Tables (7)

Table p. 55.0 — 6×2, 1 spanning cell(s)
Gravitational gauge field variables
\begin{tabular}[t]{\|l\|l\|} \hline $h_{a}{ }^{\mu}$ & translational gauge field ( $h$-field) \\ \hline $h$ & determinant of $h_{a}{ }^{\mu}$ \\ \hline $b^{a}{ }_{\mu}$ & inverse $h$-field \\ \hline $A^{a b}{ }_{\mu}$ & rotational gauge field ( $A$-field) \\ \hline $B_{\mu}$ & WGT dilational gauge field ( $B$-field) \\ \hline $V_{\mu}$ & eWGT dilational gauge field ( $V$-field) \\ \hline $A^{\dagger a b}{ }_{\mu} \equiv A^{a b}{ }_{\mu}+2 \eta^{c[a} b^{b]}{ }_{\mu} h_{c}{ }^{\nu} V_{\nu}$ & eWGT extended rotational gauge field ( $A^{\dagger}$-field) \\ \hline \end{tabular}
Derivative operators \begin{tabular}[t]{\|l\|} \hline $\mathcal{D}_{a} \equiv h_{a}{ }^{\mu} D_{\mu} \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}+\frac{1}{2} A^{a b}{ }_{\mu} \Sigma_{a b}\right)$ \\ \hline $\partial_{\mu}^{} \equiv \partial_{\mu}+w B_{\mu}$ \\ \hline $\mathcal{D}_{a}^{} \equiv h_{a}{ }^{\mu} D_{\mu}^{} \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}^{}+\frac{1}{2} A^{a b}{ }_{\mu} \Sigma_{a b}\right)$ \\ \hline $\partial_{\mu}^{\dagger} \equiv \partial_{\mu}-w\left(V_{\mu}+\frac{1}{3} T_{\mu}\right)$ \\ \hline $\mathcal{D}_{a}^{\dagger} \equiv h_{a}{ }^{\mu} D_{\mu}^{\dagger} \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}^{\dagger}+\frac{1}{2} A^{\dagger a b}{ }_{\mu} \Sigma_{a b}\right)$ \\ \hline $\mathcal{D}_{a}^{\natural} \equiv h_{a}{ }^{\mu} D_{\mu}^{\natural} \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}+\frac{1}{2} A^{\dagger a b}{ }_{\mu} \Sigma_{a b}\right)$ \\ \hline \end{tabular}	PGT (generalised) covariant derivative ${ }^{\mathrm{a}}$ WGT 'augmented' partial derivative WGT (generalised) covariant derivative ${ }^{\mathrm{a}}$ eWGT 'augmented' partial derivative eWGT (generalised) covariant derivative ${ }^{\mathrm{a}}$ eWGT (generalised) semi-covariant derivative ${ }^{\mathrm{a}}$
Gauge field strengths \begin{tabular}[t]{\|l\|} \hline $\mathcal{R}^{a b}{ }_{c d}=\mathcal{R}^{* a b}{ }_{c d}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu}\left(\partial_{[\mu} A^{a b}{ }_{\nu]}+A^{a}{ }_{c[\mu} A^{c b}{ }_{\nu]}\right) \mathcal{R}^{\dagger a b}{ }_{c d}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu}\left(\partial_{[\mu} A^{\dagger a b}{ }_{\nu]}+A^{\dagger a}{ }_{c[\mu} A^{\dagger c b}{ }_{\nu]}\right) \mathcal{T}^{a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu} b^{a}{ }_{\nu]} \mathcal{T}^{* a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu}^{*} b^{a}{ }_{\nu]} \mathcal{T}^{\dagger a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu}^{\dagger} b^{a}{ }_{\nu]} \mathcal{T}^{\natural a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu}^{\natural} b^{a}{ }_{\nu]} \mathcal{H}_{a b}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu} \partial_{[\mu} B_{\nu]} \mathcal{H}_{a b}^{\dagger}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu} \partial_{[\mu}\left(V_{\nu}+\frac{1}{3} T_{\nu]}\right)$ \\ \hline \end{tabular}	PGT and WGT rotational gauge field strength ${ }^{\mathrm{a}}$ eWGT rotational gauge field strength ${ }^{\mathrm{a}}{ }^{\mathrm{b}}$ PGT translational gauge field strength ${ }^{\text {b }}$ WGT translational gauge field strength ${ }^{\mathrm{b}}$ eWGT translational gauge field strength ${ }^{\mathrm{b}}$ eWGT translational gauge 'semi' field strengtha ${ }^{\mathrm{b}}$ WGT dilational gauge field strength eWGT dilational gauge field strength
