Hawking Temperature

 

Today is Stephen Hawking’s birthday. He would be 76 + 8 years old today. He was born on 8 January 1942. He celebrated his 76th birthday during his lifetime in 2018. After Einstein, no other scientist has achieved popularity comparable to Hawking’s.

One of Stephen Hawking’s most fundamental contributions to science is Hawking radiation. The gravitational force inside a black hole is so strong that nothing can escape from it, not even light. But in 1974, Stephen Hawking showed theoretically that the emission of energy from a black hole is possible. This energy is essentially thermal energy. If thermal energy is emitted from a black hole, then the black hole must behave like a hot body. That is, a black hole has a temperature, and since thermal energy is being emitted—no matter how small the amount—the black hole is gradually evaporating. And because thermal radiation is emitted, a black hole also has entropy.

Stephen Hawking derived the equation for calculating the temperature of a black hole. This temperature is called the Hawking temperature after his name. If the mass of a black hole is , then its Hawking temperature can be found using the following equation:

Here,
, the Dirac constant,
, the speed of light in vacuum,
, the Boltzmann constant,
, the gravitational constant.

From this equation it is clear that the temperature of a black hole depends on its mass. The greater the mass, the lower the temperature. The black hole with the smallest mass will have the highest temperature. Black holes are formed from dead stars. If we take the minimum mass of a black hole to be equal to the mass of our Sun, then the black hole’s mass would be kilograms. Substituting this value into the equation, we obtain the maximum temperature of such a black hole as kelvin. This temperature is so extremely low that it is impossible to detect with the most advanced instruments we currently have for exploring the universe. That means, although the Hawking temperature can be calculated, it is not yet possible to measure it in practice.

If the theoretical value of a black hole’s temperature is correct, then it is also possible to calculate how long a black hole would take to completely evaporate by emitting thermal radiation.

If we calculate the rate of energy emission from a black hole, the total evaporation time () is given by:

If we again consider the minimum mass of a black hole, the time for complete evaporation obtained from this equation is also impossible to observe in practice. Because after substituting the values of all constants, even the smallest black hole would take about seconds to evaporate completely. The current age of the universe is approximately seconds. It is therefore obvious that observing a black hole completely evaporate solely through thermal radiation is impossible in practice.

There is no error in Stephen Hawking’s calculations. But theoretical calculations are not considered proven until experimental evidence is found. For such experimental proof, we will have to wait until some new method is discovered.

References:
[1] Physics Education, Vol. 59, 055001, 2024.
[2] arXiv:2412.12100v2