Lorentz background, particle quantities and parameter units

Assume countable choice ACω, as in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), for the smooth cotangent-bundle carrier convention used by this page; no full Axiom of Choice is assumed.

Fix a prescribed smooth time-oriented Lorentzian four-manifold $(M,g,T)$ with signature $(-+++)$, as in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model. Here $T$ denotes the supplied smooth timelike field selecting the future cone, called $N$ in that geometric definition. Use the distinction among a spacetime, a worldline, a pointwise observer, an observer congruence, a tetrad and a chart given in Particle spacetime, observers and worldlines. This definition does not make $g$ dynamical.

At each $x\in M$, define $v^{\mathrm{♭}}:=g_x(v,\cdot )$ for $v\in T_{x}M$. Nondegeneracy of $g_x$ makes ${\mathrm{♭}}_x:T_{x}M\to T_x^{\ast }M$ an isomorphism; write $p^{\mathrm{♯}}:={\mathrm{♭}}_x^{-1}(p)$ for $p\in T_x^{\ast }M$, and define $g_x^{-1}(p,r):=g_x(p^{\mathrm{♯}},r^{\mathrm{♯}})$. In a chart, if $g_{ab}:=g({\mathrm{\partial }}_a,{\mathrm{\partial }}_b)$ and $(g^{ab})$ is its inverse matrix, then $p^{\mathrm{♯}}={\sum }_{a,b=0}^3{g}^{ab}{p}_b{\mathrm{\partial }}_a$. A covector $p$ is future causal here exactly when $p^{\mathrm{♯}}$ is a nonzero future-causal vector in the sense of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model.

In this relativistic-particle model, $m>0$ is a fixed primitive mass parameter measured in kilograms, and $c>0$ is a fixed primitive conversion parameter measured in metres per second. They are parameters of the adopted physical formulation, not quantities derived by this definition. Coordinates $x^a$ in an SI chart have length units. For a cotangent vector $p=p_{a}d{x}^a$, its components $p_a$ have momentum units $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$ when all four coordinates are in metres. Thus $p$ as a one-form, and the canonical one-form $\theta =p_{a}d{x}^a$ on $T^{\ast }M$, have action units $\mathrm{J}\mathrm{s}$, consistently with Cotangent phase data, observables and action dimensions. Let $I\subseteq \mathbb{R}$ be an interval with nonempty interior and let $\gamma :I\to M$ be a $C^1$ curve; let $\lambda$ be its real parameter and write $V=d\gamma \mathrm{/}d\lambda$. When $V$ is future timelike, set $s:=\sqrt{-g(V,V)}$ and define the proper-time density by $d\tau =sd\lambda \mathrm{/}c$. The tangent is then nonzero. If $\tilde{\lambda }=f(\lambda )$ is an increasing $C^1$ reparametrization with $f^{\mathrm{\prime }}(\lambda )>0$, then $s_{\tilde{\lambda }}=s_{\lambda }\mathrm{/}{f}^{\mathrm{\prime }}(\lambda )$, so $s_{\tilde{\lambda }}d\tilde{\lambda }=s_{\lambda }d\lambda$. The length action $-mc\int sd\lambda$ has action units. For a future unit observer $n$ and future causal covector $p$, define the model normal-observer energy and spatial momentum covector by

$E_n(p):=-cp(n),q_n(p):=p+p(n)n^{\mathrm{♭}},$

where $n^{\mathrm{♭}}=g(n,\cdot )$. We have $q_n(p)(n)=0$; write $n^{\perp }:=\{v\in T_{x}M:g_x(v,n)=0\}$. To verify the norm convention, choose a $g_x$-orthonormal basis $(e_0,e_1,e_2,e_3)$ with $e_0$ future-directed (replace $e_0$ by $-e_0$ if necessary), and write $n=n^0{e}_0+{\sum }_{i=1}^3{n}^i{e}_i$ and $v=v^0{e}_0+{\sum }_{i=1}^3{v}^i{e}_i$. Use $n_s\cdot v_s:={\sum }_{i=1}^3{n}^i{v}^i$, $\mathrm{\mid }{n}_s{\mathrm{\mid }}^2:={\sum }_{i=1}^3(n^i{)}^2$, and $\mathrm{\mid }{v}_s{\mathrm{\mid }}^2:={\sum }_{i=1}^3(v^i{)}^2$. The future-cone convention gives $n^0>\mathrm{\mid }{n}_s\mathrm{\mid }$, and $g_x(n,n)=-1$ gives $(n^0{)}^2=1+\mathrm{\mid }{n}_s{\mathrm{\mid }}^2$. If $g_x(v,n)=0$, then $v^0=(n_s\cdot v_s)\mathrm{/}{n}^0$, so

$g_x(v,v)=\mathrm{\mid }{v}_s{\mathrm{\mid }}^2-\frac{(n_s\cdot v_s{)}^2}{(n^0{)}^2}\ge \frac{\mathrm{\mid }{v}_s{\mathrm{\mid }}^2}{(n^0{)}^2}\mathrm{.}$

Here $\mathrm{\mid }{n}_s{\mathrm{\mid }}^2\mathrm{\mid }{v}_s{\mathrm{\mid }}^2-(n_s\cdot v_s{)}^2={\sum }_{1\le i<j\le 3}(n^i{v}^j-n^j{v}^i{)}^2\ge 0$; if $v_s=0$, orthogonality also gives $v^0=0$. Thus $g_x$ is positive definite on $n^{\perp }$, and the isomorphism ${\mathrm{♯}}_x$ carries covectors annihilating $n$ onto $n^{\perp }$, so $g_x^{-1}$ is positive definite on that covector subspace. Define $\mathrm{\parallel }{q}_n(p){\mathrm{\parallel }}_n:=\sqrt{g_x^{-1}(q_n(p),q_n(p))}$. These are quantity definitions and unit conventions; they do not identify canonical momentum with mechanical momentum, assert that an apparatus measured these values, or establish a dynamical law.