## Einstein-scalar field action --> Einstein-scalar field equations

Dear friends,

Just a small question I do not know how to derive.

From the Einstein-scalar field action defined by

$$S\left( {g,\psi } \right) = \int_{} {\left( {R(g) - \frac{1}{2}\left| {\nabla \psi } \right|_g^2 - V\left( \psi \right)} \right)d{\eta _g}}$$

one gets the so-called Einstein-scalar field equations given by

$${\rm Eins}_{\alpha \beta} = {\nabla _\alpha }\psi {\nabla _\beta }\psi - \frac{1}{2}{g_{\alpha \beta }}{\nabla _\mu }\psi {\nabla ^\mu }\psi - {g_{\alpha \beta }}V(\psi ).$$

My question is how to derive such equations. It seems that we need to take derivative.... but how? Thanks.

 PhysOrg.com science news on PhysOrg.com >> Hong Kong launches first electric taxis>> Morocco to harness the wind in energy hunt>> Galaxy's Ring of Fire
 you will find this derivation in most test books on GR, you can derive it by varying the action and requiring that the functional remain constant i.e that the functional is zero are you familar with functionals? one must take the functional w.r.t the metric itself it is actually easier to take the functional w.r.t the metric in [1,1] form, thus write the other components in term of this