When discussing time dilation near Black Holes one normally refers to “local” time at the event horizon and remote time or “coordinate” time as experienced by a remote observer stationary relative the BH. But time dilation increases steadily as one approaches a Black Hole. For example, if the Black Hole event horizon radius is 1 million kilometers, then 1 kilometer away from the horizon the time dilation rate would be 1000:1 relative to coordinate time. At 1mm away it would be 1,000,000:1. It shoots up to infinity as one approaches the horizon. There is relative time dilation between any two places at different distances from the Black Hole.
If you were falling feet first into a Black Hole, then as you approached the event horizon your feet would be very time dilated and slow down relative to your head. Your head would catch up with your feet and you would be concertinaed, go “squish”, and arrive at the event horizon as a thin pancake. In fact this would happen so fast that you would go “splat” as you pass through the event horizon. Of course the Black Hole would inevitably be rotating and would turn the splat into a smear. But for us distant observers, if we could make any observations that close to the event horizon, it would be a slow squish.
This applies to any cloud of matter falling in. Leading particles would slow down relative to those further out, and in their local time they would all arrive together. However this is not quite true for a large mass because its trailing edge would slowly approach a new event horizon forming further out due to the added mass.
All calculations for stellar collapse and for objects falling into Black Holes have ignored pressure in the collapsing medium. But with large amounts of matter falling into a Black Hole and “squishing”, pressure becomes an important factor. The pressure of the squish would force matter out sideways, but with matter falling in all around the equatorial regions of a Black Hole it has nowhere to go except out towards the poles. The extreme pressure would force a jet stream of matter poleward at tremendous speed (relativistic?) where it would all collide creating a jet of matter spewing outwards into space. I propose this as the mechanism behind the jet emissions observed in AGN galaxies, emitted by their central Black Holes.
Now I just need some mathematical genius to do the Black Hole calculations taking pressure into account. Modern computers should help.
I have had fun with my descriptive terms “squish”, “pancake”, etc, but I am sure there must be technical terms for these conditions, probably in the field of hydraulics.