# 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.