Re: Hawking radiation and vaccum fluctuation
carlip-nospam_at_physics.ucdavis.edu
Date: 10/26/04
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Date: Tue, 26 Oct 2004 17:55:59 +0000 (UTC)
Saju <saju.pillai@gmail.com> wrote:
[...]
> This idea of a particle escaping away and hence BH loosing mass seems
> to be implying that the particle that escapes away is always the
> "positive" particle. The anti-particle always falls in and subtracts
> from the mass of the BH.
> I am not able to comprehend why the particle that escapes must always
> be positive ? Can someone please explain why this is so ?
Stealing from my own Web page:
There are a number of ways of describing the mechanism responsible
for Hawking radiation. Here's one:
The vacuum in quantum field theory is not really empty; it's filled
with ``virtual pairs'' of particles and antiparticles that pop in
and out of existence, with lifetimes determined by the Heisenberg
uncertainty principle. When such pairs forms near the event horizon
of a black hole, though, they are pulled apart by the tidal forces
of gravity. Sometimes one member of a pair crosses the horizon, and
can no longer recombine with its partner. The partner can then escape
to infinity, and since it carries off positive energy, the energy (and
thus the mass) of the black hole must decrease.
There is something a bit mysterious about this explanation: it requires
that the particle that falls into the black hole have negative energy.
Here's one way to understand what's going on. (This argument is based
roughly on section 11.4 of Schutz's book, A first course in general
relativity.)
To start, since we're talking about quantum field theory, let's
understand what "energy" means in this context. The basic answer is
that energy is determined by Planck's relation, E=hf, where f is
frequency. Of course, a classical configuration of a field typically
does not have a single frequency, but it can be Fourier decomposed into
modes with fixed frequencies. In quantum field theory, modes with positive
frequencies correspond to particles, and those with negative frequencies
correspond to antiparticles.
Now, here's the key observation: frequency depends on time, and in
particular on the choice of a time coordinate. We know this from
special relativity, of course -- two observers in relative motion
will see different frequencies for the same source. In special
relativity, though, while Lorentz transformations can change the
magnitude of frequency, they can't change the sign, so observers
moving relative to each other with constant velocities will at least
agree on the difference between particles and antiparticles.
For accelerated motion this is no longer true, even in a flat
spacetime. A state that looks like a vacuum to an unaccelerated
observer will be seen by an accelerated observer as a thermal
bath of particle-antiparticle pairs. This predicted effect, the
Unruh effect, is unfortunately too small to see with presently
achievable accelerations, though some physicists, most notably
Schwinger, have speculated that it might have something to do
with thermoluminescence. (Most physicists are unconvinced.)
The next ingredient in the mix is the observation that, as it is
sometimes put, ``space and time change roles inside a black hole
horizon.'' That is, the timelike direction inside the horizon is
the radial direction; motion ``forward in time'' is motion ``radially
inward'' toward the singularity, and has nothing to do with what
happens relative to the Schwarzschild time coordinate t.
The final ingredient is a description of vacuum fluctuations. One
useful way to look at these is to say that when a virtual particle-
antiparticle pair is created in the vacuum, the total energy remains
zero, but one of the particles has positive energy while the other
has negative energy. (For clarity: either the particle or the
antiparticle can have negative energy; there's no preference for
one over the other.) Now, negative-energy particles are classically
forbidden, but as long as the virtual pair annihilates in a time less
than h/E, the uncertainty principle allows such fluctuations.
Now, finally, here's a way to understand Hawking radiation. Picture
a virtual pair created outside a black hole event horizon. One of
the particles will have a positive energy E, the other a negative
energy -E, with energy defined in terms of a time coordinate outside
the horizon. As long as both particles stay outside the horizon, they
have to recombine in a time less than h/E. Suppose, though, that in
this time the negative-energy particle crosses the horizon. The
criterion for it to continue to exist as a real particle is now that
it must have positive energy relative to the timelike coordinate inside
the horizon, i.e., that it must be moving radially inward. This can
occur regardless of its energy relative to an external time coordinate.
So the black hole can absorb the negative-energy particle from a vacuum
fluctuation without violating the uncertainty principle, leaving its
positive-energy partner free to escape to infinity. The effect on the
energy of the black hole, as seen from the outside (that is, relative
to an external timelike coordinate) is that it decreases by an amount
equal to the energy carried off to infinity by the positive-energy
particle. Total energy is conserved, because it always was, thoughout
the process -- the net energy of the particle-antiparticle pair was zero.
Note that this doesn't work in the other direction -- you can't have
the positive-energy particle cross the horizon and leaves the negative-
energy particle stranded outside, since a negative-energy particle
can't continue to exist outside the horizon for a time longer than h/E.
So the black hole can lose energy to vacuum fluctuations, but it can't
gain energy.
Steve Carlip
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