The Hawking–Penrose Singularity Theorem for $C^1$-Lorentzian Metrics

Abstract

We extend both the Hawking–Penrose theorem and its generalisation due to Galloway and Senovilla to Lorentzian metrics of regularity $C^1$. For metrics of such low regularity, two main obstacles have to be addressed. On the one hand, the Ricci tensor now is distributional, and on the other hand, unique solvability of the geodesic equation is lost. To deal with the first issue in a consistent way, we develop a theory of tensor distributions of finite order, which also provides a framework for the recent proofs of the theorems of Hawking and of Penrose for $C^1$-metrics (Graf in Commun Math Phys 378(2):1417–1450, 2020). For the second issue, we study geodesic branching and add a further alternative to causal geodesic incompleteness to the theorem, namely a condition of maximal causal non-branching. The genericity condition is re-cast in a distributional form that applies to the current reduced regularity while still being fully compatible with the smooth and $C^{1,1}$-settings. In addition, we develop refinements of the comparison techniques used in the proof of the $C^{1,1}$-version of the theorem (Graf in Commun Math Phys 360:1009–1042, 2018). The necessary results from low regularity causality theory are collected in an appendix.

1 Introduction

The classical singularity theorems¹ of General Relativity (GR) collect sufficient and physically reasonable conditions that lead to causal geodesic incompleteness of spacetime, hence to the occurrence of a singularity in the spirit of Roger Penrose’s Nobel Prize-winning approach [40]. Being rigorous statements in pure Lorentzian geometry, they were first formulated within the smooth category. As they form a body of essential results in GR, the quest for low regularity versions of the theorems is eminent and was already explicitly discussed in [14, Sec. 8.4]. In fact, the (smooth) theorems can be read as predicting a mere drop of the differentiability of the metric below $C^2$, rather than incompleteness.

However, only with the rather recent advent of systematic studies in low regularity GR [6] and, in particular, causality theory [5, 24, 31, 41], a rigorous extension of the singularity theorems for metrics below the $C^2$-class became feasible, cf. [46, Sec. 6.2]. This applies first of all to metrics of regularity $C^{1,1}$, which model situations where the matter variables possess finite jumps. Indeed the Hawking, the Penrose, and, finally, the Hawking-Penrose theorem were proven in this regularity [8, 25, 26]. These results rule out that a smooth spacetime that is incomplete by the classical theorems, can be extended to a complete $C^{1,1}$-spacetime. Also, they imply that the spacetime either is incomplete or the differentiability of the metric is below $C^{1,1}$, and consequently the curvature becomes unbounded.

Technically, in $C^{1,1}$ the exponential map is still available [23, 31] and the curvature is still locally bounded, which allows for a natural extension of the energy and the genericity conditions. However, the curvature tensor is only defined almost everywhere, which forbids the use of Jacobi fields and conjugate points, both essential tools in the classical proofs. Instead one uses regularisation techniques which allow one to derive weakened versions of the energy conditions for approximating smooth metrics with controlled causality. Using careful comparison techniques for the (matrix) Riccati equation, it is then possible to show that these (still) lead to the occurrence of conjugate or focal points along causal geodesics for the approximating metrics. This in turn forces the geodesics of the $C^{1,1}$-metric to stop maximising. Similarly, a natural extension of the initial and the energy conditions leads to the formation of a trapped set. This, together with an extension of the causal parts of the classical proofs, allows one to finally derive the results.

The next step in lowering the regularity assumptions was undertaken by Graf [7], who extended the Hawking and the Penrose theorems to $C^1$-metrics, with the Gannon-Lee theorem following in [43]. $C^1$-regularity is at the moment the lowest possible class where classical singularity theorems have been established, and in this paper we complete this effort by proving the most refined of these statements, namely the Hawking-Penrose theorem.

In this regularity class one faces the following added severe complications arising from the fact that the Levi-Civita connection is only continuous:

• (a) The curvature is no longer locally bounded, but is merely a distribution of order one.

• (b) The initial value problem for the geodesic equation is solvable, but not uniquely so.

• (c) The exponential map is no longer defined.

The first item is especially relevant when formulating the energy conditions. While the strong energy condition can be extended in a straightforward manner, a formulation of the null energy condition is more subtle, cf. [7, Sec. 5], see also Sect. 2.3, below. The genericity condition needed for the Hawking-Penrose theorem is still more delicate. Actually it turns out to be necessary to apply the Ricci tensor to vector fields constructed via parallel transport, which means that they are merely of $C^1$-regularity, see Definition 2.15. This can only be done since the curvature is a distribution of order one, and we present the required distributional setting in Sect. 2.1, below. Whereas in the $C^{1,1}$- context the use of distributional methods could be avoided, it becomes essential in the present work, and at the same time has the benefit of providing a global framework also for the earlier results by Graf [7] on which we build.

Then, while we can use the weakened versions of the strong and null energy conditions for approximating smooth metrics derived in [7] (employing a refined version of the Friedrichs Lemma), we have to derive from the distributional genericity condition an appropriate weakened version for smooth approximations that allows us to employ an extension of the Riccati comparison techniques developed in [8], see Lemmas  2.17, and 2.18 . The key technique of regularisation by smooth metrics with adapted causality as put forward in [5] is recalled in Sect. 2.2.

The first issue connected to item (b) is that it necessitates a decision on how to extend the notion of geodesic completeness, and following [7] we say that a spacetime is complete if every inextendible solution to the geodesic equation is complete (rather than merely demanding only one complete solution for any choice of initial data).

Next, item (b) together with item (c) forbids the use of an essential argument in the context of approximating maximising causal geodesics by maximising causal geodesics of the approximating metrics. This issue has been solved in [7, Sec. 2] in the globally hyperbolic case. In the context of the present work, however, we have to establish more general results, see Sect. 3. In fact, we have to introduce an additional assumption, namely a non-branching condition for maximising causal geodesics (Definition 3.1), to derive the corresponding approximation results in Proposition 3.4. Here it is also essential to apply a classical ODE result (Proposition 2.5), which we reformulate for our purpose in Corollary 2.6. Moreover, since the interrelation between causal geodesics (i.e., solutions of the geodesic equation) and maximising curves becomes more subtle below $C^{1,1}$ [16, 42], we rely in particular on the fact that maximising causal curves are indeed (unbroken) geodesics and hence $C^2$-curves [27, 43].

The new non-branching assumption is well motivated by similar conditions used in metric geometry, where due to the lack of a differentiable structure the geodesic equation is not available. Also a null version of the condition was essential in the proof of the $C^1$-Gannon-Lee theorem, see [43, Sec. 3]. Moreover it adds a novel facet to the interpretation of the $C^1$-version of the theorem: It predicts either geodesic incompleteness or branching of maximising causal geodesics, with both alternatives physically signifying a catastrophic event for the observer corresponding to the geodesic. Of course, there is the alternative that the regularity of the metric drops below $C^1$, which renders the curvature a distribution of higher order. Again, the result also forbids the extension of the spacetime to a complete one of regularity $C^1$ without (maximal causal) geodesic branching. This once more complements the recent $C^0$-inextendibility results of [9, 45].

Next we briefly discuss the further prospects of low-regularity singularity theorems. Concerning the causality part of the results it is expected that they extend to $C^{0,1}$-metrics², while some features of causality theory fail below this class [5, 12]. Nevertheless, the causal core of some of the singularity theorems has been established in the very general setting of closed cone structures, see [35, Sec. 2.15].

On the analytic side, already in $C^{0,1}$ one faces the problem that the right hand side of the geodesic equation is merely locally bounded and one would have to resort to non-classical solution concepts. Also another analytical technique at the core of the arguments, i.e. the Friedrichs Lemma which is used to go from the distributional energy conditions of the singular metric to useful surrogate conditions for the smooth approximations, appears to be sharp, see [7, Lem. 4.8] for the currently most advanced version. This is actually a long way from the largest possible class where the curvature can be (stably) defined in an analytical (distributional) way, i.e. $H^1\cap L^\infty$ locally [11, 28, 50]. However, the quest is there to at least go into the direction of regularity classes more closely linked to the classical existence results for the field equations, $H^s$ locally for $s>5/2$, or to current formulations of cosmic censorship, i.e. the connection lying locally in $L^2$.

Very recently, also synthetic formulations of the singularity theorems have appeared: In [1], Lorentzian length spaces that are warped products are studied and a Hawking singularity theorem is derived from suitable sectional curvature bounds (implying Ricci curvature bounds in such geometries), based on triangle comparison. Moreover, Cavalletti and Mondino prove a version of Hawking’s singularity theorem in Lorentzian metric measure spaces, where Ricci curvature bounds are implemented using methods from optimal transport [4]. However, it remains unclear to date how these results precisely relate to the analytical approach to low regularity pursued in this work.

The Hawking-Penrose theorem classically comes in a causal version [17, p. 538, Thm.], which asserts the incompatibility of the following three conditions for a $C^2$-spacetime (M, g): chronology, the fact that every inextendible causal geodesic stops maximising, and the existence of a (future or past) trapped set. The analytic main result is then a corollary [17, Sec. 3, Cor.] which collects sufficient conditions (energy and genericity conditions, and initial conditions) to derive the above incompatible items under the assumption of causal geodesic completeness. We will generally follow this path, but prove a more general version of the theorem due to Galloway and Senovilla [10], which adds the existence of a trapped submanifold of arbitrary codimension to the list of initial conditions.

Our version of the causal result is Theorem 6.2 and, for the sake of completeness and for the convenience of the reader we collect all results from $C^1$-causality theory³ needed here (and elsewhere) in Appendix A. The main result is then Theorem 6.3. The results needed to infer from its analytic conditions (the distributional energy and genericity conditions) and the non-branching assumptions the inexistence of lines are proven in Sect. 4. The existence of a trapped set is derived from the various sets of initial conditions (and the energy conditions) in Sect. 5. Finally, some technical results on extending vector fields in $C^1$-spacetimes are collected in Appendix B. They allow us to put to use the distributional genericity condition in the context of regularisations.

To conclude this introduction, we fix the notations and conventions used throughout the paper. By a manifold M we will mean a smooth, connected, second countable Hausdorff manifold of dimension n (with $n\ge 3$). When such an M is endowed with a Lorentzian metric g of regularity $C^1$ of signature $(-,+\dots ,+)$ and is time-oriented via a smooth timelike vector field we call it a $C^1$-spacetime (hence lowering the regularity will always apply to the spacetime metric only, not to the underlying manifold). The Levi-Civita connection of a smooth or $C^1$-spacetime will be denoted by $\nabla$ (for its distributional definition in the $C^1$-case, see the next section). We will always assume that M is also endowed with a smooth (complete, background) Riemannian metric h with associated Riemannian distance function $d_h$ and norm $\Vert \ \Vert _h$ which we use to estimate tensor fields. Since all such estimates will be on compact sets only, they are independent of the choice of h.

A curve $\gamma : I\rightarrow M$ is called timelike (causal, null, future or past directed) if it is locally Lipschitz and ${\dot{\gamma }}(t)$ is timelike (causal, null, future or past directed) almost everywhere. Concerning causality theory we use standard notation, cf. e.g., [34, 39]. Thus $p\ll q$ (resp. $p\le q$) means that there exists a future directed timelike (resp. causal) curve from p to q, $I^+(A):=\{q\in M:\, p\ll q\ \mathrm {for\,some}\,p\in A\}$ and $J^+(A):=\{q\in M:\, p\le q\ \mathrm {for\,some}\,p\in A\}$. These definitions do not change if one defines the causality relations via piecewise smooth curves [24, 31]. The Lorentzian distance (or time-separation function) associated to a Lorentzian metric g will be denoted by $d_g$. A $C^{1}$-spacetime (M, g) is globally hyperbolic if it is causal (i.e., contains no closed causal curves) and $J(p,q):=J^+(p)\cap J^-(q)$ is compact for all $p,q\in M$. A Cauchy hypersurface in (M, g) is a closed acausal set that is met by each inextendible causal curve.

For the Riemann curvature tensor we use the convention $R(X,Y)Z=[\nabla _X,\nabla _Y]Z-\nabla _{[X,Y]}Z$ and the Ricci tensor is given by $\mathrm {Ric}(X,Y)=\sum _{i=1}^n\langle E_i,E_i\rangle \langle R(E_i,X)Y,E_i\rangle$ (again we refer to the next section for a distributional interpretation in the case of $C^1$-metrics). Here $(E_i)_{i=1}^n$ is a (local) orthonormal frame field, while by $(e_i)_{i=1}^n$ we will denote frames in individual tangent spaces $T_pM$. For an embedded submanifold N of M of codimension m we define the second fundamental form by $\mathrm {II}(V,W)=\text{ nor }(\nabla _VW)$ and the shape operator derived from a normal unit field $\nu$ by $S(X)=\nabla _X\nu$. Using (2.13) below it follows that for $g\in C^1$ and $V, W \in C^1$, $\mathrm {II}(V,W)$ is continuous.

For g smooth (resp. $C^1$), $1<m<n$ and S a smooth (resp. $C^2$) spacelike $(n-m)$-dimensional submanifold let $e_1(q),\dots ,e_{n-m}(q)$ be an orthonormal basis for $T_q S$, varying smoothly (resp. $C^1$) with q in a neighborhood (in S) of $p \in S$. Then $H_S := \frac{1}{n-m}\sum _{i=1}^{n-m}\mathrm {II}(e_i,e_i)$ denotes the mean curvature vector field of S, and ${\mathbf {k}}_S(v):=g(H_S,v)$ the convergence of $v\in TM|_S$. For g in $C^1$, both $H_S$ and ${\mathbf {k}}_S$ are continuous. A closed spacelike submanifold S is called (future) trapped if for any future-directed null vector $\nu \in T S^\perp$ the convergence ${\mathbf {k}}_S(\nu )$ is positive, or equivalently that the mean curvature vector field $H_S$ is past pointing timelike on S.

One final convention we shall require is that of tensor classes (cf. [21, Sec. 4.6.3]): Given a causal geodesic $\gamma$ in a $C^{1}$-spacetime (M, g), we set $[{{\dot{\gamma }}}(t)]^\perp :=({{\dot{\gamma }}}(t))^\perp /{\mathbb {R}}{{\dot{\gamma }}}(t)$. For $\gamma$ null, $[{{\dot{\gamma }}}(t)]^\perp$ is an $(n-2)$-dimensional subspace of the hypersurface $({{\dot{\gamma }}}(t))^\perp$. On the other hand, for $\gamma$ timelike, $[{{\dot{\gamma }}}(t)]^\perp$ equals $({\dot{\gamma }}(t))^{\perp }$. To obtain a unified notation, throughout the paper we will denote the dimension of $[{{\dot{\gamma }}}(t)]^\perp$ by d, which amounts to setting $d=n-2$ in the null case and $d=n-1$ in the timelike case. Furthermore, we set $[{{\dot{\gamma }}}]^\perp =\bigcup _t[{{\dot{\gamma }}}(t)]^\perp$. Then for every normal tensor field A along $\gamma$ we obtain a well-defined tensor class [A] along $\gamma$, as well as a well-defined tensor class representing the induced covariant derivative $\nabla _{{{\dot{\gamma }}}}$. We then set $[\dot{A}]=[\nabla _{{{\dot{\gamma }}}}A]$. The metric $g|_{[{{\dot{\gamma }}}]^\perp }$ is positive definite in both the null and the timelike case. For smooth metrics, the curvature (or tidal force) operator is given by $[R](t):\, [{{\dot{\gamma }}}(t)]^\perp \rightarrow [{{\dot{\gamma }}}(t)]^\perp$, $[v]\mapsto [R(v,{{\dot{\gamma }}}(t)){{\dot{\gamma }}}(t)]$.

2 Distributional Curvature of $C^1$-Spacetimes

2.1 Curvature tensors of $C^1$-metrics

In what follows we discuss the general distributional framework in which to understand the curvature quantities associated to metric tensors (of arbitrary signature) of regularity below $C^2$, as is required for our further analysis. The general mathematical framework goes back to [29], while [11] provided the first study specific to GR. We mainly follow [28], cf. also [13, 49], but put a special emphasis on metrics of regularity $C^1$ and distributions of finite order.

As in [13, Sec. 3.1], for $k\in {\mathbb {N}}_0\cup \{\infty \}$ we denote by $\mathrm {Vol}(M)$ the volume bundle over M, and by $\Gamma ^k_c(M,\mathrm {Vol}(M))$ the space of compactly supported one-densities on M (i.e., sections of $\mathrm {Vol}(M)$) that are k-times continuously differentiable. Then the space of distributions of order k on M is the topological dual of $\Gamma ^k_c(M,\mathrm {Vol}(M))$,

$$\begin{aligned} {\mathcal {D}}'{}^{(k)}(M) := \Gamma ^k_c(M,\mathrm {Vol}(M))'. \end{aligned}$$

For $k=\infty$ we omit the superscript (k). Clearly there are topological embeddings ${\mathcal {D}}'{}^{(k)}(M) \hookrightarrow {{\mathcal {D}}'}{}^{(k+1)}(M) \hookrightarrow {\mathcal {D}}'(M)$ for all k. As we shall see, all curvature quantities of $C^1$-metrics are distributional tensor fields of order 1. The space of distributional (r, s)-tensor fields of order k is defined as

$$\begin{aligned} {\mathcal {D}}'{}^{(k)}{\mathcal {T}}^r_s(M) \equiv {\mathcal {D}}'{}^{(k)}(M,T^r_s M) := \Gamma ^k_c(M,{\mathcal {T}}^s_r(M) \otimes \mathrm {Vol}(M))'. \end{aligned}$$

(2.1)

By [13, 3.1.15],

$$\begin{aligned} {\mathcal {D}}'{\mathcal {T}}^r_s(M) \cong {\mathcal {D}}'(M) \otimes _{C^\infty (M)} {\mathcal {T}}^r_s(M) \cong L_{C^\infty (M)}(\Omega ^1(M)^r\times {\mathfrak {X}}(M)^s; {\mathcal {D}}'(M)). \end{aligned}$$

(2.2)

This algebraic isomorphism ultimately rests on the fact that the $C^\infty (M)$-module of $C^\infty$-sections $\Gamma (M,F)$ in any smooth vector bundle $F\rightarrow M$ is finitely generated and projective, which allows one to apply [3, Ch. II §4.2, Prop. 2]. These properties persist if we consider $\Gamma _{C^k}(M,F)$ as a $C^k(M)$-module for any finite k, so we also have

$$\begin{aligned} \begin{aligned} {\mathcal {D}}'{}^{(k)}{\mathcal {T}}^r_s(M)&\cong {\mathcal {D}}'{}^{(k)}(M) \otimes _{C^k(M)} ({\mathcal {T}}^r_s)_{C^k}(M)\\&\cong L_{C^k(M)}(\Omega ^1_{C^k}(M)^r\times {\mathfrak {X}}_{C^k}(M)^s; {\mathcal {D}}'{}^{(k)}(M)). \end{aligned} \end{aligned}$$

(2.3)

In other words, distributional tensor fields of order k act as $C^k$-balanced multilinear maps on one forms and vector fields of regularity $C^k$ to give a scalar distribution of order k. For finite k, the multiplication $C^k(M)\times {\mathcal {D}}'^{(k)}(M)\rightarrow {\mathcal {D}}'^{(k)}(M)$ is jointly continuous (w.r.t. the strong topology), whereas for $k=\infty$ it is only hypocontinuous [20, p. 362], but still jointly sequentially continuous [37, p. 233] since $C^\infty (M)$ is Fréchet, hence barrelled. The corresponding continuity properties therefore also hold for the tensor operations introduced above. The isomorphisms (2.2), (2.3) are not topological, but still bornological by [36, Thm. 15] (which remains valid for spaces of $C^k$-sections with k finite).

