Generalized Painleve-Gullstrand descriptions of Kerr-Newman black holes
aa r X i v : . [ g r- q c ] S e p Generalized Painlev´e-Gullstrand descriptions of Kerr-Newman black holes
Huei-Chen Lin ∗ and Chopin Soo † Department of Physics, National Cheng Kung University, Tainan 701, Taiwan
Generalized Painlev´e-Gullstrand metrics are explicitly constructed for the Kerr-Newman family of chargedrotating black holes. These descriptions are free of all coordinate singularities; moreover, unlike the Doran andother proposed metrics, an extra tunable function is introduced to ensure all variables in the metrics remain realfor all values of the mass M , charge Q , angular momentum aM , and cosmological constant L > − a . To de-scribe fermions in Kerr-Newman spacetimes, the stronger requirement of non-singular vierbein one-forms at thehorizon(s) is imposed and coordinate singularities are eliminated by local Lorentz boosts. Other known vierbeinfields of Kerr-Newman black holes are analysed and discussed; and it is revealed that some of these descrip-tions are actually not related by physical Lorentz transformations to the original Kerr-Newman expression inBoyer-Lindquist coordinates - which is the reason complex components appear (for certain ranges of the radialcoordinate) in these metrics. As an application of our constructions the correct effective Hawking temperaturefor Kerr black holes is derived with the method of Parikh and Wilczek. PACS numbers: 04.70.Bw, 04.20.Jb, 04.70.Dy
I. OVERVIEW
As descriptions of charged rotating black holes, Kerr-Newman solutions [1, 2] are important objects of study inGeneral Relativity. In Boyer-Lindquist coordinates [3–5],these metrics exhibit coordinate singularities at the horizon(s).The Doran form of the solution [6] can be considered to bethe extension of the Painlev´e-Gullstrand(PG) [7, 8] descrip-tion of a black hole from spherically symmetric to stationaryaxisymmetric spacetime. These descriptions have the advan-tage of being free of coordinate singularities at the horizon(s).The Kerr solution, which was discovered much later, is not astraightforward generalization of the Schwarzschild solution.The Doran metric is likewise comparatively recent; neverthe-less, as “regular” descriptions of rotating black holes, the Do-ran and other proposed metrics have found their uses in blackhole investigations. Constant-time Doran slicings of the ergo-surface in non-extremal black holes have been demonstratedto be free of conical singularities at the poles [9]. Calculationsof Hawking radiation [10, 11], and also neutrino asymmetrydue to the interaction of fermions and rotating black holes [12]have also made explicit use of the Doran metric. Other authorshave proposed to utilize the Doran metric to extend or gener-alize spherically symmetric results to the context of rotatingblack holes [13–15].It is possible to eliminate the coordinate singularities at thehorizon(s) by a choice of different coordinates; but if the finalform is regular, then the coordinate transformation from theoriginal singular form must be singular at the horizon(s). Inthis work we achieve the stronger requirement of non-singularvierbein one-forms at the horizon(s). This not only guaranteesthe metric to be regular, but is in fact also a physical require-ment in the presentation, for instance in Weyl or Dirac equa-tions, of fermions in curved spacetimes. From this perspec-tive, the coordinate singularities at the horizon(s) can be elim- ∗ [email protected] † [email protected] inated by Lorentz transformations of the vierbein one-forms.Since rotations do not change the 3-geometry of constant-time slices, Lorentz boost(s) (with infinite rapidity at the hori-zon(s)) can be effective means to eliminate unphysical