![]() Since the total energy E of the black hole is related to its mass M by Einstein's mass-energy formula: Given that the power of the Hawking radiation is the rate of evaporation energy loss of the black hole: Under the assumption of an otherwise empty universe, so that no matter or cosmic microwave background radiation falls into the black hole, it is possible to calculate how long it would take for the black hole to dissipate: It is indeed an extremely good approximation to call such an object 'black'. The power in the Hawking radiation from a solar mass black hole turns out to be a minuscule 9 × 10−29 watts. It is worth mentioning that the above formula has not yet been derived in the framework of semiclassical gravity. Where is the energy outflow, is the reduced Planck constant, is the speed of light, and is the gravitational constant. Stefan–Boltzmann-Schwarzschild-Hawking power law: Stefan–Boltzmann–Schwarzschild–Hawking black hole radiation power law derivation: Schwarzschild sphere surface area of Schwarzschild radius : Hawking radiation has a black-body (Planck) spectrum with a temperature T given by: Combining the formulas for the Schwarzschild radius of the black hole, the Stefan–Boltzmann law of black-body 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), equation derivation:īlack hole surface gravity at the horizon: The power emitted by a black hole in the form of Hawking radiation can easily be estimated for the simplest case of a nonrotating, non-charged Schwarzschild black hole of mass. When particles escape, the black hole loses a small amount of its energy and therefore of its mass (mass and energy are related by Einstein's equation E = mc²).
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