ISSUE 2005 PROGRESS IN PHYSICS VOLUME 1 ISSN 1555-5534 The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics PROGRESS IN PHYSICS A quarterly issue scientific journal, registered with the Library of Congress (DC). ISSN: 1555-5534 (print) This journal is peer reviewed and included in the abstracting and indexing coverage of: ISSN: 1555-5615 (online) Mathematical Reviews and MathSciNet (AMS, USA), DOAJ of Lund University (Sweden), 4 issues per annum Referativnyi Zhurnal VINITI (Russia), etc. Electronic version of this journal: APRIL 2005 VOLUME 1 http://www.geocities.com/ptep_online To order printed issues of this journal, con- CONTENTS tact the Editor in Chief. Chief Editor D. Rabounski A New Method to Measure the Speed of Gravitation. 3 Dmitri Rabounski Quantum Quasi-Paradoxes and Quantum Sorites Paradoxes. 7 [email protected] F. Smarandache Associate Editors F. Smarandache A New Form of Matter — Unmatter, Composed of Particles Prof. Florentin Smarandache and Anti-Particles. 9 [email protected] L. Nottale Fractality Field in the Theory of Scale Relativity. 12 Dr. Larissa Borissova [email protected] L. Borissova, D. Rabounski On the Possibility of Instant Displacements in Stephen J. Crothers the Space-Time of General Relativity. 17 [email protected] C. Castro On Dual Phase-Space Relativity, the Machian Principle and Mod- Department of Mathematics, University of ified Newtonian Dynamics. 20 New Mexico, 200 College Road, Gallup, NM 87301, USA C. Castro, M. Pavsiˇ cˇ The Extended Relativity Theory in Clifford Spaces. 31 K. Dombrowski Rational Numbers Distribution and Resonance. 65 Copyright c Progress in Physics, 2005 S. J. Crothers On the General Solution to Einstein’s Vacuum Field and Its All rights reserved. Any part of Progress in Implications for Relativistic Degeneracy. 68 Physics howsoever used in other publica- tions must include an appropriate citation S. J. Crothers On the Ramifications of the Schwarzschild Space-Time Metric..74 of this journal. S. E. Shnoll, I.A. Rubinshtein, K.I. Zenchenko, V.A. Shlekhtarev, A. V.Ka- Authors of articles published in Progress in Physics retain their rights to use their own minsky, A. A. Konradov, N. V.Udaltsova Experiments with Rotating articles in any other publications and in any Collimators Cutting out Pencil of α-Particles at Radioactive Decay of way they see fit. 239Pu Evidence Sharp Anisotropy of Space. 81 F. Smarandache There Is No Speed Barrier for a Wave Phase Nor for Entang- This journal is powered by BaKoMa-TEX led Particles. 85 A variety of books can be downloaded free from the E-library of Science: http://www.gallup.unm.edu/ smarandache ∼ ISSN: 1555-5534 (print) ISSN: 1555-5615 (online) Standard Address Number: 297-5092 Printed in the United States of America Information for Authors and Subscribers Progress in Physics has been created for publications on advanced studies in theoretical and experimental physics, including related themes from mathematics. All submitted papers should be professional, in good English, containing a brief review of a problem and obtained results. All submissions should be designed in LATEX format using Progress in Physics template. This template can be downloaded from Progress in Physics home page http://www.geocities.com/ptep_online. Abstract and the necessary information about author(s) should be included into the papers. To submit a paper, mail the file(s) to Chief Editor. All submitted papers should be as brief as possible. Short articles are preferable. All that has been accepted for the online issue of Progress in Physics is printed in the paper version of the journal. To order printed issues, contact Chief Editor. This journal is non-commercial, academic edition. It is printed from private donations. April, 2005 PROGRESS IN PHYSICS Volume 1 A New Method to Measure the Speed of Gravitation Dmitri Rabounski E-mail: [email protected] According to the standard viewpoint the speed of gravitation is the speed of weak waves of the metrics. This study proposes a new approach, defining the speed as the speed of travelling waves in the field of gravitational inertial force. D’Alembert’s equations of the field show that this speed is equal to the velocity of light corrected by gravitational potential. The approach leads to a new experiment to measure the speed of gravitation, which, using “detectors” such as planets and their satellites, is not linked to deviation of geodesic lines and quadrupole mass-detectors with their specific technical problems. (0) 1 Introduction the Ricci tensor for the metric gαβ = gαβ + ζαβ is 1 ∂2ζ 1 Herein we use a pseudo-Riemannian space with the signature (0)μν αβ Rαβ = g μ ν = ζαβ , (+ ), where time is real and spatial coordinates are imag- 2 ∂x ∂x 2 −−− inary, because the projection of a four-dimensional impulse which simplifies Einstein’s field equations R 1 g R = αβ − 2 αβ on the spatial section of any given observer is positive = κ T + λg , where in this case R = g(0)μν R . In the − αβ αβ μν in this case. We also denote space-time indices in Greek, absence of matter and λ-fields (Tαβ = 0, λ = 0), that is, in while spatial indices are Roman. Hence the time term in (0) emptiness, the Einstein equations for the metric gαβ = gαβ + αβ d’Alembert’s operator = g α β will be positive, while + ζ become ∇ ∇ αβ β the spatial part (Laplace’s operator) will be negative Δ = ζα = 0 . = gik . − ∇i∇k Actually, these are the d’Alembert equations of the cor- By applying the d’Alembert operator to a tensor field, (0) rections ζαβ to the metric gαβ = gαβ + ζαβ (weak waves of we obtain the d’Alembert equations of the field. The non- the metric). Taking the flat wave travelling in the direction zero elements are the d’Alembert equations containing the x1 = x, we see field-inducing sources. The zero elements are the equations 2 2 1 ∂ ∂ β without the sources. If there are no sources the field is free, 2 2 2 ζα = 0 , 2 c ∂t − ∂x giving a free wave. There is the time term 1 ∂ containing a2 ∂t2 so weak waves of the metric travel at the velocity of light in the linear velocity a of the wave. For this reason, in the empty space. case of gravitational fields, the d’Alembert equations give This approach leads to an experiment, based on the prin- rise to a possibility of calculating the speed of propagation ciple that geodesic lines of two infinitesimally close test- of gravitational attraction (the speed of gravitation). At the particles will deviate in a field of waves of the metric. A same time the result may be different according to the way system of two real particles connected by a spring (a quadru- we define the speed as the velocity of waves of the metric, pole mass-detector) should react to the waves. Most of these or something else. experiments have since 1968 been linked to Weber’s detector. The usual approach sets forth the speed of gravitation as The experiments have not been technically decisive until follows [1, 5]. One considers the space-time metric gαβ = now, because of problems with precision of measurement (0) , composed of a Galilean metric (0) (wherein = gαβ + ζαβ gαβ and other technical problems [3] and some purely theoretical g(0) = 1, g(0) = 0, g(0) = δ ) and tiny corrections ζ de- 00 0i ik − ik αβ problems [4, 5]. fining a weak gravitational field. Because the ζαβ are tiny, we Is the approach given above the best? Really, the result- can raise and lower indices with the Galilean metric tensor ing d’Alembert equations are derived from that form of the (0) αβ gαβ. The quantities ζ are defined by the main property Ricci tensor obtained under the substantial simplifications of σβ β of the fundamental metric tensor gασg = δα as follows: higher order terms withheld (i .e. to first order). Eddington (0) σβ β αβ (gασ + ζασ) g = δα. Besides this approach defines g [1] wrote that a source of this approximation is a specific and g = det gαβ to within higher order terms withheld as reference frame which differs from Galilean reference frames αβ (0)αβk αβk (0) σ g = g ζ and g = g (1 + ζ), where ζ = ζσ . Be- by the tiny corrections ζαβ, the origin of which could be very − ...σ cause ζαβ are tiny we can take Ricci’s tensor Rαβ = Rασβ different from gravitation. This argument leads, as Eddington (the Riemann-Christoffel curvature tensor Rαβγδ contracted remarked, to a “vicious circle”. So the standard approach has on two indices) to within higher order terms withheld. Then inherent drawbacks, as follows: D. Rabounski. A New Method to Measure the Speed of Gravitation 3 Volume 1 PROGRESS IN PHYSICS April, 2005 (1) The approach gives the Ricci tensor and hence the 1 ∗∂ + 2 vi are non-commutative. They are the chr.inv.-vector d’Alembert equations of the metric to within higher c ∂t of gravitational inertial force Fi, the chr.inv.-tensor of angular order terms withheld, so the velocity of waves of the velocities of the space rotation Aik, and the chr.inv.-tensor metric calculated from the equations is not an exact of rates of the space deformations Dik, namely theoretical result; (2) A source of this approximation are the tiny corrections 1 ∂w ∂vi Fi = , ζ to a Galilean metric, the origin of which may be √g ∂xi − ∂t αβ 00 very different: not only gravitation; (3) Two bodies attract one another because of the transfer 1 ∂vk ∂vi 1 Aik = + (Fivk Fkvi) , of gravitational force. A wave travelling in the field 2 ∂xi − ∂xk 2c2 − of gravitational force is not the same as a wave of the metric — these are different tensor fields.