coor-dinate singularities. Moreover deformation parameters of the3-geometries of constant-time slices can also be introducedthrough these local boosts.Recently it has been demonstrated [16] that there is an ob-struction to the implementation of flat PG slicings for spheri-cally symmetric spacetimes; and insistence on spatial flatnesscan lead to unphysical PG metrics with complex variablesin which the corresponding vierbein fields are not related tothose of the standard spherically symmetric metric by phys-ical Lorentz boosts. Since the Doran form contains spheri-cally symmetric PG solutions as special cases (the Reissner-Nordstr¨om solution is an explicit example), it will be afflictedwith similar problems [16]. In calculations of black hole evap-oration using the Parikh-Wilczek method [10], insistence onspatially flat PG coordinates can lead to spurious contributionswhich are ambiguous and problematic, both to the computa-tion of the tunneling rate and to the universality of the results.In a more general context, the appearance of complex metriccomponents causes unnecessary complications, and gives riseto difficulties and ambiguities in the physical interpretations.As we shall demonstrate, other proposals [17, 18] of “regular”Kerr-Newman black holes also suffer from similar problems.In fact these descriptions and the Doran vierbein are actuallynot always related by physical Lorentz transformations to theoriginal Kerr-Newman expression in Boyer-Lindquist coordi-nates - which is the reason complex components appear (forcertain ranges of the radial coordinate) in these metrics.These troubles can be avoided altogether by using a less re-strictive form of constant-time slicing which generalizes theDoran metric. We demonstrate how this goal can be realized,and construct a whole class of generalized (with adjustablefunction f ( r ) ) real PG metrics for Kerr-Newman black holeswhich are completely free of coordinate singularities. In con-tradistinction, in our metrics no complex components arisefor all values of r . Although it is possible to introduce manyparameters through the freedom of local Lorentz transforma-tions which relate vierbein one-forms of the same metric, ourgeneralized PG description is “optimal” in that only one ad-ditional function, f ( r ) , is needed, and introduced, to both re-veal and avoid all the troubles. Different choices of f resultin different constant- t P slices of 3-geometry. Our construc-tion also recovers other known descriptions, and the methodis used to clarify the relation between these metrics. Althoughthe discussion of Hawking radiation is not the main theme ofthis work, we apply our constructions to the computation ofHawking radiation following the work of Parikh and Wilczek[10]. The correct result is obtained for both the Eddington-Finkelstein and our generalized PG metrics. II. GENERALIZED PAINLEV ´E-GULLSTRAND METRICSFOR KERR-NEWMAN SPACETIMES
In this work we also include the contribution of non-trivialcosmological constant L in Kerr-Newman solutions of rotatingblack holes with angular momentum aM and charge Q . Ex-pressed in Boyer-Lindquist coordinates [3–5], the metric forthe Kerr-Newman black hole is [25] ds = − DX r ( dt − a sin q d f ) + r D dr + r X q d q + X q sin qX r ( R d f − adt ) , R : = r + a , r : = r + a cos q , X : = + L a , X q : = + L a cos q , D : = R (cid:0) − L r (cid:1) − Mr + Q . (1)In this form, the metric suffers from coordinate singularitiesat the horizon(s) where D = . On the other hand the singular-ity at r = is physical and the curvature diverges there. Themetric is also problematic whenever X q = is allowed by theparameters involved, but this does not arise (even for negativecosmological constant) provided L > − a ( a = ) .For Schwarzschild-(anti)de Sitter black holes, the explicitLorentz boost(s) between the singular standard sphericallysymmetric form and the regular PG-type metric have beendiscussed in Ref.