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The authors of the articles published in Progress in Physics retain their rights to use this journal as a whole or any part of it in any other publications and in any way they see fit. Any part of Progress in Physics howsoever used in other publications must include an appropriate citation of this journal. JULY 2008 CONTENTS VOLUME 3 I. I. Haranas and M. Harney Detection of the Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 R. T. Cahill Resolving Spacecraft Earth-Flyby Anomalies with Measured Light Speed Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 F. Smarandache and V. Christianto The Neutrosophic Logic View to Schro¨dinger’s Cat Paradox, Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 P. Wagener A Classical Model of Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A. A. Ungar On the Origin of the Dark Matter/Energy in the Universe and the Pioneer Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 P.-M. Robitaille A Critical Analysis of Universality and Kirchhoff’s Law: A Return to Stewart’s Law of Thermal Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 P.-M. Robitaille Blackbody Radiation and the Carbon Particle . . . . . . . . . . . . . . . . . . . . . . . . 36 A. Khazan The Roˆle of the Element Rhodium in the Hyperbolic Law of the Periodic Table of Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 V. Chrisatianto and F. Smarandache What Gravity Is. Some Recent Considerations . . . . 63 U. K.W. Neumann Models for Quarks and Elementary Particles — Part III: What is the Nature of the Gravitational Field? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 U. K.W. Neumann Models for Quarks and Elementary Particles — Part IV: How Much Do We Know of This Universe? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 A. Sharma The Generalized Conversion Factor in Einstein’s Mass-Energy Equation . . . . . 76 E. A. Isaeva On the Necessity of Aprioristic Thinking in Physics . . . . . . . . . . . . . . . . . . . . . . 84 S. M. Diab and S. A. Eid Potential Energy Surfaces of the Even-Even 230-238U Iso- topes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 LETTERS E. Goldfain A Brief Note on “Un-Particle” Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 F. Smarandache International Injustice in Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 This journal is powered by LATEX A variety of books can be downloaded free from the Digital 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 and astronomy. 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.ptep-online.com. 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July, 2008 PROGRESS IN PHYSICS Volume 3 Detection of the Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer Ioannis Iraklis Haranas and Michael Harneyy Department of Physics and Astronomy, York University, 314A Petrie Building, North York, Ontario, M3J-1P3, Canada E-mail: ioannis@yorku.ca y841 North 700 West, Pleasant Grove, Utah, 84062, USA E-mail: michael.harney@signaldisplay.com General Relativity predicts the existence of relativistic corrections to the static Newtonian potential which can be calculated and verified experimentally. The idea leading to quantum corrections at large distances is that of the interactions of massless particles which only involve their coupling energies at low energies. In this short paper we attempt to propose the Sagnac intrerferometric technique as a way of detecting the relativistic correction suggested for the Newtonian potential, and thus obtaining an estimate for phase difference using a satellite orbiting at an altitude of 250 km above the surface of the Earth. 1 Introduction The potential acting between to masses M and m that sepa- rated from their centers by a distance r is: V (r) = GM r m ; (1) where s the Newton’s constant of gravitation. This potential is of course only approximately valid [1]. For large masses and or large velocities the theory of General Relativity predicts that there exist relativistic corrections which can be calculated and also verified experimentally [2]. In the microscopic distance domain, we could expect that quantum mechanics, would predict a modification in the gravitational potential in the same way that the radiative corrections of quantum electrodynamics leads to a similar modification of the Coulombic interaction [3]. Even though the theory of General Relativity constitutes a very well defined classical theory, it is still not possible to combine it with quantum mechanics in order to create a satisfied theory of quantum gravity. One of the basic obstacles that prevent this from happening is that General Relativity does not actually fit the present paradigm for a fundamental theory that of a renormalizable quantum field theory. Gravitational fields can be successfully quantized on smooth-enough space-times [4], but the form of gravitational interactions is such that they induce unwanted divergences which can not be absorbed by the renormalization of the parameters of the minimal General Relativity [5]. Somebody can introduce new coupling constants and absorb the divergences then, one is unfortunately led to an infinite number of free parameters. In spite the difficulty above quantum gravity calculations can predict long distance quantum corrections. The main idea leading to quantum corrections at large distances is due to the interactions of massless particles which only involve their coupling energies at low energies, something that it is known from the theory of General Relativity, even though at short distances the theory of quantum grav- ity differs resulting to finite correction of the order, O cG3r3 . The existence of a universal long distance quantum correction to the Newtonian potential should be relevant for a wide class of gravity theories. It is well known that the ultraviolet behaviour of Einstein’s pure gravity can be improved, if higher derivative contributions to the action are added, which in four dimensions take the form: RR + R2; (2) where and are dimensionless coupling constants. What makes the difference is that the resulting classical and quan- tum corrections to gravity are expected to significantly alter the gravitational potential qat short distances comparable to that of Planck length `P = Gc3 = 10 35 m, but it should not really affect its behaviour at long distances. At long distances is the structure of the Einstein-Hilbert action that actually de- termines that. At this point we should mentioned that some of the calculation to the corrections of the Newtonian gravi- tational potential result in the absence of a cosmological con- stant which usually complicates the perturbative treatment to a significant degree due to the need to expand about a non- flat background. In one loop amplitude computation one needs to calculate all first order relativistic O cGo2crm2re2ctioannsdinthGe ,quwahnitcuhmwmillecinhcalnuidcael both the O Gc3 corrections to the classical Newtonian potential [6]. 2 The corrections to the potential Our goal is not to present the details of the one loop treatment that leads to the corrections of the Newtonian gravita- Ioannis I. Haranas and Michael Harney. The Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer 3 Volume 3 PROGRESS IN PHYSICS July, 2008 tional potential but rather sate the result and then use it in our calculations. Valid in order of G2 we have that the corrected potential now becomes [6]: V (r) = GM r m  1 G (M + m) 2c2r 122 G 15c3r2  : (3) Observing (3) we see that in the correction of the static Newtonian potential two diffeqrent length scales are involved. First, the Planck length `P = Gc3 =10 35 m and second the Schwarzschild radii of the heavy sources rsch = 2GcM2 n . Fur- thermore there are two independent dimensionless parame- ters which appear in the correction term, and involve the ratio of these two scales wit respect to the distance r. Presumably for meaningful results the two length scales are much smaller than r. 3 Perturbations due to oblateness J2 Because the Earth’s gravitational potential is not that of a perfect spherical body, we can approximate its potential as a spherical harmonic expansion of the following form: V (r; ) = " GM m r 1 X 1 n=2 Jn  Re r n Pn (sin # ) = = GM m r [Vo + VJ2 + VJ3 + : : : ] ; (4) where: r = geocentric distance,  = geocentric latitude. Re = means equatorial radius of the Earth, Pn = Legendre polynomial of degree n and order zero, Jn = Jn0 jonal harmonics of order zero, that depend on the latitude  only, and the first term GMm=r now describes the potential of a homogeneous sphere and thus refers to Keplerian motion, the remaining part represents the Earth’s oblateness via the zonal harmonic coefficients and [7] V0 = 1 VJ2 = J2 2  Re 2 r 3 sin2  VJ3 = J3 2  Re 3 r 5 sin3  9 1 >>>>>>= 3 sin  >>>>>>; (5) similarly [8] J2 J3 = 1,082.6 10 6 = 2.53 10 6 ) : (6) Therefore equation (4) can be further written: V (r; ) = GM m r  1 X 1 n=2  Jn Re r n Pn(sin ) G (M + 2rc2 m)  = (6a) = GM m r [Vo + VJ2 + VJ3 + ::: VRelativistic] : Since J2 is 400 larger that any other Jn coefficients, we can disregard them and write the following expression for the Earth’s potential function including only the relativistic correction and omitting the quantum corrections as being very small we have: V (r; ) = GMem r + GMemRe2J2 r3 3 2 sin2 + G2Mem (Me c2r2 + m) : 1 2  + (7) Since we propose a satellite in orbit that carries the Sagnac instrument it will of a help to express equation (7) for the potential in terms of the orbital elements. We know that sin  = sin i sin(f +!) where i is the inclination of the orbit, f is the true anomaly and ! is the argument of the perigee. Ignoring long and short periodic terms (those containing ! and f) we write (7) in terms of the inclination as follows: V (r; ) = GMem r + 3GMemRe2J2 2r3  sin2i 2 + G2Mem (Me c2r2 + m) 1 3  + (8) therefore the corresponding total acceleration that a mass m at r > Re has becomes: gtot = 1 @ m @r GMem r + 3GMemRe2J2 2r3  sin2i 2 1 3 + G2Mem (M r2c2 + m)  (9) so that: gtot = GMe r2 + 9 GMeRe2 J2 2r4  sin2 2 i 1 3  + + G2Me (Me c2r3 + m) : (10) 4 Basic Sagnac interferometric theory The Sagnac interferometer is based on the Sagnac effect, reported by G. Sagnac in 1913 [8]. Two beams are sent in opposite directions around the interferometer until they meet 4 Ioannis I. Haranas and Michael Harney. The Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer July, 2008 PROGRESS IN PHYSICS Volume 3 rs = (c2 h Rs2 2s) 1  Rc2s GrM2 e + 9 82Rs2N GM2re4Re2J2  s Rs sin2i 2 13s++Gr232Mcv2eo2rb h 1 cos h 2Rcs s 1 + Rsc s 1iii (15)  q  rs = (c2 Rs2 82Rs2N s Rs h  2s) 1 + Rc2s GrM2 e 3GM4erR4 e2J2 s + 2 Gr23Mc2e2  h (ReG+Mzoerhb) 1 cos 2Rcs s 1 + Rsc s 1iii (16) rs =h  c2 1+Rs Gc2MRe2e  1 82Rs2N s  1 + Rs2c2 2s   Rs s+ r 2 GRMee 1  2zRoerb 3G4Mc2eRRe4e2 J2  1  4zRoerb GR2e3Mc4e2  1 h 3zRoerb 1  zRoreb h cos 2Rcs s 1+ Rsc s 1iii (17) again to create a phase pattern. By rotating the interferometer in the direction of either the clockwise (CW) or counterclockwise (CCW) beam, a phase difference results between the two beams that its given by: rs = 8 2 Rs2ag N (c2 a2  2) ; (11) where is the angular velocity of the interferometer, Rsag is the radius of the interferometer, N is the number of turns of