Smooth, and in fact even $L^1_{\mathrm {loc}}$-tensor fields are continuously and densely embedded via

$$\begin{aligned} {\mathcal {T}}^r_s(M)&\hookrightarrow {\mathcal {D}}'{}^{(k)}{\mathcal {T}}^r_s(M) \\ t&\mapsto [(\theta _1,\dots ,\theta _r,X_1,\dots ,X_s) \mapsto [\omega \mapsto \int _M t(\theta _1,\dots ,\theta _r,X_1,\dots ,X_s)\omega ]]. \end{aligned}$$

Here, $\omega$ is a one-density. If M is orientable, densities can be canonically identified with n-forms. The fact that ${\mathcal {T}}^r_s(M)$ is dense in ${\mathcal {D}}'{}^{(k)}{\mathcal {T}}^r_s(M)$ uniquely fixes all the operations on distributional tensor fields to be introduced below in a way compatible with smooth pseudo-Riemannian geometry.

For any $t\in {\mathcal {T}}^r_s(M)$ there is a unique extension that accepts one distributional argument. E.g., if ${\tilde{\theta }}_1 \in {\mathcal {D}}'{\mathcal {T}}^0_1(M)$, then since $t(\,.\,,\theta _2,\dots ,X_s)\in {\mathfrak {X}}(M)$ we may set

$$\begin{aligned} t({\tilde{\theta }}_1,\theta _2,\dots ,X_s) := {\tilde{\theta }}_1(t(\,.\,,\theta _2,\dots ,X_s)) \in {\mathcal {D}}'(M), \end{aligned}$$

(2.4)

and analogously for the other slots.

Definition 2.1

A distributional connection is a map $\nabla : {\mathfrak {X}}(M)\times {\mathfrak {X}}(M) \rightarrow {\mathcal {D}}'{\mathcal {T}}^1_0(M)$ satisfying for $X,X',Y,Y'\in {\mathfrak {X}}(M)$ and $f\in C^\infty (M)$ the usual computational rules: $\nabla _{f X+X'}Y = f\nabla _XY + \nabla _{X'}Y$, $\nabla _X(Y+Y') = \nabla _X Y +\nabla _X Y'$, $\nabla _X(f Y) = X(f)Y + f\nabla _X Y$.

Using (2.4) any distributional connection can be extended to the entire tensor algebra, i.e., to a map $\nabla : {\mathfrak {X}}(M)\times {\mathcal {T}}^r_s(M)\rightarrow {\mathcal {D}}'{\mathcal {T}}^r_s(M)$, e.g. for $\Theta \in \Omega ^1(M)$ we set

$$\begin{aligned} (\nabla _X\Theta )(Y):=X(\Theta (Y))-\Theta (\nabla _X Y)\quad (X,Y\in {\mathfrak {X}}(M)). \end{aligned}$$

(2.5)

Let ${{\mathcal {G}}}$ be any of the spaces $C^k$ $(0\le k)$ or $L^p_{\mathrm {loc}}$ $(1\le p)$, then we call a distributional connection a ${{\mathcal {G}}}$-connection if $\nabla _X Y$ is a ${{\mathcal {G}}}$-vector field for any $X,Y\in {\mathfrak {X}}(M)$. A particularly important case are $L^2_{\mathrm {loc}}$-connections since they form the largest class that allows for a stable definition of the curvature tensor in distributions, cf. [11, 28, 49].

Using a local frame one easily sees that by virtue of the respective computational rules any ${{\mathcal {G}}}$-connection can be (uniquely) extended to a map $\nabla : {\mathfrak {X}}_{{\mathcal {F}}}(M)\times {\mathfrak {X}}_{{\mathcal {H}}}(M) \rightarrow {\mathfrak {X}}_{{\mathcal {G}}}(M)$, provided that ${\mathcal {F}}$ and ${\mathcal {H}}$ are function spaces such that ${\mathcal {F}}\cdot {\mathcal {G}}\subseteq {\mathcal {G}}$, $D{\mathcal {H}}\subseteq {\mathcal {G}}$, and ${\mathcal {H}}\cdot {\mathcal {G}}\subseteq {\mathcal {G}}$. Hence, in particular, each $L^2_{\mathrm {loc}}$-connection extends in this way to a map

$$\begin{aligned} \nabla : {\mathfrak {X}}_{C^0}(M)\times {\mathfrak {X}}_{C^1}(M) \rightarrow {\mathfrak {X}}_{L^2_{\mathrm {loc}}}(M). \end{aligned}$$

(2.6)

Moreover, by a similar reasoning, ${\mathcal {G}}$-connections allow one to insert even less regular vector fields in the first and second slot at the price of a less regular outcome. In particular, any $L^2_{\mathrm {loc}}$-connection can be extended to a map

$$\begin{aligned} \nabla : {\mathfrak {X}}(M) \times {\mathfrak {X}}_{L^2_{\mathrm {loc}}}(M) \rightarrow {\mathcal {D}}' {\mathcal {T}}^1_0(M). \end{aligned}$$

(2.7)

Using this extension, we have [28, Def. 3.3]:

Definition 2.2

The distributional Riemann tensor of an $L^2_{\mathrm {loc}}$-connection $\nabla$ is the map $R: {\mathfrak {X}}(M)^3 \rightarrow {\mathcal {D}}'{\mathcal {T}}^1_0(M)$,

$$\begin{aligned} R(X,Y,Z)(\theta ) \equiv (R(X,Y)Z)(\theta ) := (\nabla _X\nabla _Y Z - \nabla _Y\nabla _X Z- \nabla _{[X,Y]} Z)(\theta ) \end{aligned}$$

for $X,Y,Z\in {\mathfrak {X}}(M)$ and $\theta \in \Omega ^1(M)$,

If $F_i$ is a local frame in ${\mathfrak {X}}(U)$ and $F^j\in \Omega ^1(U)$ its dual frame, then the Ricci tensor corresponding to $\nabla$ is given by

$$\begin{aligned} \mathrm {Ric}(X,Y) := (R(X,F_i)Y)(F^i) \in {\mathcal {D}}'(U) \qquad (X,Y\in {\mathfrak {X}}(U)). \end{aligned}$$

(2.8)

Recall the Koszul formula for the Levi-Civita connection of a smooth metric g on M:

$$\begin{aligned} \begin{aligned} 2g(\nabla _X Y,Z)&= X(g(Y,Z)) + Y(g(Z,X)) -Z(g(X,Y)) \\&\quad - g(X,[Y,Z]) + g(Y,[Z,X]) + g(Z,[X,Y]) =: F(X,Y,Z) \end{aligned} \end{aligned}$$

(2.9)

Focusing now on the case at hand in this work, suppose that g is merely $C^1$ (still allowing arbitrary signature). Then F(X, Y, Z) is a well-defined element of $C^0(M)$, and

$$\begin{aligned} \nabla ^\flat _X Y := Z \mapsto \frac{1}{2} F(X,Y,Z) \in \Omega ^1_{C^0}(M) \subseteq {\mathcal {D}}'{\mathcal {T}}^0_1(M) \end{aligned}$$

(2.10)

defines the distributional Levi-Civita connection of g [28, Def. 4.2]. Note that this is not (yet) a distributional connection in the sense of Definition 2.1 since it is of order (0, 1) instead of (1, 0). In addition to the standard product rules it also possesses the properties (corresponding to [39, Thm. 3.11])

$$\begin{aligned} \begin{aligned} \nabla ^\flat _X Y - \nabla ^\flat _Y X&= [X,Y]^\flat ,\ \text {i.e.,} \ (\nabla ^\flat _X Y - \nabla ^\flat _Y X)(Z) = g([X,Y],Z),\\ X(g(Y,Z))&= (\nabla ^\flat _X Y)(Z) + (\nabla ^\flat _X Z)(Y) \end{aligned} \end{aligned}$$

(2.11)

for all $X, Y, Z\in {\mathfrak {X}}(M)$. These equalities hold distributionally, hence also in $C^0$.

In order to obtain an $L^2_{\mathrm {loc}}$-connection from $\nabla ^\flat$ (which in turn will allow us to define the curvature tensors we require) we need to raise the index via g, setting

$$\begin{aligned} g(\nabla _X Y,Z) := (\nabla ^\flat _X Y)(Z) \qquad (X, Y, Z\in {\mathfrak {X}}(M)). \end{aligned}$$

(2.12)

This defines a continuous vector field $\nabla _X Y$, hence even a $C^0$-connection. More generally, by [28, Sec. 4] this procedure yields an $L^2_{\mathrm {loc}}$-connection even for the significantly larger Geroch-Traschen class of metrics ($g\in H^1_{\mathrm {loc}}\cap L^\infty _{\mathrm {loc}}$ and locally uniformly non-degenerate).

The Riemann tensor of $g\in C^1$ is now given by Definition 2.2. Furthermore, since in this case $\nabla$ is a ${C^0}$-connection, it follows that $(R(X,Y)Z)(\theta ) \in {\mathcal {D}}'{}^{(1)}(M)$, and consequently, $R\in {\mathcal {D}}'{}^{(1)}{\mathcal {T}}^1_3(M)$. Indeed by the same reasoning that led to (2.6), (2.7) we may extend the $C^0$-Levi-Civita connection of g to a map

$$\begin{aligned} \nabla : {\mathfrak {X}}_{C^0}(M)\times {\mathfrak {X}}_{C^1}(M) \rightarrow {\mathfrak {X}}_{C^0}(M), \ \text{ and }\ \nabla : {\mathfrak {X}}_{C^1}(M)\times {\mathfrak {X}}_{C^0}(M) \rightarrow {\mathcal {D}}'^{(1)}{\mathcal {T}}^1_0(M),\nonumber \\ \end{aligned}$$

(2.13)

respectively. Here, the second equation shows that the term $([\nabla _X,\nabla _Y]Z)(\theta )$ yields an order one distribution, while $(\nabla _{[X,Y]}Z)(\theta ) \in C^0$ since $\nabla$ is a $C^0$-connection.

For $W,X,Y,Z \in {\mathfrak {X}}(M)$ we define $R(W,X,Y,Z)\in {\mathcal {D}}'{}^{(1)}(M)$ by

$$\begin{aligned} R(W,X,Y,Z)&:= X(g(W,\nabla _Y Z)) - Y(g(W,\nabla _X Z))\\&\quad - g(\nabla _X W,\nabla _Y Z) + g(\nabla _Y W,\nabla _X Z) - g( W,\nabla _{[X,Y]} Z). \end{aligned}$$

Using (2.4), (2.7) and (2.11) it is straightforward to check that cf. [28, Rem. 4.5]

$$\begin{aligned} R(W,Z,X,Y) = g(W,R(X,Y)Z). \end{aligned}$$

Alternatively, one may verify this identity in local coordinates using the fact that there is a well-defined multiplication of distributions of first order with $C^1$-functions.

The Ricci tensor of a $C^1$ metric g is given by (2.8), hence, in particular, is again a distributional tensor field of order 1. Note that by (2.3) this allows us to apply it to $C^1$-vector fields, which will be essential in the context of the genericity condition below. Similarly, since the Riemann tensor, again by (2.3), acts on $C^1$-vector fields, we may in particular use g-orthonormal frames (which generically are only $C^1$) to calculate the Ricci tensor, which is also vital below. Explicitly, (2.8) holds for $F_i$ a g-orthonormal frame and $F^i$ its g-dual. All these facts can alternatively be derived directly from the extension of $\nabla$ in (2.13).

Again, by the same observations, we have that the scalar curvature S is a distribution of order 1 (this answers a concern from [7, Sec. 3.2]) and that the standard local formulae hold in ${\mathcal {D}}'{}^{(1)}$:

$$\begin{aligned} R^m_{\,ijk}&= \partial _j\Gamma ^m_{\,ik} - \partial _k\Gamma ^m_{\,ij} + \Gamma ^m_{\,js}\Gamma ^s_{\,ik} - \Gamma ^m_{\,ks}\Gamma ^s_{\,ij}, \\ \mathrm {Ric}_{ij}&= R^m_{\,imj},\\ S&=\mathrm {Ric}^i_i. \end{aligned}$$

This shows that the local definitions given in [7, (3.2),(3.3)] are compatible with the global approach developed here.

2.2 Regularisation

As in previous works on the generalisation of singularity theorems to metrics of differentiability below $C^2$ [7, 24,25,26], an essential tool in our approach will be suitable regularisations, both of the metrics themselves and of the derived distributional curvature quantities. A general regularisation scheme based on chart-wise convolution with a mollifier $\rho \in {\mathcal {D}}(B_1(0))$, $\int \rho = 1$, $\rho \ge 0$ (cf. [13, 3.2.10], [24, Sec. 2], [7, Sec. 3.3]) is as follows: Cover M by a countable and locally finite family of relatively compact chart neighbourhoods $(U_i,\psi _i)$ ($i\in {\mathbb {N}}$). Let $(\zeta _i)_i$ be a subordinate partition of unity with $\mathrm {supp}(\zeta _i)\subseteq U_i$ for all i and choose a family of cut-off functions $\chi _i\in {\mathscr {D}}(U_i)$ with $\chi _i\equiv 1$ on a neighbourhood of $\mathrm {supp}(\zeta _i)$. Finally, for $\varepsilon \in (0,1]$ set $\rho _{\varepsilon }(x):=\varepsilon ^{-n}\rho \left( \frac{x}{\varepsilon }\right)$. Then denoting by $f_*$ (respectively $f^*$) push-forward (resp. pull-back) of distributions under a diffeomorphism f, for any ${\mathcal {T}} \in {\mathcal {D}}'{\mathcal {T}}^r_s(M)$ consider the expression

$$\begin{aligned} {\mathcal {T}}\star _M \rho _\varepsilon (x):= \sum \limits _i\chi _i(x)\,\psi _i^*\Big (\big (\psi _{i\,*} (\zeta _i\cdot {\mathcal {T}})\big )*\rho _\varepsilon \Big )(x). \end{aligned}$$

(2.14)

Here, $\psi _{i\,*} (\zeta _i\cdot {\mathcal {T}})$ is viewed as a compactly supported distributional tensor field on ${\mathbb {R}}^n$, so componentwise convolution with $\rho _\varepsilon$ yields a smooth field on ${\mathbb {R}}^n$. The cut-off functions $\chi _i$ secure that $(\varepsilon ,x) \mapsto {\mathcal {T}}\star _M \rho _\varepsilon (x)$ is a smooth map on $(0,1] \times M$. For any compact set $K \Subset M$ there is an $\varepsilon _K$ such that for all $\varepsilon <\varepsilon _K$ and all $x\in K$ (2.14) reduces to a finite sum with all $\chi _i\equiv 1$ (hence to be omitted from the formula), namely when $\varepsilon _K$ is less than the distance between the support of $\zeta _i\circ \psi _i^{-1}$ and the boundary of $\psi _i(U_i)$ for all i with $U_i\cap K\ne \emptyset$.

Just as is the case for smoothing via convolution in the local setting, ${\mathcal {T}}\star _M \rho _\varepsilon$ converges to ${\mathcal {T}}$ as $\varepsilon \rightarrow 0$ in ${\mathcal {D}}'{\mathcal {T}}^r_s$, and indeed in $C^k_{\mathrm {loc}}$ or $W^{k,p}_{\mathrm {loc}}$ $(p<\infty )$ if ${\mathcal {T}}$ is contained in these spaces [7, Prop. 3.5].

For a given $C^1$ Lorentzian metric g we will in addition require concrete regularisations that are adapted to the causal structure induced by g. This construction goes back to Chrusciel and Grant [5, Prop. 1.2], cf. also [24, Prop. 2.5]. The version we will use is [7, Lem. 4.2, Cor. 4.3]:

Lemma 2.3

Let (M, g) be a $C^1$-spacetime. Then for any $\varepsilon >0$ there exist smooth Lorentzian metrics ${\check{g}}_\varepsilon$, ${\hat{g}}_\varepsilon$ on M, time orientable by the same timelike vector field as g, and satisfying:

• (i) ${\check{g}}_\varepsilon \prec g \prec {\hat{g}}_\varepsilon$ for all $\varepsilon$.

• (ii) ${\check{g}}_\varepsilon$ and ${\hat{g}}_\varepsilon$ converge to g in $C^1_{\mathrm {loc}}$ as $\varepsilon \rightarrow 0$.

• (iii) ${\check{g}}_\varepsilon - g\star _M \rho _\varepsilon \rightarrow 0$ in $C^\infty _{\mathrm {loc}}$ and for any $K \Subset M$ there exist $c_K>0$ and $\varepsilon _0(K)$ such that $\Vert {\check{g}}_\varepsilon -g\star _M \rho _\varepsilon \Vert _{\infty ,K} \le c_K \varepsilon$ and $\Vert g - g \star _M \rho _\varepsilon \Vert _{\infty ,K} \le c_K \varepsilon$ for all $\varepsilon < \varepsilon _0$. An analogous statement holds for ${\hat{g}}_\varepsilon$ as well as for the inverse metrics $g^{-1},({\check{g}}_\varepsilon )^{-1}$ and $({\hat{g}}_\varepsilon )^{-1}$.

In this work we will generally use ${\check{g}}_\varepsilon$, ${\hat{g}}_\varepsilon$ to denote the regularisations of Lemma 2.3, which are of utmost importance throughout.

Next we recall the following essential result on convergence of the curvature under regularisations, which is a special case of several so-called stability results, see [11, Thm. 2], [28, Thm. 4.6], [50, Thm. 5.1].

Lemma 2.4

Let (M, g) be a $C^1$-spacetime, and let $g_\varepsilon$ be either ${\check{g}}_\varepsilon$ or ${\hat{g}}_\varepsilon$. Then we have for the Riemann and the Ricci tensor

$$\begin{aligned} R[g_\varepsilon ]\rightarrow R[g]\quad \text{ and }\quad \mathrm {Ric}[g_\varepsilon ]\rightarrow \mathrm {Ric}[g]\quad \text{ distributionally. } \end{aligned}$$

(2.15)

In particular, the Riemann and the Ricci tensor possess the usual symmetries.

We will frequently need the following result from ODE theory (cf. [15, Ch.2, Thm. 3.2]), which we will mainly use in the context of regularisation given below in Corollary 2.6.

