A 4D asymptotically flat rotating black hole solution including supertraslation corrections
aa r X i v : . [ h e p - t h ] F e b A 4D asymptotically flat rotating black hole solutionincluding displacement of supertraslation
Shingo Takeuchi † Phenikaa Institute for Advanced Study and Faculty of Basic Science,Phenikaa University, Hanoi 100000, Vietnam
Abstract
One of the problems in the current context of asymptotic symmetry is to extendthe black hole considered to the rotating one. Therefore, we in this paper obtaina four-dimensional asymptotically flat rotating black hole solution including thedisplacement of supertraslation. Since it has been obtained not by solving Einsteinequation but based on the already obtained supertranslated Schwarzschild blackhole solution by using coordinate transformations, it is not the general solution.However, it is sure that it is ‘a’ supertranslated rotating black hole solution as itcan satisfy the Einstein equation. Then, as one of the interesting issues on thesupertranslated rotating black hole, we analyze the classical gravitational pertur-bation with the effect of the rotating black hole by expanding with regard to r from the infinity to the order where the effect of the rotation of the black hole canfinally remain in the result. Introduction
Asymptotic symmetry (or BMS symmetry) [1, 2], the topic of this study, is a diffeo-morphism to change the shape of 4D asymptotically flat spacetime in the range thefalloff conditions allow, which is spontaneously broken and the degenerated spacetimeconfigurations for this exist infinitely.An important conjecture in the context of the 4D asymptotic symmetry was given in[3]; the symmetry of the diffeomorphism in the vicinity of the infinite null region (the re-gion prescribed by the falloff conditions) should be the direct product of the following twosymmetries: Virasolo symmetry and its antiholomorphic part on the conformal sphere(its SL(2, C ) × SL(2, C ) part corresponds to global infinitesimal Lorentz and transla-tion symmetries), and supertranslation acting on reterded/advanced time directionsto which these conformal spheres attach. Based on this conjecture, it is considered thatasymptotic symmetry is the infinite-dimensional spontaneously broken symmetry.After this conjecture is advocated, the central charges of asymptotic symmetry havebeen analyzed [4, 5]. On the other hand, the following 2 directions have been investi-gated and are still ongoing: i) relation between gravitational S -matrix and gravitationalsoft-theorem [6, 7, 8, 9, 10, 11, 12, 13, 14] and to get the CFT with the S -matrix math-ematically equivalent to the gravitational S -matrix in 4D asymptotically flat spacetime[15, 16, 17, 18, 19, 20] and ii) resolution of black hole information paradox ∗ by the con-sideration that the information of the initial stage would be finally reflected as someone of the infinitely degenerated spacetime configurations [54, 55, 56] (displacement ofsupertranslations appearing in the time-evolution of spacetime is analyzed in [57]).With these, observational studies in astrophysics based on phenomena concerningasymptotic symmetry, e.g. gravitational memory effect [58, 59], are also ongoing [60,61, 62, 63, 64, 65]. There are variation of the gravitational memory effect: spin memoryeffect [66], color memory effect [67], and electromagnetic memory effect [68, 69, 70].The black holes having been addressed in the context of the asymptotic symmetryuntil now are always not rotating, and its extension to the rotating one would be one ofthe important assignments in the current context of asymptotic symmetry.Here, expressions of the coordinates displeased by the diffeomorphism of the super-translation have been given already in [57]. Using these, we in this paper obtain arotating black hole solution with the displacement of the supertranslation.Concretely, utilizing some coordinate transformations together with these of [57], wefirst obtain ‘a’ rotating supertranslated black hole spacetime, where the effect of rotationis involved to linear order, therefore it is slowly rotating one. Then using the laws of thecoordinate transformation obtained from this, we obtain ‘a’ rotating black hole spacetimesolution where the black hole rotation is fully involved. Here, why author stresses ‘a’ is ∗ Fig.9 in [21] would be a better sketch for the whole view of current ideas for information paradox.For some of review papers, i) for introductory descriptions and comments, [22, 23, 24, 25, 26, 27], ii) forreviews qualitatively describing ideas, [21, 28], iii) for black hole complementarity and AMPS Firewall[29, 30, 31], iv) for fuzzball and string theory [32], [33, 34], [35, 36, 37, 