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Hawking-type singularity theorems for worldvolume energy inequalities

The classical singularity theorems of R. Penrose and S. Hawking from the 1960s show that, given a pointwise energy condition (and some causality as well as initial assumptions), spacetimes cannot be geodesically complete. Despite their great success, the theorems leave room for physically relevant improvements, especially regarding the classical energy conditions as essentially any quantum field theory necessarily violates them. While singularity theorems with weakened energy conditions exist for worldline integral bounds, so called worldvolume bounds are in some cases more applicable than the worldline ones, such as the case of some massive free fields. In this paper we study integral Ricci curvature bounds based on worldvolume quantum strong energy inequalities. Under the additional assumption of a - potentially very negative - global timelike Ricci curvature bound, a Hawking type singularity theorem is proven. Finally, we apply the theorem to a cosmological scenario proving past geodesic incompleteness in cases where the worldline theorem was inconclusive.


INTRODUCTION
The singularity theorems of Penrose [29] and Hawking [23] represent an important breakthrough in classical general relativity. They showed that gravitational systems will originate or terminate in a singularity, defined in that context as geodesic incompleteness, under general conditions. For a recent review article on singularity theorems, see [32]. Both theorems, along with mild causality conditions, require the non-negativity of contractions of the Ricci tensor. These conditions are assumed to represent properties of the matter content of the spacetime since, with the use of the Einstein Equation, they can be re-written as energy conditions, i.e. restrictions on contractions of the stress-energy tensor. The Hawking singularity theorem uses the timelike convergence condition where U µ is a timelike vector. With the use of the Einstein Equation it becomes the strong energy condition (SEC) where n is the spacetime dimension and T the trace of the stress-energy tensor. The quantity on the left is the effective energy density (EED), a term introduced in [4]. The SEC has an unclear physical interpretation [7,25] and is easily violated by classical fields, for example the free massive Klein-Gordon field. The situation is even worse when including quantum effects. It has long been known [8] that quantum fields can admit negative energies violating all pointwise energy conditions. Average energy conditions and quantum energy inequalities (QEIs) are weaker restrictions on the energy density (and similar quantities) or its renormalized expectation value in the case of quantum fields. Starting from the seminal work of Ford [16] QEIs have been derived for flat and curved spacetimes and a variety of mainly free fields (see Refs. [25], [9] and references within). They often take the form of where f is a smooth compactly supported real-valued function. Here ρ is the renormalized energy density, ω is a Hadamard state, and ||| · ||| a Sobolev norm. The integral is more often over a timelike geodesic but it can also be over a worldvolume.
With the use of the classical or the semiclassical Einstein Equation average energy conditions and QEIs become conditions on the Ricci tensor and may be used as alternative assumptions for singularity theorems. Starting with the work of Tipler [35] various authors have proven singularity theorems with weaker energy conditions [3,30]. First Fewster and Galloway [10] proved singularity theorems with a condition of the form of (3) using the Raychaudhuri equation. More recently Fewster and Kontou [12] presented proofs with a similar energy condition using index form methods and estimated the required initial contraction or mean normal curvature of a Cauchy surface. All of these works use energy conditions averaged over a single timelike or null geodesic. However, there is strong motivation to prove singularity theorems using worldvolume bounds instead. In the timelike case the worldvolume bounds are often more applicable than the worldline ones for both classical [4] and quantum fields [11]. In particular in the case of a massive scalar, the mass dependence can be removed in the case of worldvolume bounds. More important motivation to consider worldvolume inequalities arises in the null case. Fewster and Roman showed with an explicit counterexample [14] that QEIs over a null geodesic segment do not admit finite lower bounds. However, this is not the case if the integral is over two null directions as it was recently proven [15]. Using such an inequality as an assumption to a singularity theorem would lead to the proof of a semiclassical singularity theorem for null geodesic incompleteness. Previously only two semi-classical singularity theorems have been proven; one for the timelike incompleteness case [13] and one for the null incompleteness case [17] but the latter required the introduction of a theory dependent UV cutoff significantly weakening the result. The consequence of the classical singularity theorems is the existence of an incomplete causal geodesic. If one could simply extend the spacetime to a complete one this would in some sense signify that the incompleteness were not physically reasonable as there would be another better suited spacetime to model the situation. Of course, the singularity theorems also imply that one cannot accomplish this with a C 2 -Lorentzian metric without violating either the energy or causality condition. Therefore the question arises whether one could instead resolve this issue by extending with a lower regularity Lorentzian metric, i.e. given a C 2 -spacetime satisfying the assumptions of the singularity theorems, could there be a low regularity extension of it, satisfying appropriate generalizations of the assumptions but which does not predict a singularity. In other words could the singularity predicted by the smooth theorems merely be due to a drop in regularity (which could be akin to a jump or delta distribution in the curvature)? This question is essentially equivalent to whether singularity theorems continue to hold for lower regularity metrics which was already raised and extensively discussed in [24,Sec. 8.4].
