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