Non-singular quantum improved rotating black holes and their maximal extension
NNon-singular quantum improved rotatingblack holes and their maximal extension
R. Torres
Department of Physics, UPC, Barcelona, Spain.
Abstract
We add a prescription to the Newman-Janis algorithm in order to use it as a means of find-ing new extended (‘through r <
E-mail: [email protected] a r X i v : . [ g r- q c ] M a y . Introduction It is well-known that most astrophysically significant bodies are rotating. The collapse ofa rotating body contributes to the increase of its angular speed while maintaining constantangular momentum. In this way, if the body finally generates a black hole it will be a rotatingblack hole (RBH). This is the main reason why it is crucial to study RBHs and to analyzetheir properties.From a classical point of view, an uncharged (charged) RBH spacetime will be describedby a Kerr (Kerr-Newman, resp.) solution. This implies the existence of certain horizons, aspecific causal structure and a singular ring . However, several authors have suggested thatthe existence of singularities in the classical solutions has to be considered as a weaknessof the theory rather than as a real physical prediction. Consequently, some have tried toavoid the singularities in the models for RBHs by proposing heuristic regular spacetimesfor them (see, for instance, [1][2][3]). Other authors, inspired by the work of Bardeen, havetaken the path of nonlinear electrodynamics [4][5][6], which seems to provide the necessarymodifications in the energy-momentum tensor in order to avoid singularities in the RBH(see, for instance, [7][8][9]). Yet, another way of addressing the problem of singularities isto take into account that quantum gravity effects should play an important role in the coreof black holes, so that it would seem convenient to directly derive the black hole behaviourfrom an approach to quantum gravity.In this regard, some regular non-rotating Black Holes inspired in different approaches toQuantum Gravity have appeared in the recent literature (see, for example, [10][11][12][13][14][15]and references therein). For our purposes, let us remark the step in this direction taken byBonanno and Reuter in [16] by introducing an effective quantum spacetime for sphericallysymmetric black holes based on the Quantum Einstein Gravity (QEG) approach (see, for in-stance, [17][18][19]). The obtained quantum improved Schwarzschild solution indicates thatthe horizons and causal structure could be notably modified by quantum corrections andthat the BH spacetime could be devoid of singularities.However, this solution lacks of the rotation that one would expect for realistic black holes.If we want to test quantum improved metrics with astrophysical observations it is necessaryto have quantum corrected rotating solutions. In this line, Reuter and Tuiran [20] have trieda direct attack on the problem by using the QEG approach in order to obtain an improvedKerr solution . Nevertheless, some problems that had already appeared in the non-rotatingcase [16] become now much more important. Namely, in the QEG approach and through theuse of the Functional Renormalization Group Equation, first, one finds the running Newton1onstant G ( k ) depending on the considered energy scale k [17] G ( k ) = G ωG k , (1.1)where ω is a constant and G is the standard gravitational constant. Then, one converts theenergy scale dependence into a position dependence, what can be written as k ( P ) = ξd ( P ) , (1.2)where ξ is a constant (to be fixed) and d ( P ) is the distance scale that provides the rel-evant cutoff when a test particle is located at a point P . If the distance scale must bediffeomorphism invariant then one could write d ( P ) = (cid:90) C (cid:112) | ds | , where C is a curve from a reference point P to P . The problem