Science > Physics > Quantum Gravity 160.1: Sine-Gordon Equation Related To Cyclic Acceleration-Constant-Deceleration Universe By PI
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
30 Jun 2007 07:30:11 PM |
| Object: |
Quantum Gravity 160.1: Sine-Gordon Equation Related To Cyclic Acceleration-Constant-Deceleration Universe By PI |
From Osher Doctorow
Take a look at Wolfram's "Sine-Gordon Equation" and also Wikipedia's.
This equation is:
1) Dxx(v) - Dtt(v) = sin(x)
where Dxx is the second partial derivative with respect to x, and so
on. It has kink and anti-kink solutions with collisional properties
of solitons. The 1-soliton solutions for v are:
2) v = 4 arctan exp[mg(x - vt) + delta]
where g is given by:
3) g^2 = 1/(1 - v^2)
and g > 0 gives a kink, g < 0 an anti-kink.
Let's compare this with the Probable Causation/Influence (PI) Equation
obtained last time:
4) P(A) + P(A ' B ' ) = /sin(t)/
or:
5) 1 - P(A ' B) = /sin(t)/
where A is the acceleration (inflation, late acceleration) force and B
is gravitation.
Equations (1) and (4) have considerable similarity in form except that
sin(x) changes to sin(t) and - to + (although the - in (5) seems to
indicate the - direction). If x and t are approximately equal (which
in PI is the optimal condition), then only the conversion of - to +
need concern us. Notice that time is the Cause and space (x) the
Effect in (1), and that we can convert (1) to:
6) 1 + Dxx(v) - Dtt(v) = 1 + sin(x)
or:
7) P(Dtt(v) --> Dxx(v)) = 1 + sin(x)
with proper normalizations.
Osher Doctorow
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