metric:
${{{ g} _a} _b} = {{{{ \eta} _a} _b}{-{{{2}} {{\Phi}} \cdot {{{{ δ} _a} _b}}}}}$

metric inverse:
${{{ g} ^a} ^b} = {{{\frac{1}{{1}{-{{{4}} {{{\Phi}^{2}}}}}}}} {{\left({{{{ \eta} ^a} ^b} + {{{2}} {{\Phi}} \cdot {{{{ δ} ^a} ^b}}}}\right)}}}$
${{{{ g} ^a} ^c}} {{{{ g} _c} _b}}$
$\frac{{{{{{ \eta} ^a} ^c}} {{{{ \eta} _c} _b}}}{-{{{2}} {{\Phi}} \cdot {{{{ \eta} ^a} ^c}} {{{{ δ} _c} _b}}}} + {{{2}} {{\Phi}} \cdot {{{{ \eta} _c} _b}} {{{{ δ} ^a} ^c}}}{-{{{4}} {{{{ δ} ^a} ^c}} {{{{ δ} _c} _b}} {{{\Phi}^{2}}}}}}{{1}{-{{{4}} {{{\Phi}^{2}}}}}}$
$\frac{{{{ δ} ^a} _b}{-{{{2}} {{\Phi}} \cdot {{{{ \eta} ^a} ^c}} {{{{ δ} _c} _b}}}}{-{{{4}} {{{{ δ} ^a} ^c}} {{{{ δ} _c} _b}} {{{\Phi}^{2}}}}}}{{1}{-{{{4}} {{{\Phi}^{2}}}}}}$

metric derivative:
${{{{ g} _a} _b} _{,c}} = {{{{{ \eta} _a} _b} _{,c}}{-{{{2}} {{\Phi}} \cdot {{{{{ δ} _a} _b} _{,c}}}}}{-{{{2}} {{{ \Phi} _{,c}}} {{{{ δ} _a} _b}}}}}$
${{{{ g} _a} _b} _{,c}} = {-{{{2}} {{{ \Phi} _{,c}}} {{{{ δ} _a} _b}}}}$
connections:
${{{{ \Gamma} _a} _b} _c} = {{{\frac{1}{2}}} {{\left({{{{{ g} _a} _b} _{,c}} + {{{{ g} _a} _c} _{,b}}{-{{{{ g} _b} _c} _{,a}}}}\right)}}}$
${{{{ \Gamma} _a} _b} _c} = {{{{{ \Phi} _{,a}}} {{{{ δ} _b} _c}}}{-{{{{ \Phi} _{,c}}} {{{{ δ} _a} _b}}}}{-{{{{ \Phi} _{,b}}} {{{{ δ} _a} _c}}}}}$
${{{{ \Gamma} ^a} _b} _c} = {{{{{ g} ^a} ^d}} {{{{{ \Gamma} _d} _b} _c}}}$
${{{{ \Gamma} ^a} _b} _c} = {\frac{{{{{ \Phi} _{,d}}} {{{{ \eta} ^a} ^d}} {{{{ δ} _b} _c}}}{-{{{{ \Phi} _{,b}}} {{{{ \eta} ^a} ^d}} {{{{ δ} _d} _c}}}}{-{{{{ \Phi} _{,c}}} {{{{ \eta} ^a} ^d}} {{{{ δ} _d} _b}}}}{-{{{2}} {{\Phi}} \cdot {{{ \Phi} _{,c}}} {{{{ δ} ^a} ^d}} {{{{ δ} _d} _b}}}} + {{{2}} {{\Phi}} \cdot {{{ \Phi} _{,d}}} {{{{ δ} ^a} ^d}} {{{{ δ} _b} _c}}}{-{{{2}} {{\Phi}} \cdot {{{ \Phi} _{,b}}} {{{{ δ} ^a} ^d}} {{{{ δ} _d} _c}}}}}{{1}{-{{{4}} {{{\Phi}^{2}}}}}}}$
let ${\Phi} = {0}$ , but keep ${ \Phi} _{,a}$ to find:
${{{{ \Gamma} ^a} _b} _c} = {{{{{ \eta} ^a} ^d}} {{\left({{{{{ \Phi} _{,d}}} {{{{ δ} _b} _c}}}{-{{{{ \Phi} _{,b}}} {{{{ δ} _d} _c}}}}{-{{{{ \Phi} _{,c}}} {{{{ δ} _d} _b}}}}}\right)}}}$

