Schrodinger formalism, black hole horizons and singularity behavior
aa r X i v : . [ h e p - t h ] D ec Schrodinger formalism, black hole horizons and singularity behavior
John E. Wang
Department of Physics, Niagara University, Niagara, NY 14109-2044 andHEPCOS, Department of Physics, SUNY at Buffalo, Buffalo, NY 14260-1500
Eric Greenwood and Dejan Stojkovic
HEPCOS, Department of Physics, SUNY at Buffalo, Buffalo, NY 14260-1500
The Gauss-Codazzi method is used to discuss the gravitational collapse of a charged Reisner-Nordstr¨om domain wall. We solve the classical equations of motion of a thin charged shell movingunder the influence of its own gravitational field and show that a form of cosmic censorship applies.If the charge of the collapsing shell is greater than its mass, then the collapse does not form a blackhole. Instead, after reaching some minimal radius, the shell bounces back. The Schrodinger canonicalformalism is used to quantize the motion of the charged shell. The limits near the horizon and nearthe singularity are explored. Near the horizon, the Schrodinger equation describing evolution of thecollapsing shell takes the form of the massive wave equation with a position dependent mass. Theoutgoing and incoming modes of the solution are related by the Bogolubov transformation whichprecisely gives the Hawking temperature. Near the classical singularity, the Schrodinger equationbecomes non-local, but the wave function describing the system is non-singular. This indicates thatwhile quantum effects may be able to remove the classical singularity, it may also introduce somenew effects.
I. INTRODUCTION
Gauss-Codazzi equations are fundamental equations inthe theory of surfaces embedded in a higher dimensionalspace. They provide a very powerful tool of studyingproblems in general relativity. Most of the work in theexisting literature focused on problems with sources con-taining only mass distributions. The next natural stepis to generalize the Gauss-Codazzi method so that thestress-energy sources with both mass and charge can beincluded. First, we will setup the formalism and derivethe equations of motion for a charged two dimensionalsurface. The conserved quantities will follow from theseequations of motions. A charged shell of matter (repre-sented by a domain wall) moves under the influence of itsown gravitational and electromagnetic field. If the massof the shell is greater than its charge, the collapse willend by the formation of a black hole. When the chargeis greater than the mass parameter, the collapse will notyield a black hole, in agreement with the cosmic censor-ship conjecture. In this case, the solution becomes os-cillatory. The shell will collapse to some minimal radiusat which the electromagnetic repulsion will overcome thegravitational attraction and cause the bounce. From thatmoment on, the shell will be expanding until it reachessome maximal radius at which the gravitational force willagain dominate, and the new collapsing cycle will start.We then quantize the motion of the charged shell inthe context of the canonical formalism. Two of the mostimportant regimes will be the limits near the horizon andnear the singularity. Near the horizon, the Schrodingerequation describing evolution of the collapsing shell takesthe form of the massive wave equation in a Minkowskibackground with a position dependent mass. Incomingand outgoing modes are defined and related by the Bogol-ubov transformation. Despite the fact that the outgoing state is a pure state, it has a Boltzmann distribution atthe Hawking temperature. In the absence of the mat-ter fields propagating in the background of the collaps-ing shell, this is an intriguing result, indicating perhapsthat the collapsing shell loses its mass in the form ofemitted gravitons or pair production of shells. Near theclassical singularity, the Schrodinger equation becomesnon-local, but the wave function describing the systemis non-singular. We find that locality is recovered in thelimit of a very large domain wall tension (i.e. mass ofthe collapsing shell) but negligible gravitational interac-tion. Including gravitational interactions, the system ismanifestly non-local.
