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On the Effective Metric of a Planck Star

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On the Effective Metric of a Planck Star On the Effective Metric of a Planck Star Tommaso De Lorenzo∗ 1,3, Costantino Pacilioy 2,3, Carlo Rovelli3, and Simone Speziale3 1Università di Pisa, Dipartimento di Fisica “Enrico Fermi”, Largo Bruno Pontecorvo 3, 56127 Pisa, Italy 2SISSA, Via Bonomea 265, 34136 Trieste, Italy 3Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France & Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France. (Dated: March 11, 2015) Abstract. Spacetime metrics describing ‘non-singular’ black holes are commonly studied in the literature as effective modification to the Schwarzschild solution that mimic quantum gravity effects removing the central singularity. Here we point out that to be physically plausible, such metrics should also incorporate the 1-loop quantum corrections to the Newton potential and a non-trivial time delay between an observer at infinity and an observer in the regular center. We present a modification of the well-known Hayward metric that features these two properties. We discuss bounds on the maximal time delay imposed by conditions on the curvature, and the consequences for the weak energy condition, in general violated by the large transversal pressures introduced by the time delay. Introduction Most metrics in the literature, however, possess two characteristics which we find unphysical: firstly, Spacetime singularities are unavoidable in gravi- a clock in the regular center is not delayed with re- tational collapse, if classical general relativity is spect to a clock at infinity; secondly, they do not re- valid at all scales and the energy-momentum ten- produce the 1-loop quantum corrections computed sor of matter satisfies the classical energy condi- in [21] treating quantum general relativity as an ef- tions [1,2]. On the other hand, classical general fective field theory. In this short note, we show how relativity cannot be valid at all scales, because of it is possible to write effective line elements incor- quantum mechanics, and quantum effects such as porating these two effects. In particular, we pro- Hawking radiation do violate these classical energy pose an explicit and simple modification of the Hay- conditions. There is thus a certain expectation ward metric that achieves the desired result, and that near the center of a physical black hole quan- discuss limitations on the maximal time delay al- tum effects dominate, and prevent the formation lowed by the condition that the curvature remains of a singularity. This scenario is also supported sub-Planckian everywhere. Finally, we expose how by a result of loop cosmology [3]: when matter our modification introduces a violation of the weak arXiv:1412.6015v2 [gr-qc] 9 Mar 2015 reaches Planck density, quantum gravity generates energy condition in a small region around the in- pressure sufficient to counterbalance weight. For a ternal horizon, and write down conditions for its black hole, this implies that matter’s collapse can avoidance. We restrict our considerations to spheri- be stopped before the central singularity is formed, cally symmetric and static metrics. Applications of yielding to the formation of a central core, called our proposed modified Hayward metric to dynami- a ‘Planck star’ in [4]. This expectation motivates cal scenarios is left to future work. We use natural the study of models of non-singular black holes, of units c = G = ~ = 1 throughout the paper. which many examples exist in the literature (e.g. [5,6,7,8,9, 10, 11, 12, 13,4, 14, 15, 16, 17, 18, 19], and [20] for a review). ∗[email protected] [email protected] 1 I. Hayward Metric F (r ) m 0 Consider the spacetime of a spherically symmetric 1 = m m object described by a static metric of the type < ? m m? 2 2 1 2 2 2 = ds = F (r)dt + dr + r dΩ (1) r − F (r) m m with > ? 