June 10, 2026
We prove a singularity theorem in which the classical focusing hypothesis of Hawking–Penrose theory is replaced by a condition on asymptotic volume growth. Under the strong energy condition, we introduce asymptotic volume-expansion invariants associated with a compact Cauchy hypersurface and show that a uniform positive lower bound on these invariants implies past timelike geodesic incompleteness. More precisely, we obtain an explicit upper bound on the time-separation from the hypersurface to its chronological past. The theorem extends to globally hyperbolic Lorentzian length spaces satisfying the synthetic strong energy condition \(\mathsf{TCD}^e_p(0,N)\), yielding an inextendibility result valid without any smoothness or differentiability assumption. We also prove an area comparison theorem for equidistant hypersurfaces and a volume singularity theorem based on related asymptotic expansion invariants.
The singularity theorems of Penrose [1] and Hawking [2] are among the foundational achievements of mathematical relativity. They demonstrate that spacetime singularities — understood as causal geodesic incompleteness — are not merely artifacts of highly symmetric solutions of Einstein’s equations, but arise under broad and physically natural assumptions. Penrose’s theorem predicts singularity formation in gravitational collapse, whereas Hawking’s theorem applies to cosmological spacetimes undergoing expansion. Together with their many subsequent extensions and refinements, these results have profoundly shaped our understanding of black holes, cosmology, and the global structure of spacetime.
The underlying mechanism in both theorems is the combination of an energy condition with the focusing of causal geodesics. Concretely, one assumes that the spacetime is globally hyperbolic and that the Ricci curvature is nonnegative along timelike directions in Hawking’s theorem and along null directions in Penrose’s theorem. This is supplemented by a local geometric condition that induces the focusing of causal geodesics, typically formulated in terms of positive mean curvature of a compact Cauchy hypersurface or the existence of trapped surfaces. Under these hypotheses, one concludes that the spacetime contains an incomplete causal geodesic. In the classical framework of general relativity, such geodesic incompleteness is regarded as the hallmark of a spacetime singularity.
In this note we establish a singularity theorem of a different nature. Rather than assuming a local focusing condition on a compact Cauchy hypersurface, we impose a global condition on the asymptotic growth of volume. We show that, when combined with an appropriate lower bound on the timelike Ricci curvature, this large-scale geometric assumption alone suffices to force the existence of an incomplete timelike geodesic.
For the sake of exposition, we formulate the discussion in the introduction for \((n+1)\)-dimensional, smooth, globally hyperbolic Lorentzian manifolds \((M,g)\) satisfying the Hawking–Penrose strong energy condition: \[\label{eq:SEC-Intro} {\rm Ric}_g(v,v)\ge 0, \quad \text{for every timelike vector v\in TM}.\tag{1}\] The main results, however, hold in the substantially broader setting of Lorentzian length spaces [3]. In fact, they are proved under the synthetic strong energy condition \(\mathsf{TCD}^{e}_{p}(0,N)\) introduced by the authors in [4], which is rooted in the optimal transport characterizations of timelike Ricci curvature lower bounds established in the smooth setting by McCann [5] and by Suhr and the second author [6].
We begin by fixing notation. Let \(V\subset M\) be a smooth, compact, \(n\)-dimensional Cauchy hypersurface. In fact, the results of this paper are established under the weaker assumption that \(V\) is a compact achronal set with empty global edge (see 7). Denote by \(I^\pm(V)\) the chronological future and past of \(V\), respectively. The signed time-separation function from \(V\) is defined by \[\label{eq:deftauV-intro} \tau_V(x):= \begin{cases} \sup_{y\in V}\tau(y,x), & x\in I^+(V),\\ -\sup_{y\in V}\tau(x,y), & x\in I^-(V),\\ 0, & \text{otherwise}. \end{cases}\tag{2}\]
For the sake of exposition, let us first assume that \(\tau_V\) is smooth on \(I^\pm(V)\). Then the integral curves of the vector field \(-\nabla\tau_V\) induce a foliation of \(I^\pm(V)\). For each \(\alpha\in V\), let \(X_\alpha\) denote the unique integral curve through \(\alpha\) that we assume to be future complete. We parametrize it on its maximal interval of existence \((\mathfrak{a}_\alpha,+\infty)\), with \(\mathfrak{a}_\alpha<0\), and normalized by the condition \(X_\alpha(0)=\alpha\). When \(\tau_V\) fails to be smooth, an analogous decomposition still holds up to a set of vanishing volume measure; see the body of the paper for the precise statements.
The Fubini–Tonelli theorem, or more generally the disintegration theorem, yields a decomposition of the spacetime volume measure along this family of timelike geodesics: \[\label{eq:DisintVolvSmooth-Intro} \mathrm{Vol}_g\llcorner_{I^\pm(V)} = \int_V \mathfrak m_\alpha \,\mathrm{Vol}_{g_V}(\mathrm d\alpha),\tag{3}\] where \[\label{eq:malpha-Intro} \mathfrak m_\alpha = h_\alpha(\cdot)\,\mathcal{L}^1, \qquad h_\alpha(0)=1 \quad \text{for \mathrm{Vol}_{g_V}-a.e.\;\alpha\in V}.\tag{4}\] Here \(\mathrm{Vol}_g\) denotes the \((n+1)\)-dimensional volume measure of \((M,g)\), while \(\mathrm{Vol}_{g_V}\) is the induced \(n\)-dimensional volume measure on \(V\). The measures \(\mathfrak m_\alpha\) are supported on the geodesics \(X_\alpha\) and describe the distribution of spacetime volume along the foliation.
The strong energy condition 1 implies, via a Lorentzian analogue of the Bishop–Gromov inequality (which, in this one-dimensional setting, reduces to an elementary computation), that for each \(\alpha\) the function \[(\mathfrak{a}_\alpha,+\infty)\ni t\longmapsto \frac{h_\alpha(t)}{(n+1)t^n}\] is monotone non-increasing. We therefore define \[\label{eq:defthetaalpha-intro} \theta_\alpha := \lim_{t\to+\infty}\frac{h_\alpha(t)}{(n+1)t^n} = \inf_{t>\mathfrak{a}_\alpha}\frac{h_\alpha(t)}{(n+1)t^n}.\tag{5}\] Geometrically, \(\theta_\alpha\) represents the asymptotic volume expansion along the timelike ray \(X_\alpha\). The central observation of this note is that a uniform positive asymptotic expansion is incompatible with past timelike geodesic completeness. Indeed, if \(\theta_\alpha\) is bounded below by a positive constant, then the past of \(V\) has uniformly bounded time-separation from \(V\), yielding the existence of an incomplete past-directed timelike geodesic.
Theorem 1 (3). Let \((M,g)\) be an \((n+1)\)-dimensional, globally hyperbolic, smooth Lorentzian manifold satisfying Penrose-Hawking’s strong energy condition 1 . Let \(V\subset M\) be a smooth, compact, Cauchy hypersurface. Assume moreover that there exists a constant \(c>0\) such that \[\label{eq:thetaalphageqc-intro} \theta_\alpha \ge c >0 \qquad \text{for \(\mathrm{Vol}_V\)-a.e.\;\(\alpha \in V\)} .\tag{6}\] Then \[\sup_{x\in I^{-}(V)} |\tau_{V}(x)| \leq \left(\frac{1}{c}\right)^{\frac{1}{n}}.\] In particular, \((M,g)\) is not past timelike geodesically complete.
Moreover, for any (possibly non-smooth) past extension \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) of \((M,g)\) which is a timelike non-branching, globally hyperbolic, Lorentzian
geodesic space, satisfying the \(\mathsf{TCD}^{e}_{p}(0,n+1)\) condition, together with its causally-reversed structure, it holds that \[\label{eq:tauVleq147cCor1-Intro}
\sup_{x\in I^{-}_X(V)} |\tau_{V}(x)| \leq \left(\frac{1}{c}\right)^{\frac{1}{n}},\tag{7}\] where \(I^{-}_X(V)\subset X\) is the chronological past of \(V\) in \(X\).
In particular, \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) is not past timelike geodesically complete.
The second part of the theorem may be viewed as an inextendibility statement. It shows that the combination of the strong energy condition and the asymptotic volume growth assumption 6 enforces past timelike geodesic incompleteness not only in the smooth spacetime \((M,g)\) itself, but in every globally hyperbolic extension satisfying the same synthetic energy condition, regardless of its regularity. In this sense, the singularity predicted by the theorem cannot be removed by passing to a lower-regularity extension.
Beyond the singularity theorem stated above, the techniques introduced in this paper yield several additional results of independent interest. These include an area comparison estimate for equidistant hypersurfaces from \(V\) (see 4) and a volume singularity theorem (see 5.1) in the sense of [7], both formulated in terms of asymptotic invariants capturing the large-time expansion of the geodesic congruence generated by \(V\).
The present work is part of the emerging theory of synthetic Lorentzian geometry. Among the achievements of the theory of \(\mathsf{TCD}^{e}_{p}(K,N)\) Lorentzian length spaces there is the extension of Hawking’s singularity theorem to a fully synthetic setting, obtained in [4]; see also the survey [8].
Related developments include inextendibility theorems for Lorentzian pre-length spaces satisfying synthetic lower bounds on timelike sectional curvature [9]–[11], as well as singularity theorems in the broader framework of closed cone structures [12].
More generally, the synthetic approach has led to analogues of a number of foundational comparison, rigidity, and singularity theorems from Lorentzian geometry highlighting the flexibility of synthetic curvature-dimension conditions, see for instance [13]–[18].
A notable feature of the synthetic framework is its ability to accommodate singular spacetimes lying beyond the scope of classical Lorentzian geometry. In particular, Lorentzian pre-length spaces satisfying synthetic lower bounds on timelike Ricci curvature include physically relevant examples such as Penrose’s impulsive gravitational waves [19] and spacetimes endowed with Lipschitz Lorentzian metrics whose timelike Ricci curvature is bounded below in the sense of distributions [20].
The present paper should be viewed as a contribution to this program: it identifies a new synthetic mechanism for geodesic incompleteness based on asymptotic volume expansion rather than local focusing.
For the purpose of Open Access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.
A. M.acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme, Grant Agreement No., “CURVATURE”.
Part of this work was carried out during the Thematic Program “Nonsmooth Riemannian and Lorentzian Geometry” at the Fields Institute (Toronto), the workshop “Non-regular Spacetime Geometry” at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna, and the Trimester Program “Metric Analysis” at the Hausdorff Institute for Mathematics (HIM) in Bonn. The authors are grateful to these institutions and to the organisers of the respective programs for their hospitality and for providing inspiring and productive research environments.
