From Osher Doctorow
Now take a look at "Spacetime Symmetries" in Wikipedia, according to
which zero Lie derivatives "preserve" their arguments and thereby give
rise to a Symmetry. For example:
1) Killing Symmetry: L_X(gab) = 0 (L_X is L with respect to vector
field X)
2) Homothetical Symmetry: L_X(gab) = 2kgab, k real constant
3) Affine Symmetry: (L_X(gab))_;c = 0 (;c expresses covariant
derivative wrt c)
4) Matter Symmetry: L_X(Tab) = 0, Tab energy-momentum tensor
(components)
Here gab is the metric tensor with ab subscript indices.
When we move to the real double arguments:
5) L(x, y) = 2xy
we no longer see any connection with 0, and when we move even further
to identical arguments:
6) L(y, y) = 2y^2
we get a contribution to Riccati Differential Equation change dy/dt:
7) dy/dt = A(t) + B(t)y + C(t)y^2 = A(t) + L(y, 1)B(t)/2 + L(y, y)C(t)/
2
and of course L(y, 1) = 2y has this property also.
We are arguably seeing a different "dimension" of Lie Derivatives when
we generalize them to arguments that are couples of real variables
with real values, namely a dimension in which instead of preserving
variables as with Symmetry and Conservation, the emphasis is on
increasing variables as with expansion and (Probable) Influence/
Causation.
The surprise is that for both Energy and Force, at least in classical
formulations, the second interpretation of Lie Derivatives holds
also! While ordinary Lie Derivatives are related to Conservation of
Energy, the Double Generalized Lie Derivative L(v, v) = 2v^2 just
differs by a constant m from Kinetic Energy, and Force = mDtt(y) for
displacement/position y is an operator analog of this. They do not
represent "Conservations" but pushes and pulls and work or readiness
for work.
The view of GR, and classical physics, also implicitly adopted in much
of Quantum Theory, that various things are preserved, must now be seen
as only one side of a duality in which, via generalized Lie
Derivatives, things are just as plausibly increasing! And from this
viewpoint, the speed of light being a finite upper bound to speed and
the Planck constant being a finite lower bound no longer seem
intuitive to say the least in the most general scenarios.
The Riccati Differential Equation is now seen as the opposite of a
"Conservation" scenario, but rather as an Expansion or "Push/Pull"
scenario, or as a Contraction scenario.
This is why, incidentally, I think that Hoyle beat Einstein to the
expansion of the Universe, because Einstein was not oriented toward
expansion but toward Conservation. It was not a coincidence but a
misreading of Fundamentals.
Osher Doctorow
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