Simultaneity slices and rest distance relative to a future unit timelike vector

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.