Sources:
2010 Alcubierre, Mendez, "Formulations of the 3+1 evolution equations in curvilinear coordinates"
2017 Escorihuela-Tomàs, Sanchis-Gual, Degollado, Font, "Quasistationary solutions of scalar fields aroundcollapsing self-interacting boson stars"


ADM metric:

${{{ g} _{\mu}} _{\nu}} = {\left[\begin{array}{cc} {-{{\alpha}^{2}}} + {{{{ \beta} ^k}} {{{ \beta} _k}}}& { \beta} _j\\ { \beta} _i& {{ \gamma} _i} _j\end{array}\right]}$

${{{ g} ^{\mu}} ^{\nu}} = {\left[\begin{array}{cc} -{\frac{1}{{\alpha}^{2}}}& {{\frac{1}{{\alpha}^{2}}}} {{{ \beta} ^j}}\\ {{\frac{1}{{\alpha}^{2}}}} {{{ \beta} ^i}}& {{{ \gamma} ^i} ^j}{-{{{\frac{1}{{\alpha}^{2}}}} {{{ \beta} ^i}} {{{ \beta} ^j}}}}\end{array}\right]}$

${{ n} _{\mu}} = {\left[\begin{array}{cc} -{\alpha}& 0\end{array}\right]}$

${{ n} ^{\mu}} = {\left[\begin{array}{cc} \frac{1}{\alpha}& {-{\frac{1}{\alpha}}} {{{ \beta} ^i}}\end{array}\right]}$

Also let $\frac{d}{dx^0} = \partial_0 - \mathcal{L}_\vec\beta$

From 2008 Alcubierre "Introduction to 3+1 Numerical Relativity" Appendix B
${{ \bar{\Gamma}} ^0} = { {-{\frac{1}{{\alpha}^{3}}}} {{\left({{{ \alpha} _{,0}}{-{{{{ \beta} ^m}} {{{ \alpha} _{,m}}}}} + {{{{\alpha}^{2}}} {{K}}}}\right)}}}$
${{ \bar{\Gamma}} ^i} = {{{{ \Gamma} ^i} + {{{\frac{1}{{\alpha}^{3}}}} {{{ \beta} ^i}} {{\left({{{ \alpha} _{,0}}{-{{{{ \beta} ^m}} {{{ \alpha} _{,m}}}}} + {{{{\alpha}^{2}}} {{K}}}}\right)}}}}{-{{{\frac{1}{{\alpha}^{2}}}} {{\left({{{{ \beta} ^i} _{,0}}{-{{{{ \beta} ^m}} {{{{ \beta} ^i} _{,m}}}}} + {{{\alpha}} \cdot {{{ \alpha} _{,j}}} {{{{ \gamma} ^i} ^j}}}}\right)}}}}}$

Let ${\bar{\Gamma}^\alpha}_{\mu\nu}$ is the 4-metric connection.
Let $\bar{\Gamma}^\alpha = {\bar{\Gamma}^\alpha}_{\mu\nu} g^{\mu\nu}$

wave equation in spacetime:
${{{ \Phi} _{;\mu}} ^{;\mu}} = {f}$
${{{{{ g} ^{\mu}} ^{\nu}}} {{{{ \Phi} _{;\mu}} _{;\nu}}}} = {f}$
${{{{{ g} ^{\mu}} ^{\nu}}} {{\left({{{{ \Phi} _{,\mu}} _{,\nu}}{-{{{{{{ \bar{\Gamma}} ^{\alpha}} _{\mu}} _{\nu}}} {{{ \Phi} _{,\alpha}}}}}}\right)}}} = {f}$
${{{{{{ g} ^{\mu}} ^{\nu}}} {{{{ \Phi} _{,\mu}} _{,\nu}}}}{-{{{{ \bar{\Gamma}} ^{\alpha}}} {{{ \Phi} _{,\alpha}}}}}} = {f}$
split space and time:
${{{{{{{{ g} ^0} ^0}} {{{{ \Phi} _{,0}} _{,0}}}} + {{{2}} {{{{ g} ^0} ^i}} {{{{ \Phi} _{,0}} _{,i}}}} + {{{{{ g} ^i} ^j}} {{{{ \Phi} _{,i}} _{,j}}}}}{-{{{{ \bar{\Gamma}} ^0}} {{{ \Phi} _{,0}}}}}}{-{{{{ \bar{\Gamma}} ^i}} {{{ \Phi} _{,i}}}}}} = {f}$
substitute ADM metric components into wave equation:
${{{{-1}} {{{{ \Phi} _{,0}} _{,0}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{2}} {{{ \beta} ^i}} {{{{ \Phi} _{,0}} _{,i}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} {{{ \Phi} _{,0}}} {{{ \bar{\Gamma}} ^0}}} + {{{-1}} {{{ \Phi} _{,i}}} {{{ \bar{\Gamma}} ^i}}} + {{{-1}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{ \Phi} _{,i}} _{,j}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{{{ \Phi} _{,i}} _{,j}}} {{{{ \gamma} ^i} ^j}}}} = {f}$
solve for $\Phi_{,00}$:
${{{ \Phi} _{,0}} _{,0}} = {{-{{{f}} {{{\alpha}^{2}}}}} + {{{2}} {{{ \beta} ^i}} {{{{ \Phi} _{,0}} _{,i}}}}{-{{{{ \Phi} _{,0}}} {{{ \bar{\Gamma}} ^0}} {{{\alpha}^{2}}}}}{-{{{{ \Phi} _{,i}}} {{{ \bar{\Gamma}} ^i}} {{{\alpha}^{2}}}}}{-{{{{ \beta} ^i}} {{{ \beta} ^j}} {{{{ \Phi} _{,i}} _{,j}}}}} + {{{{{ \Phi} _{,i}} _{,j}}} {{{{ \gamma} ^i} ^j}} {{{\alpha}^{2}}}}}$

