${{{ \Gamma} ^t} _t} _t$
factor out index raise
${{{{ g} ^t} ^a}} {{{{{ \Gamma} _a} _t} _t}}$
substitute definition of connection
${\frac{1}{2}} {{{{{ g} ^t} ^a}} {{\left({{-{{{{ g} _t} _t} _{,a}}} + {{{2}} {{{{{ g} _a} _t} _{,t}}}}}\right)}}}$
split the index a into t and j
${\frac{1}{2}}{\left({{-{{{{{ g} ^t} ^j}} {{{{{ g} _t} _t} _{,j}}}}} + {{{{{ g} ^t} ^t}} {{{{{ g} _t} _t} _{,t}}}} + {{{2}} {{{{ g} ^t} ^j}} {{{{{ g} _j} _t} _{,t}}}}}\right)}$
replace ADM metric definitions
${\frac{1}{2}}{\left({{-{{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{\left( {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} _k} _l}}}\right)} _{,j}}}}} + {{{{\left( {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} _k} _l}}}\right)} _{,t}}} {{\frac{-1}{{\alpha}^{2}}}}} + {{{2}} \cdot {{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{ { \beta} _j} _{,t}}}}}\right)}$
raise $\beta_j$
${\frac{1}{2}}{\left({{-{{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{\left( {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} _k} _l}}}\right)} _{,j}}}}} + {{{{\left( {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} _k} _l}}}\right)} _{,t}}} {{\frac{-1}{{\alpha}^{2}}}}} + {{{2}} \cdot {{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{\left( {{{{ \gamma} _j} _k}} {{{ \beta} ^k}}\right)} _{,t}}}}}\right)}$
simplify...
${{{{ \alpha} _{,t}}} {{\frac{1}{\alpha}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,t}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{{ \gamma} _k} _l} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,t}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{{ \alpha} _{,j}}} {{{ \beta} _j}} {{\frac{1}{\alpha}}}} + {{{{ \beta} _j}} {{{ \beta} ^k}} {{{{{ \gamma} _j} _k} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{{ \beta} _j}} {{{{ \beta} ^k} _{,t}}} {{{{ \gamma} _j} _k}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _j}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _j}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{{ \gamma} _k} _l} _{,j}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _j}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,j}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}}$
relabel...
${{{{ \alpha} _{,t}}} {{\frac{1}{\alpha}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{{ \gamma} _k} _l} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,t}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,t}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{{ \alpha} _{,j}}} {{{ \beta} _j}} {{\frac{1}{\alpha}}}} + {{{{ \beta} _j}} {{{ \beta} ^k}} {{{{{ \gamma} _j} _k} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{{ \beta} _j}} {{{{ \beta} ^k} _{,t}}} {{{{ \gamma} _j} _k}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _j}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{{ \gamma} _k} _l} _{,j}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _j}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _j}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,j}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}}$
for ${{ \beta} ^i} = {0}$
${\frac{1}{\alpha}} {{ \alpha} _{,t}}$
${{{ \Gamma} ^t} _t} _i$
factor out index raise
${{{{ g} ^t} ^a}} {{{{{ \Gamma} _a} _t} _i}}$
substitute definition of connection
${\frac{1}{2}} {{{{{ g} ^t} ^a}} {{\left({{{{{ g} _a} _t} _{,i}} + {{{{ g} _a} _i} _{,t}}{-{{{{ g} _t} _i} _{,a}}}}\right)}}}$
split the index a into t and j
${\frac{1}{2}}{\left({{{{{{ g} ^t} ^j}} {{{{{ g} _j} _t} _{,i}}}} + {{{{{ g} ^t} ^j}} {{{{{ g} _j} _i} _{,t}}}}{-{{{{{ g} ^t} ^j}} {{{{{ g} _t} _i} _{,j}}}}} + {{{{{ g} ^t} ^t}} {{{{{ g} _t} _t} _{,i}}}}}\right)}$
replace ADM metric definitions
${\frac{1}{2}}{\left({{{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{ { \beta} _j} _{,i}}}} + {{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{ {{ \gamma} _j} _i} _{,t}}}}{-{{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{ {{ \gamma} _t} _i} _{,j}}}}} + {{{{\left( {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} _k} _l}}}\right)} _{,i}}} {{\frac{-1}{{\alpha}^{2}}}}}}\right)}$
raise $\beta_j$
${\frac{1}{2}}{\left({{{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{\left( {{{{ \gamma} _j} _k}} {{{ \beta} ^k}}\right)} _{,i}}}} + {{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{ {{ \gamma} _j} _i} _{,t}}}}{-{{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{ {{ \gamma} _t} _i} _{,j}}}}} + {{{{\left( {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} _k} _l}}}\right)} _{,i}}} {{\frac{-1}{{\alpha}^{2}}}}}}\right)}$
simplify...
${{{\frac{1}{2}}} {{{ \beta} _j}} {{{{{ \gamma} _j} _i} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _j}} {{{{{ \gamma} _t} _i} _{,j}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{{ \alpha} _{,i}}} {{\frac{1}{\alpha}}}} + {{{\frac{1}{2}}} {{{ \beta} _j}} {{{ \beta} ^k}} {{{{{ \gamma} _j} _k} _{,i}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{{ \gamma} _k} _l} _{,i}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,i}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _j}} {{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _j} _k}} {{\frac{1}{{\alpha}^{2}}}}}$
relabel...
