Cantorian Fractal Spacetime, Quantum-like Chaos and
Scale Relativity in Atmospheric Flows
A. M. Selvam and Suvarna Fadnavis
Indian Institute of Tropical Meteorology,
Pune 411008, India
Abstract
Cantorian fractal spacetime fluctuations characterize
quantumlike chaos in atmospheric flows. The macroscale atmospheric flow
structure behaves as a unified whole quantum system, where, the superimposition
of a continuum of eddies results in the observed global weather patterns
with long-range spatiotemporal correlations such as that of the widely
investigated El Nino phenomenon. Large eddies are visualised as
envelopes enclosing smaller eddies, thereby generating a hierarchy of eddy
circulations, originating initially from a fixed primary small scale energising
perturbation, e.g., the frictional upward momentum flux at the boundary
layer of the earth's surface. In this paper it is shown that the relative
motion concepts of Einstein's special and general theories of relativity
are applicable to eddy circulations originating from a constant primary
perturbation.
1. Physical Concepts in Space-Time Relativity
The equations of motion enunciated by Newton in 1687
[1] were believed to describe nature correctly for over 200 years. The
ideas of Newton involve the assumption that the laws of motion, and indeed
all the laws of physics, are the same for an observer at "rest" as for
an observer moving with uniform velocity with respect to the "rest" system.
This symmetry principle is sometimes called the principle of relativity.
The principle of relativity in Newton's and Einstein's theories
of mechanics differs only in the way that the speed of the observer affects
observations of positions and times in the two theories [2]. If an inertial
reference system is defined as one in which Newton's laws describe the
behavior of bodies, any other reference system, which moves with constant
velocity with respect to this first inertial system, is also an inertial
system. Time and space seem to be independent of the particular frame used
[3]. The concept of relativity (Galilean), a symmetry principle, has been
used in mechanics for a long time. By symmetry is meant an invariance against
change, something stays the same in spite of some potentially consequential
alteration [4]. Investigations into the phenomenon of electricity and magnetism
culminated in 1860 in Maxwell's equations of the electromagnetic field,
which describe electricity, magnetism and light in one uniform system.
However during the period 1890 - 1905 it was recognized that Maxwell equations
did not seem to obey the inherent symmetries present in the laws of motion
of Galileo and Newton. One of the consequences of Maxwell's equations is
that if there is disturbance in the field such that light is generated,
these electromagnetic waves go out in all directions equally at the same
speed c, equal to about 3x105
km/sec. Another consequence of the
equations is that if the source of the disturbance is moving, the light
emitted goes through space at the same speed c. This is analogous
to the case of sound, the speed of sound waves being likewise independent
of the motion of the source [5]. Incidentally, the constant c happened
to be first discovered by workers in the field of electricity, long before
electromagnetic waves were known to exist [1].
A number of experiments based
on the general idea of Galilean relativity were performed to determine
the speed of light. Michelson and Morley, in 1887 found that the velocity
of a beam of light moving from east to west is the same as that of a beam
of light moving from north to south. The east-west velocity might have
been expected to be influenced by the velocity of the earth, but such was
not the case. About 20 years later, H. A. Lorentz provided the solution
by suggesting that material bodies contract when they are moving and that
this foreshortening is only in the direction of the motion and also that
if the length is Lo when body is at rest, then
when it moves with speed u parallel to its length, the new
length L1 is given as
Although the contraction hypothesis
successfully accounted for the negative result of the experiment, it was
open to the objection that it was invented for the express purpose of explaining
away the difficulty and was too artificial [5]. The contraction in length
is concomitant with modification in time elapsed by the factor,
i.e., moving clocks run slower.
Based on the above hypothesis
of linear contraction and time dilation in moving objects, Lorentz showed
that Maxwell's equations retain their symmetry, i.e., remain unchanged
when the following Lorentz transformations are applied.
y'=y
z'=z
(1)
Lorentz's transformations
introduced into the laws of mechanics, the speed of light, basically an
electromagnetic constant.