fiber around the radius and  is the frequency of light in the fiber. Let us now assume that the Sagnac interferometer and its light laser beams are in the region of space around the Earth where the gravitational potential is given by equation (3) and let us further assume that the quantum correction to the potential is really negligible. If the Sagnac light loop area has a unit vector that is perpendicular to the acceleration of gravity vector, then the motion of the interferometer will exhibit a red-shift that will be given by: frs = 1 f c2V =1 f gcco2r z ; (12) where V is the difference in the potential between to differ- ent points P1 and P2, and gcor is the corrected or total acceleration of gravity and z is the difference in vertical distance between the two beams as the interferometer coil rotates. This distance z that the laser beams see is given by: 8 2 39 < z = Rsag 1 : cos 4 c 2 1+ Rsag Rsag c = 5 : ; (13) This Sagnac effect can also be amplified by an interferometer that is in orbit, where the orbital velocity of the interferometer with respect to the Earth’s surface produces an increased phase shift. Both terms involved in the acceleration of gravity in the first one: rs = 82Rs2agN  c2 Rs2ag Rsag 21 + vor2b gtco2tz (14)  using (14) and taking into account that M m we further obtain (15), where M is the source of the gravitational field = the mass of the Earth in our case Me, and R is the radius of the massive body = Re, and r = Re + zorb it’s orbital height plus Earth radius for an Earth-based satellite. This Sagnac effect can also be amplified by an interfer- ometer that is in orbit, where the orbital velocity of the interferometer with respect to the Earth’s surface produces an increased phase shift. Both terms involved in the acceleration of gravity in the first one: 5 Sagnac in circular orbit of known inclination Let now a Sagnac interferometer be aboard a satellite in a circular polar orbit of inclination i = 90 degrees. If the incli- nation is 90 degrees the velocityqat some height zterambosvien2th2ie 1 3 = 1 6 surface and the orbital of the Earth is vorb = (ReG+MZeorb) and (6) takes the form (16) can be finally written as (17). 6 Sagnac in elliptical orbit of known inclination If now a satellite is carrying a Sagnac device is in an elliptical orbit of eccentricity e and semi-major axis a we have that the radial orbital vector and the orbital velocity are given by: v2 =  GMe 2 r r (f ) = a1 1+e e2 cos f ; 1 a = GMe a 2 (1+e cos f) (1 e2) (18)  1 ; (19) where f is the true anomaly of the orbit. Substituting now in (8) we obtain (20). If we use the fact that GMe = n2a3 where n is the mean motion of the satellite, equation (20) can be further written as (21). When the satellite approaches perigee its orbital velocity will increase, so we will expect to see a higher phase difference than any other point of the orbit, and similarly the effect Ioannis I. Haranas and Michael Harney. The Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer 5 Volume 3 PROGRESS IN PHYSICS July, 2008 82Rs2N s  1 + Rs2c2 2s   Rs s+ 2 r GMa e  1+e2+2e cos 1 e2 f  rs =h c2 1+Rs  GMc2ea(21(+1e cos f)2 e2)2 3GM4ecR2e2aJ42((11+ee2 cos )4 f )4 G2 Mc4ea23(1(1+eec2o)s3f )3 h 1 cos h 2Rcs s 1+ Rsc s 1iii (20) 82Rs2N s  1 + Rs2c2 2s  Rs s+ 2na r 1+e21+2ee2cos f  rs =h c2 1 + Rs  n2a(1+e cos f)2 c2(1 e2)2 3n2R4ce22Ja2((11+ee2c)o4s f)4 n4a3(1+e cos c4(1 e2)3 f )3 h 1 cos h 2Rcs s 1 + Rsc s 1iii (21) 82Rs2N s  1 + Rs2c2 2s   Rs s+ r   2 GMa e 1+e 1e rs (perigee) = c2 h 1 + Rs  c2aG2(M1 ee)2 43cG2aM4(e1Re2eJ)24 c4 aG32(1Me2e)3  h 1 h cos 2Rcs s 1 + Rsc s 1iii (24) rs (perigee) = c2 h 1 + Rs  82Rs2N s 1 +  c2(n12ae)2 4c32na2(R1e2Je2)4 Rs2c2c42sn(14a3e)R3 s s+ h 1 q 2 cos nh a2Rcs11+s ee 1 + Rsc s 1iii (25) 82Rs2N s  1 + Rs2c2 2s   Rs s+ 2 r GMa e   1e 1+e rs (apogee) = c2 h 1 + Rs  c2 aG2 (M1+e e)2 43cG2aM4(e1R+e2eJ)24 c4 aG32(1M+e2e)3  h 1 cos h 2Rcs s 1 + Rsc s 1iii (26) rs (apogee) = c2 h 1 + Rs  82Rs2N s 1 +  c2(n12+ae)2 4c32na2(R1+e2Je2)4 Rs2c2c42sn(14+a3e)R3 s s+ h 1 q  2na h 1+e 1e cos 2Rcs s 1 + Rsc s 1iii (27) will be minimum at the point of apogee because the satellite’s velocity is minimal. The distance at perigee and apogee are given by the equations below: rpg = a (1 e) ) (22) rapg = a (1 + e) also the corresponding velocities are: vp2g = GM a 1 1 + e e 9 >>>= va2pg = GMe a 1 e 1+e ; >>>; (23) therefore the phase difference detected by the Sagnac due to the contribution of the Earth’s oblateness plus relativistic correction to the potential at perigee and apogee can be written as (24) or again (25). Similarly the phase difference at apogee can be written as (26) or again (27). For this last case of the elliptical orbit in (25) and (26) where the Sagnac interferometer is on the satellite and we as- sume a=8 R1s06=m1,me,= =2 0.2, 1014 Hz, N Re = 6.378 =10160m6,etesrs=w4e00arraridv/eseact, the following values for :  (perigee) = 3.57 10 16 radians,  (apogee) = 2.44 10 16 radians. These values are based on the dominant potential correction in (11) of section 3 which is the first term in (11) or the Newtonian correction: Newtonian correction = 2.17 10 16 radians. In comparison, the second and third terms in (11) are the oblateness and relativistic corrections respectively and they produce the following values based on the given parameters: Oblateness correction = 8.52 10 20, Relativistic correction = 7.91 10 26. So by comparison of the values above, the Newtonian correction is much easier to measure. 6 Ioannis I. Haranas and Michael Harney. The Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer July, 2008 PROGRESS IN PHYSICS Volume 3  q  rs = (c2 a2s  2s) 1 + as 82a2sN s as s +  2 1+(e2+e 1) cos 2acs s (1+ asc s ) 1+e cos 2acs s (1+ asc s ) 1 (ReG+Mzoerb) 1 Gr2Mc2e 3G4Mre4Rc2e2J2 Gr23Mc4e2  (30) rs =  c2 1+ as  82a2sN s 1 + 1+(e2+e 1) cos 2acs s (1+ asc s ) 1+e cos 2acs s (1+ asc s ) 1 a2sc2 2   as s+ r 2 GRMee 1  zRoreb 1  GRe2Mc2e 1  2zRoerb 3G4MRee4Rc2e2J2  1  4zRoerb GR2e3Mc4e2  1  3zRoerb (31) 82a2sN s  1 + a2sc2 2s   as s+ 2 r GMa e  1+e2+2e cos f 1 e2  rs=  c2 1+ as 1+(e2+e 1) cos 1+e cos 2acs 2acs s (1+ asc s ) s (1+ asc s ) 1 1  GMc2ea(21(+1e cos f e2)2 )2 3GM4ecR2e2aJ42((11+ee2)c4os f)4 G2Mc4ea23(1(1+eec2o)s3f)3  (32) 82a2sN s  1 + a2sc2 2s   as s+ r   2 GMa e 1+e 1e rs (perigee)=  c2 1 + aa 1+(e2+e 1) cos 1+e cos 2acs 2acs s (1+ asc s ) s (1+ asc s ) 1 1  c2aG2(M1 ee)2 43cG2aM4(e1Re2eJ)24 c4aG32(1Me2e)3  (33) 82a2sN s  1 + a2sc2 2s   as s+ r   2 GMa e 1e 1+e rs (apogee)=  c2 1 + as 1+(e2+e 1) cos 1+e cos 2acs 2acs s (1+ asc s ) s (1+ asc s ) 1 1  c2 aG2 (M1+e e)2 43cG2aM4(e1R+e2eJ)24 c4aG32(1M+e2e)3  (34) The  values given above may be more easily measured using a QPSK-modulator inserted in the CCW or CW beam path to improve phase resolution. Also, the use of higher wavelengths (factor of 10 higher in frequency) will increase resolution. we can finally write for (13): z = h as 1+(e2 +e  1) n n cos 2acs s 1+ 1 + e cos 2acs s 1 + asc s as1c os  1oi : (29) 7 We suggest a Sagnac with an elliptic fiber loop To attempt increasing the resolution of the phase difference of the Sagnac interferometer let us now propose a Sagnac loop, that has the shape of an ellipse that rotates with an angular velocity . In this case it can be shown that the height dif- ference between two points on the ellipse can be given by: " z=a 1+ e2 + e 1 1 + e cos  # cos  : (28) To check the validity of the formula we derived we can set e=0 which is the case of a circular Sagnac fiber optical path we can see that the (13) in now retrieved since Rsag = aloop(sag) = as is the semi major axis of the ellip- tical fiber loop. When the ellipse spins with angular velocity that would force it to trace out a circle whose radius r, will be that of the semi-major axis a of the ellipse, and therefore 8 Circular orbit formula for the phase difference of the Sagnac Let now as before have a Sagnac interferometer be aboard a satellite in a circular polar orbit of inclination i = 90 degrees. ItEhfaetrhtoherbiisintcavlloirnvbae(tlcioiorcnci)tyi=sa9q t 0so(dRmeeGge+Mrhzeoeeersibg)thhatenzdtea(rb6mo)vtseainkthe2es2isthuerf13afoc=remo16f(a3tnh0de) that can be finally written as (31). 9 Sagnac in elliptical orbit of known inclination If now a satellite is carrying a Sagnac device is in an elliptical orbit of eccentricity e and semi-major axis a we have that the radial orbital vector and the orbital velocity are given by (32). At perigee the equation (32) becomes (33) and also (34). Ioannis I. Haranas and Michael Harney. The Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer 7 Volume 3 PROGRESS IN PHYSICS For (33) and (34) above the following values are com- puted assuming e = 0.2,  = 2 1014 Hz, a = 8 106 meters, N = 1 (because the orbit is the Sagnac loop), Rsag = Rperigee or Rapogee as and apogee = d6et1e0rm4inreadd/bseyc(w22e),findp,erigee = 0.001 rad/sec,  (perigee) = 6.05 1010 radians,  (apogee) = 2.36 1010 radians. These values are for measuring the dominant Newtonian contribution as described in Section 6. To detect relativistic contribution which is 3.64 10 10 smaller than the Newtonian contribution the corresponding phase-shifts from (33) and (34) are:  (perigee) = 22 radians,  (apogee) = 8.59 radians. Thus, the relativistic contribution in (11) of Section 3 is easily measurable using a Sagnac interferometer where the satellites in orbit are the Sagnac loop. In this scenario, the light path can be implemented by transmitting laser beams from one satellite to the next satellite in orbit ahead of it. Also, by using the maximum spacing possible between satellites in orbit this will allow line of site transmission while reducing the number of satellites required for the Sagnac loop. With the potential to measure such small relativistic corrections, the merit of using satellites to implement a large Sagnac loop of radius Rs = Rap or Rper is well worth considering. Submitted on March 27, 2008 Accepted on April 04, 2008 References 1. Donoghue J. F. arXiv: gr-qc/9310024. 2. Bjorken J. D. and Drell S. Relativistic quantum mechanics and relativistic quantum field theory. Mc Graw-Hill, New York 1964. 3. t’ Hooft G. and Veltman M. Ann. Inst. H. Poincare, 1974, v. 20, 69. 4. Capper D. M., Leibrand G. and Medrano R. M. Phys. Rev. D, 1973, v. 8, 4320. 5. Goroff M. and Sagnotti A. Nucl. Phys. B, 1986, v. 22, 709. 6. Hamber H. W. and Liu S. arXiv: hep-th/9505128. 7. Blanco V. M. and Mc Cuskey S. W. Basic physics of the Solar system. Addison-Wesley Publishing Inc., 1961. 8. Sagnac G. R. Acad. Sci., 1913, v. 157, 708. 9. Camacho A. General Relativity and Gravitation, 2004, v. 36, no. 5, 1207–1211. 