Re: White Holes are time-reversed black holes?

Greg Egan

In article <1156788189.062953.291550@p79g2000cwp.googlegroups.com>, Igor
<thoovler@excite.com> wrote:
[snip]
>Prior to the discovery of Hawking radiation, white holes were
> potential candidates for quasars and other powerful emission sources in
> the universe. But Hawking radiation now seems to be the dominant model
> for such things and white holes are hardly talked about any more.

Hawking radiation is not, and never has been, the "dominant model"
for "powerful emission sources in the universe".

The total power emitted in Hawking radiation takes the form

P_H = C / M^2

where C is a constant and M is the mass of the black hole.
In MKS units:

C = 3.5x10^33 watts kg^2

(Source: "Particle emission rates from a black hole: Massless
particles from an uncharged, nonrotating hole" by Don N. Page,
Physical Review D, Vol 13, No 2, 15 January 1976.)

The total power emission from a stellar mass black hole, say
M=2x10^30kg, is then:

P_H = 8.75x10^{-28} watts

i.e. utterly negligible. The time it takes for a black hole to evaporate
from Hawking radiation is approximately:

T_H = 8.6x10^{-18} seconds kg^{-3} M^3
= 2.2x10^66 years per solar mass cubed

Obviously no stellar mass black hole can be a source of substantial power
via Hawking radiation. Hypothetical primordial black holes of much lower
than stellar mass would have greater power output, but then there is no
plausible mechanism for them to *sustain* that output over an extended
period.

For example, suppose we want a black hole to emit 1 solar luminosity of
Hawking radiation, i.e. we require

P_H = 3.8x10^26 watts.

Then its mass would be of the order of 3,000 kg, and its lifetime would be
of the order of 230 nanoseconds. Of course in those 230 nanoseconds it
would radiate its entire rest mass, so its luminosity would undergo a
rapid spike, but the spike would be *extremely brief*.

There are corrections to the power output that need to be applied when
the black hole becomes very small and starts emitting massive particles,
but if we assume for the sake of simplicity that P_H = C/M^2 all the way
down to zero mass for the hole, then:

P_H = 1.46x10^22 watts t^{-2/3}

when t is the time in seconds remaining until the hole has
completely evaporated. If we turn this formula around, the longest
a black hole's Hawking radiation could sustain a luminosity in
excess of a given power would be:

t = (1.46x10^22 watts / P_H)^{3/2}

or in terms of solar luminosity L_s,

t = (3.8x10^{-5} L_s / P_H)^{3/2}

So even a luminosity of .01 L_s would only be exceeded for 230
microseconds.

I am unaware of any observed source whose time profile (or spectrum)
matches the characteristics of Hawking radiation from primordial
black holes. Not even gamma ray bursts come close, let alone the
sustained output from a quasar.