# Black Hole Interior from Loop Quantum Gravity

Authors: Leonardo Modesto
Journal: Advances in High Energy Physics
 (2008)

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## Abstract

We calculate modifications to the Schwarzschild solution by using a semiclassical
analysis of loop quantum black hole. We obtain a metric inside the event horizon that coincides
with the Schwarzschild solution near the horizon but that is substantially different at the Planck
scale. In particular, we obtain a bounce of theS2 sphere for a minimum value of the radius and
that it is possible to have another event horizon close to the r=0 point.


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## Body

## 1. Introduction

Quantum gravity, the theory that wants to reconcile
general relativity and quantum mechanics, is one of the major
problems  in theoretical physics today.  General
relativity tells that because also the space-time is dynamical, it is not
possible to study other interactions on a fixed background. The background
itself is a dynamical field.Among the quantum   gravity theories, a theory
called “loop quantum gravity”  [1–5] is the most widespread nowadays. This is one of the
nonperturbative and background independent approaches to quantum gravity
(another nonperturbative approach to quantum gravity is called “asymptotic
safety quantum gravity”  [6]). In the last years, the applications of loop quantum
gravity ideas to minisuperspace models lead to some interesting results to
solve the problem of space-like singularity in quantum gravity.  As shown in cosmology
[7–10], and recently, in black hole
physics [11–16], it is possible to solve the
cosmological singularity problem and the black hole singularity problem by
using the tools and ideas developed in full loop quantum gravity theory. In the
other well-known approach to quantum gravity, the called “asymptotic safety
quantum gravity,”  authors [17, 18], using the GN running coupling constant obtained in
“asymptotic safety quantum gravity,”  have showed that nonperturbative
quantum gravity effects give a much less singular Schwarzschild metric and that
for particular values of the black hole mass it is possible to have the
formation of another event horizon.In this paper,   we study the space time inside the
event horizon at the semiclassical level using a constant polymeric parameterδ (see [19] for an analysis of the black hole interior using a
nonconstant polymeric parameter). We consider the Hamiltonian constraint
obtained in [15, 16]. In
particular, we study the Hamiltonian constraint introduced in the first paper
of reference [15, 16],
where the authors have taken the general version of the constraint for real
values of the Immirzi parameter γ.This paper is organized as follows. In
Section2,  we briefly recall the Schwarzschild solution inside
the event horizon (r<2MGN) of   [15, 16]. In Section 3, we
introduce the Hamiltonian constraint in terms of holonomies and then the
relative trigonometric form solving the Hamilton equations of motion. In
Section 4,  we give the metric form of the solution,
and we discuss the new physics suggested by loop quantum gravity.


## 2. Schwarzschild solution inside the
event horizon in Ashtekar variables

We recall the
classical Schwarzschild solution inside the event horizon [15, 16]. For the homogeneous but
nonisotropic Kantowski-Sachs space-time Ashtekar's, connection and density
triads  are (after the fixing of a residual global SU(2) gauge symmetry on the spherically reduced
phase space [15, 16])
(2.1)A=cτ3dx+bτ2dθ−bτ1sinθdϕ+τ3cosθdϕ,E=pcτ3sinθ∂∂x+pbτ2sinθ∂∂θ−pbτ1∂∂ϕ.The components variables in the
phase space can be read from the symmetric-reduced connection and density triad
we can read the components variables in the phase space: (b,pb), (c,pc).
The Poisson algebra is {c,pc}=2γGN, {b,pb}=γGN.
Following [15, 16], we
recall that the classical Hamiltonian constraint in terms of the components
variables is(2.2)𝒞H=−12γGN[(b2+γ2)pbb+2cpc]in the gauge N=γsgn(pc)|pc|/16πGNb.
Hamilton equations of motion are(2.3)b˙={b,𝒞H}=−b2+γ22b,p˙b={pb,𝒞H}=12[pb−γ2pbb2],c˙={c,𝒞H}=−2c,p˙c={pc,𝒞H}=2pc.Solutions of (2.3) using the time
parameter t≡eT and redefining the integration constant ≡eT0=m (see the first of papers in [15, 16])
are(2.4)b(t)=±γ2m/t−1,pb(t)=pb(0)t(2m−t)c(t)=∓γmt−2,pc(t)=±t2.This is exactly the
Schwarzschild solution inside the event horizon as you can verify passing to
the metric form defined by hab=diag(pb2/pc,pc,pcsin2θ)  (m contains the gravitational constant parameter GN).


