Re: Density falls at the event horizon?

Ben Rudiak-Gould wrote: (Thanks for your comments. )

> I'm pretty sure that tidal forces always preserve density,
> even at the event horizon.

That is true, but only initially. For instance Penrose "Road to
Reality" Fig 17.8 and Sec 17.5 writes "The initial distortion preserves
volume." For a detailed analysis see my paper; if you drop a cuboid,
the change in volume due to tidal forces depends on the relative size
of the upper and lower faces and the relative velocities at which the
upper and lower faces are falling. When radii are parallel, the upper
and lower faces have the same size, so only the relative velocities
matter--and they tend to reduce density by pulling the astronaut's head
and feet apart.

> > The asymptotically flat
> > interior (Kruskal-Szekeres Region III) is singularity-free and
> > isometric across r=2M to the exterior; topology does not change.
>
> Whoa, whoa... why do you call region III the interior? It's region II
that's
> normally called the interior, and it's certainly not
singularity-free. You
> can't get there (region III) from here (region I).

This is the main conclusion of my paper: the decline in density at the
horizon mean that the asymptotically flat Region III is the interior,
not Region II. Regions I and III are joined at r=2M, so you can get
there from here. This is difficult to visualize in three dimensions,
but an equatorial slice looks like an Einstein-Rosen bridge, with the
lower half below the neck representing the interior. As I show in my
paper, it is regions II and IV which singularities that are
inaccessible, given that density falls at the horizon.

> > Radii diverge in the interior, and at lower radii spheres have
larger surface
> > area, implying r=2M (Mars et al [32]).

> This makes no sense. Region III has the same geometry as region I.

Yes the same geometry but Region III is a white hole region, and matter
is repelled from r=2M. As you fall from r>2M, you read smaller and
smaller spheres until you reach r=2M, and then you fall toward larger
and larger spheres. That is, you are in an expanding universe.

> You seem
> to be talking about tachyonic worldlines lying on the embedding
diagram,
> which converge until they reach the throat and then diverge.

No; imagine particles or photons falling two nearby radii, and the
question is whether they converge or diverge.

> I assume you're no longer talking about the Schwarzschild geometry?

This is still Schwarzschild, but with Regions I and III joined at r=2M.

Hunter

.

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