Partition Function

of a Black Hole

n

En

neEgZ )()(

(A talk based on the paper:

J. Mäkelä, “Partition Function of the Schwarzschild Black Hole”,

Entropy 2011, 13, 1324-1354 (arXiv:1107.3975))

Stephen Hawking in 1976: Black hole emits thermal radiation with a characteristic temperature

2

HT (Hawking temperature)

= surface gravity at the event horizon.

Schwarzschild BH: M41

MGkcTB

H1

8

3

(SI)

Bekenstein-Hawking entropy law: Black hole with event horizon area HA possesses entropy

.41 3

HB

BH AGckS

The Hawking radiation law and the Bekenstein-Hawking entropy law raise a number of questions:

a) What happens, when a black hole radiates? b) What happens to the black hole entropy, when

?HTT c) Is the radiation of a black hole a somewhat similar

process as is the radiation of ordinary matter, where the atoms and molecules perform transitions between different quantum states and photons are emitted?

d) What are the “atoms of spacetime”?

To address these questions we need a microscopic model of a black hole which, even if not necessarily correct, at least allows us to consider these questions in precise terms. The resulting model may be used as a starting point for more advanced models.

In the process we should find the partition function of the black hole.

Schwarzschild line element:

222222

22 sin21)21( drdr

rM

drdtrMds

The proper acceleration of an observer with constant :,, r

22/1)21(:

rM

rMaaa

(blue-shifted Newtonian acceleration)

We consider the thermodynamics of the SBH from the point of view of an observer on a stretched horizon, where

.consta

.consta

This means that

0

dM

Madr

rada

in the infinitesimal changes of r and .M

r = 2M

One may show that

2)(lim2

dM

drMr

on the stretched horizon .consta

a sretched horizon originally close to the event horizon Mr 2 will stay close to the event horizon, no matter what

may happen to the mass M of the hole.

We take the energy of the black hole from the point of view of an observer on the stretched horizon to be:

AaE8

( A = the area of the stretched horizon; for a justification see the

Technical Summary)

We construct the stretched horizon out of discrete constituents, each of them contributing an area, which is an integer times a constant, to the stretched horizon.

The total area of the stretched horizon takes the form:

,2Pln nA

where

,...21 nnnn N

N the number of the constituents,

mcG

Pl35

3 106.1: (the Planck length)

a pure number to be determined later.

The quantum numbers Nnnn ,...,, 21 are non-negative integers determining the areas contributed by the constituents. Constituent j ),...,2,1( Nj is in vacuum, if .0jn

The possible energies of the hole are

,88

naAaE nn

where ,...2,1,0n We shall assume that the quantum states of the hole are encoded in the quantum states of its stretched horizon which, in turn, are determined by the quantum numbers

.jn

We take

)( nEg the number of ways of expressing n as a sum of at

most N positive integers .,...,, 21 Nnnn

The partition function

1

)(:)(n

En

neEgZ

of the Schwarzschild black hole takes, from the point of view of an observer on the stretched horizon .consta the form:

,12

1122

1)(1

N

TT CCZ

when CTT /1: ( 1)( NZ , when ).CTT

2ln8

:

aTC

is the characteristic temperature of the hole. It plays an important role in the thermodynamics of the Schwarzschild black hole.

The average energy of a system in a given temperature 1T

is:

)(ln)(

ZE .

We consider the average energy per a constituent:

.)(:)(N

EE

The average of the quantum numbers Nnnn ,...,, 21 in a given temperature is:

.2ln)(...:)( 21

C

N

TE

Nnnnn

For an astrophysical black hole N is around 8010 (very large!).

We note:

1) When ,CTT 0)()( TETn as an excellent approximation. This means that the constituents of the hole are, in effect, in vacuum, and we have no black hole.

22/1)21(

2ln82ln8:

rM

rMaTC

is the lowest temperature a black hole may have from the point of view of an observer on the stretched horizon.

2) An observer at the asymptotic infinity measures for the

hole the minimum temperature

.12ln32

)21(lim 2/1

2 MT

rMT C

Mr

Putting 2ln4 we find:

HTM

T81

Hawking temperature!

We have obtained the Hawking radiation law from our model!

3) The latent heat per a constituent associated with the phase transition, when CTT is:

2ln2 CTL .

During the phase transition the constituents jump, in average, from the vacuum, where ,0jn to the second excited states,

where .2jn

4) When a black hole radiates, the constituents of its event horizon descend, in average, from the second excited states to the vacuum, and radiation is emitted. (Explanation to the Hawking effect)

We note:

1) Entropy is given by an expression

,41)( AAS

when critAA (which means that CTT ). This is the

Bekenstein-Hawking entropy law without corrections!

2) When critAA (which means that CTT ), the entropy is given by an expression:

),2ln()2

2ln()2ln(4

1)(crit

crit

crit AAAN

AAAAAS

and the Bekenstein-Hawking law no more holds.

Similar results may be obtained for the Reissner-Nordström black holes as well.