# The four laws of black hole mechanics

@article{Bardeen1973TheFL, title={The four laws of black hole mechanics}, author={James M. Bardeen and Brandon D. Carter and Stephen William Hawking}, journal={Communications in Mathematical Physics}, year={1973}, volume={31}, pages={161-170} }

Expressions are derived for the mass of a stationary axisymmetric solution of the Einstein equations containing a black hole surrounded by matter and for the difference in mass between two neighboring such solutions. Two of the quantities which appear in these expressions, namely the area A of the event horizon and the “surface gravity” κ of the black hole, have a close analogy with entropy and temperature respectively. This analogy suggests the formulation of four laws of black hole mechanics…

## 2,587 Citations

### The first law of soliton and black hole mechanics in five dimensions

- Physics
- 2014

We derive a mass formula and a mass variation law for asymptotically flat, stationary spacetimes, invariant under two commuting rotational symmetries, in a general five dimensional theory of gravity…

### Thermodynamics of accelerating and rotating black holes

- Physics
- 2013

Thermodynamics of a large family of black holes from electrovacuum solutions of Einstein’s equations is studied. This family includes rotating and non-accelerating black holes with NUT charge, and…

### The first law of black-hole thermodynamics for black holes in string theory

- Physics
- 1998

We investigate thermodynamical properties of four- and five-dimensional black-hole solutions of toroidally compactified string theory. We find an explicit expression for the first law of black-hole…

### The first law of black hole mechanics in the Einstein-Maxwell theory revisited

- PhysicsJournal of High Energy Physics
- 2020

Abstract
We re-derive the first law of black hole mechanics in the context of the Einstein-Maxwell theory in a gauge-invariant way introducing “momentum maps” associated to field strengths and the…

### Entropy for the Interior of a Schwarzschild Black Hole Assuming the Mass is Increasing With Time

- Physics
- 2017

Black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. Here we calculate the entropy corresponding to the…

### Mass formulae of four-dimensional dilaton black holes

- Physics
- 1994

Integral and differential mass formulae of four-dimensional stationary and axisymmetric Einstein--Maxwell dilaton systems are derived. The total mass (energy) of these systems are expressed in terms…

### Thermodynamics of two black holes

- PhysicsJournal of High Energy Physics
- 2020

We study a system of two charged non-rotating black holes separated by a strut. Using the exact solution of the Einstein-Maxwell equations, which describes this system, we construct a consistent form…

### No evidence for violation of the second law in extended black hole thermodynamics

- PhysicsPhysical Review D
- 2019

Recently a number of papers have claimed that the horizon area - and thus the entropy - of near extremal black holes in anti-de Sitter spacetimes can be reduced by dropping particles into them. In…

## References

SHOWING 1-10 OF 18 REFERENCES

### Mass Formula for Kerr Black Holes

- Physics
- 1973

A new mass formula for Kerr black holes is deduced, and is constrasted to the mass formula which is obtained by integrating term by term the mass differential and which consists of three terms…

### The conservation of matter in general relativity

- Physics
- 1970

It is shown that in classical general relativity, if space-time is nonempty at one time, it will be nonempty at all times provided that the energy momentum tensor of the matter satisfies a physically…

### Energy and angular momentum flow into a black hole

- Physics
- 1972

The area of the event horizon round a rotating black hole will increase in the presence of a non-axisymmetric or time dependent perturbation. If the perturbation is a matter field, the area increase…

### Black holes in general relativity

- Physics
- 1972

It is assumed that the singularities which occur in gravitational collapse are not visible from outside but are hidden behind an event horizon. This means that one can still predict the future…

### Kerr Metric Black Holes

- PhysicsNature
- 1970

LYNDEN-BELL has suggested1 that a massive collapsed object may lie at the centre of many galaxies, including our own, and that a continuing flux of matter into such a collapsed object, or “black…

### Axisymmetric Black Hole Has Only Two Degrees of Freedom

- Physics
- 1971

A theorem is described which establishes the claim that in a certain canonical sense the Kerr metrics represent "the" (rather than merely "some possible") exterior fields of black holes with the…

### The Large Scale Structure of Space-Time

- Physics
- 1973

The theory of the causal structure of a general space-time is developed, and is used to study black holes and to prove a number of theorems establishing the inevitability of singualarities under certain conditions.

### Reversible and Irreversible Transformations in Black-Hole Physics

- Physics
- 1970

The concepts of irreducible mass and of reversible and irreversible transformations in black holes are introduced, leading to the formula…

### Killing Horizons and Orthogonally Transitive Groups in Space‐Time

- Mathematics
- 1969

Some concepts which have been proven to be useful in general relativity are characterized, definitions being given of a local isometry horizon, of which a special case is a Killing horizon (a null…

### Commun

- math. Phys. 25, 152—166
- 1972