On the Effective Metric of a Planck Star

Tommaso De Lorenzo∗ 1,3, Costantino Pacilio† 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 ) large∞ 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 gravitational 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 horizons 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 L√. 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 horizons 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. The time dependence of the internal hori- gravity effects. On the other hand, restrictions zon, and hence the radius of the ‘explosion’ event on the metric come from the requirement that its P depends on the quantum gravitational dynamics curvature has to be sub-Planckian everywhere, as of the Planck star. Because of the expected huge discussed for instance in [24], so that details of time dilation inside the gravitational potential well quantum gravity (such as the spacetime foam pro- of the star, the bounce is seen in extreme slow mo- posed by loop quantum gravity [25]) do not mat- tion from the outside, appearing as a nearly station- ter, and the effective description is meaningful. As ary black hole. Furthermore, the core (the Planck a criterium for maximal curvature, we consider the star) retains memory of the initial collapsed mass m 2 µνρσ Kretschmann scalar = RµνρσR , where Rµνρσ and the final exploding objects is much larger than K 3 i + frequency. This is a physically unmotivated restriction, and directly related to the fact that F (0) = 1. Notice that this feature is shared by most models in the literature, one exception being the gravastar of I [8]. + core There is a second limitation of (3) that we P would like to point out. While this effective metric r − is mainly motivated by including strong quantum gravity effects, it should not neglect the inclusion of r 0 0 = i those weak, but solid, quantum gravity effects that are commonly agreed upon, such as the 1-loop quan- r + tum corrections to the Newton potential obtained using effective field theory [21, 31]. The latter read, I reintroducing the Planck length, − 2 ! m `planck Φ(r) = 1 + β + o(r−4) , (9) − r r2

i − where β is a numerical constant of order 1. By a Figure 5. Penrose diagram of a collapsing and evaporat- standard derivation of the Newton potential from ing non-singular black hole. The thick line is the external the g00 component of the Schwarzschild metric, boundary of the star, while the trapping region is shaded, bounded by the two trapping horizons: the external evap- 1 Φ(r) = 1 + g00(r) , (10) orating one, and the internal expanding one. −2 the 1-loop effect is immediately related to the metric. Such an effect cannot be described by Hayward Planckian. See [17, 29, 30] for developments and metric (3), whose large scale behaviour is applications to observations of this model. The pro- 2 2 cess is illustrated by the conformal diagram of Fig- 2m 4L m −5 g00 = 1 + + o(r ) (11) ure5. While the static metric plays an important − r − r4 role in this compelling picture, it has two shortcom- and lacks the appropriate r−3 term.1 ings that we want now to point out. We now introduce a minimal generalisation of the Hayward metric that allows to fit in these two re- Two Shortcomings quests. Gravitational time dilatation slows down clocks in II. Modified Hayward Metric a gravitational potential well, compared to clocks in an asymptotically flat region. A clock kept in The most general spherically symmetric, static met- the centre of a dust cloud, for example, shows an ric can be parametrized adding an arbitrary function elapsed time shorter than that of a clock at infinity, G(r) to the 00 component of the metric, when the two clocks are moved together and com- 1 pared. Since the Hayward metric is regular at the ds2 = G(r)F (r)dt2 + dr2 + r2dΩ2. (12) origin, we can imagine a clock sitting at the centre − F (r) of the collapsed object. The time measured by this In the following we take the same F (r) as Hayward, clock is easy to compute: during the static phase and use G(r) to introduce the desired modifications. F (r = 0) = 1, Eq. (1) shows that this is equal to From the previous discussion, the physical require- the coordinate time t. The same is true for a clock ments we wish to impose on G(r) are: at infinity. Therefore a clock at the centre of the star 1 suffers no time delay with respect to a clock at infin- It is worth mentioning that the metric proposed by Bardeen in [5] does reproduce the required behaviour of the ity, and a signal sent from infinity with a given fre- newtonian potential. On the other hand, as well as all the quency will be received at the center with the same line elements proposed, it suffers for the time delay problem.