Proposition 2.5

Let f and $f_m$, $m \in {\mathbb {N}}$ be continuous functions on an open set $E \subseteq {\mathbb {R}}\times {\mathbb {R}}^n$ with $f_m \rightarrow f$ in $C^0_{\mathrm {loc}}$. Let $y_m=y_m(t)$ be a solution to the initial value problem

$$\begin{aligned} y_m'(t) = f_m(t,y_m(t)), \quad y_m'(t_m) = y_{m_0} \end{aligned}$$

with data $(t_m,y_{m_0}) \in E$ and maximal interval of existence $(\omega ^-_m,\omega ^+_m)$. Suppose that the sequence of initial data converges,

$$\begin{aligned} (t_m,y_{m_0}) \rightarrow (t_0,y_0) \in E. \end{aligned}$$

Then there exist a solution $y=y(t)$ for the initial value problem

$$\begin{aligned} y'(t) = f(t,y(t)), \quad y(t_0)=y_0, \end{aligned}$$

defined on a maximal interval $(\omega ^-,\omega ^+)$, and a subsequence $m_l$ with the property that for any $s_1<s_2$ with $(s_1,s_2) \subseteq (\omega ^-,\omega ^+)$ we have $(s_1,s_2) \subseteq (\omega ^-_{m_l},\omega ^+_{m_l})$ for all large l, and $y_{m_l} \rightarrow y$ in $C^1_{\mathrm {loc}}(\omega ^-,\omega ^+)$.⁴ In particular,

$$\begin{aligned} \limsup _m \omega ^-_m \le \omega ^- < \omega ^+ \le \liminf _m \omega ^+_m. \end{aligned}$$

We will mostly use the previous result to conclude that $g_{\varepsilon }$-geodesics, where $g_{\varepsilon } \in \{{\check{g}}_{\varepsilon },\hat{g}_{\varepsilon }\}$, converge uniformly on compact intervals to a g-geodesic:

Corollary 2.6

Let (M, g) be a $C^1$-spacetime and let $g_m = {\check{g}}_{\varepsilon _m}$ or $g_m = \hat{g}_{\varepsilon _m}$ with $\varepsilon _m \rightarrow 0$. Let $\gamma _m:(a_m,b_m) \rightarrow M$ be inextendible $g_m$-geodesics. Suppose that $\gamma _m(t_0) \rightarrow p \in M$ and ${\dot{\gamma }}_m(t_0) \rightarrow v\in T_pM$, where $t_0 \in \bigcap (a_m,b_m)$. Then there exist an inextendible g-geodesic $\gamma :(a,b) \rightarrow M$ with $\gamma (t_0)=p$, ${{\dot{\gamma }}}(t_0)=v$ and a subsequence $m_l$ such that whenever $(s_1,s_2) \subseteq (a,b)$, we have $(s_1,s_2) \subseteq (a_{m_l},b_{m_l})$ for all large l and $\gamma _{m_l}$ converges in $C^2_{\mathrm {loc}}(a,b)$ to $\gamma$. Moreover,

$$\begin{aligned} \limsup _m a_m \le a < b \le \liminf _m b_m. \end{aligned}$$

Proof

As this is a local statement, we may assume $M = {\mathbb {R}}^n$. We rewrite the geodesic equations for $g_m$ as a first order initial value problem (with $\Gamma _m$ the Christoffel symbols of $g_m$) and $1\le i \le n$:

$$\begin{aligned} {\dot{\gamma }}_m^i(t)&= \eta _{m}^i(t),\\ {\dot{\eta }}_m^i&= - (\Gamma _m)^i_{jk}(\gamma _m(t))\eta _m^j(t) \eta _m^k(t),\\ \gamma _m^i(t_0)&= p_m^i,\\ \eta _m^i(t_0)&= v_m^i. \end{aligned}$$

We define the continuous functions $f_m:{\mathbb {R}}^{2n} \rightarrow {\mathbb {R}}$, $f_m(x,y):=(y,-(\Gamma _m)^i_{jk}(x) y^j y^k)$, and similarly $f:{\mathbb {R}}^{2n} \rightarrow {\mathbb {R}}$, $f(x,y):=(y,-\Gamma _{jk}(x) y^j y^k)$. Then $f_m \rightarrow f$ uniformly on compact subsets of ${\mathbb {R}}^{2n}$, and by assumption $(p_m,v_m) \rightarrow (p,v) \in {\mathbb {R}}^{2n}$. The existence of an inextendible g-geodesic sublimit $\gamma$ of the sequence $\gamma _m$ now follows from Proposition 2.5, and the $C^2_{\mathrm {loc}}$-convergence follows suit. $\square$

In the case of only future or past inextendibility, an obvious analogue of Corollary 2.6 holds.

Remark 2.7

(Parallel transport) Under the assumptions of Corollary 2.6, let $w_m\in T_{\gamma _m(t_0)}M$, $w_m \rightarrow w \in T_pM$. Then since the ODEs governing parallel transport are linear, there are unique smooth vector fields $W_m$ along $\gamma _m$ that are global solutions of the corresponding initial value problem, and by the standard results on continuous dependence of solutions on the right hand side and the data, they converge in $C^1_{\mathrm {loc}}$ to the unique $C^1$-vector field W along $\gamma$ resulting from g-parallel transport of w along $\gamma$.

2.3 Energy conditions

Let us briefly recall the timelike and null energy conditions for $C^1$-spacetimes. They are formulated in a way so that they imply corresponding conditions for the curvatures of approximating metrics. If the metric is $C^{1,1}$ or better, then the energy conditions presented here are equivalent to the usual pointwise (or pointwise a.e.) energy conditions. We follow [7, Sec. 3-5], to which we also refer for proofs.

We start with the timelike energy condition, whose distributional generalisation is straightforward, cf. [7, Def. 3.3].

Definition 2.8

(Distributional timelike energy condition). Let (M, g) be a $C^1$-spacetime. We say that (M, g) satisfies the distributional timelike energy condition if for any timelike $X \in {\mathfrak {X}}(M)$, $\mathrm {Ric}(X,X) \in {\mathcal {D}}^{'(1)}(M)$ is a nonnegative scalar distribution on M.

Recall that a scalar distribution $u\in {\mathcal {D}}'(M)$ is nonnegative, $u\ge 0$, if $u(\omega )\equiv \langle u,\omega \rangle \ge 0$ for all nonnegative test densities $\omega \in \Gamma _c(M,\mathrm {Vol}(M))$. A nonnegative distribution is always a measure [19, Thm. 2.1.7] and hence a distribution of order 0. Moreover, non-negativity is stable with respect to regularisation [19, Thm. 2.1.9]. Finally, for $u,v\in {\mathcal {D}}'(M)$ we write $u\ge v$ if $u-v\ge 0$.

Remark 2.9

We note that an equivalent formulation of the distributional timelike energy condition consists in imposing that, for any $U\subseteq M$ open and any $X\in {\mathfrak {X}}(U)$ timelike, $\mathrm {Ric}(X,X)\ge 0$ in ${\mathcal {D}}^{'(1)}(U)$. Indeed, given a timelike $X\in {\mathfrak {X}}(U)$, let $V\Subset U$ be open and choose a cut-off function $\chi \in {\mathcal {D}}(U)$ that equals 1 on a neighbourhood of ${\overline{V}}$. By time-orientability of M there exists some timelike $T\in {\mathfrak {X}}(M)$ of the same time-orientation as X, so $Y:=\chi X +(1-\chi )T$ is a global timelike vector field on M. Then for any non-negative $\omega \in \Gamma _c(M,\mathrm {Vol}(M))$ with support in V we have $0\le \langle \mathrm {Ric}(Y,Y),\omega \rangle = \langle \mathrm {Ric}(X,X),\omega \rangle$, and by the sheaf property of ${\mathcal {D}}^{'(1)}$, we obtain $\mathrm {Ric}(X,X)\ge 0$ in ${\mathcal {D}}^{'(1)}(U)$.

The usefulness of Definition 2.8 is highlighted by its implications for the curvatures of approximating metrics, cf. [7, Lem. 4.1, Lem. 4.6].

Lemma 2.10

Let (M, g) be a $C^1$-spacetime satisfying the distributional timelike energy condition and let K be compact in M. Then

$$\begin{aligned}&\forall C>0 \ \forall \delta> 0 \ \forall \kappa< 0 \ \exists \varepsilon _0> 0 \ \forall \varepsilon < \varepsilon _0 \ \forall X \in TM|_K\\&\text {with } g(X,X) \le \kappa \text { and } \Vert X\Vert _h \le C: \ \mathrm {Ric}[{\check{g}}_{\varepsilon }](X,X) > - \delta . \end{aligned}$$

The following is an immediate consequence of the previous result.

Corollary 2.11

Let (M, g) be a $C^1$-spacetime satisfying the distributional timelike energy condition and let K be compact in M. Then

$$\begin{aligned} \forall \delta> 0 \ \exists \varepsilon _0> 0 \ \forall \varepsilon < \varepsilon _0 \ \forall X \in TM|_K \text { with } {\check{g}}_{\varepsilon }(X,X) = -1:\ \mathrm {Ric}[{\check{g}}_{\varepsilon }](X,X) > - \delta . \end{aligned}$$

Let us now turn to the proper distributional formulation of the null energy condition. The naive approach of defining it in analogy to Definition 2.8 does not yield analogues of Lemma 2.10 and Corollary 2.11 because vector fields that are ${\check{g}}_{\varepsilon }$-null are only almost g-null. This necessitates a definition of the following form.

Definition 2.12

(Distributional null energy condition, [7, Def. 5.1]). Let (M, g) be a $C^1$-spacetime. We say (M, g) satisfies the distributional null energy condition if for any compact set K and any $\delta > 0$ there exists $\varepsilon = \varepsilon (\delta ,K) > 0$ such that $\mathrm {Ric}(X,X) > - \delta$ (as distributions) for any local smooth vector field $X \in {\mathfrak {X}}(U)$ ($U \subseteq K$ open) with $\Vert X\Vert _h = 1$ and $|g(X,X)| < \varepsilon$ on U.

The above requirement $|g(X,X)| < \varepsilon$ may equivalently be replaced by $\Vert X - N\Vert _h < \varepsilon$, where N is a $C^1$ g-null vector field on U, cf. [7, Lem. 5.2]. Yet another useful equivalent reformulation is the following rescaled version, cf. [7, Def. 5.3].

Lemma 2.13

A $C^1$-spacetime (M, g) satisfies the distributional null energy condition if and only if for any compact $K\subseteq M$, any $c_1,c_2 > 0$ and any $\delta > 0$ there is $\varepsilon =\varepsilon (\delta ,K,c_1,c_2) > 0$ such that $\mathrm {Ric}(X,X) > - \delta$ (as distributions) for any local smooth vector field $X \in {\mathfrak {X}}(U)$, $U \subseteq K$, with $0 < c_1 \le \Vert X\Vert _h \le c_2$ and $|g(X,X)| < \varepsilon$ on U.

The following analogue of Lemma 2.10 and Corollary 2.11 shows that the above definition of the null energy condition is the correct one in the $C^1$-case (see [7, Lem. 5.5]).

Lemma 2.14

Let (M, g) be a $C^1$-spacetime satisfying the distributional null energy condition. Let $K\subseteq M$ be compact and let $c_1,c_2 > 0$. Then for all $\delta > 0$ there is $\varepsilon _0 = \varepsilon _0(\delta ,K,c_1,c_2) > 0$ such that

$$\begin{aligned}&\forall \varepsilon< \varepsilon _0 \ \forall X \in TM|_K \text { with } 0 < c_1 \le \Vert X\Vert _h \le c_2 \text { and }\\&{\check{g}}_{\varepsilon }(X,X) = 0:\ \mathrm {Ric}[{\check{g}}_{\varepsilon }](X,X) > - \delta . \end{aligned}$$

The above distributional versions of the timelike and null energy conditions were successfully used in [7] to prove the singularity theorems of Hawking and of Penrose for $C^1$-spacetimes. For our $C^1$-proof of the Hawking-Penrose singularity theorem, we need a distributional version of the genericity condition along geodesics.

Definition 2.15

(Distributional genericity condition). Let (M, g) be a $C^1$-spacetime and let $\gamma :I \rightarrow M$ be a causal geodesic. We say that the distributional genericity condition holds at $\gamma (t_0)$, $t_0 \in I$, if there is a neighbourhood U of $\gamma (t_0)$, $C^1$-vector fields X, V on U with the property that $(X \circ \gamma )(t) = {\dot{\gamma }}(t)$, $(V \circ \gamma )(t) \perp {\dot{\gamma }}(t)$ for all $t \in I$ with $\gamma (t) \in U$, and there exist $c > 0$ and $\delta > 0$ such that for all $C^1$-vector fields ${\tilde{X}}$, ${\tilde{V}}$ on U with $\Vert X - {\tilde{X}}\Vert _h < \delta$ and $\Vert V - {\tilde{V}}\Vert _h < \delta$ we have

$$\begin{aligned} g(R({\tilde{X}},{\tilde{V}}){\tilde{V}},{\tilde{X}}) > c \quad \text {in } {\mathcal {D}}^{'(1)}(U). \end{aligned}$$

(2.16)

This condition extends the one used in the $C^{1,1}$-case in [8, Def. 2.2] as we shall discuss in Remark 2.19, below. Again it allows us to derive an approximate genericity condition for regularisations. Note, however, that it will turn out to be essential to use $C^1$-vector fields (and their approximations), which can be inserted into $R\in {{\mathcal {D}}'}^{(1)}{{\mathcal {T}}}^1_3(M)$, by (2.3).

Also the distributional genericity condition appropriately restricts to smaller sets, a technical point which we clarify first.

Remark 2.16

In Definition 2.15 we may shrink the neighbourhood U while retaining the constants c and $\delta$. More precisely, let U, X, V, c, and $\delta$ be as in the definition and let $W\Subset U$, then (2.16) holds for all $C^1$-vector fields ${{\tilde{X}}}_W$, ${{\tilde{V}}}_W$ on W which satisfy $\Vert X|_W-{{\tilde{X}}}_W\Vert _h<\delta$ and $\Vert V|_W-{{\tilde{V}}}_W\Vert _h<\delta$. Indeed, by the sheaf property of distributions, it suffices to establish that (2.16) then holds locally around each point $x\in W$. To establish this let $\chi \in {\mathcal {D}}(W)$ be a plateau function that equals 1 in a neighbourhood $W'$ of x and set ${{\tilde{X}}}=X+\chi ({{\tilde{X}}}_W- X)$ and likewise for ${{\tilde{V}}}$. Then ${{\tilde{X}}}$ and ${{\tilde{V}}}$ are $C^1$-vector fields on U with $\Vert X - {\tilde{X}}\Vert _h < \delta$ and $\Vert V - {\tilde{V}}\Vert _h < \delta$ (2.16). So we obtain on $W'$

$$\begin{aligned} c< g(R({\tilde{X}},{\tilde{V}}){\tilde{V}},{\tilde{X}}) = g(R({\tilde{X}}_W,{\tilde{V}}_W){\tilde{V}}_W,{\tilde{X}}_W). \end{aligned}$$

(2.17)

Lemma 2.17

Let (M, g) be a $C^1$-spacetime and let $g_{\varepsilon }$ be either ${\check{g}}_{\varepsilon }$ or $\hat{g}_{\varepsilon }$. Suppose $\gamma$ is a g-causal g-geodesic and assume that the distributional genericity condition holds at $\gamma (t_0)$ (with neighbourhood U, vector fields X, V and constants $c,\delta$). Suppose that $X_{\varepsilon }$, $V_{\varepsilon }$ are $C^1$-vector fields on U with $X_{\varepsilon }, V_{\varepsilon } \rightarrow X,V$ in $C^1_{{\mathrm {loc}}}(U)$. Then for any compact subset $K\Subset U$ there is $\tilde{\varepsilon } > 0$ such that for all $\varepsilon < \tilde{\varepsilon }$:

$$\begin{aligned} g_{\varepsilon }(R[g_{\varepsilon }](X_{\varepsilon },V_{\varepsilon }) V_{\varepsilon },X_{\varepsilon }) > \frac{c}{2} \end{aligned}$$

on K.

Proof

We may take U to be relatively compact. By assumption, for all $C^1$-vector fields ${\tilde{X}},{\tilde{V}}$ on U with $\Vert X - {\tilde{X}}\Vert _h < \delta$ and $\Vert V - {\tilde{V}}\Vert _h < \delta$, we have

$$\begin{aligned} g(R({\tilde{X}},{\tilde{V}}){\tilde{V}},{\tilde{X}}) > c \quad \text {in } {\mathcal {D}}^{'(1)}(U). \end{aligned}$$

Since $X_{\varepsilon } \rightarrow X$ and $V_{\varepsilon } \rightarrow V$ in $C^1_{{\mathrm {loc}}}$, there is $\varepsilon _0 > 0$ such that for all $\varepsilon < \varepsilon _0$, $\Vert X_{\varepsilon } - X\Vert _h < \delta$ and $\Vert V_{\varepsilon } - V\Vert _h < \delta$. Let from now on $\varepsilon < \varepsilon _0$. Hence, the assumption gives

$$\begin{aligned} g(R(X_{\varepsilon },V_{\varepsilon })V_{\varepsilon },X_{\varepsilon }) > c \quad \text {in } {\mathcal {D}}^{'(1)}(U). \end{aligned}$$

Since $\star _M$-convolution with a non-negative mollifier $\rho _{\varepsilon }$ respects positivity, we have

$$\begin{aligned} g(R(X_{\varepsilon },V_{\varepsilon })V_{\varepsilon },X_{\varepsilon }) \star _M \rho _{\varepsilon } > c. \end{aligned}$$

We claim that

$$\begin{aligned} g(R(X_{\varepsilon },V_{\varepsilon })V_{\varepsilon },X_{\varepsilon }) \star _M \rho _{\varepsilon } - (g \star _M \rho _{\varepsilon }) (R[g\star _M \rho _{\varepsilon }](X_{\varepsilon },V_{\varepsilon })V_{\varepsilon },X_{\varepsilon }) \rightarrow 0 \quad \text {in } C^0_{\mathrm {loc}}(U). \end{aligned}$$

This is seen by first showing that

$$\begin{aligned} g(R(X_{\varepsilon },V_{\varepsilon })V_{\varepsilon },X_{\varepsilon }) \star _M \rho _{\varepsilon } - g((R[g] \star _M \rho _{\varepsilon })(X_{\varepsilon },V_{\varepsilon })V_{\varepsilon },X_{\varepsilon }) \rightarrow 0 \quad \text {in } C^0_{\mathrm {loc}}(U). \end{aligned}$$

The latter locally corresponds to the question whether given a net $h_{\varepsilon }$ of $C^1$-functions that converges in $C^1_{\mathrm {loc}}$ to $h \in C^1$ and a first order distribution T we can show $(h_{\varepsilon }T) *\rho _{\varepsilon } - h_{\varepsilon } (T * \rho _{\varepsilon }) \rightarrow 0$ in $C^0_{\mathrm {loc}}$. This is shown by arguing along the lines of [7, Lem. 4.8] and using that T can locally be written as a first order distributional derivative of a $C^0$-function. By a similar argument (noting the fast convergence speed of $g \star _M \rho _{\varepsilon } \rightarrow g$), the right hand side above is $C^0_{\mathrm {loc}}$-equivalent (i.e. the difference goes to 0 in $C^0_{\mathrm {loc}}$) to

$$\begin{aligned} (g \star _M \rho _{\varepsilon })(R[g] \star _M \rho _{\varepsilon }(X_{\varepsilon },V_{\varepsilon })V_{\varepsilon },X_{\varepsilon }). \end{aligned}$$

This is $C^0_{\mathrm {loc}}$-equivalent to

$$\begin{aligned} (g \star _M \rho _{\varepsilon })(R[g \star _M \rho _{\varepsilon }](X_{\varepsilon },V_{\varepsilon })V_{\varepsilon },X_{\varepsilon }) \end{aligned}$$

by the same arguments as in [7, Lem. 4.6] and uniform boundedness of $X_\varepsilon$, $V_{\varepsilon }$.

Next, we show that

$$\begin{aligned} (g \star _M \rho _{\varepsilon })(R[g \star _M \rho _{\varepsilon }](X_{\varepsilon },V_{\varepsilon })V_{\varepsilon },X_{\varepsilon }) - g_{\varepsilon }(R[g_{\varepsilon }](X_{\varepsilon },V_{\varepsilon })V_{\varepsilon }, X_{\varepsilon }) \rightarrow 0 \end{aligned}$$

in $C^0_{\mathrm {loc}}(U)$. Upon writing everything out in coordinates, we see that only terms of the form

$$\begin{aligned} (g \star _M \rho _{\varepsilon }) \partial \Gamma [g \star _M \rho _{\varepsilon }] X_{\varepsilon } V_{\varepsilon } V_{\varepsilon } X_{\varepsilon } - g_{\varepsilon } \partial \Gamma [g_{\varepsilon }] X_{\varepsilon } V_{\varepsilon } V_{\varepsilon } X_{\varepsilon } \end{aligned}$$

are of interest, because for any of the other terms in the difference, both summands converge in $C^0_{\mathrm {loc}}(U)$ to the respective expressions for g. Since the $X_{\varepsilon }$, $V_{\varepsilon }$ are uniformly bounded, we may omit them in our estimates without loss of information. But for terms as above, we only need to consider those expressions where we have second derivatives of the metrics, since $\partial _j (g \star _M \rho _{\varepsilon })^{mk} \rightarrow \partial _j g^{mk}$ in $C^0_{\mathrm {loc}}(U)$ and similarly for $\partial _j (g_{\varepsilon })^{mk}$. Hence, we work in a chart, fix a compact set K and are left with estimating the following expression on K:

$$\begin{aligned}&|(g \star _M \rho _{\varepsilon })_{bc} (g \star _M \rho _{\varepsilon })^{mk} \partial _s \partial _r (g \star _M \rho _{\varepsilon })_{ij} - (g_{\varepsilon })_{bc} (g_{\varepsilon })^{mk} \partial _s \partial _r (g_{\varepsilon })_{ij}|\\&\le |(g \star _M \rho _{\varepsilon })_{bc} (g \star _M \rho _{\varepsilon })^{mk}| \cdot |\partial _s \partial _r (g \star _M \rho _{\varepsilon })_{ij} - \partial _s \partial _r (g_{\varepsilon })_{ij}| + II. \end{aligned}$$

The first term goes to 0 uniformly on K by Lemma 2.3(iii). We continue estimating the remaining term II:

$$\begin{aligned} II&= |(g \star _M \rho _{\varepsilon })_{bc} (g \star _M \rho _{\varepsilon })^{mk} \partial _s \partial _r (g_{\varepsilon })_{ij} - (g_{\varepsilon })_{bc} (g_{\varepsilon })^{mk} \partial _s \partial _r (g_{\varepsilon })_{ij}|\\&\le |\partial _s \partial _r (g_{\varepsilon })_{ij}| \cdot |(g \star _M \rho _{\varepsilon })_{bc} (g \star _M \rho _{\varepsilon })^{mk} - (g_{\varepsilon })_{bc} (g_{\varepsilon })^{mk}|. \end{aligned}$$

To show that this goes to 0 uniformly on K, it suffices to show that the second factor can be estimated by a constant times $\varepsilon$ by the same argument as in the proof of [7, Lem. 4.5]. Since $\Vert g_{\varepsilon } - (g \star _M \rho _{\varepsilon })\Vert _{\infty ,K}$ and $\Vert g_{\varepsilon }^{-1} - (g \star _M \rho _{\varepsilon })^{-1}\Vert _{\infty ,K}$ can both be estimated by a constant times $\varepsilon$ again by Lemma 2.3(iii), this is easily seen to hold.

