The next step is to calculate the entropy of this
quantum, micro-canonical ensemble. Note first that what matters are
only the surface states. For, the ‘bulk-part’ describes, e.g.,
states of gravitational radiation and matter fields far away from
and are irrelevant for the entropy
of the WIH. Heuristically, the idea then is to
‘trace over’ the bulk states, construct a density matrix
describing a maximum-entropy mixture of surface
states and calculate
. As is usual in entropy
calculations, this translates to the evaluation of the dimension
of a well-defined sub-space
of the surface Hilbert space, namely the linear span
of those surface states which occur in
. Entropy
is given by
.
A detailed calculation [84, 149] leads to the following expression of entropy:
whereOne adopts a ‘phenomenological’ viewpoint to fix
this ambiguity. In the infinite dimensional space of geometries
admitting as their inner boundary, one can fix
one space-time, say the Schwarzschild space-time with mass
, (or, the de Sitter space-time with the cosmological
constant
, or, …). For agreement with
semi-classical considerations in these cases, the leading
contribution to entropy should be given by the Hawking-Bekenstein
formula (87
). This can happen only
in the sector
. The quantum theory is now completely
determined through this single constraint. We can go ahead and
calculate the entropy of any other type I WIH in this theory. The result is again
. Furthermore, in this
-sector, the statistical mechanical temperature of any
type I WIH is given by Hawking’s semi-classical value
[29
, 135]. Thus, we can do one
thought experiment - observe the temperature of one large black
hole from far away - to eliminate the Barbero-Immirzi ambiguity and
fix the theory. This theory then predicts the correct entropy and
temperature for all WIHs with
, irrespective of other
parameters such as the values of the electric or dilatonic charges or the cosmological
constant. An added bonus comes from the fact that the
isolated horizon framework naturally incorporates not only black
hole horizons but also the cosmological ones for which
thermodynamical considerations are also known to apply [99]. The quantum entropy calculation is able
to handle both these horizons in a single stroke, again for the
same value
of the Barbero-Immirzi parameter. In
this sense, the prediction is robust.
Finally, these results have been subjected to
further robustness tests. The first comes from non-minimal
couplings. Recall from Section 6 that
in presence of a scalar field which is non-minimally coupled to gravity, the first
law is modified [123, 183, 184]. The modification suggests
that the Hawking-Bekenstein formula is
no longer valid. If the non-minimal coupling is dictated by the
action
Next, one can consider type II horizons which can
be distorted and rotating. In this case, all the (gravitational,
electro-magnetic, and scalar field) multipoles are required as
macroscopic parameters to fix the
system of interest. Therefore, now the appropriate ensemble is
determined by fixing all these multipoles to lie in a small range
around given values. This ensemble can be constructed by first
introducing multipole moment operators and then restricting the
quantum states to lie in the subspace of the Hilbert space spanned
by their eigenvectors with eigenvalues in the given intervals.
Again recent work shows that the state counting yields the
Hawking-Bekenstein formula (87) for minimally coupled
matter and its modification (96
) for non-minimally
coupled scalar field, for the same
value
of the
Barbero-Immirzi parameter [8, 25].
To summarize, the isolated horizon framework
serves as a natural point of departure for a statistical mechanical
calculation of black hole entropy based on quantum geometry. How
does this detailed analysis compare with the ‘It from Bit’
scenario [187] with which we
began? First, the quantum horizon boundary conditions play a key
role in the construction of a consistent quantum theory of the
horizon geometry. Thus, unlike in the ‘It from Bit’ scenario, the
calculation pertains only to those 2-spheres which are cross-sections of a WIH. One can indeed
divide the horizon into elementary cells as envisaged by Wheeler:
Each cell contains a single puncture. However, the area of these
cells is not fixed but is dictated by the
-label at the puncture. Furthermore, there are not
just 2 but rather
states associated with each
cell. Thus, the complete theory is much more subtle than that envisaged in the
‘It from Bit’ scenario.