The laws of black-hole mechanics Black-hole thermodynamics

1 laws of black-hole mechanics

1.1 statement of laws
1.2 zeroth law
1.3 first law
1.4 second law
1.5 third law
1.6 discussion of laws

1.6.1 zeroth law
1.6.2 first law
1.6.3 second law
1.6.4 third law


1.7 interpretation of laws

the laws of black-hole mechanics

the 4 laws of black-hole mechanics physical properties black holes believed satisfy. laws, analogous laws of thermodynamics, discovered brandon carter, stephen hawking, , james bardeen.

statement of laws

the laws of black-hole mechanics expressed in geometrized units

the zeroth law

the horizon has constant surface gravity stationary black hole.

the first law

for perturbations of stationary black holes, change of energy related change of area, angular momentum, , electric charge by

d
e
=


κ

8
π




d
a
+
Ω

d
j
+
Φ

d
q
,


{\displaystyle de={\frac {\kappa }{8\pi }}\,da+\omega \,dj+\phi \,dq,}

where



e


{\displaystyle e}

energy,



κ


{\displaystyle \kappa }

surface gravity,



a


{\displaystyle a}

horizon area,



Ω


{\displaystyle \omega }

angular velocity,



j


{\displaystyle j}

angular momentum,



Φ


{\displaystyle \phi }

electrostatic potential ,



q


{\displaystyle q}

electric charge.

the second law

the horizon area is, assuming weak energy condition, non-decreasing function of time:

d
a


d
t



≥
0.


{\displaystyle {\frac {da}{dt}}\geq 0.}

this law superseded hawking s discovery black holes radiate, causes both black hole s mass , area of horizon decrease on time.

the third law

it not possible form black hole vanishing surface gravity.



κ
=
0


{\displaystyle \kappa =0}

not possible achieve.

discussion of laws
the zeroth law

the zeroth law analogous zeroth law of thermodynamics, states temperature constant throughout body in thermal equilibrium. suggests surface gravity analogous temperature. t constant thermal equilibrium normal system analogous



κ


{\displaystyle \kappa }

constant on horizon of stationary black hole.

the first law

the left side, de, change in energy (proportional mass). although first term not have obvious physical interpretation, second , third terms on right side represent changes in energy due rotation , electromagnetism. analogously, first law of thermodynamics statement of energy conservation, contains on right side term t ds.

the second law

the second law statement of hawking s area theorem. analogously, second law of thermodynamics states change in entropy in isolated system greater or equal 0 spontaneous process, suggesting link between entropy , area of black-hole horizon. however, version violates second law of thermodynamics matter losing (its) entropy falls in, giving decrease in entropy. however, generalizing second law sum of black-hole entropy , outside entropy, shows second law of thermodynamics not violated in system including universe beyond horizon.

the third law

extremal black holes have vanishing surface gravity. stating



κ


{\displaystyle \kappa }

cannot go 0 analogous third law of thermodynamics, states entropy of system @ absolute 0 defined constant. because system @ 0 temperature exists in ground state. furthermore, Δs reach 0 @ 0 temperature, s reach zero, @ least perfect crystalline substances. no experimentally verified violations of laws of thermodynamics known.

interpretation of laws

the 4 laws of black-hole mechanics suggest 1 should identify surface gravity of black hole temperature , area of event horizon entropy, @ least multiplicative constants. if 1 considers black holes classically, have 0 temperature and, no-hair theorem, 0 entropy, , laws of black-hole mechanics remain analogy. however, when quantum-mechanical effects taken account, 1 finds black holes emit thermal radiation (hawking radiation) @ temperature

t

h


=


κ

2
π



.


{\displaystyle t_{\text{h}}={\frac {\kappa }{2\pi }}.}

from first law of black-hole mechanics, determines multiplicative constant of bekenstein–hawking entropy, is

s

bh


=


a
4


.


{\displaystyle s_{\text{bh}}={\frac {a}{4}}.}






^ kallosh, renata (1992). supersymmetry cosmic censor . physical review d. 46 (12): 5278–5302. arxiv:hep-th/9205027 . bibcode:1992phrvd..46.5278k. doi:10.1103/physrevd.46.5278. 
^ cite error: named reference arxiv.org invoked never defined (see page).