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2021 | 152 | 82-110

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Splitting 4D spacetime into 3D space and 1D time involves operationally needed presence of two 3D spaces enclosed inside two 4D paired dual reciprocal spatial structures



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The anticipated separation of the 4D spacetime into 3D length-based space and distinct abstract 1D space of elapsing time, which would vary independently of the usual variables of the length-based space, is not reasonable in the formerly unspoken single space reality (SSR) framework, because the separation would lead to inadmissible identification of multiplicatively inverse/reciprocal derivative taken with respect to elapsing time parameter with the derivative taken with respect to the 1D time variable. It is thus unacceptable because 1/(∂/∂t)=∂t/∂ is a contravariant expression posing as derivative (with respect to some unspecific variable) in the SSR setting and as such cannot be equated to the regular covariant derivative ∂/∂t taken with respect to the independently varying parameter of the elapsing time t. Nevertheless, the separation of spacetime is feasible within the multispatial reality (MSR) framework that allows the ensuing two separated spaces to reside in twin distinct spatial structures each of which is equipped with different orthogonal homogeneous basis, provided the twin spatial structures are equidimensional 3D spaces that are mutually paired to form a hierarchical hyperspatial structure comprising mutually dual reciprocal spaces. The conclusion is supported by special relativity. In a sense thus, the process of splitting of spacetime apparently demands a paradigm shift: from the formerly unspoken and thus unchallenged SSR paradigm to a certain MSR paradigm.







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  • Science/Mathematics Education Department, Southern University and A&M College, Baton Rouge, LA 70813, USA


