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Combining the formulas for the Schwarzschild radius of the black hole, the Stefan–Boltzmann law of blackbody radiation, the above formula for the temperature of the radiation, and the formula for the surface area of a sphere (the black hole's event horizon), several equations can be derived. The Hawking radiation temperature is: [2] [22] [23]
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 ...
As the Schwarzschild radius is linearly related to mass, while the enclosed volume corresponds to the third power of the radius, small black holes are therefore much more dense than large ones. The volume enclosed in the event horizon of the most massive black holes has an average density lower than main sequence stars.
For example, Schwarzschild spacetime contains an event horizon and so can be associated a temperature. In the case of Schwarzschild spacetime this is the temperature T {\displaystyle T} of a black hole of mass M {\displaystyle M} , satisfying T ∝ M − 1 {\displaystyle T\propto M^{-1}} (see also Hawking radiation ).
The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon.The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which makes exact solutions very difficult to find.
A cosmic event horizon is a real event horizon because it affects all kinds of signals, including gravitational waves, which travel at the speed of light. More specific horizon types include the related but distinct absolute and apparent horizons found around a black hole.
The Hubble parameter can change over time if other parts of the equation are time dependent (in particular the mass density, the vacuum energy, or the spatial curvature). Evaluating the Hubble parameter at the present time yields Hubble's constant which is the proportionality constant of Hubble's law .
The temperature of the small and large horizon in the de Sitter–Schwarzschild can be calculated as the period in imaginary time of the solution, or equivalently as the surface gravity near the horizon. The temperature of the smaller black hole is relatively larger, so there is heat flow from the smaller to the larger horizon.