Nevertheless it has only recently become rigorously tractable mathematically (cf. [19,21,[26][27][28]31]) and trying to resolve it fully continues to be an active pursuit of major physical importance because a plethora of physical models involve spacetime metrics which are not C 2 (e.g., the Oppenheimer-Snyder model of a collapsing star, matched spacetimes, gravitational shock waves, thin mass shells, . . . ). For a recent comprehensive review of this research area see [34]. The current proofs of low-regularity singularity theorems commonly employ an approximation based approach aiming to distill and exploit some stability properties of both the conditions in the singularity theorems and the formation or non-formation of singularities. Note however that already within the class of smooth metrics, Ricci curvature being nonnegative along timelike or null geodesics is unstable: Even if a metric g 0 satisfies the timelike or null convergence condition, metrics g ε converging to g 0 will in general violate these conditions by some small error, with the exact nature of the introduced error depending on the topology of the convergence. If g 0 is a non-smooth metric satisfying only a distributional version of the strong or null energy condition, then the possible topologies of convergence and therefore the form of the energy condition recovered for g ε are limited by the initial regularity of g 0 . The current state of the art regarding low-regularity singularity theorems [19,26] considers g 0 ∈ C 1 . It was shown that in this case -for a family of approximations g ε constructed in a very specific way -convergence is strong enough, but only barely, to recover a classical energy condition for the approximating g ε up to "pointwise small" violations. That is one obtains a condition of the form Ric[g ε ](X, X) ≥ −δ ε for all g ε -timelike/null vectors X contained in some compact subset K ⊆ T M , where 0 < δ ε ≡ δ ε (K) → 0. Such an energy condition is still amenable to classical Riccati or index form techniques to show that (for small enough δ ε and large enough K depending on the given initial condition) certain timelike geodesics must encounter focal points before a fixed time which then can be leveraged to obtain that, in the limit, a singularity must exist for the original metric.
However, if one were only to obtain weaker convergence one expects that such pointwise estimates cannot be recovered from the distributional energy conditions placed on g 0 and that one instead will have to make do with integral bounds of some form, such as the L 1 -bound we study in Theorem 4.1.
In this work we will address the issues arising from starting with a worldvolume bound by translating these bounds into bounds along single timelike geodesics by using ideas inspired by the so-called segment inequality from Riemannian geometry due to Cheeger-Colding [5] as well as an L 1 -Ricci bound based Myers theorem by Sprouse [33]. More specifically, we consider volume integral bounds of the form see Sec. 2 for definitions and Sec. 4 for precise statements of our results. We then reduce these inequalities to inequalities along single timelike geodesics: By standard results from Lorentzian geometry the normal exponential map provides a diffeomorphism from a neighbourhood of the normal bundle of Σ to a neighbourhood of Σ. In such a region the evolution of the surface is well defined and one can use its geometric properties (like the infinitesimal area element and the mean curvature). However, as soon as the normal geodesics to Σ encounter cut points, the evolution surfaces will degenerate or form caustics. Hence we will have to restrict our analysis to regions where Σ has a well defined evolution. This will in turn allow us, using exponential coordinates, to naturally split a volume integral into an iterated integral along geodesics of the congruence and over a subset of Σ. However, the volume element will not be constant in time in these coordinates so the integrand in each line integral will not only involve the Ricci curvature along the geodesic but an additional geometric factor determined by the time evolution of the area element of Σ. To be able to deal with this additional factor, we need to develop new estimates on the time evolution of the area element from below resp. backwards in time. From there on we are able to extract bounds on a single line integral and use classical index form methods to deduce geodesic focusing given an appropriate initial mean curvature condition on Σ.
This paper is organized as follows. In Sec. 2 we give a series of definitions and known results on spacetime geometry that will be needed later. In Sec. 3 we provide area comparison results that are used to derive bounds on integrals along geodesics from worldvolume integrals. In Sec. 4 we present the main result of this paper, a singularity theorem with a worldvolume integrated energy condition as an assumption. In Sec. 5 we present a physical application of our theorem for the case of the classical non-minimally coupled scalar field. In Sec. 6 we apply the theorem in the case of a simple cosmological model proving past timelike geodesic incompleteness for a few different fields. We conclude in Sec. 7 with a summary and discussion of future work.

