The Return of the Singularities: Applications of the Smeared Null Energy Condition

Abstract

The classic singularity theorems of General Relativity rely on energy conditions that can be violated in semiclassical gravity. Here, we provide motivation for an energy condition obeyed by semiclassical gravity: the smeared null energy condition (SNEC), a proposed bound on the weighted average of the null energy along a finite portion of a null geodesic. We then prove a semiclassical singularity theorem using SNEC as an assumption. This theorem extends the Penrose theorem to semiclassical gravity. We also apply our bound to evaporating black holes and the traversable wormhole of Maldacena–Milekhin–Popov, and comment on the relationship of our results to other proposed semiclassical singularity theorems.

1. Introduction

General relativity allows for any spacetime geometry, even those with exotic features such as wormholes and causality violation. Given any metric, one can simply use the Einstein equation to find the appropriate stress-energy tensor. However, early on in the history of general relativity it was hypothesized that matter obeys certain restrictions called energy conditions. These energy conditions are bounds on the contracted stress energy tensor and they express properties expected of reasonable matter such as the positivity of the energy density. The classical energy conditions, which bound the energy density and similar quantities at every point in spacetime, were proven to be violated in all quantum field theories.

Bounds on the null-contracted stress tensor are particularly useful. The singularity theorem of Penrose was proven by assuming the Null Energy Condition (NEC), equation (1): the stress tensor contracted twice with an arbitrary null vector is greater than or equal to zero. While this condition is satisfied by all sensible classical theories, all quantum field theories contain states in which the expectation value of that null-contracted stress tensor is negative.

Recently, two of us proposed that while the null energy at a point can be arbitrarily negative, the average of the null energy over a piece of a null geodesic is bounded below in semiclassical gravity. Schematically, this smeared null energy condition (SNEC) claims that the average of the null energy over an achronal portion of a null geodesic of length tau is bounded below by a quantity of order minus one divided by Newton’s constant times tau squared — equation (2).

More precisely, the claim is equation (3): for a null geodesic parametrised by an affine parameter, and a differentiable smearing function g that controls the region where the null energy is averaged, the integral over the geodesic of g squared times the expectation value of the null-contracted stress tensor is greater than or equal to minus four times a constant B, divided by Newton’s constant, times the integral of the squared derivative of g. The presence of Newton’s constant ensures that the bound is relevant even in theories with a large number of species. When gravity is coupled to such theories, the renormalized Newton constant at one-loop order goes as the ultraviolet length scale raised to the power d minus two, divided by the number of species N. Since the stress tensor in theories with a number of fields N is essentially N times the stress tensor of each field, the factors of N cancel out to leave the bound exactly as strong as in theories with fewer fields. Note that the stress energy tensor here is contracted with the tangent vector to the null geodesic so the expression is invariant under reparametrization of the affine parameter.

We expect SNEC to be valid in the context of semi-classical gravity. Here, by semi-classical gravity, we mean a treatment of classical gravity coupled to quantum fields, where the metric is sourced by the expectation value of the stress tensor. This description can certainly be valid in quantum states with more than two particles, but it breaks down when the geometry is highly curved, or when the stress tensor has large quantum fluctuations on the characteristic distance scales relevant to the solution. A detailed understanding of the regime of validity of the semi-classical approximation is an interesting question, but not the focus of this work.

SNEC was proven, and the coefficient B fixed, by Leichenauer and Levine within the framework of induced gravity on a brane. The condition can be obtained from the one proposed in earlier work, equation (4), by setting g squared equal to the earlier smearing function f.

In this paper, we

• Motivate the precise form of the bound shown above.
• Argue, by examining examples, that this bound may remain valid when smearing over distances of order the radius of curvature or larger. Explicit examples we consider are evaporating black holes; the traversable wormhole of Maldacena, Milekhin, and Popov; and the counter-example to bounds of this form given by Fewster and Roman.
• Prove a singularity theorem, using SNEC as the assumption, which applies to real black holes.

In addition, in Appendix A we prove the two-dimensional conformal-field-theory Quantum Energy Inequality of Fewster and Hollands, in spacetimes globally conformal to Minkowski and for arbitrary smearing functions.

We begin in Section 2, where we provide motivation for SNEC using an argument of Wall. We also show how a Lorentz invariant ultraviolet cutoff evades the Fewster-Roman no-go result for bounds of this form. In Section 3 we apply the bound to a variety of situations, starting with the ANEC limit and evaporating black holes. In Section 3.3 we closely examine the Maldacena–Milekhin–Popov wormhole and apply the SNEC bounds both in four and two dimensions. In Section 4 we prove a semiclassical singularity theorem using SNEC as an assumption and apply it to evaporating black holes, comparing our results to other approaches. We conclude in Section 5 with discussion and ideas for future work.

We work in units where the speed of light and the reduced Planck constant are both one, use metric signature plus, minus, minus, minus, and assume d spacetime dimensions unless otherwise stated.

Relation to previous work

Quantum field theories often obey energy conditions averaged over an entire geodesic and quantum energy inequalities (QEIs), lower bounds on the average renormalized stress-energy tensor in some region. The latter were introduced by Ford and since then have been derived for scalar and fermionic fields in flat and curved spacetimes. In terms of interacting fields little progress has been made with concrete examples only in two dimensions.