2M(r) F (r) = 1 : (2) − r Figure 1. Redshift factor −g00 of the Hayward metric as Here M(r) tends to a constant value m for r a function of the radius for different values of the mass m. , so to recover the Schwarzschild solution as7! a m(rH ) large1 distance approximation, but it is such that the spacetime is nowhere singular. Various choices L for M(r) have appeared in the literature (e.g. m = 2 [5, 22,8, 10]). In the following we work with the r metric originally presented by Hayward in [10] and recently reconsidered, focusing on quantum gravita- tional phenomenology, in [16,4]. For the Hayward metric, m? m r3 M(r) = ; (3) rH r3 + 2mL2 r? where L is a parameter with dimensions of a length. Figure 2. The relation between the location of the hori- The logic here is that such a metric could arise from zons and the mass, at fixed L. Asymptotic values are a low-energy limit of quantum gravity, as a solution reached for m L. to Einstein’s equations modified by an additional right-hand side coming from the fundamental quan- that is responsible for the avoidance of the singu- tum theory. In this context, L is a free parameter larity, in this logic introduced by quantum gravity that is natural to assume of the order of the Planck effects, as in the example of the bounce in loop cos- length. Specifically, the right-hand side as the form mology. of a diagonal energy-momentum tensor with The spacetime under study possesses Killing hori- zons when F (r) = 0. With M(r) given by (3), the 2 2 1 3L m existence of solutions is controlled by the parame- ρ = = pr; (4) 2π (2L2m + r3)2 − ters m and Lp. There are two Killing horizons for 2 2 2 3 3 3 1 3L m (L m r ) m > m? = 4 L, merging into one at the critical pt = 2 −3 3 : (5) −π (2L m + r ) value m = m?, see Fig.1. The position of the hori- zons is given implicitly by These expressions are compatible with the weak en- 3 ergy condition everywhere, but violate the strong rH m(rH ) = ; (7) energy condition ρ + p + 2p 0 for r3 L2m. 2(r2 L2) r t H − Indeed, this can be put in evidence≥ noticing≤ that and for m L the inner and the outer horizons near the origin the metric behaves like a de Sitter approach respectively r L and r 2m. See spacetime, − + Figure2. Therefore, for values' of the mass' greater 2 than the critical value m?, the metric features an r 3 F (r) = 1 + o(r ); (6) outer horizon and can be taken as a description of − L2 a non-singular black hole. with the effective cosmological constant Λ = 3=L2 Since the metric violates only the strong energy introducing a repulsive force and thus violating the condition, it is interesting to recall how the origi- strong energy condition. It is this repulsive force nal Penrose’s singularity theorem, based upon the 2 i− i− 0.0030 r = r − 0.0025 r = − r 0.0020 r = 0 r = 0 r 2 0.0015 = K r − = r r − i+ i+ 0.0010 I+ r I+ 0.0005 = r + r + = r 0.0000 i0 i0 0 200 400 600 800 r r r + = r = + I r I Figure 4. Kretschmann scalar curvature as a function − − of the radius for the Hayward metric; here m = 105 and i− r i− L = 10 (Planck units). The vertical line represents the = r r − = inner horizon r−. − r r = 0 r = 0 r is the Riemann tensor. Its value as a function of = r − = r for the Hayward metric is shown in Fig.4. It r − r smoothly decreases from a maximum in the origin i+ i+ to zero. Hence, the curvature will be sub-Planckian Figure 3. Penrose diagram for the static line element in everywhere provided it is so near the origin. Taylor- Eq. (1). expanding the exact expression we obtain 24 r3 weak energy condition, is also avoided. The causal 2 = 1 2 + o(r5) : (8) K L4 − mL2 structure of this metric, represented in the Penrose diagram of Figure3, is analogous to that of the Therefore, requiring 2 1 imposes a restriction K ≤ Reissner-Nordström spacetime, with the difference on L, and on L alone, m controlling only the slope that the time-like surface r = 0 is not singular any- of the curve and not its maximal value. Specifically, more. The surface r = r− is a Cauchy horizon which there is a lower bound L & 3 in Planck units. means that the spacetime is not globally hyperbolic, The static Hayward metric has been suggested hence the Penrose’s singularity theorem does not ap- as an effective metric to describe a certain stage ply and the metric can be regular in spite of sat- of the life of a black hole, occurring between the isfying the dominant energy condition. The lack initial collapse and the moment when the evapora- of global hyperbolicity can also be understood in tion becomes important [10, 16, 19], with related terms of the topology change from a compact to a applications to the information-loss puzzle (see e.g. non-compact inner region, an argument which had [26, 27, 28]). As energy is radiated away in the form already been shown to lead to avoiding the singu- of Hawking radiation, the mass m decreases until larity theorems [23]. the critical value m? and a regular gravitating ob- The violation of classical energy conditions is a ject without horizons remains. In the Planck star natural consequence of the fact that the singularity- scenario [4], the initially collapsing matter bounces free metric is supposed to include some quantum out.
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