II. THE GAUSS-CODAZZI FORMALISMA. The Equations
Here we setup the Einstein’s equations in the presenceof stress-energy sources with both mass and charge con-fined to three-dimensional time-like hypersurfaces. Wefollow the technique developed in Ref. [1]. Let S de-note a three-dimensional time-like hypersurface contain-ing stress-energy and let ζ a be its unit space-like normal( ζ a ζ a = 1). The three-metric intrinsic to the hypersur-face S is h ab = g ag − ζ a ζ b (1)where g ab is the four-metric of the space-time. Let ∇ a denote the covariant derivative associated with g ab andlet D a = h ab ∇ b (2)where D a is a projection into the hypersurface S of thecovariant derivative ∇ a of space-time, and h ac is the in-duced metric on the hypersurface S . The extrinsic cur-vature of S , denoted by π ab , is defind by π ab ≡ D a ζ b = π ba . (3)The contracted forms of the first and second Gauss-Codazzi equations are then given by R + π ab π ab − π = − G ab ζ a ζ b (4) h ab D c π ab − D a π = G bc H ba ζ c . (5)Here R is the Ricci scalar curvature of the three-geometry h ab of S , π is the trace of the extrinsic curva-ture, and G ab is the Einstein tensor in four-dimensionalspace-time.The stress-energy tensor T ab of four-dimensional space-time has a δ -function singularity on S for both the massand the charge. This in turn implies that the extrinsiccurvature has a jump discontinuity across S , since theextrinsic curvature is analogous to the gradient of theNewtonian gravitational potential. Therefore we can in-troduce γ ab ≡ π + ab − π − ab (6)and S ab ≡ Z dl T ab , (7)where l is the proper distance through S in the direc-tion of the normal ζ a , and where the subscripts ± referto values just off the surface on the side determined bythe direction of ± ζ a . Using the Einstein and the Gauss-Codazzi equations, one then has S ab = − πG N ( γ ab − h ab γ cc ) . (8)We can also introduce the “average” extrinsic curvature˜ π ab = 12 ( π + ab + π − ab ) . (9)Then, using Eq. (8), by adding and subtracting Eq. (4)and Eq. (5) on opposite sides of S we get h ac D b S cb =0 , (10) h ac D b ˜ π cb − D a ˜ π =0 , (11) R + (cid:0) ˜ π ab ˜ π ab − ˜ π (cid:1) = − π G N (cid:20) S ab S ab −
12 ( S aa ) (cid:21) . (12)These from a complete set of equations to solve Einstein’sequations in the presence of a thin wall. B. Attractive Energy
Here we derive equations for an observer who is hov-ering just above the surface S on either side. Let the vector field u a be extended off S in a smooth fashion.The acceleration u a ∇ a u b = ( h bc + ζ b ζ c ) u a ∇ a u c = h bc u a ∇ a u c − ζ b u a u c π ab (13)has a jump discontinuity across S since the extrinsic cur-vature has such a discontinuity. The perpendicular com-ponents of the accelerations of observers hovering just off S on either side satisfy ζ b u a ∇ a u b (cid:12)(cid:12)(cid:12) + + ζ b u a ∇ a u b (cid:12)(cid:12)(cid:12) − = − u a u b ˜ π ab − σ S ab ˜ π ab = − τσ ( h ab + u a u b )˜ π ab − σ S ab ˜ π ab (14)and ζ b u a ∇ a u b (cid:12)(cid:12)(cid:12) + − ζ b u a ∇ a u b (cid:12)(cid:12)(cid:12) − = − u a u b γ ab = 4 πG n ( σ − τ ) . (15) III. MODEL
We consider a spherical domain wall with the constanttension σ representing a spherical shell of collapsing mat-ter and charge. The wall is described by only the radialdegree of freedom, R ( t ). The metric is taken to be the so-lution of Einstein equations for a spherical domain wallwith charge. The metric is Reisner-Nordstr¨om outsidethe wall, as follows from spherical symmetry [2] ds = − (cid:18) − GMr + Q r (cid:19) dt + (cid:18) − GMr + Q r (cid:19) − dr + r d Ω , r > R ( t ) (16)where M is the mass and Q is the charge of the wall,respectively, and d Ω = dθ + sin θdφ . (17)In the interior of the spherical domain wall, the line ele-ment is flat, as expected by Birkhoff’s theorem, ds = − dT + dr + r dθ + r sin θdφ , r < R ( t ) (18)The equation of the wall is r = R ( t ). The interior timecoordinate, T , is related to the asymptotic observer timecoordinate, t , via the proper time of an observer movingwith the shell, τ . The relations are dTdτ = " (cid:18) dRdτ (cid:19) / (19)and dtdτ = 1 f s f + (cid:18) dRdτ (cid:19) (20)where f ≡ − GMR + Q R (21)By integrating the Gauss-Codazzi equations of motionfor the charged spherical domain wall derived in the pre-vious section we find that the mass is a constant of motion(see also [2]) and is given by M = 4 πσR hp R τ − πGσR i + Q R . (22)The proof that M is really a constant of motion is given inthe Appendix. We therefore identify M and the Hamil-tonian