Reduced $A$-fields and related quantities \begin{tabular}[t]{\|l\|} \hline $c^{a}{ }_{b c} \equiv 2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} \partial_{[\mu} b^{a}{ }_{\nu]}$ \\ \hline $c^{* a}{ }_{b c} \equiv 2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} \partial_{[\mu}^{} b^{a}{ }_{\nu]}$ \\ \hline $c^{\dagger a}{ }_{b c} \equiv 2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} \partial_{[\mu}^{\dagger} b^{a}{ }_{\nu]}$ \\ \hline ${ }^{0} A^{a b}{ }_{\mu} \equiv \frac{1}{2} b^{c}{ }_{\mu}\left(c^{a b}{ }_{c}+c^{b}{ }_{c}{ }^{a}-c_{c}{ }^{a b}\right)$ \\ \hline ${ }^{0} A^{ a b}{ }_{\mu} \equiv \frac{1}{2} b^{c}{ }_{\mu}\left(c^{* a b}{ }_{c}+c^{* b}{ }_{c}{ }^{a}-c_{c}^{* a b}\right)$ \\ \hline ${ }^{0} A^{\dagger a b}{ }_{\mu} \equiv \frac{1}{2} b^{c}{ }_{\mu}\left(c^{\dagger a b}{ }_{c}+c^{\dagger b}{ }_{c}{ }^{a}-c_{c}^{\dagger a b}\right)$ \\ \hline \end{tabular}	PGT Ricci rotation coefficients WGT Ricci rotation coefficients eWGT Ricci rotation coefficients PGT reduced $A$-field (when $\mathcal{T}^{a}{ }_{b c} \equiv 0$ ) WGT reduced $A$-field (when $\mathcal{T}^{* a}{ }_{b c} \equiv 0$ ) eWGT reduced $A^{\dagger}$-field (when $\mathcal{T}^{\dagger a}{ }_{b c} \equiv 0$ )
Currents and related quantities \begin{tabular}[t]{\|l\|} \hline $L$ \\ \hline $\mathcal{L} \equiv h^{-1} L$ \\ \hline $t^{a}{ }_{b} \equiv t^{a}{ }_{\mu} h_{b}{ }^{\mu} \equiv h_{b}{ }^{\mu} \delta \mathcal{L}_{\mathrm{G}} / \delta h_{a}{ }^{\mu}$ \\ \hline $s_{a b}{ }^{c} \equiv s_{a b}{ }^{\mu} b^{c}{ }_{\mu} \equiv b^{c}{ }_{\mu} \delta \mathcal{L}_{\mathrm{G}} / \delta A^{a b}{ }_{\mu}$ \\ \hline $j^{a} \equiv j^{\mu} b^{a}{ }_{\mu} \equiv b^{a}{ }_{\mu} \delta \mathcal{L}_{\mathrm{G}} / \delta B_{\mu}$ \\ \hline $t^{\dagger a}{ }_{b} \equiv t^{a}{ }_{b}+2\left(s_{c b}{ }^{a} \mathcal{V}^{c}-s^{a c}{ }_{c} \mathcal{V}_{b}\right)$ \\ \hline $j^{\dagger a} \equiv j^{a}-2 s^{a b}{ }_{b}$ \\ \hline $\tau^{a}{ }_{b} \equiv \tau^{a}{ }_{\mu} h_{b}{ }^{\mu} \equiv h_{b}{ }^{\mu} \delta \mathcal{L}_{\mathrm{M}} / \delta h_{a}{ }^{\mu}$ \\ \hline $\sigma_{a b}{ }^{c} \equiv \sigma_{a b}{ }^{\mu} b^{c}{ }_{\mu} \equiv b^{c}{ }_{\mu} \delta \mathcal{L}_{\mathrm{M}} / \delta A^{a b}{ }_{\mu}$ \\ \hline $\zeta^{a} \equiv \zeta^{\mu} b^{a}{ }_{\mu} \equiv b^{a}{ }_{\mu} \delta \mathcal{L}_{\mathrm{M}} / \delta B_{\mu}$ \\ \hline $\tau^{\dagger a}{ }_{b} \equiv \tau^{a}{ }_{b}+2\left(\sigma_{c b}{ }^{a} \mathcal{V}^{c}-\sigma^{a c}{ }_{c} \mathcal{V}_{b}\right)$ \\ \hline $\zeta^{\dagger a} \equiv \zeta^{a}-2 s^{a b}{ }_{b}$ \\ \hline \end{tabular}	Lagrangian Lagrangian density gravitational sector energy-momentum tensor ${ }^{\mathrm{c}}$ gravitational sector spin-angular-momentum tensor ${ }^{\mathrm{c}}$ gravitational sector dilation current ${ }^{\mathrm{c}}$ gravitational sector eWGT covariant energy-momentum tensor ${ }^{\mathrm{c}}$ gravitational sector eWGT covariant dilation current ${ }^{\text {c }}$ matter sector energy-momentum tensor ${ }^{\mathrm{c}}$ matter sector spin-angular-momentum tensor ${ }^{\mathrm{c}}$ matter sector dilation current ${ }^{\text {c }}$ gravitational sector eWGT covariant energy-momentum tensor ${ }^{\mathrm{c}}$ gravitational sector eWGT covariant dilation current ${ }^{\mathrm{c}}$