38, 39, 40], v) for remnant[41, 42, 43], vi) for page curve and island [44], vii) for AdS/CFT [45, 46, 47, 48], viii) for ER=EPR[49], ix) for the case in CGHS model, [50, 51, 52], x) for some other ones, [53]. S -matrices and gravitational memory effect will become important isthe vicinity of the infinite null region. In there, we can intuitively expect that the effectof the rotation of the black hole would no longer appear as it gets weaken as it separatesfrom the black hole (author has found this actually by expanding the Maxwell equationson the rotating supertranslated black hole spacetime we obtain by the same mannerwith [12], but author has not confirmed this in the case of analysis for the gravitationalmemory effect). If one tries to perform these studies involving the effect of the rotationof the black hole , one would have to expand the analysis to some subleading orders withregard to r from infinity. However, the analysis in that case would be quite difficult asthe equations of motion gets too long as this is basically the perturbative calculation onthe rotating black holes.Another intriguing issue would be, moving to the near-horizon, the effect of theasymptotic symmetry in the Hawking fluxes. it has been studied in the non-rotatinblack hole (by author) [73, 74], in which the effect of supertranslation would not beinvolved in the parts critical for the Hawking fluxes in the near-horizon metrices, andthe Hawking fluxes are not changed. This situation would be the same, therefore, theHawking fluxes to be obtained in this study would be the same with the one in just arotating black hole.Therefore, we in this study would like to focus on the classical gravitational perturba-tion on the rotating supertranslated black hole spacetime we obtain by expanding withregard to r from the infinity to the order where the effect of the rotation of the blackhole can finally remain in the result. This would be interesting from the viewpoint offuture work for the gravitational phenomena.We mention the organization of this paper. In Sec.2, we obtain a rotating super-translated black hole spacetime. In Fig.1, we sketch what we will do in Sec.2.In Sec.2.1, we start with the rotating black hole spacetime given in Boyer-Lindquist(BL) coordinates. Then, since the expressions of the coordinates displaced by supertrans-lation are given in the isotropic coordinates , in order to utilize these expressions ofcoordinates usable in the isotropic coordinates, we rewrite the BL coordinates into theSchwarzschild type coordinates “by taking the effect of the rotation of the black hole tolinear order”, then rewrite it into the isotropic coordinates. In Sec.2.2, we involve thesupertranslation in that isotropic coordinates. By doing so, we involve the displacementof supertranslation.Once we involved the displacement of supertranslation, in Sec.2.3 to 2.5, we back thatsupertranslated isotorppic coordinates into the Schwarzschild coordinates, then back itinto the original “rotating” spacetime. By doing so, we obtain a supertranslated slowlyrotating black hole spacetime, first. We can check it can satisfy Einstein equation.In the process above, we can get a transformation rule to transform just a Schwarzschild2o supertranslated Schwarzschild directly without going through isotropic coordinateslike the above (horizontally written “C.T. (Eq.(19) and (24))” in Fig.1). Using it in thefirst rotating black hole spacetime given in BL coordinates, we obtain a supertranslatedblack hole solution with the rotation fully involved in Sec.2.6 (vertically written “C.T.(Eq.(19) and (24))” in Fig.1). We can also check it can satisfy Einstein equation.In Sec.3, we analyze the classical gravitational perturbation on the rotating super-translated black hole spacetime we obtain, and Sec.4 is devoted to summary comment. 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As the point for that in particular, in the process to obtain a slowly rotating one,we can get a transformation rule to transform just a Schwarzschild to supertranslatedSchwarzschild directly, which is , “C.T. (Eq.(19) and (24))”. Using it in the first rotatingblack hole spacetime in BL coordinates, we obtain a supertranslated black hole solutionwith the rotation ‘fully involved’.
In this section, we first obtain a 4D slowly rotating supertranslated black hole spacetimesolution. In the process above, we can get the transformation rule to transform justa Schwarzschild to supertranslated Schwarzschild directly. Using it, we obtain a 4Dsupertranslated black hole spacetime solution with the effect of the rotating black holefully involved. 3 .1 Rewriting BL coordinates to isotoropic coordinates beforeperforming supertranslation
