The classical reality of Isaac Newton assumes that space and time are
separate, and that time ticks at the same rate for all observers. Time
flows evenly, the same for everyone, and fixed spatial distances are
identical. These ideas are still useful for calculating
nonrelativistic velocity trajectories, and low energy measurements
and calculations.
Einstein discovered that space and time are not separate and immutable
but are two aspects of one entity called "spacetime". The fundamental
postulates of special relativity concern inertial reference frames. A
reference system that is a coordinate system based on three mutually
orthogonal(perpendicular) axes,
which give coordinates x,y,z in space and associated with a system of
synchronized clocks at rest in the system giving the time coordinate
"t", and when particle motion is formulated in terms of this system of
reference, Newtons first law of motion holds true.
So if K and K' are inertial reference frames, then K' is moving
relative to K without rotation and with constant velocity. The
coordinates (t,x,y,z,) correspond to one point in spacetime, and the
point is called an "event".
The two postulates of special relativity:
1. The speed of light in vacuum, c, is the same in all inertial
reference frames.
2. The laws of nature are the same in all inertial reference frames.
Since the speed of light in vacuum, c, is "invariant" or constant,
this does strange things to space and time. Space and time must
contract and dilate, as "c" remains constant.
For two inertial reference frames K and K' distance/time = c
= dr/dt = dr'/dt' .
Now we must employ the theorem of Pythagoras to calculate an interval
of distance in space.
For two dimensions it becomes: (Dx)^2 + (Dy)^2 = (Dz)^2
For three dimensions, when dealing with the spacelike separation
between two events, we have the expression:
(distance)^2 = (Dx)^2 + (Dy)^2 + (Dz)^2 , where "D" is a means "change
in" or "difference of".
When calculating the interval between two events A and B where we
have the same y and z coordinates, the separation in space between the
two events becomes: distance = Dx . The interval is given by the
mathematical expression:
[(Dt^2)  (Dx^2)]^(1/2)
Where t is the time coordinate. The expression becomes:
(Dt^2)  (Dx^2)  (Dy^2)  (Dz^2)
is for a timelike interval.
(Dx^2) + (Dy^2) + (Dz^2)  (Dt^2)
is for a spacelike interval.
The negative sign in front of the time coordinate means that time is a
dimension, but not a dimension in the same sense as the three spatial
dimensions. We cannot take a vacation to ancient greece and talk to
Plato.
So Minkowski introduces a new way to measure time with Dw =
[(1)^(1/2)]*Dt*c
(distance)^2 = (Dx)^2 + (Dy)^2 + (Dz)^2 + (Dw)^2
Time and space become parts of a larger unity,
[Dx^2 + Dy^2 + Dz^2  Dt^2]^(1/2)
Again, the minus sign shows how time is not quite the same as space
and marks
their difference in character. with the imaginary number (1)^(1/2) .
Spatial distance or "length" in the direction of motion becomes
contracted:
L = L'sqrt(1v^2/c^2) , where "sqrt" means square root,
and v is the velocity of the object in motion, L' is the length of an
interval the "at rest frame", and L is the length that is contracted
as measured by the person in the "at rest frame".
Mass is increased:
m = m'/sqrt(1v^2/c^2)
Time is dilated:
t = t'/sqrt(1v^2/c^2)
In Newtonian physics, two velocities can be added with simple
addition, to give the relative velocity:
v1 + v2 = vtotal. For example, two objects moving away from each
other in a straight line, diametrically opposed and from an initial
point of origin, their relative velocities are: 1m/sec + 1m/sec =
2m/sec.
In relativity, nothing outruns the speed of light in vacuum, so
velocities are added such, that for two emmitted photons of light,
travelling away from each other from an initial point of origin, their
relative velocity is not equal to twice the speed of light, or "2c",
but equal to c only.
vtotal = (v1 + v2)/[1 + (v1v2)/c^2]
vtotal = (c + c)/[1 + (c^2)/(c^2)] = 2c/2 = c .
General relativity is based on the equivalence of inertial mass and
gravitational mass. If one was in a closed room far away from any
gravitational influences, and the room was constantly accelerated at
one "g" , 9.8 meters/sec^2 , that person could not tell if he or she
was in an
accelerated frame of reference or at rest in a gravitational field.