## 3. Semiclassical dynamics from loop
quantum gravity

We recall now the Hamiltonian constraint coming from
“loop quantum black hole”  [15, 16]  in terms of the explicit trigonometric form of holonomies.
The Hamiltonian constraint depends explicitly on the parameter δ that defines the length of the curves along
which  we integrate the connections to define the
holonomies [15, 16]. We use
the notation 𝒞δ for the hamiltonian constraint to stress the
dependence on the parameter δ.
The Hamiltonian constraint in terms of holonomies is(3.1)𝒞δ=−N(8πGN)2γ3δ3Tr[∑ijkϵijkhi(δ)hj(δ)hi(δ)−1hj(δ)−1hk(δ){hk(δ)−1,V}+2γ2δ2τ3h1(δ){h1(δ)−1,V}]=−N2GNγ2{2sinδcδsinδbδ|pc|+(sin2δbδ2+γ2)pbsgn(pc)|pc|},where V=4π|pc|pb is the spatial section volume, and we have
calculated the Poisson brackets using the symplectic structure given in
Section 2. The holonomies are(3.2)h1δ=cosδc2+2τ3sinδc2,h2δ=cosδb2−2τ1sinδb2,h3δ=cosδb2+2τ2sinδb2.Now,  we can solve exactly the new Hamilton equations of
motion if we take a gauge, where the equations for the canonical pairs (b,pb) and (c,pc) are decoupled. A useful gauge is N=γ|pc|sgn(pc)δ2/sinδb  and in this particular gauge, the Hamiltonian
constraint becomes(3.3)𝒞δ=−12γGN{2sinδcpc+(sinδb+γ2δ2sinδb)pb}.From (3.3), we obtain two
independent  sets  of equations of motion on the
phase space:(3.4)c˙=−2sinδc,p˙c=2δpccosδc,b˙=−12(sinδb+γ2δ2sinδb),p˙b=δ2cosδb(1−γ2δ2sin2δb)pb.Solving the first two equations
for c(T) and pc(T), we obtain(3.5)c(T)=2δarctan(∓γδmpb(0)2e−2δT),pc(T)=±e−2δT[(γδmpb(0)2)2+e4δT].Introducing a new time
parameterization t≡eδT,
we obtain(3.6)c(t)=2δarctan(∓γδmpb(0)2t2)→∓γmpb(0)t2,pc(t)=±1t2[(γδmpb(0)2)2+t4]→±t2.In (3.6), we have calculated the
small δ limit for the solutions  c(t) and pc(t),
obtaining the Schwarzschild solution of paragraph one in (2.4) and calculated in
[15, 16]. A substantial
difference between the Schwarzschild solution and the solution (3.6) is that in
the second case, there is an absolute minimum in tmin=(γδmpb(0)/2)1/2,
where pc  assumes the value pc(tmin)=γδmpb(0)>0.
In  Section 4, we will analyze the new physics
coming from loop quantum gravity Hamiltonian constraint.At this point, we integrate the equation of motion forb(t), obtaining the following solution (we write the
solution in the time coordinate t):(3.7)cosδb=1+γ2δ2[1+γ2δ2+1−(2m/t)1+γ2δ2(1+γ2δ2−1)1+γ2δ2+1+(2m/t)1+γ2δ2(1+γ2δ2−1)].To calculate pb(t), we introduce the solutions c(t),pc(t),b(t) in the Hamiltonian constraint, and we obtain pb(t) from the algebraic constraint equation Cδ=0.
The solution of this equation gives pb(t) as function of the other phase space
functions:(3.8)pb(t)=−2sinδcsinδbpcsin2δb+γ2δ2.To obtain the explicit form of pb(t) in terms of the time coordinate t, it is sufficient to introduce in (3.8)
the
solution cosδb calculated in (3.7).We note that the solution is homogeneous until
it satisfied  the trigonometric propertycosδb≥−1.
Using (3.7), we can calculate the variable t value (we define this t∗) until the solution is of Kantowski-Sachs-type,
and we obtain(3.9)t∗=2m(1+γ2δ2−11+γ2δ2+1)2/1+γ2δ2.However,  we are interested in
the semiclassical limit of the solution defined by δ≪1,
then in this particular limit t∗~0 (see also  Section
4).Following [15, 16],
we study the trajectory on the plane pc−pb, and we compare the result with the
Schwarzschild solution of Section one. In Figure 1, we have a parametric plot
of pc and pb (for m=10) and γδ~1 to amplify the quantum gravity effects in the
plot  (see Section 4). We can observe the substantial
difference with the classical case. In the classical case (red line in Figure
1), pc→0 for t→0, and this point corresponds to the classical
singularity. In the semiclassical case instead, we start from t=2m,  where pc→(2m)2 and pb→0 (this point corresponds to the Schwarzschild
horizon) and decreasing t,  we arrive to a minimum value for pc,m≡pc(tmin)>0.
From this point, pc starts to grow another time until it assumes a
maximum value for pb=0 that corresponds to a new horizon in t=t∗ localized (see  Section
4, where we study the metric form of the solution). Our analysis
refers to the region t∗≤t≤2m, and the plot in Figure 1 refers to this time
interval. The solution calculated is regular in the region t∗≤t≤2m;
in fact the cotriad ω [15, 16] defined by (it is the inverse of the triad E)(3.10)ω=sgn(pc)|pb|τ3|pc|dx+sgn(pb)|pc|τ2dθ−sgn(pb)|pc|τ1sinθdϕis regular for all pc in the region t∗≤t≤2m.Semiclassical
dynamical trajectory in the planepb−pc. The plots for pc>0 and for pc<0 are disconnected and symmetric, but we plot
only the positive values of pc.
The red trajectory corresponds to the classical
Schwarzschild solution and the green trajectory
corresponds to the semiclassical solution (the
green and red curves are continuum curves). In the plot on the right, we have
enlarged the region near the pb axis.
(a)(b)


