Observer Measurements

Work in the oriented time-oriented affine Minkowski space $(A,V,\eta ,C^{+})$ (Oriented time-oriented affine Minkowski space (the spacetime primitives)) with its inertial atlas (The affine atlas makes the Minkowski model a smooth 4-manifold diffeomorphic to R^4) and the canonical identification of tangent spaces with $V$ (The tangent space of the affine Minkowski model at any event is canonically V). The objects defined below are, in general, different objects, and the distinctions between them are part of the definition.

(a) Chart and inertial chart. A chart of the model is a diffeomorphism $u:U\to u(U)$ from a nonempty open subset $U\subseteq A$ onto an open subset of ${\mathbb{R}}^4$ (Diffeomorphisms and local diffeomorphisms of manifolds, Smooth manifolds and their smooth charts), written with components $(u^0,u^1,u^2,u^3)$. An inertial chart is the global chart $x(p)=E^{-1}(p-o)$ of an inertial frame $(o;e_0,e_1,e_2,e_3)$ (Inertial frames and inertial charts of the affine Minkowski model). A general chart need not be inertial: if $x$ is an inertial chart and $N:x(U)\to {\mathbb{R}}^4$ is a diffeomorphism onto its image which is not the restriction of an affine map, then $u:=N\circ x$ is a chart that is not inertial, because if $u$ were the inertial chart of a frame then $x\circ u^{-1}=N^{-1}$ would be affine by Inertial charts are related by proper orthochronous affine Lorentz maps, and every such map is a chart transition and hence $N=(N^{-1}{)}^{-1}$ would be affine as the inverse of an affine bijection, a contradiction; the map $N(y)=(y^0,y^1+(y^0{)}^2,y^2,y^3)$ on ${\mathbb{R}}^4$ is an explicit example.

(b) Tetrad at an event and tetrad field. A tetrad at $p\in A$ is an ordered basis $(e_0,e_1,e_2,e_3)$ of $V$ with $\eta (e_a,e_b)=\mathrm{diag}(-1,1,1,1{)}_{ab}$ and $e_0$ future, $e_0\in C^{+}$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones): this is the basis of an inertial frame without an origin and without the orientation condition. A tetrad field on a nonempty open subset $\mathrm{\Omega }\subseteq A$ is a tuple $(E_0,E_1,E_2,E_3)$ of smooth vector fields on $\mathrm{\Omega }$ (C^k tensor fields on affine Minkowski space and their inertial components, type $(1,0)$) whose values at every $p\in \mathrm{\Omega }$ form a tetrad at $p$. A tetrad field is observer-adapted to a prescribed congruence $n$ when $E_0=n$. Along a prescribed $C^1$ future timelike worldline $x:I\to A$, observer adaptation means $E_0(x(s))=x^{\mathrm{\prime }}(s)\mathrm{/}\sqrt{-\eta (x^{\mathrm{\prime }}(s),x^{\mathrm{\prime }}(s))}$ whenever $x(s)\in \mathrm{\Omega }$; this is its normalized future unit tangent (Curves, regular causal curves and future timelike worldlines in affine Minkowski space).

(c) Observer congruence. An observer congruence on a nonempty open subset $\mathrm{\Omega }\subseteq A$ is a smooth vector field $n:\mathrm{\Omega }\to V$ with $\eta (n,n)=-1$ and $n(p)\in C^{+}$ for every $p\in \mathrm{\Omega }$. Its observers are its maximal integral curves (Integral curves of a vector field, Through each point there is a unique maximal integral curve). Each observer is a future timelike worldline (Curves, regular causal curves and future timelike worldlines in affine Minkowski space): its tangent at a parameter value $s$ is $n(x(s))$, which is future timelike, and it is regular because $\eta (n,n)=-1$ makes $n$ nowhere zero. Completeness of the integral curves, a global product structure of $\mathrm{\Omega }$, and global synchronization are not asserted by this definition.

(d) Adapted chart. For a congruence $n$ on $\mathrm{\Omega }$, an $n$-adapted chart is a chart $(u^0,u^1,u^2,u^3):U\to {\mathbb{R}}^4$ with $U$ a nonempty open subset of $\mathrm{\Omega }$ whose spatial coordinates $u^1,u^2,u^3$ are constant along every integral curve of $n$ contained in $U$. No global adapted chart is asserted, and $u^0$ is not required to be affine.