Time is a process. What is a process? The most general definition
would be one that subsumes all other more specific definitions i.e.
specific would be mechanical process, mental process, natural process
...etc. The highest level of generalization for a definition of
*process* should be a mathematical definition. A process is a
transition from A to B, that such that translation from condition A to
condition B is acheived in a discrete(finite number) or
continuous(infinite number) of steps.
Time is a process, and time is a dimension, albeit a "temporal" one.
What is a dimension. The most general definition again, is
mathematical. A dimension is a variable, specifiable by a set of
numbers. What is temporal?
Temporal is a more specific element defining the definition of the
dimensionality of time as a rate of change.
What is rate? The general mathematical definition? Rate is a ratio
with time as the denominator. For example 1m/second , 5lbs/minute
3cranks/hour ...etc.
This can also be interpreted as a rise/run ratio called a slope.
In mathematics the "slope" is given by: (y2  y1)/(x2  x1) .
Is time a linear dimension? No. What is linear? The most general
definition is mathematical. For real valued functions of one
variable, i.e. f: R>R, where R is the set of all real numbers.
Such a function is called linear provided the following condition
holds:
For every two real x1 and x2, f(x1+x2) = f(x1)+ f(x2) for continuous
functions.
A first order polynomial is linear e.g. y = f(x) + b
Time is a nonlinear iterative process. A function of itself. Time is
a process of flux. Flux is the rate that that a quantity passes
through a fixed axis or boundary. Time is an iterative process of
flux. Quantum fluctuations.
Time is asymmetric with its thermodynamic arrow. A temporal stacking
via Successive endomorphisms. A function of functions.
f(z)
f[f(z)]
f{f[f(z)]} ... fn{...[f_0]}
Symmetric tensors are of the form:
A^uv = A^vu
An antisymmetric tensor is a tensor which changes sign when two
indices are switched:
A^uv =  A^vu
It is really interesting that an asymmetric tensor can be represented
as a sum of symmetric and antisymmetric tensors:
An asymmetric tensor A^uv does not equal
A^vu
A^uv = (1/2)*(A^uv + A^vu) + (1/2)*(A^uv  A^vu)
The first law of thermodynamics is time symmetric.
The second law of thermodynamics is time asymmetric.
So could it be possible to generalize an asymmetric arrow of time, by
an operation similar to the summing the symmetric and antisymmetric
tensors ?
TransEinsteinian Conjecture:
When we are observing distant stars and galaxies, we are observing the
universe in an earlier time period. That is Because a ray of light
propagates at a finite speed. When time is represented as a vertical
axis and space is represented by the two horizontal axes, "past and
future light cones" for the universe, can be graphed:
Time




Space
The present moment or relative "NOW", is the intersection of the past
and future
light cones. These light cones are really "lightspheres". Since
nothing travels faster than "c", the speed of light in vacuum,
wavefunctions can also be represented as light cones, or expanding
spheresfuture light cones, with radius R, and shrinking spherespast
light cones, with radius 1/R. As we observe the light cone cross
sections, or "sphericalwave" cross sections, of the universe, it is
again the present moment, which is relative to the observer. If we
rotate the above Cartesian diagram, so that we are staring down the
time axis, and as we are observing the light cone cross sections, the
past and future light spheres form intersecting wave fronts of the
past and future wavefunctionslight cones of the universe.
The time axis is represented as squareroot(1)*c*t or ict. The space
axes are "ct". According to relativity, time is not a linear dimension
but is a curve.
Time is nonlinear. By rotating the time axis 90 degrees as we are
observing the conic cross sections, time becomes a rate of flux for
the present moment or relative "NOW". Waves resonating "inphase".
{{}}
"U" stands for universe. T duality explains that the physics for a
circle of radius R is the same for a circle of radius 1/R.
R{L^2}/R
An isomorphism. The wavefunction of the universe must be defined for
both past and future lightconesspherical wavefunctions. The present
moment would be the intersection of R and 1/R. A multiplicative
identity.
For the universe: {R}*{1/R} = 1 = c
As we are observing the past lightcone, the oldest layers of
spacetime are the "outer" layers.
