# Superconductive Planck/Scale Space

1. May 28, 2004

### Russell E. Rierson

If the universe cannot be expanding in any "absolute" sense, then it cannot be shrinking in an absolute sense, because there is no external frame of reference. All relations would be intrinsic to the universe. There would be no extrinsic perspective. So the question becomes, What exactly is space? Does separation between objects exist in an absolute sense? A metric space is a set of points such that for every pair of points, there is a nonnegative real number called their distance that is symmetric, and satisfies the triangle inequality, which states that the sum of the measures of any two sides of any triangle is greater than the measure of the third side.

Space is then a tranformation. Two objects with relative velocity will have a relative measure that transforms into the other. In effect, the separation does not exist in an extrinsic sense. Equilateral triangle rotation: ABC = BCA = CAB... Then it is realized that an absolute spatial separation cannot exist, therefore, the EPR paradox cannot actually exist. Distance interval, which is a property of space, is a type of dynamic relation. So, relativity is really a theory of invariants. Space is a set of invariance principles which, has a boundary that is zero. Yet, with the self inclusive manifold, information[structure-complexity] is increasing as a function of time.

Information is also a type of relation, in that certain invariants must hold.. So to describe tautologies of logic e.g. X or ~X , as absolute truths would not be a complete definition. A tautology is an invariance principle. A rule that transforms according to a choice of truth value, which is an invariant, in that it is always true. Yes, the force called gravity can be elucidated as a geometric effect, a "non-Euclidean geometry", where spacetime becomes anisotropic and inhomogeneous in the presence of mass-energy. Then the question becomes "what is space?" ..."What is time?" Space is relational. Time is the manifold changes OF ...space.

Heisenberg Uncertainty: DxDp >= hbar/2

The relation becomes totally unpredictable below the Planck length. So yes, space could be described as a self similar relation which is generated by the quantum uncertainty and forms analogous Penrosian "spin networks".

The curvature of spacetime could be represented as a Gaussian distribution? If mathematics only is an approximation of reality, then the mathematics of probability corresponds "exactly" with reality. The Riemann tensor explains how a tangent vector, parallel translated around a tiny parallellogram is changed. So, to say that spacetime is "curved" means how much a tangent vector changes during parallel transport around a loop. The translation of an infinitesimal tangent vector along a geodesic. So the probability distribution should agree exactly with Einstein's relativity.

The gravitational field, described by the metric of spacetime g_uv , is generated by the stress-energy tensor T_uv of matter. Various field equations relating g_uv to T_uv have been proposed. The most succsessful have been the Einstein field equations, which are of course, the foundation of general relativity.

G_uv == R_uv - 1/2 g_uv R = 8pi T_uv

where R_uv and R are the Ricci tensor and scalar curvature derived from the metric g_uv , and G_uv is the Einstein tensor.

The equations are non-linear, since the left hand side is not a linear function of the metric.

When the gravitational field is weak, the geometry of spacetime is nearly flat and the equation is: g_uv = n_uv + h_uv where all h_uv are << 1.

This linearized theory is very interesting.

A viable option for the resolution of the problem, is that space is something analogous to homogeneously distributed probability density functions(a perfect fluid?) i.e. increasing energy-density gradients, giving the observed thermodynamic arrow of time. The observed cosmic expansion is, again, a "relative" one. A perspective effect from our local vantage point. A shrinking object gives the illusion of receding motion. Increasing *refractive* density gradients give the appearence of a doppler-red-shift. Space increases density as matter/energy refractive density gradients increase via covariant derivatives maintaining diffeomorphism invariance.

Space can be hypothesized as a type of conductor, becoming "superconductive"? on the Planck scale? e is the permittivity of free space and u is the permeability of free space, epsilon and mu respectively. E = mc^2, c^2 = 1/(e*u), E/m = 1/(e*u) The ratio of ([total energy]/mass) = 1/(e*u), 1/sqrt(e*u) would of course remain "ivariant" while observing from inertial reference frames but the E and m values would individually vary, yet vary in tandem producing a constant "c" or 1/sqrt(eu) . 1/(eu) = (Ds/Dt)^2 , Ds and Dt would also vary in tandem, depending upon relative velocity. Ds/(eu)^(1/2) = Dt It seems that the permittivity and permeability of free space would change yet maintain a constant product, such that c remains invariant, thus the "curvature" of space-time is also explainable as the result of discrete space "bits" interacting as superconductive impedance variance modes, a Lorentz invariant model of gravitation. So discrete spacetime is niether Euclidean nor non-Euclidean, but rather, space is an informational structure.

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.

2. May 31, 2004

### Russell E. Rierson

Mathematically speaking, the existence of a "dual" vector space,
abstractly reflects the relationship between row vectors (1×n) and
column vectors (n×1). The construction can also take place for
infinite-dimensional spaces and gives rise to important ways of
looking at different distributions and Hilbert space. The use of the
dual space can be a characteristic of functional analysis. It is also
built into the Fourier transform.

Because the tangent space and the cotangent space at a given point are
both real vector spaces of the same dimension, they are isomorphic to
each other. But they are not "naturally isomorphic", since, for an
arbitrary tangent covector, there is no canonical tangent vector
associated with it. With the introduction of a symplectic form, the
additional structure gives a "natural isomorphism". Consequently, the
distinction between the tangent space and the cotangent space must be
maintained. Many definitions are more natural on one space than on the
other. All the cotangent spaces of a manifold can be "glued together"
to form a new differentiable manifold of twice the dimension, the
cotangent bundle of the manifold.

The vacuum equations of Einstein, appear as the optimal minimization
at constant volume for the simplest geometric functional of curvature
via the integral of scalar curvature using the Ricci scalar; pure
mathematics. First done as the Hilbert Einstein Action. To get the
non-vacuum equations requires the stress energy tensor; pure physics.
The "Uncertainty principle" of Werner Heisenberg also exists in
Fourier Analysis: It is a statement about time-frequency distributions
and is essentially equivalent to the Cauchy Schwartz inequality from
Functional Analysis; pure math. In that particular setting, the
constant corresponding to Planck's constant is normalized to be one.

Probability interpetations of QM have been obsolete for decades !

The geometric quantization using tensor fields on fiber bundles
generally have holonomy[parallel translation on a piecewise smooth
path] and noncommutative properties, from which, a geometric
quantization algebra can be constructed, equivalent to a geometric
quantization approach to Q.M.. The metaplectic[a double covering group
of the (linear) symplectic group.] structure, used to define geometric
quantization, is given on a bundle. General relativity can be put into
the Ashtekar variables with the Sen connection, interpreted in terms
of the metapletic covering.

A quantum theory of gravity...?

3. May 31, 2004

### Antonio Lao

The theory that I am working on is based implicitly on the belief that reality is one dimensional because of the motion of time with its two directions.

Further, we can also theorize that there is no distinction between the one dimension of space with that of time if we supposed that this one dimension of space possesses an intrinsic motion with also two directions.

Further, if we give the attributes of directional invariance properties to this one dimensional motion of space, we can define two dimensional space and three dimensional space. And then we can define mass and charge, and energy.

4. May 31, 2004

### Russell E. Rierson

http://www.arxiv.org/abs/quant-ph/0404040

5. Jun 2, 2004

### Antonio Lao

When space is reduced to one dimension, the spacetime interval can be given by

$$ds^2 = dx^2 - c^2 dt^2$$