Elsevier

Chaos, Solitons & Fractals

Volume 11, Issue 13, October 2000, Pages 1983-1992
Chaos, Solitons & Fractals

On conjugate complex time—I: complex time implies existence of tangential potential that can cause some equipotential effects of gravity

https://doi.org/10.1016/S0960-0779(99)00091-0Get rights and content

Abstract

El Naschie's idea of complex time implies the possibility of spatial decomposition of rate of time flow that does not lead to the unphysical consequences for which those former attempts at multidimensional time have been criticized. The spatial decomposition of time rate suggests a spatial decomposition of energy which in turn would require presence of certain tangential component of a generalized vector potential, whether gravitational or electromagnetic. Existence of the new tangential potential could explain the observed frequency decrease of radio signals as well as the apparent anomalous radial acceleration towards the sun that was found in the case of the “runaway” spacecrafts Pioneer 10 and 11. The spatial decomposition of time rate dismisses any possibility of travel back in time and thus prohibits any acausal happenings due to reversal of time flow.

Section snippets

General ramifications of the complex time idea

I want to show in this note that El Naschie's idea of complex time (ICT) [1], [2], [3], which was discussed in [4], [5], can be understood geometrically and that existence of a second dimension of time (SDT) follows from the inverse Lorentz transformation of time rate (ILTTR). Moreover, the existence of SDT seems to imply the presence of a tangential gravitational potential, which is a very serious consequence. Therefore I will show in an accompanying paper that indeed there exists experimental

Second dimension of time

It is a well-known fact that the following (orthogonal) real ordinary differential equation [26]:dydx=−xy,determines two unique semicircles (positive and negative) which satisfy Eq. (1) y=±r2−x2⇒y=±r1−xr2,where r is a constant radius. The form of Eq. (2) resembles that of the ILTTR which can be written in terms of time rates (all rates will appear in upper cases):T′=T1−vc2,where the rate T refers to the resting frame and T′ to the moving one. Since the ILTTR refers to rates of time flow (and

Some features of the complex time

Since in the light of Eqs. (4) time is not only a fourth dimension, but also distinct planar geometric structure, its imaginary character can thus be interpreted in the very geometric sense as not directly representable in the LBS, which agrees with the geometrical “picture” of the TBS as an abstract dual vector space to the LBS. As a matter of fact, in this very sense the imaginary unit i was once introduced into geometry [38], [39], [40]. As such the imaginary unit i should be distinguished

Spatial flow of time implies equipotential effects

The ICT may affect interpretation of gauge field theories. For if Eq. (4a) is the time-based vector representation of the ICT which, being formally equivalent to the ILTTR also admits a decomposition of time rate depending on speed, then the first dimension of time is motion-independent and therefore “static”. Let us consider massless particles like photons, at first. Since time rate relates to energy via frequency, then such a static component of a moving particle's decomposed energy should

Conclusions

We have seen that the conjectured spatial structure of time flow which is implied by the ICT could be derived from the ILTTR. Unlike those previous attempts at multidimensional time, however, the ICT does not really suggest any extraneous increase of the resting resultant time, but merely a possibility of decomposition of time into a static and a kinematic time component in the inertial reference frame. Such a decomposition of time, however, implies spatialized energy and therefore also the

Acknowledgments

Thanks to Slava G. Turyshev for additional information on runaway spacecrafts.

References (91)

  • Lorentz HA. The theory of electrons. New York: Dover,...
  • Katz A. Classical mechanics, quantum mechanics, field theory. New York: Academic Press,...
  • Penfield Jr. P, Haus HA. Electrodynamics of moving media. Cambridge, MA: The MIT Press,...
  • Halmos P. Finite-dimensional vector spaces. Princeton, NJ: Van Nostrand,...
  • Beiser A. Concepts of modern physics. New York: McGraw-Hill,...
  • Einstein A. On the electrodynamics of moving bodies, 37–65 in [15],...
  • Einstein A. Relativity. The special and the general theory. New York: Crown,...
  • H.A. Lorentz et al.

    The principle of relativity

    (1923)
  • Jancewicz B. Multivectors and Clifford algebra in electrodynamics. Singapore: World Scientific,...
  • W. Sierpiński

    Sur une propriété des ensembles plans équivalente àl'hypothèse du continu

    Bull Soc Roy Sci Liege

    (1951)
  • Sierpiński W. Oeuvres choisies III. Warszawa: PWN,...
  • Destouches J-L. Corpuscules et systèmes de corpuscules, Paris,...
  • Buchwald E. Symbolische Physik. Berlin...
  • A. Schild

    Discrete space–time and Integral Lorentz transformations

    Phys Rev

    (1948)
  • H.S. Snyder

    Quantized space–time

    Phys Rev

    (1947)
  • Zharkov VI. Continual-discrete space and time of micro-objects. Novosibirsk, 1971:119 [in...
  • Einstein A. The foundation of the general theory of relativity. 109–164 in [15],...
  • Misner CW, Thorne KS, Wheeler JA. Gravitation. New York: Freeman,...
  • Nielsen KL. Differential equations. New York: Barnes & Noble,...
  • R. Schlegel

    Comments on Hafele-Keating experiment

    Am J Ph

    (1974)
  • French AP. Principles of modern physics. New York: Wiley,...
  • Bergmann PG. Introduction to the theory of relativity. New York: Prentice-Hall,...
  • Bohm D. The special theory of relativity. New York: Benjamin,...
  • Lorentz HA. Michelson's interference experiment. 3–7 in...
  • Hamermesh M. Group theory and its applications to physical problems. Reading, MA: Addison-Wesley,...
  • Moore JT. Elementary linear and matrix algebra. The viewpoint of geometry. New York: Mc-Graw-Hill,...
  • F. Solomon

    A theorem concerning Lorentz transformations

    Am Math Month

    (1984)
  • Poincaré H. Science and hypothesis. New York: Dover,...
  • Nering ED. Linear algebra and matrix theory. New York: Wiley,...
  • Geroch R. Mathematical physics. Chicago: The University of Chicago Press,...
  • Poncelet JV. Traité des propriétés projectives des figures, Paris,...
  • Marie M. Histoire des sciences mathématiques et physiques XI, Paris,...
  • Conant DR. Unified quantum field theory, Los Alamos, NM,...
  • Kantor IL, Solodovnikov AS. Hypercomplex numbers. An elementary introduction to algebras. New York: Springer,...
  • J. Strnad

    On multidimensional time

    J Phys A

    (1980)

    • Cited by (0)

    View full text
    Copyright © 2000 Elsevier Science Ltd. All rights reserved.