General class of "quantum deformed" regular black holes
GGeneral class of “quantum deformed”regular black holes
Thomas Berry ID , Alex Simpson ID , and Matt Visser ID School of Mathematics and Statistics, Victoria University of Wellington,PO Box 600, Wellington 6140, New Zealand.
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We discuss the “quantum deformed Schwarzschild spacetime” as originally introducedby Kazakov and Solodukhin in 1993, and investigate the precise sense in which it doesand does not satisfy the desiderata for being a “regular black hole”. We shall carefullydistinguish (i) regularity of the metric components, (ii) regularity of the Christoffelcomponents, and (iii) regularity of the curvature. We shall then embed the Kazakov–Solodukhin spacetime in a more general framework where these notions are clearlyand cleanly separated. Finally we analyze aspects of the classical physics of these“quantum deformed Schwarzschild spacetimes”. We shall discuss the surface gravity,the classical energy conditions, null and timelike geodesics, and the appropriate variantof Regge–Wheeler equation.
Date:
Thursday 4 February 2021; L A TEX-ed February 5, 2021
Keywords : quantum deformed spacetime; regular black hole. a r X i v : . [ g r- q c ] F e b ontents – 1 – Introduction
The unification of general relativity and quantum mechanics is of the utmost impor-tance in reconciling many open problems in theoretical physics today. One avenue ofexploration towards a fully quantised theory of gravity is to, on a case–by–case ba-sis, apply various quantum corrections to existing black hole solutions to the Einsteinequations, and thoroughly analyse the resulting geometries through the lens of stan-dard general relativity. As with the majority of theoretical analysis, to make progressone begins by applying quantum–corrections to the simplest case; the Schwarzschildsolution [1].Historically, various treatments of a quantum–corrected Schwarzschild metric have beenperformed in multiple different settings [2–10]. A specific example of such a metric isthe “quantum deformed Schwarzschild metric” derived by Kazakov and Solodukhinin reference [1]. Much of the literature sees the original metric exported from thecontext of static, spherical symmetry into something dynamical, or else it invokes adifferent treatment of the quantum–correcting process to that performed in [1] (see, e.g. ,reference [11]).The metric derived in reference [1] invokes the following change to the line element forSchwarzschild spacetime in standard curvature coordinates:1 − mr −→ (cid:114) − a r − mr , (1.1)so that d s = − (cid:32)(cid:114) − a r − mr (cid:33) d t + d r (cid:113) − a r − mr + r dΩ . (1.2)To keep the metric components real, the r coordinate must be restricted to the range r ∈ [ a, ∞ ). So the “centre” of the spacetime at r → a is now a 2-sphere of finite area A = 4 πa . The fact that the “centre” has now been “smeared out” to finite r wasoriginally hoped to render the spacetime regular.This metric was originally derived via an action principle which has its roots in the2-D, (more precisely (1+1)-D), dilaton theory of gravity [1, 12]: S = − (cid:90) d z √− g (cid:20) r R (2) − ∇ r ) + 2 κ U ( r ) (cid:21) . (1.3)Here R (2) is the two–dimensional Ricci scalar, κ is a constant with dimensions of length,and U ( r ) is the “dilaton potential”. – 2 –he action (1.3) yields two equations of motion, one of which is then used to derivethe general form of the metric:d s = − f ( r ) d t + d r f ( r ) + r dΩ , f ( r ) = − mr + 1 r (cid:90) r U ( ρ ) d ρ. (1.4)The dilaton potential U ( r ) is quantised within the context of the D = 2 σ -model [1, 12],resulting in the specific metric (1.2). Specifically, Kazakov and Solodukhin choose U ( r ) = r √ r − a . (1.5)Note that generic metrics of the formd s = − f ( r ) d t + d r f ( r ) + r dΩ , (1.6)where one does not necessarily make further assumptions about the function f ( r ), havea long and complex history [13–16].In Kazakov and Solodukhin’s original work [1], they claim the metric (1.2) is “regular”.However, by this they just mean “regular” in the sense of the metric components (inthis specific coordinate chart) being finite for all r ∈ [ a, ∞ ). This is not the meaningof the word “regular” that is usually adopted in the GR community. We find it usefulto carefully distinguish (i) regularity of the metric components, (ii) regularity of theChristoffel components, and (iii) regularity of the curvature. Indeed, within the GRcommunity, the term “regular” means that the spacetime entirely is free of curvaturesingularities [17–46], with infinities in the curvature invariants being used as the typicaldiagnostic. While the metric (1.2) is regular in terms of the metric components, it failsto be regular in terms of the Christoffel components, and has a Ricci scalar which ismanifestly singular at r = a : R = 2 r − r − a r ( r − a ) = a (2 a ) ( r − a ) −
234 (2 a ) ( r − a ) + O (1) . (1.7)The specific metric (1.2) derived by Kazakov and Solodukhin falls in to a more generalclass of metrics given byd s n = − f n ( r )d t + d r f n ( r ) + r (cid:0) d θ + sin θ d φ (cid:1) , (1.8)where now we take f n ( r ) = (cid:18) − a r (cid:19) n − mr . (1.9)– 3 –ere n ∈ { } ∪ { , , , . . . } , r ∈ [ a, ∞ ), and a ∈ (0 , ∞ ). (Note, we include n = 0as a special case since this reduces the metric to the Schwarzschild metric in standardcurvature coordinates, which is useful for consistency checks). We only consider oddvalues for n (excluding the n = 0 Schwarzschild solution) as any even value of n willallow for the r -coordinate to continue down to r = 0, and so produce a black-holespacetime which is not regular at its core and hence not of interest in this work.The class of metrics described by equations (1.8)–(1.9) has the following regularitystructure: • n = 0 (Schwarzschild): Not regular; • n ≥
1: Metric–regular; • n ≥
3: Christoffel–symbol–regular; • n ≥
5: Curvature–regular.We wish to stress that, unlike reference [1], we make no attempt to derive the classof metrics described by equations (1.8)–(1.9) from a modified action principle in thiscurrent work. We feel that there are a number of technical issues requiring clarificationin the derivation presented in reference [1], so instead, we shall simply use the resultsof Kazakov and Solodukhin’s work as inspiration and motivation for the analysis of ourgeneral class of metrics. As such, our extended class of Kazakov–Solodukhin modelscan be viewed as another set of “black hole mimickers” [47–58], arbitrarily closelyapproximating standard Schwarzschild black holes, and so potentially of interest toobservational astronomers [59].
