Max-Planck-Institutfr Astrophysik

Hawking Radiation andBlack Hole Evaporation

Wolfram Schmidtwolfram@mpa-garching.mpg.de

Advisor-Seminar AstrophysikTechnische Universitat Munchen

W. Hillebrandt & E. Muller

Advisor-Seminar Astrophysik, July 2003 p.1/21

Max-Planck-Institutfr Astrophysik

OverviewClassical black hole mechanicsThe quantum vacuum in Minkowski spacetimeThe Unruh effectHawking radiationBlack hole entropy

Advisor-Seminar Astrophysik, July 2003 p.2/21

Max-Planck-Institutfr Astrophysik

The Zeroth Law

F Zeroth Law of Black Hole MechanicsThe surface gravity is constant on the horizon of astationary black hole

F Zeroth Law of ThermodynamicsThe temperature is constant throughout a systemin thermal equilibrium

Advisor-Seminar Astrophysik, July 2003 p.3/21

Max-Planck-Institutfr Astrophysik

The First Law

F First Law of Black Hole MechanicsIf mass-energy carrying angular momentum

and zero electric charge falls into a black hole, thechange of surface area of the horizon is given by

F First Law of ThermodynamicsThe change of internal energy in a quasistatic pro-cess is the sum of the added heat and mechanicalwork done on the system:

Advisor-Seminar Astrophysik, July 2003 p.4/21

Max-Planck-Institutfr Astrophysik

The Second Law

F Second Law of Black Hole Mechanics(Area theorem by HAWKING, 1971):The surface area of the horizon can never decrease,

F Second Law of ThermodynamicsThe total entropy of an isolated system can never de-crease:

Advisor-Seminar Astrophysik, July 2003 p.5/21

Max-Planck-Institutfr Astrophysik

The Area TheoremThe total area of black hole horizons is monotonically increasing with time

Advisor-Seminar Astrophysik, July 2003 p.6/21

Max-Planck-Institutfr Astrophysik

Black Hole Thermodynamics?

Analogy to thermodynamics* Surface gravity is a measure of temperature * Horizon area

is proportional to the entropy Refutations (from the purely classical point of view):8 The temperature of a black hole vanishes!8 Entropy is dimensionless, the horizon area is a length squared!8 The area is separately non-decreasing, whereas only the total

thermodynamical entropy is non-decreasing!8 Numerically, the black hole entropy is vastly larger than the

entropy of a star from which the hole could have formed!

Advisor-Seminar Astrophysik, July 2003 p.7/21

Max-Planck-Institutfr Astrophysik

The Quantum VacuumThe definition of a particle (quantum) depends on theframe of referenceIf the frames of two observers differ only by a Lorentztransformation, then they will agree about the particlecontentIf they have relative acceleration, then they will measuredifferent particle numbers!The vacuum in Minkowski spacetime appears to be athermal state when viewed by an accelerating observer(DAVIES, 1975, and UNRUH, 1976)

Advisor-Seminar Astrophysik, July 2003 p.8/21

Max-Planck-Institutfr Astrophysik

Minkowski Spacetime

Geometric units: , ,

F spacelike infinity

F future timelike infinity

F past timelike infinity

F future null infinity !

F " past null infinity !

Advisor-Seminar Astrophysik, July 2003 p.9/21

Max-Planck-Institutfr Astrophysik

Conformal Minkowski Spacetime

Compactify the manifold to include the boundaries , ! ,

,

!

and

Advisor-Seminar Astrophysik, July 2003 p.10/21

Max-Planck-Institutfr Astrophysik

QFT in a NutshellMassless scalar quantum field in1+1 Minkowski spacetimeField equation

plane-wave solutions

Wave-packets following outgoing null rays onto

:

F

!

"

and

are, respectively, creation and annihilation operatorsF Number operator for outgoing particles of frequency # is

$

&%('

)*

+

!

"

Advisor-Seminar Astrophysik, July 2003 p.11/21

Max-Planck-Institutfr Astrophysik

Inertial vs. Accelerated ObserverBasis modes in the inertial frame

,

For a Rindler observer moving with const. acceleration

, the null coordinates transform to

and

The transformation from the inertial to the acceleratedframe is given by

The Rindler observer measures different time

and length

and has differentbasis modes

,

Advisor-Seminar Astrophysik, July 2003 p.12/21

Max-Planck-Institutfr Astrophysik

The Unruh Effect

vacuum state

,

+ There are no in/outcoming particles seen by an inertial observer

Number operator for outgoing modes in the acceleratedframe is

For the vacuum state

, the Rindler observer finds

+ The Rindler observer beholds a flux of thermal radiationpropagating to !