We start with 4D rotating balckhole spacetime given by Boyer-Lindquist (BL) coordi-nates, ds = − (1 − mrr + a cos θ ) dt + r + a cos θr − m r + a dr + ( r + a cos θ ) dθ +( r + a + 2 mra sin θr + a cos θ ) sin θdφ − mra sin θr + a cos θ dtdφ. (1)We will involve the displacement of the supertranslation in this. Then the followingreason: • one way to involve the supertranslation is given in [57] in (non-rotating) isotropiccoordinates, • isotropic coordinates can be obtained from Schwarzschild coordinates by just acoordinate transformation, • (1) can be expressed in Schwarzschild type coordinates taking a to first-order as ds in (1) = − (1 − m/r ) dt + (1 − m/r ) − dr + r dθ + r sin θdϕ + O ( a ) , (2)where dϕ = dφ − m a/r dt ,we rewrite (1) in isotropic coordinates by r = ρ (1 + m/ ρ ) as ds in (2) = − (1 − m/ ρ ) (1 + m/ ρ ) dt + (1 + m/ ρ ) ( dρ + ρ ( dθ + sin θ dϕ )) + O ( a ) , (3)where r in dϕ above is given as r = ρ (1 + m/ ρ ) . Isotropic coordinates after supertranslation are given in [57] as x s = ( ρ − C ) sin θ cos ϕ + sin ϕ csc θ ∂ ϕ C − cos θ cos ϕ ∂ θ C, (4a) y s = ( ρ − C ) sin θ sin ϕ − cos ϕ csc θ ∂ ϕ C − cos θ sin ϕ ∂ θ C, (4b) z s = ( ρ − C ) cos θ + cos θ cos ϕ ∂ θ C, (4c)where t s = t , and the function C is the Goldstone boson field for supertranslation † . Weutilize these expressions, where we take C as follows: C = mε Y . (5)We list the 3 points for (4) and (5) in what follows: † The vector field to define the diffeomorophysm of the supertranslation in the 4D asymptotically flatspacetime is generally given with arbitrary function f ( z, ¯ z ). The C is generally in the relation with f as L f C ( z, ¯ z ) = f ( z, ¯ z ). For more detail see [58]. ε is some infinitesimal dimensionless parameters, which we attach to measure theorder of our supertranslation corrections in our analysis. We perform our analysisto linear order with regard to ε .The reason of this is that generally Lie derivative to define diffeomorphysm isgiven by the linear order of the coefficients of the vector fields. Therefore, evenif one performs expansion with regard to the coefficients of the vector fields tosecond-order in some given equations with the coefficients of the vector fields, aslong as these are originally rooted in the Lie derivatives to the linear order of thecoefficients of the vector fields, it is the analysis expanding the contribution definedto linear order quadratically, in which the contribution at second-order (given byone shot) is lacked. In this sense it would be insufficient as the analysis for thesecond-order diffeomorphysm.What is being written above means explicitly the following one. Suppose anequation obtained from some analysis for diffeomorphysm with some quantity c ( c characterizes the largeness of the coefficients of the vector fields) as f ( c ) = f + f ( c ) + f ( c ), where f is the part independent of c , and f , ( c ) are the partscoming from the liner- and second- orders of c from the analysis at the stage toobtain f ( c ). If one considers f ( c ) as f + f ( c ) ignoring f ( c ) and expands it as f ( c ) = f + f (0) + f ′ (0) c + f ′′ (0) c /
2, this is obviously not sufficient as the analysisto second-order. (Symbols used here is only for here.) C is in the relation of the linear order with the coefficients of the vector fields forsupertranslation (see the footnote under (4)), and (4) are the ones having beenobtained from the analysis with C to first-order, therefore calculation with regardto C in the equations obtained from (4) can have a meaning only to the linearorder.If one tries to evaluate the effect to second-order, one would need to define thediffeomorphysm by the Lie derivative to second-order at the stage where the dif-feomorphysm is defined. • m is that in (3), which we involve to have C have the dimension same with ρ ( G/c = 1 in this paper). • Why we consider Y is that it is considered that Y is the most dominant modein the quasi-normal modes (ringdown waves) emitted from the process that a soft-hairy black hole is formed [78] ‡ . Therefore, Y is an important mode in one gravi-tational phenomenology of black hole at least.Here, we shall note that generally the supertranslation is irrelevant of gravitationalwaves. This can be gotten from Eq.(5.2.3) in [58] for example; displacements ofmetrices by the supertranslation are given in Eq.