Affine phase functions and wave covectors on affine Minkowski space

Let $(A,V,\eta ,C^{+})$ be the oriented time-oriented affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives)), with displacements $q-p\in V$ (Affine spaces over a real vector space and displacement vectors).

(a) Affine phase functions. A map $\theta :A\to \mathbb{R}$ is an affine phase function when there are $q\in A$ and a covector $k\in V^{\ast }$ (Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$) with

$\theta (p)-\theta (q)=k(p-q)\text{for all }p\in A\mathrm{.}$

The covector $k$ is then unique: if $k^{\mathrm{\prime }}$ also satisfies the identity, then $(k-k^{\mathrm{\prime }})(p-q)=0$ for all $p\in A$, and as $p$ runs over $A$ the displacements $p-q$ run over all of $V$, so $k-k^{\mathrm{\prime }}=0$ as a functional. This unique covector is the differential $\mathrm{d}\theta$ of $\theta$; when $k\mathrm{\ne }0$, it is also called the wave covector of $\theta$, as in clause (c). Conversely, for every $q\in A$, every $k\in V^{\ast }$ and every ${\theta }_0\in \mathbb{R}$, the formula

$\theta (p):=k(p-q)+{\theta }_0$

defines an affine phase function with $\mathrm{d}\theta =k$ and $\theta (q)={\theta }_0$; hence for each fixed $q$ the map $\theta \mapsto (\mathrm{d}\theta ,\theta (q))$ is a bijection from the set of affine phase functions onto $V^{\ast }\times \mathbb{R}$. In the inertial chart $x$ of an inertial frame $(o;e_0,e_1,e_2,e_3)$ (Inertial frames and inertial charts of the affine Minkowski model), writing $k_a:=k(e_a)$, one has

$\theta (p)=k_a{x}^a(p)+\theta (o)$

because $p-o={\sum }_a{x}^a(p)e_a$ with unique coordinates $x^a(p)$ (A finite list $v:n\to V$ is an ordered basis if and only if every $x\in V$ equals ${\sum }_{i<n}{\lambda }_i{v}_i$ for exactly one $\lambda :n\to F$; those scalars are the coordinates of $x$ in that ordered basis) and $k$ is linear; for a second origin $o^{\mathrm{\prime }}$ one has $\theta (o^{\mathrm{\prime }})=\theta (o)+k(o^{\mathrm{\prime }}-o)$.

(b) Constant-phase directions and level sets. For $v\in V$, $p\in A$ and $s\in \mathbb{R}$ one has $\theta (p+sv)=\theta (p)+sk(v)$ by the defining identity, so $k(v)=0$ if and only if $\theta$ is constant on the line $p+\mathbb{R}v$ for every $p\in A$. If $k\mathrm{\ne }0$, the direction subspace

$\ker k:=\{v\in V:k(v)=0\}$

is a $3$-dimensional linear subspace of $V$: it is the kernel of the nonzero linear functional $k$, and choosing $v_0$ with $k(v_0)\mathrm{\ne }0$ gives every $v\in V$ the unique decomposition $v=(k(v)\mathrm{/}k(v_0))v_0+w$ with $w\in \ker k$, so $V=\ker k\oplus \mathbb{R}{v}_0$ and $\dim \ker k=4-1=3$; every nonempty level set

${\mathrm{\Sigma }}_t:=\{p\in A:\theta (p)=t\}$

is then the affine hyperplane $p_0+\ker k$ through any of its points $p_0$, and the level set is empty when $t$ is not a value of $\theta$. If $k=0$ then $\theta$ is constant and ${\mathrm{\Sigma }}_t$ is either empty or all of $A$. For the derivative assertion, $I\subseteq \mathbb{R}$ is an order-convex set with at least two points, and a $C^1$ curve $\gamma :I\to A$ means a map whose four real components in an inertial chart are differentiable with continuous derivatives, using one-sided derivatives at included endpoints. Its tangent ${\gamma }^{\mathrm{\prime }}(s)$ is the vector with those component derivatives in the frame basis; equivalently it is the limit in frame coordinates of $(\gamma (s+h)-\gamma (s))\mathrm{/}h$. This definition and tangent are independent of the inertial chart: chart transitions are affine with invertible constant linear part, and finite linear combinations preserve derivatives and their continuity 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 $g(c)\mathrm{\ne }0$ and Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function. Finally, for every such $C^1$ curve $\gamma :I\to A$ one has $(\theta \circ \gamma {)}^{\mathrm{\prime }}(s)=k({\gamma }^{\mathrm{\prime }}(s))$ for all $s\in I$: in the coordinates of any inertial chart this is the identity $\frac{\mathrm{d}}{\mathrm{d}s}(k_a{x}^a(\gamma (s))+\theta (o))={\sum }_a{k}_a\frac{\mathrm{d}}{\mathrm{d}s}{x}^a(\gamma (s))$. The sum and scalar-multiple rules of 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 $g(c)\mathrm{\ne }0$, iterated over the four components, and the zero difference quotient of the constant $\theta (o)$ justify this equality, including one-sided endpoints. Its right-hand side is the same linear combination read in the coordinates of ${\gamma }^{\mathrm{\prime }}(s)$.

(c) Wave covectors. A wave covector on $(A,V,\eta )$ is a nonzero element $k\in V^{\ast }$. With $k^{\mathrm{♯}}$ the raised vector of The metric pairing is a linear bijection to the dual, and the raised covector in an orthonormal frame (so that $\eta (k^{\mathrm{♯}},w)=k(w)$ for all $w\in V$, and $k^{\mathrm{♯}}=0$ exactly when $k=0$), the covector $k$ is called null, timelike or spacelike when $\eta (k^{\mathrm{♯}},k^{\mathrm{♯}})$ is zero, negative or positive, and future, past, future null, past null or causal when $k^{\mathrm{♯}}$ is respectively future causal, past causal, future null, past null or causal in the sense of Timelike, null, spacelike and causal vectors of a Lorentz vector space and Time orientation, future and past timelike/null/causal vectors, and the closed cones; a future null wave covector is one that is both future and null. These terms are well posed: a wave covector has $k\mathrm{\ne }0$, hence $k^{\mathrm{♯}}\mathrm{\ne }0$, and a nonzero vector has exactly one causal character and is future or past exactly when it is causal.

(d) Mathematical scope. This item introduces mathematical objects and terminology only. It excludes the zero covector from the term wave covector, it fixes no units and no physical interpretation, and it does not claim that light or any physical wave is described by such a phase; that identification is adopted in post-sr-ideal-plane-wave-model. No choice principle is used.