Ngô Quốc Anh

December 26, 2012

The evolution equations for the constraint equations

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 1:12

Let us first recall the Gauss equation which is given by

\displaystyle\displaystyle\left\langle {\overline R (X,Y)Z,W} \right\rangle = \left\langle {R\left( {X,Y)Z} \right),W} \right\rangle - \left\langle {\mathrm{I\!I}(X,Z)} , {\mathrm{I\!I}(Y,W)} \right\rangle + \left\langle {\mathrm{I\!I}(Y,Z)} , {\mathrm{I\!I}(X,W)} \right\rangle.

This can be proved as the following

\displaystyle\begin{array}{lcl} R\left( {X,Y)Z} \right) &=&\displaystyle {\nabla _X}{\nabla _Y}Z - {\nabla _Y}{\nabla _X}Z - {\nabla _{[X,Y]}}Z \hfill \\&=& {\nabla _X}({\overline \nabla _Y}Z - \mathrm{I\!I}(Y,Z)) - {\nabla _Y}({\overline \nabla _X}Z - \mathrm{I\!I}(X,Z)) - ({\overline \nabla _{[X,Y]}}Z - \mathrm{I\!I}([X,Y],Z)) \hfill \\&=& {\nabla _X}({\overline \nabla _Y}Z) - {\nabla _X}(\mathrm{I\!I}(Y,Z)) - {\nabla _Y}({\overline \nabla _X}Z) + {\nabla _Y}(\mathrm{I\!I}(X,Z)) - {\overline \nabla _{[X,Y]}}Z + \mathrm{I\!I}([X,Y],Z) \hfill \\&=& {\overline \nabla _X}{\overline \nabla _Y}Z - \mathrm{I\!I}(X,{\overline \nabla _Y}Z) - {\overline \nabla _Y}{\overline \nabla _X}Z + \mathrm{I\!I}(Y,{\overline \nabla _X}Z) - {\nabla _X}(\mathrm{I\!I}(Y,Z)) + {\nabla _Y}(\mathrm{I\!I}(X,Z)) \hfill \\ &&- {\overline \nabla _{[X,Y]}}Z + \mathrm{I\!I}([X,Y],Z) \hfill \\&=&\overline R (X,Y)Z - \mathrm{I\!I}(X,{\overline \nabla _Y}Z) + \mathrm{I\!I}(Y,{\overline \nabla _X}Z) - {\nabla _X}(\mathrm{I\!I}(Y,Z)) + {\nabla _Y}(\mathrm{I\!I}(X,Z)) + \mathrm{I\!I}([X,Y],Z),\hfill \\ \end{array}

which implies

\displaystyle\begin{array}{lcl}\left\langle {\overline R (X,Y)Z,W} \right\rangle &=& \left\langle {R(X,Y)Z,W} \right\rangle + \left\langle {{\nabla _X}(\mathrm{I\!I}(Y,Z)),W} \right\rangle - \left\langle {{\nabla _Y}(\mathrm{I\!I}(X,Z)),W} \right\rangle \hfill \\&=& \left\langle {R(X,Y)Z,W} \right\rangle + \left\langle {{\nabla _X}(\mathrm{I\!I}(Y,Z)),W} \right\rangle - \left\langle {{\nabla _Y}(\mathrm{I\!I}(X,Z)),W} \right\rangle \hfill \\ &=& \left\langle {R(X,Y)Z,W} \right\rangle - \left\langle {\mathrm{I\!I}(Y,Z),{\nabla _X}W} \right\rangle + \left\langle {\mathrm{I\!I}(X,Z),{\nabla _Y}W} \right\rangle. \hfill \\ \end{array}

By using {\nabla _X}W = {\overline \nabla _X}W - \mathrm{I\!I}(X,W) and {\nabla _Y}W = {\overline \nabla _Y}W - \mathrm{I\!I}(Y,W), we finally obtain

\displaystyle\left\langle {\overline R (X,Y)Z,W} \right\rangle = \left\langle {R(X,Y)Z,W} \right\rangle + \left\langle {\mathrm{I\!I}(Y,Z),\mathrm{I\!I}(X,W)} \right\rangle - \left\langle {\mathrm{I\!I}(X,Z),\mathrm{I\!I}(Y,W)} \right\rangle

as claimed.

This identity is slightly different from that of the usual one since \langle n, n \rangle = -1 plays a crucial role in the above computation.

