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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 Hawking and James Bardeen. Wikipedia does not have an article with this exact name. ... A black hole is a concentration of mass great enough that the force of gravity prevents anything past its event horizon from escaping it except through quantum tunnelling behaviour (known as Hawking Radiation). ... Thermodynamics (from the Greek thermos meaning heat and dynamis meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Brandon Carter is a theoretical physicist, most famous for his work on the properties of black holes and for introducing the anthropic principle. ... Stephen Hawking in 2005 Professor Stephen William Hawking, CH, CBE, FRS, (born January 8, 1942) is considered one of the worlds leading theoretical physicists. ... James M. Bardeen is an American physicist, well known for his work in general relativity, particularly his role in formulating the laws of black hole mechanics. ...

Statement of the Laws

The Zeroth Law

The horizon has constant surface gravity for a stationary black hole κ. The surface gravity of a Killing horizon is the acceleration, as exerted at infinity, needed to keep an object at the horizon. ...

The First Law

We have

,

where M is the mass, A is the horizon area, Ω is the angular velocity, J is the angular momentum, Φ is the electrostatic potential, κ is the surface gravity and Q is the electric charge. Mass is a property of a physical object that quantifies the amount of matter it contains. ... Angular velocity describes the speed of rotation. ... In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in which, the object rotates about the reference point. ... Electric potential is the potential energy per unit charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. ... The surface gravity of a Killing horizon is the acceleration, as exerted at infinity, needed to keep an object at the horizon. ... ‹ The template below has been proposed for deletion. ...

The Second Law

The horizon area is, assuming the weak energy condition, a non-decreasing function of time, The energy conditions refer to various constraints which can be imposed on a spacetime that any physically reasonable matter distributions in physics are expected to satisfy. ...

The Third Law

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

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 equilbrium for a normal system is analogous to κ constant over the horizon of a stationary black hole The zeroth law of thermodynamics may be succintly stated as: If two thermodynamic systems A and B are in thermal equilibrium, and B and C are also in thermal equilibrium, then A and C are in thermal equilibrium. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ...

The First Law

The left hand side, , 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 thermodynamics is a statement of energy conservation, which contains on its right hand side the term . Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, which exerts a force on those particles that possess the property of electric charge, and is in turn affected by the presence and motion of such particles. ... The first law of thermodynamics, a generalized expression of the law of the conservation of energy, states: the increase in the internal energy of a system is equal to the amount of energy added to the system by heating, plus the amount added in the form of work done on... For the physical concepts, see conservation of energy and energy efficiency. ...

The Second Law

The second law is the statement of Hawking's area theorem. Analogously, the second law of thermodynamics states that the entropy of a closed 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 loosing it's entropy as it falls in, giving a decrease in entroy. Generalised second law introduced as total entropy = black hole entropy + outside entropy The second law of thermodynamics, in a concise form, states that the entropy of any totally isolated system not at thermal equilibrium will tend to increase over time, approaching a maximum value. ... The Thermodynamic entropy S, often simply called the entropy in the context of thermodynamics, is a measure of the amount of energy in a physical system that cannot be used to do work. ...

The Third Law

Extremal black holes cannot have vanishing surface gravity. Stating that κ cannot go to zero is analogous to the third law of thermodynamics which, in its weak formulation, states that it is impossible to reach absolute zero temperature 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. The third law of thermodynamics was developed by Walther Nernst and is thus sometimes referred to as Nernsts theorem. ... Absolute zero is a fundamental lower bound on the temperature of any macroscopic system. ...

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, infinite 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 In astrophysics, the no-hair theorem states that black holes are completely characterized only by three externally observable parameters: mass, electrical charge, and angular momentum. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... Thermal radiation is electromagnetic radiation from an object that is due only to the objects temperature. ... In physics, Hawking radiation is thermal radiation thought to be emitted by black holes due to quantum effects. ...

.

From the first law of black hole mechanics, this determines the multiplicative constant of the Bekenstein-Hawking entropy which is

.

References

• J. M. Bardeen, B. Carter and S. W. Hawking, "The four laws of black hole mechanics", Commun. Math. Phys. 31, 161 (1973).
• J. D. Bekenstein, "Black holes and entropy", Phys. Rev. D 7, 2333 (1973).
• S. W. Hawking, "Black hole explosions?", Nature 248, 30 (1974).
• S. W. Hawking, "Particle creation by black holes", Commun. Math. Phys. 43, 199 (1975).
• S. W. Hawking and G. F. R. Ellis, "The large-scale structure of space-time", Cambridge University Press (1973).
• S. W. Hawking, "The Nature of Space and Time", (1994) [1]