Research Article Black Hole Interior from Loop Quantum Gravity

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1 Advances in High Energy Physics Volume 008, Article ID 5990, 1 pages doi:.1155/008/5990 Research Article Black Hole Interior from Loop Quantum Gravity Leonardo Modesto Department of Physics, Bologna University, and INFN Bologna, Via Irnerio 6, 016 Bologna, Italy Correspondence should e addressed to Leonardo Modesto, lmodesto@perimeterinstitute.ca Received Septemer 008; Accepted 16 Novemer 008 Recommended y K. S. Viswanathan We calculate modifications to the Schwarzschild solution y using a semiclassical analysis of loop quantum lack hole. We otain a metric inside the event horizon that coincides with the Schwarzschild solution near the horizon ut that is sustantially different at the Planck scale. In particular, we otain a ounce of the S sphere for a minimum value of the radius and that it is possile to have another event horizon close to the r 0 point. Copyright q 008 Leonardo Modesto. This is an open access article distriuted under the Creative Commons Attriution License, which permits unrestricted use, distriution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Quantum gravity, the theory that wants to reconcile general relativity and quantum mechanics, is one of the major prolems in theoretical physics today. General relativity tells that ecause also the space-time is dynamical, it is not possile to study other interactions on a fixed ackground. The ackground 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 nonperturative and ackground independent approaches to quantum gravity another nonperturative 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 prolem of space-like singularity in quantum gravity. As shown in cosmology 7, and recently, in lack hole physics 11 16, it is possile to solve the cosmological singularity prolem and the lack hole singularity prolem y 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 G N running coupling constant otained in asymptotic safety quantum gravity, have showed that nonperturative quantum gravity effects give a much less singular Schwarzschild metric and that for particular values of the lack hole mass it is possile to have the formation of another event horizon.

2 Advances in High Energy Physics In this paper, we study the space time inside the event horizon at the semiclassical level using a constant polymeric parameter δ see 19 for an analysis of the lack hole interior using a nonconstant polymeric parameter. We consider the Hamiltonian constraint otained 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 Section, we riefly recall the Schwarzschild solution inside the event horizon r < MG N of 15, 16. InSection 3, we introduce the Hamiltonian constraint in terms of holonomies and then the relative trigonometric form solving the Hamilton equations of motion. In Section, we give the metric form of the solution, and we discuss the new physics suggested y loop quantum gravity.. Schwarzschild solution inside the event horizon in Ashtekar variales We recall the classical Schwarzschild solution inside the event horizon 15, 16. For the homogeneous ut nonisotropic Kantowski-Sachs space-time Ashtekar s, connection and density triads are after the fixing of a residual gloal SU gauge symmetry on the spherically reduced phase space 15, 16 A cτ 3 dx τ dθ τ 1 sin θdφ τ 3 cos θdφ, E p c τ 3 sin θ x p τ sin θ θ p τ 1 φ..1 The components variales in the phase space can e read from the symmetric-reduced connection and density triad we can read the components variales in the phase space:, p, c, p c. The Poisson algera is {c, p c } γg N, {, p } γg N. Following 15, 16, we recall that the classical Hamiltonian constraint in terms of the components variales is C H 1 [ ( γ )p ] γg N cp c. in the gauge N γsgn p c p c /16πG N. Hamilton equations of motion are ḃ {, C H } γ, ṗ { [ } 1 p, C H ċ { c, C H } c, ṗc { p c, C H } pc. p γ p ],.3 Solutions of.3 using the time parameter t e T and redefining the integration constant e T 0 m see the first of papers in 15, 16 are t ± γ m/t 1, p t p 0 t m t c t γmt, p c t ± t..

3 Leonardo Modesto 3 This is exactly the Schwarzschild solution inside the event horizon as you can verify passing to the metric form defined y h a diag ( p /p c,p c,p c sin θ ) m contains the gravitational constant parameter G N. 3. Semiclassical dynamics from loop quantum gravity We recall now the Hamiltonian constraint coming from loop quantum lack 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.WeusethenotationC δ for the hamiltonian constraint to stress the dependence on the parameter δ. The Hamiltonian constraint in terms of holonomies is C δ [ N 8πG N γ 3 δ Tr 3 ijk { N G N γ sin δc δ ɛ ijk h δ i sin δ δ h δ j h δ 1 i h δ 1 j h δ k { },V k h δ 1 ( pc sin ) δ p γ sgn ( ) p c δ }, p c { } ] γ δ τ 3 h δ 1 h δ 1 1,V where V π p c p is the spatial section volume, and we have calculated the Poisson rackets using the symplectic structure given in Section. The holonomies are 3.1 h δ 1 h δ h δ 3 cos δc τ 3 sin δc, cos δ τ 1 sin δ, cos δ τ sin δ. 3. Now, we can solve exactly the new Hamilton equations of motion if we take a gauge, where the equations for the canonical pairs, p and c, p c are decoupled. A useful gauge is N γ p c sgn p c δ / sin δ and in this particular gauge, the Hamiltonian constraint ecomes C δ 1 { ( sinδc p c sin δ γ δ } )p. 3.3 γg N sin δ From 3.3, we otain two independent sets of equations of motion on the phase space: ḃ 1 ċ sinδc, ṗ c δp c cos δc, ( sin δ γ δ ), ṗ δ (1 cos δ γ δ sin δ sin δ ) p. 3.

