${{{ \eta} _t} _t} = {-1}$; ${{{ \eta} _x} _x} = {1}$; ${{{ \eta} _y} _y} = {1}$; ${{{ \eta} _z} _z} = {1}$

${\rho} = {{{{x}^{2}} + {{y}^{2}} + {{z}^{2}}} - {{a}^{2}}}$
${\partial_ {{x}}\left( \rho\right)} = {{{2}} {{x}}}$
${\partial_ {{y}}\left( \rho\right)} = {{{2}} {{y}}}$
${\partial_ {{z}}\left( \rho\right)} = {{{2}} {{z}}}$

${{\rho_+}} = {\sqrt{{{\rho}^{2}} + {{{4}} {{{a}^{2}}} {{{z}^{2}}}}}}$
${\partial_ {{x}}\left( {\rho_+}\right)} = {{{\rho}} \cdot {{\frac{1}{{\rho_+}}}} {{{{2}} {{x}}}}}$
${\partial_ {{y}}\left( {\rho_+}\right)} = {{{\rho}} \cdot {{\frac{1}{{\rho_+}}}} {{{{2}} {{y}}}}}$
${\partial_ {{z}}\left( {\rho_+}\right)} = {{{\frac{1}{{\rho_+}}}} {{\left({{{{\rho}} \cdot {{{{2}} {{z}}}}} + {{{4}} {{z}} {{{a}^{2}}}}}\right)}}}$

${{r}^{2}} = {{{\frac{1}{2}}} {{\left({{\rho} + {{\rho_+}}}\right)}}}$
${\partial_ {{x}}\left( r\right)} = {\frac{{{x}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}$
${\partial_ {{y}}\left( r\right)} = {\frac{{{y}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}$
${\partial_ {{z}}\left( r\right)} = {\frac{{{z}} {{\left({{\rho} + {{\rho_+}} + {{{2}} {{{a}^{2}}}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}$

${H} = {\frac{{{{r}} {{M}}} - {{\frac{1}{2}} {{Q}^{2}}}}{{{r}^{2}} + {\frac{{{{a}^{2}}} {{{z}^{2}}}}{{r}^{2}}}}}$
${\partial_ {{x}}\left( H\right)} = {\frac{{{x}} {{\left({{{{{3}} {{M}} {{\rho}} \cdot {{{a}^{2}}} {{{r}^{2}}} {{{z}^{2}}}} - {{{M}} {{\rho}} \cdot {{{r}^{6}}}}} + {{{{{3}} {{M}} {{{\rho_+}}} \cdot {{{a}^{2}}} {{{r}^{2}}} {{{z}^{2}}}} - {{{M}} {{{\rho_+}}} \cdot {{{r}^{6}}}}} - {{{\rho}} \cdot {{r}} {{{Q}^{2}}} {{{a}^{2}}} {{{z}^{2}}}}} + {{{{\rho}} \cdot {{{Q}^{2}}} {{{r}^{5}}}} - {{{{\rho_+}}} \cdot {{r}} {{{Q}^{2}}} {{{a}^{2}}} {{{z}^{2}}}}} + {{{{\rho_+}}} \cdot {{{Q}^{2}}} {{{r}^{5}}}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}} {{\left({{{{2}} {{{a}^{2}}} {{{r}^{4}}} {{{z}^{2}}}} + {{r}^{8}} + {{{{a}^{4}}} {{{z}^{4}}}}}\right)}}}}$
${\partial_ {{y}}\left( H\right)} = {\frac{{{y}} {{\left({{{{{3}} {{M}} {{\rho}} \cdot {{{a}^{2}}} {{{r}^{2}}} {{{z}^{2}}}} - {{{M}} {{\rho}} \cdot {{{r}^{6}}}}} + {{{{{3}} {{M}} {{{\rho_+}}} \cdot {{{a}^{2}}} {{{r}^{2}}} {{{z}^{2}}}} - {{{M}} {{{\rho_+}}} \cdot {{{r}^{6}}}}} - {{{\rho}} \cdot {{r}} {{{Q}^{2}}} {{{a}^{2}}} {{{z}^{2}}}}} + {{{{\rho}} \cdot {{{Q}^{2}}} {{{r}^{5}}}} - {{{{\rho_+}}} \cdot {{r}} {{{Q}^{2}}} {{{a}^{2}}} {{{z}^{2}}}}} + {{{{\rho_+}}} \cdot {{{Q}^{2}}} {{{r}^{5}}}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}} {{\left({{{{2}} {{{a}^{2}}} {{{r}^{4}}} {{{z}^{2}}}} + {{r}^{8}} + {{{{a}^{4}}} {{{z}^{4}}}}}\right)}}}}$
${\partial_ {{z}}\left( H\right)} = {\frac{{{z}} {{\left({{{{{3}} {{M}} {{\rho}} \cdot {{r}} {{{a}^{2}}} {{{z}^{2}}}} - {{{M}} {{\rho}} \cdot {{{r}^{5}}}}} + {{{{{3}} {{M}} {{{\rho_+}}} \cdot {{r}} {{{a}^{2}}} {{{z}^{2}}}} - {{{4}} {{M}} {{{\rho_+}}} \cdot {{{a}^{2}}} {{{r}^{3}}}}} - {{{M}} {{{\rho_+}}} \cdot {{{r}^{5}}}}} + {{{{{{6}} {{M}} {{r}} {{{a}^{4}}} {{{z}^{2}}}} - {{{2}} {{M}} {{{a}^{2}}} {{{r}^{5}}}}} - {{{2}} {{{Q}^{2}}} {{{a}^{4}}} {{{z}^{2}}}}} - {{{\rho}} \cdot {{{Q}^{2}}} {{{a}^{2}}} {{{z}^{2}}}}} + {{{{\rho}} \cdot {{{Q}^{2}}} {{{r}^{4}}}} - {{{{\rho_+}}} \cdot {{{Q}^{2}}} {{{a}^{2}}} {{{z}^{2}}}}} + {{{2}} {{{\rho_+}}} \cdot {{{Q}^{2}}} {{{a}^{2}}} {{{r}^{2}}}} + {{{{\rho_+}}} \cdot {{{Q}^{2}}} {{{r}^{4}}}} + {{{2}} {{{Q}^{2}}} {{{a}^{2}}} {{{r}^{4}}}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{\left({{{{{a}^{4}}} {{{z}^{4}}}} + {{{2}} {{{a}^{2}}} {{{r}^{4}}} {{{z}^{2}}}} + {{r}^{8}}}\right)}}}}$