The general framework for the paper is the one of Lorenzian pre-length spaces introduced by Kunzinger-Sämann in [3], inspired by earlier works of Kronheimer-Penrose [21] on causal spaces and of Busemann [22] on timelike spaces. An alternative approach put forward by Minguzzi-Suhr [23] derived many topological properties out of basic assumptions on the time-separation. Variants have been studied by several authors with different motivations, let us mention [5], [14], [15], [24]–[26].
Recall that \((X,\mathsf d, \ll, \leq, \tau)\) is a Lorentzian pre-length space provided: \((X,\ll,\leq)\) is a causal space (i.e., \(\leq\) is a preorder and \(\ll\) a transitive relation contained in \(\leq\)) additionally equipped with a proper metric \(\mathsf d\) (i.e., closed and bounded subsets are compact) and a lower semicontinuous function \(\tau: X\times X\to [0,\infty]\), called time-separation function, satisfying \[\nonumber \begin{align} \tau(x,y)+\tau(y,z)\leq \tau (x,z) &\quad\forall x\leq y\leq z \quad \text{reverse triangle inequality} \\ \tau(x,y)=0, \; \text{if } x\not\leq y, & \quad \tau(x,y)>0 \Leftrightarrow x\ll y. \end{align}\] We write \(x<y\) whenever \(x\le y\) and \(x\neq y\). We say that \(x\) and \(y\) are timelike (resp.causally) related if \(x\ll y\) (resp.\(x\le y\)).
Let \(A\subset X\) be an arbitrary subset. The chronological (resp.causal) future of \(A\) is defined as \[\begin{align} I^{+}(A)&:=\{y\in X:\;\exists\, x\in A \text{ such that } x\ll y\},\\ \qquad J^{+}(A)&:=\{y\in X:\;\exists\, x\in A \text{ such that } x\le y\}. \end{align}\] Analogously, one defines the chronological (resp.causal) past of \(A\).
If \(A=\{x\}\) is a singleton, we will slightly abuse notation and write \(I^{\pm}(x)\) (resp.\(J^{\pm}(x)\)) instead of \(I^{\pm}(\{x\})\) (resp.\(J^{\pm}(\{x\})\)).
We say that \((X,\mathsf d, \ll_r, \leq_r, \tau_r)\) is the causally-reversed structure of \((X,\mathsf d, \ll, \leq, \tau)\) if \[x\ll_r y \iff y\ll x, \quad x\leq_r y\iff y\leq x, \quad \tau_r(x,y)=\tau(y,x).\] The set of (resp.timelike) geodesics is defined as: \[\begin{align} &{\rm Geo}(X):= \{ \gamma\in C([0,1], X)\,: \, \tau(\gamma_{s}, \gamma_{t})=(t-s)\, \tau(\gamma_{0}, \gamma_{1}),\; \forall s<t\},\\ & {\rm TGeo}(X):=\{ \gamma\in {\rm Geo}(X) \,: \, \tau(\gamma_{0}, \gamma_{1}) > 0\}. \end{align}\]
Definition 1 (Timelike non-branching). A Lorentzian pre-length space \((X,\mathsf d, \ll, \leq, \tau)\) is said to be forward timelike non-branching if and only if for any \(\gamma^{1},\gamma^{2} \in {\rm TGeo}(X)\), it holds: \[\exists \; \bar t\in (0,1) \text{ such that } \;\forall t \in [0, \bar t\,] \quad \gamma_{ t}^{1} = \gamma_{t}^{2} \quad \Longrightarrow \quad \gamma^{1}_{s} = \gamma^{2}_{s}, \quad \forall s \in [0,1].\] \(X\) is said to be backward timelike non-branching if the causally-reversed structure is forward timelike non-branching. In case \(X\) is both forward and backward timelike non-branching it is said timelike non-branching.
Concerning the causal ladder (see [3], [27] for more details), we will only consider Lorentzian geodesic spaces, i.e. Lorentzian pre-length spaces \((X,\mathsf d, \ll, \leq, \tau)\) that additionally are:
\(\mathsf d\)-Compatible: every \(x\in X\) admits a neighbourhood \(U\) and a constant \(C\) such that \(L_{\mathsf d}(\gamma)\leq C\) for every causal curve \(\gamma\) contained in \(U\);
Geodesic: for all \(x,y\in X\) with \(x<y\) there is a future-directed causal curve \(\gamma\) from \(x\) to \(y\) with \(\tau(x,y)= L_{\tau}(\gamma)\).
We consider the following version of global hyperbolicity that fits with the previous literature (see [27]): a Lorentzian geodesic space \((X,\mathsf d, \ll, \leq,\tau)\) is called
Causal: if \(\leq\) is also antisymmetric, i.e. \(\leq\) is an order;
Globally hyperbolic: if it is causal and for every \(x,y\in X\) the causal diamond \(J^{+}(x)\cap J^{-}(y)\) is compact in \(X\).
Let \(\mathcal{P}(X)\) be the set of Borel probability measures over \(X\). Denote by \(P_i:X\times X\to X\), \(i=1,2\)
the projection map and let \((P_i)_{\sharp}: \mathcal{P}(X\times X)\to \mathcal{P}(X)\) be the corresponding push-forward map.
Given \(\mu,\nu\in \mathcal{P}(X)\), consider \[\begin{align} \Pi(\mu,\nu)&:=\{\pi\in \mathcal{P}(X\times X) \,:\, (P_{1})_{\sharp}\pi=\mu, \, (P_{2})_{\sharp}\pi=\nu \}, \nonumber \\
\Pi_{\leq}(\mu,\nu)&:=\{\pi\in \Pi(\mu,\nu) \,:\, \pi(X^{2}_{\leq})=1 \},
\nonumber \\ \Pi_{\ll}(\mu,\nu)&:=\{\pi\in \Pi(\mu,\nu) \,:\, \pi(X^{2}_{\ll})=1 \},
\end{align}\] where \(X^{2}_{\leq}:=\{(x,y) \in X^{2}\,:\, x\leq y \}\) and \(X^{2}_{\ll}:=\{(x,y) \in X^{2}\,:\, x\ll y \}\).
We next recall the notion of \(p\)-Lorentz-Wasserstein distance, following the convention in [4]. The definition below extends to Lorentzian pre-length spaces the corresponding notion given in the smooth Lorentzian setting in [28] (see also [5], [6], and [29] for \(p=1\)).
Definition 2. Let \((X,\mathsf d, \ll, \leq, \tau)\) be a Lorentzian pre-length space and let \(p\in (0,1]\). Given \(\mu,\nu\in \mathcal{P}(X)\), the \(p\)-Lorentz-Wasserstein distance is defined by \[\label{eq:defWp} \ell_{p}(\mu,\nu):= \sup_{\pi \in \Pi_{\leq}(\mu,\nu)} \left( \int_{X\times X} \tau(x,y)^{p} \, \pi(\mathrm{d}x \mathrm{d}y)\right)^{1/p}.\tag{8}\] When \(\Pi_{\leq}(\mu,\nu)=\emptyset\) we set \(\ell_{p}(\mu,\nu):=-\infty\).
A coupling \(\pi\in \Pi_{\leq}(\mu,\nu)\) maximising in 8 is said \(\ell_{p}\)-optimal. The set of \(\ell_{p}\)-optimal couplings from \(\mu\) to \(\nu\) is denoted by \(\Pi_{\leq}^{p\text{-opt}}(\mu,\nu)\). We also set \[\Pi_{\ll}^{p\text{-opt}}(\mu,\nu):= \Pi_{\leq}^{p\text{-opt}}(\mu,\nu)\cap \Pi_{\ll}(\mu,\nu)\] the family of timelike \(\ell_{p}\)-optimal couplings.
By gluing of couplings, one can prove that the \(p\)-Lorentz-Wasserstein distance \(\ell_p\) satisfies the reverse triangle inequality on causally related measures (see [4]), thus lifting the Lorentzian metric structure to the space of probability measures.
In order to transfer the causal constraint in the optimal transport problem to the cost function (see [4] for more details), it is also useful to consider \[\label{eq:defell} \ell(x,y)^{p}:= \begin{cases} \tau(x,y)^{p} \quad & \text{if } x\leq y \\ -\infty \quad & \text{otherwise} \end{cases}.\tag{9}\]
We next recall the notion of timelike \(p\)-dualisable pairs of measures from [4], relaxing the notion of \(q\)-separated measures introduced by McCann [5] in the smooth Lorentzian setting.
Definition 3 (Timelike \(p\)-dualisable). Let \((X,\mathsf d, \ll, \leq, \tau)\) be a Lorentzian pre-length space and let \(p\in (0,1]\). We say that \((\mu,\nu)\in \mathcal{P}(X)^{2}\) is timelike \(p\)-dualisable (by \(\pi\in \Pi_{\ll}(\mu,\nu)\)) if
\(\ell_{p}(\mu,\nu)\in (0,\infty)\);
\(\pi\in \Pi_{\ll}^{p\text{-opt}}(\mu,\nu)\);
there exist measurable functions \(a,b:X\to \mathbb{R}\), with \(a\oplus b \in L^{1}(\mu\otimes \nu)\) such that \(\ell^{p}\leq a\oplus b\) on \(\text{\rm supp}\, \mu \times \text{\rm supp}\, \nu\).
The motivation for considering \(p\)-dualisable pairs of measures is three-fold: firstly the \(p\)-optimal coupling \(\pi(\mathrm{d}x \mathrm{d}y)\)
matches events described by \(\mu(\mathrm{d}x)\) with events described by \(\nu(\mathrm{d}y)\) so that \(x\ll y\), secondly Kantorovich duality holds (see
[4]; see also the earlier works [5], [29], [30] in the smooth Lorentzian setting), thirdly the class of \(p\)-dualisable pairs of measures will provide a key
building block to define the synthetic timelike Ricci lower bounds. We conclude this subsection by recalling the notion of \(\ell_p\)-geodesic \((\mu_s)_{s\in [0,1]}\subset
\mathcal{P}(X)\).
The evaluation map is defined by \[\label{def:eet}
{\rm e}_{t}: C([0,1], X) \to X,
\qquad
\gamma \mapsto {\rm e}_{t}(\gamma):=\gamma_{t},
\qquad \forall t\in [0,1].\tag{10}\]
Definition 4 (\(\ell_p\)-optimal dynamical plans and \(\ell_p\)-geodesics). Let \((X,\mathsf d, \ll, \leq, \tau)\) be a Lorentzian
pre-length space and let \(p\in (0,1]\). We say that \(\eta\in \mathcal{P}({\rm Geo}(X))\) is an \(\ell_p\)-optimal dynamical plan from \(\mu_0\in \mathcal{P}(X)\) to \(\mu_1\in \mathcal{P}(X)\) if \[\label{eq:defODP}
({\rm e}_0, {\rm e}_1)_{\sharp} \eta
\in
\Pi^{p\text{-opt}}_{\leq} \big(({\rm e}_0)_\sharp \eta, ({\rm e}_1)_\sharp \eta\big).\tag{11}\] The collection of all \(\ell_p\)-optimal dynamical plans from \(\mu_{0}\) to
\(\mu_{1}\) is denoted by \({\rm OptGeo}_{\ell_{p}}(\mu_{0}, \mu_{1})\).
