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

(Retired) email: [email protected]



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.




    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

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] .


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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