Lemma: Global graph properties

For every $\mu \in I$ and $s\in [-1,1]$:

1. ${F_s}^{(\mu )}$ is globally ${\lambda }_{\ast }$-Lipschitz in the intrinsic cone distance;

2. graph projection ${{\Sigma }_s}^{(\mu )}\to C_{\mu }$ is a homeomorphism;

3. on each regular face, the induced metric is at least $1/9$ times the spatial metric;

4. the graph is acausal.

For $s_1<s_2$,

The particle intersection is

No regular Minkowski tangent plane is asserted at the particle.

Proof. The affine spatial branches have gradient norm at most ${\lambda }_{\ast }<1$, by Lemma lem:gradients; the constant branch has norm zero. Minimum and maximum preserve a common Lipschitz bound. Integrating along paths, using continuity at sector rays, the seam, and the apex, proves the intrinsic global bound.

The inverse projection is $x\mapsto ({F_s}^{(\mu )}(x),x)$, so projection is a homeomorphism. For a graph plane with Cartesian gradient $g$, its induced metric is

This is a coordinate-independent lower bound relative to the spatial metric.

A causal curve between distinct spatial points has time difference at least their intrinsic spatial distance. The graph's time difference is strictly smaller, since ${\lambda }_{\ast }<1$. Distinct graph points therefore cannot be causally related.

For fixed $x$, the function $\min \{2,\max \{h_{\mu }(x),s+k_{\mu }(x)\}\}$ is nondecreasing and $1$-Lipschitz in $s$. Adding $s$ proves (eq:ordering). At the apex $h_{\mu }=k_{\mu }=0$, giving (eq:particle-time).