ANEC. The previous examples of QEIs are all for averages over a timelike curve. The averaged null energy condition (ANEC) — equation (5), that the integral along a complete null geodesic of the expectation value of the null-contracted stress-energy tensor is greater than or equal to zero — is the most common example of a null averaged energy condition. It is believed that the self-consistent achronal ANEC is a fundamental property of physical matter at least in the semiclassical context. All known violations of the self-consistent achronal ANEC involve Planck length distances, outside the range of validity of the semiclassical approximation. There are numerous examples of proofs of the achronal ANEC. To mention some, Kontou and Olum proved the achronal ANEC for free scalar fields in curved spacetimes while Flanagan and Wald provided the first proof of ANEC in Minkowski space that includes backreaction up to second order in perturbation theory. In a different approach, Wall derived ANEC from the generalized second law of thermodynamics and more recently Faulkner and collaborators derived ANEC using modular Hamiltonians for general fields. Using similar methods Rosso proved ANEC for general quantum field theories in the near horizon geometry of spherical extremal black holes.

QNEC. Our energy condition and resulting singularity theorem differ in nature from the beautiful results based on the generalized entropy. Bousso and collaborators introduced the quantum null energy condition (QNEC). This is a lower bound on the expectation value of the null-contracted stress tensor at a single point p; the bound is computed from the von Neumann entropy, in a region whose boundary contains the point p. In particular, equation (6) states that this expectation value is bounded below by the second functional derivative of the entropy with respect to deformations of the region in the null direction at p, divided by two pi times the transverse area element. QNEC arises from the quantum focusing conjecture and it was proven for relativistic quantum field theories in Minkowski spacetime of two or more dimensions. QNEC is a bound on a local quantity, at the cost of introducing a somewhat complicated state-dependent quantity on the right hand side of the inequality.

Our bound is simpler, in the sense that the smeared stress tensor is bounded by a c-number, while in the case of the QNEC, the quantity appearing on the right side is the derivative of the entanglement entropy, which depends on the quantum state. As a result, we can prove simpler singularity theorems: our singularity theorem says that an initial surface with sufficiently negative contraction implies the existence of a singularity, whereas the singularity theorems in the generalized entropy approach refer to the generalized contraction, which to our knowledge is not an observable.

Another important difference between QNEC and SNEC is that the operator appearing in SNEC involves averaging the stress tensor with a smearing function. If the smearing function is not smooth enough, the bound diverges and the operator is unbounded. In contrast, the integrated version of the QNEC places a bound in the case that the smearing function is a top-hat function; in this case our bound diverges. So the bottom line is that the smeared operators we consider are rather different from the quantity that is bounded by QNEC, even though they appear similar at first glance.

However, our simpler theorem also comes at a cost: for an evaporating Schwarzschild black hole, our singularity theorem is weaker, in the sense that one must go deeper inside the horizon to find a sufficiently trapped surface. The generalized entropy approach allows one to deduce a singularity theorem at the location of the quantum extremal surface, which is typically only an ultraviolet length scale inside the black hole, while our theorem requires us to go a fraction of the Schwarzschild radius inside.

Null QEIs. But what about a null-averaged QEI? Flanagan provided the first example in two dimensions in flat and curved spacetimes while Fewster and Hollands derived a similar bound for classes of interacting conformal field theories (CFTs). Unlike the timelike case it is not straightforward to generalize these results in four dimensions. In fact Fewster and Roman argued using an explicit counterexample that weighted averages of the null-contracted stress-energy tensor along null geodesics in four dimensions are unbounded from below. The authors considered a sequence of vacuum-plus-two-particle states. The three-momenta of the excited modes become more and more parallel to the spatial part of the null vector that the stress-energy tensor was contracted with, making the lower bound of the energy density divergent. The introduction of SNEC overcame that problem by introducing an ultraviolet momentum cutoff.

Applications. Energy conditions have several interesting applications. One of the most important is restrictions on exotic spacetimes, for example those with wormholes. It was shown that achronal ANEC is sufficient to rule out so called short wormholes. In those it takes longer to travel through the ambient space than the wormhole, creating a shortcut in spacetime and thus allowing for causality violations. However, long wormholes, those where it takes longer to travel through the throat than outside, are allowed. In the past few years there are examples of those kinds of wormholes, with perhaps the most famous being the one proposed by Maldacena, Milekhin and Popov. Even though achronal ANEC is obeyed in this situation it was not clear if a stronger bound like SNEC is also obeyed and whether that bound is saturated or not. We answer that question with bounds in both four and two dimensions.

A second major application of energy conditions is singularity theorems, which predict the formation of a singularity, defined in that context as the spacetime possessing at least one incomplete causal geodesic. The singularity theorems of Penrose and Hawking were the first to predict a singularity under general assumptions without restricting to symmetric spacetimes. Senovilla has described the skeleton of singularity theorems as a pattern theorem with three assumptions. An initial or boundary condition which establishes the initial focusing of a congruence of geodesics, an energy condition which ensures that the focusing effect continues, and a causality condition which removes the possibility of closed timelike curves. The contradiction of the geodesic focusing with the causality condition and their length extremization property leads to geodesic incompleteness. We will divide singularity theorems into Hawking-type and Penrose-type, depending on whether they demonstrate timelike or null geodesic incompleteness respectively.

