Abstract
As proposed by Bambi and Modesto, rotating non-singular black holes can be constructed via the Newman–Janis algorithm. Here we show that if one starts with a modified Hayward black hole with a time delay in the centre, the algorithm succeeds in producing a rotating metric, but curvature divergences reappear. To preserve finiteness, the time delay must be introduced directly at the level of the non-singular rotating metric. This is possible thanks to the deformation of the inner stationarity limit surface caused by the regularisation, and in more than one way. We outline three different possibilities, distinguished by the angular velocity of the event horizon. Along the way, we provide additional results on the Bambi–Modesto rotating Hayward metric, such as the structure of the regularisation occurring at the centre, the behaviour of the quantum gravity scale alike an electric charge in decreasing the angular momentum of the extremal black hole configuration, or details on the deformation of the ergosphere.
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Notes
The same discontinuity was previously observed in a model of regular rotating black hole proposed in [40], motivated by non-commutative geometry. In that scenario, it is possible to remove both the singularity and the discontinuity with a rotating string of Planckian tension replacing the ring singularity.
Indeed, this is a typical example in which the NJ algorithm does not work in its original meaning, in the sense that it produces a metric that does not solve the same Einstein’s equations as the seed metric.
To be precise, in [16] the authors consider a mixing of the tetrads that does not preserve the norms, and thus is not a Lorentz transformation. We believe however that their procedure amounts to a Lorentz transformation plus at most a permutation of the internal indices, and thus that their definitions of energy density and pressures coincide with ours.
References
Hayward, S.A.: Formation and evaporation of regular black holes. Phys. Rev. Lett. 96, 31103 (2006). arxiv:gr-qc/0506126
Frolov, V. P.: Information loss problem and a ‘black hole’ model with a closed apparent horizon. (2014). arXiv:1402.5446
Hossenfelder, S., Smolin, L.: Conservative solutions to the black hole information problem. Phys. Rev. D81 (2010). arxiv:0901.3156
Barrau, A., Rovelli, C.: Planck star phenomenology. Phys. Lett. B 739, 405 (2014). arXiv:1404.5821
Barrau, A., Rovelli, C., Vidotto, F.: Fast radio bursts and white hole signals. Phys. Rev. D90(12), 127503 (2014). arXiv:1409.4031
Barrau, A., Bolliet, B., Vidotto, F., Weimer, C.: Phenomenology of bouncing black holes in quantum gravity: a closer look. arxiv:1507.05424
Bardeen, J.M.: Non-singular general-relativistic gravitational collapse. In: Fock, V. A., et al. (eds.) GR5, Abstracts of the 5th international conference on gravitation and the theory of relativity. University Press, Tbilisi, 9–13 September 1968
Frolov, V.P., Vilkovisky, G.A.: Spherically symmetric collapse in quantum gravity. Phys. Lett. B 106, 307–313 (1981)
Roman, T.A., Bergmann, P.G.: Stellar collapse without singularities? Phys. Rev. D 28, 1265–1277 (1983)
Casadio, R.: Quantum gravitational fluctuations and the semiclassical limit. Int. J. Mod. Phys D9, 511–529 (2000). arXiv:gr-qc/9810073
Mazur, P.O., Mottola, E.: Gravitational condensate stars: an alternative to black holes. arxiv:gr-qc/0109035
Dymnikov, I.: Cosmological term as a source of mass. Class. Quant. Grav. 19, 725–740 (2002). arXiv:gr-qc/0112052
Visser, M., Barcelo, C., Liberati, S., Sonego, S.: Small, dark, and heavy: But is it a black hole? arXiv:0902.0346. [PoSBHGRS,010(2008)]
Falls, K., Litim, D.F., Raghuraman, A.: Black holes and asymptotically safe gravity. Int. J. Mod. Phys. A 27, 1250019 (2012). arXiv:1002.0260
Modesto, L., Nicolini, P.: Charged rotating noncommutative black holes. Phys. Rev. D 82, 104035 (2010). arXiv:1005.5605
Bambi, C., Modesto, L.: Rotating regular black holes. Phys. Lett. B 721, 329–334 (2013). arXiv:1302.6075
Bambi, C., Malafarina, D., Modesto, L.: Non-singular quantum-inspired gravitational collapse. Phys. Rev. D 88, 044009 (2013). arXiv:1305.4790
Haggard, H.M., Rovelli, C.: Black hole fireworks: quantum-gravity effects outside the horizon spark black to white hole tunneling. arxiv:1407.0989
