[TML] White Dwarfs, Black Holes & 100 Diameters

[Jerry W Barrington]

On 10/30/07 8:26 PM, Timothy Little wrote:

> However, in the usual cases we're interested in, it is vastly easier:
> Tides are a rate of change of gravitational acceleration over
> distance.  Acceleration is a rate of change of velocity over time.  So
> to determine tidal gradient we can start with the initial velocity
> vector, shift it out some short distance, shift it forward some short
> time, return it back the same distance, then compare it with the same
> vector just moved forward in time.  In other words, compare the vector
> moved around two different paths in spacetime - exactly what the
> Riemann tensor describes.

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.

> If we choose our coordinates well and make some useful assumptions, we
> can simplify the math drastically.  It turns out that of the 4x4x4x4
> components of the Riemann tensor, we're only interested in 3x3 in most
> situations.  Even better, we can arrange our coordinates so that most
> of those 3x3 components of the Riemann tensor are zeroes, and we're
> left with just 3 numbers.  These are exactly the strengths of the
> tidal gradient in the 3 principal directions.
> 
> So the much shorter version is: in most situations of interest you can
> reduce the nasty Riemann curvature tensor to just three numbers that
> describe the strength of tidal gradient in 3 directions.

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.

If in 2 tidal gradients (A & B), A has slightly larger value of the
principal direction than B, but B has slightly larger in the other 2
directions, which is "bigger".  Plus it is possible to have "odd"
configurations of space.  For instance, imagine a section of space that is
curved like the surface of a cylinder.  In a very real way, it behaves as if
not curved at all!  Lines that start parallel remain so, just like in a flat
plain.  :)