covariant derivative of velocity:
${{{ u} ^a} _{;b}} = {{{{ u} ^a} _{,b}} + {{{{{{ \Gamma} ^a} _c} _b}} {{{ u} ^c}}}}$
${{{ u} ^a} _{;b}} = {{{{ u} ^a} _{,b}}{-{{{{ \Phi} _{,b}}} {{{ u} ^c}} {{{{ \eta} ^a} ^d}} {{{{ δ} _d} _c}}}}{-{{{{ \Phi} _{,c}}} {{{ u} ^c}} {{{{ \eta} ^a} ^d}} {{{{ δ} _d} _b}}}} + {{{{ \Phi} _{,d}}} {{{ u} ^c}} {{{{ \eta} ^a} ^d}} {{{{ δ} _c} _b}}}}$
${{{ u} ^i} _{;t}} = {{{{ u} ^i} _{,t}}{-{{{{ \Phi} _{,t}}} {{{ u} ^c}} {{{{ \eta} ^i} ^d}} {{{{ δ} _d} _c}}}}{-{{{{ \Phi} _{,c}}} {{{ u} ^c}} {{{{ \eta} ^i} ^d}} {{{{ δ} _d} _t}}}} + {{{{ \Phi} _{,d}}} {{{ u} ^c}} {{{{ \eta} ^i} ^d}} {{{{ δ} _c} _t}}}}$
${{{ u} ^i} _{;j}} = {{{{ u} ^i} _{,j}}{-{{{{ \Phi} _{,j}}} {{{ u} ^c}} {{{{ \eta} ^i} ^d}} {{{{ δ} _d} _c}}}}{-{{{{ \Phi} _{,c}}} {{{ u} ^c}} {{{{ \eta} ^i} ^d}} {{{{ δ} _d} _j}}}} + {{{{ \Phi} _{,d}}} {{{ u} ^c}} {{{{ \eta} ^i} ^d}} {{{{ δ} _c} _j}}}}$

gravitational acceleration:
${-{{{{ \Gamma} ^a} _b} _c}} {{{ u} ^b}} {{{ u} ^c}}$
${-{{{{{ \eta} ^a} ^d}} {{\left({{{{{ \Phi} _{,d}}} {{{{ δ} _b} _c}}}{-{{{{ \Phi} _{,b}}} {{{{ δ} _d} _c}}}}{-{{{{ \Phi} _{,c}}} {{{{ δ} _d} _b}}}}}\right)}}}} {{{ u} ^b}} {{{ u} ^c}}$
${{{-1}} {{{ \Phi} _{,d}}} {{{ u} ^b}} {{{ u} ^c}} {{{{ \eta} ^a} ^d}} {{{{ δ} _b} _c}}} + {{{{ \Phi} _{,b}}} {{{ u} ^b}} {{{ u} ^c}} {{{{ \eta} ^a} ^d}} {{{{ δ} _d} _c}}} + {{{{ \Phi} _{,c}}} {{{ u} ^b}} {{{ u} ^c}} {{{{ \eta} ^a} ^d}} {{{{ δ} _d} _b}}}$

stress-energy:
${{{ T} ^a} ^b} = {{{{\left({{\rho} + {P}}\right)}} {{{ u} ^a}} {{{ u} ^b}}} + {{{P}} {{{{ g} ^a} ^b}}}}$
divergence-free...
${{{{ T} ^a} ^b} _{;b}} = {{{{P}} {{{{{ g} ^a} ^b} _{;b}}}} + {{{{ P} _{;b}}} {{{{ g} ^a} ^b}}} + {{{P}} {{{ u} ^a}} {{{{ u} ^b} _{;b}}}} + {{{P}} {{{ u} ^b}} {{{{ u} ^a} _{;b}}}} + {{{\rho}} \cdot {{{ u} ^a}} {{{{ u} ^b} _{;b}}}} + {{{\rho}} \cdot {{{ u} ^b}} {{{{ u} ^a} _{;b}}}} + {{{{ P} _{;b}}} {{{ u} ^a}} {{{ u} ^b}}} + {{{{ \rho} _{;b}}} {{{ u} ^a}} {{{ u} ^b}}}}$
substitute...
