Download Hawking Radiation and Black Hole Evaporation

Hawking Radiation and
Black Hole Evaporation
Wolfram Schmidt
www.astro.uni-wuerzburg.de/∼schmidt
ART Vorlesung, 9 February 2005 – p.1/21
Overview
Classical black hole mechanics
The quantum vacuum in Minkowski spacetime
The Unruh effect
Hawking radiation
Black hole entropy
ART Vorlesung, 9 February 2005 – p.2/21
The Zeroth Law
✦ Zeroth Law of Black Hole Mechanics
The surface gravity κ is constant at the horizon of a
stationary black hole
✦ Zeroth Law of Thermodynamics
The temperature T is constant throughout a system
in thermal equilibrium
ART Vorlesung, 9 February 2005 – p.3/21
The First Law
✦ First Law of Black Hole Mechanics
If mass-energy δM carrying angular momentum δJ
and zero electric charge falls into a black hole, the
change of surface area of the horizon δA is given by
κ
δA + ΩδJ
δM =
8π
✦ First Law of Thermodynamics
The change of internal energy in a quasistatic process is the sum of the added heat and mechanical
work done on the system:
δE = T δS + P δV
ART Vorlesung, 9 February 2005 – p.4/21
The Second Law
✦ Second Law of Black Hole Mechanics
(Area theorem by H AWKING, 1971):
The surface area of the horizon can never decrease,
δA ≥ 0
✦ Second Law of Thermodynamics
The total entropy of an isolated system can never decrease:
δS ≥ 0
ART Vorlesung, 9 February 2005 – p.5/21
The Area Theorem
The total area of black hole horizons is monotonically increasing with time
ART Vorlesung, 9 February 2005 – p.6/21
Black Hole Thermodynamics?
Analogy to thermodynamics
☛ Surface gravity κ is a measure of temperature T
☛ Horizon area A is proportional to the entropy S
Refutations (from the purely classical point of view):
✘ The temperature of a black hole vanishes!
✘ Entropy is dimensionless, the horizon area is a length squared!
✘ The area is separately non-decreasing, whereas only the total
thermodynamical entropy is non-decreasing!
✘ Numerically, the black hole entropy is vastly larger than the
entropy of a star from which the hole could have formed!
ART Vorlesung, 9 February 2005 – p.7/21
The Quantum Vacuum
The definition of a particle (quantum) depends on the
frame of reference
If the frames of two observers differ only by a Lorentz
transformation, then they will agree about the particle
content
If they have relative acceleration, then they will measure
different particle numbers!
The vacuum in Minkowski spacetime appears to be a
thermal state when viewed by an accelerating observer
(DAVIES, 1975, and U NRUH, 1976)
ART Vorlesung, 9 February 2005 – p.8/21
Minkowski Spacetime
Geometric units: c = 1, G = 1, kB = 1
ds2 = −dt2 + dr2 + r2 dΩ
✦ r → ∞ ➢ spacelike infinity I 0
✦ t → +∞ ➢ future timelike infinity I +
✦ t → −∞ ➢ past timelike infinity I −
ds2 = −dudv + 41 (u − v)2 dΩ
✦ u = t − r → ∞ ➢ future null infinity I +
✦ v = t + r → ∞ ➢ past null infinity I −
ART Vorlesung, 9 February 2005 – p.9/21
Conformal Minkowski Spacetime
Compactify the manifold to include the boundaries I − , I − , I 0 , I + and I +
ART Vorlesung, 9 February 2005 – p.10/21
QFT in a Nutshell
Massless scalar quantum field φ in
1+1 Minkowski spacetime
Field equation ∂µ ∂ µ φ = −φ,tt + φ,rr = 0
√
±
−iω(t∓r)
➯ plane-wave solutions fω = e
/ 2ω
Wave-packets following outgoing null rays onto I + :
Z
+
+ † + ∗
φ = dω [a+
f
+
(a
ω ω
ω ) (fω ) ]
†
+
✦ (a+
ω ) and aω are, respectively, creation and annihilation operators
✦ Number operator for outgoing particles of frequency ω is
+ † +
N+
=
(a
ω ) aω
inert
ART Vorlesung, 9 February 2005 – p.11/21
Inertial vs. Accelerated Observer
Basis modes
in the inertial frame
√
√
−iωu
−iωv
+
−
/ 2ω , fω = e
/ 2ω
fω = e
For a Rindler observer moving with const. acceleration
a, the null coordinates transform to u0 and v 0
The transformation from the inertial to 0the0 accelerated
frame is given by ds2 = −dudv = −ea(v −u ) du0 dv 0
The Rindler observer measures different time
t0 = 21 (u0 + v 0 ) and length r0 = 21 (v 0 − u0 ) and has different
0 √
0 √
−iωu
+
−iωv
−
/ 2ω , gω = e
basis modes gω = e
/ 2ω
Z
+ † + ∗
+
+
(b
g
φ = dω [b+
ω ) (gω ) ]
ω ω
ART Vorlesung, 9 February 2005 – p.12/21
The Unruh Effect
− )i = 0
vacuum state |0(I − )i, ∀ω : a−
