We concentrate on the coherent state near the apparent horizon contained in the spatial slice. We find that motivated from the apparent horizon equation the graph across the horizon can be taken to be populated by radial edges, linking vertices outside and inside the horizon. One then traces over the coherent state within the horizon. Initially we take a particular time slicing of the black hole, which has the spatial slices with zero intrinsic curvature [7]. One such metric which has the time slices as flat is the Lemaitre metric

The, (in units of c = 1) and in the constant slices one can define the induced metric in terms of a “r” coordinate defined as

() on the slice. One gets the metric of the three slice to be

The entire curvature of the space-time metric is contained in the extrinsic curvature or tensor of the constant slices. Now if there exists an apparent horizon somewhere in the above spatial slice, then that is located as a solution to the equation

where, (denote the spatial indices) is the normal to the horizon, the extrinsic curvature in the induced coordinates of the slice, and the trace of the extrinsic curvature. If the horizon is chosen to be the 2-sphere, then in the coordinates of (8), , the apparent horizon equation as a function of the metric reduces to:

Note that the first term of the equation disappears trivially as for any point in the spatial slice. Even at the operator level the can be set to the identity operator in the first approximation, as (being the volume operator) upto normalisations, and in the spherically symmetric metric (upto discretisation constants). Thus the operators in the numerator and denominator cancel and the normalisation conspire, leaving. To understand the rest of the equation in terms of the holonomy and momentum variables of LQG, which are classically measured in the same metric as (8), we use the following regularisation

(N is a constant, a function of the edge lengths and the area bits of the discretisation) and V is the volume operator.

Here has been used as a parameter to identify the operator, and this is mainly a trick. In the continuum limit

As the gauge connection is a function of the Immirzi parameter due to (2), the expectation value of this operator in a coherent state will be a function of the Immirzi parameter. By taking the derivative wrt to the Immirzi parameter we are giving the same status to the parameter as is given to “dimension” in a dimensional regularisation of Feynman diagrams. We let the parameter vary by an infinitesimal amount from its value in the particular quantisation sector, take the derivative, and put its original value in the final answer for the operator. The Formula (13) is facilitated by the fact that the dependence of on the is linear. One way to check whether this gives the proper answer is to take a solved quantum mechanical system and use a similar method there. The most useful example is the Harmonic Oscillator Hamiltonian, which can be written as

The ground state is a coherent state, so we take that as an example. We define the operator

Thus

The regularisation (13) is thus an allowed approximation.

The terms involving the Christoffel connections like include derivatives in the regularised version, the derivatives appear as difference of triads across two vertices. Thus

As a result of this if we impose restrictions on the Christoffel connections and one of the vertices is within the horizon, whereas is outside the horizon, there will be correlations across the horizon.

If one evaluates the expectation value of the apparent horizon equation using the regularised variables in the coherent states, then one would obtain

(is a constant).

The density matrix is obtained as

where is the coherent state wavefunction for the entire slice, a tensor product of coherent state for each edge.

But given this, we concentrate in a “local” region to see the behavior of the horizon

where covers a band of vertices surrounding the horizon one set on a sphere at radius and one set on a sphere at radius within the horizon, as described in [9], and in the figure enclosed. This local density matrix and the correlations due to the apparent horizon equation (19) was used to derive entropy [6]. This entropy counts the number of ways to induce the horizon area using the spin networks, though the constraints have not been appropriately imposed as was obtained using a Chern-Simons theory in [14]. However, the entropy calculation using the coherent states provides a tracing mechanism, and a method to obtain correlations across the horizon which are gravitational in origin. We will henceforth deal with, but we will drop the local label for brevity.

Before we get into the analysis of what QLE evolution means, we take a simple system made up of two subsystems, and examine the consequences of a Hamiltonian evolution. Let the density matrix be defined for a system whose states are given in the tensor product Hilbert space and given by

where is the basis in and is the basis in and are the non-factorisable coefficients of the wavefunction in this basis. Let us label the wavefunction at time to be given by the coefficients. The density matrix is

The reduced density matrix if one traces over is:

We now evolve the system using a Hamiltonian which has the matrix elements, we assume that the Hamiltonian does not factorise, that is there exists interaction terms between the two Hilbert spaces. The evolution equation is:

which in this particular basis gives the density matrix elements at a infinitesimally nearby slice to be

Thus we evolve the “unreduced” density matrix and then trace over the in the evolved slice. The reduced density matrix in the evolved slice is:

This gives:

where A represents the commutator. Clearly the entropy in the evolved slice evaluated as

can be found as

Given the definition of, one gets

In case both the Hamiltonian and the density operator are Hermitian, one obtains

This is clearly calculable, and gives the change in entropy. The term yields corrections, and we ignore it in the first approximation.

To trace the origin of Horizon fluctuations, we must take an observer who is stationed outside the horizon, or in other words is not a freely falling observer. The quasilocal energy is defined using a “surface” integral of the extrinsic curvature with which the surface is embedded in three space. In our case, we take the bounding surface to be the horizon and the quasilocal energy is given by the surface term [11,15].

where is the extrinsic curvature with which the 2- surface, which in this case is the horizon is embedded in the spatial 3-slices, and is the determinant of the two metric defined on the 2-surface. This “quasilocal energy” is measured with reference to a background metric. Thus. We concentrate on the physics observed in an observer stationed at a r = constant sphere.