Let ${\Pi} = {{{{ n} ^{\mu}}} {{{ \Phi} _{,\mu}}}}$
${\Pi} = {{{{{ n} ^0}} {{{ \Phi} _{,0}}}} + {{{{ n} ^i}} {{{ \Phi} _{,i}}}}}$
${\Pi} = {{{{{ \Phi} _{,0}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{{ \Phi} _{,i}}} {{{ \beta} ^i}} {{\frac{1}{\alpha}}}}}$ (eqn. 5)

Let $\Psi_i = \nabla^\perp_i \Phi$ (eqn. 6)
$\Psi_i = {\gamma_i}^\mu \nabla_\mu \Phi$
${{ \Psi} _i} = {{ \Phi} _{,i}}$

Solve $\Pi$ for $\Phi_{,0}$:
${{ \Phi} _{,0}} = {{{{\Pi}} \cdot {{\alpha}}} + {{{{ \Phi} _{,i}}} {{{ \beta} ^i}}}}$ (eqn. 7)
$\frac{d}{dx^0} \Phi = \alpha \Pi$
${{ \Phi} _{,0}} = {{{{\Pi}} \cdot {{\alpha}}} + {{{{ \Psi} _i}} {{{ \beta} ^i}}}}$

Solve $\Pi_{,i}$ for $\Psi_{i,0}$:
${{ \Pi} _{,i}} = {{{{{{ \Phi} _{,0}} _{,i}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{{ \Phi} _{,0}}} {{{ \alpha} _{,i}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} {{{ \Phi} _{,j}}} {{{{ \beta} ^j} _{,i}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{{ \beta} ^j}} {{{{ \Phi} _{,i}} _{,j}}} {{\frac{1}{\alpha}}}} + {{{{ \Phi} _{,j}}} {{{ \alpha} _{,i}}} {{{ \beta} ^j}} {{\frac{1}{{\alpha}^{2}}}}}}$
substitute ${{ \Phi} _{,0}} = {{{{\Pi}} \cdot {{\alpha}}} + {{{{ \Psi} _i}} {{{ \beta} ^i}}}}$ , ${{ \Phi} _{,i}} = {{ \Psi} _i}$ , ${{{ \Phi} _{,0}} _{,i}} = {{{ \Psi} _i} _{,0}}$
${{ \Pi} _{,i}} = {{{{{{ \Psi} _i} _{,0}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{\Pi}} \cdot {{{ \alpha} _{,i}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{{ \Psi} _j}} {{{{ \beta} ^j} _{,i}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{{ \beta} ^j}} {{{{ \Phi} _{,i}} _{,j}}} {{\frac{1}{\alpha}}}}}$
solve for ${{ \Psi} _i} _{,0}$
${{{ \Psi} _i} _{,0}} = {{{{\Pi}} \cdot {{{ \alpha} _{,i}}}} + {{{\alpha}} \cdot {{{ \Pi} _{,i}}}} + {{{{ \Psi} _j}} {{{{ \beta} ^j} _{,i}}}} + {{{{ \beta} ^j}} {{{{ \Phi} _{,i}} _{,j}}}}}$
${{{ \Psi} _i} _{,0}} = {{{{{ \Psi} _j}} {{{{ \beta} ^j} _{,i}}}} + {{{{ \beta} ^j}} {{{{ \Phi} _{,i}} _{,j}}}} + {{\left( {{\alpha}} \cdot {{\Pi}}\right)} _{,i}}}$ (eqn. 8)
$\frac{d}{dx^0} \Psi_i = (\alpha \Pi)_{,i}$