(e) Distinctions. A chart need not be inertial, as in (a); a chart need not be $n$-adapted for any congruence. A tetrad field is not in general the coordinate frame of a chart: if $(E_0,E_1,E_2,E_3)$ is the coordinate frame of a chart $(v^0,v^1,v^2,v^3)$, that is, $E_a=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{v}^a$, then $[E_a,E_b]=0$ for all $a,b$ (The Lie bracket of smooth vector fields, Coordinate formula for the Lie bracket), so a tetrad field whose members do not commute pairwise is not the coordinate frame of a chart; no converse is claimed here. A congruence need not be the flow of a coordinate, and its pointwise rest spaces need not assemble into hypersurfaces; that integrability question is decided later on this page. In an inertial chart $x$ of $(o;e_0,e_1,e_2,e_3)$, the coordinate vector fields $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^a=e_a$ form a constant tetrad field, with values equal to the frame's ordered basis under $T_{p}A=V$. Its time vector field $n=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^0=e_0$ alone defines the congruence of stationary observers, whose integral curves are $s\mapsto x^{-1}(s+s_0,r^1,r^2,r^3)$ for fixed $s_0,r^1,r^2,r^3$. The chart, the ordered tetrad and the single vector field $n$ remain distinct objects.

• Non-inertial charts exist, and the example is checked. The map $N(y)=(y^0,y^1+(y^0{)}^2,y^2,y^3)$ is a diffeomorphism of ${\mathbb{R}}^4$ with the smooth two-sided inverse $z\mapsto (z^0,z^1-(z^0{)}^2,z^2,z^3)$. It is not the restriction of an affine map: $N(0)=0$ and $N(2e_0)=(2,4,0,0)\mathrm{\ne }2N(e_0)=(2,2,0,0)$ for $e_0=(1,0,0,0)$, so $N$ is not linear, and with $N(0)=0$ it is therefore not affine. If $N\circ x$ were an inertial chart, the transition $x\circ (N\circ x{)}^{-1}=N^{-1}$ between inertial charts would be affine by Inertial charts are related by proper orthochronous affine Lorentz maps, and every such map is a chart transition; the inverse of an affine bijection $z\mapsto Az+b$ of ${\mathbb{R}}^4$ is again affine, $w\mapsto A^{-1}w-A^{-1}b$ (Linear and affine Lorentz maps, and the Lorentz group), so $N$ would be affine, a contradiction. Hence $N\circ x$ is a chart that is not inertial, and being $C^{\mathrm{\infty }}$ it belongs to the model's smooth structure.

• Tetrads exist at every event, and tetrad fields are smooth by components. Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1) supplies an $\eta$-orthonormal basis $(f_0,f_1,f_2,f_3)$ of $V$; since $\eta (f_0,f_0)=-1<0$ the vector $f_0$ is timelike, so exactly one of $f_0$ and $-f_0$ lies in $C^{+}$ because the timelike set is the disjoint union of the two future and past cones (Time orientation, future and past timelike/null/causal vectors, and the closed cones, Timelike, null, spacelike and causal vectors of a Lorentz vector space), and negating it preserves $\eta$-orthonormality; no orientation condition is imposed on a tetrad, so this already produces a tetrad at any prescribed event. The formulation with the additional positive-orientation condition, and with an arbitrary origin, is the inertial-frame existence statement of Existence of future-oriented inertial frames at every event. A tetrad field is componentwise smooth in an inertial chart by the definition of $C^k$ tensor fields on the affine model (C^k tensor fields on affine Minkowski space and their inertial components, The tangent space of the affine Minkowski model at any event is canonically V).

• Congruences have well-defined observers. A smooth unit field $n$ on $\mathrm{\Omega }$ is a smooth map $\mathrm{\Omega }\to V$ with values in the future timelike cone, and Through each point there is a unique maximal integral curve supplies through every point a unique maximal integral curve of $n$ (Integral curves of a vector field); along such a curve the identity $\eta (n,n)=-1$ gives $\eta (x^{\mathrm{\prime }},x^{\mathrm{\prime }})=-1$ at every parameter value, so each observer is a future timelike worldline with future unit tangent, in particular regular (Curves, regular causal curves and future timelike worldlines in affine Minkowski space). Nothing about completeness of the maximal domains is asserted.

