I claim something above that seems mathematically impossible: that the photon travels perpendicularly to the instant horizon independent of the time slice that produced that instant horizon. This is possible only of a larger yet rare class of sub manifolds that include event horizons. This is connected to the problem of considering a time slice across a region of space-time that includes a black hole. There must be a hole, it seems, in the time slice. The edge of this hole would seem to be an event horizon. Considering that the definition of a horizon seems to require the concept of “asymptotically flat”, this raises questions for non asymptotically flat spaces, such as the one we once thought we lived in.

Consider the one event horizon of a pair of two black holes at some distance. If the holes are far enough apart then the horizon is divided into two components, one about each hole. As photons just outside one component head in a direction nearly toward the other hole, they will be captured. By the Penrose definition we must adjust the slope of the first component to orient it so that no photon emitted perpendicular thereto will be captured by the other hole. It seems clear to me that this requires there to be a point on each component nearest the other component. All of this argument takes place while imagining (falsely) that the holes are not accelerating towards each other. If they spin about each other the argument is more complex but I think still applies.

Here is a little math that helps reason about the event horizon.