${{{\frac{1}{2}}} {{{ \beta} _j}} {{{{{ \gamma} _j} _i} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _j}} {{{{{ \gamma} _t} _i} _{,j}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{{ \alpha} _{,i}}} {{\frac{1}{\alpha}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,i}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _j}} {{{ \beta} ^k}} {{{{{ \gamma} _j} _k} _{,i}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _j}} {{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _j} _k}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{{ \gamma} _k} _l} _{,i}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,i}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}}$
for ${{ \beta} ^i} = {0}$
${\frac{1}{\alpha}} {{ \alpha} _{,i}}$
${{{ \Gamma} ^i} _t} _t$
factor out index raise
${{{{ g} ^i} ^a}} {{{{{ \Gamma} _a} _t} _t}}$
substitute definition of connection
${\frac{1}{2}} {{{{{ g} ^i} ^a}} {{\left({{-{{{{ g} _t} _t} _{,a}}} + {{{2}} {{{{{ g} _a} _t} _{,t}}}}}\right)}}}$
split the index a into t and j
${\frac{1}{2}}{\left({{-{{{{{ g} ^i} ^j}} {{{{{ g} _t} _t} _{,j}}}}} + {{{{{ g} ^i} ^t}} {{{{{ g} _t} _t} _{,t}}}} + {{{2}} {{{{ g} ^i} ^j}} {{{{{ g} _j} _t} _{,t}}}}}\right)}$
replace ADM metric definitions
${\frac{1}{2}}{\left({{-{{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{\left( {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} _k} _l}}}\right)} _{,j}}}}} + {{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{\left( {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} _k} _l}}}\right)} _{,t}}}} + {{{2}} {{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{ { \beta} _j} _{,t}}}}}\right)}$
raise $\beta_j$
${\frac{1}{2}}{\left({{-{{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{\left( {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} _k} _l}}}\right)} _{,j}}}}} + {{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{\left( {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} _k} _l}}}\right)} _{,t}}}} + {{{2}} {{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{\left( {{{{ \gamma} _j} _k}} {{{ \beta} ^k}}\right)} _{,t}}}}}\right)}$
simplify...
${{{-1}} {{{ \alpha} _{,t}}} {{{ \beta} _i}} {{\frac{1}{\alpha}}}} + {{{\alpha}} \cdot {{{ \alpha} _{,j}}} {{{{ \gamma} ^i} ^j}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,t}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,t}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{{ \gamma} _k} _l} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} {{{ \alpha} _{,j}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{\frac{1}{\alpha}}}} + {{{-1}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{ \beta} ^k} _{,t}}} {{{{ \gamma} _j} _k}} {{\frac{1}{{\alpha}^{2}}}}} + {{{{ \beta} ^k}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _j} _k} _{,t}}}} + {{{-1}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{ \beta} ^k}} {{{{{ \gamma} _j} _k} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{{{ \beta} ^k} _{,t}}} {{{{ \gamma} ^i} ^j}} {{{{ \gamma} _j} _k}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,j}}} {{{{ \gamma} ^i} ^j}} {{{{ \gamma} _k} _l}}} + {{{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _k} _l} _{,j}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,j}}} {{{{ \gamma} ^i} ^j}} {{{{ \gamma} _k} _l}}} + {{{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{{ \gamma} _k} _l} _{,j}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,j}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}}$
relabel...
