Timelike Rauch Comparison Theorem

• Timelike Rauch Comparison Theorem is a synthetic version of the classical Lorentzian Rauch comparison theorem, providing a lower bound on the length of timelike geodesics with symmetric conjugate endpoints.
• The theorem employs a stacking technique of comparison triangles in model spaces with curvature bound κ, using partitions and perturbation of geodesics to ensure the geodesic length meets or exceeds the model diameter.
• Its synthetic approach extends classical results to non-smooth, causally structured settings, offering new insights for global geometric analysis and applications in mathematical relativity.

The timelike Rauch comparison theorem is a synthetic analogue of the classical Rauch comparison theorem from smooth Lorentzian geometry, formulated in the context of Lorentzian pre-length spaces. It provides a lower bound on the length of timelike geodesics with symmetric conjugate endpoints, under a timelike curvature upper bound. This result is foundational in extending comparison geometry to non-smooth, causally structured settings by linking curvature conditions to global and local properties of geodesic behavior.

1. Synthetic Setting and Foundational Definitions

A Lorentzian pre-length space is a quintuple $(X, d, ≪, ≤, τ)$, where:

• $(X, d)$ is a metric space;
• $τ: X \times X \to [0, +\infty]$ is a lower semicontinuous time separation function;
• The causal relations “$≪$” (timelike) and “$≤$” (causal) encode the Lorentzian causal structure (with $τ(x, y) > 0$ iff $x ≪ y$).

Timelike curves are those locally maximizing the $τ$-length, and TGeo(X) denotes the set of equivalence classes of future-directed timelike geodesics up to reparametrization. The class of spaces considered typically satisfies:

• Chronological and regular (no causal pathologies),
• Locally timelike maximizing,
• Finitely $τ$-measurable,
• Timelike curvature bounded above by $κ$ via triangle comparison (synthetic curvature condition).

A critical concept for the theorem is that of conjugate points in the synthetic setting. Notions such as one-sided and symmetric conjugacy are introduced by examining the behavior of sequences of geodesics under perturbation of endpoints and imposing convergence in the Fréchet distance (on non-stopping curves up to reparametrization). Symmetric conjugacy strengthens this by allowing small perturbations at both endpoints. In smooth spacetimes, symmetric conjugacy is equivalent to the classical Jacobi field-based notion of conjugate points (Grant et al., 16 Sep 2025).

2. The Synthetic Timelike Rauch Comparison Theorem

The central statement (Theorem "Timelike Rauch comparison theorem" in (Grant et al., 16 Sep 2025)) is:

Let $(X, d, ≪, ≤, τ)$ be a chronological, regular, locally timelike maximizing, and finitely $τ$-measurable Lorentzian pre-length space with timelike curvature bounded above by $κ$. Let $γ ∈ \mathrm{TGeo}(X)$ connect $p = γ(0)$ and $q = γ(1)$ such that $p$ and $q$ are symmetric conjugate along $γ$. Then

$> L_{τ}(γ) ≥ D_{κ} >$

where $L_{τ}(γ)$ is the $τ$-length of $γ$ and $D_{κ}$ is the timelike diameter of the model Lorentzian space $\mathcal{L}^2(κ)$ (e.g., $D_{κ} = π/\sqrt{-κ}$ for $κ < 0$, $D_κ = ∞$ for $κ ≥ 0$).

Thus, the presence of symmetric conjugate endpoints along a timelike geodesic forces its length to reach or exceed the critical model diameter. This statement is the precise synthetic counterpart to the smooth Rauch comparison theorem for Lorentzian manifolds.

3. Proof Structure and the Role of Conjugate Points

The proof is constructed using a partitioning and stacking technique for timelike comparison triangles:

• Cover the geodesic's image, $γ([0,1])$, by finitely many locally timelike maximizing neighborhoods (which also serve as comparison neighborhoods for curvature).
• Choose a partition adapted so each segment of $γ$ lies within one such neighborhood.
• Use the symmetric conjugacy assumption to produce sequences of perturbed geodesics ($γ_n, σ_n$) connecting nearby endpoints, converging to $γ$ in a topology induced by the Fréchet distance plus length differences.
• Within each subinterval, the segments of $γ_n, σ_n$ lie in the same comparison neighborhood, so one applies synthetic comparison triangles in the model space $\mathcal{L}^2(κ)$.
• The stacking principle [Proposition 2.42 in Beran–Kunzinger–Rott, 2023] allows these triangles to be concatenated in the model geometry.
• If the total length $L_{τ}(γ) < D_{κ}$, the comparison geodesics in $\mathcal{L}^2(κ)$ must intersect, contradicting uniqueness for geodesics of length less than $D_{κ}$. Therefore, $L_{τ}(γ) ≥ D_{κ}$.

