by Hilda and Stu - August 20, 2005

Our goal is to be the physics crackpots with the SHORTEST web rant. How about that?
(This rant should eventually be folded into antrobot.com, but right now it's easier for us to
publish here).

We attempt a holistic formulation of energy dynamics and spacetime curvature in a flowing,
stretchy, compressible spacetime which is not distinct from the matter and energy in it.

Our model is based on the idea of something
like compressible fluid flow of energy over a deforming, vibrating manifold. Our goal is to write
down a replacement for the Schrodinger and Einstein equations in terms of a continuous (wavelike)
deformation of spacetime, without the need for explicit particles, fields, or forces. We then
want to show how this theory may be projected to produce Einstein's,
Heisenberg/Planck/Schrodinger's, and Maxwell's formulations over limited scales and forms.

We begin with the case of local, humanly observable phenomena:

Let t be a smooth (infinitely differentiable) path through spacetime U, which we will henceforth
call "spacetime curve t in U". This t is our fundamental parameter.

All our functions are written relative to t, which for practical calculations is redefined
in terms of a path-parameter s in the usual way. In the classical projection,
s is regular old wall-clock time. However, we omit this notation in this paper, and stick
with the loosely defined idea of a path "t".

Let U(t) be our entire universe, which is assumed to contain some hypothetically quantifiable total
energy as seen from any particular bounded subset of it at a particular time. For the moment,
we avoid the question of the dimension, boundedness, shape or form of U(t). However, we plan to
later define a spacetime form U(t) that is projectable in the following way:

Let L(t) be a sum of separable convex compact projected 4-subspaces of U understood from t,
called a spacetime trajectory region. L(t) does
not need to "include" t in a conventional projection as the path of the observer.
However, if t IS the path of an observer, then our formulation should reduce to general relativity.
L(t) may be understood to be classically "moving" or "stationary" in U. Regardless, we call
it a spacetime trajectory region L(U,t).

"Separable" here means that while L may encompass stars and black holes and molecules, it must
not contain only part of a star or black hole or molecule, and may not receive or emit any
such material during our domain of validity, unless explicitly stated otherwise.

We leave aside for the moment the definition of topological norms (i.e. distance) in U.
We assume the conventional norms within classically sized L(t).

Whenever L or L(t) or L(U,t) is referenced, it will be understood that we formally mean (t, L(t)),
within the context of U(t) (which is not fully defined).

Caveat 1: Volume calculations in U and L are NOT assumed to be the same. Volume calculations
in L are the regular ones we like to make. Volume calculations in U take into account the
folding and vibrating nature of higher-dimensional space, which gives rise to observable
energy dynamics in L.

Caveat 2: Energy values for regions of U and projected L-s ARE assumed to be compatible,
and energy is assumed to be roughly conserved in most cases.

(Where I'm going with this is that there may be 'extra' or 'missing' actual volume inside a
region of U, consistent with the energy forms therein. This extra and missing volume explain
the movements we attribute to fields and forces.)

Let TEU(U(t)) = Total Deformation Energy of Universe U - a differentiable real form on t, in
Joules. We do not attempt to further define this function here, we merely treat it as
a parameter, with the assumption it must be as compatible as possible with our best
empirical understanding at t.

Let TEL(L(t)) = Total Deformation Energy of Local region L.
relativistically defined evolving 4-space L(t) projected from U(t). L must be compact but not
necessarily connected. In the classical projection, L has the usual norm and computable volume
after correction for local curvature and background oscillation effects.

Then LER(L(t),U(t),t)=TEL(L(t)) / TEU(U(t)) evaluates to a dimensionless numeric scalar function
of a spacetime path "t", indicating the Local Energy Ratio of L relative to U at t (all as "seen
from" L + t), which is a real number between 0 and 1. We rely on Caveat 2 to make this calculation
meaningful.

Familiar types of observable energy include: gravitational and electromagnetic
potential energy of systems contained within L, kinetic energy of forms within
L, all chemical binding energy and internal nuclear/electrical energy
of what we call "matter", or atoms and molecules and fluids.

These types enumerated are overlapping, implications of which we'll take up in another
paper. The intent here is simply to include all macrosopic energy forms that we know how
to observe or predict, including matter and light, particles and liquids and plasmas.

All of that stuff is energy in a deformable manifold, which flows around and deforms+vibrates
the manifold as it flows. Stable vibrations often indicate matter concentrations. Travelling
deformation waves often indicate EM/light energy propagation.
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What does LER measure?

LER measures the total amount of deformation of local spacetime (by energy and matter) of
the region L(t), where L(t) is allowed to change.

That is, for the region contained within L at spacetime (t), we assert that LER(L,U,t) is the
ratio of the amount of energy in L(t) to the total amount of energy in the universe U(t)
due to its "total deformation" at that "moment" as perceived from the part of spacetime we
are calling t.

The primary limitation of this model is
that influence of regions outside of L on L and v.v. is not well defined. This limitation
is taken as a challenge to be addressed by suitable application of limited cases until
a sufficient pattern is established. This paper merely poses the model and attempts to
begin consideration of some cases.