10. Roy A. E. Orbital motion. Adam Hilger, Third Edition, 1988. July, 2008 8 Ioannis I. Haranas and Michael Harney. The Relativistic Corrections to the Gravitational Potential using a Sagnac Interferometer July, 2008 PROGRESS IN PHYSICS Volume 3 Resolving Spacecraft Earth-Flyby Anomalies with Measured Light Speed Anisotropy Reginald T. Cahill School of Chemistry, Physics and Earth Sciences, Flinders University, Adelaide 5001, Australia E-mail: Reg.Cahill@flinders.edu.au Doppler shift observations of spacecraft, such as Galileo, NEAR, Cassini, Rosetta and MESSENGER in earth flybys, have all revealed unexplained speed “anomalies” — that the Doppler-shift determined speeds are inconsistent with expected speeds. Here it is shown that these speed anomalies are not real and are actually the result of using an incorrect relationship between the observed Doppler shift and the speed of the spacecraft — a relationship based on the assumption that the speed of light is isotropic in all frames, viz invariant. Taking account of the repeatedly measured light-speed anisotropy the anomalies are resolved ab initio. The Pioneer 10/11 anomalies are discussed, but not resolved. The spacecraft observations demonstrate again that the speed of light is not invariant, and is isotropic only with respect to a dynamical 3-space. The existing Doppler shift data also offers a resource to characterise a new form of gravitational waves, the dynamical 3-space turbulence, that has also been detected by other techniques. The Einstein spacetime formalism uses a special definition of space and time coordinates that mandates light speed invariance for all observers, but which is easily misunderstood and misapplied. 1 Introduction Planetary probe spacecraft (SC) have their speeds increased, in the heliocentric frame of reference, by a close flyby of the Earth, and other planets. However in the Earth frame of reference there should be no change in the asymptotic speeds after an earth flyby, assuming the validity of Newtonian gravity, at least in these circumstances. However Doppler shift observations of spacecraft, such as Galileo, NEAR, Cassini, Rosetta and MESSENGER in earth flybys, have all revealed unexplained speed “anomalies” — that the Doppler-shift determined speeds are inconsistent with expected speeds [1–6]. Here it is shown that these speed anomalies are not real and are actually the result of using an incorrect relationship between the observed Doppler shift and the speed of the spacecraft — a relationship based on the assumption that the speed of light is isotropic in all frames, viz invariant. Taking account of the repeatedly measured light-speed anisotropy the anomalies are resolved ab initio. The speed of light anisotropy has been detected in at least 11 experiments [7–17], beginning with the MichelsonMorley 1887 experiment [7]. The interferometer observations and experimental techniques were first understood in 2002 when the Special Relativity effects and the presence of gas were used to calibrate the Michelson interferometer in gas-mode; in vacuum mode the Michelson interferometer cannot respond to light speed anisotropy [18, 19], as confirmed in vacuum resonant cavity experiments, a modern version of the vacuum-mode Michelson interferometer [20]. So far three different experimental techniques have given consistent results: gas-mode Michelson interferometers [7–11, 16], coaxial cable RF speed measurements [12–14], and opticalfiber Michelson interferometers [15, 17]. This light speed anisotropy reveals the existence of a dynamical 3-space, with the speed of light being invariant only with respect to that 3space, and anisotropic according to observers in motion relative to that ontologically real frame of reference — such a motion being conventionally known as “absolute motion”, a notion thought to have been rendered inappropriate by the early experiments, particularly the Michelson- Morley experiment. However that experiment was never null — they reported a speed of at least 8km/s [7] using Newtonian physics for the calibration. A proper calibration of the Michelson-Morley apparatus gives a light speed anisotropy of at least 300km/s. The spacecraft Doppler shift anomalies are shown herein to give another technique that may be used to measure the anisotropy of the speed of light, and give results consistent with previous detections. The numerous light speed anisotropy experiments have also revealed turbulence in the velocity of the 3-space relative to the Earth. This turbulence amounts to the detection of sub-mHz gravitational waves — which are present in the Michelson and Morley 1887 data, as discussed in [21], and also present in the Miller data [8, 22] also using a gas-mode Michelson interferometer, and by Torr and Kolen [12], DeWitte [13] and Cahill [14] measuring RF speeds in coaxial cables, and by Cahill [15] and Cahill and Stokes [17] using an optical-fiber interferometer. The existing Doppler shift data also offers a resource to characterise this new form of gravitational waves. There has been a long debate over whether the Lorentz 3space and time interpretation or the Einstein spacetime inter- Reginald T. Cahill. Resolving Spacecraft Earth-Flyby Anomalies with Measured Light Speed Anisotropy 9 Volume 3 PROGRESS IN PHYSICS July, 2008 W V sv Earth © V Earth c vi E i ' c + vi V ' SC Fig. 1: Spacecraft (SC) earth flyby trajectory, with initial and final asymptotic velocity V, differing only by direction. The Doppler shift is determined from Fig. 2 and (1). Assuming, as conventionally done, that the speed of light is invariant in converting measured Doppler shifts to deduced speeds, leads to the so-called flyby anomaly, namely that the incoming and outgoing asymptotic speeds appear to be differ, by V1. However this effect is yet another way to observe the 3-space velocity vector, as well as 3-space wave effects, with the speed of light being c and isotropic only with re- spect to this structured and dynamical 3-space. The flyby anomalies demonstrate, yet again, that the invariance of the speed of light is merely a definitional aspect of the Einstein spacetime formalism, and is not based upon observations. A neo-Lorentzian 3-space and time formalism is more physically appropriate. pretation of observed SR effects is preferable or indeed even experimentally distinguishable. What has been discovered in recent years is that a dynamical structured 3-space exists, so confirming the Lorentz interpretation of SR [22, 24, 25], and with fundamental implications for physics. This dynamical 3-space provides an explanation for the success of the SR Einstein formalism. Indeed there is a mapping from the physical Lorentzian space and time coordinates to the nonphysical spacetime coordinates of the Einstein formalism — but it is a singular map in that it removes the 3-space velocity with respect to an observer. The Einstein formalism transfers dynamical effects, such as length contractions and clock slowing effects, to the metric structure of the spacetime manifold, where these effects then appear to be merely perspective effects for different observers. For this reason the Einstein formalism has been very confusing. Developing the Lorentzian interpretation has lead to a new account of gravity, which turns out to be a quantum effect [23], and of cosmology [21, 22, 26, 27], doing away with the need for dark matter and dark energy. So the discovery of the flyby anomaly links this effect to various phenomena in the emerging new physics. 2 Absolute motion and flyby Doppler shifts The motion of spacecraft relative to the Earth are measured by observing the direction and Doppler shift of the transponded RF transmissions. As shown herein this data gives another technique to determine the speed and direction of the dynamical 3-space, manifested as a light speed anisotropy. Up to now the repeated detection of the anisotropy of the speed of Fig. 2: Asymptotic flyby configuration in Earth frame-of-reference, with spacecraft (SC) approaching Earth with velocity V. The de- parting asymptotic velocity will have a different direction but the same speed, as no force other than conventional Newtonian gravity is assumed to be acting upon the SC. The Dynamical 3-space ve- locity is v(r; t), which causes the outward EM beam to have speed c vi, and inward speed c + vi, where vi = v cos(i), with i the angle between v and V. light has been ignored in analysing the Doppler shift data, causing the long-standing anomalies in the analysis [1–6]. In the Earth frame of reference, see Fig. 2, let the trans- mitted signal from earth have frequency f, then the corresponding outgoing wavelength is 0 = (c vi)=f, where vi = v cos(i). This signal is received by the SC to have period Tc = 0=(c vi +V ) or frequency fc = (c vi +V )=0. The signal is re-transmitted with the same frequency, and so has wavelength i = (c + vi V )=fc, and is detected at earth with frequency fi = (c + vi)=i. Then overall we obtain fi = c c+ + vi vi V c vi + V c vi f: (1) Ignoring the projected 3-space velocity vi, that is, assum- ing that the speed of light is invariant as per the usual literal interpretation of the Einstein 1905 light speed postulate, we obtain instead fi = c+ c V V f : (2) The use of (2) instead of (1) is the origin of the putative anomalies. The Doppler shift data is usually presented in the form of speed anomalies. Expanding (2) we obtain fi f = fi f f = 2V c +::: (3) From the observed Doppler shift data acquired during a flyby, and then best fitting the trajectory, the asymptotic hy- perbolic speeds Vi1 and Vf1 are inferred, but incorrectly so, as in [1]. These inferred asymptotic speeds may be related to an inferred asymptotic Doppler shift: fi f = fi f f = 2Vi1 c +::: (4) In practice the analysis is more complex as is the doppler shift technology. The analysis herein is sufficient to isolate and quantify the light-speed anisotropy effect. 10 Reginald T. Cahill. Resolving Spacecraft Earth-Flyby Anomalies with Measured Light Speed Anisotropy July, 2008 PROGRESS IN PHYSICS Volume 3 Parameter Date V1 km/s i deg i deg f deg f deg v deg(hrs) v deg v km/s i deg f deg (O) V1 mm/s (P) V1 mm/s GLL-I Dec 8, 1990 8.949 266.76 12.52 219.97 34.15 108.8(7.25) 76 420 90.5 61.8 3.92 0.3 3.92 0.1 GLL-II Dec 8, 1992 8.877 219.35 34.26 174.35 4.87 129.0(8.6) 80 420 56.4 78.2 4.6 1.0 4.60 0.6 NEAR Jan 23, 1998 6.851 261.17 20.76 183.49 71.96 108.8(7.25) 76 450 81.8 19.6 13.46 0.01 13.40 0.1 Cassini Aug 18, 1999 16.010 334.31 12.92 352.54 20.7 45.0(3.0) 75 420 72.6 76.0 21 0.99 1.0 Rosetta Mar 4, 2005 3.863 346.12 2.81 246.51 34.29 130.5(8.7) 80 420 95.3 60.5 1.80 0.03 1.77 0.3 M’GER Aug 2, 2005 4.056 292.61 31.44 227.17 31.92 168.0(11.2) 85 420 124.2 55.6 0.02 0.01 0.025 0.03 Table 1: Earth flyby parameters from [1] for spacecraft Galileo (GLL: flybys I and II), NEAR, Cassini, Rosetta and MESSENGER (M’GER). V1 is the average osculating hyperbolic asymptotic speed, and are the right ascension and declination of the incoming (i) and outgoing (f) osculating asymptotic velocity vectors, and (O) V1 is the putative “excess speed” anomaly deduced by assuming that the speed of light is isotropic in modeling the doppler shifts, as in (4). The observed (O) V1 values are from [1], and after correcting for atmospheric drag in the case of GLL-II, and thruster burn in the case of Cassini. (P) V1 is the predicted “excess speed”, using (7), taking account of the known light speed anisotropy and its effect upon the doppler shifts, using v and v as the right ascension and declination of the 3-space flow velocity, having speed v, which has been taken to be 420 km/s in all cases, except for NEAR, see Fig. 3. The values on (P) V1 indicate changes caused by changing the declination by 5% — a sensitivity indicator. The angles i and f between the 3-space velocity and the asymptotic initial/final SV velocity V are also given. The observed doppler effect is in exceptional agreement with