Science > Physics > Quantum Gravity Via Expansion-Contraction 77.2: More Re Fundamental Equation of Quantum Gravity
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
18 Jan 2007 11:51:50 PM |
| Object: |
Quantum Gravity Via Expansion-Contraction 77.2: More Re Fundamental Equation of Quantum Gravity |
From Osher Doctorow
Let's look at the Friedmann Equations:
1) (a-dot/a)^2 = (8pi G rho + Lambda)/3 - Kc^2/a^2
2) 3a-dot-dot/a = Lambda - 4piG(rho + 3p/c^2)
where dot is another expression for d( . )/dt, and dot-dot is the
second derivative. Here rho (density of fluid) and p (pressure of
fluid) are usually functions of the scale factor or "radius" a.
The possibility that the left-hand side of (1) is approximately
(da/dt)/a suggests the Riccati Differential equation or a
generalization of it, and we have:
3) u^2 = u for real u iff u(u - 1) = 0 iff u = 1 or u = 0
So for (da/dt)/a = 1 or 0, or very near this, we get:
4) (da/dt)/a = [(da/dt)/a]^2 if (da/dt)/a = 1 or 0
5) same as (4) except "approximately", near enough to 1 or 0
Notice what (da/dt)/a = 1 or 0 involves:
6) (da/dt)/a = 1 iff da/dt = a iff a = a(0)exp(t)
7) (da/dt)/a = 0 iff da/dt = constant iff a = constant times t + a
constant
In both cases, a satisfies the Riccati Differential equation:
8) da/dt = A(t) + B(t)a + C(t)a^2
with C(t) = 0 and with respectively B(t) = 1 and A(t) = 0 or B(t) = 0
and A(t) = constant.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity Via Expansion-Contraction 77.2: More Re Fundamental Equation of Quantum Gravity |
19 Jan 2007 12:07:35 AM |
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From Osher Doctorow
Let's see what condition will yield da/dt = (da/dt)^2:
1) da/dt = (da/dt)^2 iff da/dt = 1 or 0 iff a = t + k or a = k1, k and
k1 constant
In this case, ((da/dt)/a)^2 = (da/dt)/a^2, and the first Friedmann
equation becomes:
2) ((da/dt)/a^2 = (1/3)(8piG rho + Lambda) - Kc^2/a^2 (if a is not 0)
and multiplying both sides by a^2 for a nonzero yields a Riccati
Differential equation in a with respect to change in time (t) provided
that rho, Lambda, are approximately constant or very slowly varying
functions of t compared to a.
Osher Doctorow
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