The failure of local uniqueness—formalized via symmetric conjugate points—enforces this critical bound. In the smooth setting, conjugate points reflect the vanishing of Jacobi fields and the failure of the exponential map to be a local diffeomorphism, directly related to the loss of maximality of geodesic segments.

4. Curvature Bounds, Comparison Models, and Synthetic Structures

Curvature bounds in this context are imposed via triangle comparison with model spaces $\mathcal{L}^2(κ)$. The essential property is a timelike triangle comparison: $τ(p, q) \leq \bar{τ}(\bar{p}, \bar{q})$ for all relevant pairs $(p, q)$ in the synthetic triangle and their model counterparts. The use of comparison neighborhoods ensures all local segments obey the curvature constraint.

The model spaces provide explicit control of the global causal structure: when κ < 0, the model diameter $D_{κ}$ is finite, mirroring the occurrence of conjugate points/focal points at bounded interval lengths; for κ ≥ 0, the diameter is infinite, precluding synthetic conjugate points and guaranteeing global geodesic maximality.

The stacking of comparison triangles, guided by local maximization and the curvature bound, enables the patching of local comparison data into a global statement about geodesic length and conjugacy.

5. Applications and Structural Consequences

The timelike Rauch comparison theorem has significant implications:

• Lower Bound on Geodesic Lengths: In the presence of conjugate points, the τ-length of the connecting geodesic cannot be less than the model diameter. This forms the basis of synthetic singularity theorems, echoing classical incompleteness results.
• Global Structure Control: The result underpins theorems about the global geometry of Lorentzian length spaces. For instance, in the case of nonpositive timelike curvature (κ ≤ 0), uniqueness of timelike geodesics (no conjugate points) is preserved globally. This is leveraged in synthetic versions of the Lorentzian Cartan–Hadamard theorem (Erös et al., 27 Jun 2025).
• Extension Beyond Smooth Settings: The purely synthetic approach avoids reliance on differentiability, allowing statements about coarse or low regularity spacetimes—a crucial step for the analysis of generalized spacetimes in mathematical relativity and quantum gravity.
• Stacking and Comparison Techniques: The synthetic proof framework is conducive to further extension, such as gluing or globalization arguments (for instance, in Lorentzian versions of the Reshetnyak gluing theorem or splitting theorems (Beran et al., 2022)).

6. Broader Impact and Connections with Related Synthetic Comparison Theorems

Beyond its intrinsic significance, the timelike Rauch comparison theorem integrates into a hierarchy of synthetic Lorentzian comparison results:

• Synthetic Cartan-Hadamard and Splitting Theorems: The theorem enables globalization results for unique geodesic connectivity in spaces with curvature bounded above and suitable causality properties (Erös et al., 27 Jun 2025, Beran et al., 2022).
• Angle Monotonicity and Triangle Comparison: The comparison of lengths via conjugate points complements synthetic angle and triangle comparison approaches developed for lower curvature bounds (Barrera et al., 2022).
• Volume and Area Comparison: In smooth spacetimes, analogs of the theorem via the Riccati equation relate to area and volume monotonicity—providing the geometric foundation for singularity theorems such as Hawking’s (Treude et al., 2012).

The synthetic approach harmonizes with both smooth and Alexandrov-type methodologies, supporting further development of comparison geometry in Lorentzian and low-regularity settings.

7. Summary Table: Core Elements of the Synthetic Timelike Rauch Comparison Theorem

Concept	Synthetic Notion	Smooth Notion
Geodesic	τ-maximizing curve in TGeo(X)	Maximizing timelike geodesic
Conjugate points	Symmetric conjugacy via geodesic perturbations	Jacobi field vanishing
Curvature bound	Triangle comparison with model space ℒ²(κ)	Sectional/Ricci tensor inequalities
Model diameter Dₖ	Critical τ-length threshold in ℒ²(κ)	Critical value for conjugate points
Proof technique	Stacking comparison triangles, patching	Comparison of Jacobi field growth

The timelike Rauch comparison theorem thus establishes a core structural link between curvature bounds and geodesic maximality in the synthetic Lorentzian setting, with far-reaching implications for both the analytic and geometric theory of spacetime and its generalizations (Grant et al., 16 Sep 2025).