• The commutation condition is only necessary. If $(E_0,\dots ,E_3)$ is the coordinate frame of a chart $(v^0,\dots ,v^3)$, then $[E_a,E_b]=0$ by the coordinate formula for the Lie bracket; a tetrad field with $[E_a,E_b]\mathrm{\ne }0$ for some $a,b$ therefore cannot be a coordinate frame. The converse (a commuting tetrad field is locally a coordinate frame) is a Frobenius-type statement and is not asserted by this definition.

Let $(A,V,\eta ,C^{+})$ be the affine Minkowski space with tangent spaces canonically identified with $V$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), The tangent space of the affine Minkowski model at any event is canonically V), and let $\mathrm{\Omega }\subseteq A$ be a nonempty open subset. All tensor fields below are $C^k$ fields in the sense of C^k tensor fields on affine Minkowski space and their inertial components, with $k=\mathrm{\infty }$ unless a clause says otherwise.

(a) Vector fields. For smooth vector fields $X,Y$ on $\mathrm{\Omega }$ define the covariant derivative of $Y$ in the direction $X$ at $p\in \mathrm{\Omega }$ by

$\left({\mathrm{\nabla }}_{X}Y\right)(p):=\frac{d}{dt}{\mid }_{t=0}Y(p+tX(p)),$

the derivative of the $V$-valued composite $t\mapsto Y(p+tX(p))$, which is defined for $t$ in a neighbourhood of $0$ because $\mathrm{\Omega }$ is open (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set). In the inertial chart $x$ of any inertial frame one has $({\mathrm{\nabla }}_{X}Y{)}^a=X^b{\mathrm{\partial }}_b{Y}^a$; the value does not depend on the inertial chart: for another inertial chart $x^{\mathrm{\prime }}=\mathrm{\Lambda }x+b$ one has ${\mathrm{\partial }}_a^{\mathrm{\prime }}=({\mathrm{\Lambda }}^{-1}{)}_a^b{\mathrm{\partial }}_b$ and $Y^{\mathrm{\prime }a}={\mathrm{\Lambda }}_b^a{Y}^b$ (Transformation of tensor, covector, derivative and current components under affine Lorentz maps, Inertial frames and inertial charts of the affine Minkowski model), and $\mathrm{\Lambda }$ is constant, so $({\mathrm{\Lambda }}^{-1}{)}_c^a{X}^{\mathrm{\prime }b}{\mathrm{\partial }}_b^{\mathrm{\prime }}{Y}^{\mathrm{\prime }c}=X^b{\mathrm{\partial }}_b{Y}^a$. The field ${\mathrm{\nabla }}_{X}Y$ is again a smooth vector field on $\mathrm{\Omega }$.

(b) Covector and general tensor fields. For a smooth covector field $\alpha$ set $({\mathrm{\nabla }}_X\alpha )(Y):=X(\alpha (Y))-\alpha ({\mathrm{\nabla }}_{X}Y)$ for every smooth vector field $Y$, and for a type $(r,s)$ tensor field $T$ define ${\mathrm{\nabla }}_{X}T$ by the derivation rule on the evaluation of $T$: for $T:(V^{\ast }{)}^r\times V^s\to \mathbb{R}$,

$({\mathrm{\nabla }}_{X}T)({\alpha }_1,\dots ,{\alpha }_r;Y_1,\dots ,Y_s)=X[T({\alpha }_1,\dots ,{\alpha }_r;Y_1,\dots ,Y_s)]-{\sum }_{i=1}^{r}T({\alpha }_1,\dots ,{\mathrm{\nabla }}_X{\alpha }_i,\dots ,{\alpha }_r;Y_1,\dots ,Y_s)-{\sum }_{j=1}^{s}T({\alpha }_1,\dots ,{\alpha }_r;Y_1,\dots ,{\mathrm{\nabla }}_X{Y}_j,\dots ,Y_s),$

where the vector and covector arguments are smooth fields on $\mathrm{\Omega }$. In inertial coordinates every component of ${\mathrm{\nabla }}_{X}T$ is $X^b{\mathrm{\partial }}_b$ applied to that component, with no connection coefficients.

