by amluto | 2025-05-17 22:13:45+00:00
> Because that's not how relativity works! Two observers can disagree only on the order and relative timing of events, not what the events are or the total number of events.

No, and this has nothing to do with quantum mechanics or the no-hair theorem or anything particularly fancy.

As a toy example, suppose you have a frame with a (co-moving, but it doesn’t really matter) time coordinate t. A series of events happen at the origin (x=y=z=0 in this frame) at various times t.

There’s another observer in a frame with a time coordinate t'. The frames are related by t' = t - 1/t for t<0. The t=-10 event happens at t'=-9.9. The t=-4 event happens at t’=-3.25. The t=-1 event happens at t’=0. t=-1/100 happens at t'=99.99. t=0 gets closer and closer to happening but never actually happens. t=1 doesn’t even come close.

Critically, the t' observer does not observe t=0 or t=1 in some inconsistent manner. There is no disagreement between the observers as to what happens at t>=0. To the contrary, those events are simply not present in the t' observer’s coordinate system!

Note that the transformation above isn’t about when light from an event gets to the t' observer — it’s the actual relativistic transformation between two frames.

The Schwarzchild metric has a nastier transformation than this. If you toss a rock into an isolated black hole from far away, you will see the rock get progressively closer to the event horizon, and you will never see it fall in. But the rock is in trouble: its co-movimg coordinate system ends not long after it crosses the horizon. That latter phenomenon is called the “singularity”, it’s solidly inside the event horizon, and it’s not avoidable by coordinate system trickery. While general relativity does not explain what happens when one encounters the singularity, one might imagine that it’s fatal to the rock the reaches it.

edit: FWIW, you also say:

> The current "pop science" (nearly science fiction) statement is that it is possible to fall into a black hole and there is "nothing special" about the event horizon.

I’m not sure what you’re talking about. In a pure GR model of an isolated black hole, as you fall in, you will observe tidal forces. In a smaller black hole, the tidal forces will squash you long before you reach the event horizon. In a large enough black hole, they will not! Your view of the sky would certainly look very, very distorted and delightfully and possibly dangerously blue-shifted, but we’re talking about an isolated black hole. Nothing to see in the sky, and you may well survive your visit to the event horizon. Then, dramatically less than one second later for any credibly sized black hole, you will meet the singularity, and IIRC you should probably expect to be squashed by tidal forces before that. Source: I took the class and did the math. I assume this is what the “pop science” you’re talking about is saying, and it’s not wrong.

P.S. I’ve never tried to calculate how lethal the blue-shifted sky would be. Naively considering just the time transformation, it should be infinitely lethal at the event horizon. But trying to apply intuition based on only part of a relativistic transformation is a great way to reach incorrect conclusions.