We say that a curve \([0,1]\ni t \mapsto \mu_{t}\in \mathcal{P}(X)\) is an \(\ell_p\)-geodesic if there exists \(\eta\in {\rm
OptGeo}_{\ell_p}(\mu_0,\mu_1)\) such that \[\mu_t=({\rm e}_t)_\sharp \eta,
\qquad \forall t\in [0,1].\]
Observe that if \(\eta\in {\rm OptGeo}_{\ell_p}(\mu_0,\mu_1)\), then the associated curve \[\mu_t:=({\rm e}_t)_\sharp \eta, \qquad \forall t\in [0,1],\] is continuous with respect to the narrow topology and satisfies \[\ell_p(\mu_s,\mu_t) = (t-s)\,\ell_p(\mu_0,\mu_1), \qquad \forall\, 0\le s\le t\le 1.\] Finally, recall that if \(X\) is a globally hyperbolic Lorentzian geodesic space, \(\mu_0,\mu_1\in \mathcal{P}(X)\) have compact support and \(\Pi_{\leq}(\mu_0, \mu_1)\neq \emptyset\), then there always exists an \(\ell_p\)-optimal dynamical plan \(\eta\in {\rm OptGeo}_{\ell_p}(\mu_0,\mu_1)\) (and hence an \(\ell_p\)-geodesic) connecting \(\mu_0\) to \(\mu_1\); see [4] for the proof and further properties of \(\ell_p\)-geodesics.
A measured Lorentzian pre-length space \((X,\mathsf d, \mathfrak m, \ll, \leq, \tau)\) is a Lorentzian pre-length space endowed with a Radon non-negative measure \(\mathfrak m\). We say that \((X,\mathsf d, \mathfrak m, \ll, \leq, \tau)\) is globally hyperbolic (resp. geodesic) if \((X,\mathsf d, \ll, \leq, \tau)\) is so.
We next recall the definition of the Timelike Curvature–Dimension condition, denoted by \(\mathsf{TCD}^{e}_{p}(K,N)\) (and by \(\mathsf{wTCD}^{e}_{p}(K,N)\) in its weak form), as introduced in [4] (after [5] and [6]). Here \(K\in \mathbb{R}\) stands for a synthetic lower bound on the timelike Ricci curvature, and \(N\in (0,\infty)\) stands for a synthetic upper bound on the dimension. Since in this paper we will only consider the case on non-negative timelike Ricci curvature, we will focus on the case \(K=0\). We refer the reader to [4] for more details and to [8] for a concise overview.
The definition is based on a suitable concavity property of the entropy functional, which we now briefly recall. Given \(\mu \in \mathcal{P}(X)\), its Boltzmann-Shannon entropy w.r.t. \(\mathfrak m\) is given by \[{\rm Ent}(\mu|\mathfrak m) = \int_{M} \rho \log\rho \; \mathfrak m,\] if \(\mu = \rho \, \mathfrak m\) is absolutely continuous with respect to \(\mathfrak m\) and \((\rho\log \rho)_{+}\) is \(\mathfrak m\)-integrable. Otherwise we set \({\rm Ent}(\mu|\mathfrak m) = +\infty\). We set \[{\rm Dom}({\rm Ent}(\cdot|\mathfrak m)):=\{\mu\in \mathcal{P}(X)\,:\, {\rm Ent}(\mu|\mathfrak m)\in \mathbb{R}\},\] to be the finiteness domain of the entropy.
Definition 5 (\(\mathsf{TCD}^{e}_{p}(0,N)\) condition, [4]). Fix \(p\in (0,1)\) and \(N\in (0,\infty)\). We say that a measured Lorentzian pre-length space \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) satisfies \(\mathsf{TCD}^{e}_{p}(0,N)\) if the following holds. For any pair \((\mu_{0},\mu_{1})\in ({\rm Dom}({\rm Ent}(\cdot|\mathfrak m)))^{2}\) which is timelike \(p\)-dualisable by some \(\pi\in \Pi^{p\text{-opt}}_{\ll}(\mu_{0},\mu_{1})\), there exists an \(\ell_{p}\)-geodesic \((\mu_{t})_{t\in [0,1]}\) such that the function \([0,1]\ni t\mapsto e(t) : = {\rm Ent}(\mu_{t}|\mathrm{Vol}_{g})\) is convex (and thus in particular it is locally Lipschitz in \((0,1)\)) and it satisfies \[\label{eq:conveKN} e''(t) - \frac{1}{N} e'(t)^{2 } \geq 0.\tag{12}\] in the distributional sense on \([0,1]\).
A variant on the \(\mathsf{TCD}^e_p\) condition, denoted by \(\mathsf{TCD}^{*}_{p}\) was later introduced by Braun [13]. The same paper also proved the equivalence between the \(\mathsf{TCD}^{e}_{p}\) and the \(\mathsf{TCD}^{*}_{p}\) conditions in timelike non-branching Lorentzian pre-length spaces.
Another key object in our analysis is the synthetic analogue of the gradient flow lines associated with the time-separation function from a given set \(V\). We now recall some basic constructions.
Standing assumptions. From now on we will always assume that \((X,\mathsf d, \ll, \leq, \tau)\) is a globally hyperbolic Lorentzian geodesic space (see subsection 2.1.1
for the definitions).
Under such standing assumptions, the time separation \(\tau:X\times X\to \mathbb{R}\) is continuous (see [3]). A subset \(V\subset X\) is called achronal if \(x\not \ll y\) for every \(x,y\in V\). In particular, if \(V\) is achronal, then
\(I^{+}(V)\cap I^{-}(V)= \emptyset\), so we can define the signed time-separation to \(V\), \(\tau_{V}:X\to [-\infty, +\infty]\), by
\[\label{eq:deftauV}
\tau_{V}(x):=
\begin{cases}
\sup_{y\in V} \tau(y,x), &\quad \text{ for }x\in I^{+}(V)\\
-\sup_{y\in V} \tau(x,y),& \quad \text{ for }x\in I^{-}(V) \\
0 &\quad \text{ otherwise}
\end{cases}.\tag{13}\] Note that \(\tau_{V}\) is lower semi-continuous on \(I^{+}(V)\) as supremum of continuous functions, and is upper semi-continuous on \(I^{-}(V)\).
To introduce a synthetic analogue of the integral curves of \(-\nabla \tau_{V}\), some preparatory tools are needed. We briefly recall the construction and refer to [4] for a more detailed treatment.
Definition 6 (Future timelike complete (FTC) subsets, [31]). A subset \(V\subset X\) is future timelike complete (FTC), if for each point \(x\in I^{+}(V)\), the intersection \(J^{-}(x)\cap V \subset V\) has compact closure (w.r.t. \(\mathsf d\)) in \(V\). Analogously, one defines past timelike completeness (PTC). A subset that is both FTC and PTC is called timelike complete.
Lemma 1 (Lemma 4.1, [4]). Let \((X,\mathsf d, \ll, \leq, \tau)\) be a globally hyperbolic Lorentzian geodesic space and let \(V\subset X\) be an achronal FTC (resp. PTC) subset. Then for each \(x\in I^{+}(V)\) (resp. \(x\in I^{-}(V)\)) there exists a point \(y_{x}\in V\) with \(\tau_{V}(x)=\tau(y_{x},x)>0\) (resp. \(\tau_{V}(x)=-\tau(x,y_{x})<0\)).
Moreover for all \(x,z\in I^{+}(V)\cup V\), \[\label{eq:tauvzxtau} \tau_{V}(z) - \tau_{V}(x) \geq \tau(y_{x},z)-\tau(y_{x},x) \geq \tau(x,z),\tag{14}\] provided \((x,z) \in X^{2}_{\leq}\). An analogous statement is valid for \(x\in I^{-}(V)\).
The inequality 14 can be restated by saying that \(\tau_V\) is timelike reverse 1-Lipschitz on \(I^+(V)\) and, separately, on \(I^-(V)\). As shown in [18], this permits to study the integral lines of maximal steep of \(-\tau_V\) on \(I^+(V)\) and, separately, on \(I^-(V)\) giving a disintegration formula for the reference measure \(\mathfrak m\) whose conditional measures localize the eventual Ricci curvature lower bound to the integral lines of \(-\tau_V\).
For the scope of this note, the interest will be in performing this analysis jointly on \(I^+(V)\) and \(I^-(V)\). To start, for ease of writing, we will use the following notation \[I^{\pm}(V) : = (I^{+}(V)\cup I^{-}(V)\cup V).\] In general \(\tau_V\) fails to be reverse 1-Lipschitz on \(I^\pm(V)\): consider for instance in Minkowski spacetime \(V= \{0\}\), the origin. Let \(z\) be any point in the future light cone of \(0\) and \(x\) be any point in the past light cone of \(0\) additionally being in the timelike past of \(z\). Then \[\tau_V(z)-\tau_V(x)=\tau(0,z)+\tau(x,0)=0<\tau(x,z),\] contradicting the reverse 1-Lipschitz property.
Nevertheless, as discussed below, the reverse 1-Lipschitzianity holds in some special cases of interest.
Definition 7 (Empty global edge). Let \((X,\mathsf d, \ll, \leq, \tau)\) be a globally hyperbolic Lorentzian geodesic space. Let \(V\subset X\) be an achronal set. We say that \(V\) has empty global edge if for any \(x \in I^-(V)\) and \(z\in I^+(V)\), any \(\gamma\in {\rm TGeo}(X)\) connecting \(x\) to \(z\) intersects \(V\).
7 may be viewed as a global counterpart of the condition that an achronal set has empty edge. Recall that the edge of an achronal set \(S \subset X\) consists of those points \(p \in S\) such that every neighbourhood \(U \subset X\) of \(p\) contains points \(p^+ \in I^+(p)\) and \(p^- \in I^-(p)\), together with a timelike curve connecting \(p^-\) to \(p^+\) that does not intersect \(S\). Clearly, if \(V\) has empty global edge then its edge is empty as well, however the converse implication may fail.
The notion of the edge of an achronal set is classical in Lorentzian geometry and causality theory; see, for instance, the foundational references [32], [33] or the more recent survey [34].