Hence, there is $\varepsilon _1 > 0$ such that for all $\varepsilon < \min (\varepsilon _0,\varepsilon _1)$,

$$\begin{aligned} g_{\varepsilon }(R[g_{\varepsilon }](X_{\varepsilon },V_{\varepsilon })V_{\varepsilon }, X_{\varepsilon }) > \frac{c}{2} \quad \text {on } K. \end{aligned}$$

$\square$

Next we derive from the distributional genericity condition an estimate on the tidal force operator for the regularised metrics. It is this result which allows us to enter into the Riccati comparison techniques in Sect. 4.

Lemma 2.18

Let the assumptions of Lemma 2.17 hold and let $\gamma _\varepsilon$ be $g_{\varepsilon }$-causal $g_{\varepsilon }$-geodesics, whose $g_{\varepsilon }$-causal character is the same as the g-causal character of $\gamma$ and which converge in $C^1_{\mathrm {loc}}$ to $\gamma$. Then there are ${{\tilde{c}}}>0$, $r>0$, $\varepsilon _0>0$ and a neighbourhood U of $\gamma (t_0)$ such that for any $0<\varepsilon <\varepsilon _0$ there exist vector fields $E_i^\varepsilon$ on U such that $E_i^\varepsilon \circ \gamma _\varepsilon$ constitutes a $g_\varepsilon$-orthonormal frame along $\gamma _\varepsilon$ and such that $E_i^\varepsilon \rightarrow E_i$ in $C^1_{\mathrm {loc}}(U)$, where $E_i\circ \gamma$ is an orthonormal frame along $\gamma$. Further, there exists some $C=C(\varepsilon )>0$ such that along $\gamma _\varepsilon$ the $\varepsilon$-tidal force operator $[R_\varepsilon ](t):= [R_\varepsilon (.,{\dot{\gamma }}_\varepsilon (t)){\dot{\gamma }}_\varepsilon (t)]:[{\dot{\gamma }}_\varepsilon (t)]^\perp \rightarrow [{\dot{\gamma }}_\varepsilon (t)]^\perp$ fulfills

$$\begin{aligned}{}[R_\varepsilon ](t) > \mathrm {diag}({{\tilde{c}}},-C,\ldots ,-C) \text { on } [t_0-r,t_0+r] \end{aligned}$$

(2.18)

in terms of the frame $E_i^\varepsilon$ and where we have used the shorthand $R_\varepsilon$ for $R[g_\varepsilon ]$.

Proof

Let w.l.o.g. $t_0=0$. We deal with the case of $\gamma$ being null or timelike simultaneously and construct a g-orthonormal frame for $[{\dot{\gamma }}]^\perp$ along $\gamma$. First note that, if $\gamma$ is timelike, $V\circ \gamma$ is nowhere proportional to ${\dot{\gamma }}$, but the same may not be true in the null case. First we will show that we can replace V by a vector field whose restriction to $\gamma$ is parallel in $[{\dot{\gamma }}]^\perp$ and still retain (2.16).

There must exist some t such that $V\circ \gamma (t)\not \in \text {span}({\dot{\gamma }}(t))$. Indeed, suppose to the contrary that $V(\gamma (t)) = f(t)\cdot X(\gamma (t))$ for some real-valued $C^1$ function f and all t. Then extending both $X\circ \gamma$ and $V\circ \gamma$ in a cylindrically constant fashion as given by Lemma B.1, we obtain vector fields ${{\tilde{V}}}, {{\tilde{X}}}$ on a neighbourhood W of $\gamma (0)$ that are proportional and can be made arbitrarily close to V resp. X in $C^0$ when shrinking W since they restrict to the same vector fields on $\gamma$. According to Definition 2.15 they therefore satisfy (2.16) close to $\gamma (0)$. However, by the symmetry properties of R (cf. Lemma 2.4) and the proportionality of ${{\tilde{V}}}$ and ${{\tilde{X}}}$ we must have $R({{\tilde{X}}},{{\tilde{V}}}){{\tilde{V}}} =0$ in ${\mathcal {D}}'(W)$, a contradiction.

Hence there exists some $v:=V(t_1)$ with $v\in [{\dot{\gamma }}(t_1)]^\perp$. Defining $V_1$ as the parallel translate of v along $\gamma$ we obtain $V_1\in [{\dot{\gamma }}]^\perp$ along all of $\gamma$. By the same continuity argument as above, (2.16) then still holds for the vector fields $V_1$ and X (note that a $t_1$ as above must exist arbitrarily close to 0). In the timelike case we may assume that $\gamma$ is parametrised to unit speed, and we set $e_n:={\dot{\gamma }}(0)$. If $\gamma$ is null we can write ${\dot{\gamma }}(0)=e_{n-1} + e_n$ for orthonormal vectors $e_{n-1}, e_n$ with $e_n$ timelike.

In the null case, extend $e_1:=V_1(0)\in [{\dot{\gamma }}(0)]^\perp$ to a g-orthonormal basis $\{e_i\}_{i=1}^n$ (with $e_{n-1}, e_n$ as above) and define $E_i$ to be the g-parallel translate of $e_i$ along $\gamma$ (so that $E_1=V_1$ and $X=E_{n-1}+E_n$ along $\gamma$). For $\gamma$ timelike we also construct an orthonormal frame $E_1=V_1, E_2,\dots , E_n=X$ along $\gamma$ via parallel transport. As $\gamma _\varepsilon \rightarrow \gamma$ in $C^1_{\mathrm {loc}}$, we can find a $g_\varepsilon$ orthonormal basis $\{e^\varepsilon _i\}$ at $\gamma _\varepsilon (0)$ with $e^\varepsilon _i\rightarrow e_i$ in TM, and such that ${\dot{\gamma }}_\varepsilon (0) = e^\varepsilon _n$ if $\gamma _\varepsilon$ is $g_\varepsilon$-timelike, while ${\dot{\gamma }}_\varepsilon (0) = e^\varepsilon _{n-1}+e^\varepsilon _n$ with $e^\varepsilon _n$ timelike if $\gamma _\varepsilon$ is $g_\varepsilon$-null. For every $\varepsilon$ let $E_i^\varepsilon$ be the $g_\varepsilon$-parallel translate of $e^\varepsilon _i$ along $\gamma _\varepsilon$. By Remark 2.7, $E_i^\varepsilon \rightarrow E_i$ in $C^1_{\mathrm {loc}}$ (i.e. in T(TM), uniformly on compact time intervals) and we can use Lemma B.2 to extend $E_i^\varepsilon$ to vector fields on U, converging in $C^1_{\mathrm {loc}}(U)$ to the extensions of $E_i$ to U.

As we have now found vector fields $E^\varepsilon _i$ that restrict to a $g_\varepsilon$-frame along $\gamma _\varepsilon$ and converge to the g-frame $E_i$ along $\gamma$, we are in a position to apply Lemma 2.17 and hence estimate the $g_\varepsilon$-tidal force operator along $\gamma _\varepsilon$.

For the remainder of the proof we roughly follow the layout of the proof of [8, Prop. 3.6]. There, for a $C^{1,1}$-metric g the statement is first shown for the g-tidal force operator along $\gamma$ and then carried over for small $\varepsilon$ by an approximation argument. However, in the $C^1$-setting this is no longer possible, as R is now merely distributional and for a matrix with distributional entries eigenvalues are not defined. Also, a direct approximation procedure of that form is no longer feasible. Instead we use Lemma 2.17, and argue separately for any $\varepsilon$, making the procedure (and hence the constant C) dependent on $\varepsilon$⁵. More precisely:

Let c be the constant given by the genericity condition and let ${{\tilde{c}}}:=\frac{c}{2}$ and $0<c_1<{{\tilde{c}}}$. Set $R^\varepsilon _{ij}:= \langle R_\varepsilon (E^\varepsilon _i, X_\varepsilon ) X_\varepsilon , E^\varepsilon _j \rangle$, $i,j=1,\dots ,d$, where $X_\varepsilon = E_n^\varepsilon$ in the timelike case and $X_\varepsilon = E_{n-1}^\varepsilon + E_n^\varepsilon$ in the null case. As in the proof of (3.5) in [8, Lemma 3.7], we choose $C(\varepsilon ) >0$ such that $\lambda ^\varepsilon _\text {min}(x) \ge \frac{\Vert (R^\varepsilon _{1j})_j\Vert _e}{{{\tilde{c}}}-c_1}$ for all $x\in U$, where $\lambda ^\varepsilon _\text {min}(x)$ is the smallest eigenvalue of $(R^\varepsilon _{ij})_{i,j=2}^d+C(\varepsilon ) \text {Id}_{d-1}$ at $x\in U$. By Lemma 2.17 there exists some ${\tilde{\varepsilon }}>0$ such that $R^\varepsilon _{11}>{{\tilde{c}}}$ for all $0<\varepsilon <{\tilde{\varepsilon }}$ on U (shrinking U once more if necessary). For these $\varepsilon$ it then follows that $R^\varepsilon _{ij}-\text {diag}(c_1, - C(\varepsilon ), \ldots -C(\varepsilon ))$ is positive definite on U. Then picking $r>0$ such that $\gamma _\varepsilon ([-r,r]) \subseteq U$ for all $\varepsilon$ small, evaluation along $\gamma _\varepsilon$ gives the claim. $\quad \square$

Remark 2.19

Note that while Definition 2.15 may at first sight appear to be stronger than a straightforward generalisation of the genericity condition given for $C^{1,1}$-metrics in [8], due to the local boundedness of R in that regularity it is actually equivalent:

For $C^{1,1}$-metrics Definition 2.15 clearly implies Definition 2.2 of [8]. For the reverse implication, let X, V be continuous vector fields in a neighbourhood of $\gamma (t_0)$ such that $X(\gamma (t))={\dot{\gamma }}(t)$ and $V(\gamma (t))\in ({\dot{\gamma }}(t))^\perp$ for all t and such that $\langle R(V,X)X,V\rangle>c>0$. Arguing exactly as in the proof of Lemma 2.18 we can then use parallel transport and cylindrically constant extension to find $C^1$-vector fields that are uniformly close to X and V. It therefore suffices to show that the genericity condition from [8] is stable under locally uniform perturbation. Thus let ${{\tilde{V}}}, {{\tilde{X}}}$ be $C^1$ vector fields that are uniformly close to V, X. Then one can locally in a coordinate system estimate $|{{\tilde{V}}}_i - V_i| \le \delta$, $|{{\tilde{X}}}_i - X_i| \le \delta$ for some small $\delta$. Consequently,

$$\begin{aligned} \langle R({{\tilde{V}}}, {{\tilde{X}}}){{\tilde{X}}},{{\tilde{V}}} \rangle = g_{ij}R^i_{klm} {{\tilde{V}}}^l \tilde{X}^m {{\tilde{X}}}^k {{\tilde{V}}}^j \ge g_{ij}R^i_{klm}V^l X^m X^k V^j+ {\mathcal {O}} (\delta ) \end{aligned}$$

where the last term collects all $\delta$-contributions and takes into account the local boundedness of R. Consequently, choosing $\delta$ small enough (and appropriately shrinking the neighbourhood of $\gamma (t_0)$) we can secure that the sum still remains $>c/2$.

3 Branching

In metric geometry, where geodesics are defined as (local) minimisers of the length functional, local uniqueness of geodesics is expressed in the form of a non-branching condition. Here, a branch point is defined as an element of a minimiser at which the curve splits into two minimisers that on some positive parameter interval do not have another point in common, cf., e.g., [44, 51]. Similarly, in the synthetic Lorentzian setting [22], the role of causal geodesics is taken on by maximising causal curves, and non-branching is formulated analogously. In both cases, lower synthetic sectional curvature bounds (formulated via triangle comparison in constant curvature model spaces) imply non-branching [22, 44]. Synthetic Ricci curvature bounds in metric measure spaces (curvature-dimension conditions), on the other hand, do not imply non-branching of geodesics [38]. Similarly, in recent work on synthetic Ricci curvature bounds in Lorentzian pre-length spaces [4], a timelike non-branching condition is required in addition to a timelike-curvature dimension condition to obtain a version of the Hawking singularity theorem.

In the $C^1$-setting we are concerned with in this paper, the coincidence between causal local maximisers and geodesics that is familiar from smooth Lorentzian geometry ceases to hold (cf. Example 3.2 below), although it is still true that causal maximisers are geodesics (Lemma A.4). In addition, one generically cannot expect unique local solvability of the geodesic initial value problem. Nevertheless, on physical grounds, it seems reasonable to assign a privileged role to causal geodesics that are locally maximising. We therefore introduce a (weak notion of a) non-branching condition that is intended to preclude locally maximising geodesics from branching (where we do not require the second branch to be maximising as well):

Definition 3.1

(Non-branching conditions). Let (M, g) be a $C^1$-spacetime. A geodesic $\gamma :[a,b]\rightarrow M$ branches at $t_0\in (a,b)$ if there exist $\varepsilon >0$ and some geodesic $\sigma$ with $\gamma |_{[t_0-\varepsilon , t_0]}\subseteq \sigma$, but $\gamma |_{(t_0,t_0+\varepsilon )}\cap \sigma = \emptyset$.

(M, g) is called maximally causally (resp. timelike, resp. null) non-branching (MCNB, MTNB, MNNB), if no maximal causal (resp. timelike, resp. null) geodesic branches in the above sense.

Example 3.2

A $C^1$-spacetime in which maximal causal branching occurs can be constructed from the second Riemannian example given in [16]. There, a $C^{1,\alpha }$-Riemannian metric ($\alpha <1$) in the (u, v)-plane is given with non-unique solutions to the geodesic initial value problem starting at $\{v=0\}$. Further it is shown that the geodesic boundary value problem for geodesics $\gamma$ starting at the surface $\{v=0\}$ is uniquely solvable and hence such geodesics are at least initially minimizing: If they were not, by properties of any $C^1$-Riemannian manifold as a locally compact length space, there would exist a minimiser $\sigma$ between two points on $\gamma$. However in $C^1$-Riemannian manifolds minimisers are geodesics and by uniqueness (of the boundary value problem) $\gamma =\sigma$ (cf. [42] for proofs and references).

In [18] it is shown that causal geodesics in static spacetimes are maximising if and only if their Riemannian parts are minimizing and so one can easily construct both maximising timelike and null geodesics that branch.

Lemma 3.3

If the $C^1$-spacetime (M, g) is MCNB (resp. MTNB, MNNB), then any two maximal causal (resp. timelike, null) geodesics that meet tangentially at an interior point must coincide on their maximal domain of definition.

Proof

We only show this for MCNB spacetimes, the other cases are proven in exactly the same way by replacing all occurrences of “causal" with “timelike” resp. “null”. Let $\gamma , \sigma$ be two maximising, causal geodesics meeting tangentially at p, which is not an endpoint of either curve. W.l.o.g. we may assume that $\gamma (0)=\sigma (0)=p$. It suffices to show that there is $\varepsilon >0$ such that $\gamma |_{[0,\varepsilon )}=\sigma |_{[0,\varepsilon )}$. Indeed since then $\gamma$ and $\sigma$ meet tangentially at $t=\varepsilon$, the maximal such interval coincides with the maximal domain of definition (to the right and analogously to the left).

Assume to the contrary that $\gamma |_{[0,\varepsilon )} \ne \sigma |_{[0,\varepsilon )}$ for any $\varepsilon$. As the curves both satisfy the geodesic equation at p and their tangents agree, so do their second derivatives. Thus $\gamma$ and $\gamma |_{(-\varepsilon ,0]}\cup \sigma |_{[0,\varepsilon )}$ are both (unbroken) geodesics and by MCNB $\gamma$ cannot branch at $t=0$. So for every $\varepsilon >0$ we have $\gamma |_{(0,\varepsilon )}\cap \sigma |_{(0,\varepsilon )}\ne \emptyset$. Hence there exist $t_k \downarrow 0$ such that $\gamma (t_k)=\sigma (t_k)$, and by our indirect assumption there exists $s_k \downarrow 0$ such that $\gamma (s_k) \ne \sigma (s_k)$. For any k set

$$\begin{aligned} \eta _k:= \sup \{a\in [0,s_k]: \gamma |_{(s_k-a,s_k]} \cap \sigma |_{(s_k-a,s_k]}= \emptyset \}. \end{aligned}$$

Further choose some ${\tilde{\eta }}_k>0$ such that for $I_k:=(s_k-\eta _k,s_k+{\tilde{\eta }}_k)$ we have $\gamma |_{I_k}\cap \sigma |_{I_k}= \emptyset$. Since $t_k \downarrow 0$ and $\gamma (t_k)=\sigma (t_k)$, we know that $s_k>\eta _k$. However, $\gamma (s_k-\eta _k)=\sigma (s_k-\eta _k)$ by our choice of $\eta _k$. Set $s_k-\eta _k=r_k$ and for simplicity of notation assume $r_k<t_k$. If ${\dot{\gamma }}(r_k)\ne {\dot{\sigma }}(r_k)$, the curve $\alpha := \gamma |_{[0,r_k]}\cup \sigma |_{[r_{k},t_k]}$ is a broken geodesic from p to $\gamma (t_k)$ and hence not maximising by [43, Lem. 3.2], i.e., $d(p,\gamma (t_k))> L(\alpha )$. Since $\gamma$ and $\sigma$ are maximisers we have $d(p,\gamma (r_{k}))= L(\gamma |_{[0,r_k]})=L(\sigma |_{[0,r_k]})$ Consequently,

$$\begin{aligned} d(p,\sigma (t_k))=L(\sigma |_{[0,t_k]})= L(\sigma |_{[0,r_k]})+L(\sigma |_{[r_k,t_k]})=L(\alpha )< d(p,\sigma (t_k)), \end{aligned}$$

a contradiction, and hence ${\dot{\gamma }}(r_k)={\dot{\sigma }}(r_k)$.

Again employing the geodesic equation, also the second derivatives of $\gamma$ and $\sigma$ coincide at $r_k$, and so the curves $\gamma |_{[0,s_k]}$ and $\gamma |_{[0,r_k]}\cup \sigma |_{[r_k,s_k]}$ display maximal causal branching, a contradiction. $\quad \square$

The following result shows that in maximally causally non-branching spacetimes, maximal causal geodesics can always be approximated by geodesics for the regularised spacetimes ${\hat{g}}_\varepsilon$ and ${\check{g}}_\varepsilon$. Whether such a result is true in general is questionable and very likely regularisation dependent.