(c) Properties. The operator $\mathrm{\nabla }$ is $\mathbb{R}$-bilinear, is $C^{\mathrm{\infty }}(\mathrm{\Omega })$-linear in $X$, and satisfies the Leibniz rules ${\mathrm{\nabla }}_X(fY)=X(f)Y+f{\mathrm{\nabla }}_{X}Y$ and the analogous rules for tensor fields; it is metric compatible, $\mathrm{\nabla }\eta =0$, because ${\mathrm{\partial }}_a{\eta }_{bc}=0$ in every inertial chart; it is torsion-free, ${\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X=[X,Y]$, by the coordinate formula for the Lie bracket (The Lie bracket of smooth vector fields, Coordinate formula for the Lie bracket); and it has zero curvature, ${\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z=0$, by equality of mixed partial derivatives of the components (Young's theorem: total differentiability of the first partials forces equality of mixed partials).

(d) Fields along a curve. If $\gamma :I\to A$ is a $C^1$ curve and $W:I\to V$ is a $C^1$ field along it (Vector field and section along a smooth curve), the covariant derivative of $W$ along $\gamma$ is the ordinary $V$-valued derivative $DW\mathrm{/}ds:=W^{\mathrm{\prime }}$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral), because the canonical identification $T_{p}A=V$ is independent of $p$; consequently parallel transport along any curve is the identity and a field is parallel exactly when it is constant. This derivative is the one used for the acceleration of congruences, for Fermi–Walker transport and for the kinematic decomposition below; the model needs no Christoffel symbols and no parallel-transport ODE in this formulation.

(e) First-order $C^1$ version. For $C^1$ vector fields $X,Y$ and $C^1$ covector or tensor fields on $\mathrm{\Omega }$ the same directional and evaluation definitions are well-defined and have continuous coefficients: in an inertial chart $({\mathrm{\nabla }}_{X}Y{)}^a=X^b{\mathrm{\partial }}_b{Y}^a$, and each tensor component is differentiated in the same way. The component recovery and the subtraction of all argument derivatives remain valid; $C^1$ scalar multiplication satisfies the same first-order Leibniz rules; metric compatibility remains $\mathrm{\nabla }\eta =0$; and the first-order torsion identity uses the continuous coordinate bracket $[X,Y{]}^a=X^b{\mathrm{\partial }}_b{Y}^a-Y^b{\mathrm{\partial }}_b{X}^a$. No $C^1$ closure of this bracket and no second covariant derivative of arbitrary $C^1$ data are asserted. For $C^2$ vector fields the second derivatives needed for the zero-curvature identity of (c) exist, and that identity follows by equality of the continuous mixed second partials. All smooth-field properties above and the $C^1$ along-curve derivative in (d) are retained.

• Well-definedness of (a). The composite $t\mapsto Y(p+tX(p))$ is the composition of the smooth curve $t\mapsto p+tX(p)$ (which stays in $\mathrm{\Omega }$ for small $\mathrm{\mid }t\mathrm{\mid }$ because $\mathrm{\Omega }$ is open and $X(p)\in V$ is a fixed vector) with the smooth map $Y:\mathrm{\Omega }\to V$; its derivative at $0$ exists in the finite-dimensional real vector space $V$ by The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, and the coordinate expression $({\mathrm{\nabla }}_{X}Y{)}^a=X^b{\mathrm{\partial }}_b{Y}^a$ is the chain rule of The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$. Chart independence uses the transformation laws of Transformation of tensor, covector, derivative and current components under affine Lorentz maps together with the constancy of the linear part $\mathrm{\Lambda }$ of an inertial transition (Inertial frames and inertial charts of the affine Minkowski model): the second derivatives of the transition vanish, so no inhomogeneous term appears. Smoothness of ${\mathrm{\nabla }}_{X}Y$ follows because $X^b{\mathrm{\partial }}_b{Y}^a$ is a smooth function of $x(p)$ in an inertial chart, and $C^k$ tensor fields are exactly those with $C^k$ inertial components (C^k tensor fields on affine Minkowski space and their inertial components).

• Well-definedness of (b). In an inertial chart use the constant coordinate frame and its constant dual frame; their covariant derivatives vanish, so evaluating the derivation rule on those constant bases leaves only $X$ acting on each component of $T$. For variable arguments ${\alpha }_1,\dots ,Y_s$ the ordinary product rule expressed by Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$ produces exactly the subtracted terms displayed in (b), and all their derivatives cancel; this proves the stated coordinate formula and shows that the evaluation rule is well-defined, once the covector case $({\mathrm{\nabla }}_X\alpha )(Y)=X(\alpha (Y))-\alpha ({\mathrm{\nabla }}_{X}Y)$ is fixed by the requirement that ${\mathrm{\nabla }}_X$ commute with the pairing. No connection coefficients appear because $\eta$ is constant and the inertial transition matrices are constant.