Remark 2 (Sufficient conditions for \(V\subset X\) to have empty global edge).
Let \((M,g)\) be a spacetime with a continuous Lorentzian metric. If \(V\) is a Cauchy hypersurface, then \(V\) has empty global edge. For properties of Cauchy hypersurfaces in \(C^0\)-spacetimes, see [35].
Recall the definition of future Cauchy development \(D^+(V)\) of \(V\): \[D^+(V) : = \{ p \in X \colon \text{every past inext. timelike curve through p intersects V}\}.\] If \(I^+(V) \supset D^+(V)\), then \(V\) has empty global edge.
Proposition 3. Let \((X,\mathsf d, \ll, \leq, \tau)\) be a globally hyperbolic Lorentzian geodesic space and let \(V\subset X\) be an achronal, timelike complete subset.
If \(V\) has empty global edge, then \(\tau_V\) is reverse 1-Lipschitz over \(I^\pm(V)\).
Proof. It suffices to prove that, for all \(x\leq z\) with \(x \in I^-(V)\) and \(z \in I^+(V)\): \[\tau_{V}(z) - \tau_{V}(x) \geq \tau(x,z).\] If \(\tau(x,z) = 0\), the claim is verified as \(\tau_{V}(x) \leq 0\) and \(\tau_V(z)\geq 0\). If \(\tau(x,z) > 0\), then there exists \(\gamma\in {\rm TGeo}(X)\) with \(\gamma_0=x\) and \(\gamma_1 = z\). Since \(V\) is achronal with empty geodesic boundary, there exists a unique \(s \in (0,1)\) such that \(\gamma_s \in V\). Then \(\tau(x,z) = \tau(x,\gamma_s) + \tau(\gamma_s,z) \leq \tau_V(z) - \tau_V(x)\); by definition indeed \(- \tau_V(x) = \sup_{y\in V} \tau(x,y)\). The claim is therefore proved. ◻
In [18], the localization of a general reverse 1-Lipschitz function is discussed in detail. Hence we now confine ourselves to reporting the main definitions and results.
Given \(V\subset X\), an achronal set with empty global edge, define: \[\label{E:GammaV} \begin{align} \Gamma_{V} : = &~ \{ (x,z) \in I^{\pm}(V)^{2} \cap X^{2}_{\leq} \, \colon \, \tau_{V}(z) - \tau_{V}(x) = \tau(x,z)>0 \} \\ &\qquad \cup \{(x,x) \,:\, x\in I^{\pm}(V)\}. \end{align}\tag{15}\] The monotonicity of \(\Gamma_{V}\) ensures that pairs of points belonging to \(\Gamma_{V}\) are aligned along geodesics: if \((x,z) \in \Gamma_{V}\) with \(x \neq z\) and \(x \in I^{+}(V)\), then there exist \(y \in V, \gamma \in {\rm TGeo}(X)\) and \(t \in (0,1)\) such that \(y=\gamma_0, \;x = \gamma_{t}, \; z=\gamma_1\), and \[\tau(y,\gamma_{s}) = \tau_{V}(\gamma_{s}) \quad \forall s\in [0,1], \qquad (\gamma_{s},\gamma_{t}) \in \Gamma_{V}, \quad \forall s\in [0,t].\] An analogous property holds true if \(z \in I^{-}(V)\). Next, define \[\Gamma_{V}^{-1}:=\{(x,y)\,:\, (y,x)\in \Gamma_{V}\}\] and consider the transport relation \(R_{V}\) and the transport set with endpoints \(\mathcal{T}_{V}^{end}\) \[\label{E:transport} R_{V} : = \Gamma_{V} \cup \Gamma_{V}^{-1}, \qquad \mathcal{T}_{V}^{end} : = P_{1}(R_{V}\setminus \{ x = y \}).\tag{16}\] The transport relation \(R_{V}\) will be an equivalence relation on a suitable subset of \(\mathcal{T}_{V}^{end}\), once initial and final points of the integral lines are removed. The sets \[\label{eq:defendpoints} \begin{align} \mathfrak{a}= \mathfrak{a}(\mathcal{T}_{V}^{end}) : =&~ \{ x \in \mathcal{T}_{V}^{end} \colon \nexists y \in \mathcal{T}_{V}^{end} \;s.t. \;(y,x) \in \Gamma_{V}, y\neq x \} \\ \mathfrak{b}= \mathfrak{b}(\mathcal{T}_{V}^{end}) : =&~ \{ x \in \mathcal{T}_{V}^{end} \colon \nexists y \in \mathcal{T}_{V}^{end} \;s.t. \;(x,y) \in \Gamma_{V}, y\neq x \}, \end{align}\tag{17}\] are the initial and final points, respectively. The transport set without endpoints is defined by: \[\label{eq:defTV} \mathcal{T}_{V} : = \mathcal{T}_{V}^{end} \setminus (\mathfrak{a}(\mathcal{T}_{V}^{end}) \cup \mathfrak{b}(\mathcal{T}_{V}^{end})).\tag{18}\]
Remark 4. The set \(\mathfrak{a}(\mathcal{T}_{V}^{end})\) shall be understood as the past cut-locus of \(V\) while \(\mathfrak{b}(\mathcal{T}_{V}^{end})\) the future one, being indeed those points, in the smooth scenario, where the geodesics stops to be maximizing.
If additionally \(V\subset X\) is timelike complete, repeating the argument of [4] one can prove that \[I^{+}(V) \cup I^{-}(V) = \mathcal{T}_{V}^{end} \setminus V.\] If \(X\) is timelike (both backward and forward) non-branching, then the transport relation \(R_{V}\) defines an equivalence relation on \(\mathcal{T}_{V}\). The corresponding equivalence classes are timelike geodesics that locally maximize the function \(\tau_{V}\); we denote them by \(X_{\alpha}\). Moreover, there exists a measurable quotient map \[\mathfrak Q: \mathcal{T}_{V} \to Q,\] associated with the equivalence relation \(R_{V}\) on \(\mathcal{T}_{V}\).
The corresponding disintegration of the reference measure \(\mathfrak m\) and the localization of the Ricci curvature bounds were established in [4], [18] (see also [15] for the extension to variable timelike Ricci lower bounds). We denote by \(\mathcal{M}_{+}(X)\) the space of non-negative Radon measures over \(X\).
Theorem 2. Let \((X,\mathsf d, \mathfrak m, \ll, \leq, \tau)\) be a globally hyperbolic, timelike non-branching, Lorentzian geodesic space satisfying \(\mathsf{TCD}^e_p(0,N)\), assume that the causally-reversed structure satisfies the same conditions and let \(V\subset X\) be a Borel achronal timelike complete subset with empty global
edge.
Let \(\mathcal{T}_{V}^{end}, \mathfrak{a}(\mathcal{T}_{V}^{end}), \mathfrak{b}(\mathcal{T}_{V}^{end})\) and \(\mathcal{T}_{V}\) defined in 16 , 17 and 18 .
Then \(\mathfrak m(\mathfrak{a}(\mathcal{T}_{V}^{end}))=\mathfrak m(\mathfrak{b}(\mathcal{T}_{V}^{end})=0\) and the following disintegration formula holds: \[\label{E:disintegration} \mathfrak m\llcorner_{\mathcal{T}^{end}_{V}} = \mathfrak m\llcorner_{\mathcal{T}_{V}} = \int_{Q} \mathfrak m_{\alpha}\, \mathfrak q(\mathrm{d}\alpha),\tag{19}\] where \(\mathfrak q\) is a Borel probability measure over a quotient set \(Q \subset \mathcal{T}_{V}\) such that \(\mathfrak Q_{\sharp}( \mathfrak m\llcorner_{\mathcal{T}_{V}} ) \ll \mathfrak q\) and the map \(Q \ni \alpha \mapsto \mathfrak m_{\alpha} \in \mathcal{M}_{+}(X)\) satisfies the following properties:
for any \(\mathfrak m\)-measurable set \(B\subset X\), the map \(\alpha \mapsto \mathfrak m_{\alpha}(B)\) is \(\mathfrak q\)-measurable;
for \(\mathfrak q\)-a.e. \(\alpha \in Q\), \(\mathfrak m_{\alpha}\) is concentrated on \(\mathfrak Q^{-1}(\alpha) = X_{\alpha}\) (strong consistency);
for \(\mathfrak q\)-a.e. \(\alpha \in Q\), \(\mathfrak m_{\alpha}\ll \mathcal{L}^{1}\llcorner_{X_{\alpha}}\);
for \(\mathfrak q\)-a.e. \(\alpha \in Q\), the one-dimensional metric measure space \((X_{\alpha},|\cdot|, \mathfrak m_{\alpha})\) satisfies the classical \(\mathsf{CD}(0,N)\); namely,writing \(\mathfrak m_\alpha=h(\alpha, \cdot)\mathcal{L}^{1}\llcorner_{X_{\alpha}}\), then \(h(\alpha, \cdot)\) is semi-concave (and thus twice differentiable \(\mathcal{L}^{1}\)-a.e. on \(X_{\alpha}\)) and it satisfies the differential inequality \[\label{eq:DiffIneqCDKN} \frac{\partial^2}{\partial x^2}\log h(\alpha, x)+\frac{1}{N-1}\left(\frac{\partial}{\partial x}\log h(\alpha, x)\right)^2 \leq 0,\tag{20}\] at any point \(x\) in the interior of \(X_\alpha\) where \(h(\alpha, \cdot)\) is twice differentiable.
Moreover, fixed any \(\mathfrak q\) as above such that \(\mathfrak Q_{\sharp}( \mathfrak m\llcorner_{\mathcal{T}_{V}} ) \ll \mathfrak q\), the disintegration is \(\mathfrak q\)-essentially unique in the following sense: if any other map \(Q \ni \alpha \mapsto \bar \mathfrak m_{\alpha} \in \mathcal{P}(X)\) satisfies points (1)-(2), then \(\bar \mathfrak m_{\alpha} = \mathfrak m_{\alpha}\) for \(\mathfrak q\)-a.e. \(\alpha \in Q\).
Building on 2, in the previous work [18], the authors established results concerning the area of achronal subsets of \(X\). We begin by recalling the definition of the area of an achronal set.