${{ l} _t} = {1}$; ${{ l} _x} = {\frac{{{{r}} {{x}}} + {{{a}} {{y}}}}{{{r}^{2}} + {{a}^{2}}}}$; ${{ l} _y} = {\frac{{{{r}} {{y}}} - {{{a}} {{x}}}}{{{r}^{2}} + {{a}^{2}}}}$; ${{ l} _z} = {{\frac{1}{r}} {z}}$

${{{ l} _x} _{,x}} = {\partial_ {{x}}\left({\frac{{{{r}} {{x}}} + {{{a}} {{y}}}}{{{r}^{2}} + {{a}^{2}}}}\right)}$
= $\frac{{-{{{x}} {{{r}^{2}}} {{\partial_ {{x}}\left( r\right)}}}} + {{{x}} {{{a}^{2}}} {{\partial_ {{x}}\left( r\right)}}} + {{r}^{3}} + {{{{r}} {{{a}^{2}}}} - {{{2}} {{a}} {{r}} {{y}} {{\partial_ {{x}}\left( r\right)}}}}}{{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}$
= $\frac{{-{{{x}} {{{r}^{2}}} {{\frac{{{x}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}}} + {{{x}} {{{a}^{2}}} {{\frac{{{x}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}} + {{r}^{3}} + {{{{r}} {{{a}^{2}}}} - {{{2}} {{a}} {{r}} {{y}} {{\frac{{{x}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}}}}{{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}$
= $\frac{{{-{{{\rho}} \cdot {{{r}^{2}}} {{{x}^{2}}}}} - {{{{\rho_+}}} \cdot {{{r}^{2}}} {{{x}^{2}}}}} + {{{2}} {{{\rho_+}}} \cdot {{{r}^{4}}}} + {{{{{2}} {{{\rho_+}}} \cdot {{{a}^{2}}} {{{r}^{2}}}} - {{{2}} {{\rho}} \cdot {{a}} {{r}} {{x}} {{y}}}} - {{{2}} {{{\rho_+}}} \cdot {{a}} {{r}} {{x}} {{y}}}} + {{{\rho}} \cdot {{{a}^{2}}} {{{x}^{2}}}} + {{{{\rho_+}}} \cdot {{{a}^{2}}} {{{x}^{2}}}}}{{{2}} {{{\rho_+}}} \cdot {{r}} {{\left({{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}\right)}}}$