the predictions using (7) and the previously measured 3-space velocity. The flyby doppler shift is thus a new technique to accurately measure the dynamical 3-space velocity vector, albeit retrospectively from existing data. Note: By fine tuning the v and v values for each flyby a perfect fit to the observed (O) V1 is possible. But here we have taken, for simplicity, the same values for GLL-I and NEAR. However expanding (1) we obtain, for the same Doppler shift Vi1  fi f c 2 = fi f f c 2 =  1 + vi2 c2  V + ::: (5) where V is the actual asymptotic speed. Similarly after the flyby we obtain Vf1  ff f c 2 = ff f f c 2 =  1 + vf2 c2  V +::: (6) and we see that the “asymptotic” speeds Vi1 and Vf1 must differ, as indeed first noted in the data by [3]. We then obtain the expression for the so-called flyby anomaly V1 = Vf1 Vi1 = vf2 c2 vi2 V + : : : = v2 c2 cos(f )2 cos(i)2 V1 + : : : (7) where here V  V1 to sufficient accuracy, where V1 is the average of Vi1 and Vf1, The existing data on v permits We ignore terms of order vV =c2 within the parentheses, as in practice they are much smaller than the v2=c2 terms. ab initio predictions for V1, and as well a separate least- squares-fit to the individual flybys permits the determination of the average speed and direction of the 3-space velocity, relative to the Earth, during each flyby. These results are all remarkably consistent with the data from the 11 previous laboratory experiments that studied v. Note that whether the 3-space velocity is +v or v is not material to the analysis herein, as the flyby effect is 2nd order in v. 3 Earth flyby data analaysis Eqn. (7) permits the speed anomaly to be predicted as the direction and speed v of the dynamical 3-space is known, as shown in Fig. 3. The first determination of its direction was reported by Miller [8] in 1933, and based on extensive observations during 1925/1926 at Mt.Wilson, California, using a large gas-mode Michelson interferometer. These observations confirmed the previous non-null observations by Michelson and Morley [7] in 1887. The general characteris- tics of v(r; t) are now known following the detailed analysis of the experiments noted above, namely its average speed, and removing the Earth orbit effect, is some 420 30km/s, Reginald T. Cahill. Resolving Spacecraft Earth-Flyby Anomalies with Measured Light Speed Anisotropy 11 Volume 3 PROGRESS IN PHYSICS July, 2008 Fig. 3: Southern celestial sphere with RA and Dec shown. The 4 dark blue points show the consolidated results from the Miller gas-mode Michelson interferometer [8] for four months in 1925/1926, from [22]. The sequence of red points show the running daily average RA and Dec trend line, as determined from the optical fiber interferometer data in [17], for every 5 days, beginning September 22, 2007. The lightblue scattered points show the RA and Dec for individual days from the same experiment, and show significant turbulence/wave effects. The curved plots show iso-speed V1 “anomalies”: for example for v = 420 km/s the RA and Dec of v for the Galileo-I flyby must lie somewhere along the “Galileo-I 420” curve. The available spacecraft data in Table 1, from [1], does not permit a determination of a unique v during that flyby. In the case of “Galileo-I” the curves are also shown for 420 30 km/s, showing the sensitivity to the range of speeds discovered in laboratory experiments. We see that the “Galileo-I” December flyby has possible directions that overlap with the December data from the optical fiber interferometer, although that does not exclude other directions, as the wave effects are known to be large. In the case of NEAR we must have v 440 km/s otherwise no fit to the NEAR V1 is possible. This demonstrates a fluctuation in v of at least +20 km/s on that flyby day. This plot shows the remarkable concordance in speed and direction from the laboratory techniques with the flyby technique in measuring v, and its fluctuation characteristics. The upper-left coloured disk (radius = 8 ) shows concordance for September/August interferometer data and Cassini flyby data ( MESSENGER data is outside this region — but has very small V1 and large uncertainty), and the same, lower disk, for December/January/February/March data (radius = 6 ). The moving concordance effect is undertsood to be caused by the earth’s orbit about the Sun, while the yearly average of 420 30 km/s is a galaxy related velocity. Directions for each flyby v were selected and used in Table 1. 12 Reginald T. Cahill. Resolving Spacecraft Earth-Flyby Anomalies with Measured Light Speed Anisotropy July, 2008 PROGRESS IN PHYSICS Volume 3 from direction right ascension v = 5.5 2hr, declination v = 70 10 S — the center point of the Miller data in Fig. 3, together with large wave/turbulence effects, as illustrated in Fig. 4. Miller’s original calibration technique for the interferometer turned out to be invalid [22], and his speed of approximately 208 km/s was recomputed to be 420 30 km/s in [19,22], and the value of 420 km/s is used here as shown in Table 1. The direction of v varies throughout the year due to the Earth-orbit effect and low frequency wave effects. A more recent determination of the direction was reported in [17] using an optical-fiber version of the Michelson interferometer, and shown also in Fig. 3 by the trend line and data from individual days. Directions appropriate to the date of each flyby were approximately determined from Fig. 3. The SC data in Table 1 shows the values of V1 and V1 after determining the osculating hyperbolic trajectory, as discussed in [1], as well as the right ascension and declination of the asymptotic SC velocity vectors Vi1 and Vf1. In computing the predicted speed “anomaly” V1 using (7) it is only necessary to compute the angles i and f between the dynamical 3-space velocity vector and these SC incoming and outgoing asymptotic velocities, respectively, as we assume j j here that v = 420 kms, except for NEAR as discussed in Fig. 3 caption. So these predictions are essentially ab initio in that we are using 3-space velocities that are reasonably well known from laboratory experiments. The observed Doppler effects are in exceptional agreement with the predictions using (7) and the previously measured 3-space velocity. The flyby anomaly is thus a new technique to accurately measure the dynamical 3-space velocity vector, albeit retrospectively from existing data. Fig. 4: Speeds vP , of the 3-space velocity v projected onto the hori- zontal plane of the Miller gas-mode Michelson interferometer, plotted against local sidereal time in hours, for a composite day, with data collected over a number of days in September 1925. The data shows considerable fluctuations, from hour to hour, and also day to day, as this is a composite day. The dashed curve shows the nonfluctuating best-fit variation over one day, as the Earth rotates, causing the projection onto the plane of the interferometer of the velocity of the average direction of the space flow to change. The maximum projected speed of the curve is 417 km/s, and the min/max occur at approximately 5 hrs and 17 hrs sidereal time (right ascension); see Fig. 3 for September. Analysing Millers’s extensive data set from 1925/26 gives average speed, after removing earth orbit effect, of 420 30 km/s, and the directions for each month shown in Fig. 3. metric of the induced spacetime, merely a mathematical con- struct having no ontological significance, is related to v(r; t) according to [21, 22, 27] 4 New gravitational waves ds2 = dt2 (dr v(r; t)dt)2 c2 = g dx dx : (8) Light-speed anisotropy experiments have revealed that a dy- namical 3-space exists, with the speed of light being c, in vac- uum, only with respect to to this space: observers in motion “through” this 3-space detect that the speed of light is in gen- eral different from c, and is different in different directions. The dynamical equations for this 3-space are now known and involve a velocity field v(r; t), but where only relative veloc- ities are observable locally — the coordinates r are relative to a non-physical mathematical embedding space. These dy- namical equations involve Newton’s gravitational constant G and the fine structure constant . The discovery of this dy- namical 3-space then required a generalisation of the Max- well, Schro¨dinger and Dirac equations. The wave effects al- ready detected correspond to fluctuations in the 3-space ve- locity field v(r; t), so they are really 3-space turbulence or wave effects. However they are better known, if somewhat in- appropriately, as “gravitational waves” or “ripples” in “space- time”. Because the 3-space dynamics gives a deeper under- standing of the spacetime formalism we now know that the The gravitational acceleration of matter, a quantum effect, and of the structural patterns characterising the 3-space, are given by [21, 23] g = @v @t + (v r)v (9) and so fluctuations in v(r; t) may or may not manifest as a gravitational acceleration. The flyby technique assumes that the SC trajectories are not affected — only the light speed anisotropy is significant. The magnitude of this turbulence depends on the timing resolution of each particular experiment, and was characterised to be sub-mHz in frequency by Cahill and Stokes [14]. Here we have only used asymptotic osculating hyperbolic trajectory data from [1]. Nevertheless even this data suggests the presence of wave effects. For example the NEAR data requires a speed in excess of 440 km/s, and probably closer to 450 km/s, whereas the other flybys are consistent with the average of 420 km/s from laboratory experiments. So here we see flyby evidence of fluctuations in the speed v. Reginald T. Cahill. Resolving Spacecraft Earth-Flyby Anomalies with Measured Light Speed Anisotropy 13 Volume 3 PROGRESS IN PHYSICS July, 2008 Data exists for each full flyby, and analysis of that data using the new Doppler shift theory will permit the study and characterisation of the 3-space wave turbulence during each flyby: essentially the flybys act as gravitational wave detectors. These gravitational waves are much larger than predicted by general relativity, and have different properties. 5 Pioneer 10/11 anomalies The Pioneer 10//11 spacecraft have been exploring the outer solar system since 1972/73. The spacecraft have followed escape hyperbolic orbits near the plane of the ecliptic, after earlier planet flybys. The Doppler shift data, using (2), have revealed an unexplained anomaly beyond 10 AU [28]. This manifests = (2.92 as an 0.44) unmodelled 10 18 s/s2, increasing blue shift d dt ( f f ) = corresponding to a constant in- ward 10 sun-directed acceleration of a = 8 cm/s2, averaged from Pioneer 10 dV dt = (8.74 and Pioneer 1.33) 11 data. However the Doppler-shift data from these spacecraft has been interpreted using (2), instead of (1), in determining the speed, which in turn affects the distance data. Essentially this implies that the spacecraft are attributed with a speed that is too large by vc22 VD, where VD is the speed determined using (2). This then implies that the spacecraft are actually closer to the Sun by the distance vc22 RD, where RD is the distance determined using (2). This will then result in a computed spurious inward acceleration, because the gravitational pull of the Sun is actually larger than modelled, for distance RD. However this correction to the Doppler-shift analysis appears not to be large enough to explain the above mention acceler- ation anomaly. Nevertheless re-analysis of the Pioneer 10/11 data should be undertaken using (1). 