# Timelike/null generic condition in general relativity

My question concerns the following definition

Definition: The timelike (resp. null) generic condition in GR is fulfilled if $$u_{[\alpha} R_{\rho]\mu \nu [\sigma}u_{\beta]}u^\mu u^\nu \ne 0$$ at some point of each timelike (resp. null) geodesic with tangent vector $\vec u$. ($R_{\rho \mu \nu \sigma}$ is the Riemann curvature tensor.)

It is written in many places that this is the right condition to impose if one wants to assume that every freely falling (or light) particle encounters some form of matter or radiation in its history (or something to that effect).

But I don't understand, why the particular tensor $u_{[\alpha} R_{\rho]\mu \nu [\sigma}u_{\beta]}u^\mu u^\nu \ne 0$ is the right thing to look at in this context. For example, why don't we assume that $R_{\rho\mu\nu\sigma}u^\mu u^\nu \ne 0$ at some point? Or maybe that $R_{\mu \nu}u^\mu u^\nu \ne 0$, or perhaps that $G_{\mu \nu} u^\mu u^\nu \ne 0$?

I'm guessing that the last two condition could be too weak to derive the singularity theorems we want, so that something stronger must be assumed. But the expression $u_{[\alpha} R_{\rho]\mu \nu [\sigma}u_{\beta]}u^\mu u^\nu \ne 0$ really looks a bit strange to me (i.e. I don't understand its significance). Could someone explain to me why this is the right condition to impose?

Thanks!

• Wald has on p. 227: "A spacetime ... is said to satisfy the timelike generic condition if each timelike geodesic possesses at least one point at which $R_{abcd}\xi^a\xi^d\ne 0$." I haven't dug into it enough to understand what's going on, but I think $\xi$ is just an arbitrary vector, not a tangent to the geodesic. Wald also introduces a separate condition $R_{ab}\xi^a\xi^b\ne 0$. – user4552 Jul 6 '13 at 20:34
• @BenCrowell : $\xi^a$ is the tangent (beginning of the last paragraph, page 224) – Trimok Jul 6 '13 at 20:53
• $u_{[\alpha} R_{\rho]\mu \nu [\sigma}u_{\beta]}u^\mu u^\nu \ne 0$ and $R_{\rho\mu\nu\sigma}u^\mu u^\nu \ne 0$ are equvalent. – MBN Jul 7 '13 at 10:11
• @MBN - They are not equivalent. The first one applies to timelike geodesics and the second applies to null geodesics. – Prahar Jul 8 '13 at 1:09