## 4. Metric form of the solution

In this
section, we present the metric form of the solution and we give a plot for any
component of the Kantowski-Sachs metricds2=−N2(t)dt2+X2(t)dr2+Y2(t)(dθ2+sinθdϕ2).
We start recalling the relation between connection and metric
variables:(4.1)Y2(t)=|pc(t)|,X2(t)=pb2(t)|pc(t)|,N2(t)=γ2δ2|pc(t)|(16πGN)2t2sin2δb.
We give now the explicit form of the metric
components in terms of the temporal coordinate t.
The lapse function N(t) is (4.2)(16πGN)2N2(t)=γ2δ2[(γδm/2t2)2+1]1−(1+γ2δ2)[(M+1−Q)/(M+1+Q)]2,
where M denotes 1+γ2δ2 and Q denotes (2m/t)1+γ2δ2(1+γ2δ2−1).   In Figure 2, we have compared
the classical Schwarzschild solution inside the event horizon with   the solution
(4.2)
for m=10 and γδ~1 (we have taken γδ~1 to amplify, in the plot, the loop quantum
gravity modifications at the Planck scale). We can observe that the two solutions
are identically when we approach to the event horizon (in which t=20 in the units used in the plot) but are very
different when we go toward t~0.
As we have explained in Section 3, we consider the
region t>t∗ and for t=t∗ the lapse function diverges N2(t∗)→+∞.
The semiclassical solution has a minimum before diverging in t=t∗.
In the classical solution instead (it is represented in red in Figure 2), N2(t) is very small for t=t∗ and  it  goes to
zero for t→0.Figure 2
Plot of the
lapse functionN2(t) for m=10 and γδ~1 (in the horizontal axis, we have the temporal
coordinate t and in the vertical axis, we
have  the lapse function). The red trajectory
corresponds to the classical Schwarzschild solution
inside the event horizon, and the green trajectory  corresponds  to the semiclassical solution.The anisotropy functionX2(t) is related to pb(t) and pc(t) by (4.1), then by introducing (3.8) and (3.6) in
the second relation of (4.1),
we obtain(4.3)X2(t)=(2γδm)2(1−(1+γ2δ2)[(M+1−Q)/(M+1+Q)]2)t2(1+γ2δ2)2(1−[(M+1−Q)/(M+1+Q)]2)2[(γδmpb(0)/2)2+t4],
where M  denotes 1+γ2δ2 and Q denotes(2m/t)1+γ2δ2(1+γ2δ2−1). Figure 3
represents a plot of X2(t),
in this case too, the semiclassical solution reduces to the classical solution
when t  approaches  the
horizon but it is substantially different in the Planck region (we recall that
in the plot, we have chosen γδ~1 to amplify the quantum correction  to
Schwarzschild solution, but a semiclassical analysis is correct for δ~10−33).Figure 3
Plot ofX2(t) for m=10 and γδ~1 (in the horizontal axis, we have the temporal
coordinate t, and in the vertical axis we have Y2(t)). The red trajectory
corresponds   to the classical Schwarzschild solution
and the green trajectory  corresponds  to the
semiclassical solution.(In [15, 16], the spectrum of the
operator p^c was calculated as follows:(4.4)p^c|μ,τ〉=γlP2τ|μ,τ〉.In this paper, we have used
dimensionless variables then the parameter δ,
which is related to the area eigenvalues by (4.4), is of order δ~10−33.
The correct coefficient is 23 and it is calculated in the first of [15, 16] comparing the area
eigenvalues in the reduced Kantowski-Sachs model with the minimum area
eigenvalue in full loop quantum gravity [20, 21].)For the anisotropy as well as for the lapse function,
it is important to remember that the solution refers to the regiont>t∗,
while for t=t∗,
the anisotropy goes toward zero, X(t∗)→0.
We can conclude that for t=t∗,
we have another event horizon; in fact for this particular value of the time
coordinate, the lapse function diverges and contemporary the anisotropy goes to
zero. This result is qualitatively similar to the modified Schwarzschild
solution obtained in asymptotic safe gravity [6] for particular values of the
black hole mass [17, 18].
However, t∗ is very small in our semiclassical analysis,
and in this region, a complete quantum analysis of the problem is inevitable as
developed in [15, 16].The metric componentY2(t) represents the square radius of the
two-sphere S2 and it is related to the density triad
component pc(t) by the first relation reported in (4.1). Using
the solution (3.6), we obtain(4.5)Y2(t)=1t2[(γδmpb(0)2)2+t4].In Figure 4, we have a plot of Y2(t) and we can note a substantial difference with
the classical solution. In the classical case, the S2 two-sphere goes to zero for t→0,
in our semiclassical solution instead the S2 sphere bounces on a minimum value of the
radius, which is Y2(tmin)=γδm,
and it expands again to infinity for t→0.
(We have taken the integration parameter pb(0)=1 to  match  with the
classical Schwarzschild solution near the horizon, see (2.4) and the first of
[15, 16].)
The minimum
of Y2(t) corresponds to the time coordinate tmin=(mγδ/2)1/2 and tmin≫t∗,
in fact t∗~mδ4 but tmin~(mδ)1/2,
then for δ→0 (we have showed that δ~10−33), we obtain t∗≪tmin.Figure 4