Proposition 3.4

Let (M, g) be a $C^1$-spacetime that is MCNB.

• (i) Suppose that M is globally hyperbolic and let $\varepsilon _k\searrow 0$ ($k\rightarrow \infty$). Set $g_k := {\hat{g}}_{\varepsilon _k}$ or $g_k := {\check{g}}_{\varepsilon _k}$ for all k. If $\gamma :[0,a]\rightarrow M$ is a maximising, timelike g-geodesic then for any small $\delta >0$ there exists a subsequence $g_{k_l}$ of $g_k$ and $g_{k_l}$-maximising, timelike geodesics $\gamma _l$ converging in $C^1$ to $\gamma |_{[0,a-\delta ]}$.

• (ii) Let $g_k = {\check{g}}_{\varepsilon _k}$ for all k. If M is causal and $\gamma :[0,a]\rightarrow M$ is a g-maximizing null geodesic, then for any small $\delta >0$, there exists a subsequence $g_{k_l}$ of $g_k$, $t_l\downarrow 0$ and $g_{k_l}$-null geodesics $\gamma _l:[t_l, a-\delta ]\rightarrow M$ contained in $\partial I_l^+(\gamma (0))$ (hence in particular $g_{k_l}$-maximising), which converge in $C^1_{\mathrm {loc}}$ to $\gamma |_{[0,a-\delta ]}$.

  If, furthermore, M is strongly causal and S is an acausal set such that $\gamma \subseteq E^+(S)$, then even $\gamma _l:[t_l,a-\delta ]\rightarrow E^+_l(S)$ and $\gamma _l(t_l) \in S$ with $\gamma _l(t_l)\rightarrow \gamma (0)$.

Proof

Fix $\delta \in (0,a)$ and set $p:=\gamma (0)$, $q:=\gamma (a)$, and $q_\delta := \gamma (a-\delta )$.

• (i) Since $\gamma$ is timelike, $q_\delta \in I^+(p)$ and without loss of generality we may assume that also $q_\delta \in I^+_k(p)$ for all k. Since g is globally hyperbolic we can also assume w.l.o.g. that $g_k$ are globally hyperbolic as well (see [26, Prop. 2.3 (iv)], which remains valid for $g\in C^1$). Thus there exist $g_k$-maximising, timelike geodesics $\gamma _k$ from p to $q_\delta$ [41, Prop. 6.5]. By Corollary 2.6 an affinely reparametrised subsequence, without loss of generality $\gamma _k$ itself, converges in $C^1_{\mathrm {loc}}$ to a g-causal geodesic $\sigma$. By Lemma A.13$\sigma$ is a g-maximising timelike geodesic from p to $q_\delta$, in particular $d(p,q_\delta )=L(\gamma |_{[0,a-\delta ]})= L(\sigma )$. We cannot yet apply Lemma 3.3 since it is not clear whether $\sigma$ is maximising beyond $q_\delta$. However denote by $\eta := \sigma \cup \gamma |_{[a-\delta ,a]}$ the concatenation of $\sigma$ and $\gamma$. Since $d(p,q)=L(\gamma )=L(\gamma |_{[0,a-\delta ]})+L(\gamma |_{[a-\delta ,a]})= L(\sigma )+L(\gamma |_{[a-\delta ,a]})= L(\eta )$, it is a maximising, timelike geodesic from p to q. We can now apply Lemma 3.3 to the curves $\eta$ and $\gamma$, to obtain $\eta =\gamma$ and hence $\sigma =\gamma |_{[0,a-\delta ]}$.

• (ii) A similar statement was shown in [43, Cor. 3.5], but we need to adapt some of the main points of the argument to our assumptions. Choose a sequence of points $q_\delta ^k \in \partial I^+_{g_k}(p)$ converging to $q_\delta$. There exist future directed, $g_k$-null maximising geodesics $\gamma _k$ ending at $q_\delta ^k$ and contained in $\partial I^+_{g_k}(p)$. W.l.o.g we may assume their terminal velocities to be h-normalized, so we can apply Corollary 2.6 to obtain a subsequence of $\gamma _k$ (denoted in the same way) converging to a g-null geodesic $\sigma$. Since $\partial I^+_{g_k}(p)\subseteq \overline{I^+(p)}$ (recall that $g_k=\check{g}_{\varepsilon _k}$) we know that $\sigma \subseteq \partial I^+(p)$, since otherwise $q_\delta \in I^+(p)$, hence $\sigma$ is maximising (cf. Lemma A.15). Note that $\gamma$ and $\sigma$ meet tangentially at $q_\delta$, because otherwise $\eta :=\sigma \cup \gamma |_{[a-\delta ,a]}$ would not be maximising by Lemma A.4 and so $q \in I^+(\sigma )\subseteq I^+(\overline{I^+(p)})\subseteq I^+(p)$, a contradiction. Lemma 3.3 now shows that $\gamma =\eta$ and so $\sigma =\gamma$, wherever both curves are defined. There are two possibilities for $\sigma$, either it is past inextendible or it reaches p. However, if it is past inextendible, we must have $\sigma \supseteq \gamma |_{[0,a-\delta ]}$. Hence in both cases the $\gamma _k$ converge to $\gamma$ in $C^1_\text {loc}$. Further this means that there are $t_k\in [0,a-\delta ]$ with $\gamma _k(t_k)\rightarrow p$, and we have $t_k \rightarrow t_0= 0$ since otherwise $\gamma (t_0)=p=\gamma (0)$, contradicting the causality of M. Now assume M to be strongly causal and $\gamma \subseteq E^+(S)$ for S acausal, then there exists a g-causally convex, g-globally hyperbolic neighbourhood U of p by [32, Lem. 3.21] (this result holds also for $C^1$-spacetimes). In such neighbourhoods it holds that $I_U^+(S)=I^+(S)\cap U$, $J_U^+(S)=J^+(S)\cap U$ and hence also $E_U^+(S)=\partial I_U^+(S)=\partial I^+(S)\cap U$. Clearly U is also $g_k$-causally convex and $g_k$-globally hyperbolic and so these equalities also hold for $g_k$. Note that in the same fashion as for the case of a single point p one can show that $\gamma _k \rightarrow \gamma |_{[0,a-\delta ]}$. If $\gamma _k$ reaches S, we immediately obtain $\gamma _k \subseteq E^+_k(S)$. In the other case $\gamma _k\subseteq \partial I^+_k(S)\backslash J^+_k(S)$ is past inextendible. However, as they converge to $\gamma |_{[0,a-\delta ]}$, there must be $p_k\in \gamma _k$, with $p_k\rightarrow p$. Hence, w.l.o.g. $p_k \in \partial I_k^+(S)\cap U= E_{U,k}^+(S)$ and so $p_k\in J^+_k(S)$, a contradiction. We have shown that $\gamma _k$ reach S, and it only remains to show that they do so close to 0. Again let $\gamma _k(t_k)\in S$ with $\gamma _k(t_k) \rightarrow p$. If $t_k\rightarrow t_0$, then $\gamma (t_0)\in S$ and by acausality of S we must have $t_0=0$. $\quad \square$

Remark 3.5

• (i) As the proof of Proposition 3.4 shows, for (i) it suffices to assume (M, g) to be MTNB, while for (ii) MNNB would suffice.

• (ii) We will make use of Proposition 3.4(ii) in two different scenarios: Thm. 4.3 and Proposition 5.5 below. In the first case (M causal) all we need to ensure is that $\gamma _l$ are maximizing for long enough, i.e. $a-\delta -t_l$ is close to a. In the second (M strongly causal) it is important to also obtain that $\gamma _l$ reaches S, in order to be able to use the initial conditions assumed on S.

4 No Lines

In this section we will prove that, under suitable causality and energy conditions, complete causal geodesics stop being maximising also in (MCNB) $C^1$-spacetimes. To this end we first prove the existence of conjugate points for causal geodesics in smooth spacetimes under weakened versions of the energy conditions. More precisely we will invoke the energy conditions derived for regularisations in Lemma 2.10 and in Lemma 2.14 from the distributional energy conditions, as well as in Lemma 2.18 from the distributional genericity condition. The following is a strengthened version of [8, Prop. 4.2]:

Lemma 4.1

Let (M, g) be a smooth spacetime. Given some $c>0$ and $0<r< \frac{\pi }{4 \sqrt{c}}$, there exist $\delta (c,r)>0$ and $T(c,r)>0$ such that any causal geodesic $\gamma$ defined on $[-T,T]$ for which the following hold

• (i) $\text {Ric}({\dot{\gamma }},{\dot{\gamma }})\ge - \delta$ on $[-T,T]$ and

• (ii) there exists a smooth parallel orthonormal frame for $[{\dot{\gamma }}]^\perp$ and some $C>0$ such that w.r.t. this frame the tidal force operator satisfies $[R](t) > \mathrm {diag}(c,-C,\ldots ,-C)$ on $[-r,r]$,

possesses a pair of conjugate points on $[-T,T]$.

Proof

This actually follows by a more careful choice of constants in the proof of [8, Prop. 4.2]. To see this we briefly recall the main steps of that proof. The argument proceeds indirectly, assuming that for any $\delta >0$ and $T>0$ there is some $\gamma$ satisfying (i) and (ii) without conjugate points in $[-T,T]$. Denote by [A] the unique Jacobi tensor class along $\gamma$ with $[A](-T)=0$ and $[A](0)=\mathrm {id}$. Taking $[E_1](t),\dots ,$ $[E_{d}](t)$ as in (ii), linear endomorphisms of $[{{\dot{\gamma }}}]^\perp$ are written as matrices in this basis, and we set $[{{\tilde{R}}}](t):=\mathrm {diag}(c,-C,\dots ,-C)$ with C as in (ii). Then by (ii), $[{{\tilde{R}}}](t)<[R](t)$ on $[-r,r]$.

The self-adjoint operator $[B]:=[\dot{A}]\cdot [A]^{-1}$satisfies the matrix Riccati equation

$$\begin{aligned} {[}\dot{B}] + [B]^2 + [R] = 0, \end{aligned}$$

(4.1)

and we denote by $[{{\tilde{B}}}]$ the solution to (4.1), with $[{{\tilde{R}}}]$ instead of [R] and initial value prescribed at some $t_1\in [-r,r]$. We may even choose $t_1$ in $[-r,0]$ and $[{{\tilde{B}}}](t_1) := {\tilde{\beta }}(t_1)\cdot \mathrm {id}$, where ${\tilde{\beta }}(t_1)$ is greater or equal than the largest eigenvalue of $[B](t_1)$. More precisely, examining the Raychaudhuri equation

$$\begin{aligned} {{\dot{\theta }}} + \frac{1}{d}\theta ^2 + \mathrm {tr}(\sigma ^2) + \mathrm {tr}([R]) = 0, \end{aligned}$$

(4.2)

for the expansion $\theta =\mathrm {tr}([B])$ (with $\sigma =[B]-\frac{1}{d}\theta \cdot \mathrm {id}$), it follows that one can pick ${\tilde{\beta }}(t_1):=f(\nu ,\delta ,r)$ and $[\tilde{B}](t_1):=f(\nu ,\delta ,r)\cdot \mathrm {id}$ to indeed achieve that $[B](t)\le [{{\tilde{B}}}](t)$ on $[t_1,r]$. Here, $f(\nu ,\delta ,r) = \sqrt{\frac{2\nu }{r}+\delta } + \frac{\nu }{d}$ with $\nu =4d/T$.

Moreover, due to the fact that both $[{{\tilde{R}}}]$ and $[{{\tilde{B}}}](t_1)$ are diagonal, the Riccati equation for $[{{\tilde{B}}}]$ decouples. Indeed it can be explicitly solved by

$$\begin{aligned}{}[{{\tilde{B}}}](t) = \frac{1}{d}\mathrm {diag}(H_{c,f}(t),H_{-C,f}(t),\dots ,H_{-C,f}(t)), \end{aligned}$$

where

$$\begin{aligned} H_{c,f}(t) = d \sqrt{c}\cot (\sqrt{c}(t-t_1) + \mathrm {arccot}(f/\sqrt{c})), \end{aligned}$$

and

$$\begin{aligned} H_{-C,f}(t)=d \sqrt{C}\tanh \big (\sqrt{C}(t-t_1)+\mathrm {artanh}(f/\sqrt{C})\big ). \end{aligned}$$

From this point on, the proof of [8, Prop. 4.2] proceeds by exclusively analysing the function $H_{c,f}$, and the only requirement on r turns out to be $4r\sqrt{c}<\pi$. Once this is granted, $\delta$ and T can be chosen depending only on c to arrive at the desired contradiction. $\square$

Recall that a line is an inextendible causal geodesic maximising the Lorentzian distance between any of its points.

Theorem 4.2

(No timelike lines). Let (M, g) be a globally hyperbolic, MTNB $C^1$-spacetime satisfying the distributional timelike energy condition and the distributional genericity condition along any inextendible timelike geodesic. Then there is no complete timelike line in M.

Proof

Suppose $\gamma :{\mathbb {R}}\rightarrow M$ is a complete timelike line. We may assume that distributional genericity holds at $t_0=0$ along $\gamma$. We approximate g by ${\check{g}}_{\varepsilon } \prec g$, which are hence also globally hyperbolic. By Lemma 2.18 there exist $c > 0$, $0< r < \frac{\pi }{4\sqrt{c}}$ with the following property: For any $C^1$-approximation $\gamma _{\varepsilon }$ of $\gamma$ by ${\check{g}}_{\varepsilon }$-timelike geodesics, there exists $\varepsilon _0 > 0$ such that for all $\varepsilon < \varepsilon _0$, there is $C=C(\varepsilon )>0$ so that (ii) in Lemma 4.1 is satisfied for $R_{\varepsilon }$. Now for the above choice of c and r, pick $\delta > 0$ and $T > 0$ as in Lemma 4.1, and let ${\tilde{T}} > T$. By Proposition 3.4(i), for the curve $\gamma |_{[-T,{{\tilde{T}}}]}$ and $\delta < {{\tilde{T}}}-T$, we obtain a subsequence ${\check{g}}_{\varepsilon _k}$ and ${\check{g}}_{\varepsilon _k}$-maximal timelike geodesics $\gamma _k$ from $\gamma (-T)$ to $\gamma ({{\tilde{T}}})$ converging to $\gamma |_{[-T,{{\tilde{T}}}]}$ in $C^1$.

Let K be a compact neighbourhood of $\gamma ([-T,T])$. Since $\gamma _{\varepsilon _k} \rightarrow \gamma$ in $C^1([-T,T])$, there are $k_0 \in {\mathbb {N}}$, ${\tilde{C}} > 0$ and $\kappa < 0$ such that for all $k \ge k_0$, we have $\gamma _{\varepsilon _k}([-T,T]) \subseteq K$, $\Vert {\dot{\gamma }}_{\varepsilon _k}\Vert _h \le {\tilde{C}}$ and ${\check{g}}_{\varepsilon _k}({\dot{\gamma }}_{\varepsilon _k},{\dot{\gamma }}_{\varepsilon _k}) < \kappa$ on $[-T,T]$. Hence by [7, Lem. 4.6] (and the remark preceding it) we have $\mathrm {Ric}_{\varepsilon _k}({\dot{\gamma }}_{\varepsilon _k},{\dot{\gamma }}_{\varepsilon _k})\ge - \delta$ on $[-T,T]$ for large k. Therefore, also (i) in Lemma 4.1 is satisfied for $\gamma _{\varepsilon _k}$, yielding the existence of conjugate points for any $\gamma _{\varepsilon _k}$ (k large) in $[-T,T]$, a contradiction since they are maximising even in the strictly larger interval $[-T,{\tilde{T}}]$. $\quad \square$

Theorem 4.3

(No null lines) Let (M, g) be a causal, MNNB $C^1$-spacetime that satisfies the distributional null energy condition and the distributional genericity condition along any inextendible null geodesic. Then there is no complete null line in M.

Proof

Suppose $\gamma :{\mathbb {R}}\rightarrow M$ is a complete null line. We will again make use of the approximation ${\check{g}}_{\varepsilon }$. Suppose without loss of generality that the distributional genericity condition holds along $\gamma$ at $t_0=0$. By Lemma 2.18 there exist $c > 0$ and $0< r < \frac{\pi }{4\sqrt{c}}$ satisfying: For any $C^1$-approximation $\gamma _{\varepsilon }$ of $\gamma$ by ${\check{g}}_{\varepsilon }$-null geodesics, there exists $\varepsilon _0 > 0$ such that for all $\varepsilon < \varepsilon _0$, there is $C=C(\varepsilon )>0$ so that (ii) in Lemma 4.1 is satisfied for $R_{\varepsilon }$. Choose $\delta > 0$ and $T > 0$ as in Lemma 4.1 for this pair (c, r) and let ${\tilde{T}} > T$. Using Proposition 3.4(ii) for the curve $\gamma |_{[-T,{{\tilde{T}}}]}$ and $S =\gamma (-T)$ (cf. Remark 3.5(ii)), there exists a subsequence $\check{g}_{\varepsilon _k}$ and ${\check{g}}_{\varepsilon _k}$-maximal null geodesics $\gamma _k: [-T,{{\tilde{T}}}] \rightarrow M$ converging to $\gamma |_{[-T,{{\tilde{T}}}]}$ in $C^1$.

Since $\gamma _k \rightarrow \gamma$ in $C^1([-T,T])$, once we choose a compact neighbourhood K of $\gamma ([-T,T])$, there are $k_0 \in {\mathbb {N}}$ and ${\tilde{C}}_2> {\tilde{C}}_1 > 0$ such that for all $k \ge k_0$, we have $\gamma _{\varepsilon _k}([-T,T]) \subseteq K$, ${\tilde{C}}_1< \Vert {\dot{\gamma }}_k\Vert _h < {\tilde{C}}_2$ in $[-T,T]$. Hence, Lemma 2.14 implies that $\mathrm {Ric}_{\varepsilon _k}({\dot{\gamma }}_k,{\dot{\gamma }}_k) > - \delta$ in $[-T,T]$. But then Lemma 4.1 may be invoked to give the existence of conjugate points on $\gamma _k|_{[-T,T]}$ for large k, a contradiction since they were supposed to be maximising even on $[-T,{\tilde{T}}]$. $\quad \square$

5 Initial Conditions

The classical Hawking-Penrose theorem for spacetimes of dimension 4 assumes three alternative initial conditions: the existence of a compact, spacelike hypersurface, a trapped (2-)surface, or a “trapped point”, i.e., a point from where all future (or past) null geodesics converge. The second condition was later generalised in [10] to trapped submanifolds of arbitrary codimension m with $1<m<n=\dim (M)$.

While the first condition does not need any special attention here we will start by generalising the trapped submanifold-case to $C^1$-spacetimes. As for $C^{1,1}$-spacetimes (see [8, Sec. 6.2]), trapped submanifolds are defined in the support sense. However, to show that normal null geodesics emanating from them stop being maximising is more delicate now due to the lack of an exponential map, cf. (the proof of) Proposition 5.5. At the end of this section we will deal with the case of a “trapped point”.

Definition 5.1

(Support submanifolds). Let (M, g) be a $C^1$-spacetime and let $S,{\tilde{S}} \subseteq M$ be submanifolds. We say that ${\tilde{S}}$ is a future support submanifold for S at $q \in S$ if $\dim S = \dim {\tilde{S}}$, $q \in {\tilde{S}}$, and there is a neighbourhood U of q in M such that ${\tilde{S}} \cap U \subseteq J^+_U(S)$.

Definition 5.2

(Trapped $C^0$-submanifolds). Let (M, g) be a $C^1$-spacetime of dimension n. We say that a $C^0$-submanifold $S \subseteq M$ of codimension $1 \le m<n$ is a future trapped submanifold if it is compact without boundary and for any $p \in S$ there exists a neighbourhood U of p such that $U \cap S$ is achronal in U and moreover S has past pointing timelike mean curvature in the sense of support submanifolds, i.e. for any $q \in S$ there exists a future $C^2$-support submanifold ${\tilde{S}}$ for S at q whose mean curvature vector at q is past-pointing timelike.

