Hawking Radiation as a Probe for the Interior Structure of Regular Black Holes
aa r X i v : . [ g r- q c ] J un Hawking Radiation as a Probe for the Interior Structure of Regular Black Holes
Yanbin Deng, Gerald Cleaver Department of Physics, Baylor University, Waco, Texas 76706, USA (Dated: June 7, 2016)The notion of the black hole singularity and the proof of the singularity theorem were consideredgreat successes in classical general relativity. Singularities had presented deep puzzles to physicists.Conceptual challenges were set up by the intractability of the singularity. The existence of blackhole horizons which cover up the interior, including the singularity of the black hole from outsideobservers, builds an information curtain, further hindering physicists from understanding the natureof the singularity and the interior structure of black holes. The regular black hole is a conceptproduced out of multiple attempts of establishing a tractable and understandable interior structurefor black hole as well as avoiding the singularity behind the black hole horizon. A method isneeded to check the correctness of the new constructions of black holes. After studying the Hawkingradiation by fermion tunnelling from one type of regular black hole, structure dependent resultswere obtained. The result being structure dependent points out the prospects of employing theHawking radiation as a method to probe into the structure of black holes.
I. INTRODUCTION
The establishment of general relativity(GR) by Ein-stein was a great success of physics in the last century.GR has become the standard theoretical foundation ofthe whole modern study of classical gravitational physicsfrom astrophysics to cosmology. Despite great successes,a few difficulties still remain. It is arguable whether weshould reserve GR but be foreced to answer the ques-tion about the existence and nature of dark matter anddark energy, or to invest on the tireless attempts tryingto modify GR to circumvent the concepts of dark mat-ter and dark energy. Nevertheless, the existence of thesingularity is an intrinsic problem of GR. The so-calledsingularity theorem proved by Penrose and Hawking [1–4] argues that under quite general classical assumptions,the spacetime evolution will inevitably lead to some sin-gularity. It is unfortunate that the geometric curvatureblows up and physical laws lose their predictability at thesingularities.At the fundamentally level, the resolution of the singu-larity problem lies with the expectation that under situa-tions where quantum effects become strong, the behaviorof gravity could possibly greatly deviate from that pre-dicted by the classical theory of GR. Various attemptshaven been made in exploring the collapse processes andfrom there seeking for the interior structure of thus pro-duced black holes.[5, 6] It was James Bardeen who pro-posed a regular black hole solution which possesses onemore parameter than the typical metric of the Reissner-Nordstrom black hole.[7] This new type of black hole so-lution shows the same spacetime geometry outside thehorizon as the traditional black hole, but bears no sin-gularity inside, therefore endowed the name the regular black hole.Bardeen’s initiative attracted general intreset and be-came one of the typical ways of constructing regularblack hole solutions. Several similar black hole solu-tions were proposed[8–13]. Some of those efforts offerregular black hole solutions, but without specifying the energy-momentum sources. Some are sourced by a non-liear electromagnetic field energy-momentum. The na-ture of these regular black holes in the region far fromthe center is the same as traditional black holes, but theproperty is greatly different near the position where thesingularity of traditional black holes locates. The geo-metric characteristic quantities of those black holes be-come well-behaved through out the full physical range ofparameters. Various additional regular black hole solu-tions have been constructed along the years by differentmethods.[14–24]Whether or not black hole singularities should reallyexist, they would always be covered up by black hole hori-zons. Black hole horizons serve as an information curtainhindering outside observers from directly observing theinterior structure of the black hole, and determining theeventual existence of the black hole singularities. Theso-called black hole no hair theorem served narrowingthe choices of methods for pursuing the understanding ofthe interior structure of black holes. Hawking radiationis one of the rare methods we can apply to infer somebut usually little information from the black hole.[25–30]But it has not been generally recognized yet that it canbe a successful tool for unfolding the interior structurefor black holes. We conjecture that in the case of regularblack holes, which have rich interior structures to unfold,the Hawking radiation could be a powerful probe into theblack hole interior structure.To test the above conjecture, in this article we makea check and promising results are achieved. To carryout the work, we picked as an example one type of reg-ular black hole with shell-like mass distribution withinthe horizon. We studied the Hawking radiation throughfermion tunnelling from the above type regular blackhole, to infer any information about the radiation. Theproperty of the radiation is found to be structure depen-dent - dependent on the mass distribution. The result be-ing structure dependent encourages the possibility of em-ploying the Hawking radiation as a promising method toprobe into the structure of black holes. Our result echoessome attempts in the same direction.[31] We present ourcurrent result based on a case study and look forward toa general and exact proof in the subsequent research.
II. ONE TYPE OF REGULAR BLACK HOLE
One type of method of regular black hole solution con-struction relies on the understanding that there could ex-ist a fundamental minimal length scale arising from somequantum mechanism. Due to the assumption of the exis-tence of a minimal length scale θ in the noncommutativegeometry inspired black hole theory, the mass distribu-tion inside the black hole is not singular, but smearedas[32] ρ ( r ) = M πθ / exp (cid:18) − r θ (cid:19) . (1)The Schwarzschild-like black hole in such noncommuta-tive geometry inspired black hole theory has a metricspecified by, ds = − (cid:18) − M G √ πr γ (cid:18) , r θ (cid:19)(cid:19) dt + (cid:18) − M G √ πr γ (cid:18) , r θ (cid:19)(cid:19) − dr + r d Ω , (2)where γ is the so-called lower incomplete Gamma func-tion, with the definition, γ (cid:18) , r θ (cid:19) ≡ Z r θ t e − t dt. (3)Many extensions of the above solution eq. (2) are possi-ble. One possible extension is to rotate the black hole up.We summarize such a solution of a Kerr-like black holewith a smeared mass distribution based on the noncom-mutative black hole theory and we shall use this regularblack hole as an example to reveal the structure depen-dence of Hawking radiation.[33]We take the gravitational source to be a smeared masslayer, ρ ( r ) = Ar n exp (cid:18) − r l (cid:19) , (4)with l = 4 θ as the convention in P.Nicolini’s publica-tion. A is a normalization constant for the mass to bedetermined by the mass distribution as a function of r , m ( r ) = Z r π ¯ r ρ (¯ r ) d ¯ r = 2 πAl n +3 γ (cid:18) n , r l (cid:19) . (5)The Schwarzschild-like black solution is written as, ds = − f ( r ) dt + dr f ( r ) + rd Ω , (6) where f ( r ) = 1 − M πr Γ (cid:0) n +32 (cid:1) γ (cid:18) n , r l (cid:19) . (7)The Newman-Janis algorithm is then used. This is amethod of complex coordinate transformation capable oftransforming non-rotating black hole solutions into ro-tating black hole solutions. A change of coordinates tothe outgoing Eddington-Finkelstein coordinates followedby a complex coordinate transformation produces the fol-lowing Kerr-like black hole with a smeared mass distri-bution, ds = − ∆ − a sin θ Σ dt − a sin θ (cid:18) − ∆ − a sin θ Σ (cid:19) dtdφ + Σ∆ dr + Σ dθ + (cid:20) Σ + a sin θ (cid:18) − ∆ − a sin θ Σ (cid:19)(cid:21) sin θdφ , (8)where Σ= r + a cos θ, ∆= r − m ( r ) r + a . (9)The positions of the horizons of this type of regularblack holes are given by ∆( r H ) = 0, r H − M r H π Γ (cid:0) n +32 (cid:1) γ (cid:18) n , r H l (cid:19) + a = 0 . (10)We see that this equation contains modifications to thetraditional equation for the event horizon of a traditionalblack hole. We will study the Hawking radiation by thefermion tunnelling process for this type of regular blackholes and what will turn out in the last is a non-trivial de-pendence of the Hawking radiation on the modified hori-zon radius and interior mass distribution function insidethe black hole. III. HAWKING RADIATION BY FERMIONTUNNELLING AS A PROBE FOR BLACK HOLEINTERIOR STRUCTURE
Before Hawking argued that there should be blackbody radiation emitted outward from black holes’ eventhorizon, black holes were believed to be completely black.It was Hawking who revealed to people that black holesprobably radiate like a black body[25]. In this sectionwe study the Hawking radiation by the fermion tunnel-ing process for the regular black hole whose solution issummarized in section II.The method we use was proposed, developed, and sum-marized in[26–30]. The motion of fermions is describedby the Dirac equation in curved spacetime, γ µ D µ ψ + m ~ ψ = 0 , (11) D µ = ∂ µ + i αβµ Π αβ + iqA µ ~ . (12)Here q , m and A µ are the charge, mass of the fermionand the electric potential in the background respectively.The