Plot ofY2(t) for m=10  and γδ~1 (in the horizontal axis, we have the temporal
coordinate t;
in the vertical axis, we have Y2(t)). The red trajectory
corresponds to the classical Schwarzschild
solution, and the green trajectory corresponds  to
the semiclassical solution.In Figure5, we have a plot of the spatial section
volume V~X(t)Y2(t) and we can see that the semiclassical volume
has a substantially different structure at the Planck scale, where it shows a
maximum for t>t∗ and it goes to zero for t=t∗.
The volume goes toward zero on the event horizons
but this is not a problem for the singularity resolution because the horizons
are coordinate singularities and not essential singularities.Figure 5
Plot of the
spatial section volumeV~X(t)Y2(t) for m=10 and γδ~1 (in the horizontal axis, we have the temporal
coordinate t). The red trajectory
corresponds  to the classical volume, and the green
trajectory  corresponds to the semiclassical one.
From the pictures, it is possible to note that the semiclassical volume (green
line) is zero for t=t∗.Quantum Ambiguities and Semiclassical Solution
In this paragraph, we want to compare the quantum
spectrum of the operator1/|pc|^ with the semiclassical solution (4.5).  At the
quantum level, the spectrum of 1/|pc|^,
for a generic SU(2) representation j is [22](4.6)1|pc,j|^|μ,τ〉=(3γ1/2δlPj(j+1)(2j+1)∑k=−jk=j[k(|τ|−|τ−2kδ|)])2|μ,τ〉.To compare the quantum spectrum
with the semiclassical solution, we must have a relation between the eigenvalue τ and the temporal coordinate t.
We calculate this relation comparing the large τ limit of (4.6) and the semiclassical solution
near the horizon. The limit of (4.6) for large eigenvalues
gives(4.7)1|pc,j|^|μ,τ〉→τ1γlP2|τ||μ,τ〉,and on the other side, we know
that near the event horizon 1/|pc|→1/t2,
then comparing with (4.7), we obtain τ=t2/γlP2.
At this point, we have all the ingredients to compare the quantum operator
spectrum with the semiclassical solution.   From the plot in Figure 6, it is natural to
interpret the semiclassical solution as the smooth  approximation
of the quantum operator spectrum but the similarity between semiclassical and
quantum  spectra  is very stringent only if we choose
a particular relation between the black hole mass and the SU(2) representation j (in Figure 6, we have chosen m=400 and j=100). Using an heuristic argument, we can obtain
the general relation between m and j.
The relation is m=4j and now we go to show the validity of this
mass quantization formula.Figure 6
In this plot,
we compare the semiclassical solution1/Y2(t) and the spectrum of the quantum operator 1/|pc|^  for j=100 and m=400.
The semiclassical solution is represented by a red line and the quantum
spectrum by the green line one.In Figure7, we have represented with a green line the
quantum spectrum and with a red line the semiclassical solution for some values
of the representation j and of the mass m.
This plot suggests the possibility to interpret the representation ambiguities
in (4.6)
as a label for the mass m  (this idea
remembers  a recent result about the possibility to
see ordinary matter as particular states in pure loop quantum gravity [23]). In fact in the
semiclassical solution, we have a free parameter that corresponds to the black
hole mass, and on the other side in the quantum spectrum, we have the
representation j as a free parameter. If we interpret the
semiclassical solution as the smooth approximation of the quantum spectrum, it
is possible to  match   the time coordinate of the
maximum for the two solutions. This is possible only if we choose a particular
relation between m and the representation j.
To obtain this relation, we calculate the derivative of the spectrum (4.6)  with
respect to τ and we evaluate the derivative in τ=tmin2/γ=mδ/2  (t is dimensionless in our
analysis)(4.8)∂τ(1pτ,j)|τ=mδ/2=32δj(j+1)(2j+1)∑k=−jk=j[k(2m−2m−4k)],where pτ is the eigenvalue of 1/|pc|^.
Observing (4.8), we see that in the 1/|pc|^ spectrum the relative and absolute maximums
correspond to points, where the derivative is divergent. Those points are in m=4j  localized and this relation is also the mass
quantization formula in Planck units. For any fixed value of the representation j, the classical black hole mass corresponds to
the absolute maximum of the quantum spectrum in such representation.Figure 7
In this plot,
we compare the semiclassical solution1/Y2(t) and the spectrum of the quantum operator 1/|pc|^    to   three
particular  values  of the pair (j,m).
From the left to the right in the plot, we consider four particular values of
the pairs (1/2,2), (1,4), (3/2,6), (2,8), and γδ~1.
The semiclassical solution is represented by the red line and the quantum
spectrum by the green one.


