| From: | Adam Getchell <acgetchell@*******.EDU> |
|---|---|
| Subject: | A Peek Behind the Curtain: Warning!--gory details (was [OT] |
| Date: | Tue, 3 Nov 1998 11:46:14 -0800 |
>farther away (lessenign attraction and deceleration), but is being
>declerated relatively rapidly; its loosing more than 1/2 its speed in a
>given time period. In an escape situation, it loses less than 1/2 its
>speed in the same period. Then theres the funky sitution where it looses
>exactly 1/2 speed in that period; 1- 1/2 - 1/4 - 1/8 ... = 0 , but it will
>take forever for it to get to zero. The universes expansion rate would
>fall into one of those 3 categories.
> This is a horrible bastardization of the calculus involved (which I
>have not specifically studied), but it does point out the basic aspect of
>the limits / infinite series involved.
I'm not sure where you got this, but this is not the way to solve the problem.
A physicist would start with the gravitational potential written in
generalized coordinates:
ds^2 = g subscript mu, nu dx^mu dx^nu (ascii is awful for
equations) where g subscript mu, nu is the metric.
From the generalized description above one writes the connexion, or
Christoffel symbol which represents the acceleration of a free-falling
particle in a gravitational field. Implicit with the connection is the idea
of a spacetime, which is defined as a differentiable 4 dimensional manifold
possessing a Lorentzian metric (i.e. of the form diag [-,+,+,+]).
Knowing the connexion one can write the geodesic, which is the shortest
path in terms of proper distance, given that distance is usually Lorentzian
3+1 space/time. One can generally write spacelike (Riemannian) or timelike
(Lorentzian) geodesics.
With this, finally, you can write the Riemann curvature tensor which,
loosely speaking, governs the difference in acceleration of two
free-falling particles (e.g. the Moon's contribution to the Earth's Riemann
tensor causes the dominant time dependent piece at the surface known as
tides).
Contracting the Riemann tensor in various ways gives you the Ricci tensor,
Ricci scalar, and Einstein tensor that you use to then write the full
Einstein field equations which gives you the stress energy tensor used to
(theoretically) solve for the curvature of space at any specified points.
Still with me?
Most of the time the Einstein equations are nonlinear and therefore
intractable, barring special solutions (Schwarzschild, Reissner-Nordstrom,
Kerr). To work with space effectively therefore one either assumes weak
gravitational fields and uses the Minkowski four vector (diag[-1,1,1,1] or
flat spacetime; unsuitable for black holes and the universal expansion but
sufficient for Special Relativity) or, if one must deal with strong gravity
such as the end of the universe, assumes a well-behaved manifold and
employs the Arnowitt-Deser-Misner split to decompose the 3+1 metric into a
geometry of space, lapse, and shift functions.
Within the bounds of the ADM split you can solve most problems such as
wormholes, black holes, and gravitational collapse of the universe.
>Mongoose
--Adam
acgetchell@*******.edu
"Invincibility is in oneself, vulnerability in the opponent." --Sun Tzu