is that there is a great dealof freedom in choosing C for the RBH case and that there is not a unique natural choicefor the distance scale. Nevertheless, If we write the RBH spacetime in Boyer-Lindquist-like coordinates ( t, r, θ, φ ), one can restrict the dependence of d on the coordinates simply bytaking into account that for a stationary and axially symmetric spacetime d = d ( r, θ ) only (sothat G = G ( r, θ )). In [20] it is argued that the dependence of d on θ should be asymptoticallysubdominant (i.e., negligible for r → ∞ ). It is also argued that the dependence on θ shouldnot be too important for r of the order of Planck length. However, a specific expression forthe angular dependence was not found.Our aim in this article is to obtain a quantum improved rotating black hole by using asalternative approach the Newman-Janis (NJ) algorithm [21], which allows to get a rotatingsolution from a static spherically symmetric one. The use of the standard NJ algorithm withthe goal of obtaining non-singular black hole solutions was suggested by Bambi and Modestoin [1]. Here, we will see that, in general, the strict standard approach consisting of five steps[22][1] must be supplemented with an extra prescription if we want to get a well-behaved extended (‘through r < r <
2. Newman-Janis algorithm and maximally extendedspacetimes
The standard Newman-Janis algorithm is a five-step procedure for generating new solutionsof Einstein’s equations by using as a seed solution a static spherically symmetric one [21][22].The seed solution can always be written as ds = − f ( r ) dt + g ( r ) dr + r d Ω . (2.1)The five steps are [22]:1. Rewrite the seed line element in advanced null coordinates.2. Express the contravariant form of the metric in terms of a null tetrad Z µa .3. Extend the coordinates x ρ to a new set of complex coordinates x ρ → ˜ x ρ = x ρ + iy ρ ( x σ )and let the null tetrad vectors Z µa undergo a transformation Z µa → ˜ Z µa (˜ x ρ , ¯˜ x ρ ) . Require that the transformation recovers the old tetrad and metric when ˜ x ρ = ¯˜ x ρ .4. Obtain a new metric by making a complex coordinate transformation˜ x ρ = x ρ + iγ ρ ( x σ )3. Apply a coordinate transformation u = t + F ( r ), φ = ϕ + H ( r ) to transform the metricto Boyer-Lindquist-type coordinates.As it is well-known this procedure has been successfully applied to the Schwarzschildsolution in order to get (in a straightforward way) Kerr’s solution. It is convenient toremark here that, as Kerr’s solution reveals, even if the final rotating solution of this generalprocedure can be singular this does not mean that r = 0 cannot be traversed. On thecontrary, consider Kerr’s case in which r = 0 represents a whole disk [23][24]. Only theboundary of the disk ( r = 0, θ = π/
2) is singular so that r = 0 can be traversed through θ (cid:54) = π/
2. In this way, the disk can be considered as a two-sided aperture to a second sheeton which r is negative, what provide us with an analytic extension of the solution. In Kerr’ssolution the standard NJ algorithm can be applied directly obtaining a natural extension for r < f ( r ) = 1 − m/r = g − ( r ). In this way,when one considers negative values for r this is clearly mathematically feasible and physicallyequivalent to deal with the geometry generated by a negative mass. As a consequence, whenthe function f undergoes the process of complexification becoming ¯ f = ¯ f ( r, θ ) this eventuallyprovide us with a RBH spacetime that is well-behaved for all r ∈ (cid:60) . In other words, wedirectly get a natural extension through r < r and the seed spacetime covered with negative values of r ,were both well-behaved.Now, in general, when one uses the NJ-algorithm one would like to extend the newfound solution (even more, if the considered solution is regular) beyond r = 0. However,if one insists in emulating the ‘ r < r < f ( r ) and g ( r ) could have problems from amathematical point of view (for example, do they exist and are real?) and from a physicalpoint of view (is the found solution meaningful for r < r < Deduce, if possible, the correct behaviour (both from a mathematical and a physicalpoint of view) of f ( r ) and g ( r ) in the spacetime covered with negative values of r . In practice, this often requires rethinking the method used to reach the original seed(2.1), but now considering that r <