(5.2.3), in which we can see Bondinews tensor C zz is not included. In addition, we shall also note that since the Y inthis study is involved into the whole diffeomorphism (including supertranslation)by the relation L f C = f , where C is our Y and f is some coefficients of vectorfields, therefore our Y does not directly link to the gravitational waves. ‡ Author has heard opinion on this point from Vitor Cardoso in [78]. ρ s and a part in (3) after supertranslation as ρ s = ( x s + y s + z s ) / = ρ − ε r π m (3 cos(2 θ ) + 1) + O ( ε ) ≡ ρ + δρ, (6) dρ s + ρ s ( dθ s + sin ϕ s dϕ s ) = dx s + dy s + dz s , (7)where x s , y s and z s are function of ρ , θ and ϕ . With these, we formally write (3) withsupertranslation correction of ε to linear order as ds in (3) = − (1 − m/ ρ s ) (1 + m/ ρ s ) dt + (1 + m/ ρ s ) ( dx s + dy s + dz s ) + O ( ε ) + O ( a ) ≡ g tt dt + g ρρ dρ + g θθ dθ + g ϕϕ dϕ , (8)where • g tt = − ( m − ρ ) ( m + 2 ρ ) − r π εm ( m − ρ )( m + 2 ρ ) (3 cos(2 θ ) + 1) + O ( ε ) , • g ρρ = ( m + 2 ρ ) ρ + r π εm ( m + 2 ρ ) ρ (3 cos(2 θ ) + 1) + O ( ε ) , • g θθ = ( m + 2 ρ ) ρ + r π εm ( m + 2 ρ ) θ )(5 m + 6 ρ ) + m − ρ ρ + O ( ε ) , • g φφ = ( m + 2 ρ ) sin ( θ )16 ρ + r π εm ( m + 2 ρ ) ρ sin ( θ )(cos(2 θ )(9 m + 6 ρ ) + 7 m + 10 ρ ) . We use the following notations in what follows: · g µν for the metrices of the supertranslated isotropic coordinates · j µν for the metrices of the supertranslated Schwarzschild coordinates · J MN for the metrices of the slowly rotating supertranslated BL coordinates · K MN for the metrices of a rotating supertranslated BL coordinateswhere µ, ν = t, ρ, θ, ϕ and M, N = t, ρ, θ, φ . In addition, the coordinates with the lowerindex ‘ s ’ (e.g. ‘ x s ’ ) mean the coordinates displaced for supertranslation. Since we have obtained the metrices in the supertranslated isometric coordinates, wewant to rewrite these to the following Schwarzschild coordinates: ds = − (1 − µ ( ρ ) /r ) dt + (1 − µ ( ρ ) /r ) − dr + j θθ dφ + j φφ dφ . (11)In what follows, we obtain a relation between r and ρ , and µ ( ρ ) as the solution,by solving the following relations: • − (1 − µ ( ρ ) /r ) = g tt , (12a) • − µ ( ρ ) /r (cid:16) drdρ (cid:17) = g ρρ , (12b)6here g tt and g ρρ above are given in (8). The argument in µ ( ρ ) should be ρ (if we willexpress µ ( ρ ) in terms of r , (17) would be plugged in).We can obtain the r satisfying (12a) to ε -order as r = µ ( ρ )( m + 2 ρ ) mρ + ε ρ r π (3 cos(2 θ ) + 1) µ ( ρ )( m − ρ ) + O ( ε ) . (13)Let us obtain the µ ( ρ ). For this, look (12b), then plugging (13) into the r in (12b),solve it for µ ( ρ ) order by order to ε -order. As a result we can obtain µ ( ρ ) = m + c ρε ( m + 2 ρ ) + O ( ε ) , (14)where we took the integral constant at ε -order so that ε -order becomes m . c is theintegral constants at ε -order, which we put as c = 0 . (15)As a result, µ ( ρ ) is given just m , and we denote µ ( ρ ) as µ in what follows.Now we have obtained the relation “ r = · · · ” as in (13), with which, it is now possibleto rewrite the supertranslated Schwarzschild to the supertranslated isotropic coordinatesto ε -order as − (1 − µ/r ) dt + (1 − µ/r ) − dr + j θθ dθ + j φφ dφ → g tt dt + g ρρ dρ + (cid:16) j θθ + 11 − µr (cid:16) ∂r∂θ (cid:17) (cid:17) dθ + 21 − µr ∂r∂ρ ∂r∂θ dρdθ + j φφ dφ . (16)However what is needed for us is rewriting from the supertranslated isotropic to super-translated Schwarzschild coordinates. Using the things obtained in this subsection, inthe next subsection, we obtain it. We will obtain the relation between ρ and r in the form “ ρ = · · · ” to ε -order to makepossible to rewrite (16) in the opposite direction. For this, there are two ways: tosolve (12b) or to solve (13). As a result of our try, we can check that if we solveto ε -order, the same ρ can be obtained from either of them (checked this samenessnumerically).Writing what we did, plugging µ ( ρ ) in (14) into the µ in (13), then expanding it to ε -order, we can obtain ρ order by order. As a result, we obtain as ρ ( ± ) = 12 ( ± p r ( r − m ) − m + r ) + 18 r π εm (3 cos(2 θ ) + 1) + O ( ε ) , (17)We discard ρ ( − ) and adopt ρ (+) from the situation taking large r . We denote ρ (+) justas ρ in what follows. 7sing this and µ in (14), we can rewrite the supertranslated isotropic coordinates tothe supertranslated Schwarzschild coordinates to ε -order as g tt dt + g ρρ dρ + g θθ dθ + g φφ dφ → − (cid:16) − µ ( ρ ) r (cid:17) dt + (cid:16) − µ ( ρ ) r (cid:17) − dr + (cid:16) g θθ + g ρρ (cid:16) ∂ρ∂θ (cid:17) (cid:17) dθ + 2 g ρρ ∂ρ∂r ∂ρ∂θ dρdθ + g φφ dφ ≡ j tt dt + j rr dr + j θθ dθ + 2 j rθ drdθ + j φφ dφ , (18)where ρ = ρ ( r, θ ) and g MN are given in (8) and • j tt = − (1 − mr ) + O ( ε ) , • j rr = (1 − mr ) − + O ( ε ) , • j θθ = r + 3 q π εm cos(2 θ )( p r ( r − m ) + r ) p r ( r − m ) − m + r ) + O ( ε ) , • j rθ = − q π εm sin(2 θ )( r − p r ( r − m )) p r ( r − m )( p r ( r − m ) − m + r ) + O ( ε ) , • j ϕ s ϕ s = r sin ( θ ) + 3 q π εm sin (2 θ )( p r ( r − m ) + r ) p r ( r − m ) − m + r ) + O ( ε ) , where ϕ s is defined under (2). a to linear order In this section, we obtain a slowly rotating supertranslated black hole solution with a to linear order by backing the coordinate ϕ s to φ to include the effect of the black holerotation explicitly.First, when supertranslation is performed, ρ and r get displaced for that. The dis-placed ρ has been already given in (6). On the other hand, the displaced r can beobtained as r s = ρ s ( ρ )(1 + m/ ρ s ( ρ )) | eq.