In this section we shall analyse the metric (1.8), its associated Christoffel symbols, andthe various curvature tensor quantities derived therefrom.
We immediately enforce a (cid:54) = 0 since a = 0 is trivially Schwarzschild, and in fact weshall specify a > a is typically to be identified with the Planck scale. At large r and/or small a we have: f n ( r ) = (cid:18) − a r (cid:19) n − mr = 1 − mr − na r + O (cid:18) a r (cid:19) . (2.1)– 4 –o the spacetime is asymptotically flat with mass m for any fixed finite value of n . As r → a we note that for n ≥ r → a f n ( r ) = − ma . (2.2)This is enough to imply metric–regularity. Note however that for the radial derivativewe have f (cid:48) n ( r ) = na r (cid:18) − a r (cid:19) n − + 2 mr , (2.3)and that only for n ≥ r → a f (cid:48) n ( r ) = 2 ma . (2.4)Similarly for the second radial derivative f (cid:48)(cid:48) n ( r ) = na ( na + a − r ) r (cid:18) − a r (cid:19) n − − mr , (2.5)and only for n ≥ r → a f (cid:48)(cid:48) n ( r ) = − ma . (2.6)This ultimately is why we need n ≥ n ≥ Event horizons (Killing horizons) may be located by solving g tt ( r ) = f n ( r ) = 0, and soare implicitly characterized by r H = 2 m (cid:18) − a r H (cid:19) − n . (2.7)This is not algebraically solvable for general n , though we do have the obvious boundsthat r H > m and r H > a .Furthermore, for small a we can use (2.7) to find an approximate horizon location byiterating the lowest-order approximation r H = 2 m + O ( a /m ) to yield r H = 2 m (cid:26) na m + O (cid:18) a m (cid:19)(cid:27) . (2.8)– 5 –terating a second time r H = 2 m (cid:26) na m − n (3 n − a m + O (cid:18) a m (cid:19)(cid:27) . (2.9)We shall soon find that taking this second iteration is useful when estimating thesurface gravity. As usual, while event horizons are mathematically easy to work with,one should bear in mind that they are impractical for observational astronomers to dealwith — any physical observer limited to working in a finite region of space+time canat best detect apparent horizons or trapping horizons [60], see also reference [61]. Inview of this intrinsic limitation, approximately locating the position of the horizon isgood enough for all practical purposes. Up to the usual symmetries, the non-trivial non-zero coordinate components of theChristoffel connection in this coordinate system are:Γ ttr = − Γ rrr = 2 m/r + n ( a /r )(1 − a /r ) n − r { (1 − a /r ) n − m/r } ;Γ rtt = { m/r + n ( a /r )(1 − a /r ) n − }{ (1 − a /r ) n − m/r } r ;Γ rθθ = Γ rφφ sin θ = 2 m − r (1 − a /r ) n . (2.10)The trivial non-zero components areΓ θrθ = Γ φrφ = 1 r ;Γ θφφ = − sin θ cos θ ;Γ φθφ = cot θ. (2.11)Inspection of the numerators of Γ ttr , Γ rrr , and Γ rθθ shows that (in this coordinatesystem) the Christoffel symbols are finite at r = a so long as n ≥
3. Indeed as r → a we see Γ ttr = − Γ rrr → − a ; Γ rtt → − m a ;Γ rθθ = Γ rφφ sin θ → m ; Γ θrθ → Γ φrφ = 1 a . (2.12)– 6 – .4 Orthonormal components When a metric g ab is diagonal then the quickest way of calculating the orthonormalcomponents of the Riemann and Weyl tensors is to simply set R ˆ a ˆ b ˆ c ˆ d = R abcd | g ac | | g bd | ; C ˆ a ˆ b ˆ c ˆ d = C abcd | g ac | | g bd | . (2.13)When a metric g ab is diagonal and a tensor X ab is diagonal then the quickest way ofcalculating the orthonormal components is to simply set X ˆ a ˆ b = X ab | g ab | . (2.14)In both situations some delicacy is called for when crossing any horizon that might bepresent. Let us (using − + ++ signature and assuming a diagonal metric) define S = sign( − g tt ) = sign( g rr ) . (2.15)Then S = +1 in the domain of outer communication (above the horizon) and S = − We shall now analyse what values of n result in non-singular components of variouscurvature tensors in an orthonormal basis (ˆ t, ˆ r, ˆ θ, ˆ φ ). First, the non-zero orthonormalcomponents of the Riemann tensor are: R ˆ r ˆ t ˆ r ˆ t = − mr − na (cid:2) − ( n + 1) a /r (cid:3) (1 − a /r ) n − r ,R ˆ r ˆ θ ˆ r ˆ θ = R ˆ r ˆ φ ˆ r ˆ φ = − R ˆ θ ˆ t ˆ θ ˆ t = − R ˆ φ ˆ t ˆ φ ˆ t = − S (cid:26) mr + na (1 − a /r ) n − r (cid:27) ,R ˆ θ ˆ φ ˆ θ ˆ φ = 2 mr + 1 − (1 − a /r ) n r . (2.16)Analysis of the numerator of R ˆ r ˆ t ˆ r ˆ t shows that all of the orthonormal components of theRiemann tensor remain finite at r = a if and only if n ≥