columns: Gravitational gauge field variables | Gravitational gauge field variables \begin{tabular}[t]{|l|l|} \hline $h_{a}{ }^{\mu}$ & translational gauge field ( $h$-field) \\ \hline $h$ & determinant of $h_{a}{ }^{\mu}$ \\ \hline $b^{a}{ }_{\mu}$ & inverse $h$-field \\ \hline $A^{a b}{ }_{\mu}$ & rotational gauge field ( $A$-field) \\ \hline $B_{\mu}$ & WGT dilational gauge field ( $B$-field) \\ \hline $V_{\mu}$ & eWGT dilational gauge field ( $V$-field) \\ \hline $A^{\dagger a b}{ }_{\mu} \equiv A^{a b}{ }_{\mu}+2 \eta^{c[a} b^{b]}{ }_{\mu} h_{c}{ }^{\nu} V_{\nu}$ & eWGT extended rotational gauge field ( $A^{\dagger}$-field) \\ \hline \end{tabular}

Table p. 55.1 — 7×2, 0 spanning cell(s)
$h_{a}{ }^{\mu}$	translational gauge field ( $h$-field)
$h$	determinant of $h_{a}{ }^{\mu}$
$b^{a}{ }_{\mu}$	inverse $h$-field
$A^{a b}{ }_{\mu}$	rotational gauge field ( $A$-field)
$B_{\mu}$	WGT dilational gauge field ( $B$-field)
$V_{\mu}$	eWGT dilational gauge field ( $V$-field)
$A^{\dagger a b}{ }_{\mu} \equiv A^{a b}{ }_{\mu}+2 \eta^{c[a} b^{b]}{ }_{\mu} h_{c}{ }^{\nu} V_{\nu}$	eWGT extended rotational gauge field ( $A^{\dagger}$-field)

columns: $h_{a}{ }^{\mu}$ | translational gauge field ( $h$-field)

Table p. 55.2 — 6×1, 0 spanning cell(s)
$\mathcal{D}_{a} \equiv h_{a}{ }^{\mu} D_{\mu} \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}+\frac{1}{2} A^{a b}{ }_{\mu} \Sigma_{a b}\right)$
$\partial_{\mu}^{*} \equiv \partial_{\mu}+w B_{\mu}$
$\mathcal{D}_{a}^{} \equiv h_{a}{ }^{\mu} D_{\mu}^{} \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}^{*}+\frac{1}{2} A^{a b}{ }_{\mu} \Sigma_{a b}\right)$
$\partial_{\mu}^{\dagger} \equiv \partial_{\mu}-w\left(V_{\mu}+\frac{1}{3} T_{\mu}\right)$
$\mathcal{D}_{a}^{\dagger} \equiv h_{a}{ }^{\mu} D_{\mu}^{\dagger} \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}^{\dagger}+\frac{1}{2} A^{\dagger a b}{ }_{\mu} \Sigma_{a b}\right)$
$\mathcal{D}_{a}^{\natural} \equiv h_{a}{ }^{\mu} D_{\mu}^{\natural} \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}+\frac{1}{2} A^{\dagger a b}{ }_{\mu} \Sigma_{a b}\right)$

columns: $\mathcal{D}_{a} \equiv h_{a}{ }^{\mu} D_{\mu} \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}+\frac{1}{2} A^{a b}{ }_{\mu} \Sigma_{a b}\right)$

