Scalar Field Solution to Einsteins Equations

[loops496]

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.

 

[mark726]

Hello, let me help you with this problem.

Firstly, let's start by writing the energy-momentum tensor for a massless scalar field in the specified metric:
$$T_{\mu\nu} = \partial_\mu \Phi \partial_\nu \Phi - \frac{1}{2} g_{\mu\nu} \partial^\alpha \Phi \partial_\alpha \Phi$$

Next, let's compute the contraction ##T_{\mu\nu} T^{\mu\nu}##:
$$T_{\mu\nu} T^{\mu\nu} = \left( \partial_\mu \Phi \partial_\nu \Phi - \frac{1}{2} g_{\mu\nu} \partial^\alpha \Phi \partial_\alpha \Phi \right) \left( \partial^\mu \Phi \partial^\nu \Phi - \frac{1}{2} g^{\mu\nu} \partial_\alpha \Phi \partial^\alpha \Phi \right)$$
$$= \partial_\mu \Phi \partial^\mu \Phi \partial_\nu \Phi \partial^\nu \Phi - \frac{1}{2} \partial_\mu \Phi \partial^\mu \Phi \partial^\nu \Phi \partial_\nu \Phi - \frac{1}{2} \partial_\mu \Phi \partial^\mu \Phi \partial^\nu \Phi \partial_\nu \Phi + \frac{1}{4} g_{\mu\nu} g^{\mu\nu} \partial_\alpha \Phi \partial^\alpha \Phi \partial_\beta \Phi \partial^\beta \Phi$$
$$= \partial_\mu \Phi \partial^\mu \Phi \partial_\nu \Phi \partial^\nu \Phi - \partial_\mu \Phi \partial^\mu \Phi \partial^\nu \Phi \partial_\nu \Phi + \frac{1}{4} \partial_\alpha \Phi \partial^\alpha \Phi \partial_\beta \Phi \partial^\beta \Phi$$
$$= \left( \partial_\mu \Phi \partial^\mu \Phi \right)^2 - \left( \partial_\mu \Phi \partial^\mu \Phi \right) \left( \partial_\nu \Phi \partial^\nu \Phi \right) + \