Science > Physics > Quantum Gravity Via Expansion-Contraction 27.0: y = exp(kx)/x and x/exp(kx) Types
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
19 Oct 2006 11:20:27 PM |
| Object: |
Quantum Gravity Via Expansion-Contraction 27.0: y = exp(kx)/x and x/exp(kx) Types |
From Osher Doctorow
We've seen that the Riccati Differential equation and its Logistic
Differential equation and Exponential Growth/Decay/Contraction equation
subtypes of most useful form have solutions:
1) y = k1exp(kt), k and k1 constant
2) y = kt + k1
3) y = exp(kt)/(1 + exp(kt))
4) y = k/t
In looking for Quantum Gravity, one plausible possibility is finding
functional forms which explain the Acceleration periods of the Universe
and/or expansion vs contraction of the Universe.
Two useful additional solutions are suggested by (1) and (4):
5) y = exp(kt)/t
6) y = t/exp(kt)
Their respective derivatives are:
7) dy/dt = (kt exp(kt) - exp(kt))/(t^2) = exp(kt)(kt - 1)/t^2 =
kexp(kt)/t - (1/t)(exp(kt)/t) = ky - y/t = y(k - 1/t)
8) dy/dt = (y/t)(1 - kt)
Notice that if we take k > 0, then in (7) we have dy/dt > 0 iff t >
1/k, while in (8) we have dy/dt > 0 iff t < 1/k.
So equations (5) and (7) describe contraction followed by expansion at
a later time if y is (principal) radius or some other positive function
of distance, while equations (6) and (8) describe expansion followed by
contraction at a later time. In both cases, the later time is t = 1/k.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity Via Expansion-Contraction 27.0: y = exp(kx)/x and x/exp(kx) Types |
19 Oct 2006 11:37:53 PM |
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From Osher Doctorow
Let's examine the acceleration of the Universe if the expansion has the
form of (5) and (7), that is to say:
1) y = exp(kt)/t, dy/dt = y(k - 1/t), k > 0 constant
Then we obtain:
2) y" (or Dtt(y)) = (k - 1/t)y' + y(1/t^2) > 0 iff k > 1/t iff t > 1/k
So the Universe starts accelerating when it starts expanding according
to (2) and (1), recalling from last time that dy/dt in (1) > 0 iff k >
1/t.
Readers can try various modifications to move acceleration to a time
somewhat later or earlier than expansion.
Osher Doctorow
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