${{{-1}} {{{ \alpha} _{,t}}} {{{ \beta} _i}} {{\frac{1}{\alpha}}}} + {{{\alpha}} \cdot {{{ \alpha} _{,j}}} {{{{ \gamma} ^i} ^j}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,t}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,t}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{{ \gamma} _k} _l} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} {{{ \alpha} _{,j}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{\frac{1}{\alpha}}}} + {{{{ \beta} ^k}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _j} _k} _{,t}}}} + {{{-1}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{ \beta} ^k}} {{{{{ \gamma} _j} _k} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{ \beta} ^k} _{,t}}} {{{{ \gamma} _j} _k}} {{\frac{1}{{\alpha}^{2}}}}} + {{{{{ \beta} ^k} _{,t}}} {{{{ \gamma} ^i} ^j}} {{{{ \gamma} _j} _k}}} + {{{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,j}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _k} _l} _{,j}}}} + {{{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{{ \gamma} _k} _l} _{,j}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,j}}} {{{{ \gamma} ^i} ^j}} {{{{ \gamma} _k} _l}}} + {{{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,j}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,j}}} {{{{ \gamma} ^i} ^j}} {{{{ \gamma} _k} _l}}}$
for ${{ \beta} ^i} = {0}$
${{\alpha}} \cdot {{{ \alpha} _{,j}}} {{{{ \gamma} ^i} ^j}}$
${{{ \Gamma} ^i} _m} _t$
factor out index raise
${{{{ g} ^i} ^a}} {{{{{ \Gamma} _a} _m} _t}}$
substitute definition of connection
${\frac{1}{2}} {{{{{ g} ^i} ^a}} {{\left({{{{{ g} _a} _m} _{,t}} + {{{{ g} _a} _t} _{,m}}{-{{{{ g} _m} _t} _{,a}}}}\right)}}}$
split the index a into t and j
${\frac{1}{2}}{\left({{{{{{ g} ^i} ^j}} {{{{{ g} _j} _m} _{,t}}}}{-{{{{{ g} ^i} ^j}} {{{{{ g} _m} _t} _{,j}}}}} + {{{{{ g} ^i} ^j}} {{{{{ g} _j} _t} _{,m}}}} + {{{{{ g} ^i} ^t}} {{{{{ g} _t} _t} _{,m}}}} + {{{{{ g} ^i} ^t}} {{{{{ g} _t} _m} _{,t}}}}{-{{{{{ g} ^i} ^t}} {{{{{ g} _m} _t} _{,t}}}}}}\right)}$
replace ADM metric definitions
${\frac{1}{2}}{\left({{{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{ {{ \gamma} _j} _m} _{,t}}}}{-{{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{ {{ \gamma} _m} _t} _{,j}}}}} + {{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{ { \beta} _j} _{,m}}}} + {{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{\left( {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} _k} _l}}}\right)} _{,m}}}} + {{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{ {{ \gamma} _t} _m} _{,t}}}}{-{{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{ {{ \gamma} _m} _t} _{,t}}}}}}\right)}$
raise $\beta_j$
${\frac{1}{2}}{\left({{{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{ {{ \gamma} _j} _m} _{,t}}}}{-{{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{ {{ \gamma} _m} _t} _{,j}}}}} + {{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{\left( {{{{ \gamma} _j} _k}} {{{ \beta} ^k}}\right)} _{,m}}}} + {{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{\left( {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} ^l}} {{{{ \gamma} _k} _l}}}\right)} _{,m}}}} + {{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{ {{ \gamma} _t} _m} _{,t}}}}{-{{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{ {{ \gamma} _m} _t} _{,t}}}}}}\right)}$
simplify...
${{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _i}} {{{{{ \gamma} _m} _t} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{{{ \gamma} _t} _m} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{{ \gamma} _m} _t} _{,j}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{{ \gamma} _j} _m} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _j} _m} _{,t}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _m} _t} _{,j}}}} + {{{-1}} {{{ \alpha} _{,m}}} {{{ \beta} _i}} {{\frac{1}{\alpha}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{{ \gamma} _k} _l} _{,m}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{ \beta} ^k}} {{{{{ \gamma} _j} _k} _{,m}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{ \beta} ^k} _{,m}}} {{{{ \gamma} _j} _k}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,m}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,m}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} ^k}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _j} _k} _{,m}}}} + {{{\frac{1}{2}}} {{{{ \beta} ^k} _{,m}}} {{{{ \gamma} ^i} ^j}} {{{{ \gamma} _j} _k}}}$
relabel...
${{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _i}} {{{{{ \gamma} _m} _t} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{{{ \gamma} _t} _m} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{{ \gamma} _j} _m} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{{ \gamma} _m} _t} _{,j}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _j} _m} _{,t}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _m} _t} _{,j}}}} + {{{-1}} {{{ \alpha} _{,m}}} {{{ \beta} _i}} {{\frac{1}{\alpha}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{ \beta} ^l}} {{{{ \beta} ^k} _{,m}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} ^k}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _j} _k} _{,m}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{ \beta} ^k}} {{{{ \beta} ^l} _{,m}}} {{{{ \gamma} _k} _l}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{ \beta} ^k}} {{{{{ \gamma} _j} _k} _{,m}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{ \beta} ^k} _{,m}}} {{{{ \gamma} _j} _k}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{ \beta} ^k}} {{{ \beta} ^l}} {{{{{ \gamma} _k} _l} _{,m}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{{ \beta} ^k} _{,m}}} {{{{ \gamma} ^i} ^j}} {{{{ \gamma} _j} _k}}}$