The corresponding Galilean transformations are
x'=x - ut
y'=y
z'=z
t'=t
which relates the space and time coordinates
(x, y, z and t) of a system at
rest to those (x', y', z', and t')
of a system in uniform relative motion of speed u in the
x
direction.
Einstein, following a suggestion
originally made by Poincare, then proposed in his special theory of
relativity that all physical laws should be of such a kind that they
remain unchanged under a Lorentz transformation [5]. Applying Lorentz transformations
to Newton's laws of motion, Einstein, in 1905, showed that the mass m
in Newton's laws of motion should now be written as
where m0 is the rest mass
and c is the speed of light equal to about 3x105km
sec-1. Einstein's special theory of relativity proposed
in 1905 introduced modification of laws of motion, related to how physical
observers measure spatial displacements and time intervals.
The two basic postulates of Einstein's special
theory of relativity are as follows [3].
(1) The laws of electrodynamics and of mechanics
are the same in all inertial frames. This includes the requirement that
c,
the velocity of light in free space, is invariant.
(2) It is impossible to devise an experiment, which
defines a state of absolute motion. There is no special "rest" frame of
reference.
1.1 Special relativity to general relativity
After proposing the special theory of relativity, Einstein
discovered that the inverse square law of gravitation could not co-exist
consistently with the special theory of relativity [6, 7]. One of
the major achievements of the special theory of relativity was the
demonstration that the speed of light is a constant and imposes a limit
to the maximum attainable speed of any material object or electromagnetic
wave. This limiting speed c plays a crucial role in relating
space-time measurements of inertial observers, i. e., observers moving
under no forces. The gravitational attraction as postulated by Newton clearly
exceeded this speed limit, since it was instantaneous. Further, the phenomenon
of gravity prevents us from defining the inertial observers in the first
place. The inertial observers so fundamental to special relativity
does not exist because of ever-present force of gravity.
In his general theory of relativity
proposed in 1915, Einstein gave an ingenious interpretation to this property
of gravity. Realizing that gravity is permanently attached to space, he
argued that it, in fact describes an intrinsic property of space and time,
viz., its geometry. The geometry of space and time must be of a curved
non-Euclidean type, i.e., spacetime is curved. By treating spacetime as
curved Einstein eliminated gravity as a physical force. In his general
theory of relativity Einstein gave a set of equations which relate
the geometrical properties of spacetime to the distribution of gravitating
matter within it. Special relativity brought into physics the important
notion that space and time together form a joint entity. The measurements
of spatial distances and time intervals in the special theory are performed
according to flat space geometry. The notion of curvature of spacetime
and its relation to gravity is the remarkable new feature of the general
theory.
There is only a minute difference between the predictions
of general relativity and Newtonian gravity [6,7].
Nottale (1996) has discussed the implications of
the fractal spacetime characteristics on fundamental physical laws
[8].
2. Scale Relativity and Fractal Space-Time Structures
in Atmospheric Flows
Dynamical systems in nature exhibit selfsimilar
spacetime fractal fluctuations of all scales down to the microscopic
scales of the subatomic world, i.e., the vacuum zero point energy fluctuations.
Selfsimilarity implies long-range correlations, i.e., nonlocal connections
in space and time. The ubiquitous fractal spacetime structures found
in nature imply a self-organization or self-assembly process which is independent
of microscopic details such as physical, chemical, physiological, etc.,
of the dynamical system. Selvam and Fadnavis [9,10] have proposed a cell
dynamical system model for atmospheric flows which may be directly applicable
to all dynamical systems in general and in particular to the subatomic
dynamics of quantum systems. The model is based on the concept [11] that
spacetime integration of enclosed small scale (turbulent) fluctuations
results in the formation of large scale (eddy) circulations (Fig 1.).
Figure 1: Physical concept of eddy growth process
by the self-sustaining process of ordered energy feedback between the larger
and smaller scales, the smaller scales forming the internal circulations
of larger scales. The figure shows a uniform distribution of dominant turbulant
scale eddies of length scale 2r. Larger-eddy circulations such as ABCD
form as coherent structures sustained by the enclosed turbulent eddies.
