Finite event collisions and observer quantities

Use the flat oriented, time-oriented affine Minkowski model, signature $(-,+,+,+)$, and SI convention $x^0=ct$ of Particle spacetime, observers and worldlines. A relativistic-particle-mechanics collision configuration is a mass-specified causal collision datum of Collision data and four-momentum conservation: an event $p\in A$, a finite nonempty incoming list $P_1,\dots ,P_N\in V$, and a finite nonempty outgoing list $P_1^{\mathrm{\prime }},\dots ,P_K^{\mathrm{\prime }}\in V$, where every leg is a nonzero future-causal vector in the same displacement space $V$. A massive leg has a recorded mass $m>0$ and the massive-sector momentum of Mechanical momentum and observer energy; a null leg has recorded mass $0$ and is a nonzero future-null vector with its scale specified independently. The definition does not identify a null momentum with $mU$ at $m=0$.

The side totals are the finite vector sums $Q_{\mathrm{i}\mathrm{n}}:={\sum }_{i=1}^N{P}_i$ and $Q_{\mathrm{o}\mathrm{u}\mathrm{t}}:={\sum }_{j=1}^K{P}_j^{\mathrm{\prime }}$. All legs are attached to the same event, so the sums compare vectors in one affine displacement space; no transport between different events is part of the datum. The adjective isolated is reserved for a model condition used by the postulate below: no external four-momentum transfer during the idealized event and no omitted incoming or outgoing leg.

At an event, an observer is a future unit timelike vector $n$ with $\eta (n,n)=-1$, and its rest space is $n^{\perp }=\{v\in V:\eta (v,n)=0\}$, as in Particle spacetime, observers and worldlines. Observer energy $E_n(P):=-c\eta (P,n)$ and spatial momentum $q_n(P):=P+\eta (P,n)n\in n^{\perp }$ are those of Mechanical momentum and observer energy. Their units are joules and kilogram metres per second. An inertial laboratory means one named inertial chart and its time-axis observer; it is not a general observer congruence or an arbitrary coordinate chart.