fermion wave function is given by,Ψ = ψ ( t, r, x µ ) e i ~ S ( t,r,x µ ) . (13)We insert the wave function into the Dirac equation, fac-tor out the exponential terms, and multiply by ~ . Thenusing the semiclassical approximation (that is, keepingterms only to the leading order in ~ ) produces, γ µ (cid:18) ( ∂ µ ψ ) e i ~ S + ψe i ~ S i ~ ∂ µ S + i αβµ Π αβ ψe i ~ S + iqA µ ~ ψe i ~ S (cid:19) + m ~ ψe i ~ S = 0 ,γ µ (cid:18) ( ∂ µ ψ ) + ψ i ~ ∂ µ S + i αβµ Π αβ ψ + iqA µ ~ ψ (cid:19) + m ~ ψ = 0 . This simplifies to, iγ µ (cid:18) ∂S∂x µ + qA µ (cid:19) ψ + mψ = 0 . (14)Multiplying both sides of this equation by thematrix − iγ ν (cid:0) ∂S∂x ν + qA ν (cid:1) , and noticing that − iγ ν (cid:0) ∂S∂x ν + qA ν (cid:1) ψ = mψ yields, γ ν (cid:18) ∂S∂x ν + qA ν (cid:19) γ µ (cid:18) ∂S∂x µ + qA µ (cid:19) ψ + m ψ = 0 . (15)Exchanging the index µ ←→ ν , gives γ µ (cid:18) ∂S∂x µ + qA µ (cid:19) γ ν (cid:18) ∂S∂x ν + qA ν (cid:19) ψ + m ψ = 0 . (16)Adding above two equations up, and using the commu-tative relationship for ferminonic fields { γ µ , γ ν } = 2 g µν then produces, { γ µ , γ ν } (cid:18) ∂S∂x µ + qA µ (cid:19) (cid:18) ∂S∂x ν + qA ν (cid:19) ψ + 2 m ψ = 0 . Or equivalently,2 h g µν (cid:18) ∂S∂x µ + qA µ (cid:19) (cid:18) ∂S∂x ν + qA ν (cid:19) + m i ψ = 0 . (17)The phase part is separated from the wave function. Forthe wave function to have a non-trivial solution, we re-quire the vanishing of the phase part of the above equa-tion. This produces the Hamilton-Jacobi equation, g µν (cid:18) ∂S∂x µ + qA µ (cid:19) (cid:18) ∂S∂x ν + qA ν (cid:19) + m = 0 . (18)We solve this equation in the spacetime of the regularblack hole with the smeared mass distribution given inthe previous section to study the Hawking radiation of the black hole. We know that this black hole is Kerr-like from outside, but has no singularity inside becauseof the smeared mass distribution. Reciting the metricof this regular Kerr-like black hole in Boyer-Lindquistcoordinates, same as in eq. (8) ds = − ∆ − a sin θ Σ dt − a sin θ (cid:18) − ∆ − a sin θ Σ (cid:19) dtdφ + Σ∆ dr + Σ dθ + (cid:20) Σ + a sin θ (cid:18) − ∆ − a sin θ Σ (cid:19)(cid:21) sin θdφ , where Σ = r + a cos θ, ∆ = r − m ( r ) r + a . Simplification yields, ds = − ∆ − a sin θ Σ dt − a sin θ r + a − ∆Σ dt dφ + Σ∆ dr + Σ dθ + ( r + a ) − ∆ a sin θ Σ sin θ dφ . (19)In the matrix form, the metric is, g µν = − ∆ − a sin θ Σ − a sin θ ( r + a − ∆ ) Σ Σ∆ − a sin θ ( r + a − ∆ ) Σ (( r + a ) − a △ sin θ ) sin θ Σ . (20)The inverse metric is then, g µν = − ( r + a ) − a ∆ sin θ ∆Σ − a ( r + a − ∆ ) ∆Σ ∆Σ − a ( r + a − ∆ ) ∆Σ ∆ − a sin θ ∆Σ sin θ . (21)Now we are ready to expand the Hamilton-Jacobiequation out in the spacetime metric. We also choose asuitable form for the undetermined solution of Hamilton-Jacobi equation. Considering the spherical symmetry ofKerr-like black hole event horizon, separating the vari-ables for the action S as, S = − ωt + jφ + R ( r ) + P ( θ ) , (22)where ω and j are energy and angular momentum for theparticle.The black hole is uncharged, assuming any electric po-tential being aroused by any other causes is negligible, A µ = 0. Then the Hamilton-Jacobi equation simplifies, g µν ∂S∂x µ ∂S∂x ν + m = 0 . (23)Plugging in the inverse metric, with the undeterminedexpression for the action S , we get the following g tt ω + 2 g tφ ωj + g rr (cid:18) ∂R∂r (cid:19) + g θθ (cid:18) ∂P∂θ (cid:19) + g φφ j + m = 0 . (24)Completely expanding all the terms yields, − ω ( r + a ) ∆Σ + ω a sin θ Σ + 2 ωja r + a Σ∆ − ωja
1Σ + j
