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Kurzgesagt - In a Nutshell


Kurzgesagt is generally very nice. For this episode in particular, there are some statements that are basically correct but potentially misleading unless one already knows a lot more to fill in some caveats—but most of it far from any sort of problem to fuss over in a neat pop-sci presentation, particularly over the more cutting-edge ideas.

However, there are some things that I think are a bit too careless. One part around 3:00, says that due to light being able to (near-circularly?) orbit the black hole,
“If you hover... at the photonsphere, in any direction, you just see yourself.”​
But this is very obviously mistaken, because at best, it could only be true for looking in directions that are orthogonal to the radial, rather than “any direction.” And since those light orbits are highly unstable, it's very dubious how much one would notice anyway.

Another part around 5:25, which claims that:
“Trying to go in any direction only brings you to the centre faster. To survive the longest, you must do nothing.”​
This is a commonly repeated statement, but it's missing too much. Take a very idealised case of radial motion into a non-rotating, uncharged black hole. In the freefall, acceleration in terms of Schwarzschild radial coordinate r and proper time τ (i.e. measured by the infalling duck) happens to have the Newtonian form, d²r/dτ² = -m/r², while the event horizon is that r = 2m.

Put that way, because this special case of radial freefall is mathematically identical to Newtonian gravity, it becomes obvious that there are many radially freefalling trajectories that take different amounts of proper time to go from r=2m to r=0, and that there is a particular trajectory for which that time is maximised (the one that's at rest at r=2m), Because of this, even if is below r=2m, almost all infalling trajectories can fire rockets to increase their time to demise, by bringing themselves closer to that special maximising trajectory. In other words, Kurzgesagt should have said something like:
“If you were hovering very close to the event horizon and fell in, trying to go in any direction only brings you to the centre faster. ...”​
This is an important specification of the initial condition that would make the statement correct, but for most radially infalling trajectories, it's not actually true. Again, there's a special radial one for which it is, but for most other initial trajectories, the duck could extend its lifetime somewhat.

I think this confusion is related to another claim: that as one has crossed the horizon, one is enveloped by blackness with the image of the universe concentrated into a tiny spot of light. The problem is that once again it's basically true in the limit of hovering a small distance above the horizon, which involves huge outward acceleration (e.g. rocket thrust to keep from falling in), and so it's not what observer in freefall would see. Instead, there should be an interplay of relativistic beaming and tidal forces that distorts everything in a complicated way, but generally concentrates images towards one's sides, closer to a plane (with the sky above still there, but stretched out and increasingly redshifted).
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