

8.1 The model
In this model the black hole spacetime is described by a
spherically symmetric static metric with a line element of the
following general form written in advanced time
Eddington-Finkelstein coordinates,
where
and
,
, and
is the line element on the two-sphere. Hawking
radiation is described by a massless, conformally coupled quantum
scalar field
with the classical action
where
(
is the dimension of
spacetime), and
is the curvature scalar of the
spacetime it lives in.
Let us consider linear perturbations
off a background Schwarzschild metric
,
with standard line element
We look for this class of perturbed metrics in the form given by
Equation (153) (thus restricting our
consideration only to spherically symmetric perturbations),
and
where
with
and
, and where
is the Hawking temperature. This particular
parametrization of the perturbation is chosen following York’s
notation [297
, 298
, 299
]. Thus the only
non-zero components of
are
and
So this represents a metric with small static and radial
perturbations about a Schwarzschild black hole. The initial quantum
state of the scalar field is taken to be the Hartle-Hawking vacuum,
which is essentially a thermal state at the Hawking temperature and
it represents a black hole in (unstable) thermal equilibrium with
its own Hawking radiation. In the far field limit, the
gravitational field is described by a linear perturbation of
Minkowski spacetime. In equilibrium the thermal bath can be
characterized by a relativistic fluid with a four-velocity
(time-like normalized vector field)
, and temperature in
its own rest frame
.
To facilitate later comparisons with our program
we briefly recall York’s work [297, 298, 299]. (See also the work by
Hochberg and Kephart [135] for a massless
vector field, by Hochberg, Kephart, and York [136] for a
massless spinor field, and by Anderson, Hiscock, Whitesell, and
York [8] for a
quantized massless scalar field with arbitrary coupling to
spacetime curvature.) York considered the semiclassical Einstein
equation,
with
, where
is the Einstein tensor for the background spacetime.
The zeroth order solution gives a background metric in empty space,
i.e, the Schwarzschild metric.
is the linear
correction to the Einstein tensor in the perturbed metric. The
semiclassical Einstein equation in this approximation therefore
reduces to
York solved this equation to first order by using the expectation
value of the energy-momentum tensor for a conformally coupled
scalar field in the Hartle-Hawking vacuum in the unperturbed
(Schwarzschild) spacetime on the right-hand side and using
Equations (159) and (160) to calculate
on the left-hand side. Unfortunately, no exact
analytical expression is available for the
in a Schwarzschild metric with the quantum field in
the Hartle-Hawking vacuum that goes on the right-hand side. York
therefore uses the approximate expression given by Page [231
] which is known to
give excellent agreement with numerical results. Page’s approximate
expression for
was constructed using a
thermal Feynman Green’s function obtained by a conformal
transformation of a WKB approximated Green’s function for an
optical Schwarzschild metric. York then solves the semiclassical
Einstein equation (162) self-consistently to
obtain the corrections to the background metric induced by the
backreaction encoded in the functions
and
. There was no mention of fluctuations or its
effects. As we shall see, in the language of Sec. (4),
the semiclassical gravity procedure which York followed working at
the equation of motion level is equivalent to looking at the
noise-averaged backreaction effects.