${{{{ T} ^a} ^b} _{;b}} = {{{{{ P} _{,b}}} {{{{ \eta} ^a} ^b}}} + {{{P}} {{{ u} ^b}} {{{{ u} ^a} _{;b}}}} + {{{P}} {{{ u} ^a}} {{{{ u} ^b} _{;b}}}} + {{{\rho}} \cdot {{{ u} ^b}} {{{{ u} ^a} _{;b}}}} + {{{\rho}} \cdot {{{ u} ^a}} {{{{ u} ^b} _{;b}}}} + {{{{ P} _{,b}}} {{{ u} ^a}} {{{ u} ^b}}} + {{{{ \rho} _{,b}}} {{{ u} ^a}} {{{ u} ^b}}}}$
separate b index into t and j:
${{{{ T} ^a} ^b} _{;b}} = {{{{{ P} _{,t}}} {{{{ \eta} ^a} ^t}}} + {{{P}} {{{ u} ^t}} {{{{ u} ^a} _{;t}}}} + {{{P}} {{{ u} ^a}} {{{{ u} ^t} _{;t}}}} + {{{\rho}} \cdot {{{ u} ^t}} {{{{ u} ^a} _{;t}}}} + {{{\rho}} \cdot {{{ u} ^a}} {{{{ u} ^t} _{;t}}}} + {{{{ P} _{,t}}} {{{ u} ^a}} {{{ u} ^t}}} + {{{{ \rho} _{,t}}} {{{ u} ^a}} {{{ u} ^t}}} + {{{{ P} _{,j}}} {{{{ \eta} ^a} ^j}}} + {{{P}} {{{ u} ^j}} {{{{ u} ^a} _{;j}}}} + {{{P}} {{{ u} ^a}} {{{{ u} ^j} _{;j}}}} + {{{\rho}} \cdot {{{ u} ^j}} {{{{ u} ^a} _{;j}}}} + {{{\rho}} \cdot {{{ u} ^a}} {{{{ u} ^j} _{;j}}}} + {{{{ P} _{,j}}} {{{ u} ^a}} {{{ u} ^j}}} + {{{{ \rho} _{,j}}} {{{ u} ^a}} {{{ u} ^j}}}}$
look at t component of a:
${{{{ T} ^t} ^b} _{;b}} = {{{{{ P} _{,t}}} {{{{ \eta} ^t} ^t}}} + {{{P}} {{{ u} ^t}} {{{{ u} ^t} _{;t}}}} + {{{P}} {{{ u} ^t}} {{{{ u} ^t} _{;t}}}} + {{{\rho}} \cdot {{{ u} ^t}} {{{{ u} ^t} _{;t}}}} + {{{\rho}} \cdot {{{ u} ^t}} {{{{ u} ^t} _{;t}}}} + {{{{ P} _{,t}}} {{{ u} ^t}} {{{ u} ^t}}} + {{{{ \rho} _{,t}}} {{{ u} ^t}} {{{ u} ^t}}} + {{{{ P} _{,j}}} {{{{ \eta} ^t} ^j}}} + {{{P}} {{{ u} ^j}} {{{{ u} ^t} _{;j}}}} + {{{P}} {{{ u} ^t}} {{{{ u} ^j} _{;j}}}} + {{{\rho}} \cdot {{{ u} ^j}} {{{{ u} ^t} _{;j}}}} + {{{\rho}} \cdot {{{ u} ^t}} {{{{ u} ^j} _{;j}}}} + {{{{ P} _{,j}}} {{{ u} ^t}} {{{ u} ^j}}} + {{{{ \rho} _{,j}}} {{{ u} ^t}} {{{ u} ^j}}}}$
substitute...
