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From: OsherD on 11 May 2010 21:35
From Osher Doctorow

Robert D. Bock of Propagation Research Associates, Marietta Georgia
USA, in "Dynamics of the Universe in the Modified Unimodular Theory of
Gravity," arXiv: 0704.2406 v1 [gr-qc] 18 Apr 2007, 4 pages (2 columns
per page) describes the Unimodular Theory of Gravity, which involves
some very important solutions which are Riccati-like Differential
Equations for dy/dt where y is energy density and which dispense
entirely with the cosmological constant so that there are only two
epochs: the radiation-dominated epoch followed by the matter-dominated
epoch.

Jose M. Gracia-Bondia of u. de Zaragoza Spain and U. de Costa Rica in
Costa Rica, in "Notes on 'quantum gravity' and noncommutative
geometry," arXiv: 1005.1174 v1 [hep-th] 7 May 2010, 54 pages, tends to
agree with Bock above, and points out 7 major problems with "classical
gravitation" including Pioneer 10 and 11 anomalies and the slingshot
maneuver in the flight of probe Galileo by Earth and so on. He
concludes among other things that in the present form noncommutative
geometry favors the Unimodular Theory of Gravity.

In the Unimodular Theory of Gravity (UTG for short), the metric tensor
reduces under the general coordinate category into 2 important
geometric objects: g (determinant of metric tensor), and gamma_uv (the
relative tensor g_uv/(sqrt(-g)) with determinant-1. The relative
tensor alone determines the null-cone or causal structure, and the
determinant determines the measure structure of spacetime. UTG
assumes a background measure field sqrt(-g) = sigma(x^i) or
sigma(x^alpha).

I will try to discuss these papers further later. The Riccati
Differential Equations referred to are ones of the exponential decay/
growth type, and several important scenarios for exponential energy
creation or growth occur with the independent variable time being
either linear or quadratic and Hubble H being either constant or
linearly time dependent, while Gaussian curvature approaches 0 as t --
> infinity which also follows from standard inflation models. Future
work may involve variable rather than constant coefficients and vacuum
fluctuations cancel out in the field equations as well as not
requiring postulating a false vacuum state in the first 10^(-35)
seconds after a Big Bang.

Riccati Differential Equations are closely related to Probable
Causation/Influence (PI) from this thread, although most closely if
one or more past times or past time variables are included rather than
only "present time".

Osher Doctorow
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