(17) = r + O ( ε ) ≡ r + δr, (19)where we denote the r after supertranslation as r s , and in the above we have written δr formally, but δr = 0 + O ( ε ).Denoting the displaced dϕ by the supertranslation as dϕ s , it can be obtained as dϕ s = dφ − ma ( r + δr ) dt ≡ dφ + Θ r dt, (20)= dφ − ma ( ρ s (1 + m/ ρ s ) ) dt ≡ dφ + Θ ρ dt, (21)8here the top and below are respectively in terms of r and ρ andΘ r ≡ − ar + O ( ε ) , Θ ρ ≡ − amρ ( m + 2 ρ ) + 48 q π aεm ρ (3 cos(2 θ ) + 1)( m − ρ )( m + 2 ρ ) + O ( ε ) . Plugging (20) into (18), we can get a slowly rotating black holes with the displacementof the supertranslation to first-order as ds = J tt dt + J rr dr + J θθ dθ + J φφ dφ + 2 J rθ drdθ + O ( a ) + O ( ε ) , (22)where J MN are given using J tt and Θ as J MN = j tt j ϕ s ϕ s Θ0 j rr j r θ j r θ j θθ j ϕ s ϕ s Θ 0 0 j ϕ s ϕ s + O ( a ) + O ( ε ) . • J tt = − m/r, • J tϕ = − m/r − q π am sin (2 θ )( p r ( r − m ) + r ) r ( p r ( r − m ) − m + r ) ε, • J rr = (1 − m/r ) − , • J rθ = − q π m sin(2 θ )( p r ( r − m ) + r ) p r ( r − m )( p r ( r − m ) − m + r ) ε, • J θθ = r + 3 q π εm cos(2 θ )( p r ( r − m ) + r ) p r ( r − m ) − m + r ) , • J ϕ s ϕ s = r sin ( θ ) + 3 q π m sin (2 θ )( p r ( r − m ) + r ) p r ( r − m ) − m + r ) ε. We can check these can satisfy Einstein equation to ε - and a -orders § . Denoting z , θ and ϕ with the displacement of supertranslation as z s , θ s and ϕ s respec-tively, let us obtain θ s in this section. θ s can be written in terms of z s as θ s = 2 cot − | z s | , § Though the calculation in this study is to ε -order, indeed obtaining J MN to ε -order (the order of a is first) using the results in [74] I have checked these can satisfy Einstein equation to ε -order. s = i ln ¯ z s z − s , and expression of z s in terms of z (coordinate before supertranslation)is given in (26) in [57] as z s = ( z ¯ z − ρ − C ) + ( z ¯ z + 1)( ρ s − ¯ z∂ ¯ z C − z∂ z C )2¯ z ( ρ − C ) + ( z ¯ z + 1)(¯ z ∂ ¯ z C − ∂ z C ) , (23)where C and ρ s are (5) and (19) in the case of this study.We will plug ρ s in (19) and C in (5) into z s above. Here, these ρ s and C are rewrittenin terms of z by rewriting θ and ϕ of these in terms of z in advance, then expandingsuch a ρ s to linear order of ε , we can get θ s as θ s = θ + 3 m r + p ( r − m ) r − m ) r π ε sin(2 θ ) + O ( ε ) ≡ θ + δθ. (24)Similarly, we can get ϕ s as ϕ + O ( ε ) (no corrections at ε - and ε -orders).Then, in the previous section, we have reached J MN above from (2) via the isotropiccoordinates (3), however we can reach J MN from (2) directly by substituting for r and θ in (2) as ( r, θ ) → ( r s , θ s ) , (25)where r s is given in (19) and θ s is given above. Then, replacing r and θ in (1) with these,we can obtain a rotating black hole spacetime with the supertranslation correction of ε to first-order but a fully involved as ds in (1) → K MN dx M dx N + O ( ε ) , (26)where • K tt = − mra cos ( θ ) + r + 3 q π a mr sin (2 θ )( r + p ( r − r − a cos ( θ ) + r ) ε, • K tφ = − amr sin ( θ ) a cos ( θ ) + r − q π amr ( a + r ) sin (2 θ )( r + p ( r − r − a cos ( θ ) + r ) ε, • K rr = a cos ( θ ) + r a − mr + r + − q π a sin (2 θ )2( r + p ( r − r − a + r ( r − m )) ε, • K rθ = − q π sin(2 θ )( a cos ( θ ) + r )2 p ( r − r ( r + p ( r − r − ε, • K θθ = a cos ( θ ) + r + 3 q π ( a (3 cos(4 θ ) + 1) + 4( a + 2 r ) cos(2 θ ))4( r + p ( r − r − ε, • K rθ = − q π ε sin(2 θ )( a cos ( θ ) + r )2 p ( r − r ( r + p ( r − r − , K φφ = sin ( θ )( 2 a mr sin ( θ ) a cos ( θ ) + r + a + r )+ 3 q π sin (2 θ ) ε r + p ( r − r − a cos ( θ ) + r ) (3 a + a r (10 m + 11 r )+16 a r ( m + r ) + a ( a + r ( r − m ))( a cos(4 θ ) + 4( a + 2 r ) cos(2 θ ))+8 r ) . We can check that this can satisfy the Einstein equation and can agree with the slowlyrotating metrices given in (22) if expanding these to the first-order of a .Since (26) is not obtained by solving Einstein equation but obtained based on the su-pertranslated Schwarzschild black hole solution utilizing the coordinate transformation,this is not be the general solution. However, since (26) can satisfy the Einstein equationto ε - and a -orders, it is sure that this is the metrices for a rotating supertranslatedblack hole spacetime. In this section we check the classical gravitational