Since the value of LER is defined as a scalar "fraction of total energy in the universe",
the number is known to always be bounded between 0 and 1, given our other assumptions.

Take a simple case, where L1(t) and L2(t) are adjoining one metre cubes in mostly-empty spacetime
as seen from a simple timelike trajectory t near them.
Each of L1 and L2 has a very small LER value, certainly less than 10^-15, probably more
like 10^-150 or 10^-1500 or whatever, without thinking about it too hard for now.
But whatever LER(L1(t)) and LER(L2(t)) are, we can agree that they are both functions and
their sum is reasonably well defined as the LER of the 2x1x1 merged region of spacetime
(L1+L2)(t), assuming our other problems are tractable.

Note that since L
is a function of spacetime we have not assumed that the total conventional volume of the space
described by L is constant. This allows us to write certain expressions describing the movement
of some energy forms as "movement" of L(t).

That is, suppose K is the LER of energy in some brick L(t) in Newtonian motion through
uninfluenced space. Then we can write
K = LER(L(t))

...and thereby "know the energy" for the "moving region" L.

Mathematically, LER would be a functional if U were properly defined.
Movement of LER is defined as an energy signal between deforming regions of U,
which can be described with a kind of wavelet fractal analysis over parameter space R(L),
which is the set of regions in folding wavespace U that influence a particular L.

On familiar scales, LER(L,U,t) means simply aggregate local energy density: "how much total
matter/energy is inside the shape L at time t?" We can compute approximate LER by integrating the
energy density as we classically know it, and dividing by the constant EU. Then more useful
statements can be made in terms of the relative amount of LER in one region compared with that
in another, based on our classical understanding of energy flow, e.g. electrical energy.

For example, a charged battery has more LER than a comparable but dead battery. We also know
that, relative to a t-trajectory at the center of the earth, that same battery can receive
an increase in LER by being tossed upwards. Usually that LER is dissipated when the battery
is caught or strikes the ground. So LER just means energy, but defined as a ratio rather than
an amount of joules or calories. Based on our choice of TEU(U(t)), we may however hypothesize
a ratio of LER units to joules to enable practical calculations, at the end of the day. In fact,
the convergence of this ratio may be seen as the yardstick for progress in applying and verifying
the model we are discussing.

We make the shift to relative energy in order to more easily work backwards towards
the shape of U and the forms of its deformations without being hamstrung by interpretation
of electron-Volts in a light-year context, which is the predicament of modern physics.
Specifically, we are untroubled by both "dark matter" and "subatomic" phenomena,
these are simply deformations of spacetime by the galaxy form, expressable as their
actual "volume" in U, which we work backwards to find from their most apparent dynamics,
without undue assumptions..

Another example, the sun has more LER than the earth. The sun also has more LER per "unit volume"
as we understand it, than earth. Informally, we call this LERD, for Local Energy Ratio Density,
with the understanding that the volume for this density is subject to interpretation from L(t).

Without saying that we know the "mass" of the sun or the "mass" of the earth, we know that the
earth is in the Sun's gravity well. We know that somehow, the sun "eats" part of space
so that smaller objects slope down into it. Now, imagine the region of space included
by a conventional sphere of radius 1.1 AU, centered at the sun. Imagine it WITH the sun and
three inner planets, and WITHOUT the sun and three inner planets. The WITH case obviously
has higher LER. But what about volume? The WITHOUT case has no significant sloping, but
the WITH case has a huge downslope into the center, offset somewhat by radiation pressure and
finally the internal dynamics of the sun. Einstein tells us the curvature in U causes this
slope.

Imagine now a rubber sheet stretched flat. Grab part of that sheet in your fist and compress
it into a ball. What happens? The rest of the sheet is stretched toward your fist and pulls
out on your fist. Consider the two dimensional view from above of this sheet, with and without
your clenched ball. In the WITH case, points that were farther away are now closer, and in
fact more of the rubber is now locally contained in the center of the sheet. The more of the
sheet you ball up, the higher the ratio of rubber in the ball region to rubber in the far
region, and the higher the force of mutual tension between the center and the far region.
Is this tension not somewhat like gravitational force?

Without trying to apply the analogy too literally, perhaps we can geometrically infer that the
total volume (in some higher dimensional, folded U which resembles vibrating brain tissue) of
the Einstein-curved (more bunched up) WITH case is higher than the total volume of the
relatively flat (less folded) WITHOUT case?

That is, we say that the true spacetime volume of the sun in U is higher
(per 3-space "cubic meter" as we conventionally measure it) than an 'equivalent and empty'
space in U. ("Equivalent and empty" to be made more formal as we go along).

Carelessly generalizing, we suppose that presence of any energy represents
volume-consuming deformation, a 'crumpling of stretchy space', if you will. The extreme
case is, of course, a black hole, which is a mostly permanent volume/energy
singularity, as far as we know.