Singularities in Quantum Corrected Space-times

In this paper we calculate the leading quantum corrections to locally spherically symmetric and time-independent space-times to first order in curvature. The corrections are calculated in the effective field theory for quantum gravity and arise due to quantization of the graviton. We find (non-trivial) conditions under which the non-local corrections at this order of perturbative expansion in curvature remain finite when computed on regular backgrounds. When the conditions are not met, it is hard to draw strong conclusions, but one expects that the chosen backgrounds are significantly affected by quantum corrections, likely in a non-perturbative way.


Introduction
Black holes are static vacuum solutions of Einstein's field equations. Despite the simplicity of these solutions, rotating black holes which are described by the Kerr metric appear to account for the numerous observations of astrophysical black holes with a remarkable accuracy. Furthermore, it is known that massive stars collapse at the end of their lifetime and form black holes. The idea that gravitational collapse leads to a black hole is strengthened by singularity theorems that prove geodesic incompleteness when a trapped surface is formed [1,2]. However, the collapse picture may not be fully realistic. Solutions like the Kerr metric are in fact vacuum solutions, and delta-like sources are not well defined in general relativity [3]. The collapse of a star to a Kerr black hole thus requires the destruction of the matter that made up the star, while some information like the mass and angular momentum is conserved. Indeed this is the reason why general relativity is considered to break down at singularities.
It is expected that a theory of quantum gravity will provide a physical mechanism to avoid these singularities. Although many candidate theories of quantum gravity have been developed, an ultra-violet complete theory of quantum gravity is still illusive. There exists however a unique infra-red theory of quantum gravity [4][5][6][7][8][9][10] that is valid up to the scale where the new physics necessary for an ultra-violet completion kicks in. This scale is known to be far beyond the reach of current experiments and is assumed to be at the Planck scale unless there is a very large number of fields in the model.
An important prediction of the effective field theory of quantum gravity is the leading quantum correction to the Newtonian potential [11][12][13]. This correction can equivalently be described by the introduction of two classical fields. Recently it has been shown that these fields can lead to the violation of assumptions of the singularity theorems and that singularities can therefore be avoided before Planck scale energies are reached [14,15]. Moreover the possibility of the singularity avoidance in an hypothetical big crunch was shown earlier in the same framework [16].
It is thus possible that a space-time that is classically singular becomes regular, when quantum gravitational effects are taken into account. In the specific case studied in Ref. [16], singularities avoidance happens at energy densities that do not exceed the Planck scale and can thus be described by the effective action approach to quantum gravity. It is not unthinkable, however, that the opposite can also occur: a classically regular space-time might become singular, when the quantum corrections are taken into account. Close to such a new singularity, the truncated effective field theory approach breaks down. It is thus not possible to determine within the effective field theory approach whether such a singularity is physical or a spurious effect that is resolved, when the full effective field theory is taken into account. The first case would be very interesting, since it introduces a new kind of singularity that would need to be resolved in a full theory of quantum gravity. The second case would also be interesting, if it occurs for matter distributions with classical energy densities below the Planck scale, as in this case there is a priori no reason to expect a breakdown of the effective field theory approach. This paper is organized as follows: in the next section we introduce the general form for the metric considered in this paper; in section 3 we discuss some properties of this metric, related to the pressure and density of its matter source; section 4 discusses the leading quantum corrections to this metric and derives conditions for which a classical regular metric remains regular, if the leading quantum corrections are taken into account.

A general metric
We consider a Lorentzian 3+1 dimensional space-time. We assume the space-time to be locally spherically symmetric and time-independent around the origin at r = 0. The line element can then be written as 1 in which we employ the usual areal radial coordinate r. Furthermore, we assume the space-time to be regular everywhere, which we define in this paper by |R|, |R µν R µν |, |R µνρσ R µνρσ | < ∞. 2 If we impose these conditions, we can expand f and g around r = 0 in the following way: where the Lorentzian signature requires a 0 > −1. Furthermore, the non-analytic parts f ∞ and g ∞ have the property If apart from regularity, we also imposed smoothness of the curvature invariants, we would find and the extra property for f ∞ and g ∞ : Since we only want to perform a local analysis, we can truncate the series so that

Energy conditions
For the line-element introduced in the previous section, the energy-density, radial and transversal pressure can be calculated from the Einstein tensor and read A positive energy density thus requires b n < 0. Moreover, a non-zero energy density and pressure at r = 0 requires n = 2. Furthermore, |p ⊥ | ≥ |p |, and a necessary but not sufficient requirement for equality (hence isotropy) is that one of the following is true: We can write, Depending on the parameter of the model one can find various values for w. In particular several cases can be distinguished: • If 2 ≤ m = n < ∞ and a m < b n (1 + a 0 ), • If m = n = ∞, knowledge of f ∞ and g ∞ is required to determine w.
In particular a de Sitter core with w = −1 requires a non-relativistic matter (dust) core with w = 0 requires and an ultra-relativistic (radiation) core with w = 1 3 requires Finally an asymptotically Minkowski core with ρ = 0 and |w| < ∞ requires m ≥ n > 2.

