# relation between mass of black holes and tempereture?

As we know, by increasing the mass or energy of a black hole, its event horizon radius will increase, but why its temperature should increase too?
Really I want the relation between mass and temperature of black hole!

• $T=\frac{1}{8 \pi M}$. Check this Wiki article on Hawking radiation. Jan 31, 2017 at 12:58
• Really I want the physical meaning of that. Is it related to the curvature of space which occurs because of this mass? Jan 31, 2017 at 13:11
• This temperature is the temperature of BH radiation in its usual sense. Have you read the article? Jan 31, 2017 at 13:16

## 1 Answer

Higher temperature means it's hotter for a lower mass, i.e., it emits more and higher freq radiation. Micro black holes which evaporate away quickly do so at the end in a burst as they are the hottest. You have to get the details from the paper.

The temperature is the one that would define that of a black body radiator, with the same probability density function for the radiation radiated. For black holes Hawking found that it is inversely proportional to the mass. He also determined that it is proportional to the surface gravity - defined as the acceleration that you'd have to give a body at the horizon (or right outside it) to keep it there without going in. Small black holes have a very high curvature of spacetime near them, thus a very high surface gravity and temperature, and emit at higher frequencies as they get smaller, till they dissappear in a burst.

As they get smaller they will send out more gamma ray radiation.

See about black holes temperature and thermodynamics at https://en.m.wikipedia.org/wiki/Black_hole_thermodynamics

• Thak you, but as I know, curvature of spacetime depends on the mass of objects, so for small BHs, curvature should be smaller that a large BHs, since their mass is smaller. so surface gravity should be smaller, I mean a large black hole should have large temperature, so when a BH evaporates, its temperature should decrease! Feb 2, 2017 at 7:09
• Yes, a bit counterintuitive but still right. The thing is you need to also account for distance and closer in gravity is larger. Remember at r=0 it'd be infinite and and small masses have small schwarzchild radius. So temperature definitely goes higher. Feb 2, 2017 at 15:27