0, what could be nontrivial in most cases. In order toexemplify this, let us now find an extended (through r <
0) non-singular quantum improvedsolution for a rotating black hole spacetime. Note that the method usually will not imply the use of GR. . Improved rotating solution The renormalization group improved
Schwarzschild solution found by Bonanno and Reuter[16] can be written as ds = − f ( r ) dt + f ( r ) − dr + r d Ω . (3.1)where f ( r ) = 1 − G ( r ) mr and G ( r ) = G r r + ˜ ωG ( r + γG m ) , (3.2) G is Newton’s universal gravitational constant, m is the mass measured by an observerat infinity and ˜ ω and γ are constants coming from the non-perturbative renormalizationgroup theory and from an appropriate “cutoff identification”, respectively. The preferredtheoretical value of γ is γ = 9 / ω is˜ ω = 167 / π . In fact, the properties of the solution do not rely on their precise values aslong as they are strictly positive. A relevant fact with regard to ˜ ω is that it carries thequantum modifications. In effect, if we make explicit Planck’s constant in (3.2), one gets˜ ω = 167 (cid:126) / π and, thus, ˜ ω = 0 would turn off the quantum corrections.Now, in order to see the problems with the standard NJ-algorithm [22][1], we can try toblindly apply it to the solution (3.1) in order to get a quantum improved RBH spacetime.The five steps for this case would be:1. The coordinate change du = dt − dr/f ( r ) allows us to write the metric in advancednull coordinates as ds = − f ( r ) du − dudr + h ( r ) d Ω , where h ( r ) = r .2. The null tetrad Z µa = ( l µ , n µ , m µ , ¯ m µ ) satisfying l µ n µ = − m µ ¯ m µ = − l µ m µ = n µ m µ = 0 can be chosen as l µ = δ µr , n µ = δ µu − f ( r )2 δ µr , m µ = 1 (cid:112) h ( r ) (cid:18) δ µθ + i sin θ δ µφ (cid:19) so that g µν = − l µ n ν − l ν n µ + m µ ¯ m ν + m ν ¯ m µ .3. We perform the standard coordinate change r (cid:48) = r + i a cos θ, u (cid:48) = u − i a cos θ. r (cid:48) and u (cid:48) to be real. In this way the null tetrad transforms into ( Z (cid:48) µa = Z νa ∂x µ (cid:48) /∂x ν ) l (cid:48) µ = δ µr , n (cid:48) µ = δ µu − ¯ f ( r (cid:48) )2 δ µr , m (cid:48) µ = 1 (cid:112) h ( r (cid:48) ) (cid:18) δ µθ + i sin θ δ µφ + i a sin θ ( δ µu − δ µr ) (cid:19) The functions ¯ f and ¯ h come from the complexification of f and h and, for the moment,we only know that they must be real and that they must reproduce Kerr solution ifthe quantum effects are turned off (˜ ω = 0). This is possible if the functions are chosenin the usual manner [21][22][1], i.e., by using the complexification1 r → (cid:18) r (cid:48) + 1¯ r (cid:48) (cid:19) , r → r (cid:48) ¯ r (cid:48) that provide us with ¯ h = r + a cos θ = Σ¯ f = 1 − Gmr Σ , (3.3)where there is still some freedom in choosing the function ¯ G .4. The new non-zero metric coefficients can be computed to be g uu = − ¯ f ( r, θ ) , g ur = − , g uφ = − a sin θ [1 − ¯ f ( r, θ )] (3.4) g rφ = a sin θ, g θθ = Σ , g φφ = sin θ [Σ + a sin θ (2 − ¯ f )]5. In order to get the metric in Boyer-Lindquist type coordinates we perform the coordi-nate change u = t + F ( r ), φ = ϕ + H ( r ) where F ( r ) = r + a ¯ f ( r, θ )Σ + a sin θ , H ( r ) = a ¯ f ( r, θ )Σ + a sin θ (3.5)and ¯ f and ¯ h are such that F and H must be functions of r alone. In principle, onecould conceive a general ¯ G (thus, ¯ f ) with the form¯ G ( r, θ ; α, β, δ ) = G r α Σ − α/ r α Σ − α/ + ˜ ωG ( r β Σ − β/ + γG mr δ Σ − δ/ )where α , β and δ are parameters. However, note that (3.5) impliesΣ ¯ f ( r, θ ) + a sin θ = D ( r ) , f using (3.3) one immediately sees that ¯ G = ¯ G ( r ). In other words,¯ G cannot depend on θ . Thus, α = β = δ = 0 and one is left with the straightforwardcase in which ¯ G ( r ) = G ( r ) = G r r + ˜ ωG ( r + γG M ) . (3.6)In effect, in this case F and H are really functions of r alone since F ( r ) = r + a r + a − m ¯ G ( r ) r and H ( r ) = ar + a − m ¯ G ( r ) r and, therefore, it is possible to write the solution in Boyern-Lindquist type coordinates.Let us stop here the algorithm in order to consider what would happen to the metriccoefficients (3.4) if, following Kerr’s example, one tries to analytically extend the solu-tion through r = 0 by just considering negative values for r . Clearly, one gets that thisextension is not admissible from a physical point of view. It suffices to consider (1.1)in which G ( k ) ≥ G ( k ) k →∞ →
0) and compare it with (3.6) that takes negativevalues around r = 0 for negative values of r . This is due to the fact that when one goesfrom (1.1) to (3.2) [16] one assumes that r is non-negative. Therefore, as argued in theprevious section, we should have first computed the correct behaviour of the improvedSchwarzschild solution for negative values of r (what requires rethinking the derivationof the improved solution). We have done this in the appendix obtaining G ( r ) for all r ∈ (cid:60) as G ( r ) = G | r | | r | + ˜ ωG ( | r | + γG m ) . (3.7)This running G is a non-negative, even and C function (see fig.1). Only if one appliesthe N-J algorithm using this running G (and, thus, defining sensible functions f ( r )and g ( r ) in the seed spacetime covered with r < g tt = − (cid:18) − G ( r ) mr Σ (cid:19) , g tϕ = − G ( r ) mr Σ a sin θ, g rr = Σ∆ ˜ ω g θθ = Σ , g ϕϕ = sin θ (cid:18) r + a + 2 G ( r ) mr Σ a sin θ (cid:19) where ∆ ˜ ω ≡ r + a − G ( r ) mr. and G ( r ) is defined in (3.7). Thus, the line element can be written in the familiarBoyer-Lindquist form as ds = − ∆ ˜ ω Σ ( dt − a sin θdφ ) + Σ∆ ˜ ω dr + Σ dθ + sin θ Σ ( adt − ( r + a ) dφ ) , (3.8)7 igure 1: A plot of G ( r ) /G for a BH mass m =
10 Planck masses. where the quantum corrections are all included in ∆ ˜ ω (which explains why we havechosen the subindex ˜ ω ).
4. Regularity
In order to this spacetime to be devoid of scalar curvature singularities one should proof thatall the algebraically independent second order curvature scalars in this spacetime are finite.While the metric (3.8) is singular at ∆ ˜ ω = 0 and at Σ = 0 it is easy to check (see [25] for thegeneral case) that ∆ ˜ ω = 0 is just a coordinate singularity and that it defines horizons in thespacetime (which will be analyzed later). With regard to Σ = 0, the regularity checking ismore involved. On the one hand, it is easy to see [25] that this spacetime is Petrov type Dand Segre type [(1, 1) (1 1)]. This implies that the spacetime has only six real algebraicallyindependent second order curvature scalars [26] that are collected in {R , I, I , K } , where R is the curvature scalar and the rest of the invariants are defined as I ≡ S αβ S βα ,I ≡
124 ¯ C αβγδ ¯ C αβγδ ,K ≡
