Acceleration, expansion and the along-congruence derivative of an observer congruence

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, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), let $U\subseteq M$ be a nonempty open subset carrying the restricted smooth structure of An open subset of a smooth manifold has a canonical restricted smooth structure, and let $n$ be a smooth observer congruence on $U$ (Observer congruences, their observer curves and adapted charts), so that $g_p(n_p,n_p)=-1$ and $n_p$ is future-directed at every $p\in U$.

(a) Acceleration. The acceleration field of $n$ is the smooth vector field

$a:={\mathrm{\nabla }}_{n}n,$

the covariant derivative of $n$ along itself; componentwise $a^b=n^a{\mathrm{\nabla }}_a{n}^b$. The item defines the field; that it is spatial and that it is the acceleration of the observer curves is proved in Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative ↗.

(b) Expansion. The expansion of $n$ is the smooth function

$\theta :={\mathrm{\nabla }}_a{n}^a=g^{ab}{\mathrm{\nabla }}_a{n}_b\mathrm{.}$

Its chart formula and its identification with $h^{ab}{\mathrm{\nabla }}_a{n}_b$ are proved in Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative ↗.

(c) Along-congruence derivative. For a smooth function $f:U\to \mathbb{R}$ ($C^r$ and smooth maps between smooth manifolds) define

$Df:=n^a{\mathrm{\nabla }}_{a}f,$

the derivative of $f$ along $n$; $Df$ is smooth and $D$ is a derivation of the smooth functions on $U$ (Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative ↗ clause (d)).

(d) Scope. The acceleration and the expansion are geometric data of the congruence $n$ alone, computed with the connection $\mathrm{\nabla }$. This item asserts neither $a=0$ (a geodesic congruence) nor $\theta =0$, and it defines no shear, vorticity, measurement, clock reading or physical interpretation; the reading of observer curves as ideal-clock worldlines belongs to the framework's postulates and is not asserted here.

• Why only two kinematic fields. On this page the expansion enters the local energy balance through the identity ${\mathrm{\nabla }}_a(\mu n^a+{\mathrm{\Phi }}^a)=D\mu +\mu \theta +{\mathrm{\nabla }}_a{\mathrm{\Phi }}^a$ of Divergence of the measured energy-momentum current and the finite-chain balance identity. The shear and vorticity tensors are not needed for the statements proved here and are therefore not introduced; no decomposition of $\mathrm{\nabla }n$ beyond the trace $\theta$ and the acceleration $a$ is asserted.

• Choice convention. The smooth structure of the tangent bundle underlying $n$ assumes the axiom ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) (A smooth vector field is a smooth section of the tangent bundle); this is the only choice principle used in this item.