmb

)

radians

= 8.488×10−6(

MM⊙

)(R⊙b

)

radians.

For a photon grazing the surface of the Sun,M = 1M⊙andb = 1R⊙, which givesδϕ ≃ 1.75 arcseconds.

Observation of this deflection during a total solar eclipse catapultedEinstein to worldwide fame almost overnight in the early 1920s.


18.2 Schwarzschild Black Holes

There is an event horizon in the Schwarzschild spacetimeat rS = 2M, which implies a black hole inside the eventhorizon (escape velocity exceedsc).

Place analysis on firmer ground by considering a spacecraft approach-ing the event horizon in free fall (engines off).

• For simplicity, we assume the trajectory to be radial.

• Consider from two points of view

1. From a very distant point at constant distance from the blackhole (professors sipping martinis).

2. From a point inside the spacecraft (students).

• Use the Schwarzschild solution (metric) for analysis.

18.2. SCHWARZSCHILD BLACK HOLES 805

18.2.1 Approaching the Event Horizon: Outside View

We consider onlyradial motion. Settingdθ = dϕ = 0 in the lineelement gives

ds2 =−dτ2

=−(

1− 2Mr

)

dt2+

(

1− 2Mr

)−1

dr2+ r2 dθ2︸︷︷︸

=0

+r2sin2θ dϕ2︸︷︷︸

=0

=−(

1− 2Mr

)

dt2+

(

1− 2Mr

)−1

dr2

=−(

1− rS

r

)

dt2 +(

1− rS

r

)−1dr2.

• As the spacecraft approaches the event horizon its velocity asviewed from far outside in a fixed frame isv = dr/dt.

• Light signals from spacecraft travel on the light cone(ds2 = 0)and thus from the line element

ds2 = 0 =−(

1− rS

r

)

dt2+(

1− rS

r

)−1dr2

Solve fordrdt to give

v≡ drdt

=(

1− rS

r

)

.

• As viewed from a distance outsiderS, the spacecraft appears toslow as it approachesrS and eventually stops asr → rS.

• Thus, from the exterior we would never see the spacecraft crossrS: its image would remain frozen atr = rS for all eternity.


But let us examine what this means a little more carefully. Rewrite

drdt

=(

1− rS

r

)

→ dt =dr

1− rS/r.

• As r→ rS time between successive wave crests for the light wavecoming from the spacecraft tends to infinity and therefore

λ → ∞ ν → 0 E→ 0.

• The external observer not only sees the spacecraft slow rapidlyas it approachesrS, but the spacecraft image is observed tostronglyredshiftat the same time.

• This behavior is just that of the coordinate time already seen fora test particle in radial free fall:

r /M

Proper

time τ

Schwarzschild

coordinate time t

-Time/M

rS = 2M

• Therefore, more properly, the external observation is that thespacecraft approachingrS rapidly slows and redshifts until theimage fades from view before the spacecraft reachesrS.


18.2.2 Approaching the Event Horizon: Spacecraft View

Things are very different as viewed by the (doomed) studentsfromthe interior of the spacecraft.

• The occupants will use their own clocks (measuring proper time)to gauge the passage of time.

• Starting from a radial positionr0 outside the event horizon, thespacecraft will reach the origin in a proper time

τ =−23

r3/20

(2M)1/2.

• The spacecraft occupants will generally noticeno spacetime sin-gularity at the horizon.

• Any tidal forces at the horizon may be very large but will remainfinite (Riemann curvature is finite at the Schwarzschild radius).

• The spacecraft crossesrS and reaches the (real) singularity atr = 0 in a finite amount of time, where it would encounter in-finite tidal forces (Riemann curvature has components that be-come infinite at the origin).

• The trip fromrS to the singularity is very fast (Exercise):

1. Of order10−4 secondsfor stellar-mass black holes.

2. Of order10 minutesfor a billion solar mass black hole.


18.2.3 Lightcone Description of a Trip to a Black Hole

It is highly instructive to consider a lightcone description of a trip intoa Schwarzschild black hole. First we consider the region outside theSchwarzschild radius.

• Assumingradial light rays

dθ = dϕ︸︷︷︸

radial

= ds2︸︷︷︸

light

= 0

the line element reduces to

ds2 =−(

1− 2Mr

)

dt2 +

(

1− 2Mr

)−1

dr2 = 0,

• Thus theequation for the lightconeat some local coordinater inthe Schwarzschild metric can be read directly from the metric

dtdr

=±(

1− 2Mr

)−1

.

where

··· The plus sign corresponds to outgoing photons (r increasingwith time for r > 2M)

··· The minus sign to ingoing photons (r decreasing with timefor r > 2M)

• For r → ∞dtdr

=±(

1− 2Mr

)−1

.

becomes equal to±1, as for flat spacetime.

• However asr→ rS the forward lightcone opening angle tends tozero becausedt/dr→ ∞.


0 1 2 3

r/2M

Tim

e

rs=2M

Figure 18.11:Photon paths and lightcone structure of the Schwarzschild space-time.

Integratingdtdr

=±(

1− 2Mr

)−1

.

gives

t =

−r−2M ln |r−1|+ constant (Ingoing)

r +2M ln |r−1|+ constant (Outgoing)

• The null geodesics defined by this expression are plotted inFig. 18.11.

• The tangents at the intersections of the dashed and solid lines de-fine local lightcones corresponding todt/dr, which are sketchedat some representative spacetime points.


t = t0

tA

0 rS

A

B

C

D

E

Singularity

worldline

Interior of

black hole

Event horizon

Sin

gu

larity

Observer

worldline

Exterior

universe

Spacecraft

worldline

r robs

Figure 18.12:Light cone description of a trip into a Schwarzschild black hole.

• The worldline of a spacecraft is illustrated in Fig. 18.12,startingwell exterior to the black hole. The gravitational field there isweak and the light cone has the usual symmetric appearance.

• As illustrated by the dotted line from A, a light signal emittedfrom the spacecraft can intersect the worldline of an observerremaining at constant distancerobs at a finite timetA > t0.

• As the spacecraft falls toward the black hole on the worldlineindicated the forward light cone begins to narrow since

dtdr

=±(

1− 2Mr

)−1

.


t = t0

tA

0 rS

A

B

C

D

E

Singularity

worldline

Interior of

black hole

Event horizon

Sin

gu

larity

Observer

worldline

Exterior

universe

Spacecraft

worldline

r robs

• Now, at B a light signal can intersect the external observerworld-line only at a distant point in the future (arrow on light coneB).

• As the spacecraft approachesrS, the opening angle of the for-ward light cone tends to zero and a signal emitted from thespacecraft tends towardinfinite time to reach the external ob-server’s worldline atrobs (arrow on light cone C).

• The external observer sees infinite redshift as the spacecraft ap-proaches the Schwarzschild radius.


r/Mgµν

r =

2M

g11

g11

g00

+

−

Figure 18.13:Spacelike and timelike regions forg00 andg11 in the Schwarzschildmetric.

Now consider light conesinterior to the Schwarzschild radius.

• From the structure of the radial and time parts of the Schwarzschildmetric illustrated in Fig. 18.13, we observe thatdr anddt reversetheir character at the horizon (r = 2M) because the metric coef-ficientsg00 andg11 switch signsat that point.

1. Outside the event horizon thet direction,∂/∂ t, is timelike(g00 < 0) and ther direction,∂/∂ r , is spacelike (g11 > 0).

2. Inside the event horizon,∂/∂ t is spacelike (g00 > 0) and∂/∂ r is timelike (g11 < 0).

• Thus inside the event horizon the lightcones get rotated byπ/2relative to outside (space↔ time).


t = t0

tA

0 rS

A

B

C

D

E

Singularity

worldline

Interior of

black hole

Event horizon

Sin

gula

rity

Observer

worldline

Exterior

universe

Spacecraft

worldline

r robs

• The worldline of the spacecraft descends insiderS because thecoordinate time decreases (it is now behaving liker) and thedecrease inr represents the passage of time, but theproper timeis continuously increasing in this region.

• Outside the horizonr is aspacelike coordinateand applicationof enough rocket power can reverse the infall and maker beginto increase.

• Inside the horizonr is a timelike coordinateand no applicationof rocket power can reverse the direction of time.

