The oriented volume form epsilon and the permutation symbols of an inertial frame

Let $(A,V,\eta ,C^{+})$ be an oriented time-oriented affine Minkowski space and let $(o;e_0,e_1,e_2,e_3)$ be an inertial frame with its inertial chart (Inertial frames and inertial charts of the affine Minkowski model). Every $v\in V$ has a unique coordinate vector $(v^0,v^1,v^2,v^3)$ with $v={\sum }_a{v}^a{e}_a$. For an ordered $4$-tuple $(v_0,v_1,v_2,v_3)$ of vectors of $V$ let

$[v_j^a]:=\left(\begin{array}{cccc}{v}_0^0& v_1^0& v_2^0& v_3^0\\ v_0^1& v_1^1& v_2^1& v_3^1\\ v_0^2& v_1^2& v_2^2& v_3^2\\ v_0^3& v_1^3& v_2^3& v_3^3\end{array}\right)$

be the $4\times 4$ real matrix whose $j$-th column lists the coordinates of $v_j$ in the ordered basis $(e_0,e_1,e_2,e_3)$, and define

$ϵ(v_0,v_1,v_2,v_3):=\det ([v_j^a])$

by the determinant of a square matrix (For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix). Then $ϵ$ is an alternating $4$-linear form on $V$ (Alternating $k$-linear maps): the determinant is multilinear and alternating in the columns, so replacing one $v_j$ by $\lambda v_j+\mu w$ replaces the $j$-th column correspondingly and $ϵ$ is linear in each argument, and $ϵ(v_0,\dots ,v_3)=0$ whenever two arguments coincide, since then two columns coincide (The determinant is the unique normalized alternating multilinear function on the columns). The form $ϵ$ is the oriented volume form of the model. Its well-definedness as a form of the model, independent of the inertial frame used in the display, is proved in The oriented volume form is independent of the positively oriented inertial frame ↗.

Components. Write ${ϵ}_{abcd}:=ϵ(e_a,e_b,e_c,e_d)$ for the components of $ϵ$ in the inertial frame. Evaluating the determinant on the coordinate matrix of $(e_a,e_b,e_c,e_d)$, which is the permutation matrix of the tuple, gives

${ϵ}_{0123}=+1,{ϵ}_{abcd}=\mathrm{sgn}(\sigma )\text{ if }(a,b,c,d)=(\sigma (0),\sigma (1),\sigma (2),\sigma (3))\text{ for a permutation }\sigma ,$

and ${ϵ}_{abcd}=0$ whenever two of the indices coincide; these are the permutation symbols of the frame. Indeed the coordinate vector of $e_a$ is the $a$-th standard basis vector, so the coordinate matrix of $(e_a,e_b,e_c,e_d)$ is the permutation matrix of the tuple and its determinant is $+1$ on the identity tuple, the sign of the permutation otherwise, and $0$ when two indices repeat.

Raised components. The inverse metric ${\eta }^{ab}$ is the inverse of the matrix ${\eta }_{ab}:=\eta (e_a,e_b)=\mathrm{diag}(-1,1,1,1{)}_{ab}$, which exists because $\eta$ is nondegenerate (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space); it is again ${\eta }^{ab}=\mathrm{diag}(-1,1,1,1{)}^{ab}$. The raised components of the volume form are

${ϵ}^{abcd}:={\eta }^{a{a}^{\mathrm{\prime }}}{\eta }^{b{b}^{\mathrm{\prime }}}{\eta }^{c{c}^{\mathrm{\prime }}}{\eta }^{d{d}^{\mathrm{\prime }}}{ϵ}_{a^{\mathrm{\prime }}{b}^{\mathrm{\prime }}{c}^{\mathrm{\prime }}{d}^{\mathrm{\prime }}}$

(summation over repeated primed indices). Since the inverse metric is diagonal with diagonal entries $-1,1,1,1$, only one term contributes to ${ϵ}^{0123}$ and

${ϵ}^{0123}={\eta }^{00}{\eta }^{11}{\eta }^{22}{\eta }^{33}{ϵ}_{0123}=(-1)(+1)(+1)(+1)(+1)=-1.$

Spatial symbol. The spatial permutation symbol ${ϵ}_{ijk}$ for $i,j,k\in \{1,2,3\}$ is defined by ${ϵ}_{123}=+1$ together with alternation in the three indices; explicitly ${ϵ}_{ijk}=+1$ for even permutations of $(1,2,3)$, ${ϵ}_{ijk}=-1$ for odd permutations, and ${ϵ}_{ijk}=0$ whenever two of $i,j,k$ coincide.