Definition 8 (Timelike Minkowski content). Let \(A \subset X\) be a Borel achronal set and consider the signed time-separation function \(\tau_{A}\) from \(A\), see 13 . We define the future Minkowski content of \(A\) by \[\label{E:future} \begin{align} \mathfrak m^{+}(A) &: = \inf_{U \in \mathcal{U}} \limsup_{\varepsilon\to 0} \frac{\mathfrak m( \tau_{A}^{-1}((0,\varepsilon)) \cap U) }{\varepsilon}, \\ \mathcal{U}&: = \{ U \subset X \colon U \text{ open}, \;A\subset U \}. \end{align}\tag{21}\] Analogously, the past Minkowski content of \(A\) is defined by \[\label{E:pastMink} \begin{align} \mathfrak m^{-}(A) &: = \inf_{U \in \mathcal{U}} \limsup_{\varepsilon\to 0} \frac{\mathfrak m( \tau_{A}^{-1}((-\varepsilon,0)) \cap U) }{\varepsilon}. \end{align}\tag{22}\]
Remark 5. If \(X\) is a smooth, globally hyperbolic, Lorentzian manifold and \(A\subset X\) is a smooth, achronal, future causally complete hypersurface, then \(\mathfrak m^+(A)\) coincides with the standard area of \(A\) computed with respect to the restriction of the ambient Lorentzian metric; see [18].
In particular, as shown in [36], in this smooth setting the time separation function \(\tau_A\) is smooth on \(I^+(A)\setminus C_+(A)\), where \(C_+(A)\) is the future cut-locus of \(A\) that is a closed set of zero measure. This implies that each level set \(\tau_A^{-1}(s)\) is a smooth submanifold of \(I^+(A)\setminus C_+(A)\) having therefore a well-defined volume measure induced by the ambient metric. Finally the coarea formula [36] gives the claim.
As shown in [18], a convenient framework for studying the area of a given achronal set \(A\) is obtained by assuming that \(A \subset I^{+}(V)\), where \(V \subset X\) is a Borel, achronal, and timelike complete set. One may then exploit the disintegration formula associated with the function \(\tau_V\), recalled above, to derive an upper bound for the area of \(A\) in terms of its traces along the integral curves of \(\tau_V\).
Moreover, when \(A\) is given by a level set of \(\tau_V\), namely \[V_t := \{\tau_V = t\},\] the corresponding inequality becomes an identity. We recall below [18].
Proposition 6. Let \((X,\mathsf d, \mathfrak m, \ll, \leq, \tau)\) be a timelike non-branching, globally hyperbolic, Lorentzian geodesic space satisfying \(\mathsf{TCD}^{e}_{p}(0,N)\) and assume that the causally-reversed structure satisfies the same conditions. Let \(V\subset X\) be a Borel, achronal, timelike complete subset and consider the disintegration given by Theorem 2.
Then \[\label{E:V} \mathfrak m^+(V) \geq \int_Q \mathfrak m^+_\alpha(V \cap \overline{X_\alpha})\,\mathfrak q(\mathrm{d}\alpha).\qquad{(1)}\] Moreover for every \(t >0\): \[\label{E:identityslicesV} \begin{align} \mathfrak m^{+}(V_{t}) &= \int_{Q_{t}} \mathfrak m^{+}_{\alpha}(V_{t}\cap \overline{X_{\alpha}}) \,\mathfrak q(\mathrm{d}\alpha), \quad \text{for all }t>0, \\ \mathfrak m^{-}(V_{t}) &= \int_{Q_t} \mathfrak m^{-}_{\alpha}(V_{t}\cap \overline{X_{\alpha}}) \,\mathfrak q(\mathrm{d}\alpha), \quad \text{for all }t<0, \end{align}\qquad{(2)}\] where \[\label{eq:defQt} Q_{t}:= \begin{cases} \{ \alpha \in Q \colon \sup_{x \in X_{\alpha}} \tau_{V}(x) > t \}, \quad \text{ for } t>0, \\ \{ \alpha \in Q \colon \inf_{x \in X_{\alpha}} \tau_{V}(x) < t \}, \quad \text{ for }t<0. \end{cases}\qquad{(3)}\]
Finally a Bishop-Gromov type theorem is valid for \(\mathfrak m^{+}(V_{t})\), and analogously for \(\mathfrak m^-(V_t)\) in case \(t<0\); for the proof see [18].
Theorem 3 (Monotonicity formula for the area). Let \((X,\mathsf d, \mathfrak m, \ll, \leq, \tau)\) be a timelike non-branching, globally hyperbolic, Lorentzian geodesic space satisfying \(\mathsf{TCD}^{e}_{p}(0,N)\) and assume that the causally-reversed structure satisfies the same conditions. Let \(V\subset X\) be a Borel, achronal, timelike complete subset. Then \[\begin{align} (0,\infty) \ni t \longmapsto \frac{\mathfrak m^{+}(V_{t})}{t^{N-1}}, \end{align}\] is monotone non-increasing.
Thanks to 3 and 6, a comparison of areas in the future of \(V\) with the one in its past will be possible.
Leaving the Lorentzian setting for this short section, we will prove some one-dimensional inequalities for \(\mathsf{CD}(0,N)\) spaces, i.e.for \((I,|\cdot|, h \,dx)\), where \(I\subset \mathbb{R}\) is an open interval, \(h:I\to (0,\infty)\) is semi-concave and satisfies the differential inequality 20 in distributional sense.
The Bishop-Gromov inequality (which, in this one-dimensional setting, reduces to an elementary computation) yields that the function \[(0,\infty) \ni R\mapsto \frac{h(R)}{NR^{N-1}}\searrow\] is monotone non-increasing. Assuming that \(\sup I=\infty\), we denote \[\label{eq:deftheta} \theta=\theta_{h,N}:=\lim_{R\to \infty} \frac{h(R)}{NR^{N-1}}=\inf_{R>0} \frac{h(R)}{NR^{N-1}}.\tag{23}\]
In the next lemma, we relate the asymptotic “area ratio” 23 to the more familiar asymptotic volume ratio, \[{\rm{AVR}}_N(h):=\lim_{R\to\infty}\frac{1}{R^N}\int_0^R h(t)\,dt\] which appeared in the recent metric geometry literature (see for instance [37], [38]).
Lemma 7. Let \(N\ge 1\) and let \(h:[0,\infty)\to[0,\infty)\) be a continuous function such that \[t\mapsto \frac{h(t)}{Nt^{\,N-1}}\] is decreasing on \((0,\infty)\) and \[\lim_{t\to\infty}\frac{h(t)}{Nt^{\,N-1}}=\theta\] for some \(\theta\ge 0\). Then \[\lim_{R\to\infty}\frac{1}{R^N}\int_0^R h(t)\,dt=\theta.\]
Proof. Set \[f(t):=\frac{h(t)}{Nt^{\,N-1}}.\] By assumption, \(f\) is decreasing on \((0,\infty)\) and \(f(t)\to \theta\) as \(t\to\infty\). Fix \(\varepsilon>0\). Then there exists \(T>0\) such that \[\theta \le f(t)\le \theta +\varepsilon, \qquad \text{for all } t\ge T.\] For \(R>T\), we write \[\frac{1}{R^N}\int_0^R h(t)\,dt = \frac{1}{R^N}\int_0^T h(t)\,dt + \frac{1}{R^N}\int_T^R Nt^{N-1}f(t)\,dt.\] The first term in the right hand side tends to \(0\) as \(R\to\infty\). For the second term, the bounds on \(f\) give \[\theta \cdot \frac{1}{R^N}\int_T^R t^{N-1}\,dt \le \frac{1}{R^N}\int_T^R t^{N-1}f(t)\,dt \le (\theta +\varepsilon)\cdot \frac{1}{R^N}\int_T^R t^{N-1}\,dt.\] Since \[\frac{1}{R^N}\int_T^R Nt^{N-1}\,dt = \frac{R^N-T^N}{R^N}\longrightarrow 1 \qquad\text{as }R\to\infty,\] it follows that \[\lim_{R\to\infty}\frac{1}{R^N}\int_T^R t^{N-1}f(t)\,dt=\theta,\] completing the proof. ◻
Lemma 8. Let \(X = (I,|\cdot|, h \,dx)\) be a one-dimensional m.m.s. verifying the \(\mathsf{CD}(0,N)\) condition for some \(N\in (1,\infty)\), where \(I\subset \mathbb{R}\) is an open interval such that \(h(t) > 0\) for all \(t\in I\). Then: \[\label{E:1dimequalityStat} h^{\frac{1}{N-1}}(-s) + \frac{s}{R} h^{\frac{1}{N-1}}(R) \leq \left(1+\frac{s}{R}\right) h^{\frac{1}{N-1}}(0),\tag{24}\] provided \(-s, 0,R\) lie in the interior of \(I\) and \(R,s >0\).
In particular:
If \(N\in [2,\infty)\), then, for all \(R,s>0\): \[\label{eq:hsh0hR} h(-s) \leq \left(1+\frac{s}{R} \right)^{N-1} h(0)- \left(\frac{s}{R}\right)^{N-1} h(R);\tag{25}\]
If \(\sup I=\infty\), then \[\label{E:AVR2} I \subset \left[ - \left(\frac{h(0)}{N\theta}\right)^{\frac{1}{N-1}}, +\infty \right).\tag{26}\] Moreover, assuming that \(0\in I\) and denoting \(-a:=\inf I\), \(\mathfrak m= h \cdot \mathcal{L}^{1}\), then \[\label{E:AVR3} \mathfrak m([-a,0]) \leq h(0)\left(\frac{h(0)}{N\theta}\right)^{\frac{1}{N-1}}.\tag{27}\]
Proof. Let \(s, R>0\). The fact that \(h:I\to (0,\infty)\) is semi-concave and satisfies the differential inequality 20 in distributional sense is equivalent to the concavity of the function \(t\mapsto h^{1/N-1}(t)\). Such a concavity property implies: \[\frac{h^{\frac{1}{N-1}}(0) - h^{\frac{1}{N-1}}(-s)}{s} \geq \frac{h^{\frac{1}{N-1}}(R) - h^{\frac{1}{N-1}}(0)}{R}. \\None\] Hence \[h^{\frac{1}{N-1}}(-s) + \frac{s}{R} \left( h^{\frac{1}{N-1}}(R) - h^{\frac{1}{N-1}}(0)\right) \leq h^{\frac{1}{N-1}}(0),\] or \[h^{\frac{1}{N-1}}(-s) + \frac{s}{R} h^{\frac{1}{N-1}}(R) \leq h^{\frac{1}{N-1}}(0) \left( 1 + \frac{s}{R} \right),\] giving 24 . The inequality 25 follows directly from 24 once recalling that the convexity of the function \(t\mapsto t^{N-1}\), for \(N\in [2,\infty)\), ensures that \[\label{eq:anbnleqabn} a^{N-1}+b^{N-1}\leq (a+b)^{N-1}, \quad \text{for all } a,b\geq 0.\tag{28}\] Passing to the limit as \(R\to \infty\) in 24 yields: \[\label{E:AVR} h^{\frac{1}{N-1}}(-s) + s (N\theta)^{\frac{1}{N-1}} \leq h^{\frac{1}{N-1}}(0).\tag{29}\] We deduce two consequences from 29 . Firstly, that \(s \leq (h(0)/N\theta)^{\frac{1}{N-1}}\) and therefore \[I\subset \{ h >0 \} \subset \left[ - \left(\frac{h(0)}{N\theta}\right)^{\frac{1}{N-1}}, +\infty \right).\] Secondly, 29 implies trivially that: \[h(-s) \leq h(0).\] Integrating from \(-a\) to \(0\) yields: \[\mathfrak m([-a,0]) \leq a h(0),\] which, combined with 26 , implies that \[\label{E:AVR3PPf} \mathfrak m([-a,0]) \leq h(0)\left(\frac{h(0)}{N\theta}\right)^{\frac{1}{N-1}}.\tag{30}\] ◻
Definition 9. Let \((X,\mathsf d, \mathfrak m, \ll, \leq, \tau)\) be a timelike non-branching, globally hyperbolic, Lorentzian geodesic space satisfying \(\mathsf{TCD}^{e}_{p}(0,N)\) and assume that the causally-reversed structure satisfies the same conditions. Let \(V \subset X\) be an achronal, timelike complete set having finite area, i.e. \(\mathfrak m^{+}(V) <\infty\), and empty global edge.