[16] . Our aim here is to obtain, for themore intricate case of Kerr-Newman black holes, a class ofvierbein one-forms (hence metrics) which are real and regulareverywhere, except at the physical singularity r = . To wit,we seek generalized Painlev´e-Gullstrand descriptions of Kerr-Newman solutions by boosting the original singular vierbeinby e trial = L (cid:16) p f − D f (cid:17) · e BL . (2)The simplified notation above denote ( e A m ) trial dx m =[ L ( √ f − D f (cid:17) ] A B ( e B n ) BL dx n with the Boyer-Lindquist vierbienone-forms of Eq. (1) being { ( e A = , , , m ) BL dx m } = { √ DX r ( dt − a sin q d f ) , r √ D dr , r √ X q d q , √ X q sin qX r ( R d f − adt ) } ; (3) and L ( b ) represents a Lorentz boost (Lorentz indices are de-noted by uppercase Latin letters) in the first (A=1) directionwith rapidity x = tanh − b . Consequently, { e A = , , , trial } = { f X r ( dt − a sin q d f + r X p f − DD f dr ) , p f − DX r ( dt − a sin q d f ) + f rD dr , r √ X q d q , √ X q sin qX r ( R d f − adt ) } = { f X r ( dt P − a sin q d f P ) , p f − DX r ( dt P − a sin q d f P )+ r f dr , r √ X q d q , √ X q sin qX r ( R d f P − adt P ) } ; (4)wherein the PG time and azimuthal coordinates are defined as dt P : = dt + R X p f − DD f dr , d f P : = d f + a X p f − DD f dr . (5)Provided f depends only on r , these are exact differentials.To further eliminate dt P in e trial we can apply anotherLorentz boost L (cid:16) a √ X q sin q f (cid:17) resulting in e GPG = L (cid:16) a √ X q sin q f (cid:17) · L (cid:16) p f − D f (cid:17) · e BL = { p f − a X q sin qX r dt P + a sin q [ X q R − f ] X r p f − a X q sin q d f P , p f − DX r ( dt P − a sin q d f P ) + r f dr , r √ X q d q , r √ X q sin q f X p f − a X q sin q d f P } . (6)Note that the vierbein is now regular at horizon(s) D = , real ,and the metric Lorentzian ( − , + , + , +) in signature provided theadjustable parameter f ( r ) satisfies, at each value of r , the cri-terion f ( r ) > max { D ( r ) , a X } . (7)This criterion is also precisely the condition which guaran-tees that the Lorentz boosts in Eq.(6) have physical real ra-pidity parameters; and we have thus obtained a class of gen-eralized (with adjustable parameter f ) regular PG metrics forKerr-Newman solutions [26] . Different choices of f result indifferent constant- t P slices of 3-geometry. Moreover, as dis-cussed, the Lorentz boost becomes infinite (with b = tanh x = )precisely at the horizon(s) (this is also the motivation for ourparticular parametrization of b in Eq.(2)).An explicit, but by no means the only, choice for f whichsatisfies (7) is f = q R + Q + L a , L ≥ q R ( − L r ) + Q , L < . (8)The explicit vierbein for positive cosmological constant isthen e GPG = { q r + Q + L a ( − sin q cos q ) X r dt P + a sin q [ X q R − ( R + Q )] X r q r + Q + L a ( − sin q cos q ) d f P , q Mr + L ( a + R r ) X r ( dt P − a sin q d f P )+ r q R + Q + L a dr , r √ X q d q , r √ X q sin q q R + Q + L a X q r + Q + L a ( − sin q cos q ) d f P } . (9)Our construction recovers several known solutions: When f = R , L = , we recover the Doran metric [6] which is compat-ible with the vierbein e Do = { dt P , r R dr + √ R − Dr ( dt P − a sin q d f P ) , r d q , R sin q d f P } (10) e Do = L (cid:16) a sin q R (cid:17) · L (cid:16) √ R − D R (cid:17) · e BL . (11)However, for the Doran metric unphysical complex t P and f P arise for values of r such that R − D = Mr − Q < when Q = .Also, in agreement with our earlier general understanding, theDoran form is then related to the Kerr-Newman solution inBoyer-Lindquist