Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum

Throughout, $(A,V,\eta ,C^{+})$ is the oriented time-oriented affine Minkowski model of Oriented time-oriented affine Minkowski space (the spacetime primitives), $c>0$, and $n\in V$ is future unit timelike, $\eta (n,n)=-1$ (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones). Write $n^{\perp }=\{v\in V:\eta (n,v)=0\}$ for the rest space of $n$, $h(v):=v+\eta (n,v)n$ for the rest projection and $\mathrm{\mid }w\mathrm{\mid }:=\sqrt{\eta (w,w)}$ for the norm on $n^{\perp }$ (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c)); recall from that clause that $h$ is linear with image $n^{\perp }$ and kernel $\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(n)$, that $V=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(n)\oplus n^{\perp }$, and that the restriction of $\eta$ to $n^{\perp }$ is positive definite.

(a) Observer energy, spatial momentum and invariant mass. For every $P$ in the future closed cone $\mathrm{c}\mathrm{l}(C^{+})$ define

$E_n(P):=-c\eta (P,n),q_n(P):=h(P),$

and, when $P$ is nonzero future causal, define the invariant mass of $P$ by

$m(P):=\frac{\sqrt{-\eta (P,P)}}{c}\mathrm{.}$

Then $q_n(P)\in n^{\perp }$, $\eta (n,P)=-E_n(P)\mathrm{/}c$, and

$P=\frac{E_n(P)}{c}n+q_n(P)$

is the unique decomposition of $P$ as a real multiple of $n$ plus a vector of $n^{\perp }$: a vector $P$ has $P=\lambda n+w$ with $\lambda \in \mathbb{R}$ and $w\in n^{\perp }$ if and only if $\lambda =E_n(P)\mathrm{/}c$ and $w=q_n(P)$. Equivalently, $E_n(P)$ and $q_n(P)$ are the unique pair consisting of a real number and a vector of $n^{\perp }$ with that property. The pair $(E_n,q_n)$ is the restriction to $\mathrm{c}\mathrm{l}(C^{+})$ of the pair of linear maps $P\mapsto -c\eta (P,n)$ and $P\mapsto h(P)$ on $V$. For future null $P$ the formula for $m(P)$ gives $m(P)=0$, because $\eta (P,P)=0$.

(b) Rest-mass parameter, rest energy and kinetic energy. Let $m>0$ and let $P=mU$ with $U$ future timelike and $\eta (U,U)=-c^2$, the four-velocity normalisation of Four-velocity and four-acceleration of a future timelike worldline. Then call $m$ the rest-mass parameter of $P$ as written, $m{c}^2$ its rest energy, and

$T_n(P):=E_n(P)-m{c}^2$

its kinetic energy relative to the observer $n$. Bilinearity gives $\eta (P,P)=m^2\eta (U,U)=-m^2{c}^2$, so $m=m(P)>0$ is determined by $P$. Thus $T_n(P)=E_n(P)-m(P)c^2$ is a function of $(P,n)$ on the future timelike cone; recording $m$ supplies no independent parameter.

(c) No positivity, inequality or measurement claim. This definition claims no positivity of $E_n$, no inequality or vanishing property of $T_n$, and no physical measurement. The names observer energy, spatial momentum, invariant mass, rest energy and kinetic energy, and the units joule, kilogram metre per second and kilogram, belong to the unit conventions of SI units and coordinate conventions of the special-relativity framework and to the physical postulates of this framework; the mathematical content asserted here is only the explicit dependence of the five quantities on $P$, $n$ and $m$.

(d) What the objects depend on. $E_n$ and $q_n$ are functions of the pair $(P,n)$: $q_n(P)$ is an element of the vector space $V$, not a coordinate triple, and no basis of $V$ enters its definition, while $E_n(P)=-c\eta (P,n)$ uses the Lorentz form alone. $m(P)$ depends on the scalar $\eta (P,P)$ alone and hence on no observer and no basis; $T_n(P)$ uses the rest mass $m=m(P)$ fixed by that scalar. No choice principle is used.