The metric in static observer’s frame is

The where is the Schwarzschild radius. If we take to be the space-like vector, normal to the 2-surface, then the extrinsic curvature is given by:

and the trace is obviously

In the special slicing of the of the stationary observer the normal to the horizon 2-surface is given by . However, we built the coherent state on the Lemaitre slice. The Lemaitre and the Schwarzschild observer’s coordinates are related by the following coordinate transformations,

The r = const cylinder of the Schwarzschild coordinate corresponds to of the Lemaitre coordinates, and for these. Thus unit translation in the t coordinate coincides with unit translation in the coordinate. Further, the intersection of the r = constant cylinder with a t = constant surface coincides with the intersection of r = constant and the = constant surface. Thus in the initial slice, the QLE Hamiltonian can be written as

The reference frames’ quasilocal energy is a number, it just defines the zero point Hamiltonian. Thus, we replace the classical expressions by operators evaluated at the = constant slice. In the first approximation we simply take the as classical

as this arises due to the coordinate transformation and the norm of the vector in the previous frame. In the rewriting of (37) in regularised LQG variables the Hamiltonian appears rather complicated.

One can rewrite these in a much simpler form, using the apparent horizon equation. Since the Hamiltonian is an integral over the horizon, the variables will satisfy the apparent horizon Equation (10) upto quantum fluctuations. Thus the Hamiltonian operator is then re-written as

where we have used the classical apparent horizon equation (10) (with).

where consists of some dimensionless constants is the 2-dimensional area bit over which is smeared, is a dimensionfull constant which appears to get the dimension less. is the length for the angular edge over which the gauge connection is integrated to obtain the holonomy. The sum over is the set of vertices immediately outside the horizon. The (39) can then be lifted to an operator.

This regularised expression for QLE is for the horizon 2-surface only and would not apply for any other spherical surface in the Schwarzschild space-time.

Let us take the U(1) case to make the calculations easier and observe the action of the QLE Hamiltonian on the evolution of the coherent state. The spin network states are replaced by, , n is an integer and the coherent states are:

is the complexified phase space element in the “n-th” representation.

The QLE operator also takes the simplified form

The prefactors have been clubbed into.

In the calculation of the matrix elements, we drop the label of the edges for the Hamiltonian.

This calculation can be done by putting an assumption that the. In this are completely independent of. It is an allowed assumption, and identifies the dependence of the operator matrix elements, which are otherwise “hidden”. The calculation however introduces an arbitrariness in the formula, which can be fixed by requiring that the expectation value of the Hamiltonian agrees with the classical QLE [10]. However, in this paper we use the “annihilation” operators defined in [16].

This is done by observing that the U(1) coherent states are eigenstates of an annihilation operator defined thus:

The holonomy operator can thus be written as

And the derivative wrt Immirzi parameter of the holonomy which appears in the definition of the Hamiltonian replaced by

The dependence of the operator on the Immirzi parameter is known (2), and thus we could evaluate the derivative

The term

is then computable. Let us take the first term of (41) and find (46). As, (46) gives simply (we drop the “e” label for brevity)

We then concentrate on the 2nd term of the above

where we have used the fact that coherent states resolve unity. It can be shown that the expectation value of the operators in the collapses the integral to point [16]. Thus one obtains from the above

which is real, and thus

this is actually the classical QLE as it should be from.

This is obvious, as the way the Hamiltonian is defined, this is simply a function of the Hilbert space outside the horizon, and the matrix elements of this will not yield anything new. We approximated the horizon sphere by summing over vertices immediately outside the horizon. We could do the same by summing over vertices immediately within the horizon. For the Lemaitre slice, the metric is smooth at the horizon, and one can take the “quantum operators” evaluated at the vertex. In this case however, as the region is within the classical horizon, the norm of the Killing vector is negative, and has components which are imaginary. The.

Thus. In the evaluation of the QLE, the energy would emerge correct in the limit as The regularised Hamiltonian is not Hermitian, and the evolution equation is

And thus the operator which appears in the change of entropy equation is

The “rate of change” of entropy is thus

we extracted the from to get and rewrote the rest of the constants as.

Thus there is a net change in entropy, but, to see if this is Hawking radiation, we have to couple matter to the system.

The SU(2) case is easily reduced to the U(1) case in the actual calculation due to the gauge fixing. This is achieved by making the following observations: To retain the metric as in the same form as the classical metric, we impose the conditions at the operator level

such that the corresponding metric has only the diagonal terms as non-zero. With these additional “constraints” on the operators, we can put the such that each has only one component surviving, let’s say. This also makes the holonomy restricted to the U(1) case, as the gauge connection gets restricted to the and other directions can be put to zero. Thus we can take the holonomy to be diagonal. If the holonomy matrix is off-diagonal the U(1) projection still works out to be the same

The operator is then obtained as

This is same as the U(1) Hamiltonian (upto normalisations). The spin network states also project on to U(1) subgroup, thus giving us the same techniques to use in the calculation of the U(1) states as for this one. To observe this, the non-zero elements for the holonomy matrix

in the j-th representation is given by:

Clearly in the particular case we are considering, the, and m, n = –j and j. Thus the two non-zero elements are

The sum over j in the with the coherent state defined in (3) thus reduces to the U(1) case in the computation of the change in entropy. Thus the rate of change in entropy of a classically spherically symmetric black hole is given by

where the classical holonomies. If we plug in the actual values, we get this to be

where the area element at vertex on the sphere. This change in entropy is totally gravitational in origin, and seems to signify the emergence of “geometry” from within the horizon.

In fact, if we some over the area, we get the (if we set), which would be the change in entropy when the radius of the horizon changes by!