The original singularity theorems used pointwise energy conditions such as the NEC, equation (1). To prove a singularity theorem semiclassically, it is necessary to have an energy condition obeyed by quantum fields. Early work generalized Penrose’s theorem using averaged energy conditions. Fewster and Galloway presented proofs of singularity theorems with conditions inspired by QEIs. More recently Fewster and Kontou proved singularity theorems with similar conditions using index form methods. Additionally they estimated the required initial contraction on a Cauchy surface to guarantee the formation of a focal point both in the case of timelike and null geodesics. However, their conditions are still not derived from a quantum field theory. In the timelike case the relevant condition is the quantum strong energy inequality, derived for the quantum scalar field by Fewster and Kontou. The same authors use this inequality to derive the first semiclassical singularity theorem in the timelike case. In this work we focus on null geodesics where the relevant condition is SNEC.

2. Motivation for SNEC

In this section we provide some motivation for SNEC, additional to that given in the two papers that introduced and proved it. First we sketch a proof for the quantum-field-theory version of SNEC in arbitrary dimensional Minkowski spacetime. Then we revisit the Fewster-Roman counterexample to show that the Lorentz invariant SNEC bound is obeyed in the class of vacuum plus two-particle states.

2.1 Derivation of SNEC using pencils

A nice motivation for the field theory version of SNEC comes from an argument due to Jackson Fliss. Here we give a quick overview since it motivates the precise version of the bound we will use.

Wall showed that free field theory in higher-dimensional Minkowski space factorizes on the lightsheet into a collection of two-dimensional theories living on pencils. More precisely, the null plane is divided into pencils with area a. The pencils are a null line regularized by a transversal area of dimension d minus two. Effectively, they are only one-dimensional objects. We assume that a is the smallest scale in the problem; modes with wavelength shorter than a raised to the power one over d minus two cannot be excited.

The precise constants in the bound we motivate here will depend on the details of the cutoff scheme, so we do not keep track of order one constants in this section. The theory on each pencil is a chiral two-dimensional CFT with central charge proportional to the number of fields in the higher dimensional theory; for N free scalars, the relation is equation (7), central charge equals N.

The higher dimensional stress tensor is then related to the two-dimensional stress tensor by equation (8), the four-dimensional component being the two-dimensional one divided by the pencil area a, where the equality is up to order one numbers. The SNEC quantity is then related to a two-dimensional CFT quantity, equation (9). Now we can use the two-dimensional CFT result of Fewster and Hollands, equation (10) — the smeared two-dimensional null energy is bounded below by minus the central charge divided by twelve pi, times the integral of the squared derivative of the smearing function — to obtain equation (11), in which the same bound appears with the central charge replaced by N and an extra factor of one over the pencil area a.

Since a raised to the power one over d minus two is the smallest allowed wavelength, it should be identified with the ultraviolet cutoff. Using the lore relating the ultraviolet cutoff to the Planck length, equation (12) states that Newton’s constant is smaller than or of order the ultraviolet length to the power d minus two divided by N, which is itself smaller than or of order a divided by N. The above relation implies the gravitational form of our bound, equation (13), which is the SNEC statement with the four B over Newton’s constant coefficient.

Note that this argument is not a general proof, since the decomposition into pencils works only in Minkowski spacetime for free or super-renormalizable theories. However, it does motivate the particular form of the right hand side. Other possible quantities could appear on the right hand side of the bound, including higher derivatives of the function g; we will focus on this proposal since it is the correct form for the cases where the pencil decomposition is valid. The extension of the proof to interacting quantum field theories is beyond the scope of this work but it could be possible using methods developed elsewhere.

2.2 The physical meaning of B

As we showed in the previous section, when we consider free fields on Minkowski spacetime the undefined overall constant is an order one number. The constant B in this case arises from the relation between the ultraviolet cutoff and the Planck length. To have B be an order one number we need to saturate the inequality of equation (14), namely that N times Newton’s constant is of order the ultraviolet length to the power d minus two. This was the case for the induced gravity proof of Leichenauer and Levine where they derived B equal to one over thirty-two pi.

However, it is well motivated to consider a B much smaller than one, as equation (14) is typically not saturated in controlled constructions. For example, in controlled string theory constructions the string scale and the Kaluza-Klein scale are both well below the Planck scale, and these scales instead set the ultraviolet cutoff of the theory.

Additionally, a controlled construction in an effective field theory with a given cutoff should not excite modes near the ultraviolet cutoff, and so should not saturate the field theory SNEC, and therefore cannot saturate the gravity SNEC. To summarize: when semi-classical gravity is well under control, B is much smaller than one.

Since SNEC is only proven for free fields in Minkowski we can use an undefined B in order to have an inequality valid for other fields and curvature. In these cases we can consider B as the smallest possible number in order to have a generally obeyed bound.

2.3 Fewster-Roman counterexample

Any proposed bound must first address the argument of Fewster and Roman. They constructed an explicit family of states in free quantum field theory that have arbitrarily negative values for the smeared null energy.

This counter-example is given in quantum field theory without gravity, so it is appropriate to compare their construction to the field theory version of our bound, equation (15): the smeared null energy is bounded below by minus a scheme-dependent number times N divided by the ultraviolet length to the power d minus two, times the integral of the squared derivative of the smearing function. Since the ultraviolet cutoff appears explicitly, the number on the right side depends on the precise cutoff scheme. However, it is still well-defined to ask whether a bound of this form holds. It is also interesting to look for bounds that survive the limit of vanishing ultraviolet length; we hope to return to this in the future.

In this subsection, we show that when an ultraviolet cutoff is imposed on the states of Fewster and Roman, our bound is respected. This point was also addressed in earlier work, but there the ultraviolet cutoff was imposed in a way that was not manifestly Lorentz-invariant. Here we improve the analysis by imposing a manifestly Lorentz-invariant ultraviolet cutoff.