[S/T_0[[[[[[[[[[[S/T_1]]]]]]]]]]]]]]
[distance/time] "S/T" gives the velocity for a photon of light
and is invariant. S/T_0 = S/T_1. The increased layering of spacetime
would be analogous to a pressure force. The present theory for the
universe explains that matter remains constant while distances between
galaxies and clusters of galaxies expand.
Likewise, since all motion is relative, we could just as well say that
spacetime is a constant and matterenergy is uniformly shrinking. The
intersecting lightcones of the universe and objectsevents within the
universe. This is also in agreement with the TDuality of Mtheory and
possibly the Chris Langan conspansion model of the CTMU theory, at
www.ctmu.org
CONIC SECTION DIAGRAM:
{{}}
When two waves are the same frequency, in phase, and are moving
towards each other from opposed directions, they are at resonance.
This wavefunction resonance would be analogous to a type of
phasefiltering process as some waves in phase, would be reenforced,
and waves that are out of phase would cancell out by other
diametrically opposed waves, 180 degrees out of phase. A resonant
system corresponds to the Schrodinger wavefunction collapse of the
Copenhagen interpretation of quantum mechanics and the Lagrangian
pathintegral approach of Richard Feynman.
The "relative" present moment can also be represented as the
Pythagorean theorem
(ict)^2 + (ct)^2 = 0.
Since each object within the universe has its own light cone, each
object has its own resonance, depending on momentum.
wavelength = [Planck's constant]/[momentum] = h/p
According to the relativity of Einstein, increased momentum results in
relativistic effects such as time dilation and length contraction.
This can also be explained as the rotation of the objectevent's light
cone. In a gravitational field the rotation of light cones would also
correspond to increased wavefunction densities, or wavefunction
density gradients. For a massive object such as the sun, it would be
in agreement with nonEuclidean geonetry and be describable as a
varying index of refraction for a grazing ray of light. The ray of
light would appear to follow a curved path in space time.
Russell E. Rierson
analog57@yahoo.com
Copyright © 2003 Russell E. Rierson . All Rights Reserved
Reality could be a system at resonance.
Definitions:
A diffeomorphism is basically a map between manifolds with the term
"manifold" a topological space which is locally Euclidean. An
infinitely differentiable bijection also with a differentiable
inverse.
An "inverse expansion" would be material and radiative
contractioncompression. A type of mathematical "inverse" of spatial
expansion. Similar to graphing successive moments in the past light
cone cross sections of the universe. Since nothing travels faster than
"c", the speed of light in vacuum, it is possible to represent wave
functions as expanding and contracting probabilitywavesspheres.
Expansion and inverse expansion are both defined by the wavefunction
of the universe.
(())
Wave expansion with radius "R" AND inverse wave expansion with radius
1/R shoild be logically valid explanations. Defined by the
wavefunction and inverse wavefunction for the Lorentzian cross
sections is the intersection or dot product, with i the sqrt(1), =
* = x^2 + y^2 = r^2 = 0. The Pythagorean theorem =
hypotenuse, is approximately zero for the "relative" present moment,
for a
collapsed wavefunction, down to the localization boundary permitted by
Planck's constant. 10^(35) meters A resonant system.
The "TDuality" of string theory, is a type of isomorphism.
R[1/R]
Really TDuality says:
R[{L_st}^2]/R
L =~ 10^(35) meters.
The physics for a circle of radius R is the same for a circle of
radius 1/R
So a type of diagram for the self organizing universe, would be:
{{U}}
Infinity and continuity would be explained as quantum wavefunction
"potential". Real continuous waves. Discrete finite particles
(qwf intersection) would be explained as quantum wavefunction
"actualizationcollapse". The wave function collapse is equivalent to
the path integral approach of Richard Feynman. Where some histories
are 180 degrees out of phase and self cancel the other histories that
are in phase are reenforced. The reenforced waves have a type of standing wave resonance.
How can the "relativistic effects" be described by quantum
wavefunctions, when the wavefunctions describe the position and
momentum of particles in a backround OF space?
