- #1

Xander314

- 4

- 0

## Homework Statement

Let [itex]g_{\mu\nu}[/itex] be a static metric, [itex]\partial_t g_{\mu\nu}=0[/itex] where t is coordinate time. Show that the metric induced on a spacelike hypersurface [itex]t=\textrm{const}[/itex] is given by

[tex]

\gamma_{ij} = g_{ij} - \frac{g_{ti} g_{tj}}{g_{tt}} .

[/tex]

## Homework Equations

Let [itex]y^i[/itex] be the coordinates on the hypersurface and [itex]x^\mu[/itex] the spacetime coordinates. The induced metric on a generic hypersurface defined by the embedding [itex]x^\mu = X^\mu(y^i)[/itex] is given by

[tex]

\gamma_{ij} = g_{\mu\nu} \partial_i X^\mu \partial_j X^\nu .

[/tex]

## The Attempt at a Solution

I really don't see how this can work. Since it is a hypersurface of constant coordinate time, the embedding is given by [itex]X^\mu = (t_0, X^i)[/itex] so that [itex]\partial_i X^\mu = (0,\partial_i X^j)[/itex]. Then it immediately follows that

[tex]

\gamma_{ij} = g_{kl} \partial_i X^k \partial_j X^l .

[/tex]

There are no [itex]g_{ti}[/itex] cross terms in my answer, nor is it clear to me that [itex]\partial_i X^k=\delta_i{}^k[/itex]. What am I doing wrong?