Notes on Relativity and Cosmology - Physics Department, UCSB

9.5. BLACK HOLE ASTROPHYSICS AND OBSERVATIONS 267
(
)
∆E = mc 2 1
√
1 − Rs /r − 1 . (9.46)
as measured at r. This is how much energy can be put into x-ray photons and
sent back out. But, on it’s way back out, such photons will decrease in energy by
a factor of √ 1 − R s /r −1. So, the final energy that gets out of the gravitational
field is:
(
)
∆E ∞ = mc 2√ 1
1 − R s /r √
1 − Rs /r − 1
= mc 2 (1 − √ 1 − R s /r). (9.47)
In other words, the total energy released to infinity is a certain fraction of the
energy in the rest mass that fell toward the black hole. This fraction goes to
1 if the mass fell all the way down to the black hole horizon. Again, so long
as r was within a factor of 100 or so of the Schwarzschild radius, this gives an
efficiency comparable to thermonuclear reactions.
I can now say something about the question that we asked at the beginning of
this section. Using direct observations, how strongly can we bound the size of
a black hole candidate? It turns out that one can study the detailed properties
of the spectrum of radiation produced by an accretion disk, and that one can
match this to what one expects from an accretion disk living in the Schwarzschild
geometry. Current measurements focus on a particular (x-ray) spectral line
associated with iron. In the best case, the results show that the region emitting
radiation is within 25R s .
9.5.4 So, where does all of this energy go, anyway?
This turns out to be a very interesting question. There is a lot of energy being
produced by matter falling into a black hole or a neutron star. People are
working very hard with computer models to figure out just how much matter
falls into black holes, and therefore just how much energy is produced. Unfortunately,
things are sufficiently complicated that one cannot yet state results
with certainty. Nonetheless, some very nice work has been done in the last few
years by Ramesh Narayan and his collaborators showing that in certain cases
there appears to be much less energy coming out than there is going in. Where
is this energy going? It is not going into heating up the object or the accretion
disk, as such effects would increase the energy that we see coming out (causing
the object to shine more brightly). If their models are correct, one is forced to
conclude that the energy is truly disappearing from the part of the spacetime
that can communicate with us. In other words, the energy is falling behind the
horizon of a black hole. As the models and calculations are refined over the
next five years or so, it is likely that this missing energy will be the first ‘direct
detection’ of the horizon of a black hole.