Notation and Conventions
Let us collect some remarks on the notation and conventions that we will use throughout. First, a spacetime is a pair (M, g) where M is a 2nd countable, Hausdorff smooth manifold, and g is a (smooth) metric on M with signature (−, +, . . . , +). The D'Alembertian operator with respect to the metric g is defined as g := −g µν ∇ µ ∇ ν . Note that this sign convention for g is consistent with the physics literature references of Sec. 5-6, but differs from the one used in most of the references for Sec. 2-4. The Riemann curvature tensor of g is A Cauchy surface in a spacetime is a set which is uniquely met by each inextendible causal curve. Given a spacelike hypersurface Σ of a spacetime (M, g), the second fundamental form is the map II : For the mean curvature of a spacelike hypersurface, we adopt the conventions in [37], i.e. if n is the future unit normal to the hypersurface, then the mean curvature is given by H = tr(∇ n).
We use geometric units throughout this work (i.e. G = c = 1) except for Section 6. Let us note that we will employ index notation in Section 5, as is customary in the physical literature.

GEOMETRIC BACKGROUND
In this section, we collect several well-known results from spacetime geometry which will be needed for what follows. Throughout, let (M, g) be a smooth spacetime of dimension n and let Σ ⊂ M be a smooth, spacelike Cauchy surface with future timelike unit normal n. We will often only consider future versions in definitions and results, past versions are obtained by duality. As we will see in this section the exponential map not only yields a diffeomorphism from a neighbourhood of its normal bundle to a neighbourhood of our spacelike Cauchy surface Σ, it also provides us with a foliation of that domain by hypersurfaces perpendicular to the congruence of normal vector fields emanating from Σ. This is guaranteed as long as the geodesic congruence does not encounter cut points, as then the exponential map no longer is a diffeomorphism. We will mostly reference [36] for proofs of the results presented in this section, for an alternative textbook reference see [2,Sec. 9]. Since we will have to deal with the case where M is not future geodesically complete, let [0, s + (x)) denote the maximal interval of definition for the unique geodesic γ with initial data γ(0) = x,γ(0) = n x . Note that the future normal bundle of Σ is of the form Let us now recall the definition of (future) cut points and the (future) cut function.
If c + Σ (x) < s + (x) for some x ∈ Σ, we call the point exp x (c + Σ (x) n x ) a future cut point of Σ. We write Cut + (Σ) for the set of future cut points of Σ. Remark 2.5. As shown in the reference given above, the set D + is explicitly given by Lemma 2.6. Consider the diffeomorphism exp + Σ : D + → M + . Then for any t ≥ 0 the sets ({t} × Σ) ∩ D + ⊂ D + and for any t ≥ 0, S t := {τ Σ = t} ∩ M + ⊂ M + are either empty or embedded hypersurfaces (in the respective supersets). Moreover, if nonempty, S t is spacelike and the restriction of the exponential map exp Proof. Since τ is smooth on M + with timelike gradient by [36, Prop. 3.2.33], S t is an embedded, spacelike hypersurface. Since exp + Σ : Lemma 2.7. Consider the diffeomorphism exp + Σ : D + → M + . Then the pullback measure on D + of the volume measure vol g on M + under this diffeomorphism (i.e. the standard pushforward measure under its inverse) is of the form where σ is the Riemannian measure on Σ. The infinitesimal area A = A(t, x) : Proof. This is basically the "measure version" of the coarea formula, cf. [37,Prop. 2.10]. The coarea formula implies that for any f ∈ L 1 (J + (Σ)), By definition of the pushforward of a measure, we get that Since the volume form is locally an n-fold wedge, the pushforward of Reinserting this into the previous calculation, and then that back into the coarea formula, we get Since this holds for all integrable f , the claim about the measures follows.
Let us recapitulate: For any f ∈ L 1 (J + (Σ)), we have The same relation is true if we integrate over any measurable subset C ⊂ D + and exp + Σ (C) ⊂ M + .
Remark 2.8. If we write g(t, x) = g(exp + Σ (t, x)), then, upon choosing a chart in Σ, it is easily seen that , in particular A is smooth in both variables. Later, we will be interested in proving comparison results for the infinitesimal area A(t, x), which then directly imply more or less identical results for det g(t, x).
We are interested in regions of Σ such that the normal geodesics emanating from there are free of cut points up to a certain length. Working with such regions, rather than Σ, will allow us to use the normal exponential map in our integral calculations.
2. For T, η > 0, the set of T + η-regular points Reg + η (T ) ⊂ Σ is defined to be the set of Usually, η will be much smaller than T and is there to guarantee a lack of cut points for a short time beyond T .
2. It is easy to see that for any B ⊆ Reg + η (T ), i.e. U assigns to p ∈ M + the tangent vector (at p) of the unique normal unit speed geodesic from Σ that passes through p.

