Re: The Raychaudhuri Equation and Expansion-Contraction

>>From Osher Doctorow

whopk said:

>If it holds true, then Quantum Gravity suddenly >appears through the
>back door: a quantum theory established on locally >hyperbolic
spacetime
>regions, with a requirement for compatibility between >regions
includes
>as a corollary Einstein's field equations.


You are going in the correct direction in my opinion because of
something about "coarse graining" that I want to discuss (among other
things).

Coarse graining groups microstates together (makes them "equivalent")
to form countable sets, say on the basis of their variations because
close in position and momentum.

When we examine why "symmetries" such as the rotations work, we tend to
concentrate on definitions that leave geometric objects invariant in
the intuitive sense of looking the same. But from a dynamic or
kinematic perspective, what about "acting the same"?

Let's consider a circle. You can rotate it about its center ad
infinitum and get the same circle, so all these rotations form an
infinite symmetry rotation group, but what if you're a grinder and your
"circle" gets dented? You could always save yourself by calling the
other rotations that don't end on the dents "symmetry transformations",
but what if you're an "infinite grinder" with dents in all sorts of
places where there was originally a circle or cylinder? What do you
do?

Actually, what you can do is to notice something that includes both the
undented and the dented circle or cylinder: expansion-contraction.
The original circle or cylinder when described by its equation can be
regarded as referring to expansion from the axis in any direction
perpendicular to the axis. The dented circle or cylinder needs an
equation describing different amounts or distances of the dents from
the axis in each cross-section in different directions. It may need
different equations at each point on the axis from which expansion is
geneated.

Coarse-graining, in other words, can be reformulated in the language of
Expansion-Contraction and is much more effective there because it is
generalizable to more scenarios than before as with this example.

Spheres are even better for examples of expansion with "regularity"
versus irregular dents. And of course (solid) balls and solid
cylinders and so on are in some ways ideal for visualizing some of
these things.

In response to the question that I asked earlier, to simplify the study
of Expansion-Contraction, by all means start with symmetry and groups
and so algebra, but then move like heck fast to kinematics and
dynamics. Locally hyperbolic geometries and spacetime regions are nice
ways of doing this, and GR as compatibility between regions is a
fascinating idea, but then we see GR as thereby very possibly encoding
something that Expansion-Contraction encodes, and we learn more about
both.

Osher Doctorow

.

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