In quantizing spacetime geometry, we won't get
wavefunctions based on a background space. The space of wavefunctions
can be thought of the space of squareintegrable wavefunctions over
classical configuration space. In ordinary quantum mechanics,
configuration space is space itself {i.e.,to describe the
configuration of a particle, location in space is specified}. In
general relativity, there is a more general kind of configuration
space: taken to be the space of 3metrics {"superspace", not to be
confused with supersymmetric space} in the geometrodynamics
formulation,{or the space of connections of an appropriate gauge
group)in the Ashtekar/loop formulation. So the wavefunctions will be
functions over these abstract spaces, not space itself the
wavefunction _defines_ "space itself".
The *process* is the "function of functions".
Time is a function OF time. With flat sheets, foliations of space, or
Lorentzian cross sections, the light cone cross section corresponding
to a circle would be a "rotated" light cone near a massive object.
The two light cones form a relationship, describing degrees of
rotation and circularelliptic cross sections. It should be possible
to derive a set of equations from these
rotational perspective effects.
The overall spacetime structure must be stable and symmetric. The
stability and symmetry are ultimately related to the existence of a
type of juxtapositionaxis of symmetry. A function and its inverse,
generate this axisjuxtaposition of symmetry, which can be
hypothesized as a type of interactive temporally stratified sequence,
the eigenfunction diffeomorphism.
The laws of physics would be distributed over spacetime. Thus the
equivalence principle results from the law: conservation of energy.
The gravitational field, described by the metric of spacetime g_uv ,
is generated by the stressenergy 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 nonlinear, 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
(mass)>time
>
Near a massive object, space is contracted or "compressed" while time
is dilated. This corresponds to rotated lightcones or increased wave
function densities. The wave function densities give a varying index
of refraction equivalent to the nonEuclidean geometry. A dual
refractive model. A unification of QM and GR.
Sequential layers of space create the illusion of "movement" along the
fourth (temporal) dimension.
[S/T_0[[[[[[[[[[S/T]]]]]]]]]]]]]
S/T is distance/time for a photon of light. "c" the speed of light in
vacuum, is invariant for the two reference frames T_0 and T_1. Since
the time axis is rotated 90 degrees, the present moment becomes
approximately distance "zero" as described by the Pythagorean theorem:
(ct)^2 + (ict)^2 = 0, down to the localization boundary
10^(35) meters
Quantized space... as a discrete sequence of continuous manifoldswave
forms, or "pbranes" defined as quantum super membranes, could unite
gravity and quantum theory.
Time is defined as an iterative process. Fractal behaviour in the
complex number plane is produced by iterating a nonlinear function
whose variables include its own result.
Let z = 0+0i , f{z} = z^2+c
f{z} = f{0+0i} = {0+0i}^2+c = c
f[f{z}] = f{c} = c^2+c
f[f[f{z}]] = f{c^2+c} = {c^2+c}^2+c
Time becomes an iterative "nonlinear" process.
"Reality at Resonance" was(is) my initial hypothesis, resulting from
the research of various physics topics, and "chaos theory".
A stochastic resonance is a phenomenon in which a nonlinear "random"
system, is subjected to a periodic modulated signal so very weak as
to be normally undetectable, but it becomes detectable due to
resonance between the weak deterministic signal and stochastic noise.
The universe is postulated as a result of "temporal feedback" A
stochastic system that "self organizes, and the ranom noise, which is
a resultant system in equilibrium, or an infinitely symmetrical "phase
space".
Of course, the geometrical description of gravitation has ample
justification.
But what about the electromagnetic "force"??? The electromagnetic
field is different in character from gravitation. Must we consider
electromagnetism as an independent physical field, with its own
characteristic dynamics?
Must the electromagnetic field be forever described as
nongeometrical? Or is it possible to describe both the
electromagnetic and gravitational fields as aspects of the curvature
of spacetime? Or a complementary formulation that describes
gravitation as an aspect of electromagnetism? The variation of the
curvature of spacetime at one point can also be correlated with the
electromagnetic field at one point. The electromagnetic field can be
described relativistically by the Maxwell tensor F^uv with electric
and magnetic field strengths E and B.
I seem to recall reading Einstein postulated that mass generates
spacetime? with spacetime as inhomogeneous and anisotropic in the
neighborhood of mass.
Space becomes contracted in the neighborhood of massive objects such
as neutron stars according to the formula r' = (1rS/r)^(1/2) Where rS
= 2GM/c^2 .