FORWARD AND BACKWARD COMPARISON RESULTS
In the following, we discuss comparison results for the infinitesimal area A(t, x) as well as the d'Alembert operator of the (signed) time separation. As they will be used in various estimates, we give a detailed treatment of all the comparison constants. In addition to the assumptions from the previous section, we assume that (M, g) satisfies Ric g (X, X) ≥ nκ for all unit timelike vectors X and some κ < 0. Further the mean curvature H of Σ with respect to the future normal n satisfies H ≤ β < 0.
We start by proving an infinitesimal version of an area comparison result given in [37,Thm. 4.6], which allows for a direct estimation of the time evolution of the area element by the modified distance functions of constant curvature spaces. The result shows that given a lower Ricci bound and a bound on the mean curvature of the Cauchy surface Σ, the area element of the evolving surface can be given an upper bound.
where H(x) denotes the forward mean curvature of Σ at x (i.e. the mean curvature with respect to the future normal n x ) and f κ,β is the comparison warping function (cf. [20, Tab. 1]).
Proof. The second inequality is obvious since H ≤ β. In the following, denote by S A,t the whereβ is any bound of H on A and σ t is the Riemannian measure on S t . The proof of Now suppose (t, x) ∈ D + . Choose a sequence of open, relatively compact Σ-balls U i around x that collapse to {x}, then eventually {t} × U i ⊂ D + and by continuity which, together with the comparison result referenced above (takingβ i = max x∈U i H(x)), proves the claim.
1. The proof of Lemma 3.1 is based on a result in [37,Thm. 4.6], which relies on future causal completeness of the surface Σ in order to conclude that any point in the evolution S A,t is reached by a maximizing timelike geodesic from A. In our case the latter is guaranteed by Lemma 2.6 so the result holds without the assumptions of Σ being future causally complete or Cauchy. This will be essential in our backwards comparison result later, as the future comparison will be applied to the evolution of Σ, which may not be Cauchy or FCC even if Σ is.
In order to apply the area comparison results and prove a segment type inequality, we need to be able to estimate the area element of S t from below. This is done in the next Lemma and uses what we call backwards comparison. This simply means that instead of estimating the area element from above in terms of κ and the mean curvature of Σ, we use the argument backwards (time reversed) from the evolution of Σ at time T . As we still have control over the mean curvature at that time, we are able to estimate the area. Importantly, this procedure will result in a (non-trivial) bound on the area element from below in terms of κ, η and T .
For the following let H : M + → R denote the map assigning to each p ∈ M + the future mean curvature of the smooth spacelike hypersurface S τ Σ (p) at the point p. Clearly, H| Σ is then the mean curvature of Σ, hence there is no confusion in denoting both by the same symbol H.
Proof. The first inequality follows from changing the time orientation and running the argument of Lemma 3.1 backwards starting at exp x (t n x ).
Closely connected to the evolution of the area element is the d'Alembertian of the time separation function τ Σ . Heuristically speaking it measures the expansion of the volume spanned by neighbouring geodesics of the congruence and usually comparison results for the d'Alembertian are established first and then used to show area comparison results. However, as we are basing our arguments upon already established comparison results we can treat both cases independently. As was the case for the area, it allows for an estimation from above with respect to the constant curvature comparison space which we include and which was shown in [37,Thm. 4.5]. The lower bound again follows from a backwards comparison argument.
Lemma 3.5. (Backward d'Alembert comparison) For any T, η > 0, t ∈ [0, T ] and x ∈ Reg + η (T ), we have that where τ S T is the time separation from the spacelike hypersurface S T measured backwards.
Proof. This was shown in the proof of Lemma 3.3.
For convenience, by making the bound on H weaker if necessary, we can exclude the case β = −(n − 1) |κ| as well. Checking [20, Tab. 1] yields Hence, after some manipulations, the forward infinitesimal area comparison constant C A+ = C A+ (n, κ, β, T ) can be written as Next, we want to derive an explicit form of the infinitesimal backwards area comparison constant C A− = C A− (n, κ, T, η), cf. Lemma 3.3. First note that β(n, κ, η) Insertingβ = (n − 1) |κ| coth η |κ| , we get Next we consider the forward d'Alembert comparison constant C + = C + (n, κ, β, T ). By monotonicity of the comparison function H κ,β , we get The backward d'Alembert comparison constant C − = C − (n, κ, η) was already given in Lemma 3.5, namely Note that, contrary to C A+ , C A− and C − which are always strictly positive, C + can be negative, zero or positive depending on n, κ, β, T .
where g t is the restriction of g to the spacelike hypersurface S t ⊂ M + . We choose coordinates x = (x j ) on Σ, and thus also coordinates (t, x j ) = (y µ ) on M + . Now suppose F is a smooth function on M + . We writeḞ = ∂ t F . Then For the time separation, we have that One then immediately checks Similarly, the backward time separation τ S T on the T -slice satisfies (for t ≤ T ) As before, Hence, d'Alembert comparison yields on Ω + T (Reg + η (T )).