for ${{ \beta} ^i} = {0}$
${\frac{1}{2}} {{{{{ \gamma} ^i} ^j}} {{\left({{-{{{{ \gamma} _m} _t} _{,j}}} + {{{{ \gamma} _j} _m} _{,t}}}\right)}}}$
${{{ \Gamma} ^t} _i} _m$
factor out index raise
${{{{ g} ^t} ^a}} {{{{{ \Gamma} _a} _i} _m}}$
substitute definition of connection
${\frac{1}{2}} {{{{{ g} ^t} ^a}} {{\left({{{{{ g} _a} _i} _{,m}} + {{{{ g} _a} _m} _{,i}}{-{{{{ g} _i} _m} _{,a}}}}\right)}}}$
split the index a into t and j
${\frac{1}{2}}{\left({{{{{{ g} ^t} ^j}} {{{{{ g} _j} _i} _{,m}}}}{-{{{{{ g} ^t} ^j}} {{{{{ g} _i} _m} _{,j}}}}} + {{{{{ g} ^t} ^j}} {{{{{ g} _j} _m} _{,i}}}} + {{{{{ g} ^t} ^t}} {{{{{ g} _t} _m} _{,i}}}} + {{{{{ g} ^t} ^t}} {{{{{ g} _t} _i} _{,m}}}}{-{{{{{ g} ^t} ^t}} {{{{{ g} _i} _m} _{,t}}}}}}\right)}$
replace ADM metric definitions
${\frac{1}{2}}{\left({{{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{ {{ \gamma} _j} _i} _{,m}}}}{-{{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{ {{ \gamma} _i} _m} _{,j}}}}} + {{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{ {{ \gamma} _j} _m} _{,i}}}} + {{{{ {{ \gamma} _t} _m} _{,i}}} {{\frac{-1}{{\alpha}^{2}}}}} + {{{{ {{ \gamma} _t} _i} _{,m}}} {{\frac{-1}{{\alpha}^{2}}}}}{-{{{{ {{ \gamma} _i} _m} _{,t}}} {{\frac{-1}{{\alpha}^{2}}}}}}}\right)}$
raise $\beta_j$
${\frac{1}{2}}{\left({{{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{ {{ \gamma} _j} _i} _{,m}}}}{-{{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{ {{ \gamma} _i} _m} _{,j}}}}} + {{{\frac{{ \beta} _j}{{\alpha}^{2}}}} {{{ {{ \gamma} _j} _m} _{,i}}}} + {{{{ {{ \gamma} _t} _m} _{,i}}} {{\frac{-1}{{\alpha}^{2}}}}} + {{{{ {{ \gamma} _t} _i} _{,m}}} {{\frac{-1}{{\alpha}^{2}}}}}{-{{{{ {{ \gamma} _i} _m} _{,t}}} {{\frac{-1}{{\alpha}^{2}}}}}}}\right)}$
simplify...
${{{\frac{1}{2}}} {{{{{ \gamma} _i} _m} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{{ \gamma} _t} _i} _{,m}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{{ \gamma} _t} _m} _{,i}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _j}} {{{{{ \gamma} _j} _m} _{,i}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _j}} {{{{{ \gamma} _i} _m} _{,j}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _j}} {{{{{ \gamma} _j} _i} _{,m}}} {{\frac{1}{{\alpha}^{2}}}}}$
relabel...
${{{\frac{1}{2}}} {{{{{ \gamma} _i} _m} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{{ \gamma} _t} _m} _{,i}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{{ \gamma} _t} _i} _{,m}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _j}} {{{{{ \gamma} _j} _m} _{,i}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _j}} {{{{{ \gamma} _i} _m} _{,j}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _j}} {{{{{ \gamma} _j} _i} _{,m}}} {{\frac{1}{{\alpha}^{2}}}}}$
for ${{ \beta} ^i} = {0}$
$\frac{{-{{{{ \gamma} _t} _i} _{,m}}}{-{{{{ \gamma} _t} _m} _{,i}}} + {{{{ \gamma} _i} _m} _{,t}}}{{{2}} {{{\alpha}^{2}}}}$
${{{ \Gamma} ^i} _m} _n$
factor out index raise
${{{{ g} ^i} ^a}} {{{{{ \Gamma} _a} _m} _n}}$
substitute definition of connection
${\frac{1}{2}} {{{{{ g} ^i} ^a}} {{\left({{{{{ g} _a} _m} _{,n}} + {{{{ g} _a} _n} _{,m}}{-{{{{ g} _m} _n} _{,a}}}}\right)}}}$
split the index a into t and j
${\frac{1}{2}}{\left({{{{{{ g} ^i} ^j}} {{{{{ g} _j} _m} _{,n}}}}{-{{{{{ g} ^i} ^j}} {{{{{ g} _m} _n} _{,j}}}}} + {{{{{ g} ^i} ^j}} {{{{{ g} _j} _n} _{,m}}}} + {{{{{ g} ^i} ^t}} {{{{{ g} _t} _n} _{,m}}}} + {{{{{ g} ^i} ^t}} {{{{{ g} _t} _m} _{,n}}}}{-{{{{{ g} ^i} ^t}} {{{{{ g} _m} _n} _{,t}}}}}}\right)}$
replace ADM metric definitions
${\frac{1}{2}}{\left({{{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{ {{ \gamma} _j} _m} _{,n}}}}{-{{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{ {{ \gamma} _m} _n} _{,j}}}}} + {{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{ {{ \gamma} _j} _n} _{,m}}}} + {{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{ {{ \gamma} _t} _n} _{,m}}}} + {{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{ {{ \gamma} _t} _m} _{,n}}}}{-{{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{ {{ \gamma} _m} _n} _{,t}}}}}}\right)}$
raise $\beta_j$
${\frac{1}{2}}{\left({{{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{ {{ \gamma} _j} _m} _{,n}}}}{-{{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{ {{ \gamma} _m} _n} _{,j}}}}} + {{{\left({{{{ \gamma} ^i} ^j}{-{\frac{{{{ \beta} ^i}} {{{ \beta} ^j}}}{{\alpha}^{2}}}}}\right)}} {{{ {{ \gamma} _j} _n} _{,m}}}} + {{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{ {{ \gamma} _t} _n} _{,m}}}} + {{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{ {{ \gamma} _t} _m} _{,n}}}}{-{{{\frac{{ \beta} _i}{{\alpha}^{2}}}} {{{ {{ \gamma} _m} _n} _{,t}}}}}}\right)}$
simplify...