perturbation on the supertranslatedrotating black hole spacetime solution in (26).However, if we perform the analysis actually, we find that the number of termsgets enormous as this is the analysis for the perturbations on the rotating black hole.Therefore we will perform our analysis by performing expansion with regard to r frominfinity to r − order, the first order where the effect of the rotation of the black hole canfinally remain in the result.We will obtain the solutions of the perturbation with the effect of the rotation of theblack hole in the form without ambiguities, integrals and derivatives as much as possible.The result finally obtained in this study may not be much interesting, however fromthe viewpoint as a research result, the finally obtained result and the process to reachthese would have a certain amount of meaning toward future works.Let us write the metrices we consider as G MN : G MN = K MN + h MN , (27)where K MN are the ones in (26) and h MN represent perturbations, which we expandwith regard to r from infinity as h MN ( t, r, θ ) = ∞ X n =1 ¯ h ( n ) MN ( t, θ ) r n . (28)In the ones above, constants and log order are not included for the consideration that h MN should be zero at large r and not get too large in small r region. Why φ is not11ncluded in the arguments of ¯ h ( n ) MN is that the Einstein equation given just in what followsis independent of φ , therefore we could put φ to zero in all of h MN uniquely.Then, from the Einstein equation R MN − G MN R = 0, we can obtain the followingequations: • a (01)0 r + O ( r − ) , ¯ a (01)0 ≡ ∂ t ¯ h (1) rr ( t, θ ) , (29a) • a (02)1 r + ¯ a (02)0 r + O ( r − ) , (29b) · ¯ a (02)1 ≡ − ∂ t ∂ θ ¯ h (1) rr ( t, θ ) , ¯ a (02)0 ≡ m∂ t ∂ θ ¯ h (1) rr ( t, θ ) − ∂ t ∂ θ ¯ h (2) rr ( t, θ ) , • a (03)0 r + O ( r − ) , ¯ a (03)0 ≡ am sin ( θ ) ∂ t ¯ h (1) rr ( t, θ ) , (29c) • a (11)0 r + O ( r − ) , ¯ a (11)0 ≡ ∂ t ¯ h (1) tr ( t, θ ) , (29d) • a (12)1 r + ¯ a (12)0 r + O ( r − ) , (29e) · ¯ a (12)1 ≡
12 ( ∂ t ¯ h (1) rθ ( t, θ ) − ∂ t ∂ θ ¯ h (1) tr ( t, θ )) , · ¯ a (12)0 ≡
12 (2 m ¯ h (1) rθ (2 , ( t, θ ) + ∂ t ¯ h (2) rθ ( t, θ ) + 2 ∂ θ ¯ h (1) rr ( t, θ ) + 3 ∂ t ¯ h (1) tθ ( t, θ ) − m∂ t ∂ θ ¯ h (1) tr ( t, θ ) − ∂ t ∂ θ ¯ h (2) tr ( t, θ ) − ∂ θ ¯ h (1) tt ( t, θ )) , • a (13)1 r + ¯ a (13)0 r + O ( r − ) , (29f) · ¯ a (13)1 ≡ ∂ t ¯ h (1) rφ ( t, θ ) , ¯ a (13)0 ≡
12 (2 m∂ t ¯ h (1) rφ ( t, θ ) + ∂ t ¯ h (2) rφ ( t, θ ) + 3 ∂ t ¯ h (1) tφ ( t, θ )) , • a (22)3 r + ¯ a (22)2 r r + O ( r − ) , ¯ a (22)3 ≡ − ∂ t ¯ h (1) rr ( t, θ ) ¯ a (22)2 ≡ − ∂ t ¯ h (2) rr ( t, θ ) , (29g) • a (23)1 r + ¯ a (23)0 r + O ( r − ) , (29h) · ¯ a (23)1 ≡
12 ( ∂ t ¯ h (1) θφ ( t, θ ) − ∂ t ∂ θ ¯ h (1) tφ ( t, θ ) + 2 cot( θ ) ∂ t ¯ h (1) tφ ( t, θ )) , · ¯ a (23)0 ≡ −
12 ( − m ( a sin(2 θ ) ∂ t ¯ h (1) tt ( t, θ ) + ∂ t ¯ h (1) θφ ( t, θ ) − ∂ t ∂ θ ¯ h (1) tφ ( t, θ )) − ∂ t ¯ h (2) θφ ( t, θ )+ ∂ θ ¯ h (1) rφ ( t, θ ) + 2 cot( θ )¯ h (1) rφ ( t, θ ) − θ )( ∂ t ¯ h (2) tφ ( t, θ ) + ∂ t ¯ h (3) tφ ( t, θ ))+ ∂ t ∂ θ ¯ h (2) tφ ( t, θ ) + ∂ t ∂ θ ¯ h (3) tφ ( t, θ )) , a (33)3 r + ¯ a (33)2 r r + O ( r − ) , (29i) · ¯ a (33)3 ≡ −
12 sin ( θ ) ∂ t ¯ h (1) rr ( t, θ ) , ¯ a (33)2 ≡ −
12 sin ( θ ) ∂ t ¯ h (2) rr ( t, θ ) . We list the points in the ones above: • Since tt -component (00-component) is zero in the order to O ( r − ), it is not includedin the list above. • ¯ h MN have been taken to linear order. • Equations above have been obtained by performing the expansion with regardto r from infinity to the first order where the effect of rotation can appear andfinally remain in the result, which is r − order (if we got the Einstein equationsabove without expansion (28), these would get long as these are the equations forperturbation on rotating black hole spacetime).Therefore, the equations above will describe the shape of the spacetime from theinfinite far region to r − order. • We can see there is no symmetries in t -, r - and θ -directions in the equations above.Therefore, we have taken t , r and θ as the arguments of h MN , not performingFourier expansions. In the expressions above, we have written the argumentsexplicitly. • Why φ is not included in the arguments of ¯ h ( n ) MN has been written under (28). · We can see ¯ h (1) rr , ¯ h (2) rr , ¯ h (1) tr , ¯ h (1) rθ and ¯ h (1) rφ have no t -dependence, therefore we put as · ¯ h (1) rr ( t, θ ) → ¯ h (1) rr ( θ ), from (29a) , (30a) · ¯ h (2) rr ( t, θ ) → ¯ h (2) rr ( θ ), from (29b) , (30b) · ¯ h (1) tr ( t, θ ) → ¯ h (1) tr ( θ ), from (29d) , (30c) · ¯ h (1) rθ ( t, θ ) → ¯ h (1) rθ ( θ ), from (29e) , (30d) · ¯ h (1) rφ ( t, θ ) → ¯ h (1) rφ ( θ ), from (29f) , (30e) · As for (30b), if it follows (30a), supposing θ -dependence exists generally, (29a) wouldbe concluded. · As for (30d), if it follows (30c), the form of ¯ h (1) tr would be c t + c ( c , are used onlyhere), however, supposed perturbations disappear at far future (large t ), it is concludedsuch a form is not allowed and reach (30c). · As for (30e), as well as (30d), the form, c t + c , is not allowed, therefore (30e) isconcluded.Imposing (30), we can get ¯ h tr most appears in (29h). Therefore, from the standpointto make the equations simple as much as possible, it would be reasonable to impose thefollowing gauge condition: h tφ ( t, r, θ ) → , (31)13here basically we have 4 gauge fixing condition corresponding to the existence of 4coordinates in our spacetime (we comment on the problem of if we can impose the gaugefixing conditions we impose actually at last in this section). From (29f) and (29h), wecan get that ¯ h (1) rφ and ¯ h (2) rφ also have no t -dependence, therefore, in addition to (30), weput as ¯ h (1) rφ ( t, θ ) → ¯ h (1) rφ ( θ ) , ¯ h (2) rφ ( t, θ ) → ¯ h (2) rφ ( θ ) . (32)With (30), (31) and (32), all the equations of (29) except for (29e) and (29h) vanish.(29e) and (29h) remain as · ∂ t ¯ h (2) rθ ( t, θ ) + 3 ∂ t ¯ h (1) tθ ( t, θ ) − ∂ t ∂ θ ¯ h (2) tr ( t, θ ) − ∂ θ ¯ h (1) tt ( t, θ ) + 2 ∂ θ ¯ h rr ( θ ) , (33a) · am sin(2 θ ) ∂ t ¯ h (1) tt ( t, θ ) + ∂ t ¯ h (2) θφ ( t, θ ) − ∂ θ ¯ h rφ ( θ ) − θ )¯ h (1) rφ ( θ ) . (33b)Looking (33b), from the consideration that we want to obtain the perturbations asthe solution involving the effect of the rotation of the black hole in the form with lessambiguities, derivatives and integrals as much as possible, we would come to employeither of the following gauge fixing conditions:1) h rφ → , h θφ → . (34)Let us employ the 1) in what following.In this case, again from the viewpoint to obtain the perturbation as the solutioninvolving the effect of the rotation of the black hole in the form with less ambiguities,derivatives and integrals as much as possible, we would come to employ the following 2gauge conditions: h tθ ( t, r, θ ) → , h rθ ( t, r, θ ) → , (35)where we have used up the all the 4 degrees of gauge freedom. Using (34) and (35) inaddition to (30), (31) and (32), we can obtain the following 2 equations from (33) as · ¯ h (2) θφ ( t, θ ) = − am sin(2 θ ) Z dt ¯ h (1) tt ( t, θ ) , (36a) · ¯ h (1) rr ( θ ) = ¯ h (1) tt ( t, θ ) + 12 ∂ t ¯ h (2) tr ( t, θ ) , (36b)The point in (36b) is that l.h.s. is independent of t , while r.h.s. has dependence on t , which leads to the following 2 options as1) no t -dependence in ¯ h (1) tt and ¯ h (2) tr : ¯ h (1) tt ( t, θ ) → ¯ h (1) tt ( θ ) , ¯ h (2) tr ( t, θ ) → ¯ h (2) tr ( θ ) , (37a)2) suppose ¯ h (1) rr ( θ ) = 0, (37b)where the case that the form of ¯ h (2) tr is given by c t + c ( c , are used only here) cannotbe allowed from the physical consideration that perturbations should disappear at large t , therefore the case that ¯ h (1) tt ( t, θ ) → ¯ h (1) tt ( θ ) and ∂ t ¯ h (2) tr = 0 is not considered.14n the case of 1), since “ ∂ t ¯ h (2) tr ” in (36b) vanishes, it follows ¯ h (1) rr ( θ ) = ¯ h (1) tt ( θ ) from(36b), which leads to ¯ h (2) θφ ( θ ) = − amt sin(2 θ )¯ h (1) rr ( θ ) + c ( θ ) , (38)where c ( θ ) is the integral constant for the integration with regard to t . Therefore, for thereason that generally perturbation should disappear at large t , the part “ − amt sin(2 θ )¯ h (1) rr ( θ )”proportinal to t should vanish. For this purpose, ¯ h (1) rr should vanish. Finally, as the con-clusion obtained from the option 1) can be summarized as¯ h (1) rr ( θ ) = ¯ h (1) tt ( θ ) = 0 , ¯ h (2) θφ ( θ ) = c ( θ ) . (39)On the other hand, in the case of the option 2), from r.h.s. of (36b), it follows¯ h (1) tt ( t, θ ) = 2¯ h (1) tt ( θ ) t p +1 , ¯ h (2) tr ( t, θ ) = ¯ h (2) tr ( θ ) t p , (40)where p is arbitrary integers. From (40), we can finally obtain the following relations: · h (1) tt ( θ ) − p ¯ h (2) tr ( θ ) , (41a) · h (2) θφ ( t, θ ) = 2 amp sin(2 θ ) ¯ h (1) tt ( θ ) t p , (41b)where (41b) can be obtained from (36a) with (41a).We summarize what we have obtained in this section. In the range of the spacetimefrom the infinite far region to r − order, if one takes the 4 gauge conditions (31), (34)and (35), with assumption that there is no constants and log part in the perturbations(28), some of coefficients have no t -dependence like (30) and (32), and we will have 2options like (37). Depending on which one we chooses, we get the results (39) or (41).Finally we would like to consider about the possibility of the 4 gauge fixing conditionswe have used, (31), (34) and (35). These mean that the amounts of the deformationsof the metrices by the