Kinematics of an observer congruence: acceleration, expansion, shear, vorticity and Born rigidity

Let $n$ be an observer congruence on a nonempty open subset $\mathrm{\Omega }\subseteq A$ (Charts, tetrads, observer congruences and adapted charts in affine Minkowski space), let $\mathrm{\nabla }$ be the flat covariant derivative of the affine model (The flat covariant derivative on affine Minkowski space and fields along curves), and write $n_b:={\eta }_{bc}{n}^c$ for the associated covector field (C^k tensor fields on affine Minkowski space and their inertial components).

(a) Rest metric and projection. Define $h(v,w):=\eta (v,w)+\eta (n,v)\eta (n,w)$ for vectors at the same point, so that $h=\eta +n^{\mathrm{♭}}\otimes n^{\mathrm{♭}}$ in components. Then $h(v,w)=\eta (h(v),h(w))$ for the orthogonal projection $h(v):=v+\eta (n,v)n$ of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition clause (c). At every point $h$ is symmetric with radical spanned by $n(p)$, and its restriction to $n(p{)}^{\perp }$ is positive definite (eta-orthogonality and orthogonal complements in a Lorentz vector space).

(b) Acceleration. The acceleration of the congruence is $a:={\mathrm{\nabla }}_{n}n$. Then $\eta (n,a)=0$ pointwise, because $2\eta (n,{\mathrm{\nabla }}_{n}n)=n(\eta (n,n))=0$; consequently $a(p)$ is spacelike or zero by the causal classification (Timelike, null, spacelike and causal vectors of a Lorentz vector space). In the model's physical interpretation the four-acceleration of the congruence observers is $c^{2}a$ and its magnitude is $c^2\sqrt{\eta (a,a)}$; the numerical units belong to the model's SI conventions and are not part of the affine geometry (Four-velocity and four-acceleration of a future timelike worldline).

(c) Kinematic decomposition. Set ${\theta }_{ab}:=h_a{}^c{h}_b{}^d{\mathrm{\nabla }}_{(c}{n}_{d)}$ and ${\omega }_{ab}:=h_a{}^c{h}_b{}^d{\mathrm{\nabla }}_{[c}{n}_{d]}$, where ${\mathrm{\nabla }}_{(c}{n}_{d)}=\frac{1}{2}({\mathrm{\nabla }}_c{n}_d+{\mathrm{\nabla }}_d{n}_c)$ and ${\mathrm{\nabla }}_{[c}{n}_{d]}=\frac{1}{2}({\mathrm{\nabla }}_c{n}_d-{\mathrm{\nabla }}_d{n}_c)$. Then

${\mathrm{\nabla }}_a{n}_b=-n_a{a}_b+{\theta }_{ab}+{\omega }_{ab},$

with $\theta$ symmetric, $\omega$ antisymmetric, both horizontal (${\theta }_{ab}{n}^b={\omega }_{ab}{n}^b=n^a{\theta }_{ab}=n^a{\omega }_{ab}=0$), and this decomposition is unique with these properties. The tensor $\theta$ is the expansion (and shear) tensor, $h^{ab}{\theta }_{ab}$ is the expansion scalar (write it as $\vartheta :=h^{ab}{\theta }_{ab}$ to distinguish it from the tensor), the shear tensor is ${\sigma }_{ab}:={\theta }_{ab}-(\vartheta \mathrm{/}3)h_{ab}$, and $\omega$ is the vorticity 2-tensor. Here $h^{ab}:={\eta }^{ac}{\eta }^{bd}{h}_{cd}$ and all indices are moved with $\eta$.

(d) Born rigidity. For a smooth covariant two-tensor $T$, the Lie derivative used here is $({\mathcal{L}}_{n}T)(X,Y):=n(T(X,Y))-T([n,X],Y)-T(X,[n,Y])$; brackets have the meaning given in the flat-derivative supplier. The congruence is Born rigid (it describes a rigid motion) when ${\mathcal{L}}_{n}h=0$. Since $({\mathcal{L}}_{n}h{)}_{ab}=2{\theta }_{ab}$, Born rigidity is equivalent to ${\theta }_{ab}=0$ for every $a,b$, equivalently to the projected symmetrized derivative of $n^{\mathrm{♭}}$ vanishing.

(e) Scope. These are kinematic definitions for a prescribed smooth field $n$ on the affine model; no dynamical equation, matter model, global synchronization or completeness is assumed, and $a$, $\theta$, $\omega$ and $h$ are local objects on $\mathrm{\Omega }$.

• Clause (a) is an expansion identity. Using $\eta (n,n)=-1$ and bilinearity, $\eta (h(v),h(w))=\eta (v,w)+2\eta (n,v)\eta (n,w)+\eta (n,v)\eta (n,w)\eta (n,n)=\eta (v,w)+\eta (n,v)\eta (n,w)=h(v,w)$; the radical and positive-definiteness statements are clause (c) of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition together with eta-orthogonality and orthogonal complements in a Lorentz vector space.

