# Can a black hole have negative temperature?

Stephen Hawking said that black hole also have temperature and it is related to its mass so in other words a black hole can also be shown to have a negative temperature! I know that nothing is colder than absolute zero and negative temperature are hotter than all positive temperature including infinity, in the case of a black hole is it possible for it to have a negative temperature?

Lab experiments can create negative temperature by using laser cooling on the atoms which are trapped using magnetic field as a cup, next is evaporative cooling which get rid of those warmer atoms. So far so good thereafter they use the magnetic field to make the atoms attract each other and suddenly they are switch to the highest possible energy state and remain stable...

I think I must have misread something but anyway can nature produce negative temperature especially in black hole?

• Stephen Hawking said that black hole also have temperature and it is related to its mass so in other words a black hole can also be shown to have a negative temperature! This is a non sequitur. – Ben Crowell Jun 25 at 16:22

## 3 Answers

The temperature of a black hole cannot be negative. The temperature is given by:

$$T = \frac{\hbar c^3}{8\pi kGM} \tag{1}$$

and obviously this cannot be negative as all the quantities on the right hand side are greater than zero.

However the specific heat is negative and I wonder if this is what you are thinking of. If we differentiate equation (1) with respect to mass we get:

$$\frac{\mathrm dT}{\mathrm dM} = -\frac{\hbar c^3}{8\pi kGM^2}$$

If the black hole gains some heat dQ then it gains a mass given by $c^2\mathrm dM = \mathrm dQ$, and substituting this into the above equation gives:

$$\mathrm dT = -\frac{\hbar c^3}{8\pi kGM^2} \frac{\mathrm dQ}{c^2} = -\frac{\hbar c}{8\pi kGM} \frac{\mathrm dQ}{M}$$

If we write:

$$\mathrm dT = \frac{\mathrm dQ}{MC}$$

where $C$ is the specific heat we get:

$$C = -\frac{8\pi kGM}{\hbar c}$$

The specific heat is negative because if you add energy to a black hole its temperature decreases and if it loses energy by radiation its temperature increases.

• I thought specific heat is measured by the amount of energy needed to raise the temperature by 1℃, and it doesn't make sense for specific heat to have negative value. I think you are talking about hawking radiation that negative matter falls into the black hole! Thanks. – user6760 Dec 14 '15 at 12:27
• @user6760 Specific heat is commonly negative for gravitationally bound systems that expand in size (and get colder) when you add energy. It isn't a special thing for black holes, its common in the outer layers of many stars, and our sun could develop an outer layer with a negative specific heat when it expands. Rather than assuming it doesn't make sense you could study it to see how it works. – Timaeus Dec 14 '15 at 19:29

Stephen Hawking said that black hole also have temperature and it is related to its mass so in other words a black hole can also be shown to have a negative temperature!

I'm under the impression that your reasoning is the following: black holes have negative mass, mass is related to temperature, therefore, black holes have negative temperature! However, black holes do not have negative mass. A black hole is basically a collapsed star so it's mass is the same as the mass of the star before the contraction, minus, of course, the amount of mass lost by the star during the supernova stage.

Also, the concept of negative temperature doesn't really make sense. Temperature is basically the amount of kinetic energy the atoms of a given body have. So, it doesn't make sense to talk about negative temperature, unless you're talking in Celsius or Fahrenheit.

• No I didn't said black hole have negative mass :) – user6760 Dec 18 '15 at 1:04
• Sorry, I misunderstood your reasoning then. – Zek Dec 18 '15 at 1:10

A temperature below absolute zero: Atoms at negative absolute temperature are the hottest systems in the world On the absolute temperature scale, which is used by physicists and is also called the Kelvin scale, it is not possible to go below zero – at least not in the sense of getting colder than zero kelvin. According to the physical meaning of temperature, the temperature of a gas is determined by the chaotic movement of its particles – the colder the gas, the slower the particles. At zero kelvin (minus 273 degrees Celsius) the particles stop moving and all disorder disappears. Thus, nothing can be colder than absolute zero on the Kelvin scale. Physicists have now created an atomic gas in the laboratory that nonetheless has negative Kelvin values. These negative absolute temperatures have several apparently absurd consequences: although the atoms in the gas attract each other and give rise to a negative pressure, the gas does not collapse – a behavior that is also postulated for dark energy in cosmology.

• The question is about black holes. This answer is about negative temperature in general. – Ben Crowell Jun 25 at 16:21