Using the comparison results from the previous section one can derive estimates for integrals along geodesics from estimates on volume integrals. The approach taken is similar in spirit to the segment inequality used in the almost splitting theorem by Cheeger-Colding ( [5]; see also [33]). This technique is then used to obtain Ricci bounds along single geodesics from world volume inequalities, i.e. certain curvature integrals over a spacetime volume. To that end we define a map F T f assigning to a point x on Σ the integral of a non-negative f over the future directed normal geodesic to Σ starting at x up to a time T .
Next, we give a segment type inequality, i.e. a result inferring bounds on certain finite line integrals of f from a region B ⊆ Reg + η from bounds on the integral of f over the "tube" Ω + T (B).
where C A− = C A− (n, κ, T, η) is the backward area comparison constant and σ is the (Riemannian) volume measure on Σ.
Proof. Using everything discussed so far in a straightforward calculation:

THE SINGULARITY THEOREMS
We are now ready to present a typical singularity theorem which can be proven using the techniques we have developed. We decompose a function f into a sum of its positive and negative parts: f = f + + f − where f + ≥ 0 and f − ≤ 0 and write Ric − (U, U ) for the negative part of p → Ric(U p , U p ). We shall first state a singularity-theorem adjacent result which has the advantage of following very directly from Proposition 3.9.
Theorem 4.1. Let (M, g) be a globally hyperbolic spacetime with a smooth, spacelike for any T > 0, η > 0, K > 0 satisfying K < |β| − n−1 T , then B ⊆ Reg + η (T ). Proof. Assume to the contrary that B ⊂ Reg + η (T ). Set f (p) := | Ric − (U p , U p )|, then Proposition 3.9 and the assumed estimate (47) imply that Let ε > 0 be so small that K + ε < |β|, then there exists x ∈ B such that Now the result follows directly from classical variational methods: Denoting by c(t) := exp x (t n x ) the normal geodesic starting in x, by assumption c maximizes the Σ-time separation up to p := exp x (T n x ) ∈ I + (Σ). Let E i (t) be spacelike parallel orthonormal vector fields along c such that {ċ, E 1 , . . . , E n−1 } is an orthonormal frame along c. Consider variations of c starting in Σ and ending in p defined by variational vector fields V i (t) : Denote by L i (t) the length functional of the corresponding variation. Using the second variation formula it is a straightforward calculation (done e.g. in section 2 of [12] with a different sign convention) to arrive at Note that the first term on the right hand side is equal to −H(x) (recall that we follow the conventions of [37]). Since c maximizes the Σ-time separation up to p, we get which implies that Taking ε → 0 we see that this contradicts our assumption on the relationship between T , K and |β|.
While this is not a singularity theorem per se, as the conclusion of B ⊆ Reg + η only tells us that either a geodesic stops existing or maximizing the distance at or before T + η, its structure is very similar: Given the usual assumptions of singularity theorems (on Ric, causality and the mean curvature of a Cauchy surface) it implies bounds on regions evolving from Σ on which exp stays injective (even a diffeomorphism).
Let us now state a singularity theorem which still employs the segment inequality type methods, but assumes energy conditions inspired by QEIs. It will be used to prove geodesic incompleteness for the case of the non-minimally coupled classical scalar field in Sec. 5. (i) There is κ < 0 such that Ric(v, v) ≥ nκ for all unit timelike v ∈ T M .
(ii) There exist Q 1 , Q 2 > 0 such that for all F ∈ C ∞ c (M + ), we have the following integral bound on the Ricci tensor: Then no future-directed timelike curve emanating from Σ has length greater than τ and (M, g) is future timelike geodesically incomplete.
Let us therefore assume that min((n−1) |κ| coth |κ|τ , ν * ) = ν * . Now assume for a contradiction that there is a future-directed timelike curve emanating from Σ of length greater than τ = T + η. Then Reg + η (T ) is nonempty. Given any f ∈ C ∞ c (R) with support contained in [0, T ], we pick an arbitrary, non-zero h ∈ C ∞ c (Σ) with support contained in Reg + η (T ) and consider the function By d'Alembert comparison, we can estimate |∂ t A · A −1 | ≤ C max (n, κ, η) := max(|C + (n, κ, η)|, C − (n, κ, η)) = C − (n, κ, η), as in Remark 3.7. Thus, dropping the dependence of C max on n, κ, η for now, and using the classical inequality ab ≤ εa 2 + 1/(4ε)b 2 for a, b ≥ 0 and ε := T /2 On the other hand, . Since h has compact support in Reg + η (T ), the supremum on the right hand side is actually a maximum. Canceling the h 2 L 2 (Σ) , by (53) we are left with Fix x 0 ∈ supp(h) where the minimum is achieved and let c : [0, T ] → M be the distance-to-Σ-maximizing geodesic c x 0 starting orthogonally to Σ at x 0 . Since f is still free (the only restriction is that its support is contained in [0, T ]), we fix any τ 0 ∈ (0, T ) and -following the definitions in [12,Lem. 4 With this choice we closely follow the methods used in [12], especially Proposition 2.2 and partly Lemma 4.1 there and will refer to that article for details of the index form methods applied. Note that f is piece-wise linear vanishing at 0 and T , so even though f is not smooth all our above calculations, in particular (53) and (60), still hold (this is easily seen by approximating f with compactly supported smooth functions whose derivatives converge to the derivative of f in L 2 ). The next step is to estimate from above (note the different sign convention for Ric in [12]). Using the identityf 2 = (φf ) 2 + (1 − φ) 2 , we obtain: where we estimated Ric(ċ,ċ) ≥ nκ and integrated the remaining terms involving φ.