Table p. 55.3 — 1×1, 0 spanning cell(s)
$\mathcal{R}^{a b}{ }_{c d}=\mathcal{R}^{* a b}{ }_{c d}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu}\left(\partial_{[\mu} A^{a b}{ }_{\nu]}+A^{a}{ }_{c[\mu} A^{c b}{ }_{\nu]}\right) \mathcal{R}^{\dagger a b}{ }_{c d}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu}\left(\partial_{[\mu} A^{\dagger a b}{ }_{\nu]}+A^{\dagger a}{ }_{c[\mu} A^{\dagger c b}{ }_{\nu]}\right) \mathcal{T}^{a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu} b^{a}{ }_{\nu]} \mathcal{T}^{* a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu}^{*} b^{a}{ }_{\nu]} \mathcal{T}^{\dagger a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu}^{\dagger} b^{a}{ }_{\nu]} \mathcal{T}^{\natural a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu}^{\natural} b^{a}{ }_{\nu]} \mathcal{H}_{a b}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu} \partial_{[\mu} B_{\nu]} \mathcal{H}_{a b}^{\dagger}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu} \partial_{[\mu}\left(V_{\nu}+\frac{1}{3} T_{\nu]}\right)$

columns: $\mathcal{R}^{a b}{ }_{c d}=\mathcal{R}^{* a b}{ }_{c d}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu}\left(\partial_{[\mu} A^{a b}{ }_{\nu]}+A^{a}{ }_{c[\mu} A^{c b}{ }_{\nu]}\right) \mathcal{R}^{\dagger a b}{ }_{c d}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu}\left(\partial_{[\mu} A^{\dagger a b}{ }_{\nu]}+A^{\dagger a}{ }_{c[\mu} A^{\dagger c b}{ }_{\nu]}\right) \mathcal{T}^{a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu} b^{a}{ }_{\nu]} \mathcal{T}^{* a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu}^{*} b^{a}{ }_{\nu]} \mathcal{T}^{\dagger a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu}^{\dagger} b^{a}{ }_{\nu]} \mathcal{T}^{\natural a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu}^{\natural} b^{a}{ }_{\nu]} \mathcal{H}_{a b}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu} \partial_{[\mu} B_{\nu]} \mathcal{H}_{a b}^{\dagger}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu} \partial_{[\mu}\left(V_{\nu}+\frac{1}{3} T_{\nu]}\right)$

Table p. 55.4 — 6×1, 0 spanning cell(s)
$c^{a}{ }_{b c} \equiv 2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} \partial_{[\mu} b^{a}{ }_{\nu]}$
$c^{* a}{ }_{b c} \equiv 2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} \partial_{[\mu}^{*} b^{a}{ }_{\nu]}$
$c^{\dagger a}{ }_{b c} \equiv 2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} \partial_{[\mu}^{\dagger} b^{a}{ }_{\nu]}$
${ }^{0} A^{a b}{ }_{\mu} \equiv \frac{1}{2} b^{c}{ }_{\mu}\left(c^{a b}{ }_{c}+c^{b}{ }_{c}{ }^{a}-c_{c}{ }^{a b}\right)$
${ }^{0} A^{* a b}{ }_{\mu} \equiv \frac{1}{2} b^{c}{ }_{\mu}\left(c^{* a b}{ }_{c}+c^{* b}{ }_{c}{ }^{a}-c_{c}^{* a b}\right)$
${ }^{0} A^{\dagger a b}{ }_{\mu} \equiv \frac{1}{2} b^{c}{ }_{\mu}\left(c^{\dagger a b}{ }_{c}+c^{\dagger b}{ }_{c}{ }^{a}-c_{c}^{\dagger a b}\right)$

columns: $c^{a}{ }_{b c} \equiv 2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} \partial_{[\mu} b^{a}{ }_{\nu]}$