Large eddies are visualised as
envelopes enclosing inherent small scale eddies, thereby generating a continuum
of eddies, the spatial integration at each level generating the next level
(large scale) and so on. The relationship between the root mean square
(r. m. s. ) circulation speeds W and w*
respectively of large and small eddies and their respective radii R
and r is given as
The primary perturbation w*
is constant and generates a continuum of eddies of progressively increasing
radii R. The r. m. s. circulation speed W at
each level represents the mean as well as standard deviation, i.e., it
is a relative velocity with respect to the constant generating perturbation
w*
Therefore the factor
in the Lorentz transformation at Eq.(1) becomes equal
to 1, since the relative velocity u = 0. Also, w*
is equivalent to c and is a constant primary perturbation.
The Lorentz transformations when applied to eddy dynamics reduce to classical
mechanics of Galileo and Newton, since, by concept the eddy r. m. s. circulation
speeds W are relative to the constant primary perturbation
speed w* .
Einstein's principles of special
relativity and general relativity are applicable to eddy dynamics
as summarised in the model predictions [9,10] in the following
(1) Spacetime fractal structures are signatures
of string-like energy flow in a hierarchy of vortices tracing an overall
logarithmic spiral trajectory with the quasiperiodic Penrose tiling
pattern for the internal structure.
(2) The logarithmic spiral energy flow structure
can be resolved as a continuum of eddy circulations, which follow Kepler's
laws of planetary motion, in particular the third law. The inertial masses
of eddies representing gravitational masses, therefore follow Newton's
inverse square law of gravitation. Fractal spacetime fluctuations
are related to gravity and is consistent with El Naschie's [12] conjecture
that gravity is caused by 'fractal' fluctuations of time.
(3) Instantaneous non-local connection, prohibited
in Einstein's special theory of relativity, is possible and consistent
in the context of eddy circulations which are considered as extended objects
as explained in the following. The bidirectional energy flow intrinsic
to eddy circulations is associated with bimodal, i.e., formation and dissipation
respectively of phenomenological form for manifestation of energy such
as the formation of clouds in updrafts associated with simultaneous dissipation
of clouds in adjacent downdrafts, thereby generating discrete cellular
geometry to cloud structure.
Gravitation is defined as a property
of spacetime geometry in Einstein's general theory of relativity. The concept
of pointlike particle of zero dimensions in classical physics introduces
infinities or singularities in the smooth spacetime fabric representing
gravitational field [13]. Further, pointlike particles are associated with
trajectories, where, the speed of the particle cannot exceed the speed
of light according to Einstein's special theory of relativity.
The cell dynamical system model
[9,10] discussed in this paper introduces the concept of extended objects
thereby avoiding singularities and also possessing instantaneous nonlocal
connection.
3. Conclusion
Cantorian fractal spacetime structure
to atmospheric flows patterns is a result of the superposition of a continuum
of eddies, which function as a unified whole quantum system. Wave-particle
duality in the quantum system of atmospheric flows is a result of bimodal
(formation and dissipation) form for manifestation of energy in the bidirectional
energy flow intrinsic to eddy circulations, such as the formation of clouds
in updrafts and dissipation of clouds in adjacent downdrafts, manifested
in the commonplace occurrence of clouds in a row.
The eddy circulations follow
Kepler's
laws of planetary motion, in particular, the third law and therefore Newton's
inverse square law for gravitation is applicable to eddy masses. The root
mean square (r. m. s) circulation speeds of the eddies are relative to
and less than the primary constant perturbation. Therefore, the basic criteria
invoked in Einstein's special and general theories of relativity
are incorporated in the concept for generation of eddy continuum. It is
possible that the vacuum zero point electromagnetic field fluctuations,
may self-organize to generate particles such as the electrons, protons,
etc., in a manner similar to the formation of atmospheric eddy continuum.
The vacuum may be a permanent nonzero source of energy in the universe
[14] .
Acknowlegments
The authors are grateful to Dr. A. S. R. Murty for his
keen interest and encouragement during the course of the study. The authors
are indebted to Professor M. S. El Naschie for inspiration and guidance
in this field of study.
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