Cooking with black holes

kw: musings, black holes

A book I am about halfway through discusses black holes, and a lot of it has to do with Hawking radiation and the implications of that. See the link just above for a detailed treatment in Wikipedia. A briefer explanation is thus:

• Richard Feynmen's work on quantum electrodynamics showed that virtual particles are continually created and destroyed, in complementary pairs, everywhere and always. They provide the properties we associate with "the vacuum."
• Stephen Hawking showed that when a pair of virtual particles gets created just next to the event horizon of a black hole, during the tiny fraction of a second that the particles are physically separated, one may pass through the event horizon.
• The extreme "strain" on space-time caused by the proximity of the black hole reifies the particles, such that one is captured by the black hole and the other escapes, becoming a particle that is seemingly emitted spontaneously by the black hole's event horizon.

This means that a black hole is a black body, in the physics sense, with a temperature and a perfect black-body thermal spectrum. However, for the black holes we expect to find astronomically, this temperature is very low, and the particles emitted have very low energies. How low? The Wikipedia article I linked above has the equations to figure it out. I made a table to help wrap my mind around the concepts. First, the temperature is extremely low for stellar-mass and larger black holes, but can get rather higher for lighter ones. The table will include a 1kg mass to illustrate how high. I also use "E" notation for large and small numbers: 2e14 means 2•10¹⁴.

(I apologize in advance for Blogger's tendency to put too much white space before a table.)
Table: Black Hole Sizes and Radiation Characteristics

Object	Mass, kg	Temp, K	Radius, m	Power, W
Solar mass	2E30	7E-9	2,980	1.5E-32
Earth mass	6E24	0.0024	0.009 (9mm)	1.9E-21
Moon mass	7.4E22	0.19	1.1E-4 (0.11mm)	1.1E-17
Asteroid	1E14	1.4E8	1.5E-13	6.1
1 km3 H2O	1E12	1.4E10	1.5E-15	61,000
1 kg	1	1.4E22	1.5E-22	6.1E38

We expect black holes produced by natural processes to be the mass of the Sun or heavier. The top row of the table shows that the temperature and thermal output of such an object are too low to measure, particularly when the background temperature of the universe is, at present, about 3K. Even the temperature of a Lunar-mass black hole is in a realm that is difficult to measure, less than 1/5 of a Kelvin (Note, a Kelvin is what a degree of Centigrade size is called when the zero point is absolute zero, called zero Kelvins or 0K).

Should any black holes lighter than the Moon exist, the temperature is higher, and can get very high indeed, as seen by the last three rows of the table. I picked a mass for the asteroid to be about the size of the one that clobbered the Dinosaurs. What does a temperature of 1.4E8 degrees mean? 140 million degrees! Such an object, the mass of a mountain, the size of a proton, radiating 6 watts of mainly gamma rays and x-rays, would be a truly dangerous item. (The x-ray machine used to take a chest x-ray emits about a tenth of a watt of x-rays, but only about a tenth of that passes through your chest. A dental x-ray machine is considerably lower power than that. Six watts of such radiation is a lot!)

Comparing the data in these three rows, the regularities are clear. Temperature is exactly proportional to 1/Mass; radius is proportional to Mass; and total energy output is proportional to the square of 1/Mass.

Back to the radiation emission: A 1-kg black hole radiates so furiously, at a temperature of 14 billion trillion degrees, that all its mass would be emitted in less than 1E-21 second. Black holes in the range of a few kilograms and smaller are simply energy bombs! A 3-megaton thermonuclear weapon converts a gram of mass to energy in a microsecond or so. Imagine the conversion of a number of kilograms in less than a nanosecond.

What is my conclusion? Firstly, as to my title: a black hole that emits sufficient thermal energy to "cook" with, does so with penetrating radiation that would cut right through any lead shielding. Plus, your "stove" would tend to swallow up the food if it is put close enough to be heated!

Secondly, no black holes are currently evaporating, because thermal energy is entering them from the universe at a rate much greater than they emit their own. If the universe really does get stretched out by accelerating cosmic expansion until any large (stellar or galactic) black holes that now exist can begin to evaporate, not much will seem to happen for times so long they are hard to imagine: 10⁶⁰ to 10¹⁰⁰ years. But once a large black hole slowly evaporates down to a few million tons, things go rather faster, and the last ton will "evaporate" explosively. It would be a fatal mistake to be within a light-year or less of such an explosion!