• Thus, the radial coordinate of the spacecraft must decrease onceinside the horizon, for the same reason that time flows into thefuture in normal experience (whatever that reason is!).


t = t0

tA

0 rS

A

B

C

D

E

Singularity

worldline

Interior of

black hole

Event horizon

Sin

gula

rity

Observer

worldline

Exterior

universe

Spacecraft

worldline

r robs

• Inside the horizon there are no paths in the forward light cone ofthe spacecraft that can reach the external observer atr0 (the rightvertical axis)—see the light cones labeled D and E.

All timelike and null paths inside the hori-zon are bounded by the horizon and must en-counter the singularity atr = 0.

• This illustrates succinctly the real reason that nothing can escapethe interior of a black hole.Dynamics (building a better rocket)are irrelevant:once insiderS the geometry of spacetime permits

– no forward light cones that intersect exterior regions, and

– no forward light cones that can avoid the origin.

Spacetime geometry is destiny; resistance is futile!


Thus, there is no escape from the classical Schwarzschildblack hole because

1. There are literally no paths in spacetime that go fromthe interior to the exterior.

2. All timelike or null paths within the horizon lead tothe singularity atr = 0.

But notice the adjective “classical” . . . More later.


The preceding discussion is illuminating but the interpre-tation of the results is complicated by the behavior nearthe coordinate singularity atr = 2M.

• We now discuss alternative coordinate systems thatremove the coordinate singularity at the horizon.

• Although these coordinate systems have advan-tages for interpreting the interior behavior of theSchwarzschild geometry, the standard coordinatesremain useful for describing the exterior behavior be-cause of their simple asymptotic behavior.


18.2.4 Eddington–Finkelstein Coordinates

In the Eddington–Finkelstein coordinate systema new variablev isintroduced through

t = v︸︷︷︸new−r−2M ln

∣∣∣

r2M−1∣∣∣ ,

wherer , t, andM have their usual meanings in the Schwarzschildmetric, andθ andϕ are assumed to be unchanged. For eitherr > 2Mor r < 2M, insertion into the standard Schwarzschild line elementgives (Exercise)

ds2 =−(

1− 2Mr

)

dv2+2dvdr+ r2dθ2+ r2sin2θdϕ2.


• The Schwarzschild metric expressed in these new coordinates

ds2 =−(

1− 2Mr

)

dv2+2dvdr+ r2dθ2+ r2sin2θdϕ2.

is manifestly non-singular atr = 2M

• The singularity atr = 0 remains.

• Thus the singularity at the Schwarzschild radius is a coordinatesingularity that can be removed by a new choice of coordinates.

Let us consider behavior of radial light rays expressed in these coor-dinates.

• Setdθ = dϕ = 0 (radial motion)

• Setds2 = 0 (light rays).

Then the Eddington–Finkelstein line element gives

−(

1− 2Mr

)

dv2 +2dvdr= 0.


Sin

gula

rity

(r

= 0

)

Horizon

rr = 2M

1

2

34

v - 2(r + 2M ln|r/2M - 1|) = constantv -

rS

ingula

rity

OutgoingIngoing

dv = 0 (Ingoing)

dvdt

= 0

r0 0r = 2M

v - r

(a) (b)

Figure 18.14:(a) Eddington–Finkelstein coordinates for the Schwarzschild blackhole with r on the horizontal axis andv− r on the vertical axis. Only two coordi-nates are plotted, so each point corresponds to a 2-sphere ofangular coordinates.(b) Light cones in Eddington–Finkelstein coordinates.

This equation has two general solutions and one special solution [seeFig. 18.14(a)]:

−(

1− 2Mr

)

dv2 +2dvdr= 0.

• General Solution 1:dv = 0, sov = constant.→ Ingoing lightrays on trajectories of constantv (dashed lines Fig. 18.14(a)) .


Sin

gula

rity

(r

= 0

)

Horizon

rr = 2M

1

2

34


rS

ingula

rity

OutgoingIngoing

dv = 0 (Ingoing)

dvdt

= 0

r0 0r = 2M

v - r

(a) (b)

• General Solution 2:If dv 6= 0, then divide bydv2 to give

−(

1− 2Mr

)

dv2 +2dvdr= 0 → dvdr

= 2

(

1− 2Mr

)−1

,

which yields upon integration

v−2(

r +2M ln∣∣∣

r2M−1∣∣∣

)

= constant.

This solution changes behavior atr = 2M:

1. Outgoing forr > 2M.

2. Ingoingfor r < 2M (r decreases asv increases).

The long-dashed curves in Fig. 18.14(a) illustrate both ingoingand outgoing world-lines corresponding to this solution.


Sin

gula

rity

(r

= 0

)

Horizon

rr = 2M

1

2

34


rS

ingula

rity

OutgoingIngoing

dv = 0 (Ingoing)

dvdt

= 0

r0 0r = 2M

v - r

(a) (b)

• Special Solution:In the special case thatr = 2M, the differentialequation reduces to

−(

1− 2Mr

)

dv2+2dvdr= 0 → dvdr= 0,

which corresponds to light trapped at the Schwarzschild radius.The vertical solid line atr = 2M represents this solution.


Sin

gula

rity

(r

= 0

)

Horizon

rr = 2M

1

2

34


rS

ingula

rity

OutgoingIngoing

dv = 0 (Ingoing)

dvdt

= 0

r0 0r = 2M

v - r

(a) (b)

For every spacetime point in Fig. 18.14(a) there are twosolutions.

• For the points labeled 1 and 2 these correspond to aningoing and outgoing solution.

• For point 3 one solution is ingoing and one corre-sponds to light trapped atr = rS.

• For point 4 both solutions are ingoing.


Sin

gula

rity

(r

= 0

)

Horizon

rr = 2M

1

2

34

v - 2(r + 2M ln|r/2M - 1|) = constant

v -

rS

ingula

rity

OutgoingIngoing

dv = 0 (Ingoing)

dvdt

= 0

r0 0r = 2M

v - r

(a) (b)

The two solutions passing through a point determine thelight cone structure at that point (right side of figure).

• The light cones at various points are bounded by the two solu-tions, so they tilt “inward” asr decreases.

• The radial light ray that defines the left side of the light cone isingoing (general solution 1).

• If r 6= 2M, the radial light ray defining the right side of the lightcone corresponds to general solution 2.

1. These propagate outward ifr > 2M.

2. Forr < 2M they propagateinward.

• For r < 2M the light cone is tilted sufficiently that no light raycan escape the singularity atr = 0.

• At r = 2M, one light ray moves inward; one is trapped atr = 2M.


The horizon may be viewed as a null surface generated bythe radial light rays that can neither escape to infinity norfall in to the singularity.


18.2.5 Kruskal–Szekeres Coordinates

The Kruskal–Szekeres coordinatesexhibit no singularityat the Schwarzschild radiusr = 2M.

• Introduce variables(v,u,θ ,ϕ), whereθ andϕ have their usualmeaning and the new variablesu andv are defined through

u=( r

2M−1)1/2

er/4M cosh( t

4M

)

(r > 2M)

=(

1− r2M

)1/2er/4M sinh

( t4M

)

(r < 2M)

v=( r

2M−1)1/2

er/4M sinh( t

4M

)

(r > 2M)

=(

1− r2M

)1/2er/4M cosh

( t4M

)

(r < 2M)

• The corresponding line element is

ds2 =32M3

re−r/2M(−dv2 +du2)+ r2dθ2+ r2sin2θdϕ2,

wherer = r(u,v) is defined through( r

2M−1)

er/2M = u2−v2.

• This metric is manifestly non-singular atr = 2M, but singular atr = 0.


Timelike

worldline

Figure 18.15:Kruskal–Szekeres coordinates. Only the two coordinatesu andvare displayed; each point is really a 2-sphere in the variablesθ andϕ .

Kruskal diagram:lines of constantr andt plotted on auandv grid. Figure 18.15 illustrates.


Timelike

worldline

• From the form of( r

2M−1)

er/2M = u2−v2.

lines of constantr arehyperbolae of constantu2−v2.

• From the definitions ofu andv

v= utanh( t

4M

)

(r > 2M)

=u

tanh(t/4M)(r < 2M).

Thus, lines of constantt are straight lines with slope

· tanh(t/4M) for r > 2M

· 1/ tanh(t/4M) for r < 2M.