14 ¯ C αγδβ S γδ S αβ , Here the invariants are written in tensorial form. See [26] for their spinorial form. S αβ ≡ R αβ − δ αβ R / C αβγδ ≡ ( C αβγδ + i ∗ C αβγδ ) / ∗ C αβγδ ≡ (cid:15) αβµν C µνγδ / .The fact that G ( r ) is not a C function indicates that we cannot directly apply thegeneral result of regularity in [25]. However, the proof of regularity can be carried out insimilar terms. By computing the curvature scalar R for our BH one finds R = 2 m (2 G (cid:48) + rG (cid:48)(cid:48) )Σ . In order to see that this is finite along any path approaching ( r = 0 , θ = π/ ξ ≡ a cos θ/r and ξ ∗ , its value in the limit along a chosenpath approaching r = 0. Taking into account that G (cid:48) (0) = G (cid:48)(cid:48) (0) = 0, one finds R → mG (cid:48)(cid:48)(cid:48) (0 + )1 + ξ ∗ if ξ ∗ finite and we approach r = 0 from positive values of r , R → mG (cid:48)(cid:48)(cid:48) (0 − )1 + ξ ∗ if ξ ∗ finite and we approach r = 0 from negative values of r , R → ξ ∗ infinite . Since G (cid:48)(cid:48)(cid:48) (0 + ) = − G (cid:48)(cid:48)(cid:48) (0 − ) = 6 / ( γ ˜ ωG m ), R would be finite along any path.On the other hand, I → { mG (cid:48)(cid:48)(cid:48) (0 + ) } ξ ∗ ξ ∗ ) if ξ ∗ finite and we approach r = 0 from positive values of r , I → { mG (cid:48)(cid:48)(cid:48) (0 − ) } ξ ∗ ξ ∗ ) if ξ ∗ finite and we approach r = 0 from negative values of r , I → ξ ∗ infinite , what again is finite along any path.With regard to I , I → { mG (cid:48)(cid:48)(cid:48) (0 + ) } ξ ∗ − iξ ∗ ) (1 + ξ ∗ ) if ξ ∗ finite and we approach r = 0 from positive values of r , I → { mG (cid:48)(cid:48)(cid:48) (0 − ) } ξ ∗ − iξ ∗ ) (1 + ξ ∗ ) if ξ ∗ finite and we approach r = 0 from negative values of r , I → ξ ∗ infinite , so that I is finite along any path reaching r = 0. Note that R and I are real, while I and K are complex. Therefore, there are, indeed, only 6 independentreal scalars. K → { mG (cid:48)(cid:48)(cid:48) (0 + ) } ξ ∗ − iξ ∗ ) (1 + ξ ∗ ) if ξ ∗ finite and we approach r = 0 from positive values of r , K → { mG (cid:48)(cid:48)(cid:48) (0 − ) } ξ ∗ − iξ ∗ ) (1 + ξ ∗ ) if ξ ∗ finite and we approach r = 0 from negative values of r , K → ξ ∗ infinite , what is finite along any path reaching r = 0.Therefore, we conclude that there are not scalar curvature singularities in the spacetime.
5. Effective energy-momentum and energy conditions
The spacetime metric (3.8) has not been obtain by using Einstein’s equations. However, itis still possible to consider an effective energy-momentum tensor defined through8 πG T µν ≡ R µν − R g µν . For this spacetime it is easy to show that the effective energy-momentum tensor is type I[24] with µ = − p ⊥ = mr G (cid:48) πG Σ p (cid:107) = − a cos θG (cid:48) + r Σ G (cid:48)(cid:48) πG Σ m. where µ , p ⊥ and p (cid:107) are the (effective) vacuum energy density, radial and tangential pressures,respectively, in the orthonormal basis in which T diagonalizes.The weak energy conditions require µ ≥ , µ + p (cid:107) ≥ , and µ + p ⊥ ≥ . This is violated for r < µ < G (cid:48) < µ > r > µ = 0for r = 0 and θ (cid:54) = π/
2. In this way, an observer can cross r = 0 with θ (cid:54) = π/ µ reaches its absolute maximum value when approaching r = 0, θ = π/ | µ | = 3 / (4 πγ ˜ ωG ). This is of the order of the Planck energy density,10 igure 2: A plot of the effective vacuum energy-density µ for a black hole with m = a = 7 . Note that µ concentrates around θ = π/ with a maximumat r = 0 . As explained, Kerr’s singular ring is replaced by a regular belt . i.e., around 10 J/m (in the International System of Units). A plot of the the effectivevacuum energy density around r = 0 is shown in figure 2.The spacetime also violates the weak energy conditions in the region with r > r = 0. Specifically, the inequality that is not satisfied is µ + p (cid:107) ≥
0. In order to check thisit suffices to consider its expression around r = 0: µ + p (cid:107) = − θ a G γ ˜ ω r + O ( r ) , which satisfies µ + p (cid:107) < r >
6. Global structure
As stated in section 4, there is a coordinate singularity at ∆ ˜ ω = 0. As usual [23], it is possibleto extend the coordinate system beyond ∆ ˜ ω = 0 using the coordinate change in section 3 (firststep in the NJ algorithm) with straightforward predictable consequences. The coordinate r igure 3: A plot of ∆ ˜ ω (quantum corrected case) versus ∆ (classical case) for m = a = 7 . Note that the differences between both cases are smalleras r grows. changes its character from spacelike when ∆ ˜ ω > ˜ ω <
