Observer-relative energy, spatial momentum and rest mass of a tangent vector

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold, let $p\in M$, let $n$ be an observer at $p$, and let $c>0$ be a real parameter (Observers, rest spaces and the spatial projection at an event, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

(a) Observer energy and spatial momentum. For every $P\in T_{p}M$ define the observer energy and the spatial momentum of $P$ relative to $n$ by

$E_n(P):=-c{g}_p(P,n),q_n(P):=h_n(P),$

where $h_n$ is the spatial projection of the observer $n$. Both maps are $\mathbb{R}$-linear in $P$: the first because $g_p$ is bilinear, the second by The observer rest-space decomposition and the spatial projection. The second takes values in the rest space $n^{\perp }$, and by the rest-space decomposition of that lemma every $P$ has the unique decomposition

$P=\frac{E_n(P)}{c}n+q_n(P),\frac{E_n(P)}{c}\in \mathbb{R},q_n(P)\in n^{\perp }\mathrm{.}$

Equivalently $E_n(P)=c\lambda$ and $q_n(P)=w$ for the unique pair $(\lambda ,w)$ with $P=\lambda n+w$, $\lambda \in \mathbb{R}$, $w\in n^{\perp }$.

(b) Rest mass and four-velocity normalization. For $P\in T_{p}M$ with $g_p(P,P)\le 0$ define the rest mass

$m(P):=\frac{\sqrt{-g_p(P,P)}}{c}\ge 0,$

where the square root is the nonnegative root of Rational powers $a^r$ of a positive base. By strict positivity of the square root of a positive number, $m(P)=0$ exactly when $P=0$ or $P$ is null, and $m(P)>0$ exactly when $P$ is timelike (Timelike, null, spacelike and causal vectors of a Lorentz vector space). When $m(P)>0$, define the normalized four-velocity direction

$U(P):=\frac{P}{m(P)},$

so that $g_p(U(P),U(P))=g_p(P,P)\mathrm{/}m(P{)}^2=-c^2$. Conversely, if $m>0$ and $U\in T_{p}M$ satisfy $g_p(U,U)=-c^2$, then $P:=mU$ has $g_p(P,P)=-m^2{c}^2$ and $m(P)=m$, $U(P)=U$; the pair $(m,U)$ with $U$ future-directed determines and is determined by its future timelike $P=mU$.

(c) Scope and reading. $E_n$ and $q_n$ depend on the observer $n$: for the same nonzero $P$, different observers $n,n^{\mathrm{\prime }}$ generally give different values, and no observer-independent splitting of $P$ into "energy" and "momentum" is defined by this item. The identities above are algebraic; this definition asserts no positivity of $E_n$ (that is proved separately), no conservation, no dynamics and no measurement. The names observer energy, spatial momentum and rest mass, and their physical units and interpretation, belong to the physical reading of the framework's unit conventions and postulates and are not asserted here; this item fixes only the mathematical dependence on the objects $P$, $n$, $g_p$ and the parameter $c$.