4 (i) preserve the Schwarzschild behaviour at large r; 0.5

(ii) include the 1-loop quantum corrections (9); 0.4

(iii) allow for a ﬁnite time dilation between the cen- 0.3 2

ter and infinity. K 0.2 A time delay between the center and infinity can be p seen from (δt∞ δt0)/δt∞ = 1 g00(r = 0) 0.1 − − | | ∈ [0, 1), and we parametrise 0.0 0 10 20 30 40 50 g00(r = 0) = 1 α, α [0, 1). (13) − − ∈ r The larger α, the greater the time delay. Figure 6. Kretschmann scalar curvature as a function The first two conditions are satisfied by of the radial coordinate r for the corrected Hayward metric. A peak features inside the inner horizon (marked by 2 the vertical line), whose value increases beyond the Planck m `planck lim G(r) = 1 β (14) scale if the required time delay is too large. In this plot r→∞ 3 −5 5 − r 1 − α = 7 × 10 , m = 10 mPlanck, L = 10`Planck and the 4 peak is shown at 1/(2`Planck). Notice also that the modi- where β is the same of Eq. (9). In the remaining of fication introduces a decreasing of the curvature near the 2 this paper, we consider it fixed once and for all. At inner horizon, and before the peak. r = 0, condition (iii) gives

G(0) = 1 α. (15) Bounds − An additional useful restriction – albeit not manda- Next, we check what restrictions apply to the new tory – is to demand that (iv) near the center, the metric. In particular, we will see that it is not pos- equation of state of the derived energy momentum sible to arbitrarily increase the time delay between tensor is still de Sitter. Since the center and infinity: an upper bound comes from 2 the requirement that the curvature is sub-Planckian r 0 g00(r) = G(0) 1 G (0)r (16) everywhere. The Kretschmann invariant associated − − L2 − − with (18) has a rather long expression which pre- G00(0) r2 + o(r3), vents a purely analytic study of its properties. How- − 2 ever, numerical investigations clearly show that for matching the de Sitter behaviour (cf. (6)) gives α > 0 a peak in curvature develops inside the inner horizon, see Fig.6. The value of the peak can 0 00 G (0) = G (0) = 0 . (17) become arbitrarily large as α approaches 1, and ex- ceeds the Planckian value 1/`4 at a value that G(0) can then be absorbed rescaling t, introducing Planck depends on m and L. Imposing that the maximal in this way the desired time delay. Conditions (i) value of the curvature stays always below the Planck and (iv), and their associated expressions (14) and scale thus introduces an upper bound on α. (17), suggest to look for solutions as rational func- Indeed, let us call 2 the maximum value of the tions of r3. Taking the simplest case, and using (ii) max Kretschmann scalarK curvature; in general it will be and (iii) to fix the coefficients, we find a function of the three free parameters of the model, βm α i.e. m, L and α. The numerical analysis, see Fig.7, G(r) = 1 . (18) − α r3 + βm shows a monotonically increasing behaviour in α for 2 , namely This example shows how it is possible to improve the Kmax metric proposed by Hayward to take into account 2 ∂ max the 1-loop quantum corrections and a time delay in K > 0, m > m?. (19) ∂α ∀ the central core. Therefore, we can impose a bound on α by requiring 2In the numerical plots, we use the value β = 41/10π. the maximum curvature to be smaller than unity,

5 1.0

107 0.8 K2 2R Rμν 0.6 μ⋁ 104 - 1 R2 0.4 3 max 2 W 2 K 10 0.2

0.0 0.01 -0.2

0.9980 0.9985 0.9990 0.9995 1.0000 -0.4 α 0 1 2 3 4 5 6 Figure 7. Maximum value of the Kretschmann scalar r 5 as a function of α. Here again m = 10 mPlanck and Figure 9. Comparison between the diﬀerent factor con- L = 10`Planck, but the same behaviour is reproduced for tributing to the Kretschmann scalar K2. The Weyl’s tensor all values of L and m > m?. contribution is always small. Same parametrical values as in the previous ﬁgure.