Now, to show that lightrays from trapped submanifolds stop maximising, we reduce the problem to a question about smooth approximating metrics, which are understood much better. The essential results that deal with the corresponding situations for smooth metrics are [8, Lem. 6.4] and [7, Lem. 5.6]. To give a precise formulation we first introduce some notation.

Suppose $S \subseteq M$ is a spacelike $C^2$-submanifold of codimension $1<m<n$, $p \in S$ and $\nu \in T_p M$ is a future null normal to S. Let $\gamma$ be a geodesic with $\gamma (0) = p$, ${{\dot{\gamma }}}(0)=\nu$. Let $e_1,\dots ,e_{n-m}$ be an ON-basis on S around p. Further let $E_1,\dots ,E_{n-m}$ be the parallel translates of $e_1(p),\dots ,e_{n-m}(p)$ along $\gamma$, which are $C^1$ since they satisfy the parallel transport equation. We will use this notation for the following results. It will be clear from the context which surface the $E_i$ refer to.

Lemma 5.3

Let (M, g) be a smooth spacetime and let S be a codimension-m $(1<m<n)$ spacelike $C^2$ submanifold of M. Let $\gamma$ be a geodesic starting in S such that $\nu :={\dot{\gamma }}(0) \in TM|_S$ is a future null normal to S. Suppose that $c:={\mathbf {k}}_S(\nu ) > 0$ and let $b>1/c$. Then there is $\delta =\delta (b,c) > 0$ such that, if for $E_i$ as described above

$$\begin{aligned} \sum _{i=1}^{n-m} g(R(E_i,{\dot{\gamma }}){\dot{\gamma }},E_i) \ge - \delta \end{aligned}$$

along $\gamma$, then $\gamma |_{[0,b]}$ is not maximising from S, provided that $\gamma$ exists up to $t=b$.

Lemma 5.4

Let (M, g) be a smooth spacetime and let S be a codimension-2 spacelike $C^2$ submanifold of M. Let $\gamma$ be a geodesic starting in S such that $\nu :={{\dot{\gamma }}}(0)$ is a future null normal to S. Suppose that $c:={\mathbf {k}}_S(\nu ) > 0$ and let $b>1/c$. Then there is $\delta = \delta (b,c) > 0$ such that, if

$$\begin{aligned} \mathrm {Ric}({\dot{\gamma }},{\dot{\gamma }}) \ge -\delta \end{aligned}$$

along $\gamma$, then $\gamma |_{[0,b]}$ is not maximising from S, provided that $\gamma$ exists up to $t=b$.

The following result is the $C^1$-analogue of [8, Prop. 6.5]. As was the case for the distributional genericity condition, cf. Definition 2.15, also here we have to assume that the distributional curvature condition at hand is stable under $C^0$-perturbations of the vector fields involved. This ensures that we can derive the necessary curvature conditions for the smooth approximating metrics in order to be able to use Lemmas 5.3 and 5.4 .

Proposition 5.5

(Light rays from a submanifold). Let (M, g) be a strongly causal, MNNB $C^1$-spacetime and let ${\tilde{S}} \subseteq M$ be a $C^2$-spacelike submanifold of codimension $1<m<n$. Suppose ${\mathbf {k}}_{{\tilde{S}}}(\nu ) = g(\nu ,H_p)> c > 0$ and let $b>1/c$. Suppose there is a null geodesic $\gamma$ with ${\dot{\gamma }}(0)=\nu$, a neighbourhood U of $\gamma |_{[0,b]}$ and $C^1$-extensions $\overline{E}_i$ of $E_i$ and $\overline{N}$ of ${\dot{\gamma }}$ to U such that for each $\delta >0$ there exists $\eta > 0$ such that for all collections of $C^1$-vector fields $\{{\tilde{E}}_1,\dots ,{\tilde{E}}_{n-m},{\tilde{N}}\}$ on U with $\Vert {\tilde{E}}_i - \overline{E}_i\Vert _h < \eta$ for all i and $\Vert {\tilde{N}}-\overline{N}\Vert _h < \eta$, we have

$$\begin{aligned} \sum _{i=1}^{n-m} g(R({\tilde{E}}_i,{\tilde{N}}){\tilde{N}},{\tilde{E}}_i) \ge -\delta \quad \text {in } {\mathcal {D}}^{'(1)}(U). \end{aligned}$$

(5.1)

Then $\gamma |_{[0,b]}$ is not maximising from ${\tilde{S}}$.

Note that condition (5.1) localises in a similar manner as the genericity condition. For the latter see Remark 2.16.

Proof

Let $g_{\varepsilon } = {\check{g}}_{\varepsilon }$, $R_{\varepsilon }=R[{\check{g}}_{\varepsilon }]$. Clearly, ${\mathbf {k}}_{\tilde{S}}$ is continuous and ${\mathbf {k}}_{\tilde{S}, \varepsilon } \rightarrow {\mathbf {k}}_{\tilde{S}}$ uniformly on compact sets. Hence, there is a neighbourhood V of $\nu$ in $TM|_{\tilde{S}}$ and $\varepsilon _0$ such that $\forall \varepsilon \le \varepsilon _0$ and $\forall v \in V$: ${\mathbf {k}}_{\tilde{S},\varepsilon }(v) > c$. Note that g-spacelikeness implies $g_{\varepsilon }$-spacelikeness. Hence $U \cap \tilde{S}$ is $g_{\varepsilon }$-spacelike and we may assume that $W := \pi (V)$ is contained in $U \cap \tilde{S}$.

Suppose now to the contrary that $\gamma |_{[0,b]}$ maximises the distance from $\overline{U}\cap \tilde{S}$. Let $1/c<b'<b''<b$ and let $q=\gamma (b'')$. Find $q_k \in \partial J_k^+(\overline{U} \cap \tilde{S})$ (where $\partial J_k:=\partial J_{g_{\varepsilon _k}}$ and $\varepsilon _k\rightarrow 0$) with $q_k \rightarrow q$. By Corollary A.12 there are $g_{\varepsilon _k}$-null geodesics $\gamma _k: I_k \rightarrow M$ (in future directed parametrisation) with $\gamma _k(b'') = q_k$ either intersecting $\overline{U} \cap \tilde{S}$ or past inextendible. We may assume that $\Vert {\dot{\gamma }}_k(b'')\Vert _h$ are all equal to $\Vert {\dot{\gamma }}(b'')\Vert _h$, hence we may assume that the ${\dot{\gamma }}_k(b'')$ converge to a g-null vector v and the geodesics converge to a g-null geodesic $\gamma _v$ in $C^2_{\mathrm {loc}}$ by Corollary 2.6. Strong causality allows us to invoke Lemma A.30, so we can find a neighbourhood W of $\gamma (0)$ with $W \subseteq U$ such that $W \cap {{\tilde{S}}}$ is acausal in M. By (the proof of) Proposition 3.4(ii) (see also Remark 3.5(i)) we obtain that $\gamma = \gamma _v$, and that (up to picking a subsequence) $\gamma _k$ reaches ${{\tilde{S}}}$ for k large at $t_k \downarrow 0$.

Since any of the $g_{\varepsilon _k}$-geodesics $\gamma _k$ reaches ${{\tilde{S}}}$ for all $\varepsilon _k<\varepsilon _0$ (for some suitable $\varepsilon _0>0$) at points $\gamma _k(t_k) \in {{\tilde{S}}}$, by $C^1_{\mathrm {loc}}$ convergence of $\gamma _k$ to $\gamma$ we also have ${\dot{\gamma }}_k(t_k)\in V$. Note that, due to $\gamma _k(t_k)\rightarrow \gamma (0)$, we can pick $g_{\varepsilon _k}$-orthonormal bases for $T_{\gamma _k(t_k)}{{\tilde{S}}}$ converging to a g-orthonormal basis for $T_{\gamma (0)}{{\tilde{S}}}$. Denote by $E_i^{\varepsilon _k}$ and $E_i$ the $g_{\varepsilon _k}$- and g-parallel transports of these bases along $\gamma _k$ and $\gamma$, respectively. By Lemma B.2 there exist $C^1$-extensions ${{\tilde{E}}}_i^{\varepsilon _k}$ and $\tilde{E_i}$ to the (possibly shrunk) neighbourhood U. By a similar argument one can extend the velocity vector fields along $\gamma _k$ and $\gamma$ to all of U, they will be denoted by ${\tilde{N}}^\varepsilon$ and ${\tilde{N}}$, respectively.

Assume also without loss of generality that U is relatively compact and that $\gamma _k([t_k,b]) \subseteq U$ for large k. Pick $\delta =\delta (b',c)$ according to Lemma 5.3. Since they restrict to the same vector fields along $\gamma$, by shrinking U and by continuity of $\overline{E_i}$ and ${{\tilde{E}}}_i$ resp. $\overline{N}$ and ${{\tilde{N}}}$, we can assume these vector fields to be arbitrarily close in $\Vert \,.\,\Vert _h$ on U. Moreover, by the convergence of ${{\tilde{E}}}_i^{\varepsilon _k}$ and ${{\tilde{N}}}^{\varepsilon _k}$ to ${{\tilde{E}}}_i$ and ${\tilde{N}}$, respectively, also these vector fields can be made arbitrarily $\Vert \,.\,\Vert _h$-close to $\overline{E_i}$ and $\overline{N}$, respectively, on U for large k. The assumption of the Proposition therefore implies that

$$\begin{aligned} \sum _{i=1}^{n-m} g(R({{\tilde{E}}}_i^{\varepsilon _k},{\tilde{N}}^{\varepsilon _k}) {\tilde{N}}^{\varepsilon _k},{{\tilde{E}}}_i^{\varepsilon _k}) \ge -\delta /2 \end{aligned}$$

in ${\mathcal {D}}'(U)$ for k large. Since $\star _M \rho _{\varepsilon _k}$ respects inequalities, we conclude that also

$$\begin{aligned} \sum _{i=1}^{n-m} g(R({{\tilde{E}}}_i^{\varepsilon _k},{\tilde{N}}^{\varepsilon _k}) {\tilde{N}}^{\varepsilon _k},{{\tilde{E}}}_i^{\varepsilon _k}) \star _M \rho _{\varepsilon _k} \ge -\delta /2 \end{aligned}$$

on U for large k.

By the same reasoning as in Lemma 2.17 we obtain (once again writing $R_{\varepsilon _k}$ for $R[g_{\varepsilon _k}]$)

$$\begin{aligned} g(R({{\tilde{E}}}_i^{\varepsilon _k},{\tilde{N}}^{\varepsilon _k}) {\tilde{N}}^{\varepsilon _k},{{\tilde{E}}}_i^{\varepsilon _k}) \star _M \rho _{\varepsilon _k} - g_{\varepsilon _k}(R_{\varepsilon _k}({{\tilde{E}}}_i^{\varepsilon _k}, {\tilde{N}}^{\varepsilon _k}) {\tilde{N}}^{\varepsilon _k},{{\tilde{E}}}_i^{\varepsilon _k}) \rightarrow 0 \end{aligned}$$

uniformly on U as $\varepsilon _k \rightarrow 0$, $i=1,\dots ,n-m$. But then, for large k,

$$\begin{aligned} \sum _{i=1}^{n-m} g_{\varepsilon _k}(R_{\varepsilon _k}({{\tilde{E}}}_i^{\varepsilon _k}, {\tilde{N}}^{\varepsilon _k}) {\tilde{N}}^{\varepsilon _k},{{\tilde{E}}}_i^{\varepsilon _k}) \ge -\delta . \end{aligned}$$

In particular, the above holds along $\gamma _k$, so

$$\begin{aligned} \sum _{i=1}^{n-m} g_{\varepsilon _k}(R_{\varepsilon _k}(E_i^{\varepsilon _k}(t), {\dot{\gamma }}_k(t)){\dot{\gamma }}_k(t),E_i^{\varepsilon _k}(t)) \ge -\delta \end{aligned}$$

for all $t \in [t_k,b']$, where we may assume that k is so large that all $\gamma _k$ are defined on $[t_k,b']$.

Due to our choice of $\delta$, Lemma 5.3 implies that, for k sufficiently large, each $\gamma _k$ stops maximising at parameter $b'+t_k$ at the latest (if $\gamma _k$ is not $g_{\varepsilon }$-normal to ${\tilde{S}}$, then $\gamma _k$ stops maximising the distance immediately, see Remark 5.9 below). By construction the $\gamma _k$ maximize from $t_k$ to $b''$, hence if k is so large that $b'+t_k<b''$ we obtain a contradiction.$\square$

Remark 5.6

By arguments analogous to those given for the genericity condition, cf. Remark 2.19, namely essentially by boundedness of R on compact sets, one can see that if $g \in C^{1,1}$, then the curvature condition (5.1) in Proposition 5.5 is equivalent to the one given in [8, Prop. 6.5].

Proposition 5.7

(The case of trapped surfaces). Let (M, g) be a strongly causal, MNNB $C^1$-spacetime satisfying the distributional null energy condition. Let ${\tilde{S}} \subseteq M$ be a codimension-2 $C^2$-spacelike submanifold such that ${\mathbf {k}}_{{\tilde{S}}}(\nu )> c > 0$, where $\nu$ is a null normal to ${\tilde{S}}$, and let $b>1/c$. If $\gamma$ is a null geodesic with ${\dot{\gamma }}(0) = \nu$, then $\gamma |_{[0,b]}$ is not maximising from ${\tilde{S}}$.

Proof

The proof is completely analogous to that of Proposition 5.5, with the sole difference that one needs to use Lemma 5.4 instead of Lemma 5.3. Let us give a brief outline: Approximate g again by $g_{\varepsilon _k} \equiv {\check{g}}_{\varepsilon _k}$ and the compact segment $\gamma |_{[0,b]}$ as before by $g_{\varepsilon _k}$-null geodesics $\gamma _k$. Then by Lemma 2.14 (see also the proof of [7, Thm. 5.7]), for all $\delta > 0$ and large k we have $\mathrm {Ric}_{\varepsilon _k}({{\dot{\gamma }}}_k,{{\dot{\gamma }}}_k) > - \delta$. Using this, combined with Lemma 5.4 and proceeding as in Proposition 5.5, one arrives at a contradiction as before. $\quad \square$

Remark 5.8

The corresponding $C^{1,1}$-version of Proposition 5.7 was dealt with in [8, Remark 6.6(i)], however, a similar line of reasoning is not available in our case of $C^1$-metrics: While it is still possible to extend an ON-frame along a curve to a neighbourhood, this does not suffice to derive condition (5.1) of Proposition 5.5, as the needed rigidity of the condition cannot be guaranteed. More precisely: For any ON-frame $\{{\bar{E}}_i, {\bar{N}} \}$ on U the equality (as before ${\bar{N}}$ denotes the extension of $\gamma '$ in the frame)

$$\begin{aligned} \sum _{i=1}^{n-m} g(R(\bar{E}_i,\bar{N}){\bar{N}},\bar{E}_i) = \mathrm {Ric}[g]({\bar{N}}, {\bar{N}}) \ge - \delta \end{aligned}$$

still holds and the final inequality is due to the distributional null energy condition. However, to carry out a proof similar to the one of Proposition 5.5, this estimate would need to hold for all collections of vector fields close to the frame in the sense given in the Proposition. This cannot be guaranteed for $C^1$-metrics anymore as R is, in general, unbounded. So instead, the slightly different focusing result used in Proposition 5.7 is required.

Remark 5.9

If, in the situation of Proposition 5.5, $\nu$ is null but not a null normal, then geodesics in that direction immediately enter $I^+$: This is shown in the same way as in the smooth case, e.g. by using ideas from [39, Lem. 10.45, Lem. 10.50]: Let c be a geodesic in direction $\nu$, hence in particular $C^2$. By [43, Lem. 3.1] for a (piecewise) $C^1$-vector field X along c with $g(X',{\dot{c}})<0$, any variation $c_s$ with variation field X is timelike and longer than c (for small s).

Now for any vector v at p not a null normal to a spacelike surface S, w.l.o.g. there exists $y\in T_pS$ with $g(y,v)> 0$. Define $V(t) =(1- \frac{t}{b}) Y(t)$, where Y is the parallel translate of y along c. A straightforward calculation shows $g(V',{\dot{c}})<0$. Now choose a variation $c_s$ of c with variational vector field V such that $c_s(0)\in S$. We obtain that $c_s$ is timelike from S to c(b) and as b was arbitrary the statement follows.

The main result on trapped submanifolds in $C^1$-spacetimes now is the following:

Proposition 5.10

Let (M, g) be a strongly causal, MNNB $C^1$-spacetime and let $S \subseteq M$ be a trapped $C^0$-submanifold of codimension m, $1<m<\dim (M)=n$. If $m = 2$, suppose that (M, g) satisfies the distributional null energy condition, and if $m \ne 2$, suppose that any support submanifold ${\tilde{S}}$ of S satisfies the condition in Proposition 5.5 in any null normal direction $\nu$ and for any null geodesic $\gamma$ with ${\dot{\gamma }}(0)=\nu$. Then $E^+(S)$ is compact or (M, g) is null geodesically incomplete.

Proof

Suppose (M, g) is null geodesically complete.

$E^+(S)$ is relatively compact: Assume, to the contrary, that there are $q_k \in E^+(S)$ with $q_k \rightarrow \infty$. Lemma A.5 guarantees the existence of future directed maximising null geodesics $\gamma _k:[0,t_k] \rightarrow M$ connecting $p_k \in S$ to $q_k$, which we may assume to be parametrised by h-unit speed. By compactness we may further assume that $\gamma _k(0)=p_k \rightarrow p \in S$, and we write $\gamma _k(t_k)= q_k.$ Due to $q_k \rightarrow \infty$, only finitely many $\gamma _k$ are contained in a fixed neighbourhood U of p. So we may apply Theorem A.10, implying that (a subsequence, not relabeled, of) $\gamma _k$ converges to a g-null curve $\gamma :[0,\infty )\rightarrow M$ (in h-unit speed parametrisation) from p. Since the $\gamma _k$ are S-maximising, so is $\gamma$. In particular, $\gamma$ is a g-null pre-geodesic. Reparametrising $\gamma$ as a geodesic, it is still defined on $[0,\infty )$ due to null geodesic completeness and so $\gamma :[0,\infty ) \rightarrow M$ is a future complete g-null S-ray. Let $\tilde{S}$ be a future support submanifold of S at p, then $\gamma$ maximises the distance from $\tilde{S}$ everywhere, because otherwise there would be some T such that $\gamma (T) \in I^+(\tilde{S}) \subseteq I^+(S)$, contradicting the fact that $\gamma$ is an S-ray. As this cannot happen due to Proposition 5.5 resp. Proposition 5.7, $E^+(S)$ is relatively compact.

$E^+(S)$ is closed: Let $q_k \in E^+(S)$, $q_k \rightarrow q$. Let $\gamma _k:[0,t_k] \rightarrow M$ be null pre-geodesics (in h-unit speed parametrisation) from some $p_k \in S$ to $q_k$. By compactness of S, we may assume that $p_k \rightarrow p \in S$. If $q = p$, then we are done. Otherwise, there are two possibilities: If the $t_k$ are bounded, then by going over to subsequences, $t_k \rightarrow t \in (0,\infty )$. Since $q \ne p$, we are in a position to invoke Theorem A.11 to get a maximising causal limit geodesic $\gamma :[0,t] \rightarrow M$ connecting p to q. $\gamma$ cannot be timelike, as this would imply $q \in I^+(S)$ and hence $q_k \in I^+(S)$ for large k. Hence $q\in J^+(S)\setminus I^+(S) = E^+(S)$. The other possibility is that the $t_k$ are unbounded, i.e. w.l.o.g. $t_k \rightarrow \infty$. Since (M, g) is strongly causal, this means that the $\gamma _k$ leave every compact set, i.e., $\gamma _k(t_k) \rightarrow \infty$ (after possibly relabelling $t_k$). However, as $E^+(S)$ is contained in some compact set, this is not possible. $\quad \square$

Corollary 5.11

Let the assumptions be as in Proposition 5.10. Then $E^+(S) \cap S$ is achronal, and $E^+(E^+(S) \cap S)$ is compact or M is null geodesically incomplete.