Timelike

worldline

• For radial light rays in Kruskal–Szekeres coordinates (dθ = dϕ =ds2 = 0), and the line element

ds2 =32M3

re−r/2M(−dv2+du2)+ r2dθ2 + r2sin2θdϕ2

yieldsdv=±du

45 degree lightcones in theuv parameters,just like flat space.

• Over the full range of Kruskal–Szekeres coordinates(v,u,θ ,ϕ),the metric componentg00 = gvv remains negative andg11 = guu,g22 = gθθ , andg33 = gϕϕ remain positive.

• Therefore, thev direction isalways timelikeand theu directionis always spacelike,in contrast to the normal Schwarzschild co-ordinates wherer andt switch their character at the horizon.


Schwarzschild

coordinates Kruskal-Szekeres

coordinates

Figure 18.16:A trip to the center of a black hole in standard Schwarzschildcoor-dinates and in Kruskal–Szekeres coordinates.

The identification ofr = 2M as an event horizon is partic-ularly clear in Kruskal–Szekeres coordinates (Fig. 18.16).

• The light cones make 45-degree angles with the ver-tical and the horizon also makes a 45-degree anglewith the vertical.

• Thus, for any point within the horizon, its forwardworldlinemustcontain ther = 0 singularity andcan-notcontain ther = 2M horizon.


u

v

Hor

izon

r =

2M

Singularity (

r = 0

)

External

observer

(fixed r)

External

observer

(fixed r)

Kruskal-Szekeres coordinates Eddington-Finkelstein coordinates

Ho

rizo

n r

= 2

M

Sin

gu

larity

r =

0

r

v - r

Surface of collapsing star

Surface of collapsing star

Figure 18.17:Collapse to a Schwarzschild black hole.

Figure 18.17 illustrates a spherical mass distribution (a star, for ex-ample) collapsing to a black hole as represented in Kruskal–Szekerescoordinates and in Eddington–Finkelstein coordinates.

• A distant observer remains at fixedr (hyperbola) and observesperiodic light signals sent from the surface of the collapsing star.

• Light pulses, propagating on the dashed lines, arrive at longerand longer intervals as measured by the outside observer.

• At the horizon, light signals take an infinite length of timetoreach the external observer.

• Once the surface is inside the horizon, no signals can reachtheoutside observer and the entire star collapses to the singularity.

• Note: the Schwarzschild solution is valid onlyoutsidethe star.Inside GR applies but the solution is not Schwarzschild.

18.3. HAWKING BLACK HOLES: BLACK HOLES ARE NOT REALLY BLACK!831

18.3 Hawking Black Holes: Black Holes Are Not Really Black!

Classically, the fundamental structure of curved space-time ensures that nothing can escape from within theSchwarzschild event horizon. That is an emphaticallyde-terministicstatement. But what about quantum mechan-ics, which is fundamentallyindeterminate?

18.3.1 Deterministic Geodesics and Quantum Uncertainty

• Our discussion to this point has been classical in that it assumesthat free particles follow geodesics appropriate for the space-time.

• But the uncertainty principle implies that microscopic particlescannot be completely localized on classical trajectories becausethey are subject to a spatial coordinate and 3-momentum uncer-tainty of the form∆pi∆xi ≥ h̄;

• Neither can energy conservation be imposed except with an un-certainty∆E∆t ≥ h̄, where∆E is an energy uncertainty and∆t isthe corresponding time period during which this energy uncer-tainty

• This implies an inherentquantum fuzzinessin the 4-momenta as-sociated with our description of spacetime at the quantum level.


For the Killing vector ξ ≡ ξt = (1,0,0,0) in theSchwarzschild metric,

ξ ·ξ = gµνξ µξ ν = g00ξ 0ξ 0 = −(

1− 2Mr

)

︸︷︷︸

changes sign atrs

,

from which we conclude that

ξ is

timelike outside the horizon,since thenξ ·ξ < 0,

spacelike inside the horizon,since thenξ ·ξ > 0.

As we now make plausible, this property ofthe Killing vectorξ

• permits a virtual quantum fluctuation ofthe vacuum to be converted into real par-ticles, and

• these particles are detectable at infinityas emission of mass from the black hole.


Particle-antiparticle

fluctuation

Figure 18.18:Hawking radiation in Kruskal–Szekeres coordinates.

18.3.2 Hawking Radiation

Assume a particle–antiparticle pair created by vacuum fluctuation nearthe horizon of a Schwarzschild black hole, such that the particle andantiparticle end up on opposite sides of the horizon (Fig. 18.18).

• If the particle–antiparticle pair is created in a small enough re-gion of spacetime, there is nothing special implied by this regionlying at the event horizon: the spacetime is indistinguishablefrom Minkowski space because of the equivalence principle.

• Therefore, the normal principles of (special) relativistic quan-tum field theory will be applicable to the pair creation processin a local inertial frame defined at the event horizon (even ifthegravitational field is enormous there).



fluctuation

• In the Schwarzschild geometry, the conserved quantity analo-gous to the total energy in flat space is the scalar product of theKilling vector ξ ≡ ξt = (1,0,0,0) with the 4-momentump.

• Therefore, if a particle–antiparticle pair is produced near thehorizon with 4-momentap and p̄, respectively, the condition

ξ ·p+ξ · p̄= 0,

must be satisfied (to preserve vacuum quantum numbers).

• For the particle outside the horizon,−ξ ·p > 0 (it is proportionalto an energy that is measureable externally).

• If the antiparticle were also outside the horizon,it too musthave−ξ · p̄ > 0, in which case

1. The conditionξ ·p+ξ · p̄= 0 cannot be satisfied.

2. The particle–antiparticle pair can have only a fleeting exis-tence of duration∆t ∼ h̄/∆E (Heisenberg).

• So far, no surprises . . .



fluctuation

HOWEVER . . .

If instead (as in the figure) the antiparticle isinside the horizon,

• The Killing vectorξ is spacelike(ξ ·ξ > 0)

• The scalar product−ξ · p̄ is not an energy(for any observer)

• In fact, the product−ξ · p̄ is a3-momentum component.

• Therefore−ξ · p̄ can bepositive or negative (!!).

There is magic afoot:

• If −ξ · p̄ is negative, thenξ ·p+ξ · p̄= 0 can be satisfied.

• Then the virtual particle created outside the horizon can propa-gate to infinity as areal, detectable particle,while the antiparti-cle remains trapped inside the event horizon.

• The black holeemits its mass as a steady stream of particles (!).


Therefore, quantum effects imply that a black hole canemit its mass as a flux of particles and antiparticles createdthrough vacuum fluctuations near its event horizon. Theemitted particles are termedHawking radiation.


The Hawking mechanism has been described by loose analogy with arather nefarious financial transaction.

• Suppose that I am broke (a money vacuum), but I wish to giveto you a large sum of money (in Euros,E).

• Next door is a bank—Uncertainty Bank and Trust—that has lotsof money in its secure vault but has shoddy lending practices.Then

1. I borrow a large sum of money∆E from the UncertaintyBank, which will take a finite time∆t to find that I have nomeans of repayment.

We hypothesize∆E ·∆t ∼ h, whereh is con-stant, since the bank will be more diligent ifthe amount of money is larger.

2. I transfer the money∆E to your account.

3. I declare bankruptcy within the time∆t, leaving the bank onthe hook for the loan.

A virtual fluctuation of the money vacuum has causedreal money to be emitted (and detected in your distantaccount!) from behind the seemingly impregnable eventhorizon of the bank vault.


18.3.3 Mass Emission Rates and Black Hole Temperature

The methods for obtaining quantitative results from the Hawking the-ory are beyond our scope, but the results can be stated simply:

• The rate of mass emission can be calculated using relativisticquantum field theory:

dMdt

=−λh̄

M2,

whereλ is a dimensionless constant.

• The distribution of energies emitted in the form of Hawkingra-diation is equivalent to ablackbody with temperature

T =h̄c3

8πkBGM= 6.2×10−8

(M⊙M

)

K.

Advanced methods of quantum field theory are required to provethis, but there are suggestive hints from the observations that

1. A black hole acts as a perfect absorber of radiation (as woulda blackbody).

2. Hawking radiation originates in random fluctuations, as wewould expect for a thermal emission process.

• IntegratingdM/dt =−λ h̄/M2 for a black hole assumed to emitall of its mass by Hawking radiation in a timetH, we obtain

M(t) = (3λ h̄(tH− t))1/3

for its mass as a function of time.