${{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _i}} {{{{{ \gamma} _m} _n} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{{{ \gamma} _t} _n} _{,m}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{{{ \gamma} _t} _m} _{,n}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{{ \gamma} _m} _n} _{,j}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{{ \gamma} _j} _n} _{,m}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{{ \gamma} _j} _m} _{,n}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _j} _m} _{,n}}}} + {{{\frac{1}{2}}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _j} _n} _{,m}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _m} _n} _{,j}}}}$
relabel...
${{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} _i}} {{{{{ \gamma} _m} _n} _{,t}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{{{ \gamma} _t} _m} _{,n}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} _i}} {{{{{ \gamma} _t} _n} _{,m}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{{ \gamma} _m} _n} _{,j}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{{ \gamma} _j} _n} _{,m}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{{ \gamma} _j} _m} _{,n}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\frac{1}{2}}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _j} _m} _{,n}}}} + {{{\frac{1}{2}}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _j} _n} _{,m}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \gamma} ^i} ^j}} {{{{{ \gamma} _m} _n} _{,j}}}}$
for ${{ \beta} ^i} = {0}$
${\frac{1}{2}} {{{{{ \gamma} ^i} ^j}} {{\left({{-{{{{ \gamma} _m} _n} _{,j}}} + {{{{ \gamma} _j} _n} _{,m}} + {{{{ \gamma} _j} _m} _{,n}}}\right)}}}$
${{{{ R} ^t} _t} _t} _i$
substitute definition of Riemann curvature
${{ {{{ \Gamma} ^t} _t} _i} _{,t}}{-{{ {{{ \Gamma} ^t} _t} _t} _{,i}}} + {{{{{{ \Gamma} ^t} _e} _t}} {{{{{ \Gamma} ^e} _t} _i}}}{-{{{{{{ \Gamma} ^t} _e} _i}} {{{{{ \Gamma} ^e} _t} _t}}}}$
split the index e into t and m
${{ {{{ \Gamma} ^t} _t} _i} _{,t}}{-{{ {{{ \Gamma} ^t} _t} _t} _{,i}}} + {{{\left({{{{{ \Gamma} ^t} _t} _t} + {{{{ \Gamma} ^t} _m} _t}}\right)}} {{\left({{{{{ \Gamma} ^t} _t} _i} + {{{{ \Gamma} ^m} _t} _i}}\right)}}}{-{{{\left({{{{{ \Gamma} ^t} _t} _i} + {{{{ \Gamma} ^t} _n} _i}}\right)}} {{\left({{{{{ \Gamma} ^t} _t} _t} + {{{{ \Gamma} ^n} _t} _t}}\right)}}}}$
${{{{{ \Gamma} ^t} _t} _i} _{,t}}{-{{{{{ \Gamma} ^t} _t} _t} _{,i}}} + {{{{{{ \Gamma} ^t} _m} _t}} {{{{{ \Gamma} ^t} _t} _i}}} + {{{{{{ \Gamma} ^m} _t} _i}} {{{{{ \Gamma} ^t} _t} _t}}} + {{{{{{ \Gamma} ^m} _t} _i}} {{{{{ \Gamma} ^t} _m} _t}}}{-{{{{{{ \Gamma} ^t} _n} _i}} {{{{{ \Gamma} ^t} _t} _t}}}}{-{{{{{{ \Gamma} ^n} _t} _t}} {{{{{ \Gamma} ^t} _t} _i}}}}{-{{{{{{ \Gamma} ^n} _t} _t}} {{{{{ \Gamma} ^t} _n} _i}}}}$
