Rod Near Event Horizon: What Would Happen?

[Cato]

If I passed within one meter of the event horizon of a supermassive black hole (where the tidal forces are trivial) and stuck out a two meter rod such that one meter of the rod was inside the event horizon, what would I see happen to the rod?

 

[PeterDonis]

If I passed within one meter of the event horizon

You can't. The event horizon is not a place in space; it's an outgoing light ray. So your implicit mental model of what is happening in this thought experiment is not correct.

Also, if by "passed within one meter" you are thinking that you could somehow pass one meter above the horizon on a free-fall trajectory that then escaped back to infinity, that's not possible. AFAIK all such trajectories get pulled into the hole. So if you're imagining being within one meter of the hole without falling in, you have to "hover" using rocket power.

what would I see happen to the rod?

Under the conditions you've specified, you can model things as if you were a Rindler observer with a Rindler horizon one meter below you. (Note that this assumes you are "hovering" above the hole, not passing close to it--see my second note above.) So in a spacetime diagram of a local inertial frame in which the origin $t, x = 0, 0$ is an event on the horizon, your worldline would be a hyperbola $x^2 - t^2 = 1$, where the units of $x$ and $t$ are meters. The worldline of the horizon in this frame is the 45 degree line $x = t$.

If we now imagine the rod as being two meters long, and you extend it below you, you have a problem: as soon as the lower end of the rod is 1 meter below you, it is on your Rindler horizon and would have to move at the speed of light to keep up with you. Which it can't. So at some point before that, the rod will break. The reason it breaks is simple: you are exerting enormous rocket power in order to "hover" 1 meter above the horizon. That rocket power is pulling extremely strongly on the rod--so strongly that when the lower tip of the rod is 1 meter below you, the force exerted on the rod can't travel along the rod fast enough to keep the lower tip attached to the rest of the rod. So basically your rocket power pulls the rod apart.

In the spacetime diagram, you would draw the worldline of the rod's lower tip as gradually diverging from yours--say it starts at $t, x = 0, 1$ (where the units are meters), i.e., on your worldline, but then moves in the $x$ direction slower than you do, so its worldline is a shallower curve than the hyperbola you are following. At some point, the tip will cross the line $x = t$, i.e., the horizon, and at that point the tip can't keep up with you, no matter how hard you pull on it.

 

[Cato]

OK, thanks so much for the detailed and really useful discussion.