TimetH

Mass

Temperature

Ma

ss, Tem

pera

ture

Figure 18.19:Evaporation of a Hawking black hole.

From the results

T =h̄c3

8πkBGMdMdt

=−λh̄

M2 M(t) = (3λ h̄(tH− t))1/3

the mass and temperature of the black hole behave as in Fig. 18.19.Therefore,

• The black hole evaporates at an accelerating rate as it loses mass.

• Both the temperature and emission rate of the black hole tend toinfinity near the end.

• Observationally we may expect a final burst of very high energy(gamma-ray) radiation that would be characteristic of Hawkingevaporation for a black hole.


18.3.4 Miniature Black Holes

How long does it take a black hole to evaporate by the quantum Hawk-ing process? From the mass emission rate formula we may estimatea lifetime for complete evaporation as

M(t) = (3λ h̄(tH− t))1/3 → tH ≃M3

3λ h̄≃ 10−25

(M1g

)3

s.

• A one solar mass black hole would then take approximately1053

times the present age of the Universe to evaporate (with a cor-responding blackbody temperature of order10−7 K). It’s prettyblack!

• However, black holes of initial mass∼ 1014 g or less would haveevaporation lifetimes less than or equal to the present age of theUniverse, and their demise could be detectable through a char-acteristic burst of high-energy radiation.

• The Schwarzschild radius for a1014 g black hole is approxi-mately1.5× 10−14 cm, which is about15 the size of a proton.To form such a black hole one must compress1014 grams(massof a large mountain) into a volume less than the size of a proton!

• Early in the big bang there could have been such densities. There-fore, a population of miniature black holes could have formed inthe big bang and could be decaying in the present Universe witha detectable signature.

• No experimental evidence has yet been found for such minia-ture black holes and their associated Hawking radiation. (Theywould have to be relatively nearby to be seen easily.)


18.3.5 Black Hole Thermodynamics

The preceding results suggest that the gravitationalphysics of black holes and classical thermodynamics areclosely related. Remarkably, this has turned out to be cor-rect.

• It had been noted prior to Hawking’s discovery thatthere were similarities between black holes and blackbody radiators.

• The difficulty with describing a black hole in ther-modynamical terms was that a classical black holepermits no equilibrium with the surroundings(it ab-sorbs but cannot emit radiation).

• Hawking radiation supplies the necessary equilib-rium that ultimately allows thermodynamics and atemperature to be ascribed to a black hole.


Hawking Area Theorem:If A is the surface area (area ofthe event horizon) for a black hole, Hawking proved a the-orem thatA cannot decreasein any physical process in-volving a black hole horizon,

dAdt≥ 0.

• For a Schwarzschild black hole, the horizon area is

A = 16πM2 −→ dAdM

= 32πM,

which can be written (h= Planck, kB = Boltzmann).

dM =1

32πMdA −→ dM =

h8πkBM

d

(kBA4h

)

• But dE = dM is the change in total energy and thetemperature of the black hole isT = h̄/8πkBM in c=G = 1 units.

• Therefore, we may write the preceding as

dE = TdS︸︷︷︸

1st Law

dS≥ 0︸︷︷︸

2nd Law

S≡ kB

4hA

︸︷︷︸

entropy

which are just the 1st and 2nd laws of thermodynam-ics if

S∝ (surface area (!) of the black hole)

is interpreted as entropy!


Evaporation of a black hole through Hawking radiationappears to violate the Hawking area theorem in that theblack hole eventually disappears. However,

• The area theorem assumes that a local observer al-ways measures positive energy densities and thatthere are no spacelike energy fluxes.

• As far as is known these are correct assumptions atthe classical level, but they may break down in quan-tum processes.

• The correct quantum interpretation of Hawking radi-ation and the area theorem is that

1. The entropy of the evaporating, isolated blackhole decreaseswith time (because it is propor-tional to the area of the horizon).

2. The total entropy of the Universeincreasesbe-cause of the entropy associated with the Hawk-ing radiation itself.

3. That is, the area theorem is replaced by

Generalized 2nd Law:The total entropy ofthe black hole plus exterior Universe maynever decrease in any process.


18.3.6 Gravity and Quantum Mechanics: the Planck Scale

The preceding results for Hawking radiation are derived assumingthat the spacetime in which the quantum calculations are done isa fixed background that is not influenced by the propagation oftheHawking radiation.

• This approximation is expected to be valid as long asE << M,whereE is the average energy of the Hawking radiation andMis the mass of the black hole.

• This approximation breaks down on a scale given by thePlanckmass

MP =

(h̄cG

)1/2

= 1.2×1019 GeV c−2 = 2.2×10−8 kg.

• For a black hole of this mass, the effects of gravity become im-portant even on a quantum (h̄) scale, requiring a theory ofquan-tum gravity.

• We don’t yet have an adequate theory of quantum gravity.

• Note that near the endpoint of Hawking black hole evaporationone approaches the Planck scale.

• Therefore, we do not actually know yet what happens at the con-clusion of Hawking evaporation for a black hole.

18.4. THE KERR SOLUTION: ROTATING BLACK HOLES 845

18.4 The Kerr Solution: Rotating Black Holes

The Schwarzschild solution corresponds to the simplest black hole,which can be characterized by a single parameter, the massM.

• Other solutions to the field equations of general relativity permitblack holes with more degrees of freedom.

The most general black hole can possess

1. Mass

2. Charge

3. Angular momentum

as distinguishing quantities.

• Of particular interest are those solutions where we relax the re-striction to spherical symmetry and thus permit the black holeto be characterized by angular momentum in addition to mass(Kerr black holes).

• Rotating black holes are of particular interest in astrophysics be-cause they are thought to power

– quasars and other active galaxies,

– X-ray binaries, and

– gamma-ray bursts.

• Unlike for Schwarzschild black holes,energy and angular mo-mentum can be extractedfrom a (classical) rotating black hole.


18.4.1 The Kerr Metric

The Kerr metric corresponds to the line element

ds2 =−(

1− 2Mrρ2

)

dt2− 4Mrasin2θρ2 dϕdt+

ρ2

∆dr2+ρ2dθ2

+

(

r2+a2+2Mra2sin2θ

ρ2

)

sin2θdϕ2

with the definitions

a≡ J/M ρ2≡ r2+a2cos2θ ∆≡ r2−2Mr +a2.

• The coordinates(t, r,θ ,ϕ) are calledBoyer–Lindquist coordi-nates.

• The parametera, termed theKerr parameter,has units of lengthin geometrized units.

• The parameterJ will be interpreted as angular momentum andthe parameterM as the mass for the black hole.


The Kerr line element

ds2 =−(

1− 2Mrρ2

)


ρ2

∆dr2+ρ2dθ2

+

(

r2+a2+2Mra2sin2θ

ρ2

)

sin2θdϕ2

a≡ J/M ρ2≡ r2+a2cos2θ ∆≡ r2−2Mr +a2.

has the following properties.

• Vacuum solution:the Kerr metric is a vacuum solution of theEinstein equations, valid in the absence of matter.

• Reduction to Schwarzschild metric:If the black hole is not rotat-ing (a= J/M = 0), the Kerr line element reduces to the Schwarzschildline element.

• Asymptotically flat:The Kerr metric becomes asymptotically flatfor r >> M andr >> a.

• Symmetries:The Kerr metric is independent oft andϕ, implyingthe existence of Killing vectors

ξt = (1,0,0,0) (stationary metric)

ξϕ = (0,0,0,1) (axially symmetric metric).

Unlike the Schwarzschild metric, the Kerr metric has only axialsymmetry.

• The metric has off-diagonal terms

g03 = g30 =−2Mrasin2θρ2 (inertial frame dragging).


For the Kerr line element

ds2 =−(

1− 2Mrρ2

)


ρ2

∆dr2+ρ2dθ2

+

(

r2+a2+2Mra2sin2θ

ρ2

)

sin2θdϕ2

a≡ J/M ρ2≡ r2+a2cos2θ ∆≡ r2−2Mr +a2.

• Surfaces of constant Boyer–Lindquist coordinatesr andt do nothave the metric of a 2-sphere.

• Singularity and horizon structure:∆→ 0 at

r± = M±√

M2−a2

andds→ ∞, assuminga≤M.

