# Is there a simple way to define/solve for a null cone for a general spacetime geometry?

I'm wondering if there's any simple way to define and solve for a null cone for a general spacetime geometry in $n+1$ dimensions, given its vertex $p^\mu$. I can't seem to find a simple way to do it that doesn't involve solving for null geodesics. It seems like it's something that should be in a lot of GR textbooks, but I can't seem to find a simple alternative (perhaps I'm not looking hard enough, but that's why I'm asking).

So far, the best I can do is to define it as the $n$-surface formed by the family of null geodesics $y_{\bf\Omega}^\mu$ passing through some spacetime point $p^\mu$, where $\bf\Omega$ is a combination of $n-1$ angles that distinguishes the curves (at $p^\mu$, it's the collection of angles specifying the direction of the null curve).

The problem is, I'm interested in doing a foliation of spacetime using future or past null cones with respect to some timelike curve $\gamma$, and would like to define the null cones in the form of a constraint:

$C_{\gamma}(x^\mu)=\tau$

Where $\tau$ is the proper time along the timelike curve $\gamma$. This constraint satisfies:

$U(C_{\gamma})=0$

Where $U$ is the vector field defined using the tangent vectors to $y_{\bf\Omega}^\mu$ (this can be defined for all spacetime, if you take the union of all the null cones along $\gamma$). In any case, I still have to solve for the null geodesics, given what I have so far.

I'm not a fan of solving for all the null geodesics at every point along the curve, so I'm looking for a simpler way to do it.

I have the feeling that there should be a simpler way of defining and solving for null cones--it seems like they've been extensively studied in GR. Is there any way to do this without resorting to solving for the null geodesics? In particular, has anyone written down a general constraint form for the null cone?

I'm interested in doing a foliation of spacetime using future or past null cones with respect to some timelike curve $\gamma$, and would like to define the null cones in the form of a constraint ...

That is a priori impossible. In an arbitrary space-time geometry, there can be focal and conjugate points. Unless you are willing to assume from the start some bounds on the null injectivity radius (to be infinite), you can not in general foliate an entire space-time with null cones (mostly because the cones will likely cease to be cones after some affine parameter).

With that said, if you are defining null hypersurfaces, you can solve instead for the eikonal equation. That is, let $u$ be a parameter describing your null cones (in the sense that $u:M\to \mathbb{R}$ is a real valued function on your manifold, and that its level sets are "null cones"). Then a necessary equation satisfied by the function $u$ is

$$g^{\mu\nu}\partial_\mu u\partial_\nu u = 0$$

the difficulty of this approach is to "match" the tip of the cone to be precisely along the curve $\gamma$, but regularity issues like this has to manifest itself somewhere or another.

• As an example to the first point: consider yourself in Schwarzschild, and your time-like curve $\gamma$ sits exactly on the photon sphere. Then for each point $p$ on the photon sphere, there are infinitely many times $\tau_1, \tau_2, \ldots$ such that $p$ is "on the light cone" of $\gamma(\tau_i)$. – Willie Wong Jun 28 '11 at 12:07
• Good point. I was mostly thinking about it quasi-locally, and didn't consider the global implications of what I wanted to do. – Desert Coyote Jun 28 '11 at 14:38