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From: OsherD on 20 Apr 2010 01:35
From Osher Doctorow

Look at Wikipedia's online "Raychaudhuri Equation".

The equation is:

1) d(theta)/dt or Dt(theta) = w^2 - sigma^2 - theta^2/3 - E[X]^a
_( blank a) + Dt(X)^a _ (; a)

where "blank a" signifies that a is indented one space to the right, X
is a timelike unit vector field, sigma^2 is sigma_(mn) sigma^(mn), w^2
- w_(mn)w^(mn), respectively quadratic invariants of the shear tensor
and the vorticity tensor, theta is the trace of the expansion tensor
theta_(ab) (theta is called the expansion scalar).

For exact dust solutions of the Einstein Field Equation(s), vorticity
is 0, and if the world lines do not twist about one another, then (1)
becomes:

2) Dt(theta) = -sigma^2 - (1/3)(theta)^2 - R[X]_(blank a) ^a

leading either to a singularity or to a breakdown in the mathematical
model of the dust near where the singularity would be.

Readers will recognize (2) especially as being "almost" a Riccati
Differential Equation, or in fact a generalized (and/or modified)
Riccati Differential Equation. Thus it is close to Probable
Causation/Influence (PI), especially if a past time is included in the
arguments.

Thus, the "prohibition" of tensors in PI is not an "absolute" one,
especially when scalars or invariants of tensors are used as
variables.

Osher Doctorow

From: OsherD on 20 Apr 2010 01:39
From Osher Doctorow

Below (1), w^2 - should be "w^2 = ".

Osher Doctorow
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