From Osher Doctorow
When we recall from earlier threads of mine that the Probable
Influence/Causation (PI) is a one-sided partial inverse of the
Euclidean and Euclidean-like metric or distance function, then solving
the bimetric equations of the last few Sections becomes much easier.
Here I'll list the successive equations except that (1) and (2) are the
two bimetric equations, with a little explanation right after several
of the equations.
1) dt/dp = A(p) + B(p)t + C(p)t^2
2) ds/dt = A1(t) + B1(t)s + C1(t)s^2
3) ds/dt = exp(t) at and near Inflation
4) (ds/dt)(dt/dp) = ds/dp = exp(t)[A(p) + B(p)t + C(p)t^2]
5) s = r^2 = (x - y)^2
6) p = 1 + y - x
Here for simplicity I'm considering the Euclidean distance function
squared s = r^2 = (x - y)^2 on the real line between points x and y, in
which case Probable Influence/Causation PI = p is 1 + y - x. The
value of this may become apparent from the next few equations.
7) u = (definition) x - y
8) ds/dp = (ds/du)(du/dp) = 2(x - y)(-1) = 2(y - x) = 2(p - 1)
9) (from (4) and (8)) 2(p - ) = exp(t)[A(p) + B(p)t + C(p)t^2]
10) (solve (9) for rightmost factor) [A(p) + B(p)t + C(p)t^2] = 2(p -
1)/exp(t)
11) (from (1)) dt/dp = 2(p-1)/exp(t)
12) exp(t)dt = 2(p-1)dp from (11)
13) exp(t) = p^2 - 2p + k (integrating both sides of (12), with k
constant of integration)
14) exp(t) > = 1 (true for all t > = 0)
15) p^2 - 2p + k > 1 (required from (13) and (14)) iff k > 1 + 2p - p^2
For example, if k > = 2, then k > = 2 > = 1 + 2p - p^2 = 1 + p(2 - p)
iff 1 > = p(2 - p) iff p^2 - 2p + 1 > = 0 iff (p - 1)^2 > = 0, always
true
16) For k = 2, exp(t) = (p - 1)^2 + 1 (from (13)) iff t = log{(p-1)^2 +
1}
Notice that (16) is an exponential-quadratic equation (exponential in
time t, quadratic in Probable Influence/Causation p) and that the
closer PI = p is to 0, the bigger is exp(t) and the closer is exp(t) to
its maximum which is 2 and the bigger is t and the closer is t to its
maximum which is log(2).
Therefore, in this scenario or Toy Model, Inflation tends toward
"acausality" which in Probable Influence/Causation terminology is 0
Probable Influence/Causation. At the beginning of Inflation, PI is
maximum ( = 1) and exp(t) is minimum = 1 so that time t = 0 from
equation (16). So PI goes from maximum to minimum during Inflation.
Osher Doctorow
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