Table p. 55.5 — 12×1, 0 spanning cell(s)
$L$
$\mathcal{L} \equiv h^{-1} L$
$t^{a}{ }_{b} \equiv t^{a}{ }_{\mu} h_{b}{ }^{\mu} \equiv h_{b}{ }^{\mu} \delta \mathcal{L}_{\mathrm{G}} / \delta h_{a}{ }^{\mu}$
$s_{a b}{ }^{c} \equiv s_{a b}{ }^{\mu} b^{c}{ }_{\mu} \equiv b^{c}{ }_{\mu} \delta \mathcal{L}_{\mathrm{G}} / \delta A^{a b}{ }_{\mu}$
$j^{a} \equiv j^{\mu} b^{a}{ }_{\mu} \equiv b^{a}{ }_{\mu} \delta \mathcal{L}_{\mathrm{G}} / \delta B_{\mu}$
$t^{\dagger a}{ }_{b} \equiv t^{a}{ }_{b}+2\left(s_{c b}{ }^{a} \mathcal{V}^{c}-s^{a c}{ }_{c} \mathcal{V}_{b}\right)$
$j^{\dagger a} \equiv j^{a}-2 s^{a b}{ }_{b}$
$\tau^{a}{ }_{b} \equiv \tau^{a}{ }_{\mu} h_{b}{ }^{\mu} \equiv h_{b}{ }^{\mu} \delta \mathcal{L}_{\mathrm{M}} / \delta h_{a}{ }^{\mu}$
$\sigma_{a b}{ }^{c} \equiv \sigma_{a b}{ }^{\mu} b^{c}{ }_{\mu} \equiv b^{c}{ }_{\mu} \delta \mathcal{L}_{\mathrm{M}} / \delta A^{a b}{ }_{\mu}$
$\zeta^{a} \equiv \zeta^{\mu} b^{a}{ }_{\mu} \equiv b^{a}{ }_{\mu} \delta \mathcal{L}_{\mathrm{M}} / \delta B_{\mu}$
$\tau^{\dagger a}{ }_{b} \equiv \tau^{a}{ }_{b}+2\left(\sigma_{c b}{ }^{a} \mathcal{V}^{c}-\sigma^{a c}{ }_{c} \mathcal{V}_{b}\right)$
$\zeta^{\dagger a} \equiv \zeta^{a}-2 s^{a b}{ }_{b}$

columns: $L$

Table p. None.6 — 44×2, 0 spanning cell(s)
Gravitational gauge field variables
$h_a^$	translational gauge field ($h$-field)
$h$	determinant of $h_a^$
$b^a_$	inverse $h$-field
$A^ab_$	rotational gauge field ($A$-field)
$B_$	WGT dilational gauge field ($B$-field)
$V_$	eWGT dilational gauge field ($V$-field)
$A^ ab_ A^ab_ + 2^c[ab^b]_ h_c^ V_$	eWGT extended rotational gauge field ($A^$-field)
Derivative operators
$D_a h_a^ D_ h_a^(_ + 12A^ab__ab)$	PGT (generalised) covariant derivative[1]
$^_ _ + wB_$	WGT `augmented' partial derivative
$D^_a h_a^ D^_ h_a^(^_ + 12A^ab__ab)$	WGT (generalised) covariant derivative[1]
$^_ _ - w(V_ +13T_)$	eWGT `augmented' partial derivative
$D^_a h_a^ D^_ h_a^(^_ + 12A^\,ab__ab)$	eWGT (generalised) covariant derivative[1]
$D^_a h_a^ D^_ h_a^(_ + 12A^\,ab__ab)$	eWGT (generalised) semi-covariant derivative[1]