general coordinate transformation can be written so that gravita-tional perturbations can be canceled as · δ G tθ = ∇ t ξ θ + ∇ θ ξ t = − h tθ , (42a) · δ G tφ = ∇ t ξ φ + ∇ φ ξ t = − h tφ , (42b) · δ G rθ = ∇ r ξ θ + ∇ θ ξ r = − h rθ , (42c) · δ G rφ = ∇ r ξ φ + ∇ φ ξ r = − h rφ , (42d)where ξ M mean the variational amounts of the coordinates in the general coordinatetransformations ( x M → x M + ξ M ( x M )). The problem is whether the solutions for gaugefixing conditions exist or not. Looking these, since we can see that all the kinds of four ξ M are included in the gauge fixing conditions above, we can consider the solutions of ξ M realizing the gauge fixing conditions above exist.15ctually, going with a formal calculating manner as f = Z dx M g for a given relation ∇ M f = g (just simply switching covariant derivatives to integrals) and vice verve ( f and g mean any h and ξ appearing in (42)), we can represent the solutions of ξ θ,φ,r in termsof ξ t as · ξ θ ∼ − Z dt ( ∇ θ ξ t + h tθ ) , (43a) · ξ φ ∼ − Z dt ( ∇ φ ξ t + h tφ ) , (43b) · ξ r ∼ − Z dθ ( h rθ − ∇ r Z dt ( h tθ − ∇ r Z dt ( h tθ + ∇ θ ξ t ))) , (43c) ∼ − Z dφ ( h rφ − ∇ r Z dt ( h tφ − ∇ r Z dt ( h tφ + ∇ φ ξ t ))) , (43d)where (43c) and (43d) are respectively obtained from (42c) and (42d). From the condi-tion: (43c) = (43d), we can get the expression of ξ t as the solution as ξ t ∼
12 ( ∂ t ( Z dr ( ∂ t ( Z dr A ) − Z dt A )) − A ) , (44)where A ≡ Z dθh rθ − Z dφh rφ , A ≡ Z dθh tθ − Z dφh tφ . Therefore, as we can see four ξ M can be entirely given by only h MN in principle, wecan impose the gauge fixing conditions (31), (34) and (35). In this study, firstly a supertranslated rotating black hole solution has been obtained.Since it has been obtained not by solving Einstein equation but by based on the alreadyobtained supertranslated Schwarzschild black hole solution, it is not the general solu-tion. However it is sure that it is ‘a’ supertranslated rotating black hole solution as itcan satisfy the Einstein equation. In the current context of asymptotic symmetry, theextension to rotating black hole is an assignment, and what has been obtained in thisstudy would be a solution to that.In the actual analysis, to obtain the rotating solution, one of ideas author has triedin this study is to use Newman-Janis (NJ) algorithm [71, 72]. Concretely, author hasfollowed the way given in Eq.(6) in [81]. However, it could not work in our metrices.Its reason is, concisely mentioning, there are r in the metrices, and the powers ofsome of them in the correction parts of the supertranslation are given by odd numbers.As a result, imaginary numbers appear in the metrices.16he idea to obtain the rotating solution by starting from Kerr-Schild (KS) form hasbeen also tried, however it has not gone well for the situation that author could notget the metrices from KS form which can agree to the one in BL coordinates by anymeans. author has tried many ideas for this involving various parameters to be fixedso that what has been obtained from KS form can agree to what has obtained from BLcoordinates.Essential point in this problem would be that (4) seems to be originally given in theisotropic coordinates and perhaps cannot be adapted in the KS form as it is. author hastried many ideas to modify (4) upon adapting these into the KS form, however all theidea could not go well ¶ .Interesting problems in the asymptotic symmetry firstly coming up would be issuesconcerning soft theorem, memory effect and information paradox. However, as mentionedin the Sec.1, soft theorem and memory effect are basically the problems in the region ofthe infinite null regions, in there it would be expected that the effect of the black holerotation gets disappeared.Author has confirmed this point actually after author tries to expand the Maxwellequations by the same manner with [12] on the spacetime we have obtained (author hasnot confirmed this point in the case of the analysis for gravitational memory effect).If one involves the effect of rotation, one would have to expand to some subleadingorders. However, at that time, equations gets too long. Author will take this work asthe future work.The issue this paper has finally addressed is the classical gravitational perturbationincluding the effect of the rotation of the black hole. From the viewpoint as a researchresult, the finally obtained result and the process to reach these would have a certainamount of meaning toward future works. References [1] H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, “Gravitational waves ingeneral relativity. 7. Waves from axisymmetric isolated systems,” Proc. Roy. Soc.Lond. A
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