Re: [Phys-L] Black hole and time question

[John Denker <jsd@av8n.com>]

On 10/9/20 5:51 AM, Ron Mcdermott via Phys-l wrote:

> Jack is 'near' the event horizon.  Let's say that from his pov it will take
> a year to pass that horizon (moving 'slowly'; as the gravitational effect
> on time is the issue).
>
> Jane is further away and stationary, but close enough that the time delay
> for light travel is inconsequential.  In HER timeline it takes, let's say,
> two years to see Jack cross the event horizon.
>
> The black hole evaporates at the 1.5 year mark in Jill's timeline.

I assume Jill is a nickname for Jane.

> The question boils down to: Where is Jack?
>
> My feeling is that Jack is 3 months from the previous event horizon.  His
> is that this is a paradox because, according to Jack's timeline, he OUGHT
> to have passed the event horizon.
If I understand the question, it really has nothing to do with black
holes or event horizons.  It's just a question of timing, in the presence
of a gravitational red shift.  Any sufficiently-deep gravitational potential
(singular or not) will behave the same for present purposes.

Suppose it takes Jack 12 months to build a boat.

Short answer: There is a simple linear relationship between Jack-timeand Jane-time.
  24 months@Jill = 12 months@Jack = finished boat.
  18 months@Jill =  9 months@Jack = 3 before finished.

In more detail: Let's assume Jack is hovering in a gravitational potential
at a depth where the gravitational redshift is Z=3.  Let Jane be hovering
at Z=1.5, i.e. half as deep.  The situation is shown in this diagram:
  https://www.av8n.com/physics/img48/grav-red-shift-timing.png

Because of the curvature of spacetime, we cannot faithfully represent the
situation in a flat Euclidean 2D diagram, so we have to use our map-making
skills.  This is an orange-peel projection.  The red-shaded regions exist
in spacetime, whereas the intervening gray regions do not exist.  You could
cut out the red-shaded regions and paste them together in 3D to get rid of
the discontinuities in the representation, but even that would still be
imperfect.  Getting rid of the singularity would make things easier.

Flat spacetime exists somewhere way off to the right, at large r.
The black lines are contours of constant time (simultaneity) according to
an observer out there in flat spacetime.