Quantum Gravity Via Expansion-Contraction 18.0: Time Generates Space and Expansion/Contraction via Riccati Paraboloid

**OsherD** · #1 August 29th 06 posted to sci.physics

From Osher Doctorow

Readers may begin to get the idea that Quadratic equations, whether
differential or otherwise, have something fundamental to do with the
Universe from this thread. Take a look at the keyword "quadratic" in
arXiv, which produces 223 papers, and the same keyword in Front for the
Mathematics ArXiv produces 1211 papers with over 7 pages of references
out of the first 12 pages (covering 250 papers) written in 2006.

The idea that Quadratics are of unusual importance indicates that it
may be a good idea to look at the Riccati Differential equation once
again:

1) dy/dt = A(t) + B(t)y + C(t)y^2

and see whether we might have missed something obvious. Actually, the
generalization:

2) Dt(y(x,t)) = A(t) + B(t)y + C(t)y^2

from a recent Section of this thread also helps give the clue.
Equation (2) for Dt(y(x,t)) = 0, 0, 0 respectively give a parabola
and its 2 bounding regions ("inside" vs "outside" would be appropriate
descriptions of these two if the parabola had been an ellipse, for
example) in the (y, Dt(y)) plane, although the parabola changes for
different times.

But we missed one obvious generalization he

3) x^2/a^2 + y^2/b^2 = ct

Or, to put it another way, let's rewrite (2) as:

4) Dt(y) = C(t)y^2

for simplicity and then generalize to a second (x) term with C(t)
written as a(t):

5) Dt(y) = a(t)y^2 + b(t)x^2

The three cases Dt(y) = ct, ct, ct respectively generate a
paraboloid in 3 dimensions (x, y, and t) and its 2 bounding regions or
2 regions on either side of it.

I'll try to take a further look at this, which is indicated by (3),
next time.

Osher Doctorow

**OsherD** · #2 August 29th 06 posted to sci.physics

From Osher Doctorow

Look at the following sequence of equations generalizing from a simple
Riccati Differential equation parabola and its two "outside" regions to
a 3-dimensional paraboloid (infinite thimble) and its two "outside"
regions:

1) dy/dt = C(t)y^2 in the (y, dy/dt) plane (and = replaced by ,
etc.)
2) dy/dt = x^2/a^2 + y^2/b^2 - ct = 0 (and = replaced by , , etc.)

The second equation only has two spatial variables, x and y, but this
is OK with 2+1 dimensional GR as we've seen in recent Sections.

Now let's consider:

3) x^2/a^2 + y^2/b^2 = ct

and add 1 - ct to both sides:

4) 1 + x^2/a^2 + y^2/b^2 - ct = 1

which says for x, y in [0, 1]:

5) P(ct -- x^2/a^2 + y^2/b^2) = 1

which says that time t generates the two spatial x and y coordinates
with probability 1 (roughly speaking but not exactly "certainty").
Note that the sum of squares in (5) can be regarded as deriving from
elliptic cross-sections of paraboloids.

Perhaps most interesting, if c = 0 we get a point (0, 0) in the x-y
plane, while if c 0 the paraboloid opens upward in time (expansion)
while if c 0 it opens downward in time (contraction). So we have
here a model of a cyclic Universe generated first from contraction to a
singularity at (0, 0) and then expanding, with c now becoming a
"piecewise constant" quantity to reflect the three different eras of a
typical cycle.

Osher Doctorow

**fred botjon** · #3 August 29th 06 posted to sci.physics

"OsherD" wrote in message
oups.com...
From Osher Doctorow

Readers may begin to get the idea that Quadratic equations, whether
differential or otherwise, have something fundamental to do with the
Universe from this thread. Take a look at the keyword "quadratic" in
arXiv, which produces 223 papers, and the same keyword in Front for the
Mathematics ArXiv produces 1211 papers with over 7 pages of references
out of the first 12 pages (covering 250 papers) written in 2006.

The idea that Quadratics are of unusual importance indicates that it
may be a good idea to look at the Riccati Differential equation once
again:

1) dy/dt = A(t) + B(t)y + C(t)y^2

and see whether we might have missed something obvious. Actually, the
generalization:

2) Dt(y(x,t)) = A(t) + B(t)y + C(t)y^2

the left side is wrong.

from a recent Section of this thread also helps give the clue.
Equation (2) for Dt(y(x,t)) = 0, 0, 0 respectively give a parabola
and its 2 bounding regions ("inside" vs "outside" would be appropriate
descriptions of these two if the parabola had been an ellipse, for
example) in the (y, Dt(y)) plane, although the parabola changes for
different times.

But we missed one obvious generalization he

3) x^2/a^2 + y^2/b^2 = ct

Or, to put it another way, let's rewrite (2) as:

4) Dt(y) = C(t)y^2

left side is wrong again, and how did that come from (2)?

**OsherD** · #4 August 31st 06 posted to sci.physics

From Osher Doctorow

fred botjon (whose only previous postings on usenet are listed as
occurring this August, most on Austin.com, where most sci.physics
spammers hang out) wrote:

left side is wrong again, and how did that come from (2)?

I guess Fred doesn't understand the word "generalize". Well, maybe
it's hard to generalize if you've got "is wrong" on the brain.

Osher

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