Since this works for any τ 0 ∈ (0, T ) and any η, T > 0 such that T + η = τ , we may chooseτ 0 andη + T such that ντ 0 ,η becomes minimal (note that such a minimizing τ 0 exists in the open interval because the expression goes to infinity as τ 0 → 0, T ). This leads to J[f ] ≤ ντ 0 ,η = ν * ≤ −H| x 0 by the assumption (55) and there must be a focal point along c which in turn cannot be maximizing up to time τ = T + η, contradicting x 0 ∈ Reg + η (T ). Remark 4.3.
1. The proof of Theorem 4.2 really only needs the integral bound (53) for all F ∈ C ∞ c (Ω + T (Reg + η (T ))). However, as that set is in general hard to control we chose to formulate Theorem 4.2 as is.
2. In the formulation of Theorem 4.2 higher derivatives of F could be accommodated for at the price of imposing bounds on the positive and negative parts individually (writing a function b as Then one can proceed as above to prove the existence of a singularity: Indeed, choosing F to be of the form F (t, x) = f (t)h(x) as before -just without the extra factor of A(t, x) − 1 2 -, one estimates Ric + F 2 dvol g forwards (using forward comparison) and backwards (using the assumption as well as backward comparison), and similarly for Ric − F 2 dvol g , to get an estimate on inf(− Ric f 2 ) as before. From these estimates, it is then clear by following [12] how ν * has to be chosen, and the rest of the proof is completely analogous to Theorem 4.2.

THE STRONG ENERGY INEQUALITY
In this section we want to show that the inequality (53) is satisfied by the non-minimally coupled classical scalar field, an example that violates the SEC. We additionally discuss the advantages of this bound compared to the worldline one.

The bound on the effective energy density
The quantity of interest is the effective energy density (EED) given in Eq. (2). The field equation for non-minimally coupled scalar fields is where ξ is the coupling constant and R is the Ricci scalar. The constant m has dimensions of inverse length, which would be the inverse Compton wavelength if one regarded (64) as the starting-point for a quantum field theory with massive particles. The Lagrangian density is where (∇φ) 2 = −g µν (∇ µ φ)(∇ ν φ). The stress energy tensor is obtained by varying the action with respect to the metric, giving where G µν is the Einstein tensor. We should observe here that the field equation and the Lagrangian reduce to those of minimal coupling for flat spacetimes but the stress-energy tensor does not.
The trace of the stress-energy tensor is given by The field equation allows us to write the stress energy tensor without mass dependence as where P ξ = g + m 2 + ξR is the Klein-Gordon operator. We also used the identity Alternatively we can remove the dependence on the D'Alembertian operator Similarly for the trace, without explicit mass dependence we have while without dependence on the D'Alembertian operator The EED without mass dependence is obtained by combining Eqs. (2), (68) and (71) where we assumed U µ is a unit timelike vector. We get the EED without dependence on the D'Alembertian by combining Eqs. (2), (70) and (72) The last term can be discarded "on shell" i.e. for φ satisfying Eq. (64). For ξ = 0 the EED further reduces to We want to bound the following quantity where x is a spacetime variable, F (x) is a smearing function with compact support and the integral is over a spacetime volume In [4] the authors derived two bounds for this quantity from Eqs. (73) and (74). Unlike the worldline case, for worldvolume averaging we can use successive integration-by-parts to derive two bounds, one of those that has no explicit mass-dependence and remains free from any field derivatives. Introducing V µ = F (x)U µ , the averaged EED for the nonminimally coupled scalar field "on shell" obeys the following theorem [4, Thm. 2].
Theorem 5.1. If M is a manifold with metric g and dimension n ≥ 3, T µν the stress-energy tensor of a scalar field with coupling constant ξ ∈ [0, ξ v ] and f a real valued function on M with compact support, then "on shell " where and Here where ξ v < 2ξ c for any spacetime dimension n > 2, while ξ c < ξ v for n ≥ 4. By ξ c we denote the conformal coupling in n dimensions .