6 Conclusions The spacecraft earth flyby anomalies have been resolved. Rather than actual relative changes in the asymptotic inward and outward speeds, which would have perhaps required the invention of a new force, they are instead direct manifestations of the anisotropy of the speed of light, with the Earth having a speed of some 420 30 km/s relative to a dynamical 3-space, a result consistent with previous determinations using laboratory experiments, and dating back to the Michelson-Morley 1887 experiment, as recently reanalysed [18, 19, 21]. The flyby data also reveals, yet again, that the 3space velocity fluctuates in direction and speed, and with results also consistent with laboratory experiments. Hence we see a remarkable concordance between three different laboratory techniques, and the newly recognised flyby technique. The existing flyby data can now be re-analysed to give a detailed charaterisation of these gravitational waves. The detection of the 3-space velocity gives a new astronomical window on the galaxy, as the observed speeds are those relevant to galactic dynamics. The dynamical 3-space velocity effect also produces very small vorticity effects when passing the Earth, and these are predicted to produce observable effects on the GP-B gyroscope precessions [29]. A special acknowledgement to all the researchers who noted and analysed the spacecraft anomalies, providing the excellent data set used herein. Thanks also to Tom Goodey for encouraging me to examine these anomalies. Submitted on March 30, 2008 Accepted on April 10, 2008 References 1. Anderson J.D., Campbell J.K., Ekelund J.E., Ellis J. and Jordan J.F. Anomalous orbital-energy changes observed during spaceraft flybys of Earth. Phys. Rev. Lett., 2008, v. 100, 091102. 2. Anderson J.D., Campbell J.K. and Nieto M.M. New Astron. Rev., 2007, v. 12, 383. 3. Antreasian P.G. and Guinn J.R. AIAA Paper No. 984287 presented at the AIAA/AAS Astrodynamics Specialist Conference and Exhibit (Boston, August 10-12, 1998), and available at http://www.issibern.ch/teams/ Pioneer/pa-literature.htm 4. Morley T. and Budnik F. Proc. Intl. Symp. on Space Technology and Science, 2006, v. 25, 593. 5. Guman M.D., Roth D.C., Ionasescu R., Goodson T.D., Taylor A.H. and Jones J.B. Adv. Astron. Sci., 2000, v. 105, 1053. 6. Williams B., Taylor A., Carranza E., Miller J., Stanbridge D., Page. B., Cotter D., Efron L., Farquhar R., McAdams J. and Dunham D. Adv. Astron. Sci., 2005, v. 120, 1233. 7. Michelson A.A. and Morley E.W. Am. J. Sc., 1887, v. 34, 333–345. 8. Miller D.C. Rev. Mod. Phys., 1933, v. 5, 203–242. 9. Illingworth K.K. Phys. Rev., 1927, v. 3, 692–696. 10. Joos G. Ann. d. Physik, 1930, v. 7, 385. 11. Jaseja T.S. et al. Phys. Rev. A, 1964, v. 133, 1221. 12. Torr D.G. and Kolen P. In: Precision Measurements and Fundamental Constants, Taylor B.N. and Phillips W.D. eds. Natl. Bur. Stand. (U.S.), Spec. Pub., 1984, 617, 675. 13. Cahill R.T. The Roland DeWitte 1991 experiment. Progress in Physics, 2006, v. 3, 60–65. 14. Cahill R.T. A new light-speed anisotropy experiment: absolute motion and gravitational waves detected. Progress in Physics, 2006, v. 4, 73–92. 15. Cahill R.T. Optical-fiber gravitational wave detector: dynamical 3-space turbulence detected. Progress in Physics, 2007, v. 4, 63–68. 14 Reginald T. Cahill. Resolving Spacecraft Earth-Flyby Anomalies with Measured Light Speed Anisotropy July, 2008 PROGRESS IN PHYSICS 16. Mune´ra H.A., et al. In: Proceedings of SPIE, 2007, v. 6664, K1-K8, eds. Roychoudhuri C. et al. 17. Cahill R.T. and Stokes F. Correlated detection of submHz gravitational waves by two optical-fiber interferometers. Progress in Physics, 2008, v. 2, 103–110. 18. Cahill R.T. and Kitto K. Michelson-Morley experiments revisited. Apeiron, 2003, v. 10(2), 104–117. 19. Cahill R.T. The Michelson and Morley 1887 experiment and the discovery of absolute motion. Progress in Physics, 2005, v. 3, 25–29. 20. Braxmaier C. et al. Phys. Rev. Lett., 2002, 88, 010401; Mu¨ller H. et al. Phys. Rev. D, 2003, 68, 116006-1-17; Mu¨ller H. et al. Phys. Rev. D, 2003, v. 67, 056006; Wolf P. et al. Phys. Rev. D, 2004, v. 70, 051902-1-4; Wolf P. et al. Phys. Rev. Lett., 2003, v. 90, no. 6, 060402; Lipa J.A., et al. Phys. Rev. Lett., 2003, v. 90, 060403. 21. Cahill R.T. Dynamical 3-space: a review. arXiv: 0705.4146. 22. Cahill R.T. Process physics: from information theory to quantum space and matter. Nova Science Pub., New York, 2005. 23. Cahill R.T. Dynamical fractal 3-space and the generalised Schro¨dinger equation: Equivalence Principle and vorticity effects. Progress in Physics, 2006, v. 1, 27–34. 24. Levy J. From Galileo to Lorentz. . . and beyond. Apeiron, Montreal, 2003. 25. Guerra V. and de Abreu R. Relativity: Einstein’s lost frame. Extramuros, 2005. 26. Cahill R.T. Dynamical 3-space: supernovae and the Hubble expansion — the older Universe without dark energy. Progress in Physics, 2007, v. 4, 9–12. 27. Cahill R.T. A quantum cosmology: no dark atter, dark energy nor accelerating Universe. arXiv: 0709.2909. 28. Nieto M.N., Turyshev S.G., Anderson J.D. The Pioneer anomaly: the data, its meaning, and a future test. arXiv: gr-qc/0411077. 29. Cahill R.T. Novel Gravity Probe B frame-dragging effect. Progress in Physics, 2005, v. 3, 30–33. Volume 3 Reginald T. Cahill. Resolving Spacecraft Earth-Flyby Anomalies with Measured Light Speed Anisotropy 15 Volume 3 PROGRESS IN PHYSICS July, 2008 The Neutrosophic Logic View to Schro¨dinger’s Cat Paradox, Revisited Florentin Smarandache and Vic Christiantoy Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA ySciprint.org — a Free E-mail: smarand@unm.edu Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: admin@sciprint.org The present article discusses Neutrosophic logic view to Schro¨dinger’s cat paradox. We argue that this paradox involves some degree of indeterminacy (unknown) which Neutrosophic logic can take into consideration, whereas other methods including Fuzzy logic cannot. To make this proposition clear, we revisit our previous paper by offering an illustration using modified coin tossing problem, known as Parrondo’s game. 1 Introduction The present article discusses Neutrosophic logic view to Schro¨dinger’s cat paradox. In this article we argue that this paradox involves some degree of indeterminacy (unknown) which Neutrosophic logic can take into consideration, whereas other methods including Fuzzy logic cannot. In the preceding article we have discussed how Neutro- sophic logic view can offer an alternative method to solve the well-known problem in Quantum Mechanics, i.e. the Schro¨- dinger’s cat paradox [1, 2], by introducing indeterminacy of the outcome of the observation. In other article we also discuss possible re-interpretation of quantum measurement using Unification of Fusion Theo- ries as generalization of Information Fusion [3, 4, 5], which results in proposition that one can expect to neglect the prin- ciple of “excluded middle”; therefore Bell’s theorem can be considered as merely tautological. [6] This alternative view of Quantum mechanics as Information Fusion has also been proposed by G. Chapline [7]. Furthermore this Information Fusion interpretation is quite consistent with measurement theory of Quantum Mechanics, where the action of measure- ment implies information exchange [8]. In the first section we will discuss basic propositions of Neutrosophic probability and Neutrosophic logic. Then we discuss solution to Schro¨dinger’s cat paradox. In subsequent section we discuss an illustration using modified coin tossing problem, and discuss its plausible link to quantum game. While it is known that derivation of Schro¨dinger’s equa- tion is heuristic in the sense that we know the answer to which the algebra and logic leads, but it is interesting that Schro¨- dinger’s equation follows logically from de Broglie’s grande loi de la Nature [9, p.14]. The simplest method to derive Schro¨dinger’s equation is by using simple wave as [9]: @2 @x2 exp(ikx) = k2 exp(ikx) : (1) By deriving twice the wave and defining: k = 2mv h = mv = px ; (2) where px, represents momentum at x direction, and ratio- nalised Planck constants respectively. By introducing kinetic energy of the moving particle, T , and wavefunction, as follows [9]: T = mv2 2 = p2x 2m = 2 2m k2; (3) and (x) = exp(ikx) : (4) Then one has the time-independent Schro¨dinger equation from [1, 3, 4]: @2 2m @x2 (x) = T (x) : (5) It is interesting to remark here that by convention physicists assert that “the wavefunction is simply the mathematical function that describes the wave” [9]. Therefore, unlike the wave equation in electromagnetic fields, one should not consider that equation [5] has any physical meaning. Born suggested that the square of wavefunction represents the probability to observe the electron at given location [9, p.56]. Although Heisenberg rejected this interpretation, apparently Born’s interpretation prevails until today. Nonetheless the founding fathers of Quantum Mechanics (Einstein, De Broglie, Schro¨dinger himself) were dissatisfied with the theory until the end of their lives. We can summarize the situation by quoting as follows [9, p.13]: “The interpretation of Schro¨dinger’s wave function (and of quantum theory generally) remains a matter of continuing concern and controversy among scientists who cling to philosophical belief that the natural world is basically logical and deterministic.” Furthermore, the “pragmatic” view of Bohr asserts that for a given quantum measurement [9, p.42]: “A system does not possess objective values of its physical properties until a measurement of one of them is made; the act of measurement is asserted to force the system into an eigenstate of the quantity being measured.” 16 F. Smarandache and V. Christianto. The Neutrosophic Logic View to Schro¨dinger’s Cat Paradox, Revisited July, 2008 PROGRESS IN PHYSICS Volume 3 In 1935, Einstein-Podolsky-Rosen argued that the axiomatic basis of Quantum Mechanics is incomplete, and subsequently Schro¨dinger was inspired to write his well-known cat paradox. We will discuss solution of his cat paradox in subsequent section. 