We define the asymptotic area growth of \(V\), denoted by \(\Theta_{V}\), as: \[\label{eq:defThetaV} \Theta_V:=\lim_{R\to +\infty} \frac{\mathfrak m^+(V_R)}{NR^{N-1}},\tag{31}\] where \(V_R := \{\tau_V = R\}\), and the limit exists thanks to Theorem 3.
Let \(s>0\). We define the asymptotic area growth of \(V\) relative to \(V_{-s}\), denoted by \(\Theta_{V}(s)\), the quantity: \[\label{clvorhjd} \Theta_V(s):=\lim_{R\to +\infty} \frac{1}{NR^{N-1}} \int_{Q_{-s}} h_{\alpha}(R) \,\mathfrak q(\mathrm{d}\alpha).\tag{32}\]
The geometric interpretation of \(\Theta_V(s)\) is to capture the asymptotic (in the future) area growth of \(V\), restricting in the directions of the rays \(X_\alpha\) that extend in the past up to \(-s\).
Theorem 4 (An area bound in terms of the asymptotic area growth).
Let \((X,\mathsf d, \mathfrak m, \ll, \leq, \tau)\) be a timelike non-branching, globally hyperbolic, Lorentzian geodesic space satisfying \(\mathsf{TCD}^{e}_{p}(0,N)\) and assume that
the causally-reversed structure satisfies the same conditions. Let \(V \subset X\) be an achronal, timelike complete set having finite area, i.e. \(\mathfrak m^{+}(V) <\infty\), and empty
global edge.
Assume moreover that the set \(V\) has empty future cut-locus i.e. \(\mathfrak{b}= \emptyset\).
Parametrize each ray \(X_\alpha\) corresponding to \(\tau_V\) so that \(X_\alpha(0)=X_\alpha\cap V\).
Then \[\label{eq:defAreaEstAAGs} \mathfrak m^-(V_{-s})\leq \mathfrak m^+(V)- N \Theta_V(s) \, s^{N-1},\quad \text{for all } s>0.\tag{33}\]
Proof. The assumption that \(\mathfrak{b}= \emptyset\) is equivalent to require that the domain of definition of each ray \(X_\alpha\) contains \((0,+\infty)\), which in turn implies that \(Q_R=Q\), for all \(R>0\). The combination of Theorem 2 with ?? ,?? and 25 implies that \[\begin{align} \mathfrak m^-(V_{-s})&=\int_{Q_{-s}} \mathfrak m^{-}_{\alpha}(V_{-s}\cap \overline{X_{\alpha}}) \,\mathfrak q(\mathrm{d}\alpha) \leq \int_{Q_{-s}} h_\alpha(-s) \,\mathfrak q(\mathrm{d}\alpha)\\ &\leq \left(1+\frac{s}{R} \right)^{N-1} \int_{Q_{-s}} h_{\alpha}(0) \,\mathfrak q(\mathrm{d}\alpha) - \left(\frac{s}{R}\right)^{N-1} \int_{Q_{-s}} h_{\alpha}(R) \,\mathfrak q(\mathrm{d}\alpha)\\ &\leq \left(1+\frac{s}{R} \right)^{N-1} \mathfrak m^+(V) - \left(\frac{s}{R}\right)^{N-1} \int_{Q_{-s}} h_{\alpha}(R) \,\mathfrak q(\mathrm{d}\alpha) \end{align}\] Sending \(R\to + \infty\) and recalling 31 gives 33 . ◻
We reinterpret the previous inequality in smooth setting.
Corollary 1 (An area estimate in terms of the asymptotic area growth – smooth setting). Let \((M,g)\) be an \((n+1)\)-dimensional, globally hyperbolic, smooth Lorentzian manifold satisfying Penrose-Hawking’s strong energy condition, i.e. \({\rm Ric}_g(v,v)\geq 0\) for all \(v\in TM\) timelike. Let \(V\subset M\) be a smooth, compact, Cauchy hypersurface.
Assume that \(V\) has empty future cut-locus, i.e.all the geodesics \(X_\alpha\), \(\alpha\in V\), maximizing the time separation \(\tau_V\) from \(V\) can be extended indefinitely in the future.
Denote by \(\mathrm{Vol}_{g_{V}}\) the \(n\)-dimensional volume measure of \((V,g|_{TV})\). Then \[\label{eq:defAreaEstAAG-Smooth} \mathrm{Vol}_{g_{V_{-s}}}(V_{-s})\leq \mathrm{Vol}_{g_V}(V)- (n+1)\Theta_V(s) \, s^{n},\quad \text{for a.e. } s>0.\tag{34}\]
Proof. Corollary 1 follows directly from Theorem 4, once noticing that:
\((M,g, \mathrm{Vol}_g)\) is, in particular, a timelike non-branching, globally hyperbolic, Lorentzian geodesic space satisfying the \(\mathsf{TCD}^{e}_{p}(0,n+1)\) condition, together with its causally-reverse structure (see [4], after [5], [6]);
the smoothness of \(V\) implies that \(\mathrm{Vol}_{g_V}(V)\) is finite and coincides with the future Minkowski content (see 5); by 4, if \(V\) has empty future cut-locus, then \(\mathfrak{b} = \emptyset\) and therefore all the hypothesis of 4 are satisfied and 33 holds true. To conlude the validity of 34 we still need to justify its left-hand side. To this end, it is enough to refer to the discussion of 5, obtaining 34 for a.e. \(-s \in \tau_V(M)\). If \(s \notin \tau_V(M)\), the inequality is trivially satisfied.
◻
Proposition 9. In the setting of 4, the area bound 33 is sharp for \(N =
2\). More precisely, let \(\mathbb{M}^{2}\) be the \(2\)-dimensional Minkowski space with metric \(g = -dy^{2} + dx^{2}\), let \(\beta>1\), and consider the cone \[X = \{ (x,y) \colon y \geq \beta \|x\| \},\] endowed with the classical Minkowski metric and Lebesgue measure.
Then 33 is an identity for \(V : = \{(x,y) \in X \colon -y^2 + \|x\|^{2} = -1\}\) and \(N=2\).
Proof. It is rather straightforward to compute the length of \[S_t : = \{(x,y) \in X \colon -y^2 + x^{2} = -t^2\}.\] Indeed, computing the intersection of \(\{ y = \beta x\}\) with \(\{-y^2 + x^2 = -t^2\}\) and using the standard parametrization of \(S_t\) given by \(x \mapsto (x,\sqrt{t^2+x^2})\) gives \[\ell(S_t) = \int^{\frac{t}{\sqrt{\beta^2 -1 }}}_{-\frac{t}{\sqrt{\beta^2 -1 }}} \frac{t}{\sqrt{x^2 +t^2}}\,dx = 2 t \int_0^\frac{1}{\beta^2 -1 } \frac{1}{\sqrt{y^2 +1}} \,dy = 2t\mathop{\mathrm{arcsinh}}\left(\frac{1}{\beta^2 -1 }\right).\] Now, for any \(s<1\), \(V_{-s} = S_{1-s}\) and \(V_r = S_{1+r}\) so that \[\Theta = \Theta(s) = \lim_{R\to\infty} \frac{2(R+1)}{2R}\mathop{\mathrm{arcsinh}}\left(\frac{1}{\beta^2 -1 }\right) = \mathop{\mathrm{arcsinh}}\left(\frac{1}{\beta^2 -1 }\right),\] giving indeed that \(\mathfrak m^-(V_{-s}) = \mathfrak m^+(V) - 2 s \Theta\), for any \(0 < s < 1\). ◻
Remark 10. In the higher-dimensional case \(N = n+1 \ge 3\), and under the same assumptions as in 9, the inequality 33 becomes strict. Consider the surface \[\label{eq:defSSharp} S_t = \{ (x,y) \in X \colon -y^{2} + \|x\|^{2} = -t^2\}.\tag{35}\] If \(y =\beta \|x \|\), and \((x,y)\in S_t\), then \(\| x\| = t/\sqrt{\beta^2-1}\). For computing the area of \(S_t\), we parametrize \(S_t\) via the graph of the function \(u(x) = \sqrt{t^2 +\|x\|^{2}}\) over \(x\in B_{t/\sqrt{\beta^2 -1}}(0) \subset \mathbb{R}^n\), giving \[\begin{align} {\rm Area}(S_t) & = \int_{B_{t/\sqrt{\beta^2 -1}}(0)} \sqrt{1 - |\nabla u|^2}\;dx_1\cdots dx_n \\ & = t \int_{B_{t/\sqrt{\beta^2 -1}}(0)} \frac{1}{\sqrt{t^2 +\|x\|^2}}\;dx_1\cdots dx_n \\ & = t n\omega_n \int_0^{t/\sqrt{\beta^2-1}} \frac{r^{n-1}}{\sqrt{t^2+r^2}}\;dr \\ & = t^n n\omega_n \int_0^{1/\sqrt{\beta^2-1}} \frac{s^{n-1}}{\sqrt{1+s^2}}\;ds. \end{align}\] As before, calling \(V : = S_1\), it follows that \(V_t = S_{1+t}\), \(V_{-s} = S_{1-s}\) and \[\Theta = \Theta(t) = \frac{n}{n+1}\omega_n \int_0^{1/\sqrt{\beta^2-1}} \frac{s^{n-1}}{\sqrt{1+s^2}} ds,\] implying that the inequality in 33 is strict.