Gauge field strengths
$R^ab_cd = R^ ab_cd = 2h_a^ h_b^(_[ A^ab_]+A^a_c[A^cb_])$	PGT and WGT rotational gauge field strength[1],[2]
$R^\,ab_cd = 2h_a^ h_b^(_[ A^\,ab_]+A^\,a_c[A^\,cb_])$	eWGT rotational gauge field strength[1],[2]
$T^a_bc = 2h_b^ h_c^ D_[ b^a_]$	PGT translational gauge field strength[2]
$ T^ a_bc = 2h_b^ h_c^ D^_[ b^a_]$	WGT translational gauge field strength[2]
$ T^ a_bc = 2h_b^ h_c^ D^_[ b^a_]$	eWGT translational gauge field strength[2]
$ T^ a_bc = 2h_b^ h_c^ D^_[ b^a_]$	eWGT translational gauge `semi' field strength[1],[2]
$ H_ab = 2h_a^ h_b^_[ B_]$	WGT dilational gauge field strength
$ H^_ab = 2h_a^ h_b^_[ (V_+13T_])$	eWGT dilational gauge field strength
Reduced $A$-fields and related quantities
$c^a_bc 2h_b^ h_c^_[ b^a_]$	PGT Ricci rotation coefficients
$c^ a_bc 2h_b^ h_c^^_[ b^a_]$	WGT Ricci rotation coefficients
$c^ a_bc 2h_b^ h_c^^_[ b^a_]$	eWGT Ricci rotation coefficients
$^0A^ab_ 12b^c_(c^ab_c + c^b_c^a - c_c^ab)$	PGT reduced $A$-field (when $ T^a_bc 0$)
$^0A^ ab_ 12b^c_(c^ ab_c + c^ b_c^a - c^_c^ab)$	WGT reduced $A$-field (when $ T^ a_bc 0$)
$^0A^ ab_ 12b^c_(c^ ab_c + c^ b_c^a - c^_c^ab)$	eWGT reduced $A^$-field (when $ T^ a_bc 0$)
Currents and related quantities
$L$	Lagrangian
$ L h^-1L$	Lagrangian density
$t^a_b t^a_ h_b^ h_b^ L_ G/ h_a^$	gravitational sector energy-momentum tensor[3]
$s_ab^c s_ab^ b^c_ b^c_ L_ G/ A^ab_$	gravitational sector spin-angular-momentum tensor[3]
$j^a j^ b^a_ b^a_ L_ G/ B_$	gravitational sector dilation current[3]
$t^ a_b t^a_b + 2(s_cb^a V^c-s^ac_c V_b)$	gravitational sector eWGT covariant energy-momentum tensor[3]
$j^ a j^a-2s^ab_b$	gravitational sector eWGT covariant dilation current[3]
$^a_b ^a_ h_b^ h_b^ L_ M/ h_a^$	matter sector energy-momentum tensor[3]
$_ab^c _ab^ b^c_ b^c_ L_ M/ A^ab_$	matter sector spin-angular-momentum tensor[3]
$^a ^ b^a_ b^a_ L_ M/ B_$	matter sector dilation current[3]
$^ a_b ^a_b + 2(_cb^a V^c-^ac_c V_b)$	gravitational sector eWGT covariant energy-momentum tensor[3]
$^ a ^a-2s^ab_b$	gravitational sector eWGT covariant dilation current[3]