For minimally coupled fields on any spacetime, For flat spacetimes the bounds of Theorem 5.1 become and We note that only in the case of the worldvolume we can have a bound without mass dependence and field derivatives. That kind of bound has another important advantage which we can see by rescaling the support of the smearing function. We first let φ max be the maximum amplitude of the field so we can take it out of the bound. Then we look at the bound B 2 in flat spacetimes 1 Next, we consider a translationally invariant unit timelike vector field U µ and define the rescaled smearing function F λ for λ > 0 to be so that its normalization is independent of the choice of λ. Then in the limit of large λ the bound goes to zero and we have thus establishing an averaged SEC (ASEC) for flat spacetimes. A similar calculation for the B 1 bound gives a weaker, negative, bound in this case. The ASEC cannot be proven with the same assumptions for the worldline bound.

The curvature bound
To use the bound for the singularity theorem, we need to apply the Einstein equation Here we will focus only on the bound B 2 and use a similar method as in [4] to derive a curvature bound. Taking the trace of the Einstein equation G µν = 8πT µν gives Using Eq. (72) 'on shell' and using a similar method as [4] we can show Using Eqs. (91) and (92) we have (93) If we take the maximum value of the field less than the critical value 1 − 8πξφ 2 ≥ 0 and ξ ≤ ξ c we have that Now we can replace the term including the Ricci scalar in the bound of Eq. (80) Using the Einstein equation and the definition of EED we have the purely geometric bound We can get a slightly different bound by expanding the last terms of Eq. (97). We have where we defined the expansion θ = ∇ µ U µ (note that this is tr(∇U ) in index-free notation). For the next term we have where ∇ U U µ = A µ is the acceleration. Assuming that U µ is tangent to a timelike geodesic parametrized by t, A = 0 and we have whereḞ (x) = ∂F (x)/∂t. Then Eq. (97) becomes For an upper bound we may use Lemma s 3.4 together with Remark 3.6 (as we assume H ≤ β < 0 and we can always assume β > −(n − 1) |κ| by making it smaller if necessary), i.e., standard mean curvature comparison, to obtain So, as in Remark 3.7, From this (106) becomes for any F ∈ C ∞ c (M ) with support entirely contained in Ω + T (Reg + η (T )), where the constants are and We note that this bound is mass independent unlike the previously derived worldline bound. The appearance of spatial derivatives of the field can be considered small or negligible in some cases, for example those of cosmological spacetimes. We note that in geometric units the constant Q 1 has dimensions of inverse time square while the constant Q 2 is dimensionless. We are ready to state a singularity theorem for the non-minimally coupled classical scalar field. Proof. First we note that (M, g, φ) obeys the worldvolume bound of Eq. (111) for any F ∈ C ∞ c (M ) with support entirely contained in Ω + T (Reg + η (T )) (which is sufficient by Remark 4.3(1)). Thus the condition (ii) of Theorem 4.2 is also obeyed with constants Q 1 and Q 2 given by Eqs. (112) and (113). The rest of the proof follows Theorem 4.2.

APPLICATION
In this section we estimate the required extrinsic curvature to have past timelike geodesic incompleteness for the non-minimally coupled scalar field in a simple cosmological model. The approach is based on the analysis performed in [4] and [12]. The results will be useful to compare our theorem with the worldline ones proven in the previous references. One important estimate that needs to be made is the maximum allowed field value φ max . To do that we follow Ref. [4] and estimate it using a hybrid model. The idea is to estimate the (classical) φ max using the Wick square of a KMS state in Minkowski spacetime. That way we can connect the field value with a temperature T . Assuming the scalar field describes an elementary particle with mass m, a maximum temperature where the model can be trusted is taken as much smaller than the Compton temperature T m = mc 2 /k where k is the Boltzmann constant. Then T max = 10 −2 T m T m . With these considerations we have [4] where K x is the modified Bessel function. On dimensional grounds the ratioφ max /φ max is proportional to c/λ where λ is taken as the Compton length of elementary particles. Theṅ Since we will consider a cosmological setting we assume the field is spatially homogeneous and so ∆φ 2 max = 0 and φ max = 0. The values of the field and its derivative only depend on the mass of an (elementary) particle. To compare our results with Refs. [4] and [12] we use the pion, the proton and the Higgs. The pion has mass 2.40 × 10 −28 kg, the proton 1.67 × 10 −27 kg and the Higgs 2.24 × 10 −25 kg. Since we have been using geometric units, we need to restore the dimensions for this physical application. The quantity C max becomes so it has dimensions of inverse time. Similarly we restore dimensions in the constants Q 1 and Q 2 . Both constants are multiplied by G/c 4 , meaning Q 2 is dimensionless and Q 1 has dimensions of inverse time square. After restoring the dimensions we proceed with an estimation and optimization of the required extrinsic curvature ν τ 0 ,η of Theorem 4.2. First we optimize in terms of τ 0 with the