We denote by K_{\alpha\beta} the components of the scalar second fundamental form K, i.e.,

\displaystyle \mathrm{I\!I}(\cdot, \cdot) = -K(\cdot, \cdot) n.

Using this and thanks to \langle n, n \rangle = -1, we obtain

\displaystyle\left\langle {\overline R(X,Y)Z,W} \right\rangle = \left\langle {R\left( {X,Y)Z} \right),W} \right\rangle + K(X,Z)K(Y,W) - K(Y,Z)K(X,W).

By using X = {\partial _\beta}, Y = {\partial _\alpha}, Z = {\partial _\delta }, W = {\partial _\eta}, we know that the Gauss equation can be rewritten as the following

\displaystyle {\overline R}_{\beta \alpha\delta}^\eta= R_{\beta\alpha\delta }^\eta+ {K_{\beta \delta }}K_\alpha ^\eta- K_\beta ^\eta{K_{\alpha \delta }}.

Therefore, by projecting the left hand side, we obtain

\displaystyle\gamma _\alpha ^\mu \gamma _\beta ^\nu \gamma _\rho ^\eta \gamma _\delta ^\sigma \overline R _{\nu\mu \sigma }^\rho = R_{\beta\alpha \delta}^\eta + K_\alpha ^\eta {K_{\delta \beta }} - K_\beta ^\eta {K_{\alpha \delta }}.

We now contract the Gauss equation on the indices \eta and \alpha and use \gamma _\eta ^\mu\gamma _\rho ^\eta = \gamma _\rho ^\eta = \delta _\rho ^\eta+ {n^\eta}{n_\rho }, we obtain

\displaystyle\gamma _\beta ^\nu (\delta _\rho ^\mu + {n^\mu }{n_\rho })\gamma _\delta ^\sigma \overline R _{\nu \mu \sigma }^\rho = {R_{\beta \delta }} + K{K_{\delta \beta }} - K_\beta ^\eta {K_{\eta \delta }},

which is equivalent to

\displaystyle\gamma _\beta ^\nu \gamma _\delta ^\sigma {\overline R _{\nu \sigma }} + \gamma _\beta ^\nu {n^\mu }{n_\rho }\gamma _\delta ^\sigma \overline R _{\nu \mu \sigma }^\rho = {R_{\beta \delta }} + K{K_{\delta \beta }} - K_\beta ^\eta {K_{\eta \delta }}.


\displaystyle\begin{array}{lcl} \gamma _\beta ^\nu {n^\mu }{n_\rho }\gamma _\delta ^\sigma \overline R _{\nu \mu \sigma }^\rho &=&\displaystyle \gamma _\beta ^\nu {n^\mu }{n^\theta }{\gamma _{\rho \theta }}{\gamma _{\delta \tau }}{\gamma ^{\tau \sigma }}\overline R _{\nu \mu \sigma }^\rho \hfill \\&=&\displaystyle {\gamma _{\delta \tau }}\gamma _\beta ^\nu {n^\mu }{n^\theta }\underbrace {{\gamma _{\rho \theta }}{\gamma ^{\tau \sigma }}\overline R _{\nu \mu \sigma }^\rho }_{\overline R _{\theta \nu \mu }^\tau } \hfill \\&=&\displaystyle {\gamma _{\delta \tau }}\gamma _\beta ^\nu {n^\mu }{n^\theta }\overline R _{\theta \nu \mu }^\tau , \hfill \\ \end{array}

we get that by changing \delta to \alpha

\displaystyle\gamma _\alpha ^\sigma \gamma _\beta ^\nu {\overline R _{\nu \sigma }} + {\gamma _{\alpha \tau }}\gamma _\beta ^\nu {n^\mu }{n^\theta }\overline R _{\theta \nu \mu }^\tau = {R_{\alpha \beta }} + K{K_{\alpha \beta }} - K_\beta ^\eta {K_{\eta \alpha }}.

In particular, we have proved that

\displaystyle {\gamma _{\alpha \mu }}{n^\sigma }\gamma _\beta ^\nu \overline R _{\rho\nu\sigma }^\mu {n^\rho } = - \gamma _\alpha ^\mu \gamma _\beta ^\nu {\overline {\text{Ric}} _{\mu \nu }} + {\text{Ric}_{\alpha \beta }} + K{K_{\alpha \beta }} - K_\beta ^\mu {K_{\mu \alpha }}.