Solve $\Pi_{,0}$
${{ \Pi} _{,0}} = {{{{{{ \Phi} _{,0}} _{,0}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{{ \Phi} _{,0}}} {{{ \alpha} _{,0}}} {{\frac{1}{{\alpha}^{2}}}}} + {{{-1}} {{{ \Phi} _{,i}}} {{{{ \beta} ^i} _{,0}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{{ \beta} ^i}} {{{{ \Phi} _{,0}} _{,i}}} {{\frac{1}{\alpha}}}} + {{{{ \Phi} _{,i}}} {{{ \alpha} _{,0}}} {{{ \beta} ^i}} {{\frac{1}{{\alpha}^{2}}}}}}$
substitute ${{ \Phi} _{,i}} = {{ \Psi} _i}$ , ${{ \Phi} _{,0}} = {{{{\Pi}} \cdot {{\alpha}}} + {{{{ \Psi} _i}} {{{ \beta} ^i}}}}$
${{ \Pi} _{,0}} = {{{{{{ \Phi} _{,0}} _{,0}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{\Pi}} \cdot {{{ \alpha} _{,0}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{{ \Psi} _i}} {{{{ \beta} ^i} _{,0}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{{ \beta} ^i}} {{{{ \Phi} _{,0}} _{,i}}} {{\frac{1}{\alpha}}}}}$
substitute ${{{ \Phi} _{,0}} _{,0}} = {{-{{{f}} {{{\alpha}^{2}}}}} + {{{2}} {{{ \beta} ^i}} {{{{ \Phi} _{,0}} _{,i}}}}{-{{{{ \Phi} _{,0}}} {{{ \bar{\Gamma}} ^0}} {{{\alpha}^{2}}}}}{-{{{{ \Phi} _{,i}}} {{{ \bar{\Gamma}} ^i}} {{{\alpha}^{2}}}}}{-{{{{ \beta} ^i}} {{{ \beta} ^j}} {{{{ \Phi} _{,i}} _{,j}}}}} + {{{{{ \Phi} _{,i}} _{,j}}} {{{{ \gamma} ^i} ^j}} {{{\alpha}^{2}}}}}$ , ${{ \Phi} _{,i}} = {{ \Psi} _i}$ , ${{{ \Phi} _{,i}} _{,j}} = {{{ \Psi} _i} _{,j}}$
${{ \Pi} _{,0}} = {{{{-1}} {{\Pi}} \cdot {{{ \alpha} _{,0}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{\alpha}} \cdot {{f}}} + {{{-1}} {{{ \Psi} _i}} {{{{ \beta} ^i} _{,0}}} {{\frac{1}{\alpha}}}} + {{{{ \beta} ^i}} {{{{ \Phi} _{,0}} _{,i}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{\alpha}} \cdot {{{ \Phi} _{,0}}} {{{ \bar{\Gamma}} ^0}}} + {{{-1}} {{\alpha}} \cdot {{{ \Psi} _i}} {{{ \bar{\Gamma}} ^i}}} + {{{-1}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{{{ \Psi} _i} _{,j}}} {{\frac{1}{\alpha}}}} + {{{\alpha}} \cdot {{{{ \Psi} _i} _{,j}}} {{{{ \gamma} ^i} ^j}}}}$
substitute ${{ \bar{\Gamma}} ^0} = { {-{\frac{1}{{\alpha}^{3}}}} {{\left({{{ \alpha} _{,0}}{-{{{{ \beta} ^m}} {{{ \alpha} _{,m}}}}} + {{{{\alpha}^{2}}} {{K}}}}\right)}}}$ , ${{ \Phi} _{,0}} = {{{{\Pi}} \cdot {{\alpha}}} + {{{{ \Psi} _i}} {{{ \beta} ^i}}}}$ , ${{{ \Phi} _{,0}} _{,i}} = {{{{\Pi}} \cdot {{{ \alpha} _{,i}}}} + {{{\alpha}} \cdot {{{ \Pi} _{,i}}}} + {{{{ \Psi} _j}} {{{{ \beta} ^j} _{,i}}}} + {{{{ \beta} ^j}} {{{{ \Psi} _j} _{,i}}}}}$