1. This is a coordinate singularity.

2. As a→ 0 we find thatr+→ 2M, which coincides with theSchwarzschild coordinate singularity.

3. Thusr+ corresponds to the horizonthat makes the Kerr so-lution a black hole.

On the other hand, the limitρ→ 0 corresponds to aphysical sin-gularity with associated infinite components of spacetime curva-ture, similar to the case for the Schwarzschild solution.


18.4.2 Extreme Kerr Black Holes

• The horizon radius

r± = M±√

M2−a2

exists only fora≤M.

• Thus, there is a maximum angular momentumJmax for a Kerrblack hole sincea = J/M andamax = M,

Jmax = amaxM = M2.

• Black holes for whichJ = M2 are termedextreme Kerr blackholes.

• Near-extreme black holes may develop in many astrophysicalsituations:

1. Angular momentum transfer through accretion disks in ei-ther binary systems or around supermassive black holes ingalaxy cores tends to spin up the central object.

2. Massive stars collapsing to black holes may have significantinitial angular momentum


18.4.3 Cosmic Censorship

That the horizon exists for a Kerr black hole only underrestricted conditions raises the question of whether a sin-gularity could exist in the absence of a horizon (“nakedsingularity”).

Cosmic Censorship Hypothesis:Nature conspires to “cen-sor” spacetime singularities in thatall such singularitiescome with event horizonsthat render them invisible to theoutside universe.

• No known violations (observationally or theoreti-cally).

• It cannot at this point be derived from any more fun-damental concept and must be viewed as only an hy-pothesis.

• All theoretical attempts to add angular momentum toa Kerr black hole in order to cause it to exceed themaximum angular momentum permissable for exis-tence of the horizon have failed.


18.4.4 The Kerr Horizon

The area of the horizon for a black hole is of consider-able importance because of the area theorem, which statesthat the horizon area of a classical black hole can neverdecrease in any physical process.

• The Kerr horizon corresponds to a constant value ofr = r+.

• Since the metric is stationary, the horizon is also asurface of constantt.

• Settingdr = dt = 0 in the Kerr line element gives theline element for the 2-D horizon,

dσ2 = ρ2+dθ2+

(2Mr+

ρ+

)2

sin2θdϕ2,

whereρ2+ is defined byρ2≡ r2

+ +a2cos2θ

• This isnot the line interval of a 2-sphere.

Thus the Kerr horizon has constant Boyer–Lindquist co-ordinater = r+ but it is not spherical.


From the line element

dσ2 = ρ2+dθ2+

(2Mr+

ρ+

)2

sin2θdϕ2,

the metric tensor for the Kerr horizon is

g =

ρ2+ 0

0

(2Mr+

ρ+

)2

sin2θ

The area of the Kerr horizon is then

AK =

∫ 2π

0dϕ∫ π

0

√

detg dθ

= 2Mr+

∫ 2π

0dϕ∫ π

0sinθ dθ

= 8πMr+ = 8πM(

M +√

M2−a2)

.

The horizon area for a Schwarzschild black hole is ob-tained by settinga = 0 in this expression, correspondingto vanishing angular momentum, in which caser+ = 2Mand

AS = 16πM2 = 4π(2M)2,

as expected for a spherical horizon of radius2M.


18.4.5 Orbits in the Kerr Metric

We may take a similar approach as in the Schwarzschild metrictodetermine the orbits of particles and photons.

• In the Kerr metric orbits are not confined to a plane (only axial,not full spherical symmetry).

• To simplify, we shall consider only motion in the equatorialplane (θ = π

2) to illustrate concepts.

The Kerr line element with that restriction is then

ds2 =−(

1− 2Mr

)

dt2− 4Mar

dϕdt

+r2

∆dr2+

(

r2+a2+2Ma2

r

)

dϕ2.

There are two conserved quantitities,

ε =−ξt ·u ℓ = ξϕ ·u,

At large distances from the gravitational source

• ε may be interpreted as the conserved energy per unit rest mass.

• ℓ may be interpreted as the angular momentum component perunit mass along the symmetry axis.

The corresponding Killing vectors are

ξt = (1,0,0,0) ξϕ = (0,0,0,1).

and the norm of the 4-velocity provides the usual constraint

u·u =−1.


From the Kerr metric

−ε = ξt ·u = g00u0+g03u

3 ℓ = ξϕ ·u = g30u0+g33u

3,

and solving foru0 = dt/dτ andu3 = dϕ/dτ gives

dtdτ

=1∆

[(

r2+a2+2Ma2

r

)

ε− 2Mar

ℓ

]

dϕdτ

=1∆

[(

1− 2Mr

)

ℓ+2Ma

rε]

.

By similar steps as for the Schwarzschild metric, this leadsto theequations of motion

ε2−12

=12

(drdτ

)2

+Veff(r,ε, ℓ)

Veff(r,ε, ℓ)≡−Mr

+ℓ2−a2(ε2−1)

2r2 −M(ℓ−aε)2

r3 ,

which are valid in the equatorial plane.


18.4.6 Frame Dragging

A striking feature of the Kerr solution isframe dragging:loosely, theblack hole drags spacetime with it as it rotates.

• This arises ultimately because the Kerr metric contains off-diagonalcomponentsg03 = g30.

• One consequence is that a particle dropped radially onto a Kerrblack hole will acquire non-radial components of motion as itfalls freely in the gravitational field.

Let’s investigate by calculatingdϕ/dr for a particle dropped from rest(ε = 1) with zero angular momentum (ℓ = 0) onto a Kerr black hole(in the equatorial plane). Sinceε = 1 andℓ = 0 are conserved on thegeodesic, from earlier equations,

dϕdτ

=1∆

[(

1− 2Mr

)

ℓ+2Ma

rε]

−→ dϕdτ

=1∆

(2Ma

r

)

,

ε2−12

=12

(drdτ

)2

+Veff(r,ε, ℓ) −→ drdτ

=

√

2Mr

(

1+a2

r2

)

,

which may be combined to give

dϕdr

=dϕ/dτdr/dτ

=−2Mar∆

[2Mr

(

1+a2

r2

)]−1/2

.

The particle is dragged in angleϕ as it falls radially in-ward,even though no forces act on it.


This effect is termed“dragging of inertial frames” and produces agyroscopic precession termed theLense–Thirring effect.

NASA’s Gravity Probe B used 4 gyroscopes aboard asatellite in an orbit almost directly over the poles to lookfor gyroscopic precession giving direct evidence of framedragging by the rotating gravitational field of the Earth.

In the Kerr metric

pϕ ≡ p3 = g3µ pµ = g33p3+g30p0,

pt ≡ p0 = g0µ pµ = g00p0+g03p3,

p0≡mdtdτ

p3≡mdϕdτ

,

and combining these relations gives an expression fordϕ/dt,

dϕdt

=dϕ/dτdt/dτ

=p3

p0 =g33p3+g30p0

g00p0+g03p3.

If we consider a zero angular momentum particle (in the equatorialplane), thenp3 = 0 and

ω(r,θ)≡ dϕdt

=g30

g00 =gϕt

gtt .

ω(r,θ) is termed theangular velocity of a zero angularmomentum particle; it measures the frame dragging.


The Kerr line element defines the covariant components of theKerrmetricgµν . To evaluate

ω(r,θ) =gϕt

gtt

we needgµν , which may be obtained as the matrix inverse ofgµν .The metric is diagonal inr andθ so

grr = g−1rr =

∆ρ2 gθθ = g−1

θθ =1

ρ2,

and we need only evaluate

g−1 =

(

gtt gtϕ

gϕt gϕϕ

)−1

to obtain the other non-zero entries forgµν . Letting

D = detg = gttgϕϕ− (gtϕ)2,

the matrix inverse is

g−1 =1D

(

gϕϕ −gtϕ

−gϕt gtt

)

.

Inserting explicit expressions for thegµν and carrying out some alge-bra yields

ω(r,θ) =gϕt

gtt =2Mra

(r2+a2)2−a2∆sin2θ.


18.4.7 Innermost Stable Circular Orbit and Binding Energy

The property of particle orbits in the Kerr metric that isprobably of most interest in astrophysics is thebindingenergy of the innermost stable circular orbit,since this isrelated to the energy that can be extracted from accretionon a rotating black hole.

If we denote the radius of the innermost stable orbit byR, for circularorbits we have thatdr/dτ = 0 and from earlier

ε2−12

=12

(drdτ

)2

+Veff(r,ε, ℓ) −→ ε2−12

= Veff(R,ε, ℓ).