Proof

This is shown as in the smooth case, see [10, Prop. 4] or [46, Prop. 4.3] and using that for any $p\in S$ there is some neighbourhood U such that $S\cap U$ is achronal in U. $\quad \square$

We now start with our discussion of “trapped points”. In [8, Sec. 6.3], a faithful way to generalise the classical condition, which uses Jacobi tensor classes—a tool no longer available already for $C^{1,1}$-metrics—is presented: It uses the mean curvature of spacelike 2-surfaces given as the level sets of the exponential map that generate the light cone. While the latter tool presently is not at our disposal, the very formulation of the corresponding condition again uses support manifolds and can be adopted verbatim. However, it is now more delicate to derive that the horismos of a trapped point is a trapped set.

Definition 5.12

(Trapped points).

A point $p \in M$ is future trapped if for any future pointing null vector $v \in T_pM$ and for any null geodesic $\gamma$ with $\gamma (0) = p$, ${\dot{\gamma }}(0)=v$, there exists a parameter t and a spacelike $C^2$-submanifold of codimension $m=2$ with $\tilde{S}\subseteq J^+(p)$, $\gamma (t) \in \tilde{S}$ and ${\mathbf {k}}_{\tilde{S}}({\dot{\gamma }}(t)) > 0$.

Proposition 5.13

Let (M, g) be a strongly causal, MNNB $C^1$-spacetime satisfying the distributional null energy condition. If $p \in M$ is a future trapped point and (M, g) is null geodesically complete then $E^+(p)$ is compact.

Proof

Since trapped points are defined by means of codimension-2 submanifolds $\tilde{S}$, we will use Proposition 5.7 for which (only) the distributional null energy condition is needed. The proof is analogous to that of Proposition 5.10.

$E^+(p)$ is relatively compact: Suppose $q_j \in E^+(p)$, $q_j \rightarrow \infty$. Let $\gamma _j:[0,t_j] \rightarrow M$ be maximising null geodesics connecting p to $q_j$, cf. Lemma A.5. Then, up to a subsequence, the $\gamma _j$ converge to a g-null ray $\gamma :[0,\infty ) \rightarrow M$ from p (the domain is $[0,\infty )$ even in g-affine parametrisation since (M, g) is null geodesically complete). By assumption, there is a $C^2$-spacelike submanifold $\tilde{S}$ of codimension 2 and a parameter t such that ${\mathbf {k}}_{\tilde{S}}({\dot{\gamma }}(t))> c > 0$ and we let $b > 1/c$. $\gamma$ is indeed an ${\tilde{S}}$-ray: If not, it would enter $I^+({\tilde{S}}) \subseteq I^+(p)$, a contradiction. But by the previous results in this section, $\gamma$ cannot maximise from ${\tilde{S}}$ past b, a contradiction since it is a ray.

$E^+(p)$ is closed: This is shown in exactly the same way as closedness was shown in the proof of Proposition 5.10. $\quad \square$

6 The Main Result

Given the results established so far, the general mechanics of the proof of the Hawking–Penrose theorem remains the same as in the smooth case. There do, however, remain notable differences in the details in this lower regularity (e.g. the use of [7, Prop. 2.13] in the proof of Theorem 6.2), so we include the full argument.

Lemma 6.1

Let (M, g) be a strongly causal $C^1$-spacetime such that no inextendible null geodesic in M is maximising. Let $A \subseteq M$ be achronal such that $E^+(A)$ is compact, and let $\gamma$ be a future inextendible timelike curve contained in $D^+(E^+(A))^{\circ }$. Then $F:=E^+(A) \cap \overline{J^-(\gamma )}$ is achronal and $E^-(F)$ is compact.

Proof

Due to Lemma A.16 we may assume that $A=\overline{A}$. Since $E^+(A)$ is always achronal and $F\subseteq E^+(A)$, it is also achronal. By compactness of $E^+(A)$, F is compact. Note that $E^-(F) \subseteq F \cup \partial J^-(\gamma )$ by similar arguments as in [21, Lem. 9.3.4], so to show compactness of $E^-(F)$, it suffices to show compactness of $E^-(F) \cap \partial J^-(\gamma )$.

Suppose now that $v \in TM|_F$ is past pointing causal and let $c_v:[0,a) \rightarrow M$ be any past inextendible, past directed geodesic with $\dot{c}_v(0)=v$ (recall that in $C^1$-spacetimes we no longer have unique solvability of the geodesic initial value problem). Then $c_v \subseteq \overline{J^-(\gamma )}$, and $\overline{J^-(\gamma )}=I^-(\gamma ) \cup \partial J^-(\gamma )$. We show that $c_v$ meets $I^-(\gamma )$: Suppose, to the contrary, that $c_v$ is a null geodesic entirely contained in $\partial J^-(\gamma )$. Since $\gamma$ is a future inextendible timelike curve, we have $J^-(\gamma ) = I^-(\gamma )$, hence $c_v$ lies entirely in $\partial J^-(\gamma )\setminus J^-(\gamma )$. By Corollary A.12, there is a future directed, future inextendible null geodesic $\lambda$ starting at $c_v(0)$ entirely in $\partial J^-(\gamma )$ ($\gamma$ is closed in M by Corollary A.33 and Lemma A.32). This means that either $c_v \lambda$ is an inextendible broken null geodesic, hence not maximising by Lemma A.4, or an inextendible unbroken maximising null geodesic. By assumption such a geodesic cannot exist, hence in either case $c_v \lambda$ cannot lie entirely in $\partial J^-(\gamma )$ and so $c_v$ must enter $I^-(\gamma )$.

Thus we have shown that for any v and any geodesic $c_v$ as above there is $t_v = t_v(v,c_v)$ (for which $c_v$ is still defined) such that $c_v(t_v) \in I^-(\gamma )$. It remains to show that $E^-(F)\cap \partial J^-(\gamma )$ is compact.

$E^-(F) \cap \partial J^-(\gamma )$ is relatively compact: Suppose there are $q_k \in E^-(F) \cap \partial J^-(\gamma )$ with $q_k \rightarrow \infty$. Then there are past-directed maximising null pre-geodesics $c_k:[0,t_k] \rightarrow M$ in h-unit speed parametrisation with $c_k(0) = p_k \in F$ and $c_k(t_k) = q_k$. Since $q_k \rightarrow \infty$, also $t_k \rightarrow \infty$ By compactness, we may assume $p_k \rightarrow p \in F$. Since $q_k\rightarrow \infty$, there is a neighbourhood U which almost all $c_k$ leave and we are in a position to apply the Theorem A.10. We get a past-directed inextendible null limit curve $c:[0,\infty ) \rightarrow M$ from p and a subsequence of the $c_k$ (w.l.o.g. $c_k$ itself) which converges to c locally uniformly. It is easy to see that all $c_k$ have to lie entirely in $\partial J^-(\gamma )$, hence so does the limit c. In particular, c is everywhere maximising and may be reparametrised as an inextendible geodesic $c:[0,a) \rightarrow M$ entirely contained in $\partial J^-(\gamma )$. This is a contradiction since we showed before that each past directed null geodesic with initial velocity in $TM|_F$ has to enter $I^-(\gamma )$.

$E^-(F) \cap \partial J^-(\gamma )$ is closed: Let $q_j \in E^-(F) \cap \partial J^-(\gamma )$, $q_j \rightarrow q$, where we may assume $q\not \in F$, otherwise there is nothing to prove. Connect $F \ni p_j \rightarrow q_j$ via maximising null geodesics. Up to a subsequence, their limit has to be a maximising null geodesic from p to q, hence $q \in E^-(F) \cap \partial J^-(\gamma )$, because otherwise one would again obtain an inextendible maximising null geodesic entirely in $\partial J^-(\gamma )$, which would lead to the same contradiction as before. $\quad \square$

Now we are ready to give the $C^1$-version of the causal Hawking-Penrose theorem. It is the direct analog of [17, p. 538, Thm.] and [8, Thm. 7.4], respectively.

Theorem 6.2

Let (M, g) be a $C^1$-spacetime. Then the following conditions cannot all hold:

• (i) (M, g) is causal.

• (ii) No inextendible timelike geodesic in an open globally hyperbolic subset is everywhere maximising.

• (iii) No inextendible null geodesic is everywhere maximising.

• (iv) There is an achronal set $A \subseteq M$ such that $E^+(A)$ or $E^-(A)$ is compact.

Proof

Suppose all of the above conditions hold. Then by Lemma A.31, (i) and (iii) imply that (M, g) is strongly causal. W.l.o.g. we assume that $E^+(A)$ is compact. By Lemma A.35 there exists a future inextendible (in M) timelike curve $\gamma \subseteq D^+(E^+(A))^\circ$. Set $F:= E^+(A)\cap \overline{J^-(\gamma )}$, which by Lemma 6.1 is achronal and $E^-(F)$ is compact. Once more by Lemma A.35 there is some past-inextendible timelike curve $\lambda \subseteq D^-(E^-(F))^\circ$. Exactly as in the smooth case, see e.g. the second paragraph of [8, Proof of Thm. 7.4], one can show $\gamma \subseteq D^+(E^-(F))^\circ$, by invoking Proposition A.20 and Lemma A.24. So $\gamma , \lambda \subseteq D(E^-(F))^\circ$ which by Proposition A.28 is globally hyperbolic. Now we pick sequences $p_k=\lambda (s_k)$ and $q_k=\gamma (t_k)$ with $t_k, s_k \nearrow \infty$, leaving every compact set, which is possible because $\gamma$ and $\lambda$ are inextendible curves in M contained in a strongly causal set (Lemma A.32). Also because $\lambda \subseteq J^-(E^-(F)) \subseteq J^-(F)\subseteq J^-(\overline{J^-(\gamma }))$ and because $\gamma$ is timelike, we can choose $p_k \in I^-(q_k)$.

By [7, Prop. 2.13] there exist maximising timelike geodesics ${\tilde{\gamma }}_k$ in $D(E^-(F))^\circ$ from $p_k$ to $q_k$, which must intersect $E^-(F)$, say at $r_k$. By compactness of $E^-(F)$ we can assume $r_k \rightarrow r$ and by Corollary 2.6 there exists an inextendible, causal limit geodesic ${\tilde{\gamma }}$, which is also maximising (as a limit of maximisers).

By (iii), ${\tilde{\gamma }}$ cannot be null, so it must be timelike. As a limit of curves in $D(E^-(F))^\circ$ it is contained in $\overline{ D(E^-(F))}$. Again using Proposition A.20 and Lemma A.24 one can now proceed as in the last step in [8, Proof of Thm. 7.4] to arrive at the desired contradiction (to (ii)). $\quad \square$

Finally, we formulate and prove the main result of this work, the analytical Hawking-Penrose theorem for maximally causally non-branching $C^1$-spacetimes.

Theorem 6.3

(The Hawking-Penrose singularity theorem for $C^1$-metrics)

Let (M, g) be a $C^1$-spacetime of dimension n with the following properties.

• (i) (M, g) is causal.

• (ii) (M, g) satisfies the distributional timelike and null energy conditions.

• (iii) (M, g) satisfies the distributional genericity condition along any inextendible causal geodesic.

• (iv) (M, g) is maximally causally non-branching.

• (v) (M, g) contains one of the following.

  • (a) a compact, achronal, edgeless set.

  • (b) a future trapped point.

  • (c) a future trapped $C^0$-submanifold of codimension 2.

  • (d) a future trapped $C^0$-submanifold of codimension $2<m<n$ such that its support submanifolds satisfy the condition in Proposition 5.5.

Then (M, g) is causally geodesically incomplete.

Proof

Assuming causal geodesic completeness, we would like to deduce a contradiction by using Theorem 6.2.

Note that due to Theorems 4.2 and 4.3 the assumptions (ii) and (iii) of Theorem 6.2 are satisfied and it remains to establish the existence of a future or past trapped set, i.e. an achronal set with compact future or past horismos. This will be implied by (v), but we have to distinguish the different cases:

By Proposition A.36, (a) implies the existence of a trapped set. Note that by Lemma A.31 (which can be applied by Theorem 4.3 ) (M, g) is strongly causal and hence Proposition 5.13 covers the case (b), whereas Proposition 5.10 covers cases (c) and (d). $\quad \square$

Appendices

Appendix: Causality Results in $C^1$-Spacetimes

In this section, we collect various results on the causality of $C^1$-Lorentzian metrics that will be needed in the proof of the main result. Unless stated otherwise, throughout this section (M, g) consists of a connected smooth (Hausdorff and second countable) manifold M, and a $C^1$-Lorentzian metric g that is time-orientable (i.e. there exists a smooth timelike vector field on M). Many of the results in this section hold in even greater generality, namely either for Lorentzian metrics that are merely Lipschitz or even in the general setting of closed cone structures [35]. Although we will occasionally note this, in proving the results in this section we will stay close to smooth causality theory (mainly based on the comprehensive reference work [34]) and only make those changes from classical proofs that are required due to the absence of certain tools (e.g., convex normal neighbourhoods) in the $C^1$-setting.

We begin with two elementary results, namely the openness of timelike futures and pasts and the push-up property. Both of these hold for causally plain spacetimes, which include spacetimes with Lipschitz continuous metrics [5, Cor. 1.17].

Lemma A.1

For any $A \subseteq M$, the sets $I^{\pm }(A)$ are open in M.

Proof

See [5, Prop. 1.21]. $\quad \square$

Lemma A.2

Let $p,q,r \in M$ such that $p \le q \ll r$ or $p\ll q \le r$. Then $p \ll r$.

Proof

See [5, Lem. 1.22]. $\quad \square$

Remark A.3

Note that from this one can show as in the smooth case, e.g. [39, Lem. 14.03]⁶, that the timelike relation is open, i.e., if $p\ll q$ then there are neighbourhoods $U_p, U_q$ of p and q respectively, such that for all $x \in U_p$ and all $y\in U_q$, we have $x \ll y$.

Lemma A.4

Let (M, g) be a $C^1$-spacetime and let $\gamma :[a,b] \rightarrow M$ be a maximising causal curve. Then $\gamma$ is a (reparametrisation of a) causal geodesic. In particular, it is a $C^2$-curve and has a causal character.

Proof

See [27, Thm. 1.1] or [43, Thm. 3.3]. $\quad \square$

For any point $q \in M$, we denote by $E^+(q):=J^+(q) \setminus I^+(q)$ its future horismos. Its past horismos $E^-(q)$ is defined analogously.

Lemma A.5

If $p \in E^+(q)$ then there exists a maximising null geodesic segment from q to p.

Proof

Let c be a future causal curve from q to p. Since $p \notin I^+(q)$, c is null and maximising from q to p (because of push-up), hence it is (a reparametrisation of) a maximising null geodesic by Lemma A.4. $\quad \square$

Lemma A.6

For any subset $A \subseteq M$, we have $I^{\pm }(A) = J^{\pm }(A)^{\circ }$ and $\partial I^{\pm }(A) = \partial J^{\pm }(A)$. Moreover, the sets $E^{\pm }(A)$ and $\partial J^{\pm }(A)$ are achronal.

Proof

This follows by the same proofs as in the smooth case (cf. [34, Thm. 2.27], [39, Cor. 14.27]). $\quad \square$

Definition A.7

(Edge)

Let $A \subseteq M$ be achronal. The edge of A, denoted by $\mathrm {edge}(A)$, is the set of all $x \in \overline{A}$ such that for any neighbourhood U of x, there is a timelike curve from $I_U^-(x)$ to $I_U^+(x)$ that does not meet A.

Lemma A.8

Let (M, g) be a $C^1$-spacetime. An achronal set $A \subseteq M$ is a topological hypersurface if and only if $A \cap \mathrm {edge}(A) = \emptyset$. Moreover, A is a closed topological hypersurface if and only if $\mathrm {edge}(A) = \emptyset$.

Proof

The proofs for smooth metrics (cf. [39, Prop. 14.25, Cor. 14.26]) still hold in $C^1$. $\quad \square$

Lemma A.9

For any $A \subseteq M$, $\partial J^+(A)$ is a (topologically) closed, achronal topological hypersurface.

Proof

This follows from the fact that $\partial J^+(A)$ is achronal and $\mathrm {edge}(\partial J^+(A)) = \emptyset$, which is proven in precisely the same way as in the smooth case. $\quad \square$

Recall the notion of a limit maximising sequence of causal curves: A sequence $\gamma _k:[a_k,b_k] \rightarrow M$ of future directed causal curves is called limit maximising if there is $\varepsilon _k \rightarrow 0$ such that

$$\begin{aligned} L(\gamma _k) \ge d_g(\gamma (a_k),\gamma (b_k)) - \varepsilon _k, \end{aligned}$$

where $d_g$ is the time separation function induced by g. In particular a sequence of maximising curves is limit maximising. We shall employ the following version of the limit curve theorem:

Theorem A.10

(Limit Curve Theorem)

Let h be a complete Riemannian metric on M and let $\gamma _k:[a_k,b_k] \rightarrow M$ be future directed, h-arc length parametrised causal curves with $0 \in [a_k,b_k]$ such that the sequence $\{\gamma _k(0)\}$ has an accumulation point y. Suppose there are $a \le 0$ and $b \ge 0$ such that $a_k \rightarrow a$, $b_k \rightarrow b$. If there is a neighbourhood U of y such that almost all $\gamma _k$ leave U, then a subsequence of $\gamma _k$ converges h-uniformly on compact sets to a future directed causal Lipschitz curve $\gamma :[a,b] \rightarrow M$. $\gamma$ is future resp. past inextendible iff $b = \infty$ resp. $a=-\infty$. If $\{\gamma _k\}$ is limit maximising, then $\gamma$ is maximising.

Proof

This follows from [30, Thm. 3.1] and [5, Thm. 1.6] by the same proof as in [8, Thm. A.6]. $\quad \square$

For a much more general version of the limit curve theorem, valid in closed cone structures, we refer to [35, Thm. 2.14]. We shall also require the following variant of the limit curve theorem for a sequence of curves with converging past and future endpoints, cf. [30, Thm. 3.1(2)].

Theorem A.11

(Limit Curve Theorem: two converging endpoints).

Let $\gamma _k:[0,b_k] \rightarrow M$ be a sequence of future-directed, causal curves in h-arc length parametrisation connecting $x_k$ to $z_k$, with $x_k \rightarrow x$ and $z_k \rightarrow z$. Suppose that $b_k \rightarrow b \in (0,\infty )$. If there is a neighbourhood U of x such that only a finite number of $\gamma _k$ are entirely contained in U, then there is a Lipschitz future-directed, causal curve $\gamma :[0,b] \rightarrow M$ connecting x to z such that a subsequence of the $\gamma _k$ converges to $\gamma$ h-uniformly on compact sets. Moreover, if $\{\gamma _k\}$ is limit maximising, then $\gamma$ is maximising.

Proof

This is contained in the statement of [30, Thm. 3.1(2)], whose proof is easily seen to hold for $C^1$-metrics once Theorem A.10 is established. $\quad \square$

Corollary A.12

Let $A \subseteq M$. For any point $x \in \partial J^+(A)\setminus \overline{A}$, there is a causal curve $\gamma \subseteq \partial J^+(A)$ with future endpoint x that is either past inextendible and does not meet $\overline{A}$ or has a past endpoint in $\overline{A}$. It is a (reparametrisation of a) maximising null geodesic. If A is closed and $x \notin J^+(A)$, then $\gamma \subseteq \partial J^+(A) \setminus J^+(A)$ and it is past inextendible.