Mersini-Houghton, L.: Backreaction of hawking radiation on a gravitationally collapsing star i: Black holes?, PLB30496 Phys. Lett. B, 16 September 2014 (06, 2014). arxiv:1406.1525
De Lorenzo, T., Pacilio, C., Rovelli, C., Speziale, S.: On the effective metric of a Planck star. Gen. Relativ. Gravit. 47(4), 41 (2015). arXiv:1412.6015
Ayon-Beato, E., Garcia, A.: The Bardeen model as a nonlinear magnetic monopole. Phys. Lett. B493, 149–152 (2000). arXiv:gr-qc/0009077
Falls, K., Litim, D.F., Raghuraman, A.: Black holes and asymptotically safe gravity. Int. J. Mod. Phys. A 27, 1250019 (2012). arXiv:1002.0260
Saueressig, F., Alkofer, N., D’Odorico, G., Vidotto, F.: Black holes in asymptotically safe gravity, PoS FFP14 (2015) 174. arxiv:1503.06472
Bambi, C., Malafarina, D., Modesto, L.: Terminating black holes in asymptotically free quantum gravity. Eur. Phys. J. C 74, 2767 (2014). arXiv:1306.1668
Frolov, V.P.: Mass-gap for black hole formation in higher derivative and ghost free gravity. Phys. Rev. Lett. 115(5), 051102 (2015). arXiv:1505.00492
Modesto, L.: Loop quantum black hole. Class. Quant. Grav. 23, 5587–5602 (2006). arXiv:gr-qc/0509078
Ashtekar, A., Bojowald, M.: Black hole evaporation: a paradigm. Class. Quant. Grav. 22, 3349–3362 (2005). arXiv:gr-qc/0504029
Ashtekar, A., Pawlowski, T., Singh, P.: Quantum nature of the big bang. Phys. Rev. Lett. 96, 141301 (2006). arXiv:gr-qc/0602086
Rovelli, C., Vidotto, F.: Planck stars. Int. J. Mod. Phys. D 23(12), 1442026 (2014). arXiv:1401.6562
Newman, E.T., Janis, A.I.: Note on the Kerr spinning particle metric. J. Math. Phys. 6, 915–917 (1965)
Caravelli, F., Modesto, L.: Spinning loop black holes. Class. Quant. Grav. 27, 245022 (2010). arXiv:1006.0232
Ghosh, S.G., Amir, M.: Horizon structure of rotating Bardeen black hole and particle acceleration. arxiv:1506.04382
Bjerrum-Bohr, N.E.J., Donoghue, J.F., Holstein, B.R.: Quantum corrections to the Schwarzschild and Kerr metrics. Phys. Rev. D 68, 084005 (2003). arXiv:hep-th/0211071. [Erratum: Phys. Rev. D71,069904(2005)]
Adamo, T., Newman, E.T.: The Kerr–Newman metric: a review. Scholarpedia 9, 31791 (2014). arXiv:1410.6626
Drake, S.P., Szekeres, P.: Uniqueness of the Newman–Janis algorithm in generating the Kerr–Newman metric. Gen. Relativ. Gravit. 32, 445–458 (2000). arXiv:gr-qc/9807001
Stephani, H., Kramer, D., MacCallum, M.A.H., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (2004)
Debnath, U.: Accretion and evaporation of modified hayward black hole. Eur. Phys. J. C75 (2015), no. 3 129. arxiv:1503.01645
Bambi, C.: Testing the Kerr black hole hypothesis. Mod. Phys. Lett. A 26, 2453–2468 (2011). arXiv:1109.4256
Bambi, C.: Testing the Bardeen metric with the black hole candidate in Cygnus X-1. Phys. Lett. B 730, 59–62 (2014). arXiv:1401.4640
Smailagic, A., Spallucci, E.: ’Kerrr’ black hole: the Lord of the String. Phys. Lett. B 688, 82–87 (2010). arXiv:1003.3918
Gibbons, G.W., Hawking, S.W.: Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D 15, 2738–2751 (1977)
Bardeen, J.M., Press, W.H., Teukolsky, S.A.: Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation. Astrophys. J. 178, 347 (1972)
Segre, C.: Sulla teoria e sulla class delle omografie in uno spazio lineare ad un numero qualunque di dimensioni. Transeunti Acc. Lincei VIII(III), 19 (1883–84)
Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space–Time, vol. 1, 20th edn. Cambridge University Press, Cambridge (1973)
Neves, J.C.S., Saa, A.: Regular rotating black holes and the weak energy condition. Phys. Lett. B 734, 44–48 (2014). arXiv:1402.2694
Perez, A.: No firewalls in quantum gravity: the role of discreteness of quantum geometry in resolving the information loss paradox. Class. Quant. Grav. 32(8), 084001 (2015). arXiv:1410.7062
Bianchi, E., De Lorenzo, T., Smerlak, M.: Entanglement entropy production in gravitational collapse: covariant regularization and solvable models. JHEP 06, 180 (2015). arXiv:1409.0144
Smerlak, M.: Black holes and reversibility. Talk given at the International Loop Quantum Gravity Seminar, 2015. web link
De Lorenzo, T.: Investigating Static and Dynamic Non-Singular Black Holes. Master thesis in theoretical physics, University of Pisa, 2014
Carlitz, R.D., Willey, R.S.: The lifetime of a black hole. Phys. Rev. D 36, 2336 (1987)
Acknowledgments
We thank Pietro Donà, Thibaut Josset and Carlo Rovelli for useful discussions.