columns: Gravitational gauge field variables |

Gravitational gauge field variables
\begin{tabular}[t]{\|l\|l\|} \hline \(h_{a}{ }^{\mu}\) & translational gauge field ( \(h\)-field) \\ \hline \(h\) & determinant of \(h_{a}{ }^{\mu}\) \\ \hline \(b^{a}{ }_{\mu}\) & inverse \(h\)-field \\ \hline \(A^{a b}{ }_{\mu}\) & rotational gauge field ( \(A\)-field) \\ \hline \(B_{\mu}\) & WGT dilational gauge field ( \(B\)-field) \\ \hline \(V_{\mu}\) & eWGT dilational gauge field ( \(V\)-field) \\ \hline \(A^{\dagger a b}{ }_{\mu} \equiv A^{a b}{ }_{\mu}+2 \eta^{c[a} b^{b]}{ }_{\mu} h_{c}{ }^{\nu} V_{\nu}\) & eWGT extended rotational gauge field ( \(A^{\dagger}\)-field) \\ \hline \end{tabular}
Derivative operators \begin{tabular}[t]{\|l\|} \hline \(\mathcal{D}_{a} \equiv h_{a}{ }^{\mu} D_{\mu} \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}+\frac{1}{2} A^{a b}{ }_{\mu} \Sigma_{a b}\right)\) \\ \hline \(\partial_{\mu}^{} \equiv \partial_{\mu}+w B_{\mu}\) \\ \hline \(\mathcal{D}_{a}^{} \equiv h_{a}{ }^{\mu} D_{\mu}^{} \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}^{}+\frac{1}{2} A^{a b}{ }_{\mu} \Sigma_{a b}\right)\) \\ \hline \(\partial_{\mu}^{\dagger} \equiv \partial_{\mu}-w\left(V_{\mu}+\frac{1}{3} T_{\mu}\right)\) \\ \hline \(\mathcal{D}_{a}^{\dagger} \equiv h_{a}{ }^{\mu} D_{\mu}^{\dagger} \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}^{\dagger}+\frac{1}{2} A^{\dagger a b}{ }_{\mu} \Sigma_{a b}\right)\) \\ \hline \(\mathcal{D}_{a}^{\natural} \equiv h_{a}{ }^{\mu} D_{\mu}^{\natural} \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}+\frac{1}{2} A^{\dagger a b}{ }_{\mu} \Sigma_{a b}\right)\) \\ \hline \end{tabular}	PGT (generalised) covariant derivative \({ }^{\mathrm{a}}\) WGT 'augmented' partial derivative WGT (generalised) covariant derivative \({ }^{\mathrm{a}}\) eWGT 'augmented' partial derivative eWGT (generalised) covariant derivative \({ }^{\mathrm{a}}\) eWGT (generalised) semi-covariant derivative \({ }^{\mathrm{a}}\)
Gauge field strengths \begin{tabular}[t]{\|l\|} \hline \(\mathcal{R}^{a b}{ }_{c d}=\mathcal{R}^{* a b}{ }_{c d}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu}\left(\partial_{[\mu} A^{a b}{ }_{\nu]}+A^{a}{ }_{c[\mu} A^{c b}{ }_{\nu]}\right) \mathcal{R}^{\dagger a b}{ }_{c d}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu}\left(\partial_{[\mu} A^{\dagger a b}{ }_{\nu]}+A^{\dagger a}{ }_{c[\mu} A^{\dagger c b}{ }_{\nu]}\right) \mathcal{T}^{a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu} b^{a}{ }_{\nu]} \mathcal{T}^{* a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu}^{*} b^{a}{ }_{\nu]} \mathcal{T}^{\dagger a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu}^{\dagger} b^{a}{ }_{\nu]} \mathcal{T}^{\natural a}{ }_{b c}=2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} D_{[\mu}^{\natural} b^{a}{ }_{\nu]} \mathcal{H}_{a b}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu} \partial_{[\mu} B_{\nu]} \mathcal{H}_{a b}^{\dagger}=2 h_{a}{ }^{\mu} h_{b}{ }^{\nu} \partial_{[\mu}\left(V_{\nu}+\frac{1}{3} T_{\nu]}\right)\) \\ \hline \end{tabular}	PGT and WGT rotational gauge field strength \({ }^{\mathrm{a}}\) eWGT rotational gauge field strength \({ }^{\mathrm{a}}{ }^{\mathrm{b}}\) PGT translational gauge field strength \({ }^{\text {b }}\) WGT translational gauge field strength \({ }^{\mathrm{b}}\) eWGT translational gauge field strength \({ }^{\mathrm{b}}\) eWGT translational gauge 'semi' field strengtha \({ }^{\mathrm{b}}\) WGT dilational gauge field strength eWGT dilational gauge field strength