Notice that, in the proof of 33 , we used inequality 28 , which is an identity when \(N = 2\), but becomes strict for \(N > 2\) whenever \(a,b \neq 0\).
Summarizing, 9 shows that 33 is sharp in the case \(N = 2\). By contrast, in light of 10, we expect the inequality 28 to be strict whenever \(N > 2\).
In this section, we establish two singularity results in the non-smooth setting of globally hyperbolic Lorentzian geodesic spaces with non-negative timelike Ricci curvature. The first concerns volume singularities in the sense of [7], while the second addresses singularities in a more classical sense.
Complementing the classical Penrose–Hawking notion of singularity via causal geodesic incompleteness, García-Heveling [7] recently introduced the notion of a volume singularity. A spacetime \((M,g)\) is said to be past volume singular if for every \(\varepsilon>0\) there exists a point \(x\in M\) such that \[\label{eq:defVolSing} \mathrm{Vol}_g\bigl(I^-(x)\bigr)<\varepsilon.\tag{36}\] In a chronological spacetime, it suffices that there exists a point \(x\in M\) whose chronological past has finite spacetime volume, \[\label{eq:defVolSing-Suff} \mathrm{Vol}_g\bigl(I^-(x)\bigr)<\infty.\tag{37}\] Indeed, finite past volume implies past volume singularity by a monotonicity argument along timelike past; see [7]. The physical motivation for this notion is that if the past of an event has spacetime volume smaller than a Planck volume, then quantum-gravitational effects are expected to dominate, causing the classical spacetime description of general relativity to break down. From this perspective, volume singularities provide an alternative manifestation of a singular behavior, complementary to geodesic incompleteness, by signaling the transition from a classical to a quantum regime.
Before stating and proving the result, we fix some notation. Provided \(\mathfrak m^+(V)<\infty\), it will be convenient to slightly change the normalizations of the conditional probabilities \(h(\alpha,\cdot)\) in the following manner.
First, since \(Q\) can be identified with a measurable subset of \(V\), we may, without loss of generality, parametrize each ray \(X_\alpha\) so that \[X_\alpha(0)=X_\alpha \cap V .\] It then follows that \[\int_Q h(\alpha,0)\,\mathfrak q(\mathrm d\alpha) \le \mathfrak m^+(V) < \infty,\] and hence \(h(\cdot,0) \in L^1(\mathfrak q)\).
For our purposes, it will suffice to consider only those rays \(X_\alpha\) for which \(X_\alpha \cap V\) is a relative interior point of the ray. We denote by \(Q^-\) the set of indices with this property. Hence, since \(0\) is an interior point of the chosen parametrization, it follows that for \(\mathfrak q\)-a.e.\(\alpha \in Q^-\), \[h(\alpha,0) > 0.\] We may therefore normalize the densities so that \[\label{eq:halpha0611} h(\alpha,0)=1 \qquad \text{for } \mathfrak q\text{-a.e. } \alpha \in Q^-,\tag{38}\] by correspondingly modifying the quotient measure according to \[\label{eq:halpha40041611} \bar{\mathfrak q} := h(\alpha,0)\,\mathfrak q\llcorner_{Q^-} + \mathfrak q\llcorner_{Q\setminus Q^-}.\tag{39}\] Note that \(\bar{\mathfrak q}\) is no longer a probability measure. Nevertheless, it satisfies \[\label{eq:barqq40Q41leqmV} \bar{\mathfrak q}(Q) \le \mathfrak m^+(V).\tag{40}\] Finally, we assume that the map \[\alpha \mapsto \theta_\alpha^{-1}\] belongs to \(L^1(\bar{\mathfrak q},Q^-)\), where \(\theta_\alpha := \theta_{h_\alpha}\) is defined as in 23 .
Observe that the normalization \(h(\alpha,0)=1\) is also needed in order to fix, once and for all, the scaling of the one-dimensional asymptotic area ratio \(\theta_\alpha\).
Theorem 5 (A synthetic volume singularity result). Let \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) be a timelike non-branching, globally hyperbolic, Lorentzian geodesic space. Let \(V \subset X\) be an achronal, timelike complete set having finite area, i.e.\(\mathfrak m^{+}(V) <\infty\), and empty global edge. Assume moreover that:
Both \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) and the causally-reverse structure satisfy the \(\mathsf{TCD}^{e}_{p}(0,N)\) condition, for some \(N\in (1,\infty)\);
the set \(V\) has empty future cut-locus, i.e. \(\mathfrak{b}= \emptyset\).
Then: \[\label{eq:volumesing} \mathfrak m(I^-(V))^{N-1} \leq \mathfrak m^+(V)^{N-2} \left( \int_{Q^{-}} \frac{1}{N\theta_\alpha} \bar \mathfrak q(\mathrm{d}\alpha)\right).\tag{41}\] In particular, if \(\alpha \mapsto\theta_\alpha^{-1} \in L^1(\bar \mathfrak q, Q^-)\) then \(\mathfrak m(I^-(V)) <\infty\).
Proof. First, consider only those rays \(X_\alpha\) for which \(X_\alpha \cap V\) is a relative interior point of the ray, and denote the corresponding set of indices by \(Q^-\). Then the disintegration formula 19 yields \[\mathfrak m(I^{-}(V))
=
\int_{Q^-} \mathfrak m_\alpha\bigl((\mathfrak{a}_\alpha,0]\bigr)\,\mathfrak q(\mathrm d\alpha).\] Since \(\mathfrak{b}= \emptyset\), it follows that all the transport rays \(X_\alpha\) are defined indefinitely in the future. In particular, each ray \(X_\alpha\) can be parametrized on an open interval containing \([0,\infty)\) so that
\(X_\alpha\cap V=X_\alpha(0)\). Since by assumption \(X_\alpha(0)\) is an interior point of \(X_\alpha\), we infer that 27 holds
true for \(\mathfrak q\)-a.e.\(\alpha \in Q\), yielding: \[\begin{align}
\mathfrak m(I^{-}(V))&= \int_{Q^-} \mathfrak m_\alpha((\mathfrak{a}_\alpha,0]) \,\bar\mathfrak q(\mathrm{d}\alpha) \\
&\leq \int_{Q^-} h(\alpha,0)^{1 + \frac{1}{N-1}} (N\theta_\alpha)^{-\frac{1}{N-1}}\,\bar \mathfrak q(\mathrm{d}\alpha)\\
&= \int_{Q^-} (N\theta_\alpha)^{-\frac{1}{N-1}}\,\bar\mathfrak q(\mathrm{d}\alpha),
\end{align}\] where, in the last identity, we used the normalization 38 .
By Jensen’s inequality, we bound the right hand side as follows: \[\begin{align}
\int_{Q^-} (N\theta_\alpha)^{-\frac{1}{N-1}}\,\bar\mathfrak q(\mathrm{d}\alpha)
= &~
\bar \mathfrak q(Q^-) \int_{Q^-} \left(\frac{1}{N\theta_\alpha}\right)^{\frac{1}{N-1}}\,\left(\frac{\bar \mathfrak q}{\bar \mathfrak q(Q^-)}\right)(\mathrm{d}\alpha) \\
\leq &~ \bar \mathfrak q(Q^-)^\frac{N-2}{N-1}
\left( \int_{Q^{-}}\frac{1}{N\theta_\alpha} \bar \mathfrak q(\mathrm{d}\alpha) \right)^\frac{1}{N-1} \\
\leq &~ \mathfrak m^+(V)^\frac{N-2}{N-1}
\left( \int_{Q^-}\frac{1}{N\theta_\alpha} \bar \mathfrak q(\mathrm{d}\alpha) \right)^\frac{1}{N-1},
\end{align}\] where, the last inequality, we used 40 . ◻
Remark 11. Finding lower bounds on the \(\theta_\alpha\) might be a challenging task. A possible approach would be to consider, for any \(W \subset V\), the set of indices \(Q_{W}\) \[Q_{W} = \{\alpha \in Q^- \colon X_\alpha \cap W \neq \emptyset \},\] and \[\Theta_{W} : = \liminf_{R \to \infty} \frac{1}{NR^{N-1}} \int_{Q_{W}} h_{\alpha}(R) \,\bar \mathfrak q(\mathrm{d}\alpha).\] If an estimate of the form \[\Theta_W \ge c\,\mathfrak m^+(W)\] holds, with \(c>0\) independent of \(W\), then monotonicity implies that \[\theta_\alpha \ge c .\]
In the case where \((M,g)\) is an \((n+1)\)-dimensional globally hyperbolic smooth Lorentzian manifold, and \(V \subset M\) is a smooth, compact, achronal hypersurface with empty global edge, every ray associated with \(\tau_V\) intersects \(V\). This allows one to take \(V\) itself as the quotient set and the induced hypersurface measure \(\mathrm{Vol}_V\) as the quotient measure. In this setting, \(X_\alpha\cap V\) is an interior point of the ray \(X_\alpha\) for every \(\alpha\). Thus, we can choose \[\label{eq:Q-61Vsmooth} Q^-=V.\tag{42}\] Moreover, the normalization introduced above appears entirely natural and yields the following disintegration formula (which, in this case, corresponds to Fubini-Tonelli’s theorem): \[\label{eq:DisintVolvSmooth} \mathrm{Vol}_g\llcorner_{I^{\pm}(V)}=\int_V h(\alpha,\cdot)\, \mathrm{Vol}_{g_V}(\mathrm{d}\alpha), \qquad h(\alpha, 0) = 1, \, \mathrm{Vol}_V-\text{a.e.} \, \alpha,\tag{43}\] where we have denoted by \(\mathrm{Vol}_g\) (resp.\(\mathrm{Vol}_{g_V}\)) the \((n+1)\)-dimensional volume measure of \((M,g)\) (resp.the \(n\)-dimensional volume measure of \((V,g|_{TV})\)). Below we report the smooth version of 5.
Corollary 2 (A volume singularity result in the smooth setting). Let \((M,g)\) be an \((n+1)\)-dimensional, globally hyperbolic, smooth Lorentzian manifold satisfying Penrose-Hawking’s strong energy condition, i.e.\({\rm Ric}_g(v,v)\geq 0\) for all \(v\in TM\) timelike. Let \(V\subset M\) be a smooth, compact, Cauchy hypersurface. Moreover:
Assume that \(V\) has empty future cut-locus.
Assume that the map \(\alpha \mapsto \theta_\alpha^{-1}\) belongs to \(L^1(V, \mathrm{Vol}_{g_V})\), where \(\theta_\alpha:=\theta_{h_\alpha}\) is defined as in 23 .