0. Therefore, theboundaries ∆ ˜ ω = 0 between these regions are horizons of the spacetime. Classically (˜ ω = 0),there are two solutions to ∆ ≡ ∆ ˜ ω =0 = 0: r ± = G o m ± (cid:113) G m − a , corresponding to an inner r − and an outer r + ( Cauchy and event , respectively) horizons.Now, in order to get the quantum corrected horizons we should solve∆ ˜ ω = r + a − G ( r ) mr = 0 , which is equivalent to finding the roots of a fifth-degree polynomial. Even if there is nota general formula for the roots in this case we can analyze the general behaviour of thehorizons by taking into account the following • Since G ( r ) ≥ r . I.e., there are nohorizons in the r < • At large distances, G ∼ G so that one recovers the behaviour for the Kerr solution.In particular, ∆ ˜ ω > r will be a spacelike coordinate. • For r (cid:39) a (cid:54) = 0) we have ∆ ˜ ω > r will be a spacelike coordinate. Note that this is what happened in the classical Kerrsolution, however now the inner region is in full quantum regime ( G ∼ igure 4: A plot of ∆ ˜ ω (with a = 7 ) as a function of the BH mass and the coordinate r (with r ≥ ). The qualitative features are independent of the specific value chosenfor a . The points with negative values for ∆ ˜ ω have not been drawn in order to theboundary of the flat region ( ∆ ˜ ω = 0 ) to indicate the position of the inner ( r ˜ ω − ) andouter ( r ˜ ω + ) horizons. In this way, we observe that the number of horizons grows withthe mass, starting from none for small masses, reaching the extreme case for a certainmass m = m ∗ ( a ) (one horizon -denoted simply by r ˜ ω ) and, from there, stabilizing totwo: One inner and one outer horizon. • As in the classical case, the number of horizons depend on the relationship between m and a and there can be just none, one or two horizons (see fig.4).As the figure suggests the value of r for the quantum corrected inner horizon stabilizesfor big enough masses satisfying m (cid:29) a . In effect, in this case one can develop G inthe form of a series and approximately solve ∆ ˜ ω = 0 to get r ˜ ω − (cid:39) (cid:113) G γ ˜ ω + (cid:112) G γ ˜ ω (8 a + G γ ˜ ω ) , that in the a = 0 case provide us with r ˜ ω − (cid:39) (cid:112) γ ˜ ωG /
2, which is the result found in[16] for the nonrotating case. Likewise, in this big mass case one finds that the outer13 igure 5:
A plot of ∆ ˜ ω (quantum corrected case) versus ∆ (classical case) for m = a (cid:39) . . As can be seen, the quantum corrected case is extremalfor these values, while the classical case predicts two horizons. horizon satisfies r ˜ ω + (cid:39) G o m + (cid:113) G m − a − (2 + γ )˜ ω m . In this way, there is a small quantum correction with respect to the classical outerhorizon and, as in the non-rotating case [16], it affects the horizon by shrinking it. • The extreme case (one horizon) was obtained in the classical case whenever a = m .However, this is now modified by the quantum effects. For instance, it is now possibleto reach the extreme case even if a = 0 (non-rotating case [16]). Nevertheless, sinceboth the quantum effects and the action of the rotation help to generate an interiorregion with ∆ ˜ ω >
0, the inner horizon (when it exists) tends to be bigger than theclassical one and, in this way, the extreme case will be always reached for a < m .(See fig.5).Let us denote by m ∗ (= m ∗ ( a )) the value of the RBH mass that is needed to make a blackhole of rotation parameter a extreme. Then, there are three possible qualitatively differentcausal structures for the BH spacetime which are represented in the Penrose diagrams offigure 6 (for the a < m ∗ case) and of figure 7 (for the a = m ∗ or extreme case and the a > m ∗ or hyperextreme case).
