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Russell E. Rierson

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[@^2y/@t^2] = v^2[ @^2 y/@x^2] , where @ denotes the partial derivative symbol.

Standing waves can be set up in an 1-dimensional string, analogous to that in a violin string. The form of the standing wave becomes y(x,t) = psi(x) sin (wt)

Two sinusoidal travelling waves with the same amplitude and wavelength moving in opposite directions on a string, become resonating "standing waves":

y(x,t) = y1(x,t) + y2(x,t) = Asin(kx - wt) + Asin(kx + wt) = [2Asin(kx)]*cos(wt).

As the entropy continues to increase in the universe, and if the universe is a closed system, the entropy may be considered to be the result of a "damping force". This damping force may also be one possible solution to the

*dark matter*enigma.

Solve the Schwarzschild solution for the entire universe, since the universe can be postulated to be a closed system with

*nothing*outside itself. The condition of "nothingness" leads one to ask "What are the properties of nothingness?" Of course there are no measurable properties, but nothingness in itself must be a type of massless solid. A condition that has no distance - metric scales. In other words, there is no outside to the universe, no measurable border between something and nothing.

Nothing then becomes analogous to a perfectly symmetrical pressure force on the surface of existence.

-(F)^2 ---->|U|<---- +(F)^2

Simple harmonic oscillation given by the equation (F)^2 = -(K*X)^2

What is K ? What is X ?

U stands for universe. So it becomes reasonable to assume that the entire universe is analogous to that which is inside the event horizon of a black hole. The cosmos becomes a quantum superposition of states, collapsing under the crushing force of "nothingness".

Analytically continue the Schwarzschild solution to the imaginary values of the time variable. The Schwarzschild solution becomes periodic in the imaginary time direction.

All waves would then be standing waves in the closed universe. A Schrodinger wave equation in one dimension is of the form:

d^2 psi/dx^2 + (2m/hbar^2) [E - U(x)] psi(x)

U(x) is the potential energy and E is the total energy.

psi(x) is the wave function for a state in which the energy E is constant in time. Such states are called

*stationary states*. Certain definite vibration frequencies are allowed multiples of fundamental wavelengths

lambda = h/p

|psi(x)|^2 dx, is the probability of finding a particle(universe) in a certain state between the region x and x+dx

psi^2 = psi psi* . When psi is complex, psi* is the complex conjugate of psi. psi^2 (x) is the probability density.

An equation for the damped oscillator in one dimension:

X = A[exp[-(b/2m)t]]*cos[wt + theta]

Why not describe Einstein's equation as a rule that tells the geometry of space how to evolve as function of time? Lorentzian manifolds M, diffeomorphic to R x S, where the manifold S represents space, and t, an element of R, represents time. So spacetime is sliced into instants of time as an arbitrary choice, or possibly boundary limits, imposed by Planck's constant.

F: M---> R x S

Spacetime becomes quantized or "sliced up" but that could be what nature really does. According to relativity, an objects position and momentum can only be defined with respect to a frame of reference, i.e. another object. Yet the universe as a whole has no frame of reference outside of itself, so how can its momentum be defined? It can only be defined with reference to itself. Worldlines fill up spacetime and the criss crossing of world lines mark events beyond the need for coordinate systems or coordinates. Points in spacetime are given the name "events" so there is a coordinate independence.

The geometric view of physics means that the laws of physics are the same in every Lorentz reference system. Local Lorentz invariance. But since the universe has no exterior reference frame, and it must refer to itself, its world line intersects with itself. This quantized-evolution of spacetime dictated by GR and QM, means that the world line of the past intersects with the world lines of the present, for the universe. A geometric stacking of space like slices, parameterized by t, The universe is a function of itself. Spacetime becomes compressed. As the time evolution proceeds in the thermodynamic direction of t, the space like sheets continually increase in density. The information storage of space time.

(->(->(->(U)<-)<-)<-)

This increasing refractive spacetime density must be background independent. The increasing density functions are, in a sense, equivalent to the non-Euclidean geometry of Riemann and Einstein.

Russell E. Rierson

analog57@yahoo.com