• Well-definedness of (c). Torsion-freeness is the coordinate identity $X^b{\mathrm{\partial }}_b{Y}^a-Y^b{\mathrm{\partial }}_b{X}^a=[X,Y{]}^a$ (Coordinate formula for the Lie bracket); metric compatibility is the constancy of the component matrix ${\eta }_{ab}$ of the metric in inertial charts; and the curvature identity follows from the equality of mixed second partial derivatives (Young's theorem: total differentiability of the first partials forces equality of mixed partials) after expanding the two iterated derivatives of the components. The Leibniz and linearity properties are immediate from the definitions.

• Well-definedness of (d) and (e). In (d) the canonical isomorphism $T_{p}A\to V$ is independent of $p$ (The tangent space of the affine Minkowski model at any event is canonically V), so a field along a curve is simply a $V$-valued map and its covariant derivative is its ordinary derivative; parallel transport is the identity because the equation $W^{\mathrm{\prime }}=0$ has exactly the constant solutions on an interval: each real component of $W$ is constant by A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$; consequently two such functions with the same derivative differ by a constant, and a constant vector-valued map has derivative zero. In (e) a $C^1$ field $Y$ has a continuous differential, and the derivative of $t\mapsto Y(p+tX(p))$ at $0$ is $d{Y}_p[X(p)]$, which is a continuous function of $p$ when $Y$ and $X$ are $C^1$; only first derivatives occur in the Leibniz, metric-compatibility and torsion assertions, and the second covariant derivatives in the curvature identity require the stated $C^2$ hypotheses. No choice principle is used anywhere; all objects are finite-dimensional or explicitly parametrized.

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.

Let $(A,V,\eta ,C^{+})$ be the affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives)) and let $n\in V$ be future unit timelike, $\eta (n,n)=-1$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones).

(a) Rest space. The rest space of $n$ is the hyperplane $n^{\perp }=\{v\in V:\eta (n,v)=0\}$; by Norms of four-velocity and four-acceleration, and the observer rest-space decomposition clause (c) it is a three-dimensional linear subspace on which $\eta$ restricts to a positive-definite symmetric form, and the orthogonal projection onto it is $h(v)=v+\eta (n,v)n$ (eta-orthogonality and orthogonal complements in a Lorentz vector space).

(b) Simultaneity slice. For $p\in A$ the simultaneity slice of $p$ relative to $n$ is the affine hyperplane $H(p,n):=p+n^{\perp }=\{q\in A:\eta (n,q-p)=0\}$; it contains $p$ and is three-dimensional, not a single point.

(c) Simultaneity relation. Define $q{\sim }_n{q}^{\mathrm{\prime }}$ if and only if $\eta (n,q^{\mathrm{\prime }}-q)=0$, that is, if and only if $q^{\mathrm{\prime }}-q\in n^{\perp }$. Then ${\sim }_n$ is an equivalence relation on $A$ whose equivalence classes are exactly the slices $H(p,n)$; reflexivity, symmetry and transitivity are consequences of $n^{\perp }$ being a linear subspace and are not physical assumptions.

(d) Rest distance. For $p,q$ with $q\in H(p,n)$ define $d_n(p,q):=\sqrt{\eta (q-p,q-p)}$. This is a real number $\ge 0$, symmetric in $p,q$, vanishes exactly when $p=q$, and satisfies the triangle inequality on each slice, because the restriction of $\eta$ to $n^{\perp }$ is a positive-definite inner product.

(e) Dependence on $n$ and scope. The relation ${\sim }_n$ and the slices $H(p,n)$ depend on the chosen future unit timelike vector $n$: an event $q$ with $\eta (n,q-p)=0$ generally has $\eta (n^{\mathrm{\prime }},q-p)\mathrm{\ne }0$ for another future unit $n^{\mathrm{\prime }}$ (an explicit witness is given on this page). Simultaneity here is defined relative to a prescribed observer vector and is not an absolute relation. For a congruence $n(\cdot )$ one may form the pointwise slices $H(p,n(p))$; the separate integrability question concerns the tangent distribution $p\mapsto n(p{)}^{\perp }$, not the assertion that these affine hyperplanes themselves are its integral hypersurfaces. Integral hypersurfaces can be curved; no global foliation is asserted.