1Σ sin θ − j a (cid:18) dRdr (cid:19) ∆Σ + 1Σ (cid:18) dPdθ (cid:19) + m = 0 . (25)We then combine some terms into complete squares,1Σ (cid:18) aω sin θ − j sin θ (cid:19) − [ ω ( r + a ) − ja ] Σ∆+ ∆Σ (cid:18) dRdr (cid:19) + 1Σ (cid:18) dPdθ (cid:19) + m = 0 . (26)Solving for dRdr , multiplying Σ, dividing by ∆, results in (cid:18) dRdr (cid:19) = [ ω ( r + a ) − ja ] ∆ − (cid:18) aω sin θ − j sin θ (cid:19) − (cid:18) dPdθ (cid:19) − m Σ∆ . (27)We care about the radial part of wave transmission factor R ( r ) because it determines the radial transmission of thefermions. Only the radially transmitted part of the wavecan propagate far enough to reach a distant observer.We will find R ( r ) by integrating across black hole’s outerevent horizon r H , thus ∆ →
0. Throwing away lowerorder terms, (cid:18) dRdr (cid:19) = [ ω ( r + a ) − ja ] ∆ . (28)Thus dRdr = ± ω ( r + a ) − ja ∆ . (29)Defining f ( r ) = ∆ r + a , and ω = jar H + a (30)produces dRdr = ± ω − ω f ( r ) . (31)We can integrate R along r . But Hawking radiationcomes from just across the outer event horizon of theblack hole, for which r = r + ± ǫ , where ǫ is infinitesimal. We care about only the integration coming from r = r H .Then by the residual theorem, R ± = ± ( ω − ω ) Z r H + ǫr H − ǫ f ( r ) dr = ± iπ ( ω − ω ) f ′ . (32) R + and R − corresponds to incoming and outgoing wavefunction respectively.The imaginary part of the action S can be find, ImS = ImR = ImR + − ImR − . (33)The tunneling rate for the fermion is:Γ= exp( − ImS ) = exp (cid:18) − π ( ω − ω ) f ′ (cid:19) = exp (cid:18) − ( ω − ω ) T H (cid:19) , (34)with the surface radiation temperature, T H = f ′ ( r H )4 π . (35)where ∆ = r − m ( r ) + a , and m ( r ) = 2 πAγ ( n +32 , r l ).Note that eq. (10) is the equation for the position of thehorizon of the regular black hole, different from that ofthe traditional black hole, r H − M r H π Γ (cid:0) n +32 (cid:1) γ (cid:18) n , r H l (cid:19) + a = 0 . (10)We see that the temperature of the Hawking radiationfor the regular black hole receives quite complex modifi-cations from that of the traditional black hole, f ′ ( r H ) = 14 π ddr (cid:18) r − m ( r ) + a r + a (cid:19) (cid:12)(cid:12)(cid:12) r = r H . (36)Our analytic analysis thus far offers us meaningful con-clusions. Because of the non-singular mass distribution,the Hawking radiation from a Kerr-like regular black holeis modified from that of the traditional Kerr black hole.Further numerical analysis of the fermion tunnelling rateand the Hawking radiation temperature would be ren-dered to computer software processing. Whether or notthere should be richer interior structure for black holesbeyond the relatively simpler traditional black holes, theHawking radiation will prove to be a meaningful methodin determining this critical question and vital in tellingwhether the singularity exactly exists inside the blackhole given that the above modifications are testable atastronomical observations. IV. SUMMARY AND CONCLUSION
The existence of singularity inside traditional blackholes sets up deep conceptual challenges to gravitationalphysics. Regular black holes are constructions of newtypes of black holes which reproduce the geometry oftraditional black holes outside the event horizon, but of-fer tractable and understandable interior structure, andavoid the singularity inside the black hole. We presentour verification to the conjecture that the Hawking ra-diation could be a suitable test for the correctness ofthe regular black hole constructions and a probe into theblack hole interior structure.Our present work provides a positive verification to theabove conjecture. Our result shows a non-trivial depen-dence of the Hawking radiation on the modified horizonradius and interior mass distribution function inside theblack hole and confirms the possibility of employing the Hawking radiation as a promising method to probe intothe structure of black holes as far as these modificationsare testable at astronomical observations.
ACKNOWLEDGMENTS