[TML] White Dwarfs, Black Holes & 100 Diameters

[Jerry W Barrington]

On 11/3/07 11:01 PM, Timothy Little wrote:

> On Wed, Oct 31, 2007 at 04:16:17AM -0400, Jerry W Barrington wrote:
>> But that measure will depend not merely on the position, but on the
>> trajectory of the test particle.  As such, it isn't really measuring
>> the curvature of space, but the effect on certain trajectories.
> 
> The curvature *is* the effect on trajectories (technically called
> geodesics).  In GR, given the Riemann tensor values over a volume of
> spacetime you can compute the gravitational effect on all possible
> trajectories within it - and vice versa.

In a philosophical sense, the curvature is the effect.  In a practical
sense, the effect is an infinite sheaf of differences, worse than just the
3-part tensor.  :)

Also, integrating the values over a volume is nice, but a point has
curvature regardless of the extended volume around.

>> Even reduced to a 3 part vector, you can't compare them like you can
>> scalars.  With scalars, you can clearly say that *this* value is
>> more than a set limit, and *that* value is less.
> 
> What I was showing was that the Riemann curvature tensor is a lot less
> scary in practice than it might look.  Also, that those 3 curvature
> numbers within the Riemann tensor are essentially *equivalent* to the
> tidal gradient, not just loosely related in some way.  If all you want
> to do is compare magnitudes, you can obtain a physically meaningful
> scalar by computing the the square root of sum of squares of those 3
> components.
> 
> It may be worth noting here that for weak gravity (i.e. just about
> anywhere but near black holes), the curvature tensor components are
> insignificantly different from the gradient given by Newton's Law of
> Gravitation.  If dominated by a single massive body, the magnitude of
> curvature (i.e. gradient) is proportional to (mass / distance^3).

This is probably the most useful thing to do in game terms.

> An interesting part comes where there are multiple significant bodies.
> No two spherical masses can cancel out curvature at any point, but
> three can if they all happen to be at exactly the right distances in
> the right directions.  This would practically never happen by chance
> in a system, though.

This could complicate the game math.  Altho complete cancellation would be
unlikely, how do we find possible reductions?  Find a safe jump point that
is closer would always be useful!