of the system, i.e. M ≡ H . The physical meaningof Eq. (22) is straightforward. For a static shell, i.e. R τ = 0, the first term in square brackets is just the totalrest mass of the shell. For a moving shell, R τ = 0 takeskinetic energy into account. The second term in squarebrackets is the self-gravity or binding energy. Finally, thelast term in (22) is the electromagnetic contribution tothe total mass (energy). In what follows, we will identifythe conserved quantity (22) with the Hamiltonian of thesystem.It is also possible to take the non-relativistic large ra-dius limit of the above hamiltonian. For the case of con-stant mass M = 4 πσR , the above hamiltonian becomes H = M + p M − GM R + Q R (23)which is the usual hamiltonian for a particle in a grav-itational and electrical potential. The extremal limit M = ± Q naturally corresponds to a free hamiltonian.In this case it is clear that the identification of the con-served quantity with the hamiltonian is justified.The collapse of the shell also obeys charge conserva-tion, which is given by D a j a = D a ( qu q ) , (24)where j a is the four-current and u a is a timelike four-vector. IV. CLASSICAL EQUATIONS OF MOTION
In this section we will consider the classical equationof motion for the Reisner-Nordstr¨om domain wall (forearlier work see e.g. [3, 4, 5, 6] and also [7, 8]). Todo so we consider an action that leads to the conservedhamiltonian. From Eq. (22) the form of the action isthen, S eff = − π Z dτ σR hp R τ − R τ sinh − ( R τ ) − πσGR + Q πσR i (25) where τ is the propertime of the observer who is fallingin with the shell and R τ = dR/dτ . Now Eq. (25) can bewritten in terms of the asymptotic time tS eff = − π Z dtσR hs f − (1 − f ) f R t − R t p f sinh − R t s ff − R t ! + (cid:18) Q πσR − πσGR (cid:19) s f − R t f i (26)where R t = dR/dt . From Eq. (26) the effective La-grangian is then L eff = − πσR hs f − (1 − f ) f R t − R t p f sinh − R t s ff − R t ! + (cid:18) Q πσR − πσGR (cid:19) s f − R t f i . (27)The generalized momentum Π can be derived from Eq.(27)Π = 4 πσR √ f h f R t ( f − R t ) p f − (1 − f ) R t + (1 − f ) R t p f − (1 − f ) R t + ( Q πσR − πσGR ) R t p f − R t + f sinh − R t s ff − R t ! i . (28)Thus the Hamiltonian in terms of R t is given by H =4 πσR h f R t ( f − R t ) p f − (1 − f ) R t + f ( Q πσR − πσGR ) √ f p f − R t + 1 √ f p f − (1 − f ) R t ! i . (29)To obtain H as a function of ( R, Π), we need to elim-inate R t in favor of Π using Eq. (28). This can, inprinciple, be done but is messy. Instead we consider the R is close to R H and hence f →
0. In the limit f → ≈ πµR R t √ f p f − R t (30)where µ ≡ Q πσR H − πσGR H (31)where R H is the horizon radius. Using Eq. (30) we canthen write the Hamiltonian as H ≈ πµf / R p f − R t = p ( f Π) + f (4 πµR ) (32)and has the form of a relativistic particle, p p + m ,with a position dependent mass term.The Hamiltonian is a conserved quantity and so, fromEq. (32), h = B / R p f − R t (33)where h = H/ πµ is a constant. Solving Eq. (33) for R t we get R t = ± f r − f R h , (34)which, in the near horizon limit takes the form R t ≈ ± f (cid:18) − f R h (cid:19) (35)since f → R → R H , where R H is the horizon radius.The dynamics for R ∼ R H can be obtained by solv-ing the equation R t = ± f . Here we will consider twodifferent cases, the non-extremal and extremal case.
1. Non-Extremal Case
For the non-extremal case, we consider the equality1 − GMR H + Q R H = 0 (36)where R H is given by R H = GM ± p ( GM ) − Q (37)The plus sign is the outer and the minus sign is the innerhorizon. To distinguish between them we write R + = GM + p ( GM ) − Q (38) R − = GM − p ( GM ) − Q (39)Therefore we can then write Eq. (21) as f = − GM + p ( GM ) − Q R ! × − GM − p ( GM ) − Q R ! (40) ≡ f + f − . (41)Since in the near horizon limit f →
0, we can work withtwo different limits, either f + → f − →
0. In both cases R t ≈ ± f + f − .For asymptotic observers watchingthe collapse, the horizon of interest is f + . For f + → f − goes to a finite constant number, thus we have R ( t ) ≈ R + + ( R − R + ) e ± f − t/R + (42)We now make some comments on Eq. (42). First, inthe limit ( GM ) > Q , the exponential term in Eq. (42)is positive definite. Thus it is easy to see that it takes theshell an infinite amount of time as seen by the asymptoticobserver to reach R + . In the limit of ( GM ) >> Q , f reduces to B , where B ≡ − GMR . (43)This is the case studied in Refs. [9] from the view pointof an asymptotic observer and in Refs. [10, 11] from theview point of an infalling observer.