The counterexample of Fewster and Roman makes use of a particular class of states, equation (16), which are a superposition of the vacuum and two-particle states. Here the momenta of the particles appearing in the two-particle state are controlled by a function of the two momenta. Fewster and Roman consider a one-parameter family of states labelled by a parameter alpha.

They consider the limit in which alpha goes to zero, and show that the smeared null energy diverges in this limit. Concretely, for a particular family of these functions, the smeared null energy takes the form of equation (17), and the state is parameterised by two further numbers which must obey the constraint of equation (18). Therefore, in the limit of small alpha, the smeared null energy behaves as equation (19), minus a constant times alpha raised to a positive power minus one half. This diverges as alpha goes to zero as long as that power is less than one half. Because the smearing function does not depend on alpha, the right hand side of SNEC is independent of alpha, so this family of states appears to violate SNEC.

We show now that imposing an ultraviolet cutoff regulates this divergence so that our bound is respected. The idea of the Fewster-Roman counterexample is to boost the momenta along the direction of the light ray of interest. The unboosted state has a maximum momentum which we call the centre of mass momentum, equation (20). As the state is boosted and alpha goes to zero, the momenta grow as the centre of mass momentum divided by alpha. The angle between the three-momentum and the null vector is given by equation (21), the cosine of the angle being one minus alpha, so that as alpha goes to zero, the momenta become nearly parallel with the null direction.

Once we have chosen a null ray, the magnitude of the transverse momentum is invariant under Lorentz transformations that preserve the null ray. Therefore, it is Lorentz invariant to impose a cutoff on the transverse momentum, equation (22): the transverse momentum is at most one over the ultraviolet length. In our earlier work the cutoff was imposed on the full three-momentum. Here, we restrict only the transverse directions of the momentum.

As alpha goes to zero, the momentum and also the negative part of the energy density go to infinity. Therefore, we are interested in small alpha. Expanding the angle relation gives equation (23), that alpha goes as the angle squared, while the transverse component of the momentum is equation (24), the momentum magnitude times the angle, which goes as the momentum times the square root of alpha. Recalling that the momentum magnitude is the centre of mass momentum divided by alpha, this becomes equation (25): the transverse momentum equals the centre of mass momentum divided by the square root of alpha. Where we previously required the full momentum to be smaller than the inverse cutoff length, now that we impose a cut-off only on the transverse momenta we obtain the relaxed condition of equation (26).

So finally, our ultraviolet cutoff imposes a cutoff on the boost parameter, equation (27): the square root of alpha must be greater than the centre of mass momentum times the ultraviolet length. Plugging this into the expression for the smeared null energy, we obtain equation (28): the smeared null energy is bounded below by minus the constant A divided by the product of the centre of mass momentum and the ultraviolet length.

Recall that the constant A and the centre of mass momentum depend only on the unboosted state. Therefore, boosting leads to a divergence going as one over the ultraviolet length, which is milder than the SNEC bound, which is proportional to one over the ultraviolet length squared. This shows that the Fewster-Roman technique of boosting does not lead to a violation of SNEC. One would still like to prove SNEC, including order one factors, in the theories considered by Fewster and Roman.

Frame independent cutoff. As an aside, note that one can derive the same condition in a way that is manifestly invariant under all Lorentz transformations, not just those that preserve the light ray of interest. Instead of bounding the transverse momentum, we impose that the centre of mass energy between any two momenta is smaller than the cutoff. Starting from the centre of mass energy of equation (29), and remembering that the cosine of the angle between the two momenta is one minus alpha, one finds equation (30), the centre of mass energy squared equal to the momentum squared times alpha; requiring this to be below the cutoff gives equation (31), the same condition on the square root of alpha as before.

3. Examples

In this section we apply the SNEC bound to various situations of interest. First we examine the infinite geodesic length limit which reduces the bound to the ANEC integral. Then we examine long segments in the case of evaporating black holes, a situation where the NEC is violated. Finally we apply SNEC to the long wormhole of Maldacena, Milekhin and Popov. This is an interesting example as the complete geodesics are chronal, so achronal ANEC cannot be applied. However, it is possible to apply SNEC on shorter achronal segments.

3.1 The ANEC limit

Here we confirm that the SNEC bound reduces to the ANEC in the limit of large support of the smearing function. We introduce a rescaled smearing function, equation (32), whose normalization does not depend on the choice of the scaling parameter. The bound then takes the form of equations (33) to (36), and taking the scaling parameter to infinity gives equation (37): the limit inferior of the smeared null energy is greater than or equal to zero. The left hand side is just the ANEC integral, and so we recover equation (38), the statement that the integral of the null energy along the complete geodesic is greater than or equal to zero.

The rescaling method was first used with specific Lorentzian functions and later for general ones to derive the averaged null, weak and strong energy conditions for classical and quantum fields. As an example we look at the normalized Lorentzian function of equations (39) to (42); using this function the bound becomes equation (43), and taking the scaling parameter to infinity gives the ANEC integral.

(The step-by-step evaluation of these integrals is omitted for length; the complete text is at the source.)

3.2 The evaporating black hole stress tensor

In previous work, we proposed that one should trust SNEC when smearing over null achronal segments of length much shorter than the radius of curvature. Here we provide some evidence supporting the generalization of our bound for longer regions of integration.

