Undergrad Temporal Gradients at Event Horizons of Black Holes

[Kostik]

TL;DR

Does the temporal gradient in the immediate (external) vicinity of an event horizon of a black hole result in "fossilization" of any object with non-zero size crossing the event horizon?

While there is much discussion about "spaghettification" when approaching a black hole (BH) singularity due to tidal forces, many discussions say, rather casually, that a hypothetical traveler would free-fall right through the event horizon (EH) of a large BH without noticing anything.

Doesn't this scenario ignore the temporal gradient at the EH? Specifically, to any observer outside the EH, a clock approaching the EH appears to slow down. And from the point of view of a traveler approaching the EH and looking backward (in the direction opposite to which he is moving), a clock located a fixed distance from the EH appears to speed up. Therefore, if a real traveler (say a six-foot human) approached the EH -- suppose oriented head-first along his trajectory -- wouldn't he see his feet age a million years in a split-second? In other words, a clock attacked to his feet would be running much faster than a clock attached to his neck. Won't this "temporal tide" essentially fossilize (i.e., infinitely age) anything larger than a point-particle as it approaches the EH? So it seems to me that the real unpleasantness of entering a black hole is not the physical spaghettification near the singularity, but the "temporal fossilization" near the EH.

Is this right?

 

[PeterDonis]

Moderator's note: Thread level changed to "I".

 

[PeroK]

Is this right?

It's not right. The infalling observer's head and feet are both on infalling trajectories.

You are trying to compare the infalling observer's feet, say, with a "stationary" observer, hovering at a certain radius outside the event horizon. Such an observer would be very different from the infalling observer's head.

 

[PeterDonis]

Is this right?

No. See below.

from the point of view of a traveler approaching the EH and looking backward (in the direction opposite to which he is moving), a clock located a fixed distance from the EH appears to speed up.

No.

To an observer hovering just above the hole's horizon, and not falling in, clocks that are hovering higher up appear to be running fast.

To an observer falling in to the hole, clocks that are hovering higher up appear to be running slow, not fast.

But to an observer falling into the hole, a clock a bit higher that is also falling in, does not appear to be running slow or fast, except to the (small) extent that the two clocks are moving apart because of tidal gravity--that makes each one see the other running a bit slow.

 

[Kostik]

Thank you both!