${{{{ R} ^t} _t} _i} _j$
substitute definition of Riemann curvature
${{ {{{ \Gamma} ^t} _t} _j} _{,i}}{-{{ {{{ \Gamma} ^t} _t} _i} _{,j}}} + {{{{{{ \Gamma} ^t} _e} _i}} {{{{{ \Gamma} ^e} _t} _j}}}{-{{{{{{ \Gamma} ^t} _e} _j}} {{{{{ \Gamma} ^e} _t} _i}}}}$
split the index e into t and m
${{ {{{ \Gamma} ^t} _t} _j} _{,i}}{-{{ {{{ \Gamma} ^t} _t} _i} _{,j}}} + {{{\left({{{{{ \Gamma} ^t} _t} _i} + {{{{ \Gamma} ^t} _m} _i}}\right)}} {{\left({{{{{ \Gamma} ^t} _t} _j} + {{{{ \Gamma} ^m} _t} _j}}\right)}}}{-{{{\left({{{{{ \Gamma} ^t} _t} _j} + {{{{ \Gamma} ^t} _n} _j}}\right)}} {{\left({{{{{ \Gamma} ^t} _t} _i} + {{{{ \Gamma} ^n} _t} _i}}\right)}}}}$
${{{{{ \Gamma} ^t} _t} _j} _{,i}}{-{{{{{ \Gamma} ^t} _t} _i} _{,j}}} + {{{{{{ \Gamma} ^t} _m} _i}} {{{{{ \Gamma} ^t} _t} _j}}} + {{{{{{ \Gamma} ^m} _t} _j}} {{{{{ \Gamma} ^t} _t} _i}}} + {{{{{{ \Gamma} ^m} _t} _j}} {{{{{ \Gamma} ^t} _m} _i}}}{-{{{{{{ \Gamma} ^t} _n} _j}} {{{{{ \Gamma} ^t} _t} _i}}}}{-{{{{{{ \Gamma} ^n} _t} _i}} {{{{{ \Gamma} ^t} _t} _j}}}}{-{{{{{{ \Gamma} ^n} _t} _i}} {{{{{ \Gamma} ^t} _n} _j}}}}$
${{{{ R} ^t} _i} _t} _j$
substitute definition of Riemann curvature
${{ {{{ \Gamma} ^t} _i} _j} _{,t}}{-{{ {{{ \Gamma} ^t} _i} _t} _{,j}}} + {{{{{{ \Gamma} ^t} _e} _t}} {{{{{ \Gamma} ^e} _i} _j}}}{-{{{{{{ \Gamma} ^t} _e} _j}} {{{{{ \Gamma} ^e} _i} _t}}}}$
split the index e into t and m
${{ {{{ \Gamma} ^t} _i} _j} _{,t}}{-{{ {{{ \Gamma} ^t} _i} _t} _{,j}}} + {{{\left({{{{{ \Gamma} ^t} _t} _t} + {{{{ \Gamma} ^t} _m} _t}}\right)}} {{\left({{{{{ \Gamma} ^t} _i} _j} + {{{{ \Gamma} ^m} _i} _j}}\right)}}}{-{{{\left({{{{{ \Gamma} ^t} _t} _j} + {{{{ \Gamma} ^t} _n} _j}}\right)}} {{\left({{{{{ \Gamma} ^t} _i} _t} + {{{{ \Gamma} ^n} _i} _t}}\right)}}}}$
${{{{{ \Gamma} ^t} _i} _j} _{,t}}{-{{{{{ \Gamma} ^t} _i} _t} _{,j}}} + {{{{{{ \Gamma} ^t} _i} _j}} {{{{{ \Gamma} ^t} _m} _t}}}{-{{{{{{ \Gamma} ^n} _i} _t}} {{{{{ \Gamma} ^t} _t} _j}}}} + {{{{{{ \Gamma} ^m} _i} _j}} {{{{{ \Gamma} ^t} _t} _t}}} + {{{{{{ \Gamma} ^m} _i} _j}} {{{{{ \Gamma} ^t} _m} _t}}}{-{{{{{{ \Gamma} ^t} _i} _t}} {{{{{ \Gamma} ^t} _n} _j}}}} + {{{{{{ \Gamma} ^t} _i} _j}} {{{{{ \Gamma} ^t} _t} _t}}}{-{{{{{{ \Gamma} ^t} _i} _t}} {{{{{ \Gamma} ^t} _t} _j}}}}{-{{{{{{ \Gamma} ^n} _i} _t}} {{{{{ \Gamma} ^t} _n} _j}}}}$
${{{{ R} ^i} _t} _t} _j$
substitute definition of Riemann curvature
${{ {{{ \Gamma} ^i} _t} _j} _{,t}}{-{{ {{{ \Gamma} ^i} _t} _t} _{,j}}} + {{{{{{ \Gamma} ^i} _e} _t}} {{{{{ \Gamma} ^e} _t} _j}}}{-{{{{{{ \Gamma} ^i} _e} _j}} {{{{{ \Gamma} ^e} _t} _t}}}}$
split the index e into t and m