The case we examine is that of an evaporating black hole. In Schwarzschild geometry the radius of curvature is the radial coordinate times the square root of the radial coordinate divided by the Schwarzschild radius. Parametrizing the null geodesic with the radial coordinate and picking a segment from r to twice r, we see that near the horizon the length of the segment is comparable to the radius of curvature.

We proceed to show that the stress tensor of an evaporating black hole satisfies SNEC, integrated over achronal, null geodesic segments. The relevant component of the stress tensor is equation (44): minus the number of species N, divided by the Schwarzschild radius squared times the outgoing null coordinate squared. For our purposes, this stress tensor represents the negative energy density encountered by an observer traveling along a radial null geodesic outside of an evaporating black hole. We formulate our derivation as a theorem since stress tensors of this form are not specific to black holes, but they are in fact quite common.

Theorem 3.1. For any smearing function of continuity class one, and a stress tensor of the form of equation (44), SNEC is satisfied.

The stress tensor of equation (44) describes black holes that lie in the regime of the semiclassical approximation, that is, black holes for which the Schwarzschild radius squared is larger than the ultraviolet length squared, which is in turn larger than the number of species times Newton’s constant. This condition becomes relevant at the end of the proof.

Proof. Using the stress tensor of equation (44) in the bound and substituting an exponential variable, equations (45) to (47) follow. The second term on the right hand side of equation (47) is a total derivative, which gives a vanishing boundary term. The third term is manifestly positive. To conclude the proof, and assuming that B is an order one number, we need to show equation (48), that the number of species times Newton’s constant divided by the Schwarzschild radius squared is at most one — which is true for semiclassical black holes.

3.3 The Maldacena-Milekhin-Popov wormhole

In their recent work, Maldacena, Milekhin and Popov presented a wormhole solution in four dimensions. It is what we call a long wormhole, meaning that it takes longer for an observer to go between two points in spacetime if they travel inside the wormhole than if they travel in ambient space. Thus this wormhole does not lead to causality violations. The wormhole is a solution of an Einstein–Maxwell theory with charged massless fermions which give rise to negative energy, necessary for the existence of a wormhole. Here we examine the wormhole in relation to the SNEC bound. In particular we are interested in whether or not the wormhole solution saturates the proposed bound in four and two dimensions.

The wormhole interior is given as a first approximation by matching an anti-de Sitter times sphere geometry, equation (49), to the charged black hole geometry, with the matching conditions of equation (50). The parameter that appears there fixes the size of the mouth of the wormhole, and a free length scale is later identified as the length of the wormhole.

3.3.1 Four-dimensional bound

The full length of the geodesic inside the wormhole is chronal, since pi times the wormhole length is greater than the approximate distance between the mouths in ambient space. But how long is the maximum achronal segment measured in the dimensionless radial coordinate? The condition, equation (51), is that the length of the achronal segment has to be shorter than the rest of the geodesic inside the wormhole added to the distance between the mouths in ambient space. Working the two lengths out, equations (52) and (53), the condition becomes equation (54), a bound on the segment length in terms of the tangent of a quarter pi plus a term set by the ratio of the ambient distance to the wormhole length.

Inserting the maximum value of the ambient distance, extracted numerically from the relationship between the wormhole length and that distance in the original paper, we have that the segment length is smaller than 4.13. If we could approach the limit of a short wormhole, the segment length would be allowed to become larger.

The quantum contribution to the energy density in the throat was calculated by Maldacena, Milekhin and Popov and has the general form of equation (55), where the first term is a contribution from the conformal anomaly, and the second is the Casimir energy. The parameter q is the number of two-dimensional fermionic fields. A key idea here is that one four-dimensional field gives rise to a q much larger than one. Evaluating this at the maximum ambient distance gives equation (56), in which the energy density is minus q divided by twelve pi squared times the mouth size squared times the wormhole length squared, times a constant of order one.

Introducing the coordinates of equation (57), the SNEC bound takes the form of equations (58) and (59). The stress-energy tensor is a constant, so we define the dimensionless number C of equation (60), which given the relation of equation (61) between the mouth size, the number of species, the gauge coupling and the Planck length, goes as one over B times q. Rearranging gives equation (62), and for a Gaussian smearing function of width sigma, equation (63).

A direct computation of the integrals gives equation (64), an expression in the error function. To investigate its behavior we examine small and large values of the width: for width much smaller than one, C is bounded by one over the width squared, equation (65); for width much larger than one, C is bounded by a number of order one divided by the width, equation (66).

The width of the Gaussian is essentially the smearing length, that is, the length of the achronal segment on which one observes the negative energy density. The length of this segment for the wormhole is bounded by 4.13. Since q is the number of species, and one needs q to be much larger than one to be in the semiclassical limit — so as not to worry about quantum corrections that would destabilize the solution — SNEC is respected. That is true, however, provided that B is an order one number; if B is instead of order one over q the bound may indeed be saturated.

3.3.2 Two-dimensional bound

Although the wormhole may not saturate SNEC for most values of q, we prove here a bound in one space and one time dimension that does. To be able to study the wormhole using such a bound, we focus on the anti-de Sitter part of the geometry. Fewster and Hollands have proven the null smeared QEI of equation (67) for certain interacting CFTs in two-dimensional Minkowski spacetime, where the coefficient is the total central charge divided by forty-eight pi. Blanco and collaborators confirmed and slightly improved that bound using modular Hamiltonians. Surprisingly, this bound has yet to be generalized to curved space. Following work of Flanagan, we provide a simple proof of it for spacetimes globally conformal to Minkowski, and arbitrary smearing functions. The detailed proof is given in Appendix A. We should note that while the anti-de Sitter geometry is not globally conformal to Minkowski, the bound can be used approximately in the wormhole case as long as we stay away from the boundaries.