2. Extremal Case
In the extremal case GM = Q , so near the horizon wecan write f = (cid:18) − QR (cid:19) ≈ (cid:18) R − QR h (cid:19) ∼ δ R h . (44)Here R h = GM = Q (45)is the position of the horizon in the extremal limit andthe small parameter δ is δ ≡ R − Q. (46)Therefore we can write R t ≈ δ t = ± δ R h . (47)Solving Eq. (47) to leading order in R − R h , the solutionis R ( t ) ≈ R h ± ( R − R h ) R h t + R h . (48)Since we are interested in the collapsing case, we takethe negative sign again. In that case, R ( t ) = R h only as t → ∞ .From Eqs. (42) and (48) we can see that again it takesan infinite amount of time for the shell to reach the hori-zon, as seen by the asymptotic observer. However, inthe extremal case the approach is not exponential, whichis the consequence of the repulsive contribution of thecharge. A. Cosmic censorship
We now analyze what happens in the case when theparameters of the collapsing shell satisfy Q > ( GM ) .A black hole with such parameters is just a naked singu-larity, since the expression for the horizon r h = GM ± p ( GM ) − Q (49)would not be real. Thus, if the collapse proceeds all theway, this would represent a violation the cosmic censor-ship conjecture. However, this is not the case here. As wecan see from Eq. (42), the solutions have a complex expo-nential, which implies oscillating solutions. The shell willcollapse to some minimal radius at which the electromag-netic repulsion will overcome the gravitational attractionand cause the bounce. From that moment on, the shellwill expand until it reaches some maximal radius at whichthe gravitational force will again dominate, and the newcollapsing cycle will start. This is in agreement with thecosmic censorship conjecture. V. QUANTUM EFFECTS FAR FROM THEHORIZON
Previously we have shown that the hamiltonian in thenon-relativistic large radius limit is given by H = M + p M − GM R + Q R . (50)As this is similar to the usual Schrodinger equation fora hydrogen atom, bound state solutions are well known;bound states do not exist in the extremal limit. Radialwavefunctions are given by Laguerre polynomials andthe ground state energy is given by E = − M ( GM − Q ) /
8. The factors of 2 are due to the fact that this shellsatisfies the Gauss-Codazzi equations. The shell self in-teraction is not due to the full gravitational and electricalforces at the shell but the average of the values inside theshell and outside the shell. The relativistic corrections tothe energy as well as to the gravitational interactions canbe further calculated perturbatively.
VI. QUANTUM EFFECTS NEAR THEHORIZONA. Near Horizon limit
The classical Hamiltonian in Eq. (32) has a squareroot and so we first consider the squared Hamiltonian H = f Π f Π + f (4 πµR ) (51)where we have made a choice for ordering f and Π inthe first term. In general, we should add terms that depend on the commutator [ f, Π]. However, in the limit R → R H , we find[ f, Π] ∼ R H (cid:18) GM − Q R H (cid:19) . (52)In the case of a charged black hole, the normal orderingambiguity arises for black holes which are small relativeto the Planck scale. For uncharged black holes, the nor-mal ordering ambiguity is less severe. In fact in the ex-tremal limit, the normal ordering ambiguity disappearscompletely and the hamiltonian is uniquely defined.We now apply the standard quantization procedure.We substitute Π = − i ∂∂R (53)in the squared Schr¨odinger equation, H Ψ = − ∂ Ψ ∂t . (54)Then we have, − f ∂∂R (cid:18) f ∂ Ψ ∂R (cid:19) + f (4 πµR ) Ψ = − ∂ Ψ ∂t . (55)To solve this equation, we define tortoise coordinates u = R + GM ln (cid:12)(cid:12)(cid:12)(cid:12) R GM R − Q − (cid:12)(cid:12)(cid:12)(cid:12) + 2( GM ) − Q p ( GM ) − Q ln( − R − GM p ( GM ) − Q ) − GM ) − Q p ( GM ) − Q ln(2 − R − GM p ( GM ) − Q ) (56)which gives f Π = − i ∂∂u . (57)Eq. (54) then gives ∂ Ψ ∂t − ∂ Ψ ∂u + f (4 πµR ) ψ = 0 . (58)This is just the wave equation in a Minkowski backgroundwith a mass that depends on the position. From thestructure of Eq. (56), care needs to be taken to choosethe correct branch since the region R ∈ ( R H , ∞ ) mapsonto u ∈ ( −∞ , ∞ ) and R ∈ (0 , R H ) onto u ∈ (0 , −∞ ),where R H is given by Eq. (37).We now turn to examine the quantization of incomingand outgoing states describing this infalling shell. Fieldspropagating in this collapsing black hole background ex-perience particle creation at a temperature given by theHawking temperature. There are many ways to derivethis result including calculating the Bogolubov transfor-mations between incoming and outgoing states at pastand future null infinity. One way to find the Bogolubovtransformation is to consider the outgoing waves at fu-ture null infinity and using the high energy approxima-tion, tracing these solutions back to past null infinity. Wenow turn to this phenomenon and find that we can ex-tract the necessary information directly from the horizonperturbations by considering a particular set of incomingand outgoing fields. B. Schwarzschild