2 Cat paradox and imposition of boundary conditions As we know, Schro¨dinger’s deep disagreement with the Born interpretation of Quantum Mechanics is represented by his cat paradox, which essentially questioning the “statistical” interpretation of the wavefunction (and by doing so, denying the physical meaning of the wavefunction). The cat paradox has been written elsewhere [1, 2], but the essence seems quite similar to coin tossing problem: “Given p = 0.5 for each side of coin to pop up, we will never know the state of coin before we open our palm from it; unless we know beforehand the “state” of the coin (under our palm) using ESP-like phenomena. Prop. (1).” The only difference here is that Schro¨dinger asserts that the state of the cat is half alive and half dead, whereas in the coin problem above, we can only say that we don’t know the state of coin until we open our palm; i.e. the state of coin is indeterminate until we open our palm. We will discuss the solution of this problem in subsequent section, but first of all we shall remark here a basic principle in Quantum Mechanics, i.e. [9, p.45]: “Quantum Concept: The first derivative of the wavefunction of Schro¨dinger’s wave equation must be single-valued everywhere. As a consequence, the wavefunction itself must be single-valued everywhere.” The above assertion corresponds to quantum logic, which can be defined as follows [10, p.30; 11]: P _Q = P +Q PQ: (6) As we will see, it is easier to resolve this cat paradox by releasing the aforementioned constraint of “singlevaluedness” of the wavefunction and its first derivative. In fact, nonlinear fluid interpretation of Schro¨dinger’s equation (using the level set function) also indicates that the physical meaning of wavefunction includes the notion of multivaluedness [12]. In other words, one can say that observation of spin-half electron at location x does not exclude its possibility to pop up somewhere else. This counter-intuitive proposition will be described in subsequent section. 3 Neutrosophic solution of the Schro¨dinger cat paradox In the context of physical theory of information [8], Barrett has noted that “there ought to be a set theoretic language which applies directly to all quantum interactions”. This is because the idea of a bit is itself straight out of classical set theory, the definitive and unambiguous assignment of an element of the set {0,1}, and so the assignment of an information content of the photon itself is fraught with the same difficulties [8]. Similarly, the problem becomes more adverse because the fundamental basis of conventional statistical theories is the same classical set {0,1}. For example the Schro¨dinger’s cat paradox says that the quantum state of a photon can basically be in more than one place in the same time which, translated to the neutrosophic set, means that an element (quantum state) belongs and does not belong to a set (a place) in the same time; or an element (quantum state) belongs to two different sets (two different places) in the same time. It is a question of “alternative worlds” theory very well represented by the neutrosophic set theory. In Schro¨dinger’s equation on the behavior of electromagnetic waves and “matter waves” in quantum theory, the wave function, which describes the superposition of possible states may be simulated by a neutrosophic function, i.e. a function whose values are not unique for each argument from the domain of definition (the vertical line test fails, intersecting the graph in more points). Therefore the question can be summarized as follows [1]: “How to describe a particle  in the infinite micro- 2 2 : universe that belongs to two distinct places P1 and P2 in the same time?  P1 and  P1 is a true con- tradiction, with respect to Quantum Concept described above.” Now we will discuss some basic propositions in Neutrosophic logic [1]. 3a Non-standard real number and subsets  Let T,I,F be standard or non-standard real subsets ] 0, 1+[, with sup T = t sup, inf T= t inf, sup I = i sup, inf I = i inf, sup F = f sup, inf F = f inf, and n sup = t sup + i sup + f sup, n inf = t inf + i inf + f inf. Obviously, t sup, i sup, f sup 1+; and t inf, i inf, f inf 0, whereas n sup 3+ and n inf 0. The subsets T, I, F are not necessarily intervals, but may be any real subsets: discrete or continuous; single element; finite or infinite; union or intersection of various subsets etc. They may also overlap. These real subsets could represent the relative errors in determining t, i, f (in the case where T, I, F are reduced to points). For interpretation of this proposition, we can use modal logic [10]. We can use the notion of “world” in modal logic, which is semantic device of what the world might have been like. Then, one says that the neutrosophic truth-value of a statement A, NLt(A) = 1+ if A is “true in all possible worlds.” (syntagme first used by Leibniz) and all conjunctures, that one may call “absolute truth” (in the modal logic F. Smarandache and V. Christianto. The Neutrosophic Logic View to Schro¨dinger’s Cat Paradox, Revisited 17 Volume 3 PROGRESS IN PHYSICS July, 2008 it was named necessary truth, as opposed to possible truth), whereas NLt(A) = 1 if A is true in at least one world at some conjuncture, we call this “relative truth” because it is related to a “specific” world and a specific conjuncture (in the modal logic it was named possible truth). Because each “world” is dynamic, depending on an ensemble of parameters, we introduce the sub-category “conjuncture” within it to reflect a particular state of the world. In a formal way, let’s consider the world W as being generated by the formal system FS. One says that statement A belongs to the world W if A is a well-formed formula (wff ) in W, i.e. a string of symbols from the alphabet of W that conforms to the grammar of the formal language endowing W. The grammar is conceived as a set of functions (formation rules) whose inputs are symbols strings and outputs “yes” or “no”. A formal system comprises a formal language (alphabet and grammar) and a deductive apparatus (axioms and/or rules of inference). In a formal system the rules of inference are syntactically and typographically formal in nature, without reference to the meaning of the strings they manipulate. Similarly for the Neutrosophic falsehood-value, NLf (A) = 1+ if the statement A is false in all possible worlds, we call it “absolute falsehood”, whereas NLf (A) = 1 if the statement A is false in at least one world, we call it “relative falsehood”. Also, the Neutrosophic indeterminacy value NLi(A) = 1 if the statement A is indeterminate in all possible worlds, we call it “absolute indeterminacy”, whereas NLi(A) = 1 if the statement A is indeterminate in at least one world, we call it “relative indeterminacy”. 3b Neutrosophic probability definition Neutrosophic probability is defined as: “Is a generalization of the classical probability in which the chance that an event A occurs is t% true — where t varies in the subset T, i% indeterminate — where i varies in the subset I, and f% false — where f varies in the subset F. One notes that NP(A) = (T, I, F)”. It is also a generalization of the imprecise probability, which is an interval-valued distribution function. The universal set, endowed with a Neutrosophic probability defined for each of its subset, forms a Neutrosophic probability space. 3c Solution of the Schro¨dinger’s cat paradox Let’s consider a neutrosophic set a collection of possible lo- cations (positions) of particle x. And let A and B be two neutrosophic sets. One can say, by language abuse, that any particle x neutrosophically belongs to any set, due to the per- centages of truth/indeterminacy/falsity involved, which varies between 0 and 1+. For example: x (0.5, 0.2, 0.3) belongs to A (which means, with a probability of 50% particle x is in a position of A, with a probability of 30% x is not in A, and the rest is undecidable); or y (0, 0, 1) belongs to A (which normally means y is not for sure in A); or z (0, 1, 0) belongs to A (which means one does know absolutely nothing about z’s affiliation with A). [ More general, x ((0.2–0.3), (0.40–0.45) [0.50–0.51], {0.2, 0.24, 0.28}) belongs to the set A, which means: — with a probability in between 20-30% particle x is in a position of A (one cannot find an exact approximate because of various sources used); — with a probability of 20% or 24% or 28% x is not in A; — the indeterminacy related to the appurtenance of x to A is in between 40–45% or between 50–51% (limits included). The subsets representing the appurtenance, indeterminacy, and falsity may overlap, and n sup = 30% + 51% + 28% > 100% in this case. To summarize our proposition [1, 2], given the Schro¨- dinger’s cat paradox is defined as a state where the cat can be dead, or can be alive, or it is undecided (i.e. we don’t know if it is dead or alive), then herein the Neutrosophic logic, based on three components, truth component, falsehood component, indeterminacy component (T, I, F), works very well. In Schro¨dinger’s cat problem the Neutrosophic logic offers the possibility of considering the cat neither dead nor alive, but undecided, while the fuzzy logic does not do this. Normally indeterminacy (I) is split into uncertainty (U) and paradox (conflicting) (P). We have described Neutrosophic solution of the Schro¨dinger’s cat paradox. Alternatively, one may hypothesize four-valued logic to describe Schro¨dinger’s cat paradox, see Rauscher et al. [13, 14]. In the subsequent section we will discuss how this Neutrosophic solution involving “possible truth” and “indeterminacy” can be interpreted in terms of coin tossing problem (albeit in modified form), known as Parrondo’s game. This approach seems quite consistent with new mathematical formulation of game theory [20]. 4 An alternative interpretation using coin toss problem Apart from the aforementioned pure mathematics-logical approach to Schro¨dinger’s cat paradox, one can use a wellknown neat link between Schro¨dinger’s equation and FokkerPlanck equation [18]: D @2p @z2 @ @z p @p @z @p @t = 0 : (7) A quite similar link can be found between relativistic classical field equation and non-relativistic equation, for it is known that the time-independent Helmholtz equation and Schro¨dinger equation is formally identical [15]. From this reasoning one can argue that it is possible to explain Aharonov effect from pure electromagnetic field theory; and therefore it seems also possible to describe quantum mechan- 18 F. Smarandache and V. Christianto. The Neutrosophic Logic View to Schro¨dinger’s Cat Paradox, Revisited July, 2008 PROGRESS IN PHYSICS Volume 3 ical phenomena without postulating the decisive role of “observer” as Bohr asserted. [16, 17]. In idiomatic form, one can expect that quantum mechanics does not have to mean that “the Moon is not there when nobody looks at”. With respect to the aforementioned neat link between Schro¨dinger’s equation and Fokker-Planck equation, it is interesting to note here that one can introduce “finite difference” approach to Fokker-Planck equation as follows. First, we can define local coordinates, expanded locally about a point (z0, t0) we can map points between a real space (z; t) and an integer or discrete space (i; j). Therefore we can sam- ple the space using linear relationship [19]: (z; t) = (z0 + i; t0 + j ) ; (8) where  is the sampling length and  is the sampling time. Using a set of finite difference approximations for the FokkerPlanck PDE: @p @z = A1 = p (z0 + ; t0  ) p (z0 2 ; t0 ) ; (9) @2p @z2 = 2A2 = = p (z0 ; t0 ) 2p (z0; t0 2  ) +p (z0 +; t0  ) ; (10) and @p @t = B1 = p (z0; t0) p (z0; t0  ) : (11) We can apply the same procedure to obtain: @ @z = A1 = (z0 +; t0  ) (z0 2 ; t0  ) : (12) Equations (9–12) can be substituted into equation (7) to yield the required finite partial differential equation [19]: p (z0; t0) = a 1 p (z0 ; t0  ) a0 p (z0; t0  ) + + a+1 p (z0 + ; t0  ) : (13) This equation can be written in terms of discrete space by using [8], so we have: pi;j = a 1 pi 1;j 1 + a0 pi;j 1 + a+1 pi+1;j 1 : (14) plied external field. With respect to