Black hole thermodynamics

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Black hole[edit]

The only way to satisfy the second law of thermodynamics is to admit that black holes have entropy. If black holes carried no entropy, it would be possible to violate the second law by throwing mass into the black hole. The increase of the entropy of the black hole more than compensates for the decrease of the entropy carried by the object that was swallowed.

Starting from theorems proved by Stephen Hawking, Jacob Bekenstein conjectured that the black hole entropy was proportional to the area of its event horizon divided by the Planck area. Bekenstein suggested (1/2\cdot\ln{2})/4\pi as the constant of proportionality, asserting that if the constant was not exactly this, it must be very close to it. The next year, Hawking showed that black holes emit thermal Hawking radiation[2][3] corresponding to a certain temperature (Hawking temperature).[4][5] Using the thermodynamic relationship between energy, temperature and entropy, Hawking was able to confirm Bekenstein's conjecture and fix the constant of proportionality at 1/4:[6]


where A is the area of the event horizon, calculated at 4\pi R^2, k is Boltzmann's constant, and \ell_{\mathrm{P}}=\sqrt{G\hbar / c^3} is the Planck length. This is often referred to as the Bekenstein–Hawking formula. The subscript BH either stands for "black hole" or "Bekenstein-Hawking". The black hole entropy is proportional to the area of its event horizon A. The fact that the black hole entropy is also the maximal entropy that can be obtained by the Bekenstein bound (wherein the Bekenstein bound becomes an equality) was the main observation that led to the holographic principle.[1]

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"[7] theorems appeared to suggest that black holes could have only a single microstate. The situation changed in 1995 when Andrew Strominger and Cumrun Vafa calculated [8] the right Bekenstein-Hawking entropy of a supersymmetric black hole in string theory, using methods based on D-branes and string duality. 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. However, for Schwartzschild black hole, viewed as the most far from the extremal black hole, the relationship between micro and macro-state is expected to be clarified from the string theoretical viewpoint. Various studies are in progress, but has not been elucidated.

In Loop quantum gravity (LQG)[9] it is possible to associate a geometrical interpretation to the microstates: these are the quantum geometries of the horizon. LQG offers a geometric explanation of the finiteness of the entropy and of the proportionality of the area of the horizon.[10][11] It is possible to derive, from the covariant formulation of full quantum theory (Spinfoam) the correct relation between energy and area (1st law), the Unruh temperature and the distribution that yields Hawking entropy.[12] The calculation makes use of the notion of dynamical horizon and is done for non-extremal black holes. There seems to be also discussed the calculation of Beckenstein-Hawking entropy from the point of view of LQG.

The laws of black hole mechanics[edit]


Statement of the laws[edit]

The laws of black hole mechanics are expressed in geometrized units.

The Zeroth Law[edit]

The horizon has constant surface gravity for a stationary black hole.

The First Law[edit]

For perturbations of stationary black holes, the change of energy is related to change of area, angular momentum, and electric charge by:

dE = \frac{\kappa}{8\pi}\,dA+\Omega\, dJ+\Phi\, dQ,

where E is the energy, \displaystyle \kappa is the surface gravity, A is the horizon area, \Omega is the angular velocity, J is the angular momentum, \Phi is the electrostatic potential and Q is the electric charge.

The Second Law[edit]


\frac{dA}{dt} \geq 0.

This "law" was superseded by Hawking's discovery that black holes radiate, which causes both the black hole's mass and the area of its horizon to decrease over time.

The Third Law[edit]

It is not possible to form a black hole with vanishing surface gravity. \displaystyle \kappa = 0 is not possible to achieve.

Discussion of the laws[edit]

The Zeroth Law[edit]

The zeroth law is analogous to the zeroth law of thermodynamics which 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 equilibrium for a normal system is analogous to \displaystyle \kappa constant over the horizon of a stationary black hole.

The First Law[edit]

The left hand side, dE, is the change in energy (proportional to mass). 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 thermodynamics is a statement of energy conservation, which contains on its right hand side the term T dS.

The Second Law[edit]

The second law is the statement of Hawking's area theorem. Analogously, the second law of thermodynamics states that the change in entropy in an isolated system will be greater than or equal to 0 for a spontaneous process, 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. Generalized second law introduced as total entropy = black hole entropy + outside entropy.

The Third Law[edit]

Extremal black holes[13] have vanishing surface gravity. Stating that \displaystyle \kappa cannot go to zero is analogous to the third law of thermodynamics which states, the entropy of a system at absolute zero is a well-defined constant. This is because a system at zero temperature exists in its ground state. Furthermore, ΔS will reach zero at 0 kelvins, but S itself will also reach zero, at least for perfect crystalline substances. No experimentally verified violations of the laws of thermodynamics are known.

Interpretation of the laws[edit]


T_{\text{H}} = \frac{\kappa}{2\pi}.


S_{\text{BH}} = \frac{A}{4}.

Beyond black holes[edit]

Hawking and Page have shown that black hole thermodynamics is more general than black holes, that cosmological event horizons also have an entropy and temperature.


See also[edit]


  2. ^ "First Observation of Hawking Radiation" from the Technology Review
  3. ^ Matson, John (Oct 1, 2010). "Artificial event horizon emits laboratory analogue to theoretical black hole radiation". Sci. Am. 
  4. ^ Charlie Rose: A conversation with Dr. Stephen Hawking & Lucy Hawking
  5. ^ A Brief History of Time, Stephen Hawking, Bantam Books, 1988.
  6. ^ Majumdar, Parthasarathi (1998). "Black Hole Entropy and Quantum Gravity". ArXiv: General Relativity and Quantum Cosmology 73: 147. arXiv:gr-qc/9807045. Bibcode:1999InJPB..73..147M. 
  7. ^ a b No hair theorems for positive Lambda
  8. ^ Strominger, A.; Vafa, C. (1996). "Microscopic origin of the Bekenstein-Hawking entropy". Physics Letters B 379: 99. arXiv:hep-th/9601029. Bibcode:1996PhLB..379...99S. doi:10.1016/0370-2693(96)00345-0.  edit
  9. ^ See List of loop quantum gravity researchers
  10. ^ Rovelli, Carlo (1996). "Black Hole Entropy from Loop Quantum Gravity". Physical Review Letters 77 (16): 3288–3291. arXiv:gr-qc/9603063. Bibcode:1996PhRvL..77.3288R. doi:10.1103/PhysRevLett.77.3288. 
  11. ^ Ashtekar, Abhay; Baez, John; Corichi, Alejandro; Krasnov, Kirill (1998). "Quantum Geometry and Black Hole Entropy". Physical Review Letters 80 (5): 904–907. arXiv:gr-qc/9710007. Bibcode:1998PhRvL..80..904A. doi:10.1103/PhysRevLett.80.904. 
  12. ^ Bianchi, Eugenio (2012). Entropy of Non-Extremal Black Holes from Loop Gravity 1204. p. 5122. arXiv:gr-qc/1204.5122. Bibcode:2012arXiv1204.5122B. 
  13. ^ Supersymmetry as a Cosmic Censor Authors: Renata Kallosh, Andrei Linde, Tomás Ortín, Amanda Peet, Antoine Van Proeyen
  14. ^ For an authoritative review, see Ofer Aharony, Steven S. Gubser, Juan Maldacena, Hirosi Ooguri and Yaron Oz (2000). "Large N field theories, string theory and gravity". Physics Reports 323 (3–4): 183–386. arXiv:hep-th/9905111. Bibcode:1999PhR...323..183A. doi:10.1016/S0370-1573(99)00083-6.  (Shorter lectures by Maldacena, based on that review.


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