Total charge, dipole moment, Cartesian moments and traceless quadrupole of a static charge distribution

For the Lebesgue branches (a) and (b) assume Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), inherited from the exact measure interface (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); the finite sums in branch (c) are defined and computed without Choice. Let the laboratory region, its chart, its spatial Euclidean structure and its units be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values, and let ${\delta }_{ij}$ be the Kronecker symbol in the $\delta$-orthonormal basis $(e_1,e_2,e_3)$ of the frame, with $\mathrm{\mid }y\mathrm{\mid }$ the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. This definition assigns source moments to a static charge distribution in the fixed-time reading of The static and pure electrostatic regimes and their fixed-time reading (d) and (e). Three branches are declared.

(a) [Absolutely continuous branch] Let $a>0$ and let $\rho \in L^1({\mathbb{R}}^3)$ be real (The class $L^1(\mu )$ of integrable functions) and vanish ${\lambda }_3$-almost everywhere outside the closed ball ${\bar{B}}_a(0)=\{y:\mathrm{\mid }y\mathrm{\mid }\le a\}$ (Open ball, closed ball and sphere in a metric space, Measure-null sets and almost-everywhere statements relative to a measure). The total charge $Q$, the dipole moment $p$, the degree-$l$ Cartesian moments $M_{i_1\dots i_l}$ and the traceless quadrupole $Q_{ij}$ are

$Q:={\int }_{{\mathbb{R}}^3}\rho d{\lambda }_3,p_i:={\int }_{{\mathbb{R}}^3}{y}_i\rho (y)d{\lambda }_3(y),M_{i_1\dots i_l}:={\int }_{{\mathbb{R}}^3}{y}_{i_1}\cdots y_{i_l}\rho (y)d{\lambda }_3(y)(l\ge 1),$

$M^{(0)}:=Q,M_i:=p_i,Q_{ij}:={\int }_{{\mathbb{R}}^3}(3y_i{y}_j-\mathrm{\mid }y{\mathrm{\mid }}^2{\delta }_{ij})\rho (y)d{\lambda }_3(y),$

with $l\ge 1$ and indices in $\{1,2,3\}$; units are C for $Q$, C m for $p$, C m${}^l$ for the degree-$l$ Cartesian moment and C m${}^2$ for $Q_{ij}$, and all integrals are in the sense of Integral over a measurable subset and Integrable real and complex functions, and their integrals. Each integral is absolutely convergent: since $\rho =0$ ${\lambda }_3$-a.e. off ${\bar{B}}_a(0)$, and the maps $y\mapsto y_{i_1}\cdots y_{i_l}$ and $y\mapsto 3y_i{y}_j-\mathrm{\mid }y{\mathrm{\mid }}^2{\delta }_{ij}$ are bounded ${\lambda }_3$-a.e. on that ball by $a^l$ and by $4a^2$ respectively, the integrands are dominated ${\lambda }_3$-a.e. by $a^l\mathrm{\mid }\rho \mathrm{\mid }$ and $4a^2\mathrm{\mid }\rho \mathrm{\mid }$, which lie in $L^1$ (The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral). The traceless quadrupole is traceless,

${\sum }_{i=1}^3{Q}_{ii}=0,$

because ${\sum }_{i=1}^3(3y_i^2-\mathrm{\mid }y{\mathrm{\mid }}^2)=0$ pointwise, and summing the three integrable functions $3y_i^2\rho -\mathrm{\mid }y{\mathrm{\mid }}^2\rho$ and applying finite linearity of the integral on $L^1$ (The Lebesgue integral is linear on $L^1(\mu )$) evaluates the sum as the integral of the pointwise-zero function; replacing $\rho$ by an equal ${\lambda }_3$-a.e. density changes no moment (Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree).