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Appendices
A. Diagonalising \(T^{\mu \nu }\)
The co-rotating tetrad \(e^I_\mu \) of [42] in the Bambi–Modesto metric has norms given by
The corresponding Einstein tensor defines a non-diagonal energy-momentum tensor of the form
We then ask whether it is possible to diagonalise this tensor with a Lorentz transformation. To that end, we can concentrate on the 2-by-2 block with \(I=0,3\). The transformation will have to be either a simple rotation, when \(\mathrm{Sign}(\tilde{\Delta })=-1\), between the event and the Cauchy horizons, or a boost, when \(\mathrm{Sign}(\tilde{\Delta })=+1\) in the rest of the spacetime. Consider first the latter case. The most general \((1+1)\) boost reads
The condition \(D^{03}=0\), where \(D^{IJ}=\Lambda ^I{}_{K} T^{KL}\Lambda ^J{}_{L}\), reduces to
and implies
in order for the transformation to be valid, which in turn imposes conditions on the parameters of the metric. In between the horizons, since the transformation is a rotation, this problem does not arise because condition (36) is replaced by
which can be always satisfied. When condition (37) is not satisfied – notice that this may also happen locally, as the coefficients in (37) are spacetime functions –, it is not possible to diagonalize the energy momentum tensor starting from the co-rotating tetrad. Therefore, we need to start from scratch to find the good orthonormal basis. This is generally a not easy task and we do not address this problem here.
When a time delay is introduced, for cases (ii) and (iii) is still possible to define the co-rotating tetrad with the norms given in Eq. (33). In these cases, however, the transformed energy-momentum tensor \(T^{IJ}\) has two non-null components out of the diagonal, namely \(T^{03}\) and \(T^{12}\). To diagonalize it, therefore, we need now to combine a boost and a rotation for the two different 2-by-2 blocks according to the sign of \(\tilde{\Delta }\). This introduces the same condition (37) when \(\mathrm{Sign}(\tilde{\Delta })=-1\), together with a similar one for the 1–2 block of the tensor in the opposite case. The additional condition imposes strong limitations on the procedure, so that for the same range of parameters we now typically have finite regions where the energy-momentum tensor is not diagonalisable with the co-rotating tetrad. This shows up in the void zones in the numerical plots. The void regions can be either in between the horizons or outside, depending on the value of the parameters.
B. Coefficients of the \(\mathcal K\) expansion
We report in this Appendix various coefficients of power series used in the main text. For the expansion (29), we have
and the other two, \(c_2\) and \(c_3\), are functions of G(0) and its derivatives up to the second order, too long to be written here. However, once the solution for \(c_1=c_2=0\) is plugged into the equation \(c_3=0\), one gets the following relatively simple equation for G(0),
where the function \(\Gamma \) is defined by
The only positive solution of Eq. (39) is \(G(0)=1\).
On the other hand, when the time delay is introduced after the NJ procedure, non-trivial solutions exist. The expansion still has the structure (29), but with different coefficients. Both cases (i) and (ii) give the same result,
which can be all made to vanish for \(G'(0)=G''(0)=0\), while keeping an arbitrary \(G(0)\ne 0\). While the coefficients are slightly different in case (iii), their vanishing leads to the same condition.
Finally, the coefficients of the equatorial expansion (31) are much longer. Explicitly for case (ii), the first three read
These vanish iff the first two derivatives vanish. With this condition, \(d_3\equiv 0\) and
which gives the additional condition \(G^{(3)}(0)=0\) for the third derivative. Finally, the three conditions all together make \(d_1\) vanish.
C. Divergence for \(\gamma \ne \delta \)
In Sect. 4 and the previous Appendix, we showed that introducing a time delay in the rotating case by applying the NJ algorithm to the metric of [20] unavoidably leads to a divergent \(\mathcal K\), for any function G(r) and for a complexification such that M(r) and G(r) are unchanged, as choosing \(\gamma =\delta \) in the Bambi–Modesto prescription. When \(\gamma \ne \delta \), the dependence on r and \(\theta \) is too complicated to be handled explicitly, and we can not derive a power series expansion near zero. However, the divergence can be established with an indirect argument. In fact, notice that at the equator, where the divergence lurks, the metric for \(\gamma \ne \delta \) coincides with the metric for \(\gamma =\delta \). Furthermore, also the first derivatives coincide, and most of the second derivatives. The only terms that differ are second derivatives in \(\theta \) of three metric components. Explicitly,
where g is the metric for \(\gamma \ne \delta \), while \(\bar{g}\) is the one with \(\gamma = \delta \).
Hence, the Riemann tensors evaluated at the equator only differ in the terms of type \(R^{\mu }{}_{\theta \theta \nu }\), and the difference of the Kretschmann invariants is
where
The latter quantity is zero except for the u and \(\phi \) components and, more importantly, it is finite. Equation (40), therefore, tells us that the divergence of one implies the divergence of the other.
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Lorenzo, T.D., Giusti, A. & Speziale, S. Non-singular rotating black hole with a time delay in the center. Gen Relativ Gravit 48, 31 (2016). https://doi.org/10.1007/s10714-016-2026-5
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DOI: https://doi.org/10.1007/s10714-016-2026-5