The tortoise coordinate u for Schwarzshild is u = R + 2 GM ln | R GM − | . (59)In the near horizon limit, R ≈ GM we expand to lowestorder f = 1 − GM/R ≈ e u/ GM (60)where u → −∞ . The equation of motion becomes aftera further scaling of the coordinate ( t, u ) → GM ( t, u ) ∂ ψ∂t − ∂ ψ∂u + m e u ψ = 0 (61)where the constant m = (8 πµGM ) . This wave equa-tion is similar to the example of a time dependent massexamined in [12, 13]. For sufficiently slowly varying ex-ponent, the solution is essentially a plane wave and thisoccurs for large GM . The general solution can be foundby expanding ψ in positive frequency plane waves ψ ( u, t ) = R ( u ) e − iωt (62)in which case the Schrodinger equation is a form of themodified Bessel’s equation[ ∂ u − m e u + ω ] R ( u ) = 0 (63)with two classes of normalizable solutions. The modifiedBessel functions R + in (2 √ ime u/ ) = ( im ) iω √ ω Γ(1 − iω ) J − iω (2 √ ime u/ )(64)in the near horizon limit u → −∞ become the positivefrequency solutions R + in ≈ e − iωu √ ω . (65)Altogether the effective potential vanishes near the hori-zon and the wave equation becomes that of a free massivefield with the standard ingoing (and outgoing) Fouriermodes. At distances far from the horizon these corre-spond to solutions which grow without bound and inde-pendently of ω R + out ≈ e − u/ √ me u/ (66) although this exponential growth in the modes is an arti-fact of our approximation Eq. 60, which no longer is validfar from the horizon. In fact in the large radius limit, thesolutions become wavelike.In addition we consider the solutions R + out = r π ie πω ) − / H − iω (2 √ i √ me u/ ) (67)which are related by a Bogolubov transformation R + out = aR in + b ¯ R in . (68)to the incoming modified Bessel functions H − iω ( x ) = J iω ( x ) − e πω J − iω ( x ) − i sin( − iπω ) . (69)Near the horizon u → −∞ , the Hankel function for com-plex argument can be expanded as H − iω (2 √ i √ me u/ ) ≈ e − iωu − e πω e iωu . (70)Relating the incoming and outgoing states we find R + out ≈ ( R in + e − πω ¯ R in ). This physically shows particle pro-duction as the shell approaches the horizon and that thedensity of particles created is given by the ratio | b/a | = e − πω (71)at temperature 1 / π . Restoring the units of temperaturewe find that the modes of the shell experience a temper-ature T = 1 / πGM (72)which is the Bekenstein-Hawking temperature of theblack hole! The Bogolubov transformation also showsthat at large ω the vacuum for the in and out states areidentical.The two sets of wavefunction solutions are related bythe Bogolubov transformation. Despite the fact that theoutgoing state is a pure state, it has a Boltzmann distri-bution at temperature T H = 1 / πGM which is preciselythe Hawking temperature of the black hole. This is a re-sult of the fact that the interaction potential is periodicin Euclidean space.Normally the Hawking temperature is measured atasymptotic infinity while the near horizon temperatureis blue-shifted and infinite. In our near horizon analysiswe extract the finite Hawking temperature for outgoingmodes although it is measured relative to that of the in-coming modes. Naturally this should be a finite quantity.If one were to take the usual temperature calculation andsimultaneously measure the outgoing radiation versus theincoming radiation at large but finite radial distance thenthe relative blueshifting would cancel out leading to thesame Hawking temperature.We note here that this result implies outgoing ther-mal radiation, though we did not consider any matterfields propagating in the background of the collapsingshell. The shell itself is both the source of gravitationalfield and matter that collapses. Ref. [14] performed ananalysis of a massless spherical shell and the radiation itemits. In their analysis the shell was an approximation ofa particle moving in the black hole background. Thus, wemay conclude that the shell itself loses its mass, perhapsin the form of emitted gravitons and pairs of sphericalshells. C. Charged Reissner-Nordstrom
For the charged black hole the analysis is very similarexcept that two logarithm terms are kept in the expan-sion near the outer horizon and the distance from theouter horizon is written in terms of the coordinate u as R − R + ≈ e u/ ( GM + GM )2 − Q √ ( GM )2 − Q ) . (73)Up to rescalings the equations of motion are the samemodified Bessel equations. Performing the same analy-sis in terms of the near horizon modes, the temperatureexperienced by the gravitational modes is the same asthe Bekenstein-Hawking temperature for a charged blackhole T = p ( GM ) − Q π ( GM + p ( GM ) − Q ) . (74) D. Scalar Fields and Temperature