The vacuum T^uv = 0 solution to the Einstein field equation, which is
spherically symmetric and static, is called the Schwarzschild
geometry. For r very large compared to rS(distances many Schwarzschild
radii from the neutron star) , rS/r will be very small and r' will be
virtually equal to r . As r shrinks toward rS/r, r' will approach 1 
rS/r = 0.
Rulers appear to get shorter, in the approach towards the neutron
star, and time intervals get longer. The anisotropy and inhomogeneity
arising from general relativity depend on the gravitational potential
energy, and
vary as 1/r, rather than varying with the strength of the
gravitational field. But since the Swarzschild radius is radially
symmetric, the curvature in the path of a ray of light can be
explained as the varying density of space which creates a region of
varying refractive index. This is what causes the ray to bend slightly
as it grazes a massive star.
As far as special relativity is concerned, the increase in mass of a
body moving at relativistic speeds is a kind of rotational
"perspective effect". When dealing with the spacelike separation
between two events we have the expression:
(distance)^2 = (Dx)^2 + (Dy)^2 + (Dz)^2
When dealing with an interval between two events A and B where we
have the same y and z coordinates, the separation in space between the
two events becomes: distance = Dx . The interval is given by the
mathematical expression:
[(Dt^2)  (Dx^2)]^(1/2)
Where t is the time coordinate. The expression becomes:
(Dt^2)  (Dx^2)  (Dy^2)  (Dz^2)
for a timelike interval.
For a spacelike interval:
(Dx^2)  (Dy^2)  (Dz^2)  (Dt^2)
So Minkowski introduces a new way to measure time with Dw =
[(1)^(1/2)]*Dt
(distance)^2 = (Dx)^2 + (Dy)^2 + (Dz)^2 + (Dw)^2
Time and space become parts of a larger unity,
[Dx^2 + Dy^2 + Dz^2  Dt^2]^(1/2)
The minus sign shows how time is not quite the same as space and marks
their difference in character. with the imaginary number (1)^(1/2) .
The light cone is an interesting feature of Lorentz geometry. A flash
of light at one moment in time. The Lorentz geometry has interesting
and important characteristics for the understanding the structure of
the physical world.
The future light cone tells the history of the expanding spherical
pulse that started at point P. Similarly the bacwards light cone tells
the history of a converging pulse of radiation collapsing at point P,
the origin, at time zero.
The light cone of the event at point P and the light cone of any other
event, has an existence in spacetime apart from any coordinates we may
use to describe it. Events that effect each other are independent of
the reference frame in which the observation of said connection of
events was observed. So the causal connection between two events is
preserved in every reference frame.
Light carries energy because E = (h*c)/w , where w is the
wavelength, c is the speed of light in vacuum, and h is Planck's
Constant. But in any local reference frame, the speed of light in a
vacuum is always fixed to be c = 299,792.5 kilometers/second. As light
escapes a gravitational field, it loses energy as it works against the
gravitational field, or spacetime curvature.
Its speed does not change in any local "freely falling" reference
frame because of special relativity, this means that for its energy to
change in the above equation, the wavelength of the light is the
factor that changes. Thus, light escaping a gravitational field will
be shifted to longer wavelengths as its energy is reduced; this is
what distant observers see as a redshift.
If matterenergy, space, and time scales, are reduced in tandem, we
would still measure the same speed of light in vacuum and it would
still be invariant. Time would be a process and "c" would be the rate
of the process.
[[[[[[[[U]]]]]]]]
The temporal stacking effect of the successive layering of space,
creates a pressure force, corresponding to a varying refractive index,
as the reciprocal transformation of the physics for a circle of radius
R = the physics for a circle of radius 1/R.
The index of refraction "n" ,
n = (w_f)/(w_g) ,
where w_f is the wavelength of light in flat spacetime and w_g is the
wavelength of light in the gravitational field.
T
..............T
/
w_f w_g
Time becomes "dilated". For a spherical object such as a neutron star,
photons emitted at successive points along radial distances from the
surface of the star give different successively lessening red shifts
for "w_g".
w_g is a function of radial distance, for emitted photons.
Russ
Copyright © 2003 Russell E. Rierson . All Rights Reserved