1.0 ●◆■ ▲ ◆●■ ▲ ◆●■▲ ◆●■▲ ◆●■▲ ◆●■▲ ◆●▲■ ◆●▲■ ◆●▲■ ▲▲●■◆▲ ▲ ▲ ●■◆ ▲▲▲▲▲ ●■ ◆ ◆ ● ▲ ◆▲ ◆▲ ◆▲ ◆▲ ◆▲ ▲ ◆◆ ▲ ◆◆ ▲▲ ◆◆◆◆ ▲▲▲▲ ■◆ 2.2 ▲ ◆▲ ◆ ▲ ▲ ◆ 0.9 ◆ ▲▲▲▲▲▲▲ ▲ ▲ ▲ ◆ ◆ ■ ■ ■ ■ ■ ■ ■ ■ ■■■ ■■■ ●■◆◆◆ ◆ ■ ■ ■ ◆ 2.0 ■ ■ ◆ ● ■ ◆ 0.8 ● ● ● ● ● ● ● ● ●●● ●●● ◆ ● ● max ■ ■ ● ◆ ● ) α ● 1.8 ◆◆ ■ ● ■

● max ■ L= 10 ● 2 ●■ ■ ■ ■ ■ 0.7 ■ L= 20 1.6 ● r(K ● ● L 10 ◆ L= 100 ■ = ● ● ■ ■ L= 20 ● ▲ L= 1000 ■ ● 1.4 ■ ● ■● ● ◆ L= 100 0.6 ● 4 5 6 ● ▲ L= 1000 10 100 1000 10 10 ● 10 1.2 ● ● m 100 1000 104 105 106 107 108

Figure 8. αmax as a function of the mass m for different m values of L (Planck units). Figure 10. The position of the peak, at its maximal allowed value 0.1, as a function of m, for various values of L. It reaches asymptotic values in both parameters, and it is say 0.1. Let us call αmax the bound value, namely always well inside the inner horizon. 2 (αmax, m, L) = 0.1 . (20) Kmax 5 arately the various Weyl and Ricci contributions to For instance m = 10 and L = 10 give 1 αmax − ∼ 2 10−4, corresponding to a time delay of 98%. We the Kretschmann scalar, showing the Weyl contri- thus× see that the bound is not very stringent, and bution is always small, and the peak comes from rather large time delays can be introduced without the Ricci tensor uniquely. violating the condition that the metric can be taken Finally, the position of the maximal peak at αmax as an effective description throughout spacetime. as a function of m and L is reported in Fig. 10. The dependence of the bound on the two other parameters of the model is reported in Fig.8. III. Energy Conditions The plots show that αmax increases with m, and (slightly) decreases with L. In physical terms, a Because the origin of the peak in curvature comes larger (non-singular) black hole allows a greater time from the effective energy-momentum tensor, it is delay, whereas stronger quantum gravity effects re- useful to investigate what happens to the weak en- duce it. ergy condition that, as we recall, was satisfied by The origin of the peak, and thus of the bound, is the Hayward metric. As it turns out, the modifica- to be found in the behaviour of the effective energy- tion (18) introduces a violation of the weak energy momentum tensor for which the modified Hayward condition in a small region confined around the in- metric is a solution. Indeed, in Fig.9 we plot sep- ner horizon. As shown in Fig. 11, the sum of the

6 0.0010 90 ▲▼ ▲▼ ▲▼ ● L= 10 ■ L= 70 ◆ L= 200 ◆ ▲▼ 80 ◆ ▲▼ ◆ ▲▼ ◆ ▲▼ ◆ ▲▼ ▲ L= 500 ▼ L=1000 0.0005 ◆ ◆ ▲▼ ◆ ▲▼ ▼ ◆ ◆ ▲ ▲▼ 70 ◆ ▲▼ ▼ ◆ ◆ ▲ ▲▼ ◆ ▲▼ ▼ ■ ◆ ◆ ▲ ▲▼ ■ ■ ◆ ▲▼ ■ ◆ ▲▼ ▼ ■ ■ ◆ ◆ ▲ ▲▼ 60 ■ ■ ◆ ◆ ▲▼ ▲▼ 0.0000 ■ ■ ◆ ▲▼ ■ ■ ◆ ◆ ▲▼ ▼ ■ ■ ◆ ▲ ▲▼ ■ ■ ◆ ◆ ▲▼ ■ ■ ◆ ▲▼ ■ ■ ◆ ■ ■ 50 ■ ■ ■ ■ ■ ■ -0.0005 ■ ■ ● ● ● ● ρ+p_t original Hayward ● ● 40 ● ● ● ● ● ● ● ● ρ+p_t modified Hayward ● ● ● ● r(Max WEC violation) ● ● -0.0010 30 ● ● ρ+p_r modified Hayward ● ● ● ● ● ● ● 20 ● -0.0015 0 100 200 300 400 500 600 0.4 0.5 0.6 0.7 0.8 0.9 r α