• Clause (b) computes the acceleration as a derivative along $n$. With metric compatibility of $\mathrm{\nabla }$ (The flat covariant derivative on affine Minkowski space and fields along curves clause (c)) one has $n(\eta (n,n))={\mathrm{\nabla }}_n(\eta (n,n))=2\eta (n,{\mathrm{\nabla }}_{n}n)$, and $\eta (n,n)=-1$ has zero derivative; hence $\eta (n,a)=0$, so $a$ is spacelike wherever it is nonzero. The identification of the observers' four-acceleration with $c^{2}a$ is the chain rule: parametrize an observer by the congruence parameter $s$, so that its tangent is $x^{\mathrm{\prime }}(s)=n(x(s))$ (Charts, tetrads, observer congruences and adapted charts in affine Minkowski space clause (c)); its four-velocity is $U=x^{\mathrm{\prime }}\mathrm{/}{\tau }^{\mathrm{\prime }}=cn$ because $\eta (x^{\mathrm{\prime }},x^{\mathrm{\prime }})=-1$ forces ${\tau }^{\mathrm{\prime }}=1\mathrm{/}c$ (Four-velocity and four-acceleration of a future timelike worldline), and the smooth components $n^a$ are totally differentiable with Jacobian $({\mathrm{\partial }}_b{n}^a)$ by If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative. Applying The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$ to $n^a\circ x$ in an inertial chart (the curve parameter domain is an open interval) gives $U^{\mathrm{\prime }}(s)=c{\mathrm{\partial }}_b{n}^a(x(s))x^{\mathrm{\prime }b}(s)=c({\mathrm{\nabla }}_{n}n{)}^a(x(s))$, so the four-acceleration $A=U^{\mathrm{\prime }}\mathrm{/}{\tau }^{\mathrm{\prime }}=c^{2}a$ (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$).

• Clause (c) is a pointwise algebra identity. Projecting ${\mathrm{\nabla }}_a{n}_b$: since $h_a{}^c{\mathrm{\nabla }}_c{n}_d={\mathrm{\nabla }}_a{n}_d+n_a{\mathrm{\nabla }}_n{n}_d={\mathrm{\nabla }}_a{n}_d+n_a{a}_d$ (because $h_a{}^c={\delta }_a^c+n_a{n}^c$) and $n^d{\mathrm{\nabla }}_a{n}_d=\frac{1}{2}{\mathrm{\nabla }}_a(\eta (n,n))=0$, one gets $h_a{}^c{h}_b{}^d{\mathrm{\nabla }}_c{n}_d={\mathrm{\nabla }}_a{n}_b+n_a{a}_b$ from the identity $\eta (n,a)=0$; hence ${\mathrm{\nabla }}_a{n}_b=-n_a{a}_b+{\theta }_{ab}+{\omega }_{ab}$. Uniqueness holds because contracting with $n^a$ recovers $a_b$, after which ${\mathrm{\nabla }}_a{n}_b+n_a{a}_b$ is a fixed tensor whose symmetric and antisymmetric parts determine $\theta$ and $\omega$. Horizontality follows from $h_b{}^d{n}^b=0$, and symmetry and antisymmetry of $\theta$ and $\omega$ are part of their definitions. All indices are raised and lowered with $\eta$ and the computations are finite-dimensional pointwise algebra.

• Clause (d) is the standard rigidity identity. For the torsion-free metric-compatible $\mathrm{\nabla }$ one has $({\mathcal{L}}_n\eta {)}_{ab}={\mathrm{\nabla }}_a{n}_b+{\mathrm{\nabla }}_b{n}_a$ and $({\mathcal{L}}_n{n}^{\mathrm{♭}}{)}_b=n^a{\mathrm{\nabla }}_a{n}_b+n_a{\mathrm{\nabla }}_b{n}^a=a_b$, since $n_a{\mathrm{\nabla }}_b{n}^a=\frac{1}{2}{\mathrm{\nabla }}_b(\eta (n,n))=0$. Hence $({\mathcal{L}}_{n}h{)}_{ab}=({\mathcal{L}}_n\eta {)}_{ab}+a_a{n}_b+n_a{a}_b=({\mathrm{\nabla }}_a{n}_b+{\mathrm{\nabla }}_b{n}_a)+a_a{n}_b+n_a{a}_b$, and substituting the decomposition of (c) cancels the four acceleration terms and leaves $2{\theta }_{ab}$; therefore ${\mathcal{L}}_{n}h=0$ if and only if the expansion tensor vanishes, ${\theta }_{ab}=0$ for every $a,b$. This is the equation recorded as Definition 7/(123) of the cited Giulini section, in the $(-+++)$ convention of this library.

• No global or dynamical assertion. The definitions are local on $\mathrm{\Omega }$; in particular the rest spaces $n(p{)}^{\perp }$ are not asserted to assemble into hypersurfaces, and the Frobenius integrability condition for that assembly is a separate statement on this page.