The two-dimensional stress tensor, equation (69), is the mouth size squared times the four-dimensional one and is proportional to q, and the bound becomes equation (70), in which the ratio q over the central charge appears. As we have mentioned, q is approximately proportional to the number of species of fields. In the CFTs of interest, this role is played by the central charge, so the ratio is of order one. As before we calculate the integrals for a Gaussian of width sigma to obtain equations (71) to (73). Since the ratio is an order one number, and the width is smaller than 4.13 since for a Gaussian it is equal to the length of the achronal segment, this two-dimensional bound can be easily saturated.

4. A Penrose-type singularity theorem

An important application of the SNEC bound is the proof of a singularity theorem. The question of whether the semi-classical effects of negative energy invalidate the singularity theorems, before quantum gravity effects become significant, was suspected to have a negative answer. Here we present a proof of a null singularity theorem using a condition obeyed by quantum fields.

Fewster and Kontou proved a singularity theorem for null geodesic incompleteness with the NEC replaced by a weaker condition inspired by QEIs using index form methods. After we state that condition we show that the SNEC bound is a bound of the same form. Then we specify the required initial condition that leads to geodesic incompleteness. Finally we examine a particular example, that of the spherically symmetric evaporating black hole.

Let P be a future converging achronal spacelike submanifold of the spacetime, of co-dimension two, with mean normal curvature vector field pointing along a future-pointing timelike unit vector. Then let gamma be a future-directed null geodesic emanating normally from P. As in all null geodesics we need to specify an affine parametrization for gamma. We extend the unit vector by parallel transporting along gamma. Next we choose an affine parameter on gamma such that the contraction of that unit vector with the tangent to gamma equals one. Then define the length of the geodesic with respect to that parameter. Now we can state the condition required by Fewster and Kontou, equation (74): the smeared null-contracted Ricci tensor is bounded below by minus a constant times the squared norm of the m-th derivative of the smearing function, minus another constant times the squared norm of the smearing function itself, where the norms are the usual square-integral norms of equation (75).

The SNEC inequality is a bound on the expectation value of components of the stress-energy tensor. But singularity theorems require a geometric assumption which, in the case of Penrose-type theorems, is a bound on the null-contracted Ricci tensor. Classically the Einstein equation connects curvature to the stress-energy tensor. Semiclassically, the semiclassical Einstein equation of equation (76) equates eight pi times Newton’s constant times the expectation value of the stress-energy tensor with the classical Einstein tensor. With the use of that equation we assume that we have a self-consistent solution, which includes a state and a metric that satisfy it. The SNEC bound can then be written as equation (77): the smeared null-contracted Ricci tensor is bounded below by minus thirty-two pi B times the squared norm of the derivative of the smearing function. This is a bound of the required form with m equal to one, the first constant equal to thirty-two pi B, and the second constant equal to zero.

4.1 Mean normal curvature

The Fewster–Kontou analysis has two scenarios to describe all possible initial conditions. In scenario 1, initially the NEC is satisfied for an affine length short compared to the one for the formation of a focal point. In scenario 2 this requirement is dropped and instead conditions are imposed on the null contracted Ricci tensor for small negative values of the affine parameter. For scenario 2, perhaps counterintuitively, negative null energy in this region leads to smaller required initial contraction because this negative energy must be over-compensated by positive energy, an effect known as quantum interest.

The goal is to find the mean normal curvature of the surface P required to have null geodesic incompleteness for our bound.

For scenario 1 we suppose that initially the NEC is satisfied, so let the null-contracted Ricci tensor be an initially positive function on an initial stretch of the geodesic. Then we can use Lemma 4.1 of Fewster and Kontou with the parameter values our bound supplies, which gives:

Lemma 4.1. For a null-contracted Ricci tensor satisfying our bound on the geodesic, if minus twice the mean normal curvature at the starting point is greater than or equal to a threshold value — equation (78), built from minus two-thirds of the initial curvature times the initial length, plus two over that length, plus a term that vanishes when the focal length is much larger than the initial length — then the geodesic contains a focal point before that focal length.

Assuming the focal length is much larger than the initial length we can discard the last term and obtain the simpler threshold of equation (79). Then for a given initial length and initial curvature we can calculate the required initial contraction to have a focal point before the focal length. As expected, for larger initial length and curvature, smaller initial contraction is required. However, when the threshold falls below three over the initial length, the original Penrose theorem should be used instead.

Turning to scenario 2 we drop the assumption that the NEC holds. We instead extend the geodesic backwards and assume that our bound holds on the extended geodesic. Defining the maximum of the null-contracted Ricci tensor on the extension, we can use Lemma 4.7 of Fewster and Kontou with the same parameter values, which gives:

Lemma 4.2. For a null-contracted Ricci tensor satisfying our bound on the extended geodesic, if minus twice the mean normal curvature is greater than or equal to the sum of two minimized functions defined by equations (81) to (83), then there is a focal point to P along the geodesic within the given range.

The first function is minimized at the full focal length. In the case where the second is minimized at the end of the backward extension, the condition becomes equation (84): minus twice the mean normal curvature is at least the first constant plus two, divided by the focal length, plus the first constant divided by the backward length, plus one third of the maximum curvature times the backward length.