coordinates by an unphysical Lorentz boost(as can be seen from complex rapidity parameter in the trans-formation (11) above). This deficiency can be overcome byour more general choice (or the explicit form in (8)) which sat-isfies criterion (7). Without the advantage of a tunable func-tion it is both hard to reveal the problem and also to guess anexact explicit form of f ( r ) which renders the metric finite andreal for all values of r > .For the case of non-rotating Schwarzschild-(anti)de Sitterblack holes ( Q = a = ), our generalized PG metric reduces tothe form found previously in Ref.[16], ds = − (cid:16) fr (cid:17) dt P + rf dr + dt P r(cid:16) fr (cid:17) − + GMr + L R ! + r d W . We can also recover the Eddington-Finkelstein descriptionof Kerr-Newman solutions by choosing a different set of timeand azimuthal coordinates, dt EF : = dt + X R D dr , d f EF : = d f + X a D dr ,after the first step of Eq. (4). This results in e EF = { f X r ( dt EF − a sin q d f EF ) + r ( − f + p f − D ) D dr , p f − DX r ( dt EF − a sin q d f EF ) + r ( f − p f − D ) D dr , r √ X q d q , √ X q sin qX r ( R d f EF − adt EF ) } . (12)The vierbein is regular at the horizon(s) provided f − p f − D = a D R i.e. f = a D R + R a . The criterion f − D > needed to ensurereality of the coordinates and the Lorentz boost can be attainedby setting a = R √ R + Q , L ≥ R q R ( − L r )+ Q , L < . (13) For positive cosmological constant, we thus obtain the regularvierbein through e EF = L (cid:16) ( R + Q ) − D ( R + Q )+ D (cid:17) · e BL , giving e EF = { ( R + Q ) + D p R + Q X r ( dt EF − a sin q d f EF ) − r p R + Q dr , ( R + Q ) − D p R + Q X r ( dt EF − a sin q d f EF ) + r p R + Q dr , r √ X q d q , √ X q sin qX r ( R d f EF − adt EF ) } . (14)This yields Kerr black holes in the form of advancedEddington-Finkelstein coordinates (for L = and Q = ) as ds = h AB e AEF e BEF = − ( − Mr r ) dt EF + dt EF dr − Mra sin q dt EF d f EF r − a sin q drd f EF + r d q + ( R sin q + Mra sin qr ) d f EF , (15)which is free of coordinate singularities and always real. III. HAWKING RADIATION
As an application of our constructions, we consider Hawk-ing radiation for Kerr black holes with the tunneling processof Parikh and Wilczek [10]. In such computations, regular-ity of the metric at the horizon(s) is essential; singular met-rics can lead to factor-of-two discrepancies and other com-plications [19]. We shall demonstrate that both the regularadvanced Eddington-Finkelstein metric and our generalizedPG metrics yield the correct results. To wit, Hawking radi-ation is treated as tunneling across the (outer) horizon from r in to r out of massless emissions carrying energy and angu-lar momentum. The black hole mass parameter M shrinksby w and the angular momentum changes by a w resultingin TdS BH = dM − W dJ = − w ( − W a ) , for the first law of blackholes [24]. W is the angular velocity at the outer horizon r + = M + √ M + a . The decay rate comes from the imaginarypart of the associated particle action [10] (for simplicity weconsider massless particles without spin) which is I = Z r out r in p · dr = Z r out r in (cid:16) Z p dp (cid:17) · dr = Z r out r in (cid:16) Z H + w ( − W a ) H dH ˙ r i (cid:17) dr i . (16)In the last step, Hamilton’s equation, dHdp i (cid:12)(cid:12)(cid:12) r = ˙ r i , for the semi-classical process is invoked. Switching the order of integra-tion, together with dH = d w ′ ( − W ′ a ) , yields I = Z w Z r out r in dr i ˙ r i ( − W ′ a ) d w ′ , W ′ = ar + ( w ′ ) + a , (17)with r + ( w ′ ) : = ( M − w ′ ) + p ( M − w ′ ) + a denoting the locationthe shifted horizon when M decreases by w ′ .With advanced Eddington-Finkelstein coordinates, the nullgeodesics corresponding to { dt EF dp , drdp , d q dp , d f EF dp } = { R D , , , a D } yield ˙ r = ( drdp ) / ( dt EF dp ) = D R , and thus I = Z w Z r out r in dr R D ( − W ′ a ) d w ′ + I f EF ; (18)wherein I f EF : = R w R r out r in [ d f EF / ( ˙ f EF )]( − W ′ a ) d w ′ does not con-tribute to the imaginary part of I as ˙ f EF = aR is finite at theouter horizon. The integral I , with pole at r + , is defined bydeforming the contour to go through a clockwise infinitesimalsemicircle r = r + + e e i q around the pole [10]. Its imaginary partis then ` I = ` lim e → Z w Z p p e e i q id q h ( R + r e e i q + ··· ) D + e e i q ¶ r D + ··· i(cid:12)(cid:12)(cid:12)(cid:12) r = r + ( w ′ ) ( − W ′ a ) d w ′ = − Z w p ( r + a ) r − r − (cid:12)(cid:12)(cid:12)(cid:12) r = r + ( w ′ ) ( − W ′ a ) d w ′ (19)For our generalized PG metrics (4), it can be verifiedthat { dt P dp , drdp , d q dp , d f P dp } = e R r g ( r ′ ) dr ′ { R , f D f + √ f − D , , a } correspond tonull geodesics if g : = f ( ¶ r D )( f + f √ f − D − D ) − ( ¶ r f ) D f D ( D − f − f √ f − D ) . Now with ˙ r =( drdp ) / ( dt P dp ) = f D R ( f + √ f − D ) , and proceeding as before, we are leadto I ( w ) = Z w Z r out r in dr R ( f + p f − D ) f D ( − W ′ a ) d w ′ + I f P . (20)Provided p f − D remains real, the imaginary part of the actionis ` I = ` lim e → Z w Z p p d ( i q )( − W ′ a ) d w ′ · e e i q h ( R + e e i q ¶ r R )[ f + p f − D + e e i q ¶ r ( f + p f − D )] f D + e e i q ( f ¶ r D + D ¶ r f ) i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r + ( w ′ ) = − Z w p h R ( f + p f − D ) f ¶ r D i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r + ( w ′ ) ( − W ′ a ) d w ′ = − Z w p ( r + a ) r − r − (cid:12)(cid:12)(cid:12)(cid:12) r = r + ( w ′ ) ( − W ′ a ) d w ′ . (21)Both Eqs. (19) and (21) give the same final result. The changeof the Bekenstein-Hawking entropy from D S BH = − D S = ` I ( w ) yields, at the lowest order (higher order corrections in ` I ( w ) indicate departures from pure thermal physics), the effectivetemperature as T ef f = h dS BH ( − dH ) i − (cid:12)(cid:12)(cid:12)(cid:12) w = = h d ` I − ( − W a ) d w i − (cid:12)(cid:12)(cid:12)(cid:12) w = = r + − r − p ( r + + a ) , which agrees with the Hawking temperature T Hawking = k p = r + − r − p ( r + + a ) . Thus both the Eddington-Finkelstein and our gen-eralized PG metrics lead to the same physical effective Hawk-ing temperature. It should be noted that for the generalized PGmetrics, the result is, reassuringly, independent of the function f , provided the physical criterion f − D > is maintained. IV. DISCUSSION AND FURTHER REMARKS
The constant- t P hypersurfaces are conformally flat iff theCotton-York tensor vanishes [20]. The explicit computation of the tensor for the constant- t P f which results in conformally flat slicings. There is analyticproof that there is no maximal, non-boosted, conformally flatslices not only in the Kerr spacetime, but also in any stationaryspacetime with non-vanishing angular momentum [21]. Forthe special case of spherically symmetric metrics, PG metricswith flat slicing can violate criterion (7). This can be remediedby appropriate choices of f which however does not in generallead to spatial flatness [16].With vanishing cosmological constant, the Doran metrichas e Do (cid:181) dt P but, as discussed, unphysical complex compo-nents appear for certain values of r whenever Q = . In the classof metrics we constructed with regular real vierbein for all val-ues of r , the requirement e GPG (cid:181) dt P is satisfied iff R X q − f = .The latter cannot be achieved for non-vanishing cosmologi-cal constant for our vierbeins since ¶ f ¶q is required to vanish;the gauge condition e ′ GPG (cid:181) dx can however be attained by afurther Lorentz boost.Other parametrizations of Kerr-Newman