${{{ l} _x} _{,y}} = {\partial_ {{y}}\left({\frac{{{{r}} {{x}}} + {{{a}} {{y}}}}{{{r}^{2}} + {{a}^{2}}}}\right)}$
= $\frac{{-{{{x}} {{{r}^{2}}} {{\partial_ {{y}}\left( r\right)}}}} + {{{x}} {{{a}^{2}}} {{\partial_ {{y}}\left( r\right)}}} + {{{a}} {{{r}^{2}}}} + {{{a}^{3}} - {{{2}} {{a}} {{r}} {{y}} {{\partial_ {{y}}\left( r\right)}}}}}{{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}$
= $\frac{{-{{{x}} {{{r}^{2}}} {{\frac{{{y}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}}} + {{{x}} {{{a}^{2}}} {{\frac{{{y}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}} + {{{a}} {{{r}^{2}}}} + {{{a}^{3}} - {{{2}} {{a}} {{r}} {{y}} {{\frac{{{y}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}}}}{{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}$
= $\frac{{{-{{{\rho}} \cdot {{x}} {{y}} {{{r}^{2}}}}} - {{{{\rho_+}}} \cdot {{x}} {{y}} {{{r}^{2}}}}} + {{{2}} {{{\rho_+}}} \cdot {{a}} {{{r}^{3}}}} + {{{{{2}} {{{\rho_+}}} \cdot {{r}} {{{a}^{3}}}} - {{{2}} {{\rho}} \cdot {{a}} {{r}} {{{y}^{2}}}}} - {{{2}} {{{\rho_+}}} \cdot {{a}} {{r}} {{{y}^{2}}}}} + {{{\rho}} \cdot {{x}} {{y}} {{{a}^{2}}}} + {{{{\rho_+}}} \cdot {{x}} {{y}} {{{a}^{2}}}}}{{{2}} {{{\rho_+}}} \cdot {{r}} {{\left({{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}\right)}}}$

${{{ l} _x} _{,z}} = {\partial_ {{z}}\left({\frac{{{{r}} {{x}}} + {{{a}} {{y}}}}{{{r}^{2}} + {{a}^{2}}}}\right)}$
= $\frac{{{\partial_ {{z}}\left( r\right)}} {{\left({{-{{{2}} {{a}} {{r}} {{y}}}} + {{{{x}} {{{a}^{2}}}} - {{{x}} {{{r}^{2}}}}}}\right)}}}{{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}$
= $\frac{{{\left({{-{{{2}} {{a}} {{r}} {{y}}}} + {{{{x}} {{{a}^{2}}}} - {{{x}} {{{r}^{2}}}}}}\right)}} {{\frac{{{z}} {{\left({{\rho} + {{\rho_+}} + {{{2}} {{{a}^{2}}}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}}{{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}$
= $\frac{{{z}} {{\left({{-{{{2}} {{\rho}} \cdot {{a}} {{r}} {{y}}}} + {{{{{\rho}} \cdot {{x}} {{{a}^{2}}}} - {{{\rho}} \cdot {{x}} {{{r}^{2}}}}} - {{{2}} {{{\rho_+}}} \cdot {{a}} {{r}} {{y}}}} + {{{{{{{\rho_+}}} \cdot {{x}} {{{a}^{2}}}} - {{{{\rho_+}}} \cdot {{x}} {{{r}^{2}}}}} - {{{4}} {{r}} {{y}} {{{a}^{3}}}}} - {{{2}} {{x}} {{{a}^{2}}} {{{r}^{2}}}}} + {{{2}} {{x}} {{{a}^{4}}}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}} {{\left({{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}\right)}}}$

${{{ l} _y} _{,x}} = {\partial_ {{x}}\left({\frac{{{{r}} {{y}}} - {{{a}} {{x}}}}{{{r}^{2}} + {{a}^{2}}}}\right)}$
= $\frac{{-{{{y}} {{{r}^{2}}} {{\partial_ {{x}}\left( r\right)}}}} + {{{{{y}} {{{a}^{2}}} {{\partial_ {{x}}\left( r\right)}}} - {{{a}} {{{r}^{2}}}}} - {{a}^{3}}} + {{{2}} {{a}} {{r}} {{x}} {{\partial_ {{x}}\left( r\right)}}}}{{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}$
= $\frac{{-{{{y}} {{{r}^{2}}} {{\frac{{{x}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}}} + {{{{{y}} {{{a}^{2}}} {{\frac{{{x}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}} - {{{a}} {{{r}^{2}}}}} - {{a}^{3}}} + {{{2}} {{a}} {{r}} {{x}} {{\frac{{{x}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}}}{{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}$
= $\frac{{{-{{{\rho}} \cdot {{x}} {{y}} {{{r}^{2}}}}} - {{{{\rho_+}}} \cdot {{x}} {{y}} {{{r}^{2}}}}} + {{{2}} {{\rho}} \cdot {{a}} {{r}} {{{x}^{2}}}} + {{{{{2}} {{{\rho_+}}} \cdot {{a}} {{r}} {{{x}^{2}}}} - {{{2}} {{{\rho_+}}} \cdot {{r}} {{{a}^{3}}}}} - {{{2}} {{{\rho_+}}} \cdot {{a}} {{{r}^{3}}}}} + {{{\rho}} \cdot {{x}} {{y}} {{{a}^{2}}}} + {{{{\rho_+}}} \cdot {{x}} {{y}} {{{a}^{2}}}}}{{{2}} {{{\rho_+}}} \cdot {{r}} {{\left({{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}\right)}}}$