5. Indeed as r → a (where S → −
1) we see R ˆ r ˆ t ˆ r ˆ t → − ma ; R ˆ θ ˆ φ ˆ θ ˆ φ → a + 2 ma .R ˆ r ˆ θ ˆ r ˆ θ = R ˆ r ˆ φ ˆ r ˆ φ = − R ˆ θ ˆ t ˆ θ ˆ t = − R ˆ φ ˆ t ˆ φ ˆ t → + ma . (2.17)– 7 –onversely at large r (where S → +1) we see R ˆ r ˆ t ˆ r ˆ t = − mr + O ( a /r ) ,R ˆ r ˆ θ ˆ r ˆ θ = R ˆ r ˆ φ ˆ r ˆ φ = − R ˆ θ ˆ t ˆ θ ˆ t = − R ˆ φ ˆ t ˆ φ ˆ t = − mr + O ( a /r ) ,R ˆ θ ˆ φ ˆ θ ˆ φ = 2 mr + O ( a /r ) . (2.18)So, as it should, the spacetime curvature asymptotically approaches that of Schwarzschild. The non-zero orthonormal components of the Ricci tensor are: R ˆ t ˆ t = − R ˆ r ˆ r = − S na r (cid:2) − ( n − a /r (cid:3) (1 − a /r ) n − ,R ˆ θ ˆ θ = R ˆ φ ˆ φ = 1 r − r (cid:2) n − a /r (cid:3) (1 − a /r ) n − . (2.19)Analysis of the R ˆ r ˆ r component shows that all of the components of the Ricci tensorremain finite at r = a so long as n ≥
5. Indeed as r → a we see R ˆ t ˆ t = − R ˆ r ˆ r → , R ˆ θ ˆ θ = R ˆ φ ˆ φ → a . (2.20)Conversely at large r we have R ˆ t ˆ t = − R ˆ r ˆ r = R ˆ θ ˆ θ = R ˆ φ ˆ φ = − na r + O ( a /r ) . (2.21) As stated in Section 1, our class of metrics is only curvature regular for n ≥
5, where n is an odd integer. Indeed, in general we have R = 2 r − (1 − a /r ) n − (cid:26) n − a /r + ( n − n − a /r r (cid:27) , (2.22)and so the spacetime is non-singular at r = a if and only if n ≥
5. Furthermore, any n ≥ r = a , where R → a .– 8 –s an explicit example, R n =5 = 2 r − √ r − a (cid:26) r + a r + 12 a r (cid:27) , (2.23)which is indeed singularity–free in the region r ∈ [ a, ∞ ) and positive at r = a . The non-zero components of the Einstein tensor are G ˆ t ˆ t = − G ˆ r ˆ r = Sr (cid:40) − (cid:20) n − a r (cid:21) (cid:18) − a r (cid:19) ( n − / (cid:41) ,G ˆ θ ˆ θ = G ˆ φ ˆ φ = − na r (cid:20) − ( n − a r (cid:21) (cid:18) − a r (cid:19) ( n − / . (2.24)Analysis of the G ˆ θ ˆ θ component reveals that the Einstein tensor remains finite in all ofits orthonormal components if and only if n ≥
5. Indeed as r → a (where S → −
1) wesee G ˆ t ˆ t = − G ˆ r ˆ r → − a , G ˆ θ ˆ θ = G ˆ φ ˆ φ → . (2.25)At large r (where S → +1) we have G ˆ t ˆ t = − G ˆ r ˆ r = G ˆ θ ˆ θ = G ˆ φ ˆ φ = − na r + O ( a /r ) . (2.26) The non-zero components of the Weyl tensor are C ˆ r ˆ t ˆ r ˆ t = 2 S C ˆ r ˆ θ ˆ r ˆ θ = 2 S C ˆ r ˆ φ ˆ r ˆ φ = − S C ˆ θ ˆ t ˆ θ ˆ t = − S C ˆ φ ˆ t ˆ φ ˆ t = − C ˆ θ ˆ φ ˆ θ ˆ φ = − mr + (1 − a /r ) n − − r − a (1 − a /r ) n − (cid:26) (5 n + 4) − ( n + 2)( n + 1) a /r r (cid:27) . (2.27)Thus, the components of the Weyl tensor remain finite at r = a so long as n ≥ r → a (where S → −
1) we see C ˆ r ˆ t ˆ r ˆ t = − C ˆ r ˆ θ ˆ r ˆ θ = − C ˆ r ˆ φ ˆ r ˆ φ = +2 C ˆ θ ˆ t ˆ θ ˆ t = +2 C ˆ φ ˆ t ˆ φ ˆ t = − C ˆ θ ˆ φ ˆ θ ˆ φ → − a − ma . (2.28)At large r (where S → +1) we find C ˆ r ˆ t ˆ r ˆ t = 2 C ˆ r ˆ θ ˆ r ˆ θ = 2 C ˆ r ˆ φ ˆ r ˆ φ = − C ˆ θ ˆ t ˆ θ ˆ t = − C ˆ φ ˆ t ˆ φ ˆ t = − C ˆ θ ˆ φ ˆ θ ˆ φ = − mr − na r + O ( a /r ) . (2.29) The Weyl scalar is defined by C abcd C abcd . In view of all the symmetries of the spacetimeone can show that C abcd C abcd = 12( C ˆ r ˆ t ˆ r ˆ t ) , so one gains no additional behaviour beyondlooking at the Weyl tensor itself. Thus, for purposes of tractability we will only displaythe result for n = 5 at r = a in order to show that the n = 5 spacetime is indeedregular at r = a : ( C abcd C abcd ) n =5 (cid:12)(cid:12) r = a = 4(6 m + a ) a . (2.30) The Kretschmann scalar is given by K = R abcd R abcd = C abcd C abcd + 2 R ab R ab − R . (2.31)The general result is rather messy and does not provide much additional insight intothe spacetime. Thus, for purposes of tractability we will only display the result for n = 5 at r = a in order to show that the n = 5 spacetime is indeed regular at r = a : K n =5 | r = a = 4 a (cid:0) a + 4 am + 12 m (cid:1) . (2.32)The fact that the Kretschmann scalar is positive definite, and can be written as a sum ofsquares, is ultimately a due to spherical symmetry and the existence of a hypersurfaceorthogonal Killing vector [62]. – 10 – Surface gravity and Hawking temperature
Let us calculate the surface gravity at the event horizon for the generalised QMSspacetime. Because we are working in curvature coordinates we always have [63] κ H = lim r → r H ∂ r g tt √ g tt g rr . (3.1)Thence κ H = 12 ∂ r f n ( r ) (cid:12)(cid:12)(cid:12)(cid:12) r H = mr H + na r H (cid:18) − a r H (cid:19) n − . (3.2)Using equation (2.7) we can also rewrite this as κ H = mr H (cid:26) na r H − a (cid:27) . (3.3)This result is, so far, exact. Given that the horizon location is not analytically known forgeneral n , we shall use the asymptotic result r H = 2 m (cid:110) na m − n (3 n − a m + O ( a m ) (cid:111) .Thence κ H = 14 m (cid:26) − n ( n − a m + O ( a /m ) (cid:27) . (3.4)Note the potential O ( a /m ) term vanishes (which is why we estimated r H up to O ( a )).As usual the Hawking temperature is simply k B T H = π (cid:126) κ H . Let us examine the Einstein field equations for this spacetime. Above the horizon, for r > r H , we have 8 π ρ = G ˆ t ˆ t ; 8 π p r = G ˆ r ˆ r . (4.1)Below the horizon, for r < r H , we have8 π ρ = G ˆ r ˆ r ; 8 π p r = G ˆ t ˆ t . (4.2)– 11 –ut then regardless of whether one is above or below the horizon one has ρ = − p r = 18 πr (cid:40) − (cid:20) n − a r (cid:21) (cid:18) − a r (cid:19) ( n − / (cid:41) ,p ⊥ = − na πr (cid:20) − ( n − a r (cid:21) (cid:18) − a r (cid:19) ( n − / . (4.3)By inspection, for n > p ⊥ ( r ) = 0 at r = √ n − a . Indeed we see that p ⊥ ( r ) > r < √ n − a and p ⊥ ( r ) < r > √ n − a . The analagous resultfor ρ ( r ) is not analytically tractable (though it presents no numerical difficulty) as byinspection it amounts to finding the roots of( r − a ) r n − ( r − a ) n ( r + ( n − a ) = 0 . (4.4)We note that asymptotically ρ = − na πr + O ( a /r ) , (4.5)and p ⊥ = − na πr + O ( a /r ) . (4.6)An initially surprising result is that the stress-energy tensor has no dependence on themass m of the spacetime. To see what is going on here, consider the Misner–Sharpquasi-local mass1 − m ( r ) r = f n ( r ) = ⇒ m ( r ) = m + r (cid:40) − (cid:18) − a r (cid:19) n (cid:41) . (4.7)Then, noting that m = m ( r ) + 4 π (cid:82) ∞ r ρ (¯ r )¯ r d¯ r above the horizon, we see4 π (cid:90) ∞ r ρ (¯ r )¯ r d¯ r = − r (cid:40) − (cid:18) − a r (cid:19) n (cid:41) . (4.8)Here the RHS is manifestly independent of m . Consequently, without need of anydetailed calculation, ρ ( r ) is manifestly independent of m . As an aside note that m ( r H ) = r H , so we could also write m ( r ) = r H + 4 π (cid:82) rr H ρ (¯ r )¯ r d¯ r .– 12 – Energy conditions
The classical energy conditions are constraints on the stress-energy tensor that attemptto keep various aspects of “unusual physics” under control [64–84]. While it can beargued that the classical energy conditions are not truly fundamental [71, 74, 80], oftenbeing violated by semi-classical quantum effects, they are nevertheless extremely usefulindicative probes, well worth the effort required to analyze them.