Figure 11. Pressure profiles for both the original Hay- Figure 13. The position of the maximum of weak energy ward metric and its modification. The radial component condition violation, as a function of α, for various values of of the weak energy condition is zero everywhere in Haw- L. yard’s case, and shown in green for the modified case. The transversal contribution, in blue for the original Hayward metric and in orange for the modified one. The latter shows to remark that the violation of the weak energy con- the violation of the condition. The violation is confined to dition is a priori avoidable, according to a theorem a small region inside the inner horizon, marked by the ver- 5 by Dymnikova [9], which we briefly review here, and tical line in the plot. Here m = 10 mPlanck, L = 10`Planck, and 1 − α = 0.99. use it to arrive at a condition on G(r). To review the derivation of the theorem, we parametrise the most general static, spherically symmetric line element as density profile with the tangential pressures there ds2 = eµ(r)dt2 + eν(r)dr2 + r2dΩ2 . (21) fails to be non-negative. The amount by which the − condition is violated increases as L increases, and as From the Einstein equations, α approaches 1. The exact location of the violating 1 region decreases as we increase both α and L, see Rµν gµνR = 8πTµν, (22) Fig. 13. Since the inner horizon is approximately lo- − 2 cated at L, the plot shows that the violation occurs we find the non-null components of the stress energy around the inner horizon for L = 10, and it remains tensor well confined inside it for higher values. 1 ν0 1 8πT 0 = 8πρ(r) = e−ν (23a) While a violation of the weak energy condition is 0 − r2 − r − r2 per se not a problem in a model which is supposed 0 1 −ν 1 µ 1 to include quantum gravity effects, it is interesting 8πT1 = 8πpr(r) = e + (23b) r2 r − r2 2 3 8πT2 = 8πT3 = 8πp⊥(r) = 100 ▼ ▼ 00 02 0 0 0 0 ▼ ▼ ▼ (23c) ▼ ▼ ▼ µ µ µ ν µ ν ▼ ▼ ▼ −ν ▼ ▼ ▼ ▼ ▼ ▼ = e + + − , ▼ ▼ ▼ ▼ ▼ ▲ ▲ ▼ ▼ ▲ ▲ ▲ 2 4 2r − 4 ▼ ▼ ▲ ▲ 10 ▼ ▼ ▲ ▲ ▲ ▼ ▼ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ◆ where ρ(r) is the energy density, while p (r) and ▲ ▲ ◆ ◆ ◆ r ▲ ▲ ◆ ◆ ◆ ▲ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ are respectively the radial and the transversal 1 ◆ ◆ ■ p⊥(r) ◆ ◆ ◆ ■ ■ ◆ ◆ ■ ■ ■ ◆ ■ ■ ● ◆ ◆ ■ ■ ■ ● ◆ ◆ ■ ■ ■ ● ● ◆ ◆ ■ ■ ■ ● ● pressures. Integration of Eq. (23a) yields ■ ■ ■ ● ● ● ■ ■ ● ● ● ■ ■ ■ ● ● ● ■ ■ ● ● ● ■ ■ ■ ● ● ● ● ● ● ● ● ● ● 0.10● ● −ν 2M(r) |Max-WECviolation |/ρ e = 1 (24) ● L= 10 ■ L= 50 ◆ L= 200 − r ▲ L= 500 ▼ L=1000 0.01 with 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Z r α M(r) = 4π dxρ(x)x2 . (25) 0 Figure 12. Maximal violation of the weak energy condi- To match with our previous notations, we call tion, normalised by the energy density ρ, as a function of the α and for different values of L. Here m = 105 (Planck Z ∞ 2 units). m = drρ(r)r (26) 0