  • [1] Czajko J. Mathematical gateway to complementary hidden variables in macrophysics. Int. Lett. Chem. Phys. Astron. 50 (2015) 117-142
  • [2] Czajko J. Flawed fundamentals of tensor calculus. World Scientific News 149 (2020) 140-165
  • [3] Czajko J. Finegrained 3D differential operators hint at the inevitability of their dual reciprocal portrayals. World Scientific News 132 (2019) 98-120
  • [4] Czajko J. Unrestricted division by zero as multiplication by the – reciprocal to zero – infinity. World Scientific News 145 (2020) 180-197
  • [5] Choquet G. Geometry in modern setting. Paris: Hermann, 1969, p.14.
  • [6] Goodstein R.L. Existence in mathematics. Compos. Math. 20 (1968) 70 see p.82.
  • [7] Bruno G. On the infinite universe and worlds. [pp. 440-452 in: Weaver J.H. (Ed.) The world of physics. A small library of the literature of physics from antiquity to the present. New York: Simon and Schuster, 1987, see p. 443].
  • [8] Jeffrey A. Handbook of mathematical formulas and integrals. San Diego: Academic Press, 2000, p.87.
  • [9] Grattan-Guinness I. Routes of learning. Highways, pathways, and byways in the history of mathematics. Baltimore: The Johns Hopkins Univ. Press, 2009, pp.308f.
  • [10] Rudin W. Real and complex analysis. New Delhi: McGraw-Hill, 2006, p.18f.
  • [11] Maurin K. Analysis II: Integration, distributions, holomorphic functions, tensor and harmonic analysis. Dordrecht: Reidel, 1980, p.65.
  • [12] Cheng E. Beyond infinity. An expedition to the outer limits of mathematics. New York: Basic Books, 2017, p.13ff.
  • [13] Czajko J. Algebraic division by zero implemented as quasigeometric multiplication by infinity in real and complex multispatial hyperspaces. World Scientific News 92(2) (2018) 171-197
  • [14] Czajko J. Quaternionic division by zero is implemented as multiplication by infinity in 4D hyperspace. World Scientific News 94(2) (2018) 190-216
  • [15] Czajko J. Multiplicative inversions involving real zero and neverending ascending infinity in the multispatial framework of paired dual reciprocal spaces. World Scientific News 151 (2021) 1-15
  • [16] Czajko J. On Conjugate Complex Time II: Equipotential Effect of Gravity Retrodicts Differential and Predicts Apparent Anomalous Rotation of the Sun. Chaos Solit. Fract. 11 (2000) 2001-2016
  • [17] Czajko J. Galilei was wrong: Angular nonradial effects of radial gravity depend on density of matter. Int. Lett. Chem. Phys. Astron. 30 (2014) 89-105
  • [18] Szekeres G. Effect of gravitation on frequency. Nature 220 (1968) 1116-1118
  • [19] Sadeh D., Knowles S.H. & Yaplee B.S. Search for a frequency shift of the 21-centimeter line from Taurus A near occultation by Sun. Science 159 (1968) 307-308
  • [20] Sadeh D., Knowles S. & Au B. The effect of mass on frequency. Science 161 (1968) 567-569
  • [21] Sadeh D., Hollinger J.P., Knowles S.H. & Youmans A.B. Search for an effect of mass on frequency during a close approach of pulsar CP 0950 to the Sun. Science 162(3856) (1968) 897-898
  • [22] Weast R.C. (Ed.) Handbook of chemistry and physics, 51st ed. Cleveland, OH: The Chemical Rubber Co., 1970, p.F145.
  • [23] Einstein A. The Foundation of the General Theory of Relativity. [pp. 109-164 in: Lorentz H.A. et al. (Eds.) The Principle of Relativity. New York: Dover, 1923, see pp.115ff,117].
  • [24] Czajko J. Equipotential energy exchange depends on density of matter. Studies in Mathematical Sciences 7(2) (2013) 40-54
  • [25] Møller C. Triumphs and limitations of Einstein’s theory of relativity and gravitation. [pp.473-492 in: De Finis F. (Ed.) Relativity, quanta and cosmology II. New York: Johnson Reprint Corp., 1979].
  • [26] Joseph D.W. Generalized covariance. Rev. Mod. Phys. 37(1) (1965) 225-227 see p.226
  • [27] Vvedensky D. Partial differential equations with Mathematica ®. Reading, MA: Addison-Wesley, 1993, pp.18ff,21ff.
  • [28] Czajko J. New product differentiation rule for paired scalar reciprocal functions. World Scientific News 144 (2020) 358-371
  • [29] Tartar L. Compacité par compensation : résultats et perspectives. [p.350-369 in : Brezis H. & Lions J.L. (Eds.) Nonlinear partial differential equations and their applications. Collège de France Seminar IV. Boston: Pitman, 1983, see p.364].
  • [30] Silvestrov P.G. Localization in imaginary vector potential. Physical Review B (Condensed Matter and Materials Physics) 58(16) (1998) R10111-R10114
  • [31] Czajko J. Operational constraints on dimension of space imply both spacetime and timespace. Int. Lett. Chem. Phys. Astron. 36 (2014) 220-235
  • [32] Freitag E. Riemannian and Lorentzian geometry. Differential geometry, Riemannian manifolds, Lorentzian manifolds, cosmology. Coppell, TX: 2020, p.80ff.
  • [33] Frankel T. The geometry of physics. An introduction. Cambridge: Cambridge Univ. Press, 1997, p.73.
  • [34] Czajko J. Operational restrictions on morphing of quasi-geometric 4D physical spaces. Int. Lett. Chem. Phys. Astron. 41 (2015) 45-72
  • [35] Birkhoff G. (Ed.) A source book in classical analysis. Cambridge, MA: Harvard Univ. Press, 1973, p.360.
  • [36] Fridman A.M. & Polyachenko V.L. Physics of gravitating systems I: Equilibrium and stability. New York: Springer, 1984, p.7.
  • [37] Czajko J. Dual reciprocal scalar potentials paired via differential operators in Frenet frames make the operators to act simultaneously in each of two paired 3D reciprocal spaces. World Scientific News 137 (2019) 96-118
  • [38] Zeidler E. (Ed.) Oxford users’ guide to mathematics. Oxford: Oxford Univ. Press, 1996, p.266ff.
  • [39] Choquet-Bruhat Y. Introduction to general relativity, black holes, and cosmology. Oxford: Oxford Univ. Press, 2015, p.5.