the aforementioned Neutrosophic solu- tion to Schro¨dinger’s cat paradox, one can introduce a new “indeterminacy” parameter to represent conditions where the outcome may be affected by other issues (let say, apparatus setting of Geiger counter). Therefore equation (14) can be written as: pi;j = 1 2 + a0  " pi;j 1 pi 1;j 1 + +  1 2 + "   pi+1;j 1; (17)  where unlike the bias parameter ( 1/200), the indeterminacy parameter can be quite large depending on the system in ques- tion. For instance in the Neutrosophic example given above, we can write that:   0.2 0.3 = k  d t  1 = k  t d  0.50: (18) The only problem here is that in original coin tossing, one cannot assert an “intermediate” outcome (where the outcome is neither A nor B). Therefore one shall introduce modal logic definition of “possibility” into this model. Fortunately, we can introduce this possibility of intermediate outcome into Parrondo’s game, so equation (17) shall be rewritten as: pi;j = 1 2 " + (2)   pi 1;j 1 + pi;j 1 +  1 2 + "   pi+1;j 1 ; (19)  For instance, by setting  0.25, then one gets the finite difference equation: pi;j = (0.25 ") pi 1;j 1 + (0.5) pi;j 1 + + (0.25 + ") pi+1;j 1 ; (20) which will yield more or less the same result compared with Neutrosophic method described in the preceding section. For this reason, we propose to call this equation (19): Neutrosophic-modified Parrondo’s game. A generalized expression of equation [19] is: Equation (14) is precisely the form required for Parron- do’s game. The meaning of Parrondo’s game can be described in simplest way as follows [19]. Consider a coin tossing prob- lem with a biased coin: phead = 1 2 "; (15) where " is an external bias that the game has to “overcome”. This bias is typically a small number, for instance 1/200. Now we can express equation (15) in finite difference equation (14) as follows: pi;j = 1 2  " pi 1;j 1 +0 pi;j 1  + 1 2 +  " pi+1;j 1: (16) Furthermore, the bias parameter can be related to an ap- pi;j = (p0 " ) pi 1;j 1 + (z) pi;j 1 + + (p0 + " ) pi+1;j 1 ; (21) where p0, z represents the probable outcome in standard coin tossing, and a real number, respectively. For the practical meaning of , one can think (by analogy) of this indetermi- nacy parameter as a variable that is inversely proportional to the “thickness ratio” (d=t) of the coin in question. Therefore using equation (18), by assuming k = 0.2, coin thickness = 1.0 mm, and coin diameter d = 50 mm, then we get d=t = 50, or  = 0.2(50) 1 = 0.004, which is negligible. But if we use a thick coin (for instance by gluing 100 coins alto- gether), then by assuming k = 0.2, coin thickness = 100 mm, F. Smarandache and V. Christianto. The Neutrosophic Logic View to Schro¨dinger’s Cat Paradox, Revisited 19 Volume 3 PROGRESS IN PHYSICS July, 2008 and coin diameter d = 50 mm, we get d=t = 0.5, or  = 0.2(0.5) 1 = 0.4, which indicates that chance to get out- come neither A nor B is quite large. And so forth. It is worth noting here that in the language of “modal logic” [10, p.54], the “intermediate” outcome described here } is given name ‘possible true’, written A, meaning that “it is not necessarily true that not-A is true”. In other word, given that the cat cannot be found in location x, does not have to mean that it shall be in y. Using this result (21), we can say that our proposition in the beginning of this paper (Prop. 1) has sufficient reasoning; i.e. it is possible to establish link from Schro¨dinger wave equation to simple coin toss problem, albeit in modified form. Furthermore, this alternative interpretation, differs appreciably from conventional Copenhagen interpretation. It is perhaps more interesting to remark here that Heisenberg himself apparently has proposed similar thought on this problem, by introducing “potentia”, which means “a world devoid of single-valued actuality but teeming with unrealized possibility” [4, p.52]. In Heisenberg’s view an atom is certainly real, but its attributes dwell in an existential limbo “halfway between an idea and a fact”, a quivering state of attenuated existence. Interestingly, experiments carried out by J. Hutchison seem to support this view, that a piece of metal can come in and out from existence [23]. In this section we discuss a plausible way to represent the Neutrosophic solution of cat paradox in terms of Parrondo’s game. Further observation and theoretical study is recommended to explore more implications of this plausible link. 5 Concluding remarks In the present paper we revisit the Neutrosophic logic view of Schro¨dinger’s cat paradox. We also discuss a plausible way to represent the Neutrosophic solution of cat paradox in terms of Parrondo’s game. It is recommended to conduct further experiments in order to verify and explore various implications of this new proposition, including perhaps for the quantum computation theory. Acknowledgment The writers would like to thank to D. Rabounski for early discussion concerning interpretation of this cat paradox. Submitted on March 31, 2008 / Accepted on April 10, 2008 References 1. Smarandache F. An introduction to the Neutrosophic probability applied in quantum physics. Bull. Pure and Appl. Sci., Physics, 2003, v. 22D, no. 1, 13–25. 2. Smarandache F. and Christianto V. The Neutrosophic logic view to Schro¨dinger’s cat paradox. Progr. in Phys., 2006, no. 2. 3. Smarandache F. and Christianto V. A note on geometric and information fusion interpretation of Bell’s theorem and quantum measurement. Progress in Physics, 2006, no. 4. 4. Smarandache F. and Christianto V. Multivalued logic, neutrosophy and Schro¨dinger equation. Hexis, Phoenix (AZ), 2006, p.52–54. 5. Smarandache F. and Dezert J. Advances and applications of DSmT for information fusion. American Research Press, Rehoboth (NM), 2004. 6. Kracklauer A. La theorie de Bell, est-elle la plus grande meprise de l’histoire de la physique? Ann. Fond. Louis de Broglie, 2000, v. 25, 193. 7. Chapline G. arXiv: adap-org/9906002; quant-ph/9912019; Granik A. and Chapline G. arXiv: quant-ph/0302013. 8. Zurek W. (ed.) Complexity, entropy and the physics of information. Addison-Wesley Publ., 1990, p.378. 9. Hunter G. Quantum chemistry: wave mechanics applied to atoms and molecules. Lecture notes. Chapter 1: Genesis of Quantum Mechanics. 2001, p.14, 26, 42. 10. deVries A. Algebraic hierarchy of logics unifying fuzzy logic and quantum logic. arXiv: math.LO/0707.2161, p.30, 54. 11. Aerts D. Description of many separated physical entities without the paradoxes encountered in Quantum Mechanics. Found. Phys., 1982, v. 12, no. 12, p.1142, 1149–1155. 12. Jin S., Liu H., Osher S., and Tsai R. Computing multivalued physical observables for the semiclassical limits of the Schro¨dinger equation. J. Comp. Phys., 2005, v. 205, 222–241. 13. Rauscher E.A. and Targ R. The speed of thought: investigation of complex spacetime metric to describe psychic phenomena. J. Scientific Exploration, 2001, v. 15, no. 3, 344–354. 14. Rauscher E.A. and Amoroso R. The physical implications of multidimensional geometries and measurement. Intern. J. Comp. Anticipatory Systems, D. Dubois (ed.), 2006. 15. Lu J., Greenleaf J., and Recami E. Limited diffraction solutions to Maxwell and Schro¨dinger equation. arXiv: physics/9610012. 16. Aharonov Y., et al. arXiv: quant-ph/0311155. 17. Goldstein S. Quantum theory without observers — part one. Physics Today, March 1998, 42–46. 18. Ho C.-L. and Sasaki R. Deformed Fokker Planck equations. arXiv: cond-mat/0612318. 19. Allison A., et al. State space visualization and fractal properties of Parrondo’s game. arXiv: cond-mat/0205536; cond-mat/ 0208470. 20. Wu J. A new mathematical representation of game theory. arXiv: quant-ph/0404159. 21. Smarandache F. Unification of fusion theories (UFT). Intern. J. Appl. Math. and Stat., 2004, v. 2, 1–14. 22. Smarandache F. An in-depth look at information fusion rules and unification of fusion theories. Invited speech at NASA Langley Research Center, Hampton, VA, USA, November 5, 2004. 23. Smarandache F., Christianto V., Khrapko R., Yuhua F., Hutchison J. Unfolding labyrinth: open problems in physics, mathematics, astrophysics, and other areas of science. Hexis, Phoenix (AZ), 2006. 20 F. Smarandache and V. Christianto. The Neutrosophic Logic View to Schro¨dinger’s Cat Paradox, Revisited July, 2008 PROGRESS IN PHYSICS Volume 3 A Classical Model of Gravitation Pieter Wagener Department of Physics, NMMU South Campus, Port Elizabeth, South Africa E-mail: Pieter.Wagener@nmmu.ac.za A classical model of gravitation is proposed with time as an independent coordinate. The dynamics of the model is determined by a proposed Lagrangian. Applying the canonical equations of motion to its associated Hamiltonian gives conservation equa- tions of energy, total angular momentum and the z component of the angular momen- tum. These lead to a Keplerian orbit in three dimensions, which gives the observed values of perihelion precession and bending of light by a massive object. An expression for gravitational redshift is derived by accepting the local validity of special relativity at all points in space. Exact expressions for the GEM relations, as well as their associated Lorentz-type force, are derived. An expression for Mach’s Principle is also derived. 1 Introduction The proposed theory is based on two postulates that respectively establish the dynamics and kinematics of a system of particles subject to a gravitational force. The result is a closed particle model that satisfies the basic experimental observations of the force. The details of applications and all derivations are included in the doctoral thesis of the author [1]. 2 Postulates The model is based on two postulates: Postulate 1: The dynamics of a system of particles subject to gravitational forces is determined by the Lagrangian, L= m0(c2 + v2) exp R r ; (1) where m0 is gravitational rest mass of a test body mov- ing at velocity v body of mass M , in =the1=vpici1nity of a massive, central v2=c2, R = 2GM=c2 is the Schwarzschild radius of the central body. Postulate 2: Special Relativity (SR) is valid instantan-eously and locally at all points in the reference system of the central massive body. This gives the kinematics of the system. 3 Conservation equations Applying the canonical equations of motion to the Hamilto- nian, derived from the Lagrangian, leads to three conservation equations: E= m0c2 eR=r 2 = total energy = constant ; (2) L = eR=r M ; (3) = total angular momentum = constant ; Lz = eR=r m0r2 sin2 _ ; (4) = z component of L = constant ; where M = (r m0v). Equations (2), (3) and (4) give the quadrature of motion: d du =  e2Ru L2 u2 EeRu  L2 1=2 ; (5) where u = 1=r, L =jLj and is defined by jMj = m0r2 d dt : (6) Expanding the exponential terms to second degree yields a differential equation of generalized Keplerian form, d du = (au2 + bu + c) 1=2; (7) where u = 1 r a = R2(4 2L2 E) b = R(2 L2 E) c = 1 E L2 9 1 >>>>>>>>>>>>>= >>>>>>>>>>>>>; ; (8) and the convention m0 = c = 1 was used. Integrating (7) gives the orbit of a test particle as a gener- alized conic, u = K(1 + cos k ) ; (9) where the angles are measured from = 0, and k = ( a) 12 ; (10) K= b 2a ; (11)  =1 4ac b  21 : (12) Pieter Wagener. A Classical Model of Gravitation 21 Volume 3 PROGRESS IN PHYSICS July, 2008 … 7 Lorentz-type force equation The corresponding force equation is found from the associated Euler-Lagrange equations: p_ = Em + m0v H ; (16) c r0  M c deflection =2 where p = m0 r_ = m0v ; (17) m = m0 2 ; (18) E= ^r GM r2 ; (19) H = GM (v c2r3 r) : (20)  Fig. 1: Deflection of light. The force equation shows the deviation from Newton’s law of gravitation. The above equations are analogous to the gravitoelectromagnetic (GEM) equations derived by Mashhoon [2] as a lowest order approximation to Einstein’s field equations for v  c and r  R. 4 Gravitational redshift Assuming the validity of d = dt of SR at each point in space and taking frequencies as the inverses of time, (2) yields  = 0 e R=2r (0 = constant), (13) which, to first approximation in exp( R=2r), gives the ob- served gravitational redshift. 