# How do we know if a Killing Spinor is Time-like or Null?

How to know whether a Killing spinor orbit is time-like or null? This is present in a paper like this 29/39 here. I'm not asking for a technical answer, just a logical cliche answer chit-chat answer.

Let me go into more details here

The KSEs for $N = 1$ backgrounds have been solved using spinor bilinears. In the spinorial approach [Gillard, Gran, GP], the KSEs are solved utilizing the representatives of the two orbits. $\epsilon = f(1+e_{12345})$: Solutions admit a time-like Killing vector field and its orbit space has a SU(5) structure.

$\epsilon = 1+e_{1234}$: Solutions admit a null Killing vector field, the spacetime has a $Spin(7)\times \mathbb{R}^9$ structure and include backgrounds like pp-wave propagating in a $Spin(7)$ manifold.

To see whether a Killing spinor is timelike or null, you need to make a following combination. $$V_\mu\epsilon^{ab}=\bar{\epsilon}^a\gamma_\mu\epsilon^b\,.$$ Then using Fierz identities, you can compute the norm of the vector $V_\mu$, and see whether it is timelike or null. For example, in five-dimensional minimal supergravity, you will find that $$V_\mu V^\mu=f^2\,,$$ where $f$ is a scalar which is also constructed by using a Killing spinor $\epsilon^a$ such as $$f\epsilon^{ab}=\bar\epsilon^a\epsilon^b\,.$$