condition T τ 0 . Then the optimal τ 0 is To proceed we will make some further approximations. First, for the rest of the calculation we assume ξ = 1/6, the conformal coupling in 4-dimensions. In this case any coupling in  the allowed range is expected to give similar results. Then we assume that With this assumption we can take cosh y ≈ 1 and coth y ≈ y −1 . The maximum allowed value of η so that the approximation is valid is 10 −2 c/ √ Gκ. We fix the value of η to be equal to that and we will revisit it to make sure η T as expected. With these assumptions, for each particle, the required extrinsic curvature is a function of T and κ. As we expect η T we will refer to T as the maximum time to singularity. To compare the results of our theorem to cosmological data we will use results from the PLANCK collaboration [1]. The Hubble parameter is the same as the extrinsic curvature of a cosmological spacelike Cauchy surface and today it has an estimated value of H 0 = 2.18 × 10 −18 s −1 while the age of the universe is t 0 = 4.35 × 10 17 s 2 . The goal is to find the largest possible κ so that ν ≤ H 0 and T is as close as possible to t 0 . We note that ν has a term of the form ∼ 3/T similar to the worldline theorem. That means T has to be larger than 1.5 × 10 18 s to have the required extrinsic curvature smaller than the measured one. Considering that we we want singularity prediction we examine values of T between this value and 10 20 s, about three orders of magnitude larger than t 0 . For each elementary particle and within this range of T we found the maximum possible value of κ for our theorem to apply. The results are presented in Tab. I. Comparing our results with the ones from Ref. [12] where the same model was examined using a worldline inequality we find several differences. First of all with the worldvolume inequality we are able to predict past geodesic incompleteness for the Higgs which was impossible in the worldline case. Second, our method doesn't require the SEC to hold near the Cauchy surface, thus this assumption was dropped and we are able to apply the theorem using the spacelike cosmological Cauchy surface today. The drawback of our theorem is the requirement for a global pointwise condition. However, we find that the negative energy density allowed at each spacetime point is very large, comparable to the energy density at the very early universe.

DISCUSSION
In this work we prove the first singularity theorems using a worldvolume inequality as an assumption. Using area comparison results in a Lorentzian manifold, we are able to derive bounds on individual timelike geodesics from worldvolume integrals. Additionally, using index form methods developed in [12] we prove a singularity theorem with a worldvolume integrated energy condition. Our results have physical applications, and as an example we apply our theorem in the case of the classical non-minimally coupled field, which violates the strong energy condition. Finally, we apply our theorem using a toy cosmological model and compare our results with the ones from worldline inequalities. We find that the worldvolume theorem could give better results especially in the case of elementary particles with large mass such as the Higgs boson. The obvious extension of this work is to apply the theorem to a quantum field theory using a quantum energy inequality as an assumption as done in Ref. [13]. There are currently a few obstacles for this application: Theorem 4.2 was proven only for one timelike derivative on the smearing function while the quantum inequality requires two. Perhaps more importantly, the theorem has as an assumption a pointwise energy condition. Even though the negative energy allowed can be large, in the context of quantum field theory there are no lower energy bounds for spacetime points. Thus the pointwise condition would need to be replaced with an integral one. The null geodesic incompleteness case seems even more interesting than the timelike one. As it is impossible to bound the null energy on single geodesic segments [14] but it is possible on finite null surfaces [15] , it would be of great interest to prove a Penrose-type theorem for worldvolume inequalities. One mathematical difficulty arising in the null case is that there are no direct Riemannian analogues to draw from, however there are null area comparison results, cf. [6,22], controlling the (n − 2)-dimensional area of constant-affine-parameter slices of the lightcone. This work is currently in progress. ACKNOWLEDGMENTS E-AK is supported by the ERC Consolidator Grant QUANTIVIOL. This work is part of the ∆-ITP consortium, a program of the NWO that is funded by the Dutch Ministry of Education, Culture and Science (OCW). A.O. is supported by the project P33594 of the Austrian Science Fund FWF. MG is supported by the German research foundation (DFG) focus program SPP2026 Geometry at Infinity and BS was supported by the Uni:Docs program of the University of Vienna. The authors would like to thank Chris Fewster, Michael Kunzinger and Roland Steinbauer for helpful discussions. They are also grateful to the Mathematical Research Institute of Oberwolfach (MFO) for hosting them for two weeks in April 2022 during the course of the "Research in Pairs" program, which contributed to this project.