# Nullity of the twist scalar and existence of wave fronts in general dimensions

Context

Consider a lightlike congruence with velocity vector $$k^\mu$$ in a $$D$$-dimensional lorentzian manifold, and the optical decomposition:

$$\nabla_\mu k_\nu = \omega_{\mu\nu}+\sigma_{\mu\nu}+\tfrac{1}{D-2}h_{\mu\nu}\theta,$$

where $$\nabla$$ is the Levi-Civita connection and $$h_{\mu\nu}$$ is the $$(D-2)$$-dimensional transversal metric. In particular, $$\omega_{\mu\nu}$$ represents the antisymmetric part (known as twist tensor):

$$\omega_{\mu\nu}:=\partial_{[\mu}k_{\nu]}.$$

It is also usual to introduce the optical scalars. One of them is the "twist" of the congruence given by:

$$\omega:=\sqrt{\tfrac{1}{D-2}\partial_{[\mu}k_{\nu]}\partial^{\mu}k^{\nu}} \equiv \sqrt{\tfrac{1}{D-2}\omega_{\mu\nu}\omega^{\mu\nu}}.$$

Question

In [1], for example, where the calculations are done in four dimensions, it is shown that the nullity of the twist scalar $$\omega$$ is equivalent to the existence of wave fronts. But they take advantage of the dimension to build the typical null tetrad $$\{t,\bar{t},m,k\}$$.

My question is: is it a general property for any $$D$$? I mean, the implication:

$$\omega=0 \qquad{} \Rightarrow \qquad{}\text{existence of wavefront}.$$

Some thoughts

I see clearly for example that $$\omega_{\mu\nu}=0$$ implies that $$k_\mu$$ is locally an exact form and then, $$k^\mu$$ is a gradient field, and we can use this to build the wave fronts. So

$$\omega_{\mu\nu}=0 \qquad{} \Rightarrow \qquad{}\text{existence of wavefront}.$$

If there were an equivalence between $$\omega_{\mu\nu}=0$$ and $$\omega=0$$ in general dimensions (for null congruences at least), everything works. However I think the implication $$\omega=0\Rightarrow \omega_{\mu\nu}=0$$ is not true in general.

[1] W. Kundt, Zeitschrift für Physics, 163 (1961), 77-86.