Furthermore, to remain circular the radial acceleration must vanish,

∂Veff

∂ r

∣∣∣∣r=R

= 0,

and to be a stable orbit the potential must be a minimum, whichre-quires the second derivative to be positive,

∂ 2Veff

∂ r2

∣∣∣∣r=R≥ 0,

with equality holding for the last stable orbit.

This set of equations may be solved to determine the in-nermost stable orbit and its binding energy.


For extremal black holes (a = M), one finds that

1. Co-rotating orbits (accretion orbits revolving in thesame sense as the rotation of the hole) are more stablethan the corresponding counter-rotating orbits.

2. For co-rotating orbits,

ε =1√3

ℓ =2M√

3R= M.

3. The quantityε is the energy per unit rest mass mea-sured at infinity, so the binding energy per unit restmass,B/M, is given by

BM

= 1− ε.

4. These equations imply that the fractionf of the restmass that could theoretically be extracted is

f = 1− ε = 1− 1√3≃ 0.42.

for a transition from a distant unbound orbit to theinnermost circular bound orbit of a Kerr black hole.


Therefore, in principle 42% of the rest mass of accretedmaterial could be extracted as usable energy by accretiononto an extreme Kerr black hole.

• One expects less efficiency than this for actual situa-tions, but optimistically one might expect as much as20–30% efficienciesin realistic accretion scenarios.

• Compare with the maximum theoretical efficiency of6% for accretion on a spherical black hole.

• Compare with the0.7% efficiency for hydrogen fu-sion in the conversion of rest mass to usable energy.

Accretion onto rotating black holes is averyefficient mechanismfor converting mass toenergy.


18.4.8 The Ergosphere

• With capable propulsion, an observer could remain stationaryat any point outside the event horizon of a Schwarzschild blackhole.

• No amount of propulsion can enable an observer to remain sta-tionary inside a Schwarzschild horizon (causality violation).

• We now demonstrate that, for a rotating black hole, even out-side the horizon it may be impossible for an observer to remainstationary.


For a stationary observer

uµobs= (u0

obs,0,0,0) =

(dtdτ

,0,0,0

)

.

Writing out the conditionuobs·uobs =−1 for the Kerr metric gives

uobs·uobs = g00(u0obs)

2 =−1

But

g00=−(

1− 2Mrρ2

)

=−(

1− 2Mrr2+a2cos2θ

)

=−(

r2+a2cos2θ −2Mrr2+a2cos2θ

)

vanishes ifr2−2Mr +a2cos2θ = 0

Generally then, solving the above quadratic forr

• g00 = 0 on the surface defined by

re(θ) = M +√

M2−a2cos2θ ,

• g00 > 0 inside this surface.

• g00 < 0 outside this surface.

Therefore, since(u0obs)

2 must be positive,the condition

uobs·uobs = g00(u0obs)

2︸︷︷︸

+

=−1

cannot be satisfiedfor any observer havingr ≤ re(θ)(since theng00 > 0).


r+

Cutaway "Side View"

Ergosphere

Black HoleBlack Hole

re(θ)

"Top View"

θ

Horizon

g00

> 0

g00

> 0g

00 < 0

Figure 18.20:The ergosphere of a Kerr black hole. Note that the horizon lies at aconstant value ofr but it is not a spherical surface.

Comparing

r+ = M +√

M2−a2 re(θ) = M +√

M2−a2cos2θ ,

we see that

• If a 6= 0 the surfacere(θ) generally lies outside the horizonr+,except at the poles, where the two surfaces are coincident (Fig-ure 18.20).

• If a = 0, the Kerr black hole reduces to a Schwarzschild blackhole, in which caser+ andre(θ) define the same surface.

The region lying betweenre(θ) and the horizonr+ istermed theergosphere.


From preceding considerations, there can beno stationaryobservers within the ergosphere.Further insight comesfrom considering the motion of photons within the ergo-sphere.

• Assume photons within the ergosphere moving tangent to a cir-cle at constantr in the equatorial planeθ = π

2 , sodr = dθ = 0.

• Since they are photons,ds2 = 0 and from the Kerr line elementwith dr = dθ = 0,

ds2 = 0 → g00dt2+2g03dtdϕ +g33dϕ2 = 0.

• Divide both sides bydt2 to give a quadratic equation indϕ/dt

g00+2g03dϕdt

+g33

(dϕdt

)2

= 0

Thus, solving the quadratic fordϕ/dt

dϕdt

=−2g03±

√

4g203−4g33g00

2g33

=−g03

g33±

√(

g03

g33

)2

− g00

g33,

where

··· + is associated with motion opposite black hole rotation.

··· − is associated with motion with black hole rotation.


• Now g00

– vanishes at the boundaryof the ergosphere and

– is positive inside the boundary.

• Settingg00 = 0 in

dϕdt

=−g03

g33±

√(

g03

g33

)2

− g00

g33,

gives two solutions,

dϕdt

= 0︸︷︷︸

opposite rotation

dϕdt

=−2g03

g33︸︷︷︸

with rotation

,

The photon sent backwards against the holerotation at the surface of the ergosphere is sta-tionary inϕ!

• Obviously a particle, which must have a velocity less than apho-ton, must rotate with the black hole irrespective of the amountof angular momentum that it has.

• Inside the ergosphereg00 > 0, so all photons and particles mustrotate with the hole.

In essence, the frame dragging forr < re(θ) is so severethatspeeds in excess of lightwould be required for an ob-server to remain at rest with respect to infinity.


r+

Cutaway "Side View"

Ergosphere

Black HoleBlack Hole

re(θ)

"Top View"

θ

Horizon

g00

> 0

g00

> 0g

00 < 0

• In the limit a = J/M→ 0, where the angular momentum of theKerr black hole vanishes,

re(θ) = M +√

M2−a2cos2θ → 2M

and becomescoincident with the Schwarzschild horizon.

• Thus, rotation

– has extended the regionr < 2M of the spherical black holewhere no stationary observers can exist to

– a larger regionr < re(θ) surrounding the rotating black holewhereno observer can remain at rest because of frame drag-ging effects.

• This ergospheric region

– lies outsidethe horizon, implying that

– a particle could enter it andstill escapefrom the black hole.


18.4.9 Extraction of Rotational Energy from Black Holes

We have seen that quantum vacuum fluctuations allowmass to be extracted from a Schwarzschild black hole asHawking radiation, but it is impossible to extract massfrom a classical Schwarzschild black hole (it all lieswithin the event horizon).

• However, the existence of separate surfaces defining the ergo-sphere and the horizon for a Kerr black hole implies the possi-bility of extracting rotational energy from aclassicalblack hole.

• The simplest way to demonstrate the feasibility of extracting ro-tational energy from a black hole is through aPenrose process.


E0

E1

E2

Black

Hole

Ergosphere

g00

> 0

g00

< 0

Figure 18.21:A Penrose process.

A Penrose process is illustrated schematically in Fig. 18.21.

• A particle falls into the ergosphere of a Kerr black hole andde-cays into two particles. One falls through the horizon; one exitsthe ergosphere and escapes to infinity.

• The decay within the ergosphere is a local process. By equiva-lence principle arguments, it may be analyzed in a freely fallingframe according to the usual rules of scattering theory.

• In the decay 4-momentump is conserved

p0 = p2 + p1

(NOTE: subscripts label particles, not components!)

• p = (−E, ppp) and ξt = (1,0,0,0), so E = −p·ξt . Thus, if theparticle 2 scattering to infinity has rest massm2, its energy is

E2 =−p2·ξt =−m2

m2p2·ξt =−m2 p2

ξt

m2︸︷︷︸−ε

= m2ε

whereε =−ξt ·u =−ξt ·(p/m) has been used.


E0

E1

E2

Black

Hole

Ergosphere

g00

> 0

g00

< 0

• Taking the scalar product ofp0 = p2+ p1 with ξt

p0 ·ξt = (p2 + p1) ·ξt

yields the requirement,

E2 = E0−E1.

sinceE =−ξt ·p.

• If particle 1 were to reach∞ instead of crossing the event hori-zon of the black hole,E1 would necessarily be positive so

1. E2 < E0

2. Less energy would be emitted than put in.


E0

E1

E2

Black

Hole

Ergosphere

g00

> 0

g00

< 0

However . . .