• Well-definedness. $n^{\perp }$ is three-dimensional because $\eta$ is nondegenerate and $n\mathrm{\ne }0$ (eta-orthogonality and orthogonal complements in a Lorentz vector space); positive definiteness of the restriction of $\eta$ to $n^{\perp }$, together with $h(n)=0$, $h\circ h=h$ and $\mathrm{im}h=n^{\perp }$, is clause (c) of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition. Consequently each slice $H(p,n)$ is the translate $p+n^{\perp }$, an affine hyperplane of dimension three, and $\eta (n,q-p)=0$ is equivalent to $q-p\in n^{\perp }$, so the equivalence classes of ${\sim }_n$ are exactly the slices: this uses linearity of $\eta$ in each variable and linearity of $n^{\perp }$ as a subspace, nothing else.

• The rest distance is an inner-product norm. On the slice through $p$ the map $(q,q^{\mathrm{\prime }})\mapsto \eta (q-p,q^{\mathrm{\prime }}-p)$ is the positive-definite symmetric form obtained by restricting $\eta$ to $n^{\perp }$, transported to the affine hyperplane by subtracting $p$; the axioms of a positive-definite inner product therefore give nonnegativity, symmetry, the vanishing condition $d_n(p,q)=0\iff p=q$ and the triangle inequality. No choice principle is used: all objects are finite-dimensional or explicitly parametrized.

• Dependence on the observer vector is not a defect. Two distinct future unit timelike vectors $n\mathrm{\ne }{n}^{\mathrm{\prime }}$ have distinct rest spaces $n^{\perp }\mathrm{\ne }{n}^{\mathrm{\prime }\perp }$ in general, and the relation ${\sim }_n$ is the relation of lying in a common slice of the foliation of $A$ by hyperplanes orthogonal to the fixed vector $n$. For a congruence the vector varies from point to point, and the affine pointwise slices need not fit together. The Frobenius statement below tests whether there are hypersurfaces with tangent spaces $n(p{)}^{\perp }$; it does not identify those hypersurfaces with the entire affine slices $H(p,n(p))$. No global simultaneity is asserted by this definition.

Adopt the ideal-clock and ideal-light assumptions of Adopted ideal-clock and ideal-vacuum-light interpretation and the SI conventions of SI units and coordinate conventions of the special-relativity framework. Let $x:I\to A$ be a future timelike worldline (Curves, regular causal curves and future timelike worldlines in affine Minkowski space) with proper-time function $\tau$ (Proper time along a future timelike worldline), and let $s_{-}<s_{+}$ be parameter values in $I$. Suppose an event $q$ and the events $e_{-}:=x(s_{-})$, $e_{+}:=x(s_{+})$ satisfy: $q-e_{-}$ is future null and $e_{+}-q$ is future null (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones, Oriented time-oriented affine Minkowski space (the spacetime primitives)). Under the adopted ideal-light assumption an ideal ray is emitted at $e_{-}$, reaches $q$, and an ideal ray emitted at $q$ reaches $e_{+}$; the event $q$ is idealized as an instantaneous reflection that changes nothing else. The radar protocol of the observer assigns to $q$

(i) the radar time ${\tau }_{\mathrm{r}\mathrm{a}\mathrm{d}}(q):=\frac{1}{2}(\tau (s_{-})+\tau (s_{+}))$, the ideal clock's reading midway between emission and reception, and

(ii) the radar distance ${\rho }_{\mathrm{r}\mathrm{a}\mathrm{d}}(q):=\frac{c}{2}(\tau (s_{+})-\tau (s_{-}))$, half the round-trip light time multiplied by $c$.

Two events measured by the same observer are radar-simultaneous when their radar times agree. The assignment does not change under increasing $C^1$ reparametrizations with everywhere positive derivative and the same clock origin of the observer's worldline, because the proper-time function does not (Length-integral invariance under monotone C^1 maps and proper-time invariance under regular increasing maps).

