# Scalar Field Solution to Einsteins Equations

## Homework Statement

Compute
$$T_{\mu\nu} T^{\mu\nu} - \frac{T^2}{4}$$

For a massless scalar field and then specify the computation to a spherically symmetric static metric
$$ds^2=-f(r)dt^2 + f^{-1}(r)dr^2 + r^2 d\Omega^2$$

## Homework Equations

$$4R_{\mu\nu} R^{\mu\nu} - R^2 = 16\pi^2 \left( T_{\mu\nu} T^{\mu\nu} - \frac{T^2}{4} \right)$$

$$R= -8\pi T$$

## The Attempt at a Solution

I've already solved the energy-momentum tensor for the scalar field and have its trace, also I already have the curvature scalar and the contraction ##R_{\mu\nu} R^{\mu\nu}##. BUT Trying to solve for ##f(r)## by solving the components of the Ricci tensor (I've assumed ##\Phi=\Phi(r,t)## by symmetry of space-time) I get,
$$R_{\mu\nu} = 2\Phi_{,\mu}\Phi_{,\nu}$$
so solving the ##\theta## part I obtain,
$$-r f'(r)-f(r)+1=0 \Rightarrow f(r)=1+\frac{C}{r}$$
But using this in the ##t## component I get ##0=(\partial_t \Phi)^2## which doesn't quite make sense. So I'm stuck. Any help or hint is appreciated.

## Answers and Replies

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?