# Age of a black hole

Is there a way to determine the age of a black hole. Suppose 100 Billion years from now, if two black holes have exactly the same mass(say 30 M☉). One of them formed 10 Billion years from now and other formed 20 billion years from now. At t = t0 + 100 Billion years, looking back into the past, Can we predict how old these black holes are ? Is rate of dissipation of Hawking radiations any different for them?

• "Is rate of dissipation of Hawking radiations any different for them?" that would seem to be the most specific question. – Fattie Oct 20 '16 at 13:13

From looking at the Black Hole alone, there is no possibility of determining its age. The state of the Black Hole is fully determined by a few fundamental variables (mass, angular momentum and electric charge). This is the statement of the famous dictum A black hole has no hair. Hawking radiation in special is only dependent on these variables.

You may be able to determine the age of a Black hole by indirect means (e.g., by looking at its surroundings and see how much it is cleaned from matter).

• That is why I specifically asked after time 100 Billions years from now. By that time, surroundings will be pretty much clean. – Knu8 Oct 20 '16 at 12:14
• that's not necessarily the case Knu8. you are out by many orders of magnitude! en.wikipedia.org/wiki/Timeline_of_the_far_future It looks like ordinary galaxies, etc, will still exist say one trillion years from now. But jkn has fully answered the question. – Fattie Oct 20 '16 at 13:10

No you can't say anything about their ages and yes, their Hawking radiation is different... not that you could detect the difference. In more detail:

If your two black holes began with identical mass, but at different times, the younger hole would have lost less mass than the other through Hawking radiation, with an age difference of only 10 Billion years the mass difference would be unmeasurable (with current technology).

If at some point in the distant future two BHs of different masses are encountered, one could not in general tell whether they had been born at the same time with different masses, or at different times with the same mass.

The intensity of Hawking radiation depends on the temperature of the black hole and as a BH evaporates it effectively heats up, and as it heats up it emits more and more Hawking radiation leading to a final burst before whatever happens right at the end.

So it is also true that the older black hole, having lost more mass, will be at a slightly (but unmeasurablly so) higher temperature.

However, I have neglected the temperature of the cosmic microwave background, so far. That is currently about 2.7 Kelvin and until the expansion of the universe has lowered it to below that of a black hole, the black hole will actually absorb more energy from the CMB than it emits through Hawking radiation and will actually have increased (immeasurably) in mass.

This is why it will take of the order of 10^100 years for all black holes to evaporate, compared to which even your nominal 100 billion year interval is a drop in the (cosmic) ocean.

• "So it is also true that the older black hole, having lost more mass, will be at a slightly (but unmeasurablly so) higher temperature." -- No. The temperature (and thus Hawking radiation) of a black hole depends on its mass, period. Two black holes of the same mass would have the same temperature, regardless of their ages. – Peter Erwin Oct 22 '16 at 13:20
• You misunderstand. The black holes in the thought experiment start out with identical masses at different times, therefore the BH that has evolved longest has (CMB etc neglected) lost more mass than the other - esp if you consider the time between the formation of the 1st and 2nd ;) – Julian Moore Oct 22 '16 at 13:54
• Of course you are correct in absolute terms re temp vs mass but the comparison being made is relative – Julian Moore Oct 22 '16 at 14:05
• @JulianMoore In my thought experiment, black holes started at different times with different masses. However at some arbitrary time Tx in timeline of the universe, they have identical masses. – Knu8 Oct 24 '16 at 7:53
• @Knu8 Ah, OK. A Schwarzschild BH is entirely characterised by its mass, so given that it is the only thing you know/can know, it can't say anything about history/age. For two BH with equal masses in the future, one might have started small and gained more mass or they might have evolved identically. Re: Hawking radiation - that is also purely a function mass/horizon area (the two are related) and though at 30Msol it's absolutely negigible would be the same for both (Schwarschild BH: a correction may be expected for charged/rotating BH's, but I don't have a specific answer for those cases) – Julian Moore Oct 24 '16 at 9:56