  • [40] Czajko J. Multiplicative counterpart of the essentially additive Borsuk-Ulam theorem as the pivoting gateway to equidimensional paired dual reciprocal spaces. World Scientific News 150 (2020) 118-131
  • [41] Penrose R. The road to reality. A complete guide to the laws of the universe. New York: Alfred A. Knopf, 2005, pp.189,503.
  • [42] Minkowski H. Space and time. [pp. 73-91 in: Lorentz H.A. et al. (Eds.) The Principle of Relativity. New York: Dover, 1923, see pp.85ff,87].
  • [43] Misner C.W., Thorne K.S. & Wheeler J.A. Gravitation. New York: Freeman, 1973, pp.50,51,54,463f.
  • [44] Ohanian H. & Ruffini R. Gravitation and spacetime. New York: W.W. Norton, 1994, p.2.
  • [45] Schutz B. Gravity from ground up. Cambridge: The Press Syndicate of the Univ. of Cambridge, 2007, p.54.
  • [46] Rindler W. Introduction to special relativity. Oxford: Clarendon Press, 1991, pp.17,52ff.
  • [47] Naber G.L. The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Mineola, NY: Dover, 1992, p.18ff.
  • [48] Pauli W. Theory of relativity. New York: Dover, 1958, p.11ff.
  • [49] Bohm D. The special theory of relativity. London: Routledge, 2006, p.194ff.
  • [50] Czajko J. Equalized mass can explain the dark energy or missing mass problem as higher density of matter in stars amplifies their attraction. World Scientific News 80 (2017) 207-238
  • [51] Thomas G.B., Jr. & Finney R.L. Calculus and analytic geometry II. Reading, MA: Addison-Wesley, 1996, p.998.
  • [52] Hawking S. & Penrose R. The nature of space and time. Princeton, NJ: Princeton Univ. Press, 1996, p.100.
  • [53] French A.P. Special relativity. New York: W.W. Norton, 1968, p.221ff.
  • [54] Katz R. An introduction to the special theory of relativity. Princeton, NJ: Van Nostrand, 1964, p.53ff.
  • [55] Dixon W.G. Special relativity. The foundation of macroscopic physics. Cambridge: Cambridge Univ. Press, 1978, p.101ff.
  • [56] Lorentz H.A. The theory of electrons and its applications to the phenomena of light and radiant heat. New York: The Columbia Univ. Press, 1909, p.198f.
  • [57] Bickerstaff R.P. & Patsakos G. Relativistic generalization of mass. Eur. J. Phys. 16 (1995) 63-66
  • [58] Czajko J. Radial and nonradial effects of radial fields in Frenet frame. Applied Physics Research 3(1) (2011) 2-7 DOI:10.5539/apr.v3n1pdoi:10.5539/apr.v3n1p
  • [59] Czajko J. With Equalized Mass, its Density of Matter can Affect Radial Gravitational Interactions too. Int. Lett. Chem. Phys. Astron. 54 (2015) 112-121
  • [60] Weinberg S. Gravitation and cosmology. Principles and applications of the general theory of relativity. New Delhi: Wiley, 2017, p.70.
  • [61] Stephani H. Relativity. An introduction to special and general relativity. Cambridge: Cambridge Univ. Press, 2007, p.237.
  • [62] Tolman R.C. The theory of relativity of motion. Berkeley, CA: Univ. of California Press, 1917, p.100.
  • [63] Stephani H. General relativity. An introduction to the theory of gravitational field. Cambridge: Cambridge Univ. Press, 1990, p.117.
  • [64] Kenyon I.R. General relativity. Oxford: Oxford Univ. Press, 1991, p.15ff.
  • [65] Born M. Einstein’s theory of relativity. New York: Dover, 1965, p.354.
  • [66] Hakim R. An introduction to relativistic gravitation. Cambridge: Cambridge Univ. Press, 1999, p.136.
  • [67] Harpaz A. Relativity theory. Concepts and basic principles. Boston: Jones and Bartlett, 1992, pp.94,120ff,124.
  • [68] Weber J. General relativity and gravitational waves. Mineola, NY: Dover, 2004, p.61.
  • [69] Wu Q. et al. Common fixed points for two pairs of compatible mappings in 2-metric spaces. [pp.239-246 in: Cho Y.J. (Ed.) Nonlinear functional analysis and applications. Volume 1. New York: Nova Science Publishers, 2010].
  • [70] Chang S.-S., Cho Y.J. & Kim J.K. Generalized contraction mapping principle in Menger probabilistic metric spaces. [pp.247-257 in: Cho Y.J. (Ed.) Nonlinear functional analysis and applications. Volume 1. New York: Nova Science Publishers, 2010, see p.253].
  • [71] Meschkowski H. Hundert Jahre Mengenlehre. München: Deutsche Taschenbuch Verlag, 1973, p.52.
  • [72] Peter R. Rekursive Funktionen. Berlin: Akademie-Verlag, 1957, p.218.
  • [73] Beiser A. Concepts of modern physics. Boston: McGraw-Hill Book Co., 1973, p.15.
  • [74] Fock V. The theory of space, time and gravitation. New York: Macmillan, 1964, p.42ff.
  • [75] Hafele J.C. & Keating R.E. Around-the-World Atomic Clocks: Observed Relativistic Time Gains. Science 177 (4044) (1972) 168–170
  • [76] Hafele J.C. & Keating R.E. Around-the-World Atomic Clocks: Predicted Relativistic Time Gains. Science 177 (4044) (1972) 166–168
  • [77] Alley C. P. Proper Time Experiments in Gravitational Fields with Atomic Clocks, Aircraft, and Laser Light Pulses. [pp. 363-427 in: Alley C.O., Meystre P. & Scully M. O. (Eds.) Quantum Optics, Experimental Gravity, and Measurement Theory. New York: Plenum, 1983].
  • [78] Czajko J. On the Hafele-Keating experiment. Ann. Phys. (Leipzig) 47 (1990) 517-518
  • [79] Czajko J. Experiments with flying atomic clocks. Exper. Tech. Phys. 39 (1991) 145-147
  • [80] Weber A. & Lide D.R., Jr. Molecular spectroscopy. [pp.1522-1600 in: Lerner R.G. & Trigg G.L. (Eds.) Encyclopedia of physics: Vol. 2: M-Z. Third, completely revised and enlarged edition. Weinheim: Wiley-VCH Verlag, 2005, see pp.1523,1532].
  • [81] Tourrenc P. Relativity and gravitation. Cambridge: Cambridge Univ. Press, 1997, p.20ff.
  • [82] Weinstein E. Wave equation. MathWorld. https://mathworld.wolfram.com/WaveEquation.html

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