Black hole thermodynamics
Thermodynamics |
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In physics, black hole thermodynamics[1] is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. As the study of the statistical mechanics of black-body radiation led to the development 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.[2]
Overview
The second law of thermodynamics requires 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.
In 1972, Jacob Bekenstein conjectured that black holes should have an entropy,[3] where by the same year, he proposed no-hair theorems.
In 1973 Bekenstein suggested as the constant of proportionality, asserting that if the constant was not exactly this, it must be very close to it. The next year, in 1974,
where is the area of the event horizon, is the Boltzmann constant, and is the
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
In
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 Jacob Bekenstein, Brandon Carter, and James Bardeen. Further considerations were made by Stephen Hawking.
Statement of the laws
The laws of black hole mechanics are expressed in
The zeroth law
The horizon has constant surface gravity for a stationary black hole.
The first law
For perturbations of stationary black holes, the change of energy is related to change of area, angular momentum, and electric charge by
where is the energy, is the surface gravity, is the horizon area, is the angular velocity, is the angular momentum, is the electrostatic potential and is the electric charge.
The second law
The horizon area is, assuming the
This "law" was superseded by Hawking's discovery that black holes radiate, which causes both the black hole's mass and the area of its horizon to decrease over time.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
It is not possible to form a black hole with vanishing surface gravity. That is, cannot be achieved.
Discussion of the laws
The zeroth law
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 constant over the horizon of a stationary black hole.
The first law
The left side, , 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 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 side the term .
The second law
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. However, generalizing the second law as the sum of black hole entropy and outside entropy, shows that the second law of thermodynamics is not violated in a system including the universe beyond the horizon.
The generalized second law of thermodynamics (GSL) was needed to present the second law of thermodynamics as valid. This is because the second law of thermodynamics, as a result of the disappearance of entropy near the exterior of black holes, is not useful. The GSL allows for the application of the law because now the measurement of interior, common entropy is possible. The validity of the GSL can be established by studying an example, such as looking at a system having entropy that falls into a bigger, non-moving black hole, and establishing upper and lower entropy bounds for the increase in the black hole entropy and entropy of the system, respectively.[17] One should also note that the GSL will hold for theories of gravity such as Einstein gravity, Lovelock gravity, or Braneworld gravity, because the conditions to use GSL for these can be met.[18]
However, on the topic of black hole formation, the question becomes whether or not the generalized second law of thermodynamics will be valid, and if it is, it will have been proved valid for all situations. Because a black hole formation is not stationary, but instead moving, proving that the GSL holds is difficult. Proving the GSL is generally valid would require using
The third law
Extremal black holes[20] have vanishing surface gravity. Stating that cannot go to zero is analogous to the third law of thermodynamics, which states that 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, will reach zero at zero temperature, but itself will also reach zero, at least for perfect crystalline substances. No experimentally verified violations of the laws of thermodynamics are known yet.
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,[11] 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 a temperature
From the first law of black hole mechanics, this determines the multiplicative constant of the Bekenstein–Hawking entropy, which is (in
which is the entropy of the black hole in Einstein's general relativity. Quantum field theory in curved spacetime can be utilized to calculate the entropy for a black hole in any covariant theory for gravity, known as the Wald entropy.[21]
Quantum gravitational corrections to the entropy
The Hawking formula for the entropy receives corrections as soon as quantum effects are taken into account. Any UV finite theory of quantum gravity should reduce at low energy to General Relativity. Works pioneered by Barvinsky and Vilkovisky [22][23][24][25] suggest as a starting point up to second order in curvature the following action, consisting of local and non-local terms:
where is an energy scale. The exact values of the coefficients are unknown, as they depend on the nature of the ultra-violet theory of quantum gravity. is an operator with the integral representation
The new additional terms in the action modify the classical Einstein equations of motion. This implies that a given classical metric receives quantum corrections, which in turn shift the classical position of the event horizon. When computing the Wald entropy, one then takes the shifted position of the event horizon into account:
Here, is the Lagrangian density of the theory, , is the Riemann tensor and is an antisymmetric tensor normalised as
This method was applied in 2021 by Calmet et al.[26] for Schwarzschild black holes. The Schwarzschild metric does not receive quantum corrections at second order in curvature and the entropy is
A generalisation for charged (Reissner-Nordström) black holes was subsequently carried out by Campos Delgado.[27]
Critique
While black hole thermodynamics (BHT) has been regarded as one of the deepest clues to a quantum theory of gravity, there remain some philosophical criticisms that it “is often based on a kind of caricature of thermodynamics” and "it’s unclear what the systems in BHT are supposed to be", leading to the conclusion -- "the analogy is not nearly as good as is commonly supposed".[28][29]
These criticisms triggered a fellow skeptic to reexamine "the case for regarding black holes as thermodynamic systems", with particular attention paid to "the central role of Hawking radiation in permitting black holes to be in thermal contact with one another" and "the interpretation of Hawking radiation close to the black hole as a gravitationally bound thermal atmosphere", ending with the opposite conclusion -- "stationary black holes are not analogous to thermodynamic systems: they are thermodynamic systems, in the fullest sense."[30]
Beyond black holes
Gary Gibbons and Hawking have shown that black hole thermodynamics is more general than black holes—that cosmological event horizons also have an entropy and temperature.
More fundamentally, Gerard 't Hooft and Leonard Susskind used the laws of black hole thermodynamics to argue for a general holographic principle of 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 is central to theories like the AdS/CFT correspondence.[31]
There are also connections between black hole entropy and fluid surface tension.[32]
See also
Notes
Citations
- S2CID 119114925.
- ^ S2CID 55096624.
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- ^ Matson, John (Oct 1, 2010). "Artificial event horizon emits laboratory analogue to theoretical black hole radiation". Sci. Am.
- ^ Charlie Rose: A conversation with Dr. Stephen Hawking & Lucy Hawking Archived March 29, 2013, at the Wayback Machine
- ^ A Brief History of Time, Stephen Hawking, Bantam Books, 1988.
- S2CID 55539246.
- Bibcode:1999InJPB..73..147M.
- ^
Van Raamsdonk, Mark (31 August 2016). "Lectures on Gravity and Entanglement". New Frontiers in Fields and Strings. pp. 297–351. S2CID 119273886.
- ^ S2CID 119496541.
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- ].
- S2CID 120254309.
- ^ S2CID 123043135.
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- ^ Dougherty, John; Callender, Craig. "Black Hole Thermodynamics: More Than an Analogy?" (PDF). philsci-archive.pitt.edu. Guide to the Philosophy of Cosmology, editors: A. Ijjas and B. Loewer. Oxford University Press.
- ^ Foster, Brendan Z. (September 2019). "Are We All Wrong About Black Holes? Craig Callender worries that the analogy between black holes and thermodynamics has been stretched too far". quantamagazine.org. Retrieved 3 September 2021.
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