4 Advances in High Energy Physics Solving the first two equations for c T and p c T, we otain c T ( δ arctan γδmp 0 e ), δt [( 0 ) ] γδmp p c T ±e δt e δt. 3.5 Introducing a new time parameterization t e δt,weotain c t δ arctan ( ) γδmp 0 γmp 0, t t p c t ± 1 [( 0 ) ] γδmp t ± t. t 3.6 In 3.6, we have calculated the small δ limit for the solutions c t and p c t, otaining the Schwarzschild solution of paragraph one in. and calculated in 15, 16. A sustantial difference etween the Schwarzschild solution and the solution 3.6 is that in the second case, there is an asolute minimum in t min γδmp 0 / 1/, where p c assumes the value p c t min γδmp 0 > 0. In Section, we will analyze the new physics coming from loop quantum gravity Hamiltonian constraint. At this point, we integrate the equation of motion for t, otaining the following solution we write the solution in the time coordinate t : 1 γ cos δ 1 γ δ δ 1 m/t 1 γ δ ( ) 1 γ δ 1 1 γ δ 1 m/t 1 γ δ ( ). 1 γ δ To calculate p t, we introduce the solutions c t, p c t, t in the Hamiltonian constraint, and we otain p t from the algeraic constraint equation C δ 0. The solution of this equation gives p t as function of the other phase space functions: p t sinδc sin δ p c sin δ γ δ. 3.8 To otain the explicit form of p t in terms of the time coordinate t,itissufficient to introduce in 3.8 the solution cos δ calculated in 3.7.

5 Leonardo Modesto 5 We note that the solution is homogeneous until it satisfied the trigonometric property cos δ 1. Using 3.7, we can calculate the variale t value we define this t until the solution is of Kantowski-Sachs-type, and we otain t 1 γ δ 1 m /1 γ δ γ δ 1 However, we are interested in the semiclassical limit of the solution defined y δ 1, then in this particular limit t 0 see also Section. Following 15, 16, we study the trajectory on the plane p c p, and we compare the result with the Schwarzschild solution of Section one. In Figure 1, we have a parametric plot of p c and p for m and γδ 1 to amplify the quantum gravity effects in the plot see Section. We can oserve the sustantial difference with the classical case. In the classical case red line in Figure 1, p c 0fort 0, and this point corresponds to the classical singularity. In the semiclassical case instead, we start from t m, where p c m and p 0 this point corresponds to the Schwarzschild horizon and decreasing t, we arrive to a minimum value for p c,m p c t min > 0. From this point, p c starts to grow another time until it assumes a maximum value for p 0 that corresponds to a new horizon in t t localized see Section, where we study the metric form of the solution. Our analysis refers to the region t t m,andtheplotinfigure 1 refers to this time interval. The solution calculated is regular in the region t t m; in fact the cotriad ω 15, 16 defined y it is the inverse of the triad E ω sgn ( p c ) p τ3 pc dx sgn ( p ) pc τ dθ sgn ( p ) pc τ1 sin θdφ 3. is regular for all p c in the region t t m.. 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 metric ds N t dt X t dr Y t dθ sin θdφ. We start recalling the relation etween connection and metric variales: Y t pc t, X t p t pc t, N t γ δ pc t ( 16πGN ) t sin δ..1 We give now the explicit form of the metric components in terms of the temporal coordinate t. The lapse function N t is ( ) 16πGN N γ δ [( γδm/t ) ] 1 t 1 ( 1 γ δ )[ M 1 Q / M 1 Q ],.

