Undergrad Spatially Homogeneous Scalar Field on Spacetime: Showing $\nabla^2 f$

[ergospherical]

If $f$ is a "spatially homogeneous" scalar field on spacetime $ds^2 = dt^2 - a^2(t) \delta_{ij} dx^i dx^j$ then show that $\nabla^2 f = \ddot{f} + 3H \dot{f}$. Should be easy if I knew what the condition on $f$ is, i.e. $\nabla^2 f = \partial_{\mu} \partial^{\mu} f = \ddot{f}- a^{-2}(t) \delta^{ij} \partial_i \partial_j f = \dots$?

 

[Orodruin]

Spatially homogeneous = does not depend on spatial position. In other words, $f$ is a function of $t$ only.

 

[ergospherical]

I messed up the double covariant, should be:\begin{align*}
\nabla^2 f &= \nabla_{\mu}(\partial^{\mu} f) \\
&= \partial_{\mu} \partial^{\mu} f + g^{\nu \rho} \Gamma_{\mu \nu}^{\mu} \partial_{\rho} f \\
&= \partial_t^2 f + 3H \partial_t f
\end{align*}($\Gamma_{0i}^{j} = H\delta^{j}_{i}$)

 

[Orodruin]

Indeed. Only the time derivatives survive since the field only depends on t. Then it is just a matter of finding the appropriate trace of Christoffel symbols.