solutions havebeen suggested, and our general understanding can also beapplied to analyze these prescriptions. An alternative to theDoran metric, proposed by Natario [17] for pure Kerr blackholes, is e Nat = { dt P , r √ s ( dr − vdt P ) , r d q , ( d f P + d d q − W C dt P ) √ s sin q } , with dt P = dt − kdr , d f P = d f − k W C dr − d d q , r v = − R √ R − D , r s = R − a D sin q , k = r v D , and d = − R ¥ r k ¶ W C ¶q dr . It is related to theKerr-Newman metric in Boyer-Lindquist coordinates by e Nat = L R r r R − Ds ! · L a √ D sin q R ! · e BL . However, without the benefit of our adjustable function f inthe boost to ensure criterion (7) is satisfied, the problem withthe above metric is again the condition R − D > for physicalLorentz boosts and real metric variables is violated for somevalues of r . Another proposed metric [18] also suffers fromsimilar problems.It is possible to discuss Kerr-Newman solutions within thecontext of general axisymmetric metrics expressed in Chan-drasekhar form [22, 23]. The latter is compatible with thevierbein e axisym = (cid:8) A s dt , B − s dr , r d q , C s ( d f − W C dt ) (cid:9) , (22)with Kerr-Newman parameters, A s = √ D r X q X q R X q − a D sin q , B s = √ Dr C s = q R X q − a D sin qr X sin q , W C = a ( R X q − D ) R X q − a D sin q . However, the Kerr-Newman solution in Chandrasekhar formhas extra coordinate singularities at S = R X q − a D sin q = .The reason is again revealed by our general understandingbetween singular and regular forms of the veirbein. ForKerr-Newman metrics, the explicit relation between the Chan-drasekhar and Boyer-Lindquist expressions is again a Lorentzboost, e axisym = L (cid:16) a √ D sin q R √ X q (cid:17) · e BL , which is infinite at S = . Thusthe above Chandrasekhar form of Kerr-Newman black holeswill have additional coordinate singularities at S = to con-tend with, in addition to the coordinate singularity of theBoyer-Lindquist expression (1) at the horizon(s) D = . Incontradistinction, our generalized PG expressions of Kerr-Newman black holes constructed and displayed in this workare free of all coordinate singularities, and real, for all valuesof r . Acknowledgments
This work has been supported in part by the National Sci-ence Council of Taiwan under Grant Nos. NSC98-2112-M-006-006-MY3 and 99-2811-M-006-015, and by the NationalCenter for Theoretical Sciences, Taiwan. Beneficial interac-tions with C. Y. Lin during the early phase of this work arealso gratefully acknowledged. [1] R. P. Kerr, Phys. Rev. Lett.
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199 (1975)[25] Throughout this work geometric units G = c = ( − +++) spacetime signature are adopted.[26] The second Lorentz boost L , performed to eliminate dt P from e GPG and also to allow comparison with various existing de-scriptions, is actually not needed to obtain a regular metric; inwhich case, following the steps leading to Eq.(4), criterion (7)can be relaxed to f ( rr
199 (1975)[25] Throughout this work geometric units G = c = ( − +++) spacetime signature are adopted.[26] The second Lorentz boost L , performed to eliminate dt P from e GPG and also to allow comparison with various existing de-scriptions, is actually not needed to obtain a regular metric; inwhich case, following the steps leading to Eq.(4), criterion (7)can be relaxed to f ( rr ) > D ( rr
199 (1975)[25] Throughout this work geometric units G = c = ( − +++) spacetime signature are adopted.[26] The second Lorentz boost L , performed to eliminate dt P from e GPG and also to allow comparison with various existing de-scriptions, is actually not needed to obtain a regular metric; inwhich case, following the steps leading to Eq.(4), criterion (7)can be relaxed to f ( rr ) > D ( rr ) ∀ rr