Proof

Using Theorem A.10 and Lemma A.5, this follows as in [8, Prop. A.7]. $\quad \square$

Lemma A.13

Let (M, g) be a $C^1$-spacetime and let $\{g_k\}$ be either the sequence $\{\hat{g}_{\varepsilon _k}\}$ or $\{{\check{g}}_{\varepsilon _k}\}$ with $\varepsilon _k \downarrow 0$. Let $p,q \in M$ and let $\gamma _k$ be a $g_k$-maximising curve from p to q, so $\gamma _k$ is in particular a $g_k$-geodesic. Suppose that $\gamma _k$ converge to a curve $\gamma$ in $C^1_{\mathrm {loc}}$. If $\{g_k\} = \{{\check{g}}_{\varepsilon _k}\}$, suppose in addition that (M, g) is globally hyperbolic and $p \ll _g q$. Then $\gamma$ is a g-maximising geodesic from p to q.

Proof

The case $\{g_k\} = \{\hat{g}_{\varepsilon _k}\}$ follows immediately from [41, Prop. 6.5], so we will only consider the case $\{g_k\} = \{{\check{g}}_{\varepsilon _k}\}$ together with the additional assumptions of (M, g) being globally hyperbolic and $p \ll _g q$ in this case. Due to global hyperbolicity, the space C(p, q) of future directed causal curves from p to q (considered up to orientation preserving reparametrisations) is compact with respect to its natural topology (see [41, Thm. 3.2]). Consequently, the h-speeds of curves in C(p, q) are uniformly bounded by some constant $C_1 >0$, where h is some complete Riemannian metric on M.

Since (M, g) is globally hyperbolic, there is an $L_g$-maximising g-geodesic $c:[0,1] \rightarrow M$ with $c(0)=p$, $c(1)=q$ [41, Prop. 6.4]. Also, $p \ll _g q$ implies that c is g-timelike. By compactness, there is a constant $C_2 > 0$ such that $g({\dot{c}},{\dot{c}}) < -C_2$ on [0, 1], and hence $g_k({\dot{c}},{\dot{c}}) < 0$ for k large, implying that c is $g_k$-timelike for such k. If we set $\delta _k:=d_h(g,g_k)$ (the h-distance of $g, g_k$ as in [5, (1.6)]), then

$$\begin{aligned} L_g(c)&= \int _0^1 \sqrt{-g({\dot{c}},{\dot{c}})} dt \le \int _0^1 \sqrt{-g_k({\dot{c}},{\dot{c}}) + C_1^2 \delta _k} dt \le L_{g_k}(c) + C_1 \sqrt{\delta _k}\\&\le d_{g_k}(p,q) + C_1 \sqrt{\delta _k} = L_{g_k}(\gamma _k) + C_1 \sqrt{\delta _k} \rightarrow L_g(\gamma ) \qquad (k\rightarrow \infty ), \end{aligned}$$

so $L_g(\gamma ) \ge L_g(c)$, which proves that also $\gamma$ is maximising from p to q. $\quad \square$

Lemma A.14

Let $\gamma _k \subseteq \overline{J^+(p)}$ be causal curves converging locally uniformly to a causal curve $\gamma$ with future endpoint q. If $d_g(p,q)=0$, then $\gamma$ is a maximising null geodesic.

Proof

Clearly $\gamma \subseteq \overline{J^+(p)}$. If $\gamma$ would meet $I^+(p)$, then there would exist some $r\in I^+(p)\cap J^-(q)$ and using push-up we could conclude that $q \in I^+(p)$, contradicting $d_g(p,q)=0$. Hence $\gamma \subseteq \partial J^+(p)$, which is achronal and $\gamma$ is a maximising null geodesic by Lemma A.4. $\quad \square$

Lemma A.15

Let $g_k={\check{g}}_{\varepsilon _k}$ and let $\gamma _k \subseteq \overline{J^+(p)}$ be $g_k$-null geodesics converging in $C^1_{\mathrm {loc}}$ to a g-geodesic $\gamma$ that ends in q. Assume that $d_g(p,q)=0$. Then $\gamma$ is a maximising g-null geodesic.

Proof

This follows immediately from Lemma A.14 upon noting that any $g_k$-null geodesic is g causal. $\quad \square$

Lemma A.16

Let A be achronal. Then so is $\overline{A}$. Moreover, if $E^+(A)$ is closed, then $E^{\pm }(A) = E^{\pm }(\overline{A})$.

Proof

This follows by the same proof as in [8, Lem. A.8]. $\quad \square$

Lemma A.17

Let (M, g) be a $C^1$-spacetime. Then every point in M has a neighbourhood base of globally hyperbolic neighbourhoods.

Proof

Fix any smooth, time-orientable Lorentzian metric $\hat{g}$ with $g \prec \hat{g}$. For $x \in M$, [34, Thm. 2.7] gives a neighbourhood base $V_m$ of neighbourhoods of x that are globally hyperbolic with respect to $\hat{g}|_{V_m}$. Any Cauchy hypersurface for ${\hat{g}}$ in $V_m$ is also a Cauchy hypersurface for g, so the claim follows from [41, Thm. 5.7]. $\quad \square$

Lemma A.18

Let S be a spacelike ($C^2$-) hypersurface and let $p\in S$. Then there exists a neighbourhood V of p in M such that $V\cap S$ is a Cauchy hypersurface in V.

Proof

The proof of the analogous result [25, Lemma A.25] also works in the $C^1$-setting. $\quad \square$

Definition A.19

(Cauchy development and Cauchy horizon)

Let A be an achronal set. The future Cauchy development of A is the set

$$\begin{aligned} D^+(A):= \{x \in M: \text { every past inextendible causal curve through }x\text { meets }A\}. \end{aligned}$$

The future Cauchy horizon of A is defined as

$$\begin{aligned} H^+(A):=\overline{D^+(A)} \setminus I^-(D^+(A)). \end{aligned}$$

The past Cauchy development and past Cauchy horizon are defined analogously.

The first part of the following result is partly a $C^1$-analogue of [39, Lem. 14.51], where it is proved using convex normal neighbourhoods, a tool that we do not have at our disposal in the $C^1$-setting. The use of such neighbourhoods can, however, be avoided, as our arguments below illustrate.

Proposition A.20

Let A be a closed, achronal set. Then

$$\begin{aligned} \overline{D^+(A)} = \{x \in M: \text {every p.i. t.l. curve through }x\text { meets }A\} \end{aligned}$$

and

$$\begin{aligned} \partial D^+(A) = A \cup H^+(A). \end{aligned}$$

Proof

Let T be the set on the right hand side in the first equation.

$\overline{D^+(A)}\subseteq T$: Suppose there is a point $p \in \overline{D^+(A)}\setminus T$. Then there is some past inextendible timelike (future directed) curve $\alpha :(a,0] \rightarrow M$ with $\alpha (0) = p$ not meeting A. Let $p_k \in D^+(A)$, $p_k \rightarrow p$ and let $U_k$ be open neighbourhoods of p with $U_k \rightarrow \{p\}$. By choosing subsequences, we may assume that $p_k \in I^+_{U_k}(\alpha (-1/k))$ and $\alpha |_{[-1/k,0]} \subseteq U_k$ for all k.

Construct now past inextendible causal curves $\alpha _k$ by connecting $\alpha (-1/k)$ to $p_k$ in a timelike way such that these curves stay in $U_k$, and let the rest of $\alpha _k$ (i.e. the part to the past of $\alpha (-1/k)$) be the original curve $\alpha$. Then the $\alpha _k$ are all past inextendible timelike, and since $p_k \in D^+(A)$ they must meet A at some $a_k$. Since $\alpha$ does not meet A, the $a_k$ have to lie on the replaced piece. But these pieces are entirely in $U_k$, which go to $\{p\}$, so we see that $a_k \rightarrow p$. Since A is closed, we have $p \in A$, which is absurd since $p \notin T \supseteq A$ by assumption.

$\overline{D^+(A)}\supseteq T:$ Suppose $q \notin \overline{D^+(A)}$, and let $r \in I^-_{M\setminus \overline{D^+(A)}}(q)$. Since in particular $r \notin D^+(A)$, there is a past inextendible causal curve $\alpha$ from r missing A. Precisely as in the smooth case [39, Lem. 14.30(1)], noting that pushup still holds in $C^1$), there is a past inextendible timelike curve from q not meeting A. Thus $q \notin T$.

Finally, $\partial D^+(A) = A \cup H^+(A)$ follows exactly as in the smooth case (cf. [39, Lem. 14.52]). $\quad \square$

We note that the previous result in fact remains true even for locally Lipschitz proper cone structures [35, Thm. 2.35, 2.36].

Lemma A.21

Let A be achronal. Then $H^+(A)$ is a closed, achronal set.

Proof

The proof of [34, Prop. 3.15] carries over unchanged to $C^1$ metrics. $\quad \square$

Lemma A.22

Let A be closed and achronal. If $x \in D^+(A) \setminus H^+(A)$, then every past inextendible causal curve through x meets $I^-(A)$. Moreover, if $x\in D(A)^\circ$, then every future (resp. past) inextendible future (resp. past) directed causal curve emanating from x intersects $I^+(A)$ (resp. $I^-(A)$).

Proof

This follows from [34, Prop. 3.27] and [34, Prop. 3.42], which still hold for $C^1$-metrics. $\quad \square$

Lemma A.23

Let A be closed, achronal. Let $x \in J^+(A) \setminus D^+(A)$ or $x \in I^+(A) \setminus D^+(A)^{\circ }$. Then every causal curve from x to A meets $H^+(A)$.

Proof

The proof for $C^{1,1}$-metrics [8, Lem. A.12] still holds in $C^1$ because of Lemma A.1, Lemma A.2, and Proposition A.20. $\quad \square$

Using these results, one establishes the following formula for the timelike future of the Cauchy horizon of a closed, achronal set precisely as in the $C^{1,1}$-case, cf. [8, Lem. A.13].

Lemma A.24

Let A be closed and achronal. Then $I^+(H^+(A)) = I^+(A) \setminus \overline{D^+(A)}$.

Lemma A.25

Let A be closed and achronal. Then $\mathrm {edge}(H^+(A)) \subseteq \mathrm {edge}(A)$.

Proof

Based on Lemmas A.23 and A.24 , the proof of [8, Lem. A.14] carries over to the $C^1$-setting. $\quad \square$

Lemma A.26

Let A be an achronal set, then $H^+(\partial J^+(A))$ is a closed, achronal, topological hypersurface.

Proof

$H^+(\partial J^+(A))$ is closed and achronal by Lemma A.21, and by the previous result, combined with Lemmas A.8 and A.9 , $\mathrm {edge}(H^+(\partial J^+(A)))\subseteq \mathrm {edge}(\partial J^+(A)) = \emptyset$, so the claim follows from Lemma A.8. $\quad \square$

Lemma A.27

Let A be closed, achronal. Then $H^+(\overline{E^+(A)})\subseteq H^+(\partial J^+(A))$.

Proof

The proof of [8, Lem. A.16] carries over, using Corollary A.12, Proposition A.20 and Lemmas A.23, A.24. $\quad \square$

Proposition A.28

Let (M, g) be a $C^1$-spacetime and let $A\subseteq M$ be a closed, achronal set. Then the interior of its Cauchy development, i.e. the set $D(A)^{\circ } = (D^+(A) \cup D^-(A))^{\circ }$, if nonempty, is a globally hyperbolic $C^1$-spacetime when considered with the induced metric.

Proof

Based on Theorem A.10 and Lemma A.22, this follows from the same proof as in [34, Thm. 3.45]. $\quad \square$

Definition A.29

(Strong causality)

A $C^1$-spacetime (M, g) is called strongly causal at $p \in M$ if for any neighbourhood U of p there is a neighbourhood V of p with $V \subseteq U$ such that any causal curve starting and ending in V is contained in U. (M, g) is called strongly causal if it is strongly causal at every point.

Lemma A.30

(M, g) is strongly causal at p if and only if for any neighbourhood U of p there is a neighbourhood V of p with $V \subseteq U$ such that no causal curve leaving V ever returns.

Proof

This follows from [33, Lem. 3.21], which continues to hold in the $C^1$-case. $\quad \square$

Lemma A.31

Let (M, g) be chronological and suppose there are no inextendible maximising null geodesics in M. Then (M, g) is strongly causal.

Proof

Using Theorem A.10, this follows by the same argument as [8, Lem. A.19]. $\quad \square$

Lemma A.32

A strongly causal $C^1$-spacetime (M, g) is non-totally and non-partially imprisoning: No future or past inextendible causal curve is contained in a compact set and no future or past inextendible causal curve returns to a compact set infinitely often.

Proof

The classical proof for the smooth case (see [39, Lem. 14.13]) holds even for continuous metrics. $\quad \square$

Corollary A.33

Let (M, g) be a non-totally and non-partially imprisoning $C^1$-spacetime. Then the image of every inextendible causal curve is closed in M.

Proof

Let $\gamma$ be any inextendible causal curve. Since M is a manifold, the topology on M is compactly generated. Let $K \subseteq M$ be any compact subset. By non-total and non-partial imprisonment, $\gamma \cap K$ is a finite union of closed segments of $\gamma$, hence $\gamma \cap K$ is closed in K. Since K was arbitrary, $\gamma$ is closed in M. $\quad \square$

Lemma A.34

Let (M, g) be a strongly causal $C^1$-spacetime and let $A \subseteq M$ be a closed, achronal subset. If $H^+(\overline{E^+(A)})$ is nonempty, then it is noncompact.

Proof

The proof of the smooth case goes through, because all the necessary preliminary results continue to hold in $C^1$ (see the outline in [8, Lem. 7.1]). $\quad \square$

Lemma A.35

Let (M, g) be a strongly causal $C^1$-spacetime. Let $A \subseteq M$ be an achronal subset such that $E^+(A)$ is compact. Then there is a future inextendible timelike curve $\gamma$ contained in $D^+(E^+(A))^{\circ }$.

Proof

The proof of the smooth case goes through in $C^1$, see e.g. [17, Lem. 2.12], or also [21, Lem. 9.3.3], combined with Lemma A.16. $\quad \square$

Proposition A.36

Let (M, g) be a $C^1$-spacetime and let $A \subseteq M$ be achronal and edgeless. Then $E^+(A) = A$, in particular if A is compact so is $E^+(A)$.

Proof

This was shown e.g. in [34, Cor. 2.145] and the proof carries over to $C^1$ spacetimes. $\quad \square$

Appendix: Extending Vector Fields in $C^1$-Spacetimes Uniformly

When applying the genericity condition (Definition 2.15) we need to extend vectors or vector fields along curves to neighbourhoods, due to the fact that distributional (or also $L^\infty$) curvature quantities are not well defined on points or along curves (as these are sets of measure zero). It is rather straightforward to extend a vector field along a curve to a neighbourhood. For $C^1$-metrics one can still use parallel transport along certain curves spanning an open set. Alternatively, one can also use “cylindrically constant” (see below) extensions in coordinates around a (part of a) curve.

In our case, we need to simultaneously extend vector fields along a sequence of converging curves. To be more precise, let $\gamma _k$ be a sequence converging in $C^1$ to some $\gamma$ and let $V_k$ be vector fields along $\gamma _k$ and V along $\gamma$ such that $V_k \rightarrow V$ in $C^1$ as $k\rightarrow \infty$. Then given a point on $\gamma$, does it possess a neighbourhood U such that all $V_k$ can be extended to vector fields on U and such that their extensions converge in $C^1$ to the extension of V?

To begin with, we consider the situation of extending vector fields around a point of a curve to a neighbourhood by cylindrically constant extension in coordinates.

Lemma B.1

(Cylindrically constant extension of vector fields on curves). Let $\gamma :I\rightarrow M$ be a $C^1$-curve which is regular at 0, i.e. ${\dot{\gamma }}(0)\ne 0$ and set $p:=\gamma (0)$. Then there exists a chart $(U,(x^1,\dots ,x^n))$ around p with the following property: For any $C^1$ vector field V along $\gamma$ there exists a $C^1$ extension ${{\tilde{V}}}$ of V to U such that, in these coordinates, ${{\tilde{V}}}$ is independent of $x^2,\dots ,x^n$.

Proof

Since the claim is local we may assume that $M={\mathbb {R}}^n$, that $p=0$, and that $\gamma =(\gamma ^1,\dots ,\gamma ^n)$ with $\gamma _1'(t)> c > 0$ for some c and all $t\in (a,b)$, where $a<0<b$. Thus $\gamma ^1$ is a $C^1$-diffeomorphism from (a, b) onto some interval J. Then we set $U:= J \times B^{n-1}_R(0)$, where $R>0$ is such that $\gamma ((a,b))\subseteq U$. Given a $C^1$-vector field V along $\gamma$, i.e., a $C^1$-map $V:(a,b)\ni t \mapsto (\gamma (t),v(t))$ with $v\in C^1((a,b),{\mathbb {R}}^n)$, set ${{\tilde{V}}}: U \rightarrow U\times {\mathbb {R}}^{n}$,

$$\begin{aligned} {{\tilde{V}}}(x) := (x,v((\gamma ^1)^{-1}(x_1))) \end{aligned}$$

to obtain the desired extension. $\quad \square$

Now that we have set up a way of extending vector fields in a cylindrically constant fashion, we can implement an analogous procedure uniformly on a sequence of curves.

Lemma B.2

Let $\gamma _k:[-1,1]\rightarrow M$ be a sequence of $C^1$-curves converging in $C^1([-1,1])$ to the regular $C^1$-curve $\gamma :[-1,1]\rightarrow M$ and let $V_k$ and V be $C^1$ vector fields along $\gamma _k$ and $\gamma$, respectively. Further, let $V_k\rightarrow V$ in $C^1$, i.e., both $V_k\rightarrow V$ and $V_k'\rightarrow V'$ in TM, uniformly on $[-1,1]$. Then there exists some open neighbourhood U of $\gamma (0)$ and $C^1$ extensions ${{\tilde{V}}}_k$ of $V_k$ and ${{\tilde{V}}}$ of V to U such that ${{\tilde{V}}}_k \rightarrow {{\tilde{V}}}$ in $C^1_{\mathrm {loc}}(U)$.

Proof

Again we may assume that $M={\mathbb {R}}^n$. As in the previous Lemma we write $\gamma =(\gamma ^1,\dots ,\gamma ^n)$ and can assume without loss of generality that $\gamma (0)=0$, and that $(\gamma ^1)'(t)> c > 0$ for some c and all $t\in (a,b)$, where $a<0<b$. Due to the assumption on the $C^1$-convergence of the $\gamma _k$, we may additionally suppose that the same inequality holds on (a, b) for each $(\gamma ^1_k)'$. Thus each $\gamma ^1_k$ and $\gamma ^1$ itself are $C^1$-diffeomorphisms from (a, b) onto their respective image. In addition (restricting to n large if necessary), there exists a nontrivial interval J around 0 that is contained in $\bigcap _k \gamma ^1_k((a,b)) \cap \gamma ((a,b))$. Then we set $U:= J \times B^{n-1}_R(0)$, where $R>0$ is chosen such that $\bigcup _k \gamma _k((a,b)) \cup \gamma ((a,b))\subseteq U$. We can write V in the form $t \mapsto (\gamma (t),v(t))$ with $v\in C^1((a,b),{\mathbb {R}}^n)$, and analogously $V_k(t) = (\gamma _k(t),v_k(t))$, with $v_k \rightarrow v$ in $C^1$. For $x\in U$ we set

$$\begin{aligned} {{\tilde{V}}}(x) := (x,v((\gamma ^1)^{-1}(x^1))) \qquad {{\tilde{V}}}_k(x) := (x,v_k((\gamma _k^1)^{-1}(x^1))), \end{aligned}$$

giving $C^1$-extensions of V resp. $V_k$ to U. By [2, Cor. 1] we have that $(\gamma _k^1)^{-1}$ converges to $(\gamma ^1)^{-1}$ locally uniformly, and that the same is true for the first derivatives. Consequently, ${{\tilde{V}}}_k \rightarrow {{\tilde{V}}}$ in $C^1_{\mathrm {loc}}(U)$. $\quad \square$

As the proof shows, if the $V_k$ converge to V merely in $C^0_{\mathrm {loc}}$ then one can still find $C^1$-extensions to U that also converge in $C^0_{\mathrm {loc}}$.