A necessary and sufficient condition for the null energy condition (NEC) to hold is thatboth ρ + p r ≥ ρ + p ⊥ ≥ r , a , m . Since ρ = − p r , the former inequality istrivially satisfied, and for all r ≥ a we may simply consider ρ + p ⊥ = 18 πr (cid:26) − (1 − a /r ) n − (cid:20) n − a /r − ( n + 2)( n − a /r (cid:21)(cid:27) . (5.1)Whether or not this satisfies the NEC depends on the value for n . Furthermore, for novalue of n is the NEC globally satisfied.Provided n ≥
5, so that the limits exist, we havelim r → a ( ρ + p ⊥ ) = + 18 πa . (5.2)So the NEC is definitely satisfied deep in the core of the system. Note that at asymp-totically large distances ρ + p t = − na πr + O ( a /r ) . (5.3)So the NEC (and consequently all the other classical point-wise energy conditions) arealways violated at asymptotically large distances. However, for some values of n , thereare bounded regions of the spacetime in which the NEC is satisfied. See Figure 1. In order to satisfy the weak energy condition (WEC) we require the NEC be satisfied,and in addition ρ ≥
0. But in view of the asymptotic estimate (4.5) for ρ we seethat the WEC is always violated at large distances. Furthermore, it can be seen fromTable 1 that the region in which the NEC is satisfied is always larger than that in which ρ is positive (this would be as good as impossible to prove analytically for general n ).– 13 – - - - - - - / an = - / an = - / an =
51 2 3 4 50.00.51.01.52.0 r / an = / an = / an = Figure 1 : Plots of the NEC for several values of n . Here the y -axis depicts 8 πr ( ρ + p ⊥ ),plotting against r/a on the x -axis. Of particular interest are the qualitative differencesin behaviour as r/a →
1; we see divergent behaviour for the n = 1 and n = 3 cases,whilst for n ≥ πr ( ρ + p ⊥ ) → r → a . This is ultimately due tothe fact that the n ≥ curvature regular , with globally finite stress-energycomponents.Thus, we can conclude (see Table 2) that the WEC is satisfied for smaller regions thanthe NEC for all values of n . In order to satisfy the strong energy condition (SEC) we require the NEC to be satisfied,and in addition ρ + p r + 2 p ⊥ = 2 p ⊥ ≥
0. But regardless of whether one is above orbelow the horizon, the second of these conditions p ⊥ ≥ < a < r ≤ a √ n − . (5.4)However, it can be seen from Table 1 that the region in which the NEC is satisfied isalways smaller than that in which p ⊥ is positive (this would be as good as impossibleto prove analytically for general n ). Thus, we can conclude (see Table 2) that the SECis satisfied in the same region as the NEC for all values of n .– 14 – .4 Dominant energy condition The dominant energy condition is the strongest of the standard classical energy condi-tions. Perhaps the best physical interpretation of the DEC is that for any observer withtimelike 4-velocity V a the flux vector F a = T ab V b is non-spacelike (timelike or null).It is a standard result that in spherical symmetry (in fact for any type I stress-energytensor) this reduces to positivity of the energy density ρ > | p i | ≤ ρ . Since in the current framework for the radial pressure we always have p r = − ρ , the only real constraint comes from demanding | p ⊥ | ≤ ρ . But this means wewant both ρ + p ⊥ ≥ and ρ − p ⊥ ≥
0. The first of these conditions is just the NEC, sothe only new constraint comes from the second condition. By inspection, it can be seenfrom Table 1 that the region in which the NEC is satisfied is always larger than that inwhich ρ − p ⊥ is positive (this would be as good as impossible to prove analytically forgeneral n ). Thus, we can conclude (see Table 2) that the DEC is satisfied for smallerregions than the NEC for all values of n . Table 1 : Regions of the spacetime where the orthonormal components of the stress-energy tensor satisfy certain inequalities. n ρ + p ⊥ ≥ ρ ≥ p ⊥ ≥ ρ − p ⊥ ≥ a < r < ∞ a < r < ∞ a < r < ∞ a < r < ∞ a < r (cid:47) . a a < r (cid:47) . a a < r (cid:47) . a globally violated5 a < r (cid:47) . a a < r (cid:47) . a a < r ≤ a a < r (cid:47) . a a < r (cid:47) . a a < r (cid:47) . a a < r (cid:47) . a a < r (cid:47) . a a < r (cid:47) . a a < r (cid:47) . a a < r (cid:47) . a < r (cid:47) . a a < r (cid:47) . a a < r (cid:47) . a a < r (cid:47) . a a < r (cid:47) . a ... ... ... ... ...