7 the ADM mass of the spacetime. Let us make the and r := (r−, r+), the trapping region where ∈1 I 0 following assumptions: ρ = T1 and pr = T0 . Let us assume that G(r) > 0, so that− the position of the horizons is still the same 1. Dominant energy condition (DEC); as before, determined by the zeroes of F (r). Then, the condition ρ 0 is given by 2. Asymptotic flatness; ≥ 1 (r F )0 0, r ; 3. Regularity of the metric in r = 0; − ≥ ∈ O (27) G1 (rF )0 rF G0 0, r . 4. Finiteness of ρ(r) for all r; − − ≥ ∈ I The condition ρ + pr 0 imposes 5. Finiteness of m. ≥ FG0 0, r ; The dominant energy condition holds if and only if ≥ ∈ O (28) FG0 0, r . ρ(r) pi for i = 1, 2, 3 [2]. As a consequence, hy- ≤ ∈ I pothesis≥ |4 |ensures that the principal pressures are Finally, the transversal part ρ p⊥ implies finite everywhere. It is clear that we could also re- ≥ quire the finiteness of the pressures together with r2FG02 + G24 4F + 2r2F 00 − − the weak energy condition, instead of asking the + rG3r F 0G0 + 2FG0 + 2r F G00 0, r , dominant energy condition to be satisfied. ≥ ∈ O r2FG02 + G24 4F + 2r2F 00 These assumptions imply the following restric- − − + rG3r F 0G0 2FG0 + 2r F G00 0, r . tions on the functions µ(r) and ν(r) that charac- − ≥ ∈ I terize the metric: (29)

1. lim µ(r) = 0; It can be easily checked that the function G(r) pro- r→∞ posed in Eq. (18) satisfies the first two conditions 2. ν(0) = 0; but not the last. Therefore, the violation of the weak energy condition associated with our proposal 3. µ(0) 0; ≤ comes from the large transversal pressures. 4. µ0(r) + ν0(r) 0 r, with µ0(0) = ν0(0) = 0. It is natural at this point to ask whether it is pos- ≥ ∀ sible to find a G(r) satisfying the previous require- The function A(r) µ(r) + ν(r) is monotonically ments and also (29), thus including the time delay ≡ increasing from a non-positive value A(0) = µ(0) without violating the weak energy condition. While ≤ 0 in the origin to the asymptotical value A( ) = 0. this is conceivably the case, we were not able to find ∞ The explicit choice A(0) = 0 entails that A(r) = 0 an explicit exemple, the difficulty being matching everywhere, i.e. µ(r) = ν(r), bringing us back with a smooth G(r) the condition (29) on its deriva- to a family of metrics of− the type in Eq. (1) and tives with the local requirements (i iii), even re- − then to the Hayward case. This is the case mostly laxing the requirement (iv) of a de Sitter behaviour discussed in [9], and indeed most non-singular black near the origin, which we remark it is not demanded hole metrics in the literature belong to this class. by the theorem above reviewed. We leave the ques- On the other hand, here we suggest to rather work tion open for future work. with A(0) < 0, as in Eq. (12), so to allow for the time delay in the center. As this behaviour is allowed IV. Conclusions by the conditions of the theorem, it means that it is possible in principle to introduce the time delay We have shown how it is possible to modify the non- without ever violating the weak energy condition. singular black hole metric proposed by Hayward as Let us see explicitly what restrictions this implies to incorporate the 1-loop quantum corrections eval- on the function G(r) parametrising the metric as in uated using the effective field theory approach to (12). Since we are using Schwarzschild coordinates, quantum gravity, as well as a non-trivial time delay we must distinguish two different cases: r := between an observer at spatial infinity and one at ∈ O [0, r−) (r+, ), that is the inner core and exterior rest in the central core, as motivated by physical re- ∪ ∞ 0 1 region, where r is space-like and ρ = T , pr = T ; quirements. We have derived an upper bound on the − 0 1 8 time delay induced by the requirement that the cur- [3] A. Ashtekar, T. Pawlowski and P. 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