• For the Killing vectorξt = (1,0,0,0),

ξt ·ξt = gµνξ µt ξ ν

t

= g00ξ 0t ξ 0

t = g00 =−(

1− 2Mrρ2

)

,

andg00 vanishes on the surfacere(θ) and is positive inside it.

• Therefore,within the ergosphereξt is a spacelike vector:

ξt ·ξt = g00 > 0 (within the ergosphere).

• By arguments similar to those concerning Hawking radiation,

1. −E1 is not an energy within the ergosphere but is a compo-nent of spatial momentum, which can have either apositiveor negative value (!).

2. For those decays where the trajectories are arranged suchthat E1 < 0, we obtain fromE2 = E0−E1 that E2 > E0 andnet energy is extractedin the Penrose process.


The energy extracted in the Penrose process comes at theexpense of the rotational energy of the Kerr black hole.

• For trajectories withE1 < 0, the captured particleadds a negative angular momentum, thus reducingthe angular momentum and total energy of the blackhole to just balance the angular momentum and totalenergy carried away by the escaping particle.

• A series of Penrose events could extract all the an-gular momentum of a Kerr black hole, leaving aSchwarzschild black hole.

• No further energy can then be extracted from theresulting spherical black hole (except by quantumHawking processes).


The Penrose mechanism establishesproof of principleina simple model that the rotational energy of Kerr blackholes is externally accessible in classical processes.

• Practically, Penrose processes are not likely to be im-portant in astrophysics because the required condi-tions are not easily realized.

• Instead, the primary sources of emitted energy fromblack hole systems likely come from

1. Complex electromagnetic coupling of rotatingblack holes to external accretion disks and jets.

2. Gravitational energy released by accretion ontoblack holes.

There is very strong observational evidence for such pro-cesses in a variety of astrophysical environments.

18.5. BLACK HOLE THEOREMS AND CONJECTURES 873

18.5 Black Hole Theorems and Conjectures

In this section we summarize (in a non-rigorous way) a set of theo-rems and conjectures concerning black holes. Some we have alreadyused in various contexts.

• Singularity theorems:Loosely, any gravitational collapse thatproceeds far enough results in a spacetime singularity.

• Cosmic censorship conjecture:All spacetime singularities arehidden by event horizons (no naked singularities).

• (Classical) area increase theorem:In all classical processes in-volving horizons, the area of the horizons can never decrease.

• Second law of black hole thermodynamics:Where quantum me-chanics is important the classical area increase theorem isre-placed by

1. The entropy of a black hole is proportional to the surfacearea of its horizon.

2. The total entropy of the Universe can never decrease an anyprocess.


• The no-hair theorem/conjecture:If gravitational collapse to ablack hole is nearly spherical,

– All non-spherical parts of the mass distribution (quadrupolemoments, . . . ) except angular momentum are radiated awayas gravitational waves.

– Horizons eventually become stationary.

– A stationary black hole is characterized by three numbers:the massM, the angular momentumJ, and the chargeQ.

– M, J, andQ are all determined by fieldsoutsidethe horizon,not by integrals over the interior.

The most general solution characterized byM, J, andQ istermed aKerr–Newman black hole.However,

– It is likely that the astrophysical processes thatcould form a black hole would neutralize any excesscharge.

– Thus astrophysical black holes are Kerr black holes(the Schwarzschild solution being a special case ofthe Kerr solution for vanishing angular momentum).

The “No Hair Theorem”:black holes destroy all details(the hair) about the matter that formed them, leaving be-hind only M, J, and possiblyQ as observable externalcharacteristics.

18.5. BLACK HOLE THEOREMS AND CONJECTURES 875

• Birkhoff’s theorem:The Schwarzschild solution is theonlyspher-ically symmetric solution of the vacuum Einstein equations. (Thestatic assumption is, in fact, a consequence of the spherical sym-metry assumption.)

These theorems and conjectures place the mathematics of black holeson reasonably firm ground. To place thephysicsof black holes onfirm grounds, these ideas must be tested by observation, to which wenow turn.


18.6 Observation of Stellar Black Holes

We have very strong reasons to believe that black holesexist and are being observed indirectly because of

1. Unseen massive companions in binary star systemsthat are

• Strong sources of X-rays

• Probably too massive to be anything other thanblack holes.

2. The centers of many galaxies where

• Masses inferred by virial theorem methods (oreven direct measurement of individual star or-bits in the center of the Milky Way) are far toolarge to be accounted for by any simple hypoth-esis other than a supermassive black hole.

• In many cases there is direct evidence of an enor-mous energy source in the center of the galaxy.

We first summarize some of the reasons why the firstclass of observations give us strong confidence thatblack holes, or at least objects that have many of thefeatures of black holes, exist.

18.6. OBSERVATION OF STELLAR BLACK HOLES 877

-1 -0.5 0 0.5 1

Orbital Phase

-500

0

+500

Helio

centr

ic R

adia

l V

elo

city (

km

/s)

A 0620 - 00

P

K

Figure 18.22:Velocity curve for the black hole binary candidate A 0620–00.

18.6.1 Black Hole Masses in X-ray Binaries

From Kepler’s laws, the mass functionf (M) for the binary may berelated to the radial velocity curve (Fig. 18.22) through

f (M) =(M sini)3

(M +Mc)2 =PK3

2πG

= 0.1036

(P

1 day

)(K

100 km s−1

)3

M⊙

where

• K is the semiamplitude andP the period of the radial velocity.

• i is the tilt angle of the orbit

• Mc is the mass of the companion star.

• M is the mass of the black hole


Table 18.1: Some black hole candidates in galactic X-ray binariesX-ray source Period (days) f (M) Mc(M⊙) M(M⊙)

Cygnus X-1 5.6 0.24 24–42 11–21V404 Cygni 6.5 6.26 ∼0.6 10–15GS 2000+25 0.35 4.97 ∼0.7 6–14H 1705–250 0.52 4.86 0.3–0.6 6.4–6.9GRO J1655–40 2.4 3.24 2.34 7.02A 0620–00 0.32 3.18 0.2–0.7 5–10GS 1124–T68 0.43 3.10 0.5–0.8 4.2–6.5GRO J0422+32 0.21 1.21 ∼0.3 6–144U 1543–47 1.12 0.22 ∼2.5 2.7–7.5

Because the anglei is often not known,

• The measured mass function places a lower limit onthe mass of the unseen component of the binary if themass of the companion star is known.

• The mass of the companion can often be estimatedreliably from systematic spectral features.

Table 18.1 illustrates some candidate binary star systems where amass function analysis suggests an unseen companion too massiveto be a white dwarf or neutron star. We assume these to be blackholes.

18.6. OBSERVATION OF STELLAR BLACK HOLES 879

18.6.2 Example: Cygnus X-1

The first strong case found, and most famous stellar black hole candi-date, is Cygnus X-1 .

1. In the early 1970s an X-ray source was discovered in Cygnusand designated Cygnus X-1.

2. In 1972, a radio source was found in the same general area andidentified optically with a blue supergiant star called HDE226868.

3. Correlations in radio activity of HDE226868 and X-ray activityof Cygnus X-1 implied that the two were probably componentsof the same binary system.

4. Doppler measurements of the radial velocity of HDE226868andother data confirmed that it was a member of a binary with aperiod of 5.6 days.

5. Detailed analysis showed that the X-ray source was fluctuatingin intensity on timescales as short as 1/1000 of a second.

• Signals controlling the fluctuation are limited by light speed,which implies that the X-ray source must be very compact,probably no more than hundreds of kilometers in diameter.

• The

– small size,

– orbital perturbation on HDE226868, and

– strong X-ray emission,

indicate that Cygnus X-1 is a compact object (white dwarf,neutron star, or black hole, with the latter two possibilitiesmore likely).


5. The mass of the blue supergiant HDE226868 was estimated fromknown properties of such stars (it is a spectrum and luminosityclass O9.7Iab star).

• This, coupled with a mass function analysis, can be used toestimate the mass of the unseen, compact companion.

• These estimates are uncertain because the geometry (tilt ofthe binary orbit) can only partially be inferred from data.

• However, all such estimates place a lower limit of about5–6 M⊙ on the unseen companion and more likely indicate amass near10M⊙.