In the above analysis we have argued that the Hawkingradiation temperature can be calculated for the quantummechanical wavefunction describing gravity and the col-lapsing shell near the horizon. In this picture it was notnecessary to invoke the asymptotically flat region of theblack hole. The relevant thermal properties could be cal-culated in the vicinity of the horizon. Furthermore noblue shift factors were needed either as both the incom-ing and outgoing states were localized to the horizon.The same analysis can also be applied to a scalar fieldtheory in the vicinity of a black hole. For simplicity wewill work with the Schwarzschild black hole. A scalarfield Φ can be decomposed into modes of the form φ = r − f ( r, t ) Y lm ( θ, φ ) so that the wave equation becomes ∂ f∂t − ∂ f∂u +(1 − GMr )[ l ( l + 1) r + 2 GMr + m ] f = 0 (75)where we have u is the tortoise coordinate. In the nearhorizon limit (1 − M/r ) ≈ e u/ GM as before and thesecond term in parenthesis becomes a constant.The mode equation of the scalar field therefore is justthe modified Bessel equation that we found for the shell.It is then possible to use the same analysis to find the in-coming and outgoing states which can be written as mod-ified Bessel functions. These outgoing states are thermal relative to the incoming states and are at the Hawkingtemperature. VII. QUANTUM EFFECTS NEAR THESINGULARITYA. Near singularity limit for the uncharged blackhole
In this section we investigate the question of quantumeffects when the collapsing shell approaches the origin(i.e. classical singularity at R → t can not study this limit, so we perform our analysisusing the time τ of an observer located on the collapsingshell. The Hamiltonian (in terms of R τ ) is just the con-served quantity in (22). After setting Q = 0, the effectiveLagrangian consistent with the conserved quantity (22)is L = − πσR hp R τ − R τ sinh − ( R τ ) − πσGR i . (76)The generalized momentum, Π, can be derived from thisLagrangian as Π = 4 πσR sinh − ( R τ ) . (77)The Hamiltonian in terms of R τ is H = 4 πσR hp R τ − πσGR i (78)From the Hamiltonian we can get R τ as R τ = ± s(cid:18) hR + 2 πσGR (cid:19) − h = H/ πσ . Here we can study two cases. First weconsider the ultra-relativistic limit near the origin where R τ is very large. Up to the leading term near the origin,the Hamiltonian is H = 4 πσR R τ . (80)Clearly, this choice eliminates the Newton’s constant Gfrom the equation, and thus important gravitational ef-fects are not included. In terms of the generalized mo-mentum (77), the Hamiltonian now is H = 4 πσR sinh( Π4 πσR ) . (81)The Schrodinger equation becomes2 πσR [ e ( Π4 πσR ) − e − ( Π4 πσR ) ] ψ ( R, τ ) = i ∂ψ ( R, τ ) ∂τ (82)Defining a new variable u = R , the equation becomes2 πσu / [ e − ( i πσ ∂∂u ) − e ( i πσ ∂∂u ) ] ψ ( u, τ ) = i ∂ψ ( u, τ ) ∂τ (83)Since the exponentials are now just the translation oper-ators, we have2 πσu / [ ψ ( u − i πσ , τ ) − ψ ( u + 3 i πσ , τ )] = i ∂ψ∂τ . (84)It is interesting to note that this equation is dependenton σ which describes the particular shell. This in generalmight be solvable from a recursion relationship. This is amanifestly non-local equation. The non-locality that wefound may be a simple consequence of the fact that we areusing the functional Schrodinger formalism which is onlyan effective theory, i.e. only an approximation of somemore fundamental local theory. The other possibility isthat the quantum description of the black hole physicsrequires inherently non-local physics. The answer to thisquestion requires further investigation.Leaving this question aside, we examine a particularlocal limit as follows. When σ → ∞ the above expressionbecomes a derivative2 πσ ( − i/ πσ ) u / ∂ u ψ ( u, τ ) = i ∂ψ∂τ (85)or after simplifying − u / ∂ u ψ = ∂ τ ψ. (86)Although the left side of this equation appears to vanishin the near singularity limit, by rewriting this equationin terms of the variable R we find the simple relationship − ∂ R ψ = ∂ τ ψ (87)which has solutions of the form ψ = ψ ( R − τ ). In order tomaintain the same position on the profile, if R becomessmaller τ has to move to smaller values as well, whichmeans that the wavefunction moves backwards in time.However inside the black hole horizon, time and spaceswitch their roles. By analyzing the conformal structureof the black hole, we find that inside the horizon τ mov-ing to smaller values corresponds to a normal infallingtrajectory. Our local equation for the evolution of thewavefunction appears simple to understand in this limit.In this case though there is no boundary condition im-posed on the wavefunction and it appears that the waveequation allows solutions to propagate through the singu-larity. This is due to the fact that this limit correspondsto the pure kinetic energy limit which neglects gravity.We now consider the case when the tension σ is verylarge and the gravitational interaction term in Eq. (79)is dominant 2 πσGR ≫ H/ πσR ≥ . (88)In this case R τ ≈ πσGR and the hamiltonian becomes H ≈ πσR /R τ . The above inequality can be written as8 πσ GR ≫ H and substituting for the hamiltonian wefind that this is the ultra-relativistic limit( σGR ) ≫ ↔ R τ >> . (89) In terms of the generalized momentum (77), the