Normal neighborhoods, convex normal sets, the position vector and the squared-displacement function

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\mathrm{\nabla }$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class), and let ${\exp }_q:{\mathcal{E}}_q\to M$ be the exponential maps of The geodesic initial-value problem and the exponential map of the metric-selected connection. For an open set $U\subseteq M$ write $TU\subseteq TM$ for the tangent vectors based at points of $U$.

(a) Normal neighborhood. An open neighborhood $U$ of $p\in M$ is a normal neighborhood of $p$ when there is an open star-shaped subset $W\subseteq {\mathcal{E}}_p$ of $T_{p}M$ such that $U={\exp }_p(W)$ and ${\exp }_p{\mathrm{\mid }}_W:W\to U$ is a diffeomorphism onto $U$ (Point-normal and exponential coordinate charts of the metric-selected connection clause (b) supplies such pairs). The exponential coordinates at $p$ on such a $U$ are $u:=\psi \circ {\exp }_p^{-1}:U\to {\mathbb{R}}^4$ for a linear isomorphism $\psi :T_{p}M\to {\mathbb{R}}^4$.

(b) Convex normal and strictly convex normal sets. An open set $C\subseteq M$ is convex normal when it is a normal neighborhood of each of its points; thus for every $q\in C$ there is an open star-shaped $C_q\subseteq {\mathcal{E}}_q$ with ${\exp }_q(C_q)=C$ and ${\exp }_q{\mathrm{\mid }}_{C_q}$ a diffeomorphism onto $C$. It is strictly convex normal when it is convex normal and every pair $q,q^{\mathrm{\prime }}\in \bar{C}$ is joined by exactly one affinely parametrized geodesic segment $\gamma :[0,1]\to \bar{C}$ with $\gamma (0)=q$ and $\gamma (1)=q^{\mathrm{\prime }}$, and, if $q\mathrm{\ne }{q}^{\mathrm{\prime }}$, its parameter interior $\gamma ((0,1))$ lies in $C$. In the degenerate case $q=q^{\mathrm{\prime }}$, the required segment is the constant geodesic at $q$; no interior-containment condition is imposed on this degenerate segment when $q\in \mathrm{\partial }C$.

(c) Position vector. Fix a normal neighborhood $U={\exp }_p(W)$ of $p$ as in (a), with its chosen inverse $({\exp }_p{\mathrm{\mid }}_W{)}^{-1}$, and let $q\in U$. Put $v:=({\exp }_p{\mathrm{\mid }}_W{)}^{-1}(q)\in T_{p}M$. Since $W$ is star-shaped, $tv\in W$ for $0\le t\le 1$, and $x(t):={\exp }_p(tv)$ is the radial geodesic segment in $U$ from $p$ to $q$. The vector $P(p,q):={\gamma }_{p,v}^{\mathrm{\prime }}(1)=d({\exp }_p{)}_v(v)\in T_{q}M$ is the position vector of $q$ with respect to $p$; here ${\gamma }_{p,v}$ is the maximal geodesic of The geodesic initial-value problem and the exponential map of the metric-selected connection. The notation $P(p,q)$ uses this chosen inverse branch; another normal neighborhood may select a different radial segment for the same endpoints. On a convex normal set $C$, use the inverse domains $C_p$ of (b) and the joining segment in $C$; on a strictly convex normal set this segment is unique by (b). By the Gauss lemma of The Lorentzian Gauss lemma and the radial maximum of proper time $g_q(P(p,q),P(p,q))=g_p({\exp }_p^{-1}(q),{\exp }_p^{-1}(q))$.

(d) Squared-displacement function. For a normal neighborhood $U$ of $p$ the squared-displacement function is $D_p^2:U\to \mathbb{R},D_p^2(q):=g_p({\exp }_p^{-1}(q),{\exp }_p^{-1}(q))\mathrm{.}$ It is smooth on $U$ because ${\exp }_p^{-1}$ is smooth and $g_p$ is bilinear; $D_p^2(p)=0$, $D_p^2(q)<0$ exactly when $P(p,q)$ is timelike, $D_p^2(q)=0$ exactly when $P(p,q)$ is null or zero, and $D_p^2(q)>0$ exactly when $P(p,q)$ is spacelike. On a convex normal $C$ we write $D^2:C\times C\to \mathbb{R}$, $D^2(p,q):=D_p^2(q)$, whenever the position vector is defined using these inverse domains.

(e) Scope. These are definitions inside the smooth Lorentzian model; they assert no existence of convex normal sets (supplied separately by Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points ↗), no completeness and no physical reading. The position vector is an element of $T_{q}M$, not a vector field and not a coordinate displacement; $D^2$ is not a metric distance and takes negative values on timelike-separated pairs.

• Well-definedness. The conditions in (a) and (b) are consistent by The geodesic initial-value problem and the exponential map of the metric-selected connection clause (d) and Point-normal and exponential coordinate charts of the metric-selected connection clause (b), which provide the local diffeomorphism domains used above; the existence of convex normal neighborhoods, asserted in the wording of (c), is proved in Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points ↗. The definition records the vocabulary only and proves nothing by itself.

• The pair map. By The joint exponential map: smoothness, invertible differential at the diagonal, and the small-velocity two-point inverse the map $(p,q)\mapsto {\exp }_p^{-1}(q)$ is smooth near the diagonal where it is defined; the position vector $P(p,q)=d({\exp }_p{)}_v(v)$ with $v={\exp }_p^{-1}(q)$ is thereby a smooth function of $(p,q)$ near such configurations, a fact used in The differential of the squared-displacement function and the position vector of the reversed geodesic.