Then: \[\label{eq:volumesingSmooth} \mathrm{Vol}_g(I^-(V))^{n} \leq \mathrm{Vol}_{g_V}(V)^{n-1} \left( \int_Q \frac{1}{(n+1)\theta_\alpha} \mathrm{Vol}_{g_V}(\mathrm{d}\alpha)\right).\tag{44}\] In particular, \(\mathrm{Vol}_g(I^-(V)) <\infty\).
Moreover, for any (possibly non-smooth) past extension \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) of \((M,g)\) which is a timelike non-branching, globally hyperbolic, Lorentzian geodesic space, satisfying the \(\mathsf{TCD}^{e}_{p}(0,n+1)\) condition, together with its causally-reversed structure, it holds that \[\label{eq:volumesingSmoothExt} \mathfrak m(I^-(V))^{n} \leq \mathrm{Vol}_{g_V}(V)^{n-1} \left( \int_Q \frac{1}{(n+1)\theta_\alpha} \mathrm{Vol}_{g_V}(\mathrm{d}\alpha)\right),\tag{45}\] in particular, \(\mathfrak m(I^-(V)) <\infty\).
Proof. Corollary 2 follows directly from Theorem 5, once noticing that
\((M,g)\) is, in particular, a timelike non-branching, globally hyperbolic, Lorentzian geodesic space satisfying the \(\mathsf{TCD}^{e}_{p}(0,n+1)\) condition, together with its causally-reverse structure (see [4], after [5], [6]);
since \(V\) is a Cauchy hypersurface, it has empty global edge;
the smoothness of \(V\) implies that all the rays \(X_\alpha\), \(\alpha\in V\), maximizing the time separation \(\tau_V\) from \(V\) can be extended across \(V\), i.e.so that \(X_\alpha(0)=X_\alpha\cap V\) is an interior point of \(X_\alpha\);
the disintegration formula \[(\mathrm{Vol}_g)|_{I^+(V)}=\int_V h_\alpha\, \mathrm{Vol}_V(\mathrm{d}\alpha)\] corresponds to choosing \(Q=V\) and \(\mathfrak q=\mathrm{Vol}_{g_V}\) in the disintegration theorem, which imply that \(h_\alpha(0)=1\) for \(\mathrm{Vol}_{g_V}\)-a.e.\(\alpha\in V\);
The compactness of \(V\) implies that it is timelike complete and has finite \(n\)-dimensional volume. Moreover, \[\mathrm{Vol}_g^+(V)=\mathrm{Vol}_{g_V}(V),\] see Remark 5.
The final claim on the non-smooth extension is a consequence of the following observations: if \(X\) is a past extension of \((M,g)\), then \((M,g)\) is isomorphically embedded into \(X\); moreover, denoting with \(I^+_M\) and \(I^+_X\) the future sets in \(M\) and \(X\) respectively, then \(I_M^+(V)=I^+_X(V)\) and \(I_M^-(V)\subset I^-_X(V)\). One can then repeat verbatim the proof of Theorem 5 with the aforementioned simplifications. ◻
We define a timelike version of the classical Riemannian asymptotic volume ratio.
Definition 10. Let \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) be a measured Lorentzian pre-length space and \(V \subset X\) be a Borel achronal set. Then the \(N\)-timelike asymptotic volume ratio of \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) w.r.t. to \(V\) is defined by \[\label{E:TAVR} \mathrm{TAVR}_{V,N}(X) : = \limsup_{R\to \infty} \frac{\mathfrak m( \{ x \in I^{+}(V) \colon \tau_{V}(x) \leq R \})}{R^{N}}.\tag{46}\]
Lemma 12. Let \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) be a timelike non-branching, globally hyperbolic, Lorentzian geodesic space satisfying \(\mathsf{TCD}^{e}_{p}(0,N)\) and assume that the causally-reversed structure satisfies the same conditions. Let \(V\subset X\) be a Borel achronal timelike complete subset. Then the \(\limsup\) defining \(\mathrm{TAVR}_{V,N}(X)\) in 46 can be promoted to a limit.
Proof. It will be enough to show a similar monotonicity property to 3. Up to a measurable shift, it is not restrictive to assume that the parametrization of the transport ray all start from \(V\) at \(s = 0\).
From 2 and the Riemannian Bishop-Gromov inequality satisfied by the conditional measures \(\mathfrak m_\alpha\), we obtain for all \(0<S<R\): \[\begin{align} \mathfrak m( \{ x \in I^{+}(V) \colon \tau_{V}(x) \leq R \} = &~ \int_{Q}\mathfrak m_{\alpha}( \{ x \in I^{+}(V) \colon \tau_{V}(x) \leq R \} \, \mathfrak q(\mathrm{d}\alpha) \\ \leq &~ \left(\frac{R}{S}\right)^{N}\int_{Q}\mathfrak m_{\alpha}( \{ x \in I^{+}(V) \colon \tau_{V}(x) \leq S \} \, \mathfrak q(\mathrm{d}\alpha) \\ =&~ \left(\frac{R}{S}\right)^{N}\mathfrak m( \{ x \in I^{+}(V) \colon \tau_{V}(x) \leq S \}, \end{align}\] proving the claim. ◻
We conjecture that 41 admits the following sharp form, valid in every dimension.
Conjecture 13. For every \(N\ge 2\) there exists a constant \(C_N>0\) such that, if \(X\) and \(V\subset X\) are as in 5, then \[C_N\;\mathrm{TAVR}_{V,N}(X)\;\mathfrak m(I^-(V))^{N-1} \le \mathfrak m^+(V)^N.\]
The volume singularity theorem can be upgraded to a classical singularity theorem, provided a “pointwise" lower bound is enforced on the \(\theta_\alpha\).
Theorem 6 (A synthetic singularity theorem, based on asymptotic volume growth). Let \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) be a timelike non-branching, globally hyperbolic, Lorentzian geodesic space. Let \(V \subset X\) be an achronal, timelike complete set having finite area, i.e.\(\mathfrak m^{+}(V) <\infty\), and empty global edge. Assume moreover that:
Both \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) and the causally-reverse structure satisfy the \(\mathsf{TCD}^{e}_{p}(0,N)\) condition, for some \(N\in (1,\infty)\);
there exists \(c>0\) such that, adopting the normalization 38 , it holds \[\label{eq:thetaalphageqc} \theta_\alpha \geq c >0 \quad \text{for \mathfrak q-a.e. \alpha \in Q^-}.\tag{47}\]
Then \[\label{eq:tauVleq147c} \sup_{x\in I^{-}(V)} |\tau_{V}(x)| \leq \left(\frac{1}{c}\right)^{\frac{1}{N-1}}.\tag{48}\] In particular, \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) is not past timelike geodesically complete.
Proof. For \(\mathfrak q\)-a.e.\(\alpha \in Q^-\), the positivity of the one-dimensional asymptotic area ratio \(\theta_\alpha > 0\) implies that the corresponding ray \(X_\alpha\) has no future endpoint.
Since \[\mathfrak Q_{\sharp}(\mathfrak m\llcorner_{\mathcal{T}_V}) \ll \mathfrak q,\] it follows that, up to a set of \(\mathfrak m\)-measure zero, the set \(I^-(V)\) is foliated by integral curves of \(\tau_V\) having no future endpoint and indexed by elements of \(Q^-\).
Hence, combining the assumption 47 with the bound 26 , we obtain that for \(\mathfrak q\)-a.e.\(\alpha\), \[\sup_{x \in X_\alpha \cap I^-(V)} \bigl(-\tau_V(x)\bigr) \le \left(\frac{1}{c}\right)^{\frac{1}{N-1}} .\] The disintegration formula 19 implies that \[-\tau_V(x) \le \left(\frac{1}{c}\right)^{\frac{1}{N-1}} , \quad \text{for \( \mathfrak m\)-a.e.\;\( x \in I^-(V) \)}.\]
Since \(-\tau_V\) is lower semicontinuous on \(I^-(V)\) and \(\text{\rm supp}\;\mathfrak m= X\) by standing assumption, the above almost-everywhere
estimate extends to all of \(I^-(V)\). This concludes the proof of 48 .
We finally show that \(X\) is not past timelike geodesically complete. Consider any timelike geodesic \(\gamma\) parametrized by arclength and defined on a maximal (on the left) interval
\((-a, 0]\subset (-\infty, 0]\) such that \(\gamma_{0} \in V\). We claim that \[a\leq \left(\frac{1}{c}\right)^{\frac{1}{N-1}}.\] Indeed, if by
contradiction for some \(s_{0}\in [0,a)\) \[\tau(\gamma_{s_0}, \gamma_{0}) =s_{0} > \left(\frac{1}{c}\right)^{\frac{1}{N-1}},\] the very definition 13 of
\(\tau_{V}\) would imply \[-\tau_{V} (\gamma_{s_{0}})> \left(\frac{1}{c}\right)^{\frac{1}{N-1}},\] contradicting 48 . ◻
Recalling 42 , 43 , and arguing as in the proof of 2, we obtain the following corollary in the smooth setting.
Corollary 3 (A singularity result, based on asymptotic volume growth – smooth setting). Let \((M,g)\) be an \((n+1)\)-dimensional, globally hyperbolic, smooth Lorentzian manifold satisfying Penrose-Hawking’s strong energy condition, i.e.\({\rm Ric}_g(v,v)\geq 0\) for all \(v\in TM\) timelike. Let \(V\subset M\) be a smooth, compact, Cauchy hypersurface. Assume moreover that there exists a constant \(c>0\) such that \[\theta_\alpha \ge c >0 \qquad \text{for \(\mathrm{Vol}_V\)-a.e.\;\(\alpha \in V\)} .\]Then \[\sup_{x\in I^{-}(V)} |\tau_{V}(x)| \leq \left(\frac{1}{c}\right)^{\frac{1}{n}}.\] In particular, \((M,g)\) is not past timelike geodesically complete.
Moreover, for any (possibly non-smooth) past extension \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) of \((M,g)\) which is a timelike non-branching, globally hyperbolic, Lorentzian
geodesic space, satisfying the \(\mathsf{TCD}^{e}_{p}(0,n+1)\) condition, together with its causally-reversed structure, it holds that \[\label{eq:tauVleq147cCor1}
\sup_{x\in I^{-}_X(V)} |\tau_{V}(x)| \leq \left(\frac{1}{c}\right)^{\frac{1}{n}},\tag{49}\] where \(I^{-}_X(V)\subset X\) is the chronological past of \(V\) in \(X\).
In particular, \((X,\mathsf d,\mathfrak m, \ll, \leq, \tau)\) is not past timelike geodesically complete.