+ The temperature of the vacuum is proportional to acceleration:

Advisor-Seminar Astrophysik, July 2003 p.13/21

Max-Planck-Institutfr Astrophysik

The Collapsing Star Spacetime

A black hole is a region in spacetime which is not in the past of of !

Advisor-Seminar Astrophysik, July 2003 p.14/21

Max-Planck-Institutfr Astrophysik

QFT in Curved SpacetimeUse a prescribed background geometry with metric :

HAWKING (1973) studied a scalar field propagating inthe background spacetime around a star collapsing to ablack holeF Towards ! , the spacetime is nearly MinkowskianF Positive frequency modes are partially converted into negative

frequency modes near the horizon (pair creation)F Whereas positive frequency modes scatter off the horizon,

negative frequency modes are absorbedF Assuming the state ! ! in the remote past (the Boulware

state), HAWKING found that in the future of the collapse, there is aflux of thermal particles to !

Advisor-Seminar Astrophysik, July 2003 p.15/21

Max-Planck-Institutfr Astrophysik

Pair Creation Near the Event HorizonVirtual pair creation close to the horizon can result in a real particleescaping to !

which carries away a fraction of the black hole mass

Advisor-Seminar Astrophysik, July 2003 p.16/21

Max-Planck-Institutfr Astrophysik

Hawking Radiation

A black hole produces thermal radiation with

The temperature is proportional to the surface gravity

, where

The wavelength of Hawking radiation is

vanishes in the classical limit

is the local acceleration of a stationaryobserverHawking radiation is a consequence of the state nearthe horizon being the vacuum as viewed by free-fallobservers

Advisor-Seminar Astrophysik, July 2003 p.17/21

Max-Planck-Institutfr Astrophysik

Black Hole Evaporation

Negative frequency (energy) particle flux into diminishes the black hole massThe the rate of positive energy emission to

is

Black hole life time is of the order

F

!

F A solar mass black hole ( ) emits thermal radiation ofroughly

and lives about

times the ageof the universe

F Only primordial mini black holes could be observedif they were around

Advisor-Seminar Astrophysik, July 2003 p.18/21

Max-Planck-Institutfr Astrophysik

Bekenstein-Hawking Entropy

There is entropy associated with a black hole horizon:

The unit of horizon area is

For a stellar mass black hole,

which is about

times the entropy of a collapsing star from whichthe hole could have formedGeneralised second law: The sum of BH entropy andthe entropy outside black holes can never decrease

Advisor-Seminar Astrophysik, July 2003 p.19/21

Max-Planck-Institutfr Astrophysik

The Meaning of Black Hole Entropy

In a large self-gravitating system (the Universe)gravitational collapse can occur The notion of entropy has to be extendedHAWKING: extra quantum uncertaintyF corresponds to the maximum information lost in the BHF Information about quantum states can not be restored

PENROSE: complementary quantum uncertaintyF Loss of information corresponds to shrinking of phase space

volume as trajectories converge towards singularitiesF This is counter-balanced by quantum measurements in which

information and, thus, phase space volume is regained

Advisor-Seminar Astrophysik, July 2003 p.20/21

Max-Planck-Institutfr Astrophysik

More to Read...

S. HAWKING, Quantum Mechanics of Black Holes,Scientific American, January 1977

S. HAWKING and R. PENROSE,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.psJ. TRASCHEN,An Introduction to Black Hole Evaporation,gr-qc/0010055

Advisor-Seminar Astrophysik, July 2003 p.21/21

OverviewThe Zeroth LawThe First LawThe Second LawThe Area TheoremBlack Hole Thermodynamics?The Quantum VacuumMinkowski SpacetimeConformal Minkowski SpacetimeQFT in a NutshellInertial vs. Accelerated ObserverThe Unruh EffectThe Collapsing Star SpacetimeQFT in Curved SpacetimePair Creation Near the Event HorizonHawking RadiationBlack Hole EvaporationBekenstein-Hawking EntropyThe Meaning of Black Hole EntropyMore to Read...