${{{ l} _y} _{,y}} = {\partial_ {{y}}\left({\frac{{{{r}} {{y}}} - {{{a}} {{x}}}}{{{r}^{2}} + {{a}^{2}}}}\right)}$
= $\frac{{-{{{y}} {{{r}^{2}}} {{\partial_ {{y}}\left( r\right)}}}} + {{{y}} {{{a}^{2}}} {{\partial_ {{y}}\left( r\right)}}} + {{r}^{3}} + {{{r}} {{{a}^{2}}}} + {{{2}} {{a}} {{r}} {{x}} {{\partial_ {{y}}\left( r\right)}}}}{{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}$
= $\frac{{-{{{y}} {{{r}^{2}}} {{\frac{{{y}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}}} + {{{y}} {{{a}^{2}}} {{\frac{{{y}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}} + {{r}^{3}} + {{{r}} {{{a}^{2}}}} + {{{2}} {{a}} {{r}} {{x}} {{\frac{{{y}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}}}{{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}$
= $\frac{{{-{{{\rho}} \cdot {{{r}^{2}}} {{{y}^{2}}}}} - {{{{\rho_+}}} \cdot {{{r}^{2}}} {{{y}^{2}}}}} + {{{2}} {{{\rho_+}}} \cdot {{{r}^{4}}}} + {{{2}} {{{\rho_+}}} \cdot {{{a}^{2}}} {{{r}^{2}}}} + {{{2}} {{\rho}} \cdot {{a}} {{r}} {{x}} {{y}}} + {{{2}} {{{\rho_+}}} \cdot {{a}} {{r}} {{x}} {{y}}} + {{{\rho}} \cdot {{{a}^{2}}} {{{y}^{2}}}} + {{{{\rho_+}}} \cdot {{{a}^{2}}} {{{y}^{2}}}}}{{{2}} {{{\rho_+}}} \cdot {{r}} {{\left({{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}\right)}}}$

${{{ l} _y} _{,z}} = {\partial_ {{z}}\left({\frac{{{{r}} {{y}}} - {{{a}} {{x}}}}{{{r}^{2}} + {{a}^{2}}}}\right)}$
= $\frac{{{\partial_ {{z}}\left( r\right)}} {{\left({{{{2}} {{a}} {{r}} {{x}}} + {{{{y}} {{{a}^{2}}}} - {{{y}} {{{r}^{2}}}}}}\right)}}}{{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}$
= $\frac{{{\left({{{{2}} {{a}} {{r}} {{x}}} + {{{{y}} {{{a}^{2}}}} - {{{y}} {{{r}^{2}}}}}}\right)}} {{\frac{{{z}} {{\left({{\rho} + {{\rho_+}} + {{{2}} {{{a}^{2}}}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}}{{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}$
= $\frac{{{z}} {{\left({{{{2}} {{\rho}} \cdot {{a}} {{r}} {{x}}} + {{{{\rho}} \cdot {{y}} {{{a}^{2}}}} - {{{\rho}} \cdot {{y}} {{{r}^{2}}}}} + {{{2}} {{{\rho_+}}} \cdot {{a}} {{r}} {{x}}} + {{{{{\rho_+}}} \cdot {{y}} {{{a}^{2}}}} - {{{{\rho_+}}} \cdot {{y}} {{{r}^{2}}}}} + {{{{4}} {{r}} {{x}} {{{a}^{3}}}} - {{{2}} {{y}} {{{a}^{2}}} {{{r}^{2}}}}} + {{{2}} {{y}} {{{a}^{4}}}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}} {{\left({{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{a}^{4}}}\right)}}}$

${{{ l} _z} _{,x}} = {\partial_ {{x}}\left({{\frac{1}{r}} {z}}\right)}$
= $-{\frac{{{z}} {{\partial_ {{x}}\left( r\right)}}}{{r}^{2}}}$
= $-{\frac{{{z}} {{\frac{{{x}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}}{{r}^{2}}}$
= $-{\frac{{{x}} {{z}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{{r}^{3}}}}}$