6 6 Advances in High Energy Physics p c p c p p a Figure 1: Semiclassical dynamical trajectory in the plane p p c. The plots for p c > 0andforp c < 0are disconnected and symmetric, ut we plot only the positive values of p c. 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 p axis. 8 N 6 5 Figure : Plot of the lapse function N t for m 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. t 15 0 where M denotes 1 γ δ and Q denotes m/t 1 γ δ 1 γ δ 1.InFigure, we have compared the classical Schwarzschild solution inside the event horizon with the solution. for m and γδ 1 we have taken γδ 1 to amplify, in the plot, the loop quantum gravity modifications at the Planck scale. We can oserve that the two solutions are identically when we approach to the event horizon in which t 0 in the units used in the plot ut 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 N t. The semiclassical solution has a minimum efore diverging in t t. In the classical solution instead it is represented in red in Figure, N t is very small for t t and it goes to zero for t 0.

7 Leonardo Modesto X t Figure 3: Plot of X t for m and γδ 1 in the horizontal axis, we have the temporal coordinate t,and in the vertical axis we have Y t. The red trajectory corresponds to the classical Schwarzschild solution and the green trajectory corresponds to the semiclassical solution. t 15 0 The anisotropy function X t is related to p t and p c t y.1, then y introducing 3.8 and 3.6 in the second relation of.1, weotain X t γδm ( 1 ( 1 γ δ )[ M 1 Q / M 1 Q ] ) t 1 γ δ ( 1 [ M 1 Q / M 1 Q ] )[( 0 γδmp /) ],.3 t where M denotes 1 γ δ and Q denotes m/t 1 γ δ 1 γ δ 1. Figure 3 represents aplotofx t, in this case too, the semiclassical solution reduces to the classical solution when t approaches the horizon ut it is sustantially different in the Planck region we recall that in the plot, we have chosen γδ 1 to amplify the quantum correction to Schwarzschild solution, ut a semiclassical analysis is correct for δ 33. In 15, 16, the spectrum of the operator p c was calculated as follows: p c μ, τ γl P τ μ, τ.. In this paper, we have used dimensionless variales then the parameter δ, which is related to the area eigenvalues y., isoforderδ 33. The correct coefficient is 3 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 0, 1. For the anisotropy as well as for the lapse function, it is important to rememer that the solution refers to the region t>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 otained in asymptotic safe gravity 6 for particular values of the lack hole mass 17, 18. However, t is very small in our semiclassical analysis, and in this region, a complete quantum analysis of the prolem is inevitale as developed in 15, 16.

8 8 Advances in High Energy Physics Y t Figure : Plot of Y t for m and γδ 1 in the horizontal axis, we have the temporal coordinate t; in the vertical axis, we have Y t. The red trajectory corresponds to the classical Schwarzschild solution, and the green trajectory corresponds to the semiclassical solution. 6 t 8 The metric component Y t represents the square radius of the two-sphere S and it is related to the density triad component p c t y the first relation reported in.1.usingthe solution 3.6,weotain Y t 1 [( 0 ) ] γδmp t..5 t In Figure, we have a plot of Y t and we can note a sustantial difference with the classical solution. In the classical case, the S two-sphere goes to zero for t 0, in our semiclassical solution instead the S sphere ounces on a minimum value of the radius, which is Y t min γδm, and it expands again to infinity for t 0. We have taken the integration parameter 1 to match with the classical Schwarzschild solution near the horizon, see. and the first of 15, 16. The minimum of Y t corresponds to the time coordinate t min mγδ/ 1/ and t min t, in fact t mδ ut t min mδ 1/, then for δ 0 we have showed that δ 33,weotaint t min. In Figure 5, we have a plot of the spatial section volume V X t Y t and we can see that the semiclassical volume has a sustantially 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 ut this is not a prolem for the singularity resolution ecause the horizons are coordinate singularities and not essential singularities. p 0 Quantum amiguities and semiclassical solution In this paragraph, we want to compare the quantum spectrum of the operator 1/ p c with the semiclassical solution.5. At the quantum level, the spectrum of 1/ p c, for a generic SU representation j is ( 1 3 μ, τ p c,j γ 1/ δl P j j 1 j 1 k j k j [ ( k τ τ kδ )] ) μ, τ..6

