# Temperature and Entropy of a Black Hole

Temperature and Entropy of a Black Hole

According to the "no hair theorems" of general relativity, the properties of an electrovac black hole (i.e., a black hole whose exterior is a vacuum and whose interior contains electromagnetic but no other charges) are completely determined by just three parameters: mass , angular momentum , and electric (or magnetic monopole) charge . While this suggests a possible violation of the second law of thermodynamics, since the entropy of the universe could be reduced by creating single-microstate black holes, the black hole region is classically experimentally inaccessible. Thus, it is possible that a quantum mechanical black hole can possess additional microstates. Moreover, in a series of papers starting in 1973, Jacob Bekenstein showed that the area of a black hole seemed to be a measure of the entropy of a black hole, in precisely the correct magnitude to preserve the second law. In 1974, Stephen Hawking proposed, based on quantum field theory (QFT), that the immense energy density of a black hole's gravitational field could give rise to continual creation and annihilation of virtual particles and antiparticles, with lifetimes consistent with the uncertainty principle, . Should one of the virtual pair be swallowed up by the black hole, its partner would become a real particle which, to an outside observer, would appear to have been radiated by the black hole. Moreover, the effective temperature of the black hole corresponded nicely with Bekenstein's classical derivation, giving a physical justification for the earlier ad-hoc identification.

M

J

Q

ΔEΔt≲ℏ

Thus, Hawking and Bekenstein worked out the principles of black hole thermodynamics. We limit our considerations to the case , a rotating Kerr black hole. The horizon has a radius =+- (for , this reduces to the Schwarzschild radius =2GM/). The spherical horizon is surrounded by an oblate spheroidal ergosphere, which corotates in the same direction, a consequence of frame dragging. The ergosphere, shown in gray, is just a region of spacetime containing no matter. According to the Bekenstein–Hawking theory, the temperature of a black hole is given by =1- and the emitted radiation follows a blackbody distribution. The temperature is inversely proportional to the mass: ≈× K, where =≈2.18× g, the Planck mass. For the Earth's mass, ≈0.02 K, while for a solar mass, ≈100 nK, both less than the temperature of the cosmic microwave background radiation (2.725 K). Evidently, smaller black holes are hotter and, when radiating into empty space, should eventually evaporate.

Q=0

r

h

GM

2

c

2

GM

2

c

2

J

Mc

J=0

r

s

2

c

T

BH

ℏ

3

c

4πGM

k

B

r

s

2

r

h

T

BH

5.64

M

M

p

30

10

M

p

1/2

(ℏc/G)

-5

10

T

BH

T

BH

In the graphic, variations in temperature at the event horizon are simulated by colors of the visible spectrum, but with the scales of space, time, and temperature being grossly exaggerated for display purposes. Black holes of the order of the Planck mass are considered.

The Bekenstein–Hawking entropy of a black hole is given by =A, where is the area of the event horizon, equal to , expressed in units of the Planck area, , where =≈1.62× cm, the Planck length. Thus each element of area on the horizon represents one Planck unit of entropy. Since a Planck mass contains over proton masses, it can be surmised that 1 Planck unit corresponds to something of the order of bits of information on the atomic level.

S

BH

1

4

A

A=4π+

2

r

h

2

(J/Mc)

2

L

p

L

p

1/2

(ℏG)

3

c

-33

10

1

4

2

L

p

19

10

40

10

While Hawking radiation provides a physical basis for black hole entropy, it also raises new questions. According to the no-hair theorems, the radiation must be perfectly blackbody and therefore devoid of any information. All research into the QFT of black holes supports this conclusion. When the black hole evaporates completely, all of the information from the particles it absorbed disappears, leading to the so-called black hole information paradox. In a more radical direction, since the state of a three-dimensional system can, in concept, be represented on its two-dimensional boundary, Gerard 't Hooft and Leonard Susskind (2001) proposed a general "holographic principle" of nature, which suggests that consistent theories of gravity and quantum mechanics can be represented by lower-dimensional structures. Both the information paradox and the holographic principle remain active areas of investigation.