## 5. Conclusions

In this paper,
we have solved the Hamilton equation of motion for the Kantowski-Sachs
space-time using the regularized Hamiltonian constraint suggested by loop
quantum gravity.  We have obtained a solution reproducing the Schwarzschild
solution near the event horizon but that is substantially different in the
Planck region near the pointr=0,
where the singularity is (classically) localized. The structure of the solution
suggests the possibility to have another event horizon near the point r=0 (this is similar to the result in “asymptotic
safety quantum gravity”  [17, 18], but the radius of such horizon is smaller than the
Planck length and in this region a complete quantum analysis of the problem
is inevitable   [15, 16]).Another interesting result is related to theS2 sphere part of the three
metrics. We obtain that in the semiclassical
analysis, the radius of the two-sphere does not
vanish, as in the classical case, but the sphere
bounces on a minimum radius and it expands again to infinity. The solution is
summarized in  Table 1.Table 1gμνSemiclassicalClassical−N2(t)−γ2δ2[(γδm2t2)2+1]1−(1+γ2δ2)[1+γ2δ2+1−(2m/t)1+γ2δ2(1+γ2δ2−1)1+γ2δ2+1+(2m/t)1+γ2δ2(1+γ2δ2−1)]2−1(2m/t)−1X2(t)(2γδm)2(1−(1+γ2δ2)[1+γ2δ2+1−(2m/t)1+γ2δ2(1+γ2δ2−1)1+γ2δ2+1+(2m/t)1+γ2δ2(1+γ2δ2−1)]2)t2(1+γ2δ2)2(1−[1+γ2δ2+1−(2m/t)1+γ2δ2(1+γ2δ2−1)1+γ2δ2+1+(2m/t)1+γ2δ2(1+γ2δ2−1)]2)2[(γδm2)2+t4]2mt−1Y2(t)1t2[(γδm2)2+t4]t2Using a heuristic
argument, we have calculated the mass quantization formula comparing the
semiclassical and quantum  spectra  of the inverse of
theS2 sphere square radius, 1/|pc|.
Our arguments suggests that the mass spectrum formula m=4j.It is possible that the semiclassical analysis
performed here will shed light on the problem of the “information loss”
in the process of black hole formation and evaporation. See, in particular
[24] for a possible
physical interpretation of the black hole information loss problem.



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