Quantum Gravity Via Expansion-Contraction 41.1: Scalar Einstein Equation in Probable Causation of Tensors, Matrices, Determinants - Physics Banter

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Quantum Gravity Via Expansion-Contraction 41.1: Scalar Einstein Equation in Probable Causation of Tensors, Matrices, Determinants

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Quantum Gravity Via Expansion-Contraction 41.1: Scalar Einstein Equation in Probable Causation of Tensors, Matrices, Determinants

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#1

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December 2nd 06 posted to sci.physics

OsherD

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Quantum Gravity Via Expansion-Contraction 41.1: Scalar Einstein Equation in Probable Causation of Tensors, Matrices, Determinants

From Osher Doctorow

Whether there exists a scalar Einstein or "Neo-Einstein" equation is
somewhat conjectural, so I'm just following this idea up to see if it
works.

The calculus of variations proof of the Einstein Field Equation(s)
involves varying the action S which in the notation of Wikipedia's
"Einstein-Hilbert Action" is:

1) S[g] = kI[R sqrt(-g)]d^4 x, I...d^4 x an integral of a region of
spacetime

But the quantity R sqrt(-g) is an invariant and in fact g = det(gab), R
= Ricci scalar, so instead of varying S with respect to the reciprocal
of the metric as usual to get the Einstein Field Equation(s), we can
arguably make contact with the scalar nature of Probable Causation with
respect to determinants and alternating series by only using the
integrand, namely:

2) R sqrt(-g)

In greater generality, the proof replaces R sqrt(-g) in (1) by [kR +
L_M]sqrt(-g), and even more generally if a Cosmological Constant Lambda
is used we replace kR by k(R - 2 Lambda).

So in the most general version, our question is whether:

3) [k(R - 2 Lambda) + L_M]sqrt(-g) = f

where f is some variable or constant or piecewise constant function
having physical meaning on its own.

The nice thing about (3) is that it seems to directly relate in scalar
fashion curvature, the cosmological constant, and the matter Lagrangian
L_M which is also a scalar. The simplest choice seems to be f =
constant, but it could also be a function of space, time, probability,
etc.

Osher Doctorow

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#2

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December 2nd 06 posted to sci.physics

OsherD

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Posts: 3,831

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Quantum Gravity Via Expansion-Contraction 41.1: Scalar Einstein Equation in Probable Causation of Tensors, Matrices, Determinants

From Osher Doctorow

For example, let's examine f = k1 (real constant scalar):

1) [k(R - 2 Lambda) + L_M]sqrt(-g) = k1

whose solution for R is:

2) R = 2Lambda + [k1/sqrt(-g) - L_M]/k

So the Ricci scalar R gets a slightly positive contribution from
positive but almost zero Lambda and depending on whether f = k1 is
positive or negative it gets either a positive or negative contribution
from 1/sqrt(-g) where g = det(g_ab) = det(gab). It always gets a
negative contribution from nonnegative L_M assuming that k is positive,
so that Ricci curvature "decreases with mass/energy" in the appropriate
range, or if the second bracket is negative (for large enough L_M
positive), R becomes more negative "with all else more or less
constant". In this last scenario, "more mass/energy makes Ricci
curvature more negative". The dependence on sqrt(-g) in the
hypothetical case that g varies based on coefficients of gab is also
noteworthy, or if g doesn't vary then k1/sqrt(-g) acts like a sort of
logistic "maximum supply" with Lambda working to deep R positive and
L_M up to the supply point adding less and less to R until over the
supply point k1/sqrt(-g) it works to make R more and more negative.

In this last scenario, the Cosmological constant Lambda works in the
opposite (positive) direction to the mass/energy Lagrangian L_M, as we
would expect in intuitive Quantum Gravity, relative to curvature.

Osher Doctorow

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#3

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December 2nd 06 posted to sci.physics

OsherD

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Posts: 3,831

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Quantum Gravity Via Expansion-Contraction 41.1: Scalar Einstein Equation in Probable Causation of Tensors, Matrices, Determinants

From Osher Doctorow

Usually one wants the Ricci curvature (scalar) to be nonnegative even
if the Gaussian curvature is to be negative, but similar ideas to the
above apply except that the intervals of positive and negative
curvature referred to earlier just get shifted upward to two positive
intervals.

Osher Doctorow

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