– 15 – able 2 : Regions of the spacetime where the energy conditions are satisfied. n NEC WEC SEC DEC0 a < r < ∞ a < r < ∞ a < r < ∞ a < r < ∞ a < r (cid:47) . a a < r (cid:47) . a same as NEC globally violated5 a < r (cid:47) . a a < r (cid:47) . a same as NEC a < r (cid:47) . a a < r (cid:47) . a a < r (cid:47) . a same as NEC a < r (cid:47) . a a < r (cid:47) . a a < r (cid:47) . a same as NEC a < r (cid:47) . a a < r (cid:47) . a a < r (cid:47) . a same as NEC a < r (cid:47) . a ... ... ... ... ... We have the generalised quantum modified Schwarzschild metricd s = − (cid:40)(cid:18) − a r (cid:19) n − mr (cid:41) d t + d r (cid:0) − a r (cid:1) n − mr + r dΩ . (6.1)Let us now find the location of both the photon sphere for massless particles, and theISCO for massive particles, as functions of the parameters m , n , and a . Consider thetangent vector to the worldline of a massive or massless particle, parameterized bysome arbitrary affine parameter, λ : g ab d x a d λ d x b d λ = − g tt (cid:18) d t d λ (cid:19) + g rr (cid:18) d r d λ (cid:19) + r (cid:40)(cid:18) d θ d λ (cid:19) + sin θ (cid:18) d φ d λ (cid:19) (cid:41) . (6.2)We may, without loss of generality, separate the two physically interesting cases (time-like and null) by defining: (cid:15) = (cid:26) − i.e. timelike worldline0 massless particle, i.e. null worldline . (6.3)That is, d s / d λ = (cid:15) . Due to the metric being spherically symmetric we may fix θ = π – 16 –rbitrarily and view the reduced equatorial problem: g ab d x a d λ d x b d λ = − g tt (cid:18) d t d λ (cid:19) + g rr (cid:18) d r d λ (cid:19) + r (cid:18) d φ d λ (cid:19) = (cid:15). (6.4)The Killing symmetries yield the following expressions for the conserved energy E andangular momentum L per unit mass: (cid:40)(cid:18) − a r (cid:19) n − mr (cid:41) (cid:18) d t d λ (cid:19) = E ; r (cid:18) d φ d λ (cid:19) = L. (6.5)Hence (cid:40)(cid:18) − a r (cid:19) n − mr (cid:41) − (cid:40) − E + (cid:18) d r d λ (cid:19) (cid:41) + L r = (cid:15), (6.6)implying (cid:18) d r d λ (cid:19) = E + (cid:40)(cid:18) − a r (cid:19) n − mr (cid:41) (cid:26) (cid:15) − L r (cid:27) . (6.7)This gives “effective potentials” for geodesic orbits as follows: V (cid:15) ( r ) = (cid:40)(cid:18) − a r (cid:19) n − mr (cid:41) (cid:26) − (cid:15) + L r (cid:27) . (6.8) For a photon orbit we have the massless particle case (cid:15) = 0. Since we are in a sphericallysymmetric environment, solving for the locations of such orbits amounts to finding thecoordinate location of the “photon sphere”. These circular orbits occur at V (cid:48) ( r ) = 0.That is: V ( r ) = (cid:40)(cid:18) − a r (cid:19) n − mr (cid:41) (cid:26) L r (cid:27) , (6.9)leading to: V (cid:48) ( r ) = L r (cid:40) m + r (cid:18) − a r (cid:19) n − (cid:20) ( n + 2) a r − (cid:21)(cid:41) . (6.10)Solving V (cid:48) ( r ) = 0 analytically is intractable, but we may perform a Taylor seriesexpansion of the above function about a = 0 for a valid approximation (recall a isassociated with the Planck length). – 17 –o fifth-order this yields: V (cid:48) ( r ) = 2 L r (3 m − r ) + 2 L na r − na L ( n − r + O (cid:0) L a /r (cid:1) . (6.11)Equating this to zero and solving for r yields: r γ = 3 m (cid:26) a n (3 m ) − n (11 n − a m ) + O ( a /m ) (cid:27) . (6.12)The a = 0, (or n = 0), Schwarzschild sanity check reproduces r γ = 3 m , the expectedresult.To verify stability, we check the sign of V (cid:48)(cid:48) ( r ): V (cid:48)(cid:48) ( r ) = − L r (cid:40) mr − (cid:18) − a r (cid:19) n − (cid:20) − (7 n + 12) a r + ( n + 2)( n + 3) a r (cid:21)(cid:41) . (6.13)We now substitute the approximate expression for r γ into Eq. (6.13) to determine thesign of V (cid:48)(cid:48) ( r γ ). We find: V (cid:48)(cid:48) ( r γ ) = − L m (cid:26) − na (3 m ) + n (67 n − a m ) + O ( a /m ) (cid:27) (6.14)Given that all bracketed terms to the right of the 1 are strictly subdominant in viewof a (cid:28) m , we may conclude that V (cid:48)(cid:48) ( r γ ) <
0, and hence the null orbits at r = r γ areunstable.Let us now recall the generalised form of equation (6.9), and specialise to n = 5 (thelowest value for n for which our quantum deformed Schwarzschild spacetime is regular ).We have: V ( r, n = 5) = L r (cid:40)(cid:18) − a r (cid:19) − mr (cid:41) ; (6.15) V (cid:48) ( r, n = 5) = L r (cid:26) m − √ r − a (cid:18) − a r + 7 a r (cid:19)(cid:27) . (6.16)Once again setting this to zero and attempting to solve analytically is an intractableline of inquiry, and we instead inflict Taylor series expansions about a = 0.– 18 –o fifth-order we have the following V (cid:48) ( r, n = 5) = − L r (cid:26) − mr − a r + 45 L a r + O ( a /r ) (cid:27) ;= ⇒ r γ = 3 m (cid:26) a (3 m ) − a m ) + O ( a /m ) (cid:27) , (6.17)which is consistent with the result for general n displayed in Eq. (6.12). For massive particles the geodesic orbit corresponds to a timelike worldline and we havethe case that (cid:15) = −