In the previous note, we derive the so-called Ricci equation for the spatial M sitting in the spacetime V. The equation simply reads as follows

\displaystyle {\gamma _{\alpha \mu }}{n^\sigma }\gamma _\beta ^\nu \overline R_{\rho \nu \sigma }^\mu {n^\rho } = \frac{1}{N}{\mathcal L_m}{K_{\alpha \beta }} + \frac{1}{N}{({\nabla ^M})_\alpha }{({\nabla ^M})_\beta }N + {K_{\alpha \mu }}K_\beta ^\mu .

Using this, we arrive at

\displaystyle\gamma _\alpha ^\mu \gamma _\beta ^\nu {\overline {\text{Ric}} _{\mu \nu }} = - \frac{1}{N}{\mathcal L_m}{K_{\alpha \beta }} - \frac{1}{N}{({\nabla ^M})_\alpha }{({\nabla ^M})_\beta }N + {\text{Ric}_{\alpha \beta }} + K{K_{\alpha \beta }} - 2{K_{\alpha \mu }}K_\beta ^\mu .

Thus, we get the following evolution equation for K_{\alpha\beta} as follows

\displaystyle {\mathcal L_m}{K_{\alpha \beta }} = - N\gamma _\alpha ^\mu \gamma _\beta ^\nu {\overline {\text{Ric}} _{\mu \nu }} - {\nabla _\alpha }{\nabla _\beta }N + N({\text{Ric}_{\alpha \beta }} + K{K_{\alpha \beta }} - 2{K_{\alpha \mu }}K_\beta ^\mu )

where we obviously have \nabla=\nabla^M, the induced connection on M. Since the vector m is nothing but \frac{\partial}{\partial t}-\vec\beta, we further have

\displaystyle \frac{\partial }{{\partial t}}{K_{\alpha \beta }} = - {\nabla _\alpha }{\nabla _\beta }N + N({\text{Ric}_{\alpha \beta }}-\gamma _\alpha ^\mu \gamma _\beta ^\nu {\overline {\text{Ric}} _{\mu \nu }} + K{K_{\alpha \beta }} - 2{K_{\alpha \mu }}K_\beta ^\mu ) + {\mathcal L_{\vec \beta}}{K_{\alpha \beta }}.

In the last part of the note, we derive the evolution equation for the metric \gamma. Thanks to the previous note, we have

\displaystyle\begin{array}{lcl}{\mathcal L_m}{\gamma _{\alpha \beta }} &=&\displaystyle {m^\mu }{\nabla _\mu }{\gamma _{\alpha \beta }} + {\gamma _{\mu \beta }}{\nabla _\alpha }{m^\mu } + {\gamma _{\alpha \mu }}{\nabla _\beta }{m^\mu } \hfill \\&=&\displaystyle N{n^\mu }{\nabla _\mu }({n_\alpha }{n_\beta }) - {\gamma _{\mu \beta }}(NK_\alpha ^\mu + {n_\alpha }{({\nabla ^M})^\mu }N - {n^\mu }{\nabla _\alpha }N) - {\gamma _{\alpha \mu }}(NK_\beta ^\mu + {n_\beta }{({\nabla ^M})^\mu }N - {n^\mu }{\nabla _\beta }N) \hfill \\&=&\displaystyle N({n_\beta }\underbrace {{n^\mu }{\nabla _\mu }{n_\alpha }}_{{a_\alpha } = {N^{ - 1}}{{({\nabla ^M})}_\alpha }N} + {n_\alpha }\underbrace {{n^\mu }{\nabla _\mu }{n_\beta }}_{{a_\beta } = {N^{ - 1}}{{({\nabla ^M})}_\beta }N}) - N{K_{\beta \alpha }} - {n_\alpha }{({\nabla ^M})_\beta }N - N{K_{\alpha \beta }} - {n_\beta }{({\nabla ^M})_\alpha }N \hfill \\&=&\displaystyle - 2N{K_{\alpha \beta }}. \hfill \\ \end{array}

Thus, we obtain

\displaystyle \frac{\partial }{{\partial t}}{\gamma _{\alpha \beta }} = - 2N{K_{\alpha \beta }} + {\mathcal L_{\vec \beta} }{\gamma _{\alpha \beta }}.

See also:

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