Hamil-tonian now is H = 4 πσR sinh( Π4 πσR ) . (90)which we write using the translation operator8 πσu / T ( − i/ πσ ) − T (3 i/ πσ ) ψ ( u, τ ) = i ∂ψ ( u, τ ) ∂τ (91)Taking the large σ limit, reduces the denominator to − i/ πσ∂ u so the equation of motion becomes an inversedifferential operator equation16 π σ u / ∂ u ψ ( u, τ ) = ∂ψ ( u, τ ) ∂τ . (92)Here, the wavefunction is non-infinite and constant atthe origin provided that the inverse differential operatorhas a finite behavior. This indicates that quantum effectsmay be able to remove the classical gravitational singu-larity at the center. To make a definite statement onewould need to calculate the conserved probability for thewhole space (not only in the near horizon and near theclassical singularity limit), which is proportional to ψ ∗ ψ but also contains a non-trivial measure term due to thecurved background. The integrated probability over thewhole space-time should then be non-singular. Withinour approximations we can not do this (we do not havesolutions which are valid everywhere), but the fact thatthe wave function is not singular at R = 0, where theclassical singularity was located, is a strong indicationthat quantum effects may make gravity non-singular.Eq. (92) also indicates that strong gravity regime ismanifestly non-local (because of the inverse differentialoperator). While in the previous ultra-relativistic limitwith the absence of G we were able to remove non-localeffects by taking large σ limit, in this case we are not ableto do the same. This indicates that gravity is inherentlynon-local, and while quantum mechanical non-localitiesmay be removed by taking an infinitely large measuringapparatus, once we turn on gravity this is no longer pos-sible [24]. B. Near singularity limit for the charged black hole
When we include charge, the full Hamiltonian from(22) is H = 4 πσR [ p R τ − πGσR ] + Q R . (93)From here we find R τ = r ( H πσR − Q πσR + 2 πGσR ) − R τ ≈ Q / πσR . In this limit it is possibleto drop the gravitational energy term in the Hamiltonianand get the relativistic kinetic energy limit H = 4 πσR R τ + Q R . (95)The generalized momentum in this limit becomesΠ = 4 πσR sinh − R τ (96)so the Hamiltonian can be written as H = 4 πσR sinh( Π4 σR ) + Q R . (97)The Schrodinger equation is the same as before but withthe addition of the Coulomb term2 πu / [ ψ ( u − i πσ , τ ) − ψ ( u + 3 i πσ , τ )] + Q u / ψ = i ∂ψ∂τ (98)and we can study this in the limit of large energy density σ → ∞ to get − ∂ R ψ − i Q R ψ = ∂ t ψ (99)which has solutions of the form ψ = ψ R − iQ e A ( R − τ ) (100)where A is an arbitrary constant. There is a phase factor R − iQ / which corresponds to an infinite number of mod-ulus one oscillations. The exponential term correspondsto the generic wave which propagates to the classical sin-gularity at R = 0. VIII. CONCLUSION
In this paper we have analyzed the collapse of a mas-sive shell of all perfect fluids with charge using the Gauss-Codazzi method. The hamiltonians and lagrangians wereconstructed for these systems and several limits were an-alyzed. The key point was to then invoke the functionalSchrodinger formalism to find key quantum features. Thenear horizon limit analysis led to a new way to determinethe Hawking temperature of black holes without refer-ence to asymptotic states. While the first order termsled to thermal radiation it may be useful to further studyhigher order terms to look for non-thermality. The nearsingularity limit showed that the behavior of the wave-function was non-singular. While the equations to solvewere inherently non-local and contained an infinite num-ber of derivatives, certain local limits (i.e. very massiveshells) were analyzed.Questions regarding the wavefunction remain. Is therea measure for the wavefunction which leads to the con-servation of probability? In particular how is the proba-bility conserved on spatial slices when timelike/spacelikenotions change through the horizon. It is also unclearif the probability of the wavefunction inside the shell is finite in the limit where the shell collapses to zero size.If it is, how do we treat this information which is withinthe shell. Can fluctuations outside the shell, propagateand stay inside the shell?It would be interesting to see if one can find numericalsolutions to the quantum wavefunction in all regions ofspacetime.We note that we worked in the context of the functionalSchrodinger formalism which in many ways resembles theWheeler-DeWitt approach to quantum gravity [15]. Re-lated work in the existing literature based on differentapproaches also indicates that quantum effects may becapable of removing classical singularity at the center[16, 17, 18, 19, 20, 21, 22]. While it has been previouslyargued that the standard Schrodinger formalism does notyield conclusive claims about the non-singularity [23], ourwork apparently gives strong indications for non-singularbehavior at the center of the black hole.