9 Leonardo Modesto 9 V t Figure 5: Plot of the spatial section volume V X t Y t for m 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 possile to note that the semiclassical volume green line is zero for t t. t 15 0 To compare the quantum spectrum with the semiclassical solution, we must have a relation etween the eigenvalue τ and the temporal coordinate t. We calculate this relation comparing the large τ limit of.6 and the semiclassical solution near the horizon. The limit of.6 for large eigenvalues gives 1 1 μ, τ pc,j τ τ, τ μ,.7 γl P and on the other side, we know that near the event horizon 1/ p c 1/t, then comparing with.7, weotainτ t /γl P. At this point, we have all the ingredients to compare the quantum operator spectrum with the semiclassical solution. From the plot in Figure 6, itis natural to interpret the semiclassical solution as the smooth approximation of the quantum operator spectrum ut the similarity etween semiclassical and quantum spectra is very stringent only if we choose a particular relation etween the lack hole mass and the SU representation j in Figure 6, we have chosen m 00 and j 0. Using an heuristic argument, we can otain the general relation etween m and j. The relation is m j and now we go to show the validity of this mass quantization formula. In Figure 7, 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 possiility to interpret the representation amiguities in.6 as a lael for the mass m this idea rememers a recent result aout the possiility to see ordinary matter as particular states in pure loop quantum gravity 3. In fact in the semiclassical solution, we have a free parameter that corresponds to the lack 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 possile to match the time coordinate of the maximum for the two solutions. This is possile only if we choose a particular relation etween m and the representation j. Tootainthis relation, we calculate the derivative of the spectrum.6 with respect to τ and we evaluate the derivative in τ t min /γ mδ/ t is dimensionless in our analysis τ ( 1 pτ,j ) τ mδ/ [ ( 3 k j k δj j 1 j 1 k j m )] m k,.8

10 Advances in High Energy Physics Figure 6: In this plot, we compare the semiclassical solution 1/Y t and the spectrum of the quantum operator 1/ p c for j 0 and m 00. The semiclassical solution is represented y a red line and the quantum spectrum y the green line one Figure 7: In this plot, we compare the semiclassical solution 1/Y t and the spectrum of the quantum operator 1/ p c 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/,, 1,, 3/, 6,, 8, and γδ 1. The semiclassical solution is represented y the red line and the quantum spectrum y the green one. where p τ is the eigenvalue of 1/ p c. Oserving.8, we see that in the 1/ p c spectrum the relative and asolute maximums correspond to points, where the derivative is divergent. Those points are in m j localized and this relation is also the mass quantization formula in Planck units. For any fixed value of the representation j, the classical lack hole mass corresponds to the asolute maximum of the quantum spectrum in such representation. 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 y loop quantum gravity. We have otained a solution reproducing the Schwarzschild solution near the event horizon ut that is sustantially different in the Planck region near the point r 0, where the

11 Leonardo Modesto 11 Tale 1 g μν Semiclassical Classical [( ) γδm γ δ 1] N t t 1 1 ( 1 γ δ ) 1 γ δ 1 m/t 1 γ δ ( ) 1 γ δ m/t γ δ 1 m/t 1 γ δ ( ) 1 γ δ 1 X t Y t γδm 1 ( 1 γ δ )[ 1 γ δ 1 m/t 1 γ δ ( ) 1 γ δ ] 1 1 γ δ 1 m/t 1 γ δ ( ) t 1 γ δ 1 ( 1 γ δ ) 1 γ δ 1 m/t 1 γ δ ( ) 1 γ δ 1 [( ) γδm ] 1 1 γ δ 1 m/t 1 γ δ ( ) t 1 γ δ 1 [( ) 1 γδm ] t t m t t 1 singularity is classically localized. The structure of the solution suggests the possiility to have another event horizon near the point r 0 this is similar to the result in asymptotic safety quantum gravity 17, 18, ut the radius of such horizon is smaller than the Planck length and in this region a complete quantum analysis of the prolem is inevitale 15, 16. Another interesting result is related to the S sphere part of the three metrics. We otain that in the semiclassical analysis, the radius of the two-sphere does not vanish, as in the classical case, ut the sphere ounces on a minimum radius and it expands again to infinity. The solution is summarized in Tale 1. Using a heuristic argument, we have calculated the mass quantization formula comparing the semiclassical and quantum spectra of the inverse of the S sphere square radius, 1/ p c. Our arguments suggests that the mass spectrum formula m j. It is possile that the semiclassical analysis performed here will shed light on the prolem of the information loss in the process of lack hole formation and evaporation. See, in particular for a possile physical interpretation of the lack hole information loss prolem. Acknowledgment The author is grateful to Roerto Balinot, Alfio Bonanno, and Eugenio Bianchi for many important and clarifying discussions. References 1 C. Rovelli, Quantum Gravity, Camridge University Press, Camridge, UK, 00. A. Ashtekar and J. Lewandowski, Background independent quantum gravity: a status report, Classical and Quantum Gravity, vol. 1, no. 15, pp. R53 R15, T. Thiemann, Loop quantum gravity: an inside view, in Approaches to Fundamental Physics, vol. 71 of Lecture Notes in Physics, pp , Springer, Berlin, Germany, 007.

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