(b) [General locally integrable branch] If instead $\rho \in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^3)$ is real and ${\int }_{{\mathbb{R}}^3}\mathrm{\mid }y{\mathrm{\mid }}^l\mathrm{\mid }\rho (y)\mathrm{\mid }d{\lambda }_3(y)<\mathrm{\infty }$ for a given integer $l\ge 0$, then the degree-$l$ Cartesian moments of the same formula are absolutely convergent and defined; if the absolute weighted moment ${\int }_{{\mathbb{R}}^3}\mathrm{\mid }y{\mathrm{\mid }}^L\mathrm{\mid }\rho (y)\mathrm{\mid }d{\lambda }_3(y)$ is finite for some real $L\ge 0$, then every integer Cartesian degree $0\le l\le L$ is defined, and in particular $Q=\int \rho d{\lambda }_3$ is absolutely convergent, so that $\rho \in L^1({\mathbb{R}}^3)$ is a conclusion of this branch and not an added hypothesis. Here $\rho$ is restricted to the closed unit ball, where it is integrable by local integrability, and outside it $\mathrm{\mid }y{\mathrm{\mid }}^l\le \mathrm{\mid }y{\mathrm{\mid }}^L$; splitting the nonnegative integral over these two measurable sets and applying monotonicity and additivity of the integral gives $\int \mathrm{\mid }y{\mathrm{\mid }}^l\mathrm{\mid }\rho \mathrm{\mid }\le {\int }_{{\bar{B}}_1(0)}\mathrm{\mid }\rho \mathrm{\mid }+{\int }_{\{\mathrm{\mid }y\mathrm{\mid }>1\}}\mathrm{\mid }y{\mathrm{\mid }}^L\mathrm{\mid }\rho \mathrm{\mid }<\mathrm{\infty }$ (Monotonicity and nonnegative homogeneity of the nonnegative integral, Open ball, closed ball and sphere in a metric space). The convention $\mathrm{\mid }y{\mathrm{\mid }}^0=1$ includes the origin, no compact support and no $L^1$-membership is assumed in this branch, and Cartesian monomials of degree $l$ have modulus at most $\mathrm{\mid }y{\mathrm{\mid }}^l$, so all lower moments, including $Q$, exist together with the degree-$l$ moment.

(c) [Finite system of stationary point charges] Let $N\ge 0$, let $x_1,\dots ,x_N\in {\mathbb{R}}^3$ and let $q_1,\dots ,q_N\in \mathbb{R}$ be charge parameters (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current (a)), read as the stationary point charges $q_k{\delta }_{x_k}$ in the fixed-time reading of The static and pure electrostatic regimes and their fixed-time reading (e); for $N=0$ the lists are empty and every sum below is the empty sum $0$. Define

$Q:={\sum }_{k=1}^N{q}_k,p:={\sum }_{k=1}^N{q}_k{x}_k,M_{i_1\dots i_l}:={\sum }_{k=1}^N{q}_k{x}_{k,i_1}\cdots x_{k,i_l}(l\ge 1),M^{(0)}:=Q,$

$Q_{ij}:={\sum }_{k=1}^N{q}_k(3x_{k,i}{x}_{k,j}-\mathrm{\mid }{x}_k{\mathrm{\mid }}^2{\delta }_{ij}),$

finite sums in the sense of Finite sums and finite products, by recursion (Laws of finite sums and finite products); again

${\sum }_{i=1}^3{Q}_{ii}=0,$

by finite distributivity. The points need not be distinct; charges at a common point add in every sum.

(d) [Reference point and scope] Every moment of (a)-(c) is computed about the chart origin of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields; this origin is part of the definition. These are source moments of a prescribed static distribution, not momenta, rest masses, intrinsic spins, self-energies or dynamical quantities, and no field, force, equation of motion, equilibrium or empirical claim is made here; the static regime in which the source is read is the adopted idealisation of The static and pure electrostatic regimes and their fixed-time reading (f). Every Cartesian degree $l$ in this definition is a nonnegative integer (the number of Cartesian indices); only the absolute weighted exponent $L$ in (b) may be any nonnegative real number.