APPENDIX A
Here we prove that the mass given by Eq. (22) is theconstant of motion. From Eqs. (14) and (15) one findsthat the acceleration for the charged domain wall is givenby 1 α R ττ = Q πσR − αR + 6 πσG (A1)where α is defined as α = p R τ . (A2)From Eq. (22) we have that the mass of the domain wallis given by M = 4 πσR ( α − πσGR ) + Q R . (A3)To show that the mass is a conserved quantity it is suf-ficient to show that M τ = 0. So using Eq. (22) we have M τ = R τ h − Q R + 8 πσR ( α − πσGR )+ 4 πσR (cid:18) α R ττ − πσG (cid:19) i Now using the acceleration we can write this as M τ = R τ h − Q R + 8 πσR ( α − πσGR )+ 4 πσR (cid:18) Q πσR − αR + 6 πσG − πσG (cid:19) i Multiplying out and canceling terms leaves M τ = R τ (0) = 0 , (A4)hence the mass is a conserved quantity.0 [1] J. Ipser and P. Sikivie, Phys. Rev. D , 712 (1984).[2] C. A. L´opez, Phys. Rev. D , 65 (2004) [arXiv:astro-ph/0307211].[4] L. Campanelli, P. Cea, G. L. Fogli and L. Tedesco, JCAP , 005 (2006) [arXiv:astro-ph/0505531].[5] S. Gao and J. P. S. Lemos, Int. J. Mod. Phys. A , 2943(2008) [arXiv:0804.0295 [hep-th]].[6] E. Hawkins, Phys. Rev. D , 6556 (1994) [Erratum-ibid.D , 7744 (1994)] [arXiv:gr-qc/9312033].[7] P. Hajicek, B. S. Kay and K. V. Kuchar, Phys. Rev. D , 5439 (1992).[8] P. Hajicek, Phys. Rev. D , 936 (1998).[9] T. Vachaspati, D. Stojkovic and L. M. Krauss, Phys. Rev.D , 024005 (2007) [arXiv:gr-qc/0609024]. T. Vachas-pati and D. Stojkovic, Phys. Lett. B , 107 (2008)[arXiv:gr-qc/0701096];[10] E. Greenwood and D. Stojkovic, arXiv:0806.0628 [gr-qc].[11] E. Greenwood and D. Stojkovic, JHEP , 042 (2008)[arXiv:0802.4087 [gr-qc]].[12] A. Strominger, “Open string creation by S-branes,”arXiv:hep-th/0209090.[13] A. Maloney, A. Strominger and X. Yin, JHEP , 048(2003) [arXiv:hep-th/0302146]. [14] P. Kraus and F. Wilczek, Nucl. Phys. B , 403 (1995)[arXiv:gr-qc/9408003].[15] B. S. DeWitt, Phys. Rev. , 1113 (1967).[16] A. Bogojevic and D. Stojkovic, Phys. Rev. D , 084011(2000) [arXiv:gr-qc/9804070].[17] L. Modesto, Phys. Rev. D , 124009 (2004)[arXiv:gr-qc/0407097].[18] E. Guendelman, A. Kaganovich, E. Nissimov andS. Pacheva, arXiv:0908.4195 [hep-th].[19] K. A. Bronnikov, V. N. Melnikov and H. Dehnen, Gen.Rel. Grav. , 973 (2007) [arXiv:gr-qc/0611022].[20] S. Shankaranarayanan and N. Dadhich, Int. J. Mod.Phys. D , 1095 (2004) [arXiv:gr-qc/0306111].[21] N. Mankoc Borstnik, H. B. Nielsen, C. D. Froggatt andD. Lukman, arXiv:hep-ph/0512061.[22] D. h. Yeom and H. Zoe, Phys. Rev. D , 104008 (2008)[arXiv:0802.1625 [gr-qc]].[23] V. Husain, Class. Quant. Grav.4