Reduced \(A\)-fields and related quantities \begin{tabular}[t]{\|l\|} \hline \(c^{a}{ }_{b c} \equiv 2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} \partial_{[\mu} b^{a}{ }_{\nu]}\) \\ \hline \(c^{* a}{ }_{b c} \equiv 2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} \partial_{[\mu}^{} b^{a}{ }_{\nu]}\) \\ \hline \(c^{\dagger a}{ }_{b c} \equiv 2 h_{b}{ }^{\mu} h_{c}{ }^{\nu} \partial_{[\mu}^{\dagger} b^{a}{ }_{\nu]}\) \\ \hline \({ }^{0} A^{a b}{ }_{\mu} \equiv \frac{1}{2} b^{c}{ }_{\mu}\left(c^{a b}{ }_{c}+c^{b}{ }_{c}{ }^{a}-c_{c}{ }^{a b}\right)\) \\ \hline \({ }^{0} A^{ a b}{ }_{\mu} \equiv \frac{1}{2} b^{c}{ }_{\mu}\left(c^{* a b}{ }_{c}+c^{* b}{ }_{c}{ }^{a}-c_{c}^{* a b}\right)\) \\ \hline \({ }^{0} A^{\dagger a b}{ }_{\mu} \equiv \frac{1}{2} b^{c}{ }_{\mu}\left(c^{\dagger a b}{ }_{c}+c^{\dagger b}{ }_{c}{ }^{a}-c_{c}^{\dagger a b}\right)\) \\ \hline \end{tabular}	PGT Ricci rotation coefficients WGT Ricci rotation coefficients eWGT Ricci rotation coefficients PGT reduced \(A\)-field (when \(\mathcal{T}^{a}{ }_{b c} \equiv 0\) ) WGT reduced \(A\)-field (when \(\mathcal{T}^{* a}{ }_{b c} \equiv 0\) ) eWGT reduced \(A^{\dagger}\)-field (when \(\mathcal{T}^{\dagger a}{ }_{b c} \equiv 0\) )
Currents and related quantities \begin{tabular}[t]{\|l\|} \hline \(L\) \\ \hline \(\mathcal{L} \equiv h^{-1} L\) \\ \hline \(t^{a}{ }_{b} \equiv t^{a}{ }_{\mu} h_{b}{ }^{\mu} \equiv h_{b}{ }^{\mu} \delta \mathcal{L}_{\mathrm{G}} / \delta h_{a}{ }^{\mu}\) \\ \hline \(s_{a b}{ }^{c} \equiv s_{a b}{ }^{\mu} b^{c}{ }_{\mu} \equiv b^{c}{ }_{\mu} \delta \mathcal{L}_{\mathrm{G}} / \delta A^{a b}{ }_{\mu}\) \\ \hline \(j^{a} \equiv j^{\mu} b^{a}{ }_{\mu} \equiv b^{a}{ }_{\mu} \delta \mathcal{L}_{\mathrm{G}} / \delta B_{\mu}\) \\ \hline \(t^{\dagger a}{ }_{b} \equiv t^{a}{ }_{b}+2\left(s_{c b}{ }^{a} \mathcal{V}^{c}-s^{a c}{ }_{c} \mathcal{V}_{b}\right)\) \\ \hline \(j^{\dagger a} \equiv j^{a}-2 s^{a b}{ }_{b}\) \\ \hline \(\tau^{a}{ }_{b} \equiv \tau^{a}{ }_{\mu} h_{b}{ }^{\mu} \equiv h_{b}{ }^{\mu} \delta \mathcal{L}_{\mathrm{M}} / \delta h_{a}{ }^{\mu}\) \\ \hline \(\sigma_{a b}{ }^{c} \equiv \sigma_{a b}{ }^{\mu} b^{c}{ }_{\mu} \equiv b^{c}{ }_{\mu} \delta \mathcal{L}_{\mathrm{M}} / \delta A^{a b}{ }_{\mu}\) \\ \hline \(\zeta^{a} \equiv \zeta^{\mu} b^{a}{ }_{\mu} \equiv b^{a}{ }_{\mu} \delta \mathcal{L}_{\mathrm{M}} / \delta B_{\mu}\) \\ \hline \(\tau^{\dagger a}{ }_{b} \equiv \tau^{a}{ }_{b}+2\left(\sigma_{c b}{ }^{a} \mathcal{V}^{c}-\sigma^{a c}{ }_{c} \mathcal{V}_{b}\right)\) \\ \hline \(\zeta^{\dagger a} \equiv \zeta^{a}-2 s^{a b}{ }_{b}\) \\ \hline \end{tabular}	Lagrangian Lagrangian density gravitational sector energy-momentum tensor \({ }^{\mathrm{c}}\) gravitational sector spin-angular-momentum tensor \({ }^{\mathrm{c}}\) gravitational sector dilation current \({ }^{\mathrm{c}}\) gravitational sector eWGT covariant energy-momentum tensor \({ }^{\mathrm{c}}\) gravitational sector eWGT covariant dilation current \({ }^{\text {c }}\) matter sector energy-momentum tensor \({ }^{\mathrm{c}}\) matter sector spin-angular-momentum tensor \({ }^{\mathrm{c}}\) matter sector dilation current \({ }^{\text {c }}\) gravitational sector eWGT covariant energy-momentum tensor \({ }^{\mathrm{c}}\) gravitational sector eWGT covariant dilation current \({ }^{\mathrm{c}}\)

\(h_{a}{ }^{\mu}\)	translational gauge field ( \(h\)-field)
\(h\)	determinant of \(h_{a}{ }^{\mu}\)
\(b^{a}{ }_{\mu}\)	inverse \(h\)-field
\(A^{a b}{ }_{\mu}\)	rotational gauge field ( \(A\)-field)
\(B_{\mu}\)	WGT dilational gauge field ( \(B\)-field)
\(V_{\mu}\)	eWGT dilational gauge field ( \(V\)-field)
\(A^{\dagger a b}{ }_{\mu} \equiv A^{a b}{ }_{\mu}+2 \eta^{c[a} b^{b]}{ }_{\mu} h_{c}{ }^{\nu} V_{\nu}\)	eWGT extended rotational gauge field ( \(A^{\dagger}\)-field)