Expansion in flat spacetime

Question

I have been studying Raychaudhuri equation and focusing theorem related to it. Focusing theorem says that if the strong energy condition is satisfied and rotation tensor vanishes $\omega_{ab}$=0 then rate of expansion is negative. Frobenius theorem for timelike vector says that timelike geodesic is hypersurface orthogonal iff $\omega_{ab}$=0.

I was wondering to apply this in flat spacetime but I can't find any suitable timelike geodesic in flat spacetime which would be hypersurface orthogonal and d\theta /d\tau is negative. Can anyone help with this?

If I have any such geodesic and as in flat spacetime Riemann curvature tensor would be 0 therefore only expansion term and shear tensor term would be left in Raychaudhuri equation which can be found through simple computation and hence focusing theorem could be satisfied in flat spacetime.

Answer 1

In flat spacetime, let $(t,x,y,z)$ be a global inertial cartesian coordinate system. Then the lines with $x,y,z$ fixed are orthogonal to the hypersurfaces of constant $t$.

This is the simplest case possible: trajectories of infinitely many inertial observers in rest relative to each other.

If you apply Raychaudhuri to this example, you will find $\dot{\theta}=0$ since $\theta_0 = 0$ initially, and thus $\theta = 0$ for any $t$.