1. Therefore: V − ( r ) = (cid:40)(cid:18) − a r (cid:19) n − mr (cid:41) (cid:26) L r (cid:27) , (6.18)and it is easily verified that this leads to: V (cid:48) − ( r ) = 2 m (3 L + r ) r + (1 − a /r ) n − r (cid:20) na + L (cid:18) ( n + 2) a r − (cid:19)(cid:21) . (6.19)For small a we have V − ( r ) = (cid:26) L r (cid:27) (cid:26) − mr − na r + n ( n − a r + O (cid:18) a r (cid:19)(cid:27) , (6.20)and V (cid:48) − ( r ) = 2( L (3 m − r ) + mr ) r + (2 L + r ) na r − (3 L + 2 r ) n ( n − a r + O (cid:18) a r (cid:19) . (6.21)Equating this to zero and rearranging for r presents an intractable line of inquiry.Instead it is preferable to assume a fixed circular orbit at some r = r c , and rearrangethe required angular momentum L c to be a function of r c , m , and a . It then followsthat the innermost circular orbit shall be the value of r c for which L c is minimised. It isof course completely equivalent to perform this procedure for the mathematical object L c , and we do so for tractability. – 19 –ence if V (cid:48) − ( r c ) = 0, we have: L c = na (cid:16) − a r (cid:17) n + 2 mr (cid:16) − a r (cid:17)(cid:0) − a r (cid:1) n (cid:2) − ( n + 2) a r (cid:3) − mr (cid:0) − a r (cid:1) . (6.22)For small a we have L c = mr r − m + nr ( r − m ) a r − m ) − n { (2 n + 4) r + (5 n − mr − n − m } a r ( r − m ) + O ( a ) . (6.23)As a consistency check, for large r c ( i.e. r c (cid:29) a, m ) we observe from the dominantterm of Eq. (6.23) that L c ≈ √ mr c , which is consistent with the expected value whenconsidering circular orbits in weak-field GR. Indeed it is easy to check that for large r we have L c = mr c + O (1). Note that in classical physics the angular momentumper unit mass for a particle with angular velocity ω is L c ∼ ωr c . Kepler’s third lawof planetary motion implies that r c ω ∼ G N m/r c . (Here m is the mass of the centralobject, as above.) It therefore follows that L c ∼ (cid:112) G N m/r c r c . That is L c ∼ √ mr c , asabove.Differentiating Eq. (6.22) and finding the resulting roots is not analytically feasible.We instead differentiate Eq. (6.23), obtaining a Taylor series for ∂L c ∂r c for small a : ∂L c ∂r c = mr c ( r c − m )( r c − m ) − mn (5 r c − m ) a r c − m ) (6.24) − n { r c + ( n − r c + 21 mr c − m r c + 27 m ) } a r c ( r c − m ) + O ( a ) . Solving for the stationary points yields: r ISCO = 6 m (cid:26) na m − n (49 n − a m + O (cid:18) a m (cid:19)(cid:27) , (6.25)and the a = 0 Schwarzschild sanity check reproduces r c = 6 m as required. Denoting r H as the location of the horizon, r γ as the location of the photon sphere, and r ISCO as the location of the ISCO, we have the following summary:– 20 – r H = 2 m × { na m ) − n (3 n − a m ) + O ( a m ) } ; • r γ = 3 m × (cid:110) a n (3 m ) − n (11 n − a m ) + O ( a m ) (cid:111) ; • r ISCO = 6 m × (cid:110) na m − n (49 n − a m + O ( a m ) (cid:111) . Now considering the Regge–Wheeler equation, in view of the unified formalism devel-oped in reference [85], (see also references [58, 86, 87]), we may explicitly evaluate theRegge–Wheeler potentials for particles of spin S ∈ { , } in our spacetime. Firstlydefine a tortoise coordinate as follows:d r ∗ = d r (cid:0) − a r (cid:1) n − mr . (7.1)This tortoise coordinate is, for general n , not analytically defined. However let us makethe coordinate transformation regardless; this yields the following expression for themetric:d s = (cid:40)(cid:18) − a r (cid:19) n − mr (cid:41) (cid:26) − d t + d r ∗ (cid:27) + r (cid:0) d θ + sin θ d φ (cid:1) . (7.2)It is convenient to write this as:d s = A ( r ∗ ) (cid:26) − d t + d r ∗ (cid:27) + B ( r ∗ ) (cid:0) d θ + sin θ d φ (cid:1) . (7.3)The Regge–Wheeler equation is [85–87]: ∂ r ∗ ˆ φ + { ω − V S } ˆ φ = 0 , (7.4)where ˆ φ is the scalar or vector field, V is the spin-dependent Regge–Wheeler potentialfor our particle, and ω is some temporal frequency component in the Fourier domain.For a scalar field ( S = 0) examination of the d’Alembertian equation