5 Perihelion precession In the case of an ellipse ( < 1), the presence of the coefficient k causes the ellipse not to be completed after a cycle of  = 2 radians, i.e. the perihelion is shifted through a cer- tain angle. This shift, or precession, can be calculated as (see Appendix 9):  = a 3R (1 2) ; (14) where a is the semi-major axis of the ellipse. This expression gives the observed perihelion precession of Mercury. 8 Mach’s Principle An ad hoc formulation for Mach’s Principle has been pre- sented as [3, 4] G Lc2 M ; (21) where: L = radius of the universe, M = mass of the universe mass of the distant stars. This relation can be found by applying the energy relation of (2) to the system of Fig. 2. distant stars M2 M1 L © 6 Deflection of light We define a photon as a particle for which v = c. From (2) it follows that E = 0 and the eccentricity of the conic section is found to be (see Appendix 9) = r0 R ; (15) where r0 is the impact parameter. Approximating r0 by the radius of the sun, it follows that > 1. From Fig. 1 we see that the trajectory is a hyperbola with total deflection equal to 2R=r0. This is in agreement with observation. Fig. 2: Mutual gravitational interaction between a central mass M1 and the distant stars of total mass M2. The potential at M2 due to M1 is and the potential of the shell at M1 is 12==GGMM21==LL==RR21cc22==22LL. Furthermore, since M1 and M2 are in relative motion, the value of will be the same for both of them. Applying (2) to the mutual gravitational interaction between the shell of distant stars and the central body then gives E = M1c2 exp R2 L = M2c2 exp R1 L : 22 Pieter Wagener. A Classical Model of Gravitation July, 2008 PROGRESS IN PHYSICS Volume 3  Since L > R2 R1 we can realistically approximate the  exponential to first order in R2=L. After some algebra we get R2 L, which gives the Mach relation, 2GM2 Lc2  1: 9 Comparison with General Relativity The equations of motion of General Relativity (GR) are approximations to those of the proposed Lagrangian. This can be seen as follows. The conservation equations of (2), (3) and (4) can also be derived from a generalized metric, ds2 = e R=rdt2 eR=r(dr2 + r2d2 + r2 sin2 d2): (22) Comparing this metric with that of GR,  ds2 = 1 R r  dt2 1 1 R r dr2 r2d2 r2 sin2 d2; (23) we note that (23) is a first order approximation to the time and radial coefficients, and a zeroth order approximation to the angular coefficients of (22). It implies that all predictions of GR will be accommodated by the Lagrangian of (1) within the orders of approximation. Comparing (5) with the corresponding quadrature of GR, d du = 1 E J2 + uRE J2  1=2 u2 + Ru3 ; (24) we note that it differs from the Newtonian limit, or the Keple- rian form of (7), by the presence of the Ru3 term. The form of this quadrature does not allow the conventional Keplerian orbit of (9). where a = (r+ + r )=2 is the semi-major axis of the approx- imate ellipse. Substituting these values in (8) gives a= 3R a(1 2) 1: (27) Substituting this value in (25) gives (14). A.2 Deflection of light We first have to calculate the eccentricity this case,  =1 4ac b2 1=2 : For a photon, setting v = c in (8) gives  2= 1 + L2 R2  : of the conic for (28) At the distance of closest approach, r = r0 = 1=u0, we have d=du = 0; so that from (5): L2 = e2Ru0 u20 = r02 e2R=r0: (29) From (28) and (29), and ignoring terms of first and higher order in R=r0, we find  r0 R : (30) For a hyperbola cos  = 1= , so that (see Fig. 1): sin = 1= )  1= ) 2  2R=r0 = total deflection. Submitted on April 07, 2008 Accepted on April 10, 2008 Appendix A.1 Precession of the perihelion After one revolution of 2 radians, the perihelion of an el- lipse given by the conic of (9) shifts through an angle  = = 2 k 2 or, from (10), as  = 2 ( a) 1=2 1 ; (25) where a is given by (8). The constants of motion E and L are found from the boundary conditions of the system, i.e. du=d = 0 at u = 1=r and 1=r+, where r+ and r are the maximum and minimum radii respectively of the ellipse. We find [1] E  1 + R 2a R2 L2  2R a (1 2) 9 >>= ; >>; (26) References 1. Wagener P. C. Principles of a theory of general interaction. PhD thesis, University of South Africa, 1987. An updated version is to be published as a book during 2008. 2. Mashhoon. Gravitoelectromagnetism: a brief review. In L. Iorio, editor, Measuring Gravitomagnetism: A Challenging Enterprise, pages 29–39. Nova Publishers, 2007. arXiv: gr-qc/ 0311030. 3. Chiu H.-Y. and Hoffmann W. P. (eds.). Gravitation and Relativity, Ch. 7., W. A. Benjamin Inc., New York, 1964. 4. Sciama D. W. The physical foundations of General Relativity. Doubleday, 1969. Pieter Wagener. A Classical Model of Gravitation 23 Volume 3 PROGRESS IN PHYSICS July, 2008 On the Origin of the Dark Matter/Energy in the Universe and the Pioneer Anomaly Abraham A. Ungar Dept. of Mathematics, North Dakota State University, Fargo, North Dakota 58105-5075, USA E-mail: Abraham.Ungar@ndsu.edu Einstein’s special relativity is a theory rich of paradoxes, one of which is the recently discovered Relativistic Invariant Mass Paradox. According to this Paradox, the relativistic invariant mass of a galaxy of moving stars exceeds the sum of the relativistic invariant masses of the constituent stars owing to their motion relative to each other. This excess of mass is the mass of virtual matter that has no physical properties other than positive relativistic invariant mass and, hence, that reveals its presence by no means other than gravity. As such, this virtual matter is the dark matter that cosmologists believe is necessary in order to supply the missing gravity that keeps galaxies stable. Based on the Relativistic Invariant Mass Paradox we offer in this article a model which quantifies the anomalous acceleration of Pioneer 10 and 11 spacecrafts and other deep space missions, and explains the presence of dark matter and dark energy in the universe. It turns out that the origin of dark matter and dark energy in the Universe lies in the Paradox, and that the origin of the Pioneer anomaly results from neglecting the Paradox. In order to appreciate the physical significance of the Paradox within the frame of Einstein’s special theory of relativity, following the presentation of the Paradox we demonstrate that the Paradox is responsible for the extension of the kinetic energy theorem and of the additivity of energy and momentum from classical to relativistic mechanics. Clearly, the claim that the acceleration of Pioneer 10 and 11 spacecrafts is anomalous is incomplete, within the frame of Einstein’s special relativity, since those who made the claim did not take into account the presence of the Relativistic Invariant Mass Paradox (which is understandable since the Paradox, published in the author’s 2008 book, was discovered by the author only recently). It remains to test how well the Paradox accords with observations. 1 Introduction Einstein’s special relativity is a theory rich of paradoxes, one of which is the Relativistic Invariant Mass Paradox, which was recently discovered in [1], and which we describe in Sec- tion 5 of this article. The term mass in special relativity usu- ally refers to the rest mass of an object, which is the Newto- nian mass as measured by an observer moving along with the object. Being observer’s invariant, we refer the Newtonian, rest mass to as the relativistic invariant mass, as opposed to the common relativistic mass, which is another name for en- ergy, and which is observer’s dependent. Lev B. Okun makes the case that the concept of relativistic mass is no longer even pedagogically useful [2]. However, T. R. Sandin has argued otherwise [3]. As we will see in Section 5, the Relativistic Invariant Mass Paradox asserts that the resultant relativistic invariant mass m0 of a system S of uniformly moving N particles ex- ceeds the 1; : : : ; N, osufmitsocfonthsetitrueelanttivpiasrttiiccliensv,amria0n>t mPasNks=es1 mk, mk, k= since the contribution to m0 comes not only from the masses mk of the constituent particles of S but also from their speeds relative to each other. The resulting excess of mass in the resultant relativistic invariant mass m0 of S is the mass of virtual matter that has no physical properties other than positive relativistic invariant mass and, hence, that reveals itself by no means other than gravity. It is therefore naturally identified as the mass of virtual dark matter that the system S possesses. The presence of dark matter in the universe in a form of virtual matter that reveals itself only gravitationally is, thus, dictated by the Relativistic Invariant Mass Paradox of Einstein’s special theory of relativity. Accordingly, (i) the fate of the dark matter particle(s) theories as well as (ii) the fate of their competing theories of modified Newtonian dynamics (MOND [4]) are likely to follow the fate of the eighteenth century phlogiston theory and the nineteenth century luminiferous ether theory, which were initiated as ad hoc postulates and which, subsequently, became obsolete. Dark matter and dark energy are ad hoc postulates that account for the observed missing gravitation in the universe and the late time cosmic acceleration. The postulates are, thus, a synonym for these observations, as C. La¨mmerzahl, O. Preuss and H. Dittus had to admit in [5] for their chagrin. An exhaustive review of the current array of dark energy theories is presented in [6]. 24 Abraham A. Ungar. On the Origin of the Dark Matter/Energy in the Universe and the Pioneer Anomaly July, 2008 PROGRESS IN PHYSICS Volume 3 The Pioneer anomaly is the anomalous, unmodelled acceleration of the spacecrafts Pioneer 10 and 11, and other spacecrafts, studied by J. D. Anderson et al in [7] and summarized by S. G. Turyshev et al in [8]. In [7], Anderson et al compared the measured trajectory of a spacecraft against its theoretical trajectory computed from known forces acting on the spacecraft. They found the small, but significant discrepancy known as the anomalous, or unmodelled, acceleration directed approximately towards the Sun. The inability to explain the Pioneer anomaly with conventional physics has contributed to the growing interest about its origin, as S. G. Turyshev, M. M. Nieto and J. D. Anderson pointed out in [9]. It is believed that no conventional force has been overlooked [5] so that, seemingly, new physics is needed. Indeed, since Anderson et al announced in [7] that the Pioneer 10 and 11 spacecrafts exhibit an unexplained anomalous acceleