Re: Path in Schwarzschild

In sci.physics.relativity, Koobee Wublee
<koobee.wublee@xxxxxxxxx>
wrote
on Thu, 20 Sep 2007 04:41:17 -0000
<1190263277.969685.180720@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>:
On Sep 19, 7:42 am, The Ghost In The Machine wrote:
In sci.physics.relativity, Koobee Wublee wrote:

This is the definition of the coordinate system. Ask Professor
Roberts to explain it to you. He can do a better job than I can on
this one.

The main issue, of course, is that one can pick a different
coordinate system and get a different result. 4 pi r^2
only works when one is working within a 3-dimensional
undistorted Euclidean space.

You are a little bit closer to the truth than the Einstein
Dingleberrie team lead by Mr. McCullough, Mr. Bielawski, Professor
Draper, and no-college-degree Gisse.

It does not matter what coordinate system you picked. Regardless how
space is curved, the observed surface area of a sphere is always (4 pi
r^2) where r is the observed radius of the sphere.

I would have to do some work to disprove this, but
presumably one can find a coordinate mapping and a metric
such that the following are true.

[1] For every point P in the embedding space E, which is
a Euclidean N-dimensional space (N >= 3), one can define
three values x'(P), y'(P), z'(P). I will call this point P'.
This embedding is bijective, at least around a certain
center point C or C'.

[2] A C1 or C2 [*] function m() between P_a and P_b is
such that m(P_a,P_c) <= m(P_a,P_b) + m(P_b,P_c), that
M(P_a,P_a)=0, M(P_a,P_b) = M(P_b,P_a), and
M(P_a,P_b) >= 0. There might be other conditions but I'd
have to look. This function can be construed as a metric
between P'_a and P'_b; define m'(P'_a,P'_b) = m(P_a, P_b).
Surface area can be defined in terms of m', though
it would take some work. Not sure how one would define
the angle between P'_a, P'_c, and P_'b either, or how it
would relate to the surface area of the eventual sphere
(which is, after all, the set of points P' such that
m'(P', C') = r' for some point C' and constant r'.)

[3] The surface area of the resultant sphere is not 4*pi*r'^2.

One extremely simple try might be P=(x,y,z), P'=(x*3,y*2,z*1),
where m'(P'_a,P'_b) is simply the distance between P_a and P_b.
Regrettably, my brain's too fried to do the calculations.

[rest snipped]

[*] C1 = continuous, C2 = continuous and once-differentiable,
if memory serves.

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