4.2 Application to evaporating black holes

In his seminal work, Penrose proved the first singularity theorem which applies to a classical black hole spacetime. In an evaporating black hole spacetime, where the NEC is violated, the original Penrose theorem cannot be used. This provides us with the perfect opportunity to test the previous two Lemmas.

We assume that the metric is well-approximated by Schwarzschild geometry near the horizon; the metric inside the horizon is equation (85). As described previously, to fix the affine parametrization of the null geodesic we use the mean normal curvature vector field of the hypersurface P. We focus on spherically symmetric hypersurfaces P, so that the hypersurface is defined by Schwarzschild coordinates. Because the metric is static and spherically symmetric, the mean normal curvature vector field is purely in the radial direction, and the affine parameter takes the form of equation (86). The mean normal curvature of hypersurfaces with constant radius and time changes as we go further inside the black hole. Inside the horizon, the mean normal curvature of our surfaces is given by equation (87): minus one over the surface radius, times the square root of the Schwarzschild radius over the surface radius minus one.

We start with scenario 2. First we assume that the maximum curvature on the backward extension is negative, so that the NEC is violated everywhere on that stretch and we can drop the last term, obtaining equation (88), a condition on the mean normal curvature in terms of the focal length and the length over which the NEC is violated.

We need to pick the point where we start having NEC violation. To include the backward stretch, all null geodesics emanating normally from P need to be able to extend to that point. While ingoing radial null geodesics can be extended backwards to large radius, outgoing geodesics can only be extended back as far as the past horizon, where the Unruh state is singular. Physically, this region of the spacetime should be replaced by whatever matter collapsed to form the black hole.

Since the congruence of outgoing geodesics cannot be extended further than the event horizon, we define equation (89), the difference between the Schwarzschild radius and the surface radius as a fraction x of the Schwarzschild radius, with x between zero and one. (Figure 3: schematic representation of a Schwarzschild black hole and the parameters in scenario 2. The dashed circle is the constant-radius, constant-time hypersurface P. One distance is measured from the point where the NEC starts being violated, and the other is from P to the singularity, pictured at zero radius.) The focal length can have as a minimum the remaining coordinate distance, meaning that the singularity is located exactly at zero radius; we can also consider sending it to infinity, meaning we have no information about the location of the singularity.

We define a second fraction y by demanding that the affine distance corresponds to a coordinate distance y times the Schwarzschild radius, equation (90). Now we can pick a point where the mean normal curvature is smaller than the one required, and equating the two expressions for the mean normal curvature gives equation (91). The two extreme cases are the case where the singularity sits at the smallest possible distance, for which there are no solutions, and the limit of an unbounded focal length, for which x equals the first constant divided by two plus that constant. (Figure 4: required value of x to have a singularity for different values of y, for two values of the constant.)

The Leichenauer–Levine value of B equal to one over thirty-two pi translates to a first constant equal to one. Using this value, we find that the minimum x is one third. Therefore, we can prove the singularity theorem for any surface P whose radius is at most two-thirds of the Schwarzschild radius — equation (92). As we discussed in Section 2.2 there is also strong motivation to use a value of B much smaller than one and so a first constant much smaller than one. For small values of that constant, we have a singularity theorem for spheres whose radius differs from the Schwarzschild radius by half that constant times the Schwarzschild radius, equation (93).

The requirement that the NEC is violated for an affine parameter length comparable to the Schwarzschild radius is a strong requirement. However, it is not necessary. First we remember that scenario 2 works for both negative and positive maximum curvature. Analytic approximations of the null energy near a Schwarzschild black hole horizon show that the maximum values, positive or negative, do not exceed of order one hundred times Newton’s constant divided by the fourth power of the Schwarzschild radius for a single field, and N times that for N fields. To be in the semiclassical regime we assume that N times Newton’s constant is below the ultraviolet length squared. Astrophysical black holes have a Schwarzschild radius far larger than that length. Then the relevant comparison shows that the condition of Lemma 4.2 holds, and its last term is orders of magnitude smaller than the other terms. So even though in the case of positive maximum curvature we need a larger mean normal curvature on P, that contribution is negligible, and the previous discussion extends to cases where the NEC is not violated everywhere on the backward stretch with very small corrections.

Turning to scenario 1 and Lemma 4.1, we first note that since the initial curvature is positive we can discard the first term, giving equation (94). Here the initial length is the distance for which the NEC is obeyed and it is unknown, so we set it to a coordinate distance x times the Schwarzschild radius. The distance from P to zero radius is also unknown here so we set it equal to z times the Schwarzschild radius. (Figure 5: schematic representation of a Schwarzschild black hole and the parameters in scenario 1.) The distance for singularity formation varies from that distance to infinity so we use the variable y as before.

To apply Lemma 4.1 we need the required mean normal curvature to be smaller in absolute value than the geometric one at the corresponding radius. This gives equation (95), a relation between one over z, the first constant over twice x, and two plus that constant over twice the difference of y and x. There are three variables to vary: z, the distance of P from zero radius; x, the region over which the NEC is obeyed; and y, the distance to the singularity. Not all values of them are possible. First we note that there is no solution when y is at most z, that is, a singularity at most at zero radius. However, there are solutions for larger values of y. Second, the NEC should not be obeyed everywhere from P to zero radius, since then the original Penrose theorem applies. Third, the location of P is between zero radius and the Schwarzschild radius. Mathematically these conditions are equation (96). At different physical situations one might want to minimize x, a small range over which the NEC is obeyed, or maximize z, placing P closer to the horizon. As expected, for smaller values of the first constant it is easier to have both small x and large z. (Figure 6: density plot of the distance from zero radius to P in terms of x and y, for two values of the constant.)