${{{{ T} ^t} ^b} _{;b}} = {{{ \rho} _{,t}} + {{{P}} {{{{ u} ^j} _{,j}}}} + {{{\rho}} \cdot {{{{ u} ^j} _{,j}}}} + {{{{ P} _{,j}}} {{{ u} ^j}}} + {{{{ \rho} _{,j}}} {{{ u} ^j}}} + {{{P}} {{{ \Phi} _{,d}}} {{{ u} ^c}} {{{{ \eta} ^j} ^d}} {{{{ δ} _c} _j}}}{-{{{P}} {{{ \Phi} _{,c}}} {{{ u} ^c}} {{{{ \eta} ^j} ^d}} {{{{ δ} _d} _j}}}}{-{{{P}} {{{ \Phi} _{,j}}} {{{ u} ^c}} {{{{ \eta} ^j} ^d}} {{{{ δ} _d} _c}}}}{-{{{\rho}} \cdot {{{ \Phi} _{,c}}} {{{ u} ^c}} {{{{ \eta} ^j} ^d}} {{{{ δ} _d} _j}}}} + {{{\rho}} \cdot {{{ \Phi} _{,d}}} {{{ u} ^c}} {{{{ \eta} ^j} ^d}} {{{{ δ} _c} _j}}}{-{{{\rho}} \cdot {{{ \Phi} _{,j}}} {{{ u} ^c}} {{{{ \eta} ^j} ^d}} {{{{ δ} _d} _c}}}}}$

look at i component of a:
${{{{ T} ^i} ^b} _{;b}} = {{{{{ P} _{,t}}} {{{{ \eta} ^i} ^t}}} + {{{P}} {{{ u} ^t}} {{{{ u} ^i} _{;t}}}} + {{{P}} {{{ u} ^i}} {{{{ u} ^t} _{;t}}}} + {{{\rho}} \cdot {{{ u} ^t}} {{{{ u} ^i} _{;t}}}} + {{{\rho}} \cdot {{{ u} ^i}} {{{{ u} ^t} _{;t}}}} + {{{{ P} _{,t}}} {{{ u} ^i}} {{{ u} ^t}}} + {{{{ \rho} _{,t}}} {{{ u} ^i}} {{{ u} ^t}}} + {{{{ P} _{,j}}} {{{{ \eta} ^i} ^j}}} + {{{P}} {{{ u} ^j}} {{{{ u} ^i} _{;j}}}} + {{{P}} {{{ u} ^i}} {{{{ u} ^j} _{;j}}}} + {{{\rho}} \cdot {{{ u} ^j}} {{{{ u} ^i} _{;j}}}} + {{{\rho}} \cdot {{{ u} ^i}} {{{{ u} ^j} _{;j}}}} + {{{{ P} _{,j}}} {{{ u} ^i}} {{{ u} ^j}}} + {{{{ \rho} _{,j}}} {{{ u} ^i}} {{{ u} ^j}}}}$
substitute...
${{{{ T} ^i} ^b} _{;b}} = {{{{P}} {{{{ u} ^i} _{;t}}}} + {{{\rho}} \cdot {{{{ u} ^i} _{;t}}}} + {{{{ P} _{,t}}} {{{ u} ^i}}} + {{{{ P} _{,j}}} {{{{ \eta} ^i} ^j}}} + {{{{ \rho} _{,t}}} {{{ u} ^i}}} + {{{P}} {{{ u} ^i}} {{{{ u} ^t} _{;t}}}} + {{{P}} {{{ u} ^j}} {{{{ u} ^i} _{;j}}}} + {{{P}} {{{ u} ^i}} {{{{ u} ^j} _{;j}}}} + {{{\rho}} \cdot {{{ u} ^i}} {{{{ u} ^t} _{;t}}}} + {{{\rho}} \cdot {{{ u} ^j}} {{{{ u} ^i} _{;j}}}} + {{{\rho}} \cdot {{{ u} ^i}} {{{{ u} ^j} _{;j}}}} + {{{{ P} _{,j}}} {{{ u} ^i}} {{{ u} ^j}}} + {{{{ \rho} _{,j}}} {{{ u} ^i}} {{{ u} ^j}}}}$