${{ \Pi} _{,0}} = {{{{-1}} {{\alpha}} \cdot {{f}}} + {{{K}} {{\Pi}} \cdot {{\alpha}}} + {{{-1}} {{{ \Psi} _i}} {{{{ \beta} ^i} _{,0}}} {{\frac{1}{\alpha}}}} + {{{{ \Pi} _{,i}}} {{{ \beta} ^i}}} + {{{-1}} {{\alpha}} \cdot {{{ \Psi} _i}} {{{ \bar{\Gamma}} ^i}}} + {{{{ \Psi} _i}} {{{ \alpha} _{,0}}} {{{ \beta} ^i}} {{\frac{1}{{\alpha}^{2}}}}} + {{{\alpha}} \cdot {{{{ \Psi} _i} _{,j}}} {{{{ \gamma} ^i} ^j}}} + {{{K}} {{{ \Psi} _i}} {{{ \beta} ^i}}} + {{{{ \Psi} _j}} {{{ \beta} ^i}} {{{{ \beta} ^j} _{,i}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{{ \Psi} _i}} {{{ \alpha} _{,j}}} {{{ \beta} ^i}} {{{ \beta} ^j}} {{\frac{1}{{\alpha}^{2}}}}}}$
substitute ${{ \bar{\Gamma}} ^i} = {{{{ \Gamma} ^i} + {{{\frac{1}{{\alpha}^{3}}}} {{{ \beta} ^i}} {{\left({{{ \alpha} _{,0}}{-{{{{ \beta} ^j}} {{{ \alpha} _{,j}}}}} + {{{{\alpha}^{2}}} {{K}}}}\right)}}}}{-{{{\frac{1}{{\alpha}^{2}}}} {{\left({{{{ \beta} ^i} _{,0}}{-{{{{ \beta} ^j}} {{{{ \beta} ^i} _{,j}}}}} + {{{\alpha}} \cdot {{{ \alpha} _{,j}}} {{{{ \gamma} ^i} ^j}}}}\right)}}}}}$
${{ \Pi} _{,0}} = {{{{-1}} {{\alpha}} \cdot {{f}}} + {{{{ \Pi} _{,i}}} {{{ \beta} ^i}}} + {{{K}} {{\Pi}} \cdot {{\alpha}}} + {{{-1}} {{\alpha}} \cdot {{{ \Gamma} ^i}} {{{ \Psi} _i}}} + {{{\alpha}} \cdot {{{{ \Psi} _i} _{,j}}} {{{{ \gamma} ^i} ^j}}} + {{{{ \Psi} _i}} {{{ \alpha} _{,j}}} {{{{ \gamma} ^i} ^j}}}}$
$\frac{d}{dx^0} \Pi = $ ${{{-1}} {{\alpha}} \cdot {{f}}} + {{{K}} {{\Pi}} \cdot {{\alpha}}} + {{{-1}} {{\alpha}} \cdot {{{ \Gamma} ^i}} {{{ \Psi} _i}}} + {{{\alpha}} \cdot {{{{ \Psi} _i} _{,j}}} {{{{ \gamma} ^i} ^j}}} + {{{{ \Psi} _i}} {{{ \alpha} _{,j}}} {{{{ \gamma} ^i} ^j}}}$

collected:
${{ \left[\begin{array}{c} \Phi\\ \Pi\\ { \Psi} _i\end{array}\right]} _{,0}} = {\left[\begin{array}{c} {{{\Pi}} \cdot {{\alpha}}} + {{{{ \Psi} _i}} {{{ \beta} ^i}}}\\ {{{-1}} {{\alpha}} \cdot {{f}}} + {{{{ \Pi} _{,i}}} {{{ \beta} ^i}}} + {{{K}} {{\Pi}} \cdot {{\alpha}}} + {{{-1}} {{\alpha}} \cdot {{{ \Gamma} ^i}} {{{ \Psi} _i}}} + {{{\alpha}} \cdot {{{{ \Psi} _i} _{,j}}} {{{{ \gamma} ^i} ^j}}} + {{{{ \Psi} _i}} {{{ \alpha} _{,j}}} {{{{ \gamma} ^i} ^j}}}\\ {{{{ \Psi} _j}} {{{{ \beta} ^j} _{,i}}}} + {{{{ \beta} ^j}} {{{{ \Phi} _{,i}} _{,j}}}} + {{\left( {{\alpha}} \cdot {{\Pi}}\right)} _{,i}}\end{array}\right]}$