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

[bookworm_vn]

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.

 

[andrew1915]

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