- Black hole thermodynamics
physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamicswith the existence of black hole event horizons. Much as the study of the statistical mechanics of black bodyradiation led to the advent of the theory of quantum mechanics, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle.
Black hole entropy
Black hole entropy is the
entropycarried by a black hole.
If black holes carried no entropy, it would be possible to violate the
second law of thermodynamicsby throwing mass into the black hole. The only way to satisfy the second law is to admit that the black holes have entropy whose increase more than compensates for the decrease of the entropy carried by the object that was swallowed.
Starting from theorems proved by
Stephen Hawking, Jacob Bekensteinconjectured that the black hole entropy was proportional to the area of its event horizondivided by the Planck area. Later, Stephen Hawking showed that black holes emit thermal Hawking radiationcorresponding to a certain temperature (Hawking temperature). Using the thermodynamicrelationship between energy, temperature and entropy, Hawking was able to confirm Bekenstein's conjecture and fix the constant of proportionality at 1/4:
where "k" is
Boltzmann's constant, and is the Planck length. The black hole entropy is proportional to its area . The fact that the black hole entropy is also the maximal entropy that can be squeezed within a fixed volume was the main observation that led to the holographic principle. The subscript BH either stands for "black hole" or "Bekenstein-Hawking".
Although Hawking's calculations gave further thermodynamic evidence for black hole entropy, until 1995 no one was able to make a controlled calculation of black hole entropy based on
statistical mechanics, which associates entropy with a large number of microstates. In fact, so called "no hair" theorems appeared to suggest that black holes could have only a single microstate. The situation changed in 1995 when Andrew Stromingerand Cumrun Vafacalculated the right Bekenstein-Hawking entropy of a supersymmetric black hole in string theory, using methods based on D-branes. Their calculation was followed by many similar computations of entropy of large classes of other extremal and near-extremal black holes, and the result always agreed with the Bekenstein-Hawking formula. Loop quantum gravity, viewed as the main competitor of string theory, also offered a slightly more heuristic calculation of the black hole entropy. This calculation confirms that the entropy is proportional to the surface area, with the proportionality constant dependent on the only free parameter in LQG, Immirzi parameter.
The laws of black hole mechanics
The four laws of black hole mechanics are physical properties that
black holes are believed to satisfy. The laws, analogous to the laws of thermodynamics, were discovered by Brandon Carter, Stephen Hawkingand James Bardeen.
tatement of the laws
The laws of black hole mechanics are expressed in
The Zeroth Law
The horizon has constant
surface gravityfor a stationary black hole.
The First Law
where is the
mass, is the horizon area, is the angular velocity, is the angular momentum, is the electrostatic potential, is the surface gravityand is the electric charge.
The Second Law
The horizon area is, assuming the weak energy condition, a non-decreasing function of time,
The Third Law
It is not possible to form a black hole with vanishing surface gravity. = 0 is not possible to achieve.
Discussion of the laws
The Zeroth Law
The zeroth law is analogous to the
zeroth law of thermodynamicswhich states that the temperature is constant throughout a body in thermal equilibrium. It suggests that the surface gravity is analogous to temperature."T" constant for thermal equilbrium for a normal system is analogous to constant over the horizon of a stationary black hole.
The First Law
The left hand side, "dM", is the change in mass/energy. Although the first term does not have an immediately obvious physical interpretation, the second and third terms on the right hand side represent changes in energy due to rotation and
electromagnetism. Analogously, the first law of thermodynamicsis a statement of energy conservation, which contains on its right hand side the term "T dS".
The Second Law
The second law is the statement of Hawking's area theorem. Analogously, the
second law of thermodynamicsstates that the entropyof a isolated system is a non-decreasing function of time, suggesting a link between entropy and the area of a black hole horizon. However, this version violates the second law of thermodynamics by matter losing (its) entropy as it falls in, giving a decrease in entropy. Generalised second law introduced as total entropy = black hole entropy + outside entropy
The Third Law
Extremal black holes have vanishing surface gravity. Stating that cannot go to zero is analogous to the
third law of thermodynamicswhich, in its weak formulation, states that it is impossible to reach absolute zerotemperature in a physical process. The strong version of the third law of thermodynamics, which states that as the temperature approaches zero, the entropy also approaches zero, does not have an analogue for black holes. However, the strong version is violated by many known systems in condensed matter physics, and has therefore been rejected as a law.
Interpretation of the laws
The four laws of black hole mechanics suggest that one should identify the surface gravity of a black hole with temperature and the area of the event horizon with entropy, at least up to some multiplicative constants. If one only considers black holes classically, then they have zero temperature and, by the
no hair theorem, zero entropy, and the laws of black hole mechanics remain an analogy. However, when quantum mechanical effects are taken into account, one finds that black holes emit thermal radiation( Hawking radiation) at temperature
From the first law of black hole mechanics, this determines the multiplicative constant of the Bekenstein-Hawking entropy which is
Beyond black holes
Hawking and Page showed that black hole thermodynamics is more general than black holes, that cosmological event horizons also have an entropy and temperature.
More fundamentally, t'Hooft and Susskind used the laws of black hole thermodynamics to argue for a general
Holographic Principleof nature, which asserts that consistent theories of gravity and quantum mechanics must be lower dimensional. Though not yet fully understood in general, the holographic principle has led to the only complete theories of quantum gravity, such as the AdS/CFT correspondence.
*cite journal |last=Bardeen |first=J. M. |authorlink= |coauthors=Carter, B.; Hawking, S. W. |year=1973 |month= |title=The four laws of black hole mechanics |journal=Communications in Mathematical Physics |volume=31 |issue=2 |pages=161–170 |doi=10.1007/BF01645742 |url= |accessdate= |quote=
*cite journal |last=Bekenstein |first=Jacob D. |authorlink= |coauthors= |year=1973 |month= |title=Black holes and entropy |journal=Physical Review D |volume=7 |issue=8 |pages=2333–2346 |doi=10.1103/PhysRevD.7.2333 |url= |accessdate= |quote=
*cite journal |last=Hawking |first=Stephen W. |authorlink= |coauthors= |year=1974 |month= |title=Black hole explosions? |journal=Nature |volume=248 |issue=5443 |pages=30–31 |doi=10.1038/248030a0 |url= |accessdate= |quote=
*cite journal |last=Hawking |first=Stephen W. |authorlink= |coauthors= |year=1975 |month= |title=Particle creation by black holes |journal=Communications in Mathematical Physics |volume=43 |issue=3 |pages=199–220 |doi=10.1007/BF02345020 |url= |accessdate= |quote=
*cite book |title=The Large Scale Structure of Space-time |last=Hawking |first=S. W. |authorlink= |coauthors=Ellis, G. F. R. |year=1973 |publisher=Cambridge University Press |location=New York |isbn=0521099064 |pages=
*cite journal |last=Hawking |first=Stephen W. |authorlink= |coauthors= |year=1994 |month= |title=The Nature of Space and Time |journal=arΧiv e-print |volume= |issue= |pages= |id=arXiv|hep-th|9409195v1 |url= |accessdate= |quote=
*cite journal |last=Meyer |first=A. J., II |authorlink= |coauthors= |year=2006 |month= |title=Black Holes, Entropy and the Third Law |journal=arΧiv e-print |volume= |issue= |pages= |id=arXiv|physics|0608080v1 |url= |accessdate= |quote=
* [http://nrumiano.free.fr/Estars/bh_thermo.html Black Hole Thermodynamics]
* [http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=hep-th&level=1&index1=3281361 Black hole entropy on arxiv.org]
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