Geodesic congruences and their expansion, shear and vorticity

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\mathrm{\nabla }$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields) and let $U\subseteq M$ be a nonempty open subset.

(a) Geodesic congruence. A smooth future unit timelike geodesic congruence on $U$ is an observer congruence $n$ on $U$ (Observer congruences, their observer curves and adapted charts, Observers, rest spaces and the spatial projection at an event), so that $g(n,n)=-1$ and $g(n,N)<0$ on $U$, which in addition is geodesic: ${\mathrm{\nabla }}_{n}n=0\text{on }U\mathrm{.}$ Thus each observer curve of the congruence is an affinely parametrized geodesic (Geodesic of an affine connection) and its flow parameter is affine; by Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed the tangent is future unit timelike at every parameter value.

(b) Observer curves. By Through each point there is a unique maximal integral curve each $q\in U$ lies on a unique maximal integral curve ${\gamma }_q:I_q\to U$ of $n$ with ${\gamma }_q(0)=q$; these are the observer curves or flow lines of the congruence, and their parameter $s$ is the affine parameter of the geodesic. The mathematical proper-time function $\tau$ of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold, normalized by $\tau (0)=0$, satisfies $\tau (s_2)-\tau (s_1)=\frac{1}{c}{\int }_{s_1}^{s_2}\sqrt{-g(n,n)}ds=\frac{s_2-s_1}{c}(s_1<s_2,s_1,s_2\in I_q)\mathrm{.}$ Thus $\tau (s)=s\mathrm{/}c$ with this normalization. This is the metric-defined mathematical proper-time parameter, not a clock reading; the ideal-clock interpretation is a separate physical assumption.

(c) Kinematic quantities. On $U$ define the smooth covariant $2$-tensor field $B_{ab}:={\mathrm{\nabla }}_b{n}_a,n_a:=g_{ab}{n}^b,$ the rest-space metric field $h_{ab}:=g_{ab}+n_a{n}_b,$ with index-raised field $h^{ab}:=g^{ab}+n^a{n}^b$ built from the inverse metric (The inverse metric, musical isomorphisms and index raising), and: $\theta :=g^{ab}{B}_{ab},{\sigma }_{ab}:=B_{(ab)}-\frac{1}{3}\theta h_{ab},{\omega }_{ab}:=B_{[ab]},$ the expansion, shear and vorticity of the congruence, where $B_{(ab)}=\frac{1}{2}(B_{ab}+B_{ba})$ and $B_{[ab]}=\frac{1}{2}(B_{ab}-B_{ba})$. The three fields $\theta$, $\sigma$, $\omega$ and the tensor $B$ are smooth in every chart (Smooth covariant 2-tensor fields on a smooth manifold), because components of $\mathrm{\nabla }n$, $g$ and $g^{-1}$ are smooth.

(d) Scope. The definition asserts no dynamical equation, no sign of $\theta$ and no physical interpretation. In particular the identities making $\sigma$ and $\omega$ rest-space tensors, their tracelessness and antisymmetry, and the decomposition $B=\frac{1}{3}\theta h+\sigma +\omega$ are proved in Rest-space character of the kinematic fields and the double contraction of B ↗, not assumed here.