6. Since we know of no conditions that would permit a neutron starto exist above about 2–3 solar masses (or a white dwarf aboveabout 1.4 solar masses), we conclude that the unseen companionmust be a black hole.

Although this chain of reasoning is indirect, it builds avery strong case that Cygnus X-1 contains a black hole.

18.7. SUPERMASSIVE BLACK HOLES IN THE CORES OF GALAXIES881

18.7 Supermassive Black Holes in the Cores of Galaxies

Many observations of star motion near the centers ofgalaxies indicate the presence of large, unseen mass con-centrations. The most direct is for our own galaxy.


Sgr A*

1992.231994.32 1995.53

1996.25

1996.43

1997.54

1998.36

1999.47

2000.47

2001.50

2002.252002.33

2002.40

2002.50

2002.58

2002.66

Right Ascension (arcsec)

Declin

ation (

arc

sec)

0.15 0.10 0.05 0 -0.05 -0.10

0

0.05

0.10

0.15

0.20

Period: 15.2 years

Inclination: 46 Degrees

Eccentricity: 0.87

Semimajor axis: 0.119"

Closest approach to the

black hole: 17 light hours

Figure 18.23:Orbit of the star S2 near the radio source SGR A∗.

18.7.1 The Black Hole at the Center of the Milky Way

The star denoted S2 is a 15 solar mass main sequence star that orbitsin the vicinity of the strong radio source SGR A∗, which is thought tolie very near the center of the Milky Way.

• The orbit has been measured precisely using adaptive optics andspeckle interferometry at near-IR wavelengths (penetratedust).

• The orbit of S2 is shown in Fig. 18.23, with dates shown in frac-tions of a year beginning in 1992.

• The orbit corresponds to the projection of the best-fit ellipse withSGR A∗ at a focus. The closest approach for the orbit corre-sponds to a distance of about 17 light hours from SGR A∗.


2 x 107

1 x 107

5 x 106

2 x 106

1 x 106

5 x 105

0.001 0.01 0.1 1 10

Radius from SgrA* (pc)

En

clo

se

d m

ass (

so

lar

ma

sse

s)

Figure 18.24:Mass distribution near Sag A∗ as inferred from stellar motion.

• From detailed fits to the Kepler orbit of S2, the mass inside theorbit of the star is estimated to be2.6×106M⊙ (see Fig. 18.24).

• From the size of the orbit, this mass is concentrated in a regionthat cannot be much larger than the Solar System, with littlelu-minous mass in that region.

• By far the simplest explanation is that the radio source SGRA∗

coincides with a nearly 3 million solar mass black hole.

• The innermost point in Fig. 18.24 is derived from the observedelliptical orbit for the star S2 that comes within 17 light hours ofthe black hole

– This distance is still well outside the tidal distortion radiusfor the star (which is about 16 light minutes).

– It is about 2100 times larger than the event horizon.

– At closest approach the separation from the black hole is notmuch larger than the radius of the Solar System.


18.7.2 The Water Masers of NGC 4258

The galaxy NGC 4258 (also called M106) is an Sb spiralvisible through a small telescope, but its nucleus is mod-erately active and it is also classified as a Seyfert 2 ActiveGalactic Nucleus (AGN). It lies in the constellation CanesVenatici (very near the Big Dipper) at a distance of around20 Mpc. This makes it one of the nearest AGNs.

• A set of masers has been observed in the central re-gion of the galaxy. (Masers are the microwave analogof a visible-light laser.) The maser emission in NGC4258 is due to clouds of heated water vapor, so theseare termed water masers.

• Because masers produce sharp spectral lines (allow-ing precise Doppler shifts), and because microwavesare not strongly attenuated by the gas and dust nearthe nucleus of the galaxy, observation of the watermasers has permitted the motion of gas near the cen-ter to be mapped very precisely using the Very LongBaseline Array (VLBA).


Figure 18.25:Water masers in NGC 4258 and evidence for a central black hole.

Water maser emission from the central region of NGC 4258 implies(Fig. 18.25).

• The masers are mapped with a precision of better than 1 milli-arcsecond and their radial velocities indicate bulk motionof thegas within several light years of the core at velocities approach-ing 1000 km s−1.

• In Fig. 18.25 masers approaching us (blue shift) are shown inblue, masers receding from us (red shift) are shown in red, andthose with no net radial velocity are shown in green. From thiswe may infer the general rotation of the gas disk, as indicated bythe arrow labeled “Keplerian motion”.


• The masers are embedded in a thin, warped, dusty, moleculargas disk revolving around the core.

• The motion of the masers is Keplerian→masers in orbit arounda large mass that is completely contained within all their orbits.

• The mass obtained is approximately3.5×106M⊙.

• Measured mass and measured size of region enclosed by maserorbits implies a minimum density of108M⊙ per cubic light year.

– 10,000 times more dense than any known star cluster.

– If such a star cluster existed, calculations indicate that colli-sions of the stars would cause the stars of the cluster to driftapart or mutually collapse to a huge black hole.

– Only plausible explanation is a supermassive black hole.


• The nucleus of the galaxy produces radio jets that appear tocome from the dynamical center of the rotating disk and are ap-proximately perpendicular to it (see fig).

• The position of the radio jets and precise location of the center ofthe disk determines the location of the central black hole enginewithin the uncertainty of the black circle shown in the figure.

– This black circle denotes the uncertainty in location of theblack hole, not its size.

– The black circle is about 0.05 ly in diameter, but the super-massive black hole would have an event horizon hundredsof times smaller than this.


These results taken together make NGC 4258 one of thestrongest cases known for the presence of supermassiveblack hole engines at the cores of active galactic nuclei.


18.7.3 Virial Methods and Central Masses

For distant galaxies we have insufficient telescopic resolu-tion to track individual stars but we may still learn some-thing about the mass contained within particular regionsby observing the average velocities of stars in that region.

• On conceptual grounds, we may expect that thelarger the gravitational field that stars feel, the fasterthey will move.

• This intuitive idea may be quantified by using thevirial theorem to show that the massM responsi-ble for the gravitational field in which stars move insome spherical region of radiusR is given by

M ≃ 5Rσ2r

G,

where the radial velocity dispersion

σ2r = 〈v2

r 〉 ≡1

3N

N

∑i=1

v2i

can be determined by averaging over the squares ofradial velocity fields determined from Doppler-shiftmeasurements.


Figure 18.26:Evidence for a central black hole in the galaxy M87.

18.7.4 Evidence for a Supermassive Black Hole in M87

The left portion of Fig. 18.26 is a Hubble Space Telescope imageshowing the center of the giant elliptical radio galaxy M87.

• The diagonal line emanating from the nucleus in the left image isa jet of high-speed electrons—a synchrotron jet—approximately6500 light years (2 kpc) long.

• This is what would be expected for matter swirling around thesupermassive black hole, with part of it falling forever into theblack hole and part of it being ejected in a high-speed jet.

• The right side of Fig. 18.26 illustrates Doppler shift measure-ments made on the central region of M87 that suggest rapid mo-tion of the matter near the center.


• The measurement was made by studying how the light from thedisk is redshifted and blueshifted by the Doppler effect.

• The gas on one side of the disk is moving away from Earth at aspeed of about 550 kilometers per second (redshift). The gasonthe other side of the disk is approaching the Earth at the samespeed (blueshift).

• These high velocities suggest a gravitational field produced by ahuge mass concentration at the center of M87.

• By applying the virial theorem to the measured radial velocitieswe infer that approximately 3 billion solar masses are concen-trated in a region at the galactic core that is only about the sizeof the Solar System.

• This inferred mass is far larger than could be accounted forbythe visible matter there and the simplest interpretation isthat a 3billion solar mass black hole lurks in the core of M87.


18.8 Summary: A Strong But Circumstantial Case

The evidence cited in this chapter is not yet iron-clad proofof the existence of black holes.

• The defining characteristic of a black hole is an eventhorizon.

• No known black hole candidates are near enough topermit imaging an angular region the size of an eventhorizon with present instrumentation.

• (However, angular resolution sufficient to resolve theevent horizon of the Milky Way’s suspected centralblack hole appears at least technically feasible on a2-decade timescale.)

Nevertheless, data on X-ray binary systems and on themotions of stars and other objects in the central regionof our galaxy and other nearby ones provide extremelystrong circumstantial evidence for the existence of blackholes (or at least of objects very much like black holes).

lecture ch18

Documents