${{ {{{ \Gamma} ^i} _t} _j} _{,t}}{-{{ {{{ \Gamma} ^i} _t} _t} _{,j}}} + {{{\left({{{{{ \Gamma} ^i} _t} _t} + {{{{ \Gamma} ^i} _m} _t}}\right)}} {{\left({{{{{ \Gamma} ^t} _t} _j} + {{{{ \Gamma} ^m} _t} _j}}\right)}}}{-{{{\left({{{{{ \Gamma} ^i} _t} _j} + {{{{ \Gamma} ^i} _n} _j}}\right)}} {{\left({{{{{ \Gamma} ^t} _t} _t} + {{{{ \Gamma} ^n} _t} _t}}\right)}}}}$
${{{{{ \Gamma} ^i} _t} _j} _{,t}}{-{{{{{ \Gamma} ^i} _t} _t} _{,j}}} + {{{{{{ \Gamma} ^i} _m} _t}} {{{{{ \Gamma} ^t} _t} _j}}}{-{{{{{{ \Gamma} ^i} _t} _j}} {{{{{ \Gamma} ^n} _t} _t}}}} + {{{{{{ \Gamma} ^i} _t} _t}} {{{{{ \Gamma} ^m} _t} _j}}} + {{{{{{ \Gamma} ^i} _m} _t}} {{{{{ \Gamma} ^m} _t} _j}}}{-{{{{{{ \Gamma} ^i} _n} _j}} {{{{{ \Gamma} ^t} _t} _t}}}} + {{{{{{ \Gamma} ^i} _t} _t}} {{{{{ \Gamma} ^t} _t} _j}}}{-{{{{{{ \Gamma} ^i} _t} _j}} {{{{{ \Gamma} ^t} _t} _t}}}}{-{{{{{{ \Gamma} ^i} _n} _j}} {{{{{ \Gamma} ^n} _t} _t}}}}$
${{{{ R} ^t} _i} _j} _k$
substitute definition of Riemann curvature
${{ {{{ \Gamma} ^t} _i} _k} _{,j}}{-{{ {{{ \Gamma} ^t} _i} _j} _{,k}}} + {{{{{{ \Gamma} ^t} _e} _j}} {{{{{ \Gamma} ^e} _i} _k}}}{-{{{{{{ \Gamma} ^t} _e} _k}} {{{{{ \Gamma} ^e} _i} _j}}}}$
split the index e into t and m
${{ {{{ \Gamma} ^t} _i} _k} _{,j}}{-{{ {{{ \Gamma} ^t} _i} _j} _{,k}}} + {{{\left({{{{{ \Gamma} ^t} _t} _j} + {{{{ \Gamma} ^t} _m} _j}}\right)}} {{\left({{{{{ \Gamma} ^t} _i} _k} + {{{{ \Gamma} ^m} _i} _k}}\right)}}}{-{{{\left({{{{{ \Gamma} ^t} _t} _k} + {{{{ \Gamma} ^t} _n} _k}}\right)}} {{\left({{{{{ \Gamma} ^t} _i} _j} + {{{{ \Gamma} ^n} _i} _j}}\right)}}}}$
${{{{{ \Gamma} ^t} _i} _k} _{,j}}{-{{{{{ \Gamma} ^t} _i} _j} _{,k}}} + {{{{{{ \Gamma} ^t} _i} _k}} {{{{{ \Gamma} ^t} _m} _j}}}{-{{{{{{ \Gamma} ^n} _i} _j}} {{{{{ \Gamma} ^t} _t} _k}}}} + {{{{{{ \Gamma} ^m} _i} _k}} {{{{{ \Gamma} ^t} _t} _j}}} + {{{{{{ \Gamma} ^m} _i} _k}} {{{{{ \Gamma} ^t} _m} _j}}}{-{{{{{{ \Gamma} ^t} _i} _j}} {{{{{ \Gamma} ^t} _n} _k}}}} + {{{{{{ \Gamma} ^t} _i} _k}} {{{{{ \Gamma} ^t} _t} _j}}}{-{{{{{{ \Gamma} ^t} _i} _j}} {{{{{ \Gamma} ^t} _t} _k}}}}{-{{{{{{ \Gamma} ^n} _i} _j}} {{{{{ \Gamma} ^t} _n} _k}}}}$
${{{{ R} ^i} _t} _j} _k$
substitute definition of Riemann curvature
${{ {{{ \Gamma} ^i} _t} _k} _{,j}}{-{{ {{{ \Gamma} ^i} _t} _j} _{,k}}} + {{{{{{ \Gamma} ^i} _e} _j}} {{{{{ \Gamma} ^e} _t} _k}}}{-{{{{{{ \Gamma} ^i} _e} _k}} {{{{{ \Gamma} ^e} _t} _j}}}}$
split the index e into t and m
${{ {{{ \Gamma} ^i} _t} _k} _{,j}}{-{{ {{{ \Gamma} ^i} _t} _j} _{,k}}} + {{{\left({{{{{ \Gamma} ^i} _t} _j} + {{{{ \Gamma} ^i} _m} _j}}\right)}} {{\left({{{{{ \Gamma} ^t} _t} _k} + {{{{ \Gamma} ^m} _t} _k}}\right)}}}{-{{{\left({{{{{ \Gamma} ^i} _t} _k} + {{{{ \Gamma} ^i} _n} _k}}\right)}} {{\left({{{{{ \Gamma} ^t} _t} _j} + {{{{ \Gamma} ^n} _t} _j}}\right)}}}}$