7. Conclusions
The standard NJ algorithm can be used as a means of obtaining rotating spacetimes fromstatic spherically symmetric ones. However, we have seen that, in general, its use does14 igure 6:
Penrose diagram for a regular rotating black hole satisfying a < m ∗ .The spacetime has been extended through r = 0 to asymptotically flat regions withnegative values for r (IV or IV’). The grey regions are the regions where the coordinate r is timelike. Starting from the asymptotically flat region I, one could enter region IIby traversing the event horizon r ˜ ω + . Region III could next be reached by traversingthe Cauchy horizon r ˜ ω − . Then, the asymptotically flat region IV could be reachedby passing through the regular r = 0 . (Note that, since there are not singularities,unlike in Kerr’s solution, the diagram is valid for all θ ). not provide us directly with a correct extended (through r <
0) spacetime, neither from amathematical point of view, nor from a physical point of view. Guided by the fact that adirect natural extension can be found in Kerr’s solution, in section 2 we have put forwarda prescription in order to obtain well-behaved natural extensions for the RBH spacetimesobtained through the use of the NJ algorithm. We have seen that we could choose to extendthe solution with negative values of r , but in order to do so, we first need to control thebehaviour of the seed spacetime covered with negative values of the coordinate r .We have shown this with a particular example in which the blind application of thestandard algorithm provide us with an extension (through r <
0) of the obtained rotatingsolution that turned out to be totally incorrect from a physical point of view. On the other15 igure 7:
Penrose diagrams for an extreme ( a = m ∗ ) regular rotating black hole(to the left) and for a hyperextreme ( a > m ∗ ) regular rotating black holes (to theright). In the extreme case there is only one horizon denoted by r ˜ ω in which thecoordinate r is lightlike. r is never timelike. r ˜ ω acts as both an event and a Cauchyhorizon. In the hyperextreme case there are no horizons and r is always spacelike. Inboth cases, the spacetime has been extended through r = 0 to an asymptotically flatregion with negative values for r . (Note that, again, since there are not singularities,the diagrams are valid for all θ ). hand, if one carries out a previous analysis of the seed spacetime covered with negative valuesof r and uses the information in the algorithm, it provide us with a direct correct extension ofthe rotating solution for negative values of r , both from a mathematical and from a physicalpoint of view. The obtained extension, however, is not an analytical extension since G is a C function. In fact, this is just another example in which the analytical extension is notthe correct option (see [27] for other cases and further clarifications).The application of the algorithm to a quantum improved solution has allowed us toobtain the extended spacetime corresponding to a regular rotating black hole that emulatesthe behaviour of the maximally extended Kerr solution in the regions where quantum effectsare negligible – what is in itself a very interesting result. Moreover, we have seen that the16lgorithm provides us with an unique running G from our chosen seed solution. We haverigourously shown that the obtained spacetime does not have scalar curvature singularitiesand that this fact is linked to its violation of the weak energy conditions (what allows thespacetime to avoid the conditions for the existence of singularities appearing in the standardsingularity theorems). In this way, while in the (classical) Kerr solution Σ = 0 ⇔ ( r = 0 , θ = π/
2) defines a singular ring, in the quantum improved spacetime Σ = 0 is just a regularbelt . The features of the obtained regular belt are similar to those heuristically described in[1] and obtained for noncommutative inspired regular RBH in [28][29]. However, they differfrom the features found for the exact regular RBH solutions in the framework of conformalquantum gravity [30], where the spacetime is inextendible beyond “ r = 0” and the curvatureinvariants are continuous.We have seen that there are three qualitatively different cases for the obtained regularrotating black hole according to the relationship between m and a what, in fact, is similar tothe situation found in Kerr’s case. In particular, we have seen that the number of horizonsand the corresponding causal structures in the classical and quantum-improved cases arestrongly related. However, the position of the horizons is modified due to the repulsivecharacter of the quantum improvements. In this way, the inner horizon is bigger than theclassical inner horizon, while the outer horizon shrinks with respect to the classical one.Related to this effect, we get that the extreme case is obtained for smaller rotations than inthe classical case when quantum improvements in the RBH spacetime are considered (i.e.,it is obtained for a < m ).It must be taken into account that the reliability of the QEG approach used to obtain theseed improved Schwarzschild solution [16] is questionable in the planckian regime, so thatthe regular belt is just suggested by the approach, but can not be guaranteed. Indeed, onlya still nonexistent full Quantum Gravity Theory could provide us with the exact descriptionin the planckian regime.Finally, it is necessary to remark that, in general, there are other possible extensions(‘beyond r = 0’) for rotating black hole spacetimes, apart from the ‘ r <
0’ extension discussedhere. In each case, every possible extension has its own mathematical and/or physical prosand cons. A full analysis of the different extensions for general rotating black hole spacetimeswill be the subject of a future work [31]. 17 cknowledgements
R Torres acknowledges the financial support of the Ministerio de Econom´ıa y Competitividad(Spain), projects MTM2014-54855-P.