${{{ l} _z} _{,y}} = {\partial_ {{y}}\left({{\frac{1}{r}} {z}}\right)}$
= $-{\frac{{{z}} {{\partial_ {{y}}\left( r\right)}}}{{r}^{2}}}$
= $-{\frac{{{z}} {{\frac{{{y}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}}{{r}^{2}}}$
= $-{\frac{{{y}} {{z}} {{\left({{\rho} + {{\rho_+}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{{r}^{3}}}}}$

${{{ l} _z} _{,z}} = {\partial_ {{z}}\left({{\frac{1}{r}} {z}}\right)}$
= $\frac{{r} - {{{z}} {{\partial_ {{z}}\left( r\right)}}}}{{r}^{2}}$
= $\frac{{r} - {{{z}} {{\frac{{{z}} {{\left({{\rho} + {{\rho_+}} + {{{2}} {{{a}^{2}}}}}\right)}}}{{{2}} {{{\rho_+}}} \cdot {{r}}}}}}}{{r}^{2}}$
= $\frac{{{{{{2}} {{{\rho_+}}} \cdot {{{r}^{2}}}} - {{{\rho}} \cdot {{{z}^{2}}}}} - {{{{\rho_+}}} \cdot {{{z}^{2}}}}} - {{{2}} {{{a}^{2}}} {{{z}^{2}}}}}{{{2}} {{{\rho_+}}} \cdot {{{r}^{3}}}}$

${{{ g} _u} _v} = {{{{ \eta} _u} _v} + {{{2}} {{H}} {{{ l} _u}} {{{ l} _v}}}}$
${{ {l_*}} ^u} = {{{{{ \eta} ^u} ^v}} {{{ l} _v}}}$
${{{ g} ^u} ^v} = {{{{ \eta} ^u} ^v} - {{{2}} {{H}} {{{ {l_*}} ^u}} {{{ {l_*}} ^v}}}}$
${{{{ g} _u} _v} _{,w}} = {{{2}} {{\left({{{{H}} {{{ l} _u}} {{{{ l} _v} _{,w}}}} + {{{H}} {{{ l} _v}} {{{{ l} _u} _{,w}}}} + {{{{ H} _{,w}}} {{{ l} _u}} {{{ l} _v}}}}\right)}}}$
${{{{ \Gamma} _a} _b} _c} = {{{\frac{1}{2}}} {{\left({{{{{{ g} _a} _b} _{,c}} + {{{{ g} _a} _c} _{,b}}} - {{{{ g} _b} _c} _{,a}}}\right)}}}$
${{{{ \Gamma} _a} _b} _c} = {{{{H}} {{{ l} _a}} {{{{ l} _b} _{,c}}}} + {{{H}} {{{ l} _a}} {{{{ l} _c} _{,b}}}} + {{{{H}} {{{ l} _b}} {{{{ l} _a} _{,c}}}} - {{{H}} {{{ l} _b}} {{{{ l} _c} _{,a}}}}} + {{{{{H}} {{{ l} _c}} {{{{ l} _a} _{,b}}}} - {{{H}} {{{ l} _c}} {{{{ l} _b} _{,a}}}}} - {{{{ H} _{,a}}} {{{ l} _b}} {{{ l} _c}}}} + {{{{ H} _{,b}}} {{{ l} _a}} {{{ l} _c}}} + {{{{ H} _{,c}}} {{{ l} _a}} {{{ l} _b}}}}$
${{ \dot{u}} _a} = { {-{{{{ \Gamma} _a} _b} _c}} {{{ u} ^b}} {{{ u} ^c}}}$
${{ \dot{u}} _a} = {{{{ u} ^b}} {{{ u} ^c}} {{\left({{{{{{{-{{{H}} {{{ l} _a}} {{{{ l} _b} _{,c}}}}} - {{{H}} {{{ l} _a}} {{{{ l} _c} _{,b}}}}} - {{{H}} {{{ l} _b}} {{{{ l} _a} _{,c}}}}} - {{{H}} {{{ l} _c}} {{{{ l} _a} _{,b}}}}} - {{{{ H} _{,b}}} {{{ l} _a}} {{{ l} _c}}}} - {{{{ H} _{,c}}} {{{ l} _a}} {{{ l} _b}}}} + {{{H}} {{{ l} _b}} {{{{ l} _c} _{,a}}}} + {{{H}} {{{ l} _c}} {{{{ l} _b} _{,a}}}} + {{{{ H} _{,a}}} {{{ l} _b}} {{{ l} _c}}}}\right)}}}$