Bekenstein-Hawking entropy

The Bekenstein-Hawking entropy of black holes assigns them a formal entropy that only depends on the surface area of ​​their event horizon and on fundamental natural constants. It was found in 1973 by Jacob Bekenstein and soon afterwards supported by Stephen Hawking through his theory of Hawking radiation . ${\displaystyle S_{\mathrm {SL} }}$

With the entropy equation of Bekenstein and Hawking, a connection can be established between thermodynamics , quantum mechanics and general relativity . A fundamental goal of a theory of quantum gravity , which has only existed in rudiments so far, is the interpretation of the Bekenstein-Hawking entropy through microscopic degrees of freedom.

The Bekenstein-Hawking entropy was a motivation for the holographic principle .

history

In 1971 Stephen Hawking established the second law of thermodynamics of black holes: The surface of black holes can never decrease during processes such as the merging or scattering of black holes, and also not when a particle falls into it. The surface corresponds to the square of the irreducible mass of the black hole (mass after reversible removal of charge and torque). This suggested the analogy of the surface of black holes with an entropy. During his doctoral thesis, Jacob Bekenstein carried out the following thought experiment: If a body with entropy falls into a black hole, an outside observer can only determine two things: The entropy outside the event horizon has decreased and the surface of the black hole has increased. To rule out a violation of the second law of thermodynamics , he must therefore interpret the surface of the black hole as a measure of the entropy contained in the black hole: ${\displaystyle S}$

${\displaystyle S_{\mathrm {SL} }={\frac {k_{\mathrm {B} }\,c^{3}\ A}{4\,\hbar \,G}}={\frac {k_{\mathrm {B} }A}{4\,l_{\mathrm {P} }^{2}}}}$,

whereby the entropy of the black hole is the Boltzmann constant , the speed of light , the surface of the event horizon, the Planck's constant divided by 2 and the gravitational constant . The second representation uses the Planck length . In the literature, the Boltzmann constant is often omitted or set. ${\displaystyle S_{\mathrm {SL} }}$ ${\displaystyle k_{\mathrm {B} }}$ ${\displaystyle c}$ ${\displaystyle A}$ ${\displaystyle \hbar }$ ${\displaystyle \pi }$ ${\displaystyle G}$ ${\displaystyle l_{\mathrm {P} }={\sqrt {G\hbar /c^{3}}}}$ ${\displaystyle k_{\mathrm {B} }=1}$

The surface of the event horizon is through for uncharged, stationary, spherically symmetrical black holes (described by a Schwarzschild metric , mass ) ${\displaystyle M}$

${\displaystyle A=4\pi r_{\mathrm {S} }^{2}=16\pi {\left({\frac {GM}{c^{2}}}\right)}^{2}}$

given with the Schwarzschild radius , and for rotating black holes (angular momentum ) by: ${\displaystyle r_{\mathrm {S} }={\frac {2GM}{c^{2}}}}$ ${\displaystyle J}$

${\displaystyle A=4\pi \left(r_{\mathrm {S} }^{2}+\left({\frac {J}{Mc}}\right)^{2}\right)}$

Stephen Hawking criticized the fact that the black hole also had to have a temperature. A body with a non-vanishing temperature, however, has black body radiation , which contradicts the common image that nothing more escapes from the black hole. Hawking resolved this paradox by pointing out that an event horizon without any expansion, assuming exact energy density at the same time , would contradict the quantum mechanical uncertainty relation. In the immediate vicinity of the event horizon, the energy density of the gravitational field is rather so great that particle pairs are formed, one of which falls into the black hole, but the other escapes. With this Hawking radiation it is possible to equate the entropy and the surface of the black hole; the entropy of the black hole therefore bears the name Bekenstein-Hawking entropy .

The temperature of the black hole is

${\displaystyle T={\frac {\hbar \ c^{3}}{8\pi \,G\,Mk_{\mathrm {B} }}}}$.

This temperature is typically on the order of a millionth of a Kelvin and decreases as the mass of the black hole increases. The black hole can dissolve if the energy of the Hawking radiation emitted exceeds the energy of the incident matter for a sufficiently long period of time.

Generalized second law of thermodynamics

The second law of thermodynamics says that for a closed system the entropy cannot decrease. Since bodies containing entropy can also fall into a black hole, the question arises whether this violates the second law. Due to the connection between the surface and the entropy of the black hole, the second law can be generalized: " The sum of" ordinary "entropy and the ( multiplied by) total area of ​​all event horizons cannot decrease over time. ${\displaystyle k_{\mathrm {B} }/(4l_{\mathrm {P} }^{2})}$ "

For example, consider the fusion of two black holes with masses M ₁ and M ₂ . The fusion process is isentropic , i.e. that is, the ordinary entropy of the system does not change. Since the area of ​​the event horizon A is proportional to the square of the mass, the result for the change is : ${\displaystyle \Delta A=A_{\text{nachher}}-A_{\text{vorher}}}$

${\displaystyle \Delta A=A(M_{1}+M_{2})-A(M_{1})-A(M_{2})\sim (M_{1}+M_{2})^{2}-M_{1}^{2}-M_{2}^{2}=2M_{1}\,M_{2}>0}$

The total area increases and the fusion of two black holes does not contradict the generalized second law. Now consider the decay of a black hole of mass M ₁ + M ₂ into two smaller black holes of mass M ₁ and M ₂ . The decay process is again isentropic. The following then applies to the change in the total area of ​​the event horizons:

${\displaystyle \Delta A=A(M_{1})+A(M_{2})-A(M_{1}+M_{2})\sim M_{1}^{2}+M_{2}^{2}-(M_{1}+M_{2})^{2}=-2M_{1}\,M_{2}<0}$

The total area would therefore decrease if a black hole were to disintegrate into two smaller ones. The generalized second law of thermodynamics forbids the disintegration of a black hole into two smaller ones.

Web links

• Black holes light , ablebnis-universum.de
• Bekenstein: Bekenstein-Hawking Entropy , Scholarpedia

Individual evidence

1. ^ ^{A b} Jacob D. Bekenstein : Black holes and entropy . In: Phys. Rev. D, no. 7 , 1973, p. 2333–2346 ( online [PDF; accessed December 9, 2014]).
2. ^ Hawking, Gravitational radiation from colliding black holes, Phys. Rev. Lett., Vol. 26, 1971, p. 1344
3. ↑ Misner, Thorne, Wheeler, Gravitation , Freeman 1973, p. 889
4. ↑ Stephen W. Hawking : Particle Creation by Black Holes . In: Commun. Math. Phys. tape 43 , 1975, p. 199-220 , doi : 10.1007 / BF02345020 .
5. ↑ ^{a b} Stephen W. Hawking : A Brief History of Time . 1st edition. Rowohlt Verlag, 1988, ISBN 3-498-02884-7 ( limited preview in Google book search).