A. Running G for r < In the introduction we stated that the Functional Renormalization Group Equation leads toa running G with the form [16][17] G ( k ) = G ωG k . Then, one converts the energy scale dependence into a position dependence, what can bewritten as k ( P ) = ξd ( P ) , where ξ is a numerical constant to be fixed and d ( P ) is the distance scale that provides therelevant cutoff when a test particle is located at a point P . Finally, if the distance scale d ( P )form the point P to the center of the black hole must be diffeomorphism invariant then onecould write d ( P ) = (cid:90) C (cid:112) | ds | , where C is a curve joining the points. In the case of a spherically symmetric BH the symmetryimposes that d = d ( r ), however one still has to find an expression for the function, whatrequires considering the different possibilities for C .So far, we have been following the procedure described in [16]. We will still do it, withthe sole difference that now we want to consider r <
0. It is straightforward to see thatSchwarzschild’s solution has no horizons for r < r and t remain spacelike and timelike, respectively, for all r <
0, whatin fact makes the computations easier than in the r > radial curve C : r = λ, t = t , θ = θ , φ = φ . We have for all r ≤ d ( r ) = (cid:90) r (cid:18) − G m,r (cid:19) − / dr == (cid:112) r ( r − G m ) − G m tanh − (cid:114) rr − G m . (A.1)18Note that d ( r < > | r | (cid:28) G m is d ( r ) (cid:39)
23 1 √ G m | r | / (A.2)while for | r | (cid:29) G m d ( r ) (cid:39) | r | . (A.3)This is exactly the behaviour obtained for r > || ) to our negative r . Likewise, following [16], it is easy to see that othercurves provide the same behaviour (A.2) for | r | (cid:28) G m , while for | r | (cid:29) G m the behaviour(A.3) provides the largest momentum scale and, therefore, the actual cutoff. In this way,even if one cannot assert that (A.1) provide us with the exact behaviour of the distancescale, one concludes that the correct qualitative behaviour should interpolate between | r | / and | r | , what suggest to use in concrete computations the interpolating distance scale d ( r ≤
0) = (cid:18) r r − γG m (cid:19) / with γ = 9 /
2. Now, using k ( r ) = ξ/d ( r ) and the expression for the running GG ( r ≤
0) = G r r + ˜ ωG ( r − γG m ) , where ˜ ω ≡ ωξ . Therefore, as stated in (3.7), the behaviour for r ≥ r ≤ G for all r as G ( r ) = G | r | | r | + ˜ ωG ( | r | + γG m ) . References [1] Bambi C and Modesto L 2013
Phys. Lett. B
Phys. Rev. D Gen. Rel. and Grav.
31. Corrigendumat
Gen. Rel. and Grav. Conference Proceedings of GR5 p.174[5] Ayon-Beato E and Garcia A 2000
Phys. Lett. B
Gen. Rel. and Grav. Phys. Rev. D Class. Quant. Grav. Eur. Phys. J. C Int. J. Mod. Phys. A JHEP
049 (arXiv:1402.5446 [hep-th])[12] Torres R 2014
Phys. Lett. B
Proceedings of Science
038 (arXiv:1408.3050 [gr-qc])[14] Haggard H M and Rovelli C 2015
Phys. Rev. D Phys. Rev. D Phys. Rev. D Phys. Rev. D JHEP Phys. Rev. D J. Math. Phys. Gen. Rel. and Grav. J. Math. Phys. The large scale structure of space-time
Cambridge:Cambridge University Press.[25] Torres R and Fayos F 2017
Gen. Rel. and Grav. Gen. Rel. and Grav. Class. Quantum Grav. Phys. Lett. B