|0(I
ω
☞ There are no in/outcoming particles seen by an inertial observer
Number operator for outgoing modes in the accelerated
+ ) † b+
frame is N+
=
(b
ω
ω
Rindler
For the vacuum state |0(I − )i, the Rindler observer finds
h0(I
−
−
)|N+
|0(I
)i
Rindler
=
1
e2πω/a − 1
☞ The Rindler observer beholds a flux of thermal radiation
propagating toward I +
☞ The temperature of the vacuum is proportional to acceleration:
T = ~a/2π
ART Vorlesung, 9 February 2005 – p.13/21
The Collapsing Star Spacetime
A black hole is a region in spacetime which is not in the past of of I +
ART Vorlesung, 9 February 2005 – p.14/21
QFT in Curved Spacetime
Use a prescribed background geometry with metric gµν :
g µν ∂µ ∂ν φ = 0
H AWKING (1973) studied a scalar field φ propagating in
the background spacetime around a star collapsing to a
black hole
✦ Towards I − , the spacetime is asymptotically Minkowskian
✦ Positive frequency modes are partially converted into negative
frequency modes near the horizon (pair creation)
✦ Whereas positive frequency modes scatter off the horizon,
negative frequency modes are absorbed
✦ Assuming the state |0(I − )i in the remote past (the Boulware
state), H AWKING found that in the future of the collapse, there is a
flux of thermal particles toward I +
ART Vorlesung, 9 February 2005 – p.15/21
Pair Creation Near the Event Horizon
Virtual pair creation close to the horizon can result in a real particle
escaping to I + which carries away a fraction of the black hole mass
ART Vorlesung, 9 February 2005 – p.16/21
Hawking Radiation
A black hole produces thermal radiation with T =
~
2π κ
The temperature is proportional to the surface gravity
κ = 12 f 0 (RH ) = ~/8πM , where f (r) = 1 − 2M/r
The wavelength of Hawking radiation is λH = 8π 2 RH
T vanishes in the classical limit ~ → 0
a = κ/f 1/2 (r) is the local acceleration of a stationary
observer
Hawking radiation is a consequence of the state near
the horizon being the vacuum as viewed by free-fall
observers
ART Vorlesung, 9 February 2005 – p.17/21
Black Hole Evaporation
Negative frequency (energy) particle flux into H+
diminishes the black hole mass
The the rate of positive energy emission to I + is ∝ M −2
Black hole life time is of the order M 3
✦ (M/MP )3 TP ∼ 10−28 M 3 [cgs]
✦ A solar mass black hole (M ∼ 1033 g) emits thermal radiation of
roughly 10−11 eV and lives about 1054 times the age
of the universe
✦ Only primordial mini black holes could be observed
if they were around
ART Vorlesung, 9 February 2005 – p.18/21
Bekenstein-Hawking Entropy
There is entropy associated with a black hole horizon:
SBH
AH
=
4~
The unit of horizon area is L2P ∼ 10−66 cm2
For a stellar mass black hole, SBH ∼ 1078 which is about
1021 times the entropy of a collapsing star from which
the hole could have formed
Generalised second law: The sum of BH entropy and
the entropy outside black holes can never decrease
δ(Soutside + SBH ) ≥ 0
ART Vorlesung, 9 February 2005 – p.19/21
The Meaning of Black Hole Entropy
H AWKING (until last year): extra quantum uncertainty
✦ SBH corresponds to the maximum information lost in the BH
✦ Information about quantum states can not be restored
P ENROSE (The Emperor’s New Mind): complementary
quantum uncertainty
✦ Loss of information corresponds to shrinking of phase space
volume as trajectories converge towards singularities
✦ This is counter-balanced by quantum measurements in which
information and, thus, phase space volume is regained
M- THEORY: holographic principle and such
✦ Information is stored in the quantised horizon and eventually will
be rejuvenated by BH evaporation
ART Vorlesung, 9 February 2005 – p.20/21
More to Read...
S. H AWKING, Quantum Mechanics of Black Holes,
Scientific American, January 1977
S. H AWKING and R. P ENROSE,
The Nature of Space and Time,
Princeton University Press, 1996
T. JACOBSON,
Introductory Lectures on Black Hole Thermodynamics,
www.glue.umd.edu/∼tajac/BHTlectures/lectures.ps
J. T RASCHEN,
An Introduction to Black Hole Evaporation,
de.arxiv.org/abs/gr-qc/?0010055
The Holographic Principle and M-theory
www.damtp.cam.ac.uk/user/gr/public/holo/
ART Vorlesung, 9 February 2005 – p.21/21