quickly yields: V S =0 = (cid:26) A B (cid:27) (cid:96) ( (cid:96) + 1) + ∂ r ∗ BB . (7.5)– 21 –or a massless vector field, ( S = 1; e.g. photon), explicit conformal invariance in 3+1dimensions guarantees that the Regge–Wheeler potential can depend only on the ratio A/B , whence normalising to known results implies: V S =1 = (cid:26) A B (cid:27) (cid:96) ( (cid:96) + 1) . (7.6)Collecting results, for S ∈ { , } we have: V S ∈{ , } = (cid:26) A B (cid:27) (cid:96) ( (cid:96) + 1) + (1 − S ) ∂ r ∗ BB . (7.7)The spin 2 axial mode is somewhat messier, and (for current purposes) not of immediateinterest.Noting that for our metric ∂ r ∗ = (cid:26)(cid:16) − a r (cid:17) n − mr (cid:27) ∂ r and B ( r ) = r we have: ∂ r ∗ BB = ∂ r ∗ (cid:26)(cid:16) − a r (cid:17) n − mr (cid:27) r = 1 r (cid:40)(cid:18) − a r (cid:19) n − mr (cid:41) (cid:40) n (cid:18) − a r (cid:19) n − a r + 2 mr (cid:41) . (7.8)For small a : ∂ r ∗ BB = 2 m (1 − m/r ) r + n ( r − m ) r a + n { n − m − n − r } r a + O (cid:18) ma r (cid:19) . (7.9)Therefore: V S ∈{ , } = 1 r (cid:34)(cid:18) − a r (cid:19) n − mr (cid:35) (cid:40) (cid:96) ( (cid:96) + 1) + (1 − S ) (cid:34) n (cid:18) − a r (cid:19) n − a r + 2 mr (cid:35) (cid:41) . (7.10)This has the correct behaviour as a →
0, reducing to the Regge–Wheeler potential forSchwarzschild: lim a → V S ∈{ , } = 1 r (cid:20) − mr (cid:21) (cid:26) (cid:96) ( (cid:96) + 1) + (1 − S ) 2 mr (cid:27) . (7.11)– 22 –n the small a approximation we have the asymptotic result V S ∈{ , } = (cid:0) − mr (cid:1) r (cid:26) (cid:96) ( (cid:96) + 1) + (1 − S ) 2 mr (cid:27) − na r (cid:26) (cid:96) ( (cid:96) + 1) + 2(1 − S ) (cid:20) mr − (cid:21)(cid:27) + na r (cid:26) ( n − (cid:96) ( (cid:96) + 1)] − (1 − S ) (cid:104) n −
1) + 5 (cid:16) − n (cid:17) mr (cid:105)(cid:27) + O (cid:18) a r (cid:19) . (7.12)The Regge–Wheeler equation is fundamental to exploring the quasi-normal modes of thecandidate spacetimes, an integral part of the “ringdown” phase of the LIGO calculationto detect astrophysical phenomena via gravitational waves. Exploring the quasi-normalmodes is, for now, relegated to the domain of future research. The original Kazakov–Solodukhin “quantum deformed Schwarzschild spacetime” [1]is slightly more “regular” than Schwarzschild spacetime, but it is not “regular” inthe sense normally intended in the general relativity community. While the metriccomponents are regular, both Christoffel symbols and curvature invariants diverge atthe “centre” of the spacetime, a 2-sphere where r → a with finite area A = 4 πa .The “smearing out” of the “centre” to r → a is not sufficient to guarantee curvatureregularity.We have generalized the original Kazakov–Solodukhin spacetime to a two-parameterclass compatible with the ideas mooted in reference [1]. Our generalized two-parameterclass of “quantum corrected” Schwarzschild spacetimes contains exemplars which havemuch better regularity properties, and we can distinguish three levels of regularity:metric regularity, Christoffel regularity, and regularity of the curvature invariants.Furthermore, our generalized two-parameter class of models distorts Schwarzschildspacetime in a clear and controlled way — so providing yet more examples of black-hole“mimickers” potentially of interest for observational purposes. In this regard we haveanalyzed the geometry, surface gravity, stress-energy, and classical energy conditions.We have also perturbatively analyzed the locations of ISCOs and photon spheres, andset up the appropriate Regge–Wheeler formalism for spin-1 and spin-0 excitations.Overall, the general topic of “quantum corrected” Schwarzschild spacetimes is certainlyof significant interest, and we hope that these specific examples may serve to encouragefurther investigation in this field. – 23 – cknowledgements TB acknowledges financial support via a MSc Masters Scholarship provided by Vic-toria University of Wellington. TB is also indirectly supported by the Marsden fund,administered by the Royal Society of New Zealand.AS acknowledges financial support via a PhD Doctoral Scholarship provided by Vic-toria University of Wellington. AS is also indirectly supported by the Marsden fund,administered by the Royal Society of New Zealand.MV was directly supported by the Marsden Fund, via a grant administered by theRoyal Society of New Zealand.
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