Quantum corrections to the metric
We shall here use the same approach as discussed in Ref. [17], for which we review the main steps. 3 The effective action is given by with S M the matter action and wherec i (µ) are renormalization scale dependent coefficients that follow from matching with an ultra-violet complete theory and experiment. Furthermore,α = α−γ,β = β+4 γ with the values of α, β, γ given in Ref. [17]. The equation of motion for the metric can then be solved perturbatively in G N , where we set = 1. The zeroth order equation is the Einstein equation and the first order equation is given by Equation (4.5) can now be solved for the leading quantum corrections to the metric such thatg µν = g µν + G N δg µν , (4.9) whereg µν is the quantum corrected metric, g µν is the solution of Eq. (4.4) and δg µν ≡ g q µν .

Local corrections
The leading local quantum corrections are given by Hence, the leading local corrections are of the order r m−2 and r n−2 . For m ≤ 3 or n ≤ 3 these corrections would make the space-time singular at r = 0. Notice however that for m = n = 2 the corrections vanish. Hence, only m = 3 or n = 3 would introduce singularities at this order. If singularities are to be avoided, one thus needs a 3 = b 3 = 0. Note that this condition is automatically satisfied, if the classical space-time is assumed to be smooth from the onset. Furthermore, the density and pressure are modified due to the quantum corrections in the following way: where m, n ≥ 2, and (4.16) Using the expressions for the density and pressure above, one can repeat the analysis of the previous section in order to derive conditions on several coefficients. Notice that, if m = n = 2, the quantum corrections to the density and pressure are perturbatively small compared to the classical counterpart, since they are suppressed by a factor of G N = l 2 p . However, if 4 ≤ m, n < ∞ the quantum corrections to the density and pressure dominate compared to the classical density and pressure for r l p , while the energy density and pressure remain small in this region. Finally, if m = n = ∞, knowledge of f ∞ and g ∞ is required to determine whether the quantum corrections can dominate.

Non-local corrections
Corrections due to the non-local terms in Eq. (4.8) are more difficult to calculate, since the ln is an infinite derivative operator. However, for a smooth time-independent and spherically symmetric function f , one can derive 4 Therefore, we find ln − µ 2 R r r = y 0 + y 2 r 2 + O r 4 , (4.21) ln − µ 2 R θ θ = z 0 + z 2 r 2 + O r 4 .
Hence, if singularities are to be avoided, it is necessary to impose Interestingly, this condition can be translated into the following condition for the pressure anisotropy: 5 If the time-independence and spherical symmetry holds globally, this is equivalent to Moreover, if p is differentiable, conservation of the energy momentum tensor, ∇ µ T µν = 0, implies that the above condition is equivalent to 6 These identities are clearly satisfied for isotropic fluids, but cause trouble for many anisotropic fluids. For instance, it can easily be verified that the condition is not satisfied for the Bardeen [18], Hayward [19] and Frolov [20] metrics, but it is satisfied for the Simpson-Visser metric [21]. 7

Discussion
In this paper, we have investigated whether quantum gravity can introduce new singularities in classically regular space-times. It is found that this can indeed be the case. Furthermore, for a locally spherically symmetric and time-independent space-time, we have found a condition, given in Eq. (4.26), on the pressure anisotropy for which this can happen. Matter distributions that satisfy this condition remain regular, when the one-loop quantum effects of the graviton are taken into account. Matter distribution that violate the condition contain a singularity at this order. We remark that this singularity is purely due to the quantization of the graviton, since the corresponding classical space-time is regular. It should be stressed that the truncation of the effective field theory breaks down at and close to the singularity. When the effective field theory is considered up to infinite order, there are two possible scenarios: either the singularity is resolved by resummation effects or the singularity persists. The first scenario is interesting, as there is a priori no reason to expect that the truncation of the effective field theory will break down. The classical energy density and curvature can after all be kept far below the Planck scale. The second scenario is even more interesting, as it introduces a new type of singular space-times, that would need to be resolved by an ultra-violet complete theory of quantum gravity.
It would be interesting to investigate the higher order corrections within this framework. Such a higher order analysis could provide new constraints, such as the one in Eq. (4.26), for classical matter distributions. Additionally, from dimensional analysis, one expects that local terms in the effective action at order l 2k p generate corrections 6 We assume p (∞) = 0. 7 It should be noted that these are all regular black hole metrics and require some form of exotic matter. If this matter is due to ultra-violet quantum gravity effects, as is often assumed, part of the local corrections are already included in the classical metric. The non-local corrections and the remaining part of the local corrections on the other hand are due to the quantization of the graviton.
to the metric at order min{r m−2k , r n−2k }. This was indeed found in Eqs. (4.10) and (4.11). For 2k ≥ min{m − 2, n − 2}, this generates terms that make the metric singular. For k = 1 we found that these dangerous terms are not present, as their coefficients vanish. It is expected that the dangerous terms also vanish for k > 1. If such a mechanism is not present, this would open up many new questions, as it would suggest that any locally spherically symmetric and time-independent matter distribution needs an ultra-violet complete theory to make sense, unless the density is more than polynomially suppressed.
Furthermore, following the same reasoning, we find that, if 2 < m, n < ∞, the quantum corrections will always generate corrections with smaller powers of r. In particular, the non-local terms are expected to generate corrections at order r 2 . A regular and smooth quantum space-time, that is locally spherically symmetric and time-independent should thus always have the form of Eq. (2.1) with f and g given by Eqs. (2.5) and (2.6), with the extra assumption a 2 , b 2 = 0. In addition, a 0 > −1, and using the analysis in section 3, one can impose b 2 < 0, and |a 2 | ≤ (1 + a 0 )|b 2 |.
Finally, one could try to generalize these results to space-times that do not have local spherical symmetry or are time-dependent. We will leave this for a future paper.