${{{{{ \Gamma} ^i} _t} _k} _{,j}}{-{{{{{ \Gamma} ^i} _t} _j} _{,k}}} + {{{{{{ \Gamma} ^i} _m} _j}} {{{{{ \Gamma} ^t} _t} _k}}}{-{{{{{{ \Gamma} ^i} _t} _k}} {{{{{ \Gamma} ^n} _t} _j}}}} + {{{{{{ \Gamma} ^i} _t} _j}} {{{{{ \Gamma} ^m} _t} _k}}} + {{{{{{ \Gamma} ^i} _m} _j}} {{{{{ \Gamma} ^m} _t} _k}}}{-{{{{{{ \Gamma} ^i} _n} _k}} {{{{{ \Gamma} ^t} _t} _j}}}} + {{{{{{ \Gamma} ^i} _t} _j}} {{{{{ \Gamma} ^t} _t} _k}}}{-{{{{{{ \Gamma} ^i} _t} _k}} {{{{{ \Gamma} ^t} _t} _j}}}}{-{{{{{{ \Gamma} ^i} _n} _k}} {{{{{ \Gamma} ^n} _t} _j}}}}$
${{{{ R} ^i} _j} _t} _k$
substitute definition of Riemann curvature
${{ {{{ \Gamma} ^i} _j} _k} _{,t}}{-{{ {{{ \Gamma} ^i} _j} _t} _{,k}}} + {{{{{{ \Gamma} ^i} _e} _t}} {{{{{ \Gamma} ^e} _j} _k}}}{-{{{{{{ \Gamma} ^i} _e} _k}} {{{{{ \Gamma} ^e} _j} _t}}}}$
split the index e into t and m
${{ {{{ \Gamma} ^i} _j} _k} _{,t}}{-{{ {{{ \Gamma} ^i} _j} _t} _{,k}}} + {{{\left({{{{{ \Gamma} ^i} _t} _t} + {{{{ \Gamma} ^i} _m} _t}}\right)}} {{\left({{{{{ \Gamma} ^t} _j} _k} + {{{{ \Gamma} ^m} _j} _k}}\right)}}}{-{{{\left({{{{{ \Gamma} ^i} _t} _k} + {{{{ \Gamma} ^i} _n} _k}}\right)}} {{\left({{{{{ \Gamma} ^t} _j} _t} + {{{{ \Gamma} ^n} _j} _t}}\right)}}}}$
${{{{{ \Gamma} ^i} _j} _k} _{,t}}{-{{{{{ \Gamma} ^i} _j} _t} _{,k}}} + {{{{{{ \Gamma} ^i} _m} _t}} {{{{{ \Gamma} ^t} _j} _k}}}{-{{{{{{ \Gamma} ^i} _t} _k}} {{{{{ \Gamma} ^n} _j} _t}}}} + {{{{{{ \Gamma} ^i} _t} _t}} {{{{{ \Gamma} ^m} _j} _k}}} + {{{{{{ \Gamma} ^i} _m} _t}} {{{{{ \Gamma} ^m} _j} _k}}}{-{{{{{{ \Gamma} ^i} _n} _k}} {{{{{ \Gamma} ^t} _j} _t}}}} + {{{{{{ \Gamma} ^i} _t} _t}} {{{{{ \Gamma} ^t} _j} _k}}}{-{{{{{{ \Gamma} ^i} _t} _k}} {{{{{ \Gamma} ^t} _j} _t}}}}{-{{{{{{ \Gamma} ^i} _n} _k}} {{{{{ \Gamma} ^n} _j} _t}}}}$
${{{{ R} ^i} _j} _k} _l$
substitute definition of Riemann curvature
${{ {{{ \Gamma} ^i} _j} _l} _{,k}}{-{{ {{{ \Gamma} ^i} _j} _k} _{,l}}} + {{{{{{ \Gamma} ^i} _e} _k}} {{{{{ \Gamma} ^e} _j} _l}}}{-{{{{{{ \Gamma} ^i} _e} _l}} {{{{{ \Gamma} ^e} _j} _k}}}}$
split the index e into t and m
${{ {{{ \Gamma} ^i} _j} _l} _{,k}}{-{{ {{{ \Gamma} ^i} _j} _k} _{,l}}} + {{{\left({{{{{ \Gamma} ^i} _t} _k} + {{{{ \Gamma} ^i} _m} _k}}\right)}} {{\left({{{{{ \Gamma} ^t} _j} _l} + {{{{ \Gamma} ^m} _j} _l}}\right)}}}{-{{{\left({{{{{ \Gamma} ^i} _t} _l} + {{{{ \Gamma} ^i} _n} _l}}\right)}} {{\left({{{{{ \Gamma} ^t} _j} _k} + {{{{ \Gamma} ^n} _j} _k}}\right)}}}}$
${{{{{ \Gamma} ^i} _j} _l} _{,k}}{-{{{{{ \Gamma} ^i} _j} _k} _{,l}}} + {{{{{{ \Gamma} ^i} _m} _k}} {{{{{ \Gamma} ^t} _j} _l}}}{-{{{{{{ \Gamma} ^i} _t} _l}} {{{{{ \Gamma} ^n} _j} _k}}}} + {{{{{{ \Gamma} ^i} _t} _k}} {{{{{ \Gamma} ^m} _j} _l}}} + {{{{{{ \Gamma} ^i} _m} _k}} {{{{{ \Gamma} ^m} _j} _l}}}{-{{{{{{ \Gamma} ^i} _n} _l}} {{{{{ \Gamma} ^t} _j} _k}}}} + {{{{{{ \Gamma} ^i} _t} _k}} {{{{{ \Gamma} ^t} _j} _l}}}{-{{{{{{ \Gamma} ^i} _t} _l}} {{{{{ \Gamma} ^t} _j} _k}}}}{-{{{{{{ \Gamma} ^i} _n} _l}} {{{{{ \Gamma} ^n} _j} _k}}}}$