Total four-momentum, invariant mass and center-of-momentum frame of a finite system

Throughout, $(A,V,\eta ,C^{+})$ is the oriented time-oriented affine Minkowski model of Oriented time-oriented affine Minkowski space (the spacetime primitives) and $c>0$.

(a) Total four-momentum. For a finite list $(P_1,\dots ,P_N)$, $N\ge 0$, of vectors of $V$, define the total four-momentum $P_{\mathrm{t}\mathrm{o}\mathrm{t}}:={\sum }_{i=1}^N{P}_i$ as the finite sum of A finite sum in a commutative monoid indexed by an arbitrary finite set in the additive group $(V,+)$ of the vector space $V$; the empty sum is $0$. The sum uses the vectors $P_i$ alone: no event, no position and no simultaneity datum enters it, and the order of the list does not affect $P_{\mathrm{t}\mathrm{o}\mathrm{t}}$ because addition in $V$ is commutative.

(b) Invariant mass of the total. If $P_{\mathrm{t}\mathrm{o}\mathrm{t}}$ is future timelike (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones), define the invariant mass of the system and its rest energy by

$M:=M(P_{\mathrm{t}\mathrm{o}\mathrm{t}}):=\frac{\sqrt{-\eta (P_{\mathrm{t}\mathrm{o}\mathrm{t}},P_{\mathrm{t}\mathrm{o}\mathrm{t}})}}{c}>0$

and its rest energy is $M{c}^2$.

The square root is a real number because $\eta (P_{\mathrm{t}\mathrm{o}\mathrm{t}},P_{\mathrm{t}\mathrm{o}\mathrm{t}})<0$ for a timelike vector, and it is strictly positive, so $M>0$ (Rational powers $a^r$ of a positive base). If $P_{\mathrm{t}\mathrm{o}\mathrm{t}}$ is future null, set $M:=0$ and say that the system has no rest frame. For $P_{\mathrm{t}\mathrm{o}\mathrm{t}}$ equal to zero, spacelike or past causal, this convention defines no invariant mass.

(c) Center-of-momentum frame. A center-of-momentum frame of a nonzero $P_{\mathrm{t}\mathrm{o}\mathrm{t}}$ is an inertial frame $(o;e_0,e_1,e_2,e_3)$ (Inertial frames and inertial charts of the affine Minkowski model) whose first vector satisfies

$h_{e_0}(P_{\mathrm{t}\mathrm{o}\mathrm{t}})=0,h_{e_0}(v):=v+\eta (v,e_0)e_0(v\in V),$

where the projection is defined on all of $V$; for a future causal total this is $q_{e_0}(P_{\mathrm{t}\mathrm{o}\mathrm{t}})=0$, meaning that its total spatial momentum vanishes for the observer stationary in the frame (Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum). The condition is equivalent to $P_{\mathrm{t}\mathrm{o}\mathrm{t}}=-\eta (P_{\mathrm{t}\mathrm{o}\mathrm{t}},e_0)e_0$, i.e. to $P_{\mathrm{t}\mathrm{o}\mathrm{t}}\in \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(e_0)$; if $P_{\mathrm{t}\mathrm{o}\mathrm{t}}$ is future timelike this holds exactly when $P_{\mathrm{t}\mathrm{o}\mathrm{t}}=\lambda e_0$ with $\lambda >0$, and then $E_{e_0}(P_{\mathrm{t}\mathrm{o}\mathrm{t}})=M{c}^2$ for $M$ as in (b).

(d) Comparison with the single-vector convention. For a single future timelike vector $P$, the invariant mass $M$ of (b) coincides with $m(P)$ of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum; in particular if $P=mU$ with $m>0$ and $\eta (U,U)=-c^2$ (the four-velocity normalisation of Four-velocity and four-acceleration of a future timelike worldline), then $M=m$. Both quantities are functions of the scalar $\eta (P,P)$ alone.

(e) Scope. This definition asserts no existence of a center-of-momentum frame (that is Center-of-momentum frames of a future timelike four-momentum and their unique time vector), no physical conservation law and no dynamics; only the total vector carries the invariant mass, and the individual summands need not be causal for the sum to be defined. No choice principle is used.