Conventions and scope. (a) The protocol is a definition of an operational assignment conditional on the existence of the emission, reflection and reception events, and it assumes an available ideal reflector and ideal apparatus. (b) The split of the round trip into a "time" and a "distance" uses the symmetric convention in which half of the round-trip time is assigned to the outgoing and half to the returning segment; the protocol determines the round-trip quantity $\tau (s_{+})-\tau (s_{-})$ operationally, while the derived one-way assignment is a convention and does not measure a one-way light speed independently of a synchronization convention. (c) The conditional assignment is defined for the stated future timelike worldline, including an accelerated one, but no general existence, uniqueness or synchronization theorem for accelerated observers is asserted. Nothing is claimed about dispersive or absorbing media, finite-resolution apparatus, global single-valuedness or detector models, and no measurement outcome, statistical inference or empirical comparison is asserted.

• Well-definedness. The data $\tau (s_{-})$ and $\tau (s_{+})$ are real numbers determined by the worldline and the two parameters (Proper time along a future timelike worldline, Curves, regular causal curves and future timelike worldlines in affine Minkowski space), so ${\tau }_{\mathrm{r}\mathrm{a}\mathrm{d}}(q)$ and ${\rho }_{\mathrm{r}\mathrm{a}\mathrm{d}}(q)$ are well defined; the existence hypotheses on $q$ are exactly the condition that $q$ is connected to the observer's history by a future null outgoing segment from $e_{-}$ and a future null returning segment to $e_{+}$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones). Since $\tau$ is invariant under increasing $C^1$ reparametrizations with everywhere positive derivative and the same clock origin of the worldline (Length-integral invariance under monotone C^1 maps and proper-time invariance under regular increasing maps), so are the radar time and radar distance.

• The two null segments are the operational content of the adopted assumption. The existence of ideal rays along the null displacements $q-e_{-}$ and $e_{+}-q$ is supplied by the ideal-light clause of Adopted ideal-clock and ideal-vacuum-light interpretation, which identifies ideal light rays in vacuum with future null straight lines; the protocol therefore describes an assignment conditional on that idealization and does not describe a physical mirror, a finite apparatus, or a medium.

• The half-split is a convention, not a derived fact. The protocol measures the round-trip time $\tau (s_{+})-\tau (s_{-})$ operationally; writing half of it as a one-way time and $c\mathrm{/}2$ times it as a distance is the symmetric convention adopted here. For inertial observers this convention is shown to agree with the simultaneity slices of the observer in Radar synchronization of an inertial observer reproduces its inertial chart, while the rotating-ring example below concerns synchronization by the congruence's local rest spaces, not a failure of a single rotating observer's radar assignment. Neither statement is part of the definition.

Let $n$ be a future unit timelike vector, $\eta (n,n)=-1$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones), let $o$ be an event, and let ${\lambda }_{-}<{\lambda }_{+}$ be real numbers; put $e_{-}:=o+{\lambda }_{-}n$ and $e_{+}:=o+{\lambda }_{+}n$ (Oriented time-oriented affine Minkowski space (the spacetime primitives)). For an event $q$ the following are equivalent:

(i) $q-e_{-}$ is future null and $e_{+}-q$ is future null;

(ii) there is $u\in n^{\perp }$ with $\mathrm{\mid }u\mathrm{\mid }:=\sqrt{\eta (u,u)}=({\lambda }_{+}-{\lambda }_{-})\mathrm{/}2$ such that $q=o+\frac{{\lambda }_{-}+{\lambda }_{+}}{2}n+u$.

In that case $q-e_{-}=\frac{{\lambda }_{+}-{\lambda }_{-}}{2}n+u$ and $e_{+}-q=\frac{{\lambda }_{+}-{\lambda }_{-}}{2}n-u$, the affine midpoint $m:=o+\frac{{\lambda }_{-}+{\lambda }_{+}}{2}n=\frac{e_{-}+e_{+}}{2}$ satisfies $\eta (n,q-m)=0$ (so $q$ lies in the slice $H(m,n)$) and $\mathrm{\mid }q-m\mathrm{\mid }=({\lambda }_{+}-{\lambda }_{-})\mathrm{/}2$ (Simultaneity slices and rest distance relative to a future unit timelike vector). In particular for fixed $e_{-},e_{+}$ all admissible reflection events $q$ have the same midpoint $m$ and the same rest distance $({\lambda }_{+}-{\lambda }_{-})\mathrm{/}2$, and differ only in the direction $u$ of $n^{\perp }$.