Mechanical momentum and observer energy

Use the massive and null sectors as separately adopted in Massive spinless test-particle action and Massless test-particle action and momentum scale. For a massive body of constant mass $m>0$, its mechanical four-momentum is $P=mU$, where along an admissible $C^2$ future timelike trajectory $x:I\to M$, with $I\subseteq \mathbb{R}$ an interval with nonempty interior and parameter $s$, put $T=dx\mathrm{/}ds$, and $U:=cT\mathrm{/}\sqrt{-g(T,T)}$ is its future tangent normalized by $g(U,U)=-c^2$. The clock convention in the massive-sector adoption gives $d\tau \mathrm{/}ds=\sqrt{-g(T,T)}\mathrm{/}c$, so $U=dx\mathrm{/}d\tau$. In the null sector, the mechanical four-momentum is the separately specified nonzero future-null vector $P$ from the einbein formulation; no null proper time or $P=mU$ is defined.

At an event, for any observer $n$ of Observers, rest spaces and the spatial projection at an event, define the spatial momentum and observer energy by

$q_n(P):=P+g(P,n)n\in n^{\perp },E_n(P):=-cg(P,n)\mathrm{.}$

The projection identity and positive-definiteness of $n^{\perp }$ are supplied by The observer rest-space decomposition and the spatial projection. With SI units, $P$ has units kg m s${}^{-1}$, $q_n$ has the same units, and $E_n$ has units joules. This definition is pointwise and does not assert conservation, positivity, or an energy interpretation for an arbitrary vector field that is not an observer.