** **

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.

** **

** **

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

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 ** L_{o}**
when body is at rest, then when it moves with speed

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.

(1)

Lorentz's transformations introduced into the laws of mechanics, the speed of light, basically an electromagnetic constant.

The corresponding Galilean transformations are

which relates the space and time coordinates
(* x, y, z* and

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 ** m_{0}** is the rest mass
and

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

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].

** **

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.).

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

The primary perturbation ** w***
is constant and generates a continuum of eddies of progressively increasing
radii

Therefore the factor

in the Lorentz transformation at Eq.(1) becomes equal
to 1, since the relative velocity ** u **=

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.

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.

** **

2. Weinberg, S., *Dreams of a Final Theory*.
Vintage, 1993.

3. Kip, A. F., *Fundamentals of Electricity and
Magnetism*. McGraw-Hill Book Company, New York, pp 630.,1969

4. Schroeder M., *Fractals, Chaos and Power Laws***,
**1990**, **W.H. Freeman and Co., N.Y.

5. Feynman, R. P., Leighton, R. B and Sands M., *Lectures
on Physics*, 1963, Volume I, pp. 52-11.

6. Narlikar, J.N., *Violent Phenomena in the Universe*.
Oxford University Press, Oxford, 1982, pp. 213.

7. Narlikar, J.N., 1996: *The Lighter Side of Gravity*.
Cambridge Univeristy Press, Cambridge,1996, pp. 217.

8. Nottale, L., Scale relativity and fractal space-time
: application to quantum physics, cosmology and chaotic systems. *Chaos
Solitons and Fractals*,1996, **7(6)**, 877-938.

9 Selvam A.M., Fadnavis Suvarna, Signatures of a
universal spectrum for atmospheric inter-annual variability in some disparate
climatic regimes, *Meteorology and Atmospheric Physics* ,1998, **66,**
87-112.

10. Selvam, A. M., and Fadnavis Suvarna, Superstrings,
cantorian-fractal spacetime and quantum-like chaos in atmospheric flows,
*Chaos Solitons and Fractals *(in press) 1998.

11. Townsend, A. A., *The Structure of Turbulent
Shear Flow*. Cambridge University Press, London, U.K.1956.

12. El Naschie, M. S., Dimensional symmetry breaking
and gravity in cantorian space. *Chaos Solitons and Fractals*,1997,
**8**, 753-759.

13. Argyris, J., Ciubotariu, C. and Andreadis, I.,
Complexity in spacetime and gravitation I: From chaos to superchaos . *Chaos,
Solitons and Fractals *(in press) 1998.

14. Mathews, R., Nothing like a vacuum. *New Scientist*
1995, **25 Feb.**, 30-33.