BSSN formalism

ADM Formalism

More information , ...

Variable	Definition
${\displaystyle \alpha }$	lapse function
${\displaystyle \beta ^{i}}$	shift vector
${\displaystyle \gamma _{ij}}$	metric tensor on a 3D-hypersurface of the foliation
${\displaystyle R}$	3D Ricci scalar
${\displaystyle K_{ij}}$	extrinsic curvature
${\displaystyle D_{i}}$	3D covariant derivative

Close

The derivation begins with the ADM formalism. The ADM metric is given by:

${\displaystyle {\begin{aligned}ds^{2}&=-(\alpha ^{2}-\beta _{i}\beta ^{i})dt^{2}+2\beta _{i}dtdx^{i}+\gamma _{ij}dx^{i}dx^{j}\end{aligned}}}$

The Hamiltonian constraint is given by:

${\displaystyle {\begin{aligned}R-K_{ij}K^{ij}+K^{2}=0\end{aligned}}}$

The momentum constraint is given by:

${\displaystyle {\begin{aligned}D_{i}K_{j}^{i}-D_{j}K&=0\end{aligned}}}$

and the evolution equations are given by:

${\displaystyle {\begin{aligned}{\dfrac {\partial }{\partial t}}\gamma _{ij}&=-2\alpha K_{ij}+D_{i}\beta _{j}+D_{j}\beta _{i}\end{aligned}}}$

${\displaystyle {\begin{aligned}{\dfrac {\partial }{\partial t}}K_{ij}&=\alpha (R_{ij}+KK_{ij}-2K_{ij}K_{j}^{\ell })-D_{i}D_{j}\alpha +(D_{j}\beta ^{m})K_{mi}+(D_{i}\beta ^{m})K_{mj}+\beta _{m}D_{m}K_{ij}\end{aligned}}}$