Is the temperature of a black hole infinite as its mass approaches zero?

[kurious]

According to Hawking, for a black hole:

T = hc^3 / 8pi^3 kGM

as M gets close to zero does the black hole have a nearly infinite temperature?

 

[NateTG]

According to Hawking, for a black hole:

T = hc^3 / 8pi^3 kGM

as M gets close to zero does the black hole have a nearly infinite temperature?

The short answer is: Yes

A slightly longer answer:

People like to use familiar words such as temperature, and rotation to describe black holes, and although those terms have some application, they have connotations that do not apply to black holes.

Specificially, in the case of black hole temperature, it's important (at least in this case) to recognize that black hole temperature is really 'radiation temperature'. When Hawking worked out his predictions for black hole radiation, he discovered that the radiation should be in the same spectrum as black body at a particular temperature. Questions like, "what is the specific heat of a black hole?" are, as far as I know meaningless.

That said, the calculations that I've been exposed to make some simplifying assumptions about the radius of the black hole being large in relation to the separation of the virtual particle pairs. As the black hole starts getting smaller, those assumptions may fail.

Since the smaller black holes radiate more rapidly, there is also a point where a black hole would have to radiate more than it's own mass in order to match the equations. IIRC this occurs at the Plank temperature and the Plank radius.

In fact, it's been speculated in some science fiction that a small black hole represents a plausible mechanism for mater to energy conversion.

 

[R0man]

Yes, as the mass of a black hole approaches zero, the temperature calculated using Hawking's formula will approach infinity. This is because the mass is in the denominator of the equation, and as it gets smaller, the temperature will increase. However, it's important to note that this is a theoretical concept and does not necessarily reflect the physical reality of a black hole. In reality, the temperature of a black hole is determined by its surroundings and the amount of matter and energy falling into it, rather than just its mass. So while the mathematical formula may suggest an infinitely high temperature, it may not necessarily be the case in actuality.