Looking at both scenarios we can see that there are several physically interesting situations where the Lemmas presented can be applied. Specifically we can have a singularity when the NEC is violated for part of the black hole spacetime, a case where the Penrose theorem does not apply. Of course one could consider a situation where the NEC is violated close to the horizon and then obeyed with very large positive energies for only a short affine parameter, where our Lemmas do not apply. We should note though that this does not necessarily mean singularity avoidance. These cases could be minimized with a stronger SNEC bound and in particular with a small B.

4.3 Comparison to quantum singularity theorem

An alternative singularity theorem that allows NEC violations was proposed by Wall. It assumes the generalized second law and the existence of a quantum trapped surface, defined in terms of the generalized entropy.

It is not straightforward to compare the assumptions of the two theorems as the generalized second law is a very different condition from SNEC. What we can do is compare the location of the quantum trapped surface with the location of the surface with the required mean normal curvature.

Penington calculated the location of the quantum extremal surface of a spherically symmetric evaporating black hole, equation (97), and the location of the horizon with respect to the apparent horizon, equation (98). With the four-dimensional evaporation rate of equation (99), the quantum extremal surface sits inside the horizon by an amount given in equation (100), the number of species times Newton’s constant divided by forty-eight pi squared times the Schwarzschild radius. We can estimate the distance of the quantum extremal surface from the event horizon for semiclassical black holes, equation (101); for astrophysical black holes the fractional offset is expected to be related to the black hole entropy as its reciprocal, equation (102). We should note that this calculation is only valid for slowly evaporating black holes.

This is a much shorter distance from the classical horizon — the proper distance lies between the ultraviolet length and the Planck scale — compared to our estimates for a first constant equal to one.

It remains an open question why the quantum singularity theorem delivers a much stronger singularity theorem than SNEC. One possibility is that the quantum focusing theorem simply requires stronger assumptions; for example, the proof relies on the generalized second law, which is violated during the later stages of Hawking radiation. Another interesting possibility is that SNEC may be a sub-optimal bound, and that a stronger bound could be proven.

One benefit of our approach is that the singularity theorem proposed here gives an estimate for the location of the singularity where no information is given in the case of the quantum singularity theorem. A further comparison of the conditions of the two theorems and specifically the requirement of the generalized second law would be valuable.

5. Conclusion

In this work we have given both motivation and useful applications for the smeared null energy condition (SNEC). We showed that a field theory version of SNEC can be derived in Minkowski spacetime for free and super-renormalizable theories. This version of the bound addresses the counterexample of Fewster and Roman. We showed that SNEC reduces to the averaged null energy condition (ANEC) at the limit of large support, and that it is a valid bound over segments comparable to the radius of curvature for evaporating black holes. After detailed analysis, we concluded that SNEC is not saturated in the case of the Maldacena-Milekhin-Popov wormhole in four dimensions. However, we showed that this wormhole does saturate the analogous dimensionally reduced two-dimensional QEI whose curved space version we proved here. Finally, we proved a Penrose-type singularity theorem using SNEC and applied this theorem to establish that spacetimes that approximate the Schwarzschild solution near the horizon must contain a singularity.

Even though there is strong motivation for it, SNEC remains unproven in the case of general curved spacetimes and interacting fields. An important open question is whether or not there are correction terms to the SNEC bound dependent on the curvature. In the case of quantum energy inequalities over timelike curves such terms appear, but they do not in the null integrated bounds in two dimensions. The interacting case is more challenging as QEI results are few, but perhaps methods such as those used for proving ANEC could be useful here.

In general, SNEC can be used in scenarios when achronal ANEC cannot be used, since it can be applied to smaller achronal pieces of chronal curves. Examples include different traversable wormholes such as the one proposed by Gao, Jafferis and Wall. A different direction is the case of the area theorem and its tension with black hole evaporation. Comparison of SNEC with different proposed bounds such as the quantum null energy condition or the generalized second law are an interesting direction. In a different application, bounce cosmology scenarios often require the violation of singularity theorems, which is attributed to the violation of the NEC by quantum fields. Using a singularity theorem obeyed by quantum fields, these models can be critically re-examined.

We do not have an example where SNEC is saturated. Therefore, it may be that a stronger bound than SNEC is true. It would be interesting to find an example saturating SNEC, or to suggest a stronger bound. One example where a stronger bound was proven was a positivity result for incomplete but maximally extended achronal null geodesics in an anti-de Sitter times sphere geometry.

Finally, it would be very nice to derive a version of the field theory SNEC that is independent of the ultraviolet cutoff. This will require smearing over additional directions.

Acknowledgements

The authors would like to thank Srivatsan Balakrishnan, Thomas Faulkner, Chris Fewster, Eanna Flanagan, Jackson Fliss, Diego Hofman, Manthos Karydas, Juan Maldacena, Ken Olum and Alex Vilenkin for useful discussions.

Funding information. BF and E-AK are supported by the ERC Consolidator Grant QUANTIVIOL. This work is part of the Delta ITP consortium, a program of the NWO that is funded by the Dutch Ministry of Education, Culture and Science.

(Appendix A, the proof of the two-dimensional curved-spacetime null quantum energy inequality for interacting conformal field theories, and the reference list are omitted for length; the complete text is at the source.)