Curvature bounds, regularity and inextendibility of spacetimes

Abstract

We provide a completely new relation between curvature bounds and definiteness of the causal character of maximizers by exploiting the robust notion of synthetic curvature. This enables us to relate low-regularity inextendibility of spacetimes to unboundedness of curvature - which is at present unattainable using classical methods - thereby strengthening and complementing the results of Grant-Kunzinger-Saemann (2019) significantly.

Figures (2)

• Figure 1: Proof of Theorem \ref{['thm-ccbb-reg']}: The dashed line from $x$ to $z_1$ represents a null portion of the maximizer $[y,z_1]$. The null edge $[x,z_1]$ of the degenerate triangle with vertices $y$, $x$, $z_1$ collapses in the comparison configuration due to the regularity of the model space. It follows that $x$ and $z_1$ cannot be locally distinguished by their futures.
• Figure 2: The space $X$ from Example \ref{['ex:non-regular']}. Two domains isometric to a Minkowski half-space are joined by a null strip. The maximizers $[y_2, z_1]$ and $[x,z_2]$ overlap in their null sections but do not combine to give a maximizer.

Theorems & Definitions (19)

• Definition 2.1: Lorentzian pre-length space
• Definition 2.2: Weakly normal neighborhood
• Lemma 2.3: Intrinsic, strongly causal and localizing implies weakly normal
• proof : Proof:
• Definition 2.4: Triangle comparison
• Definition 2.5: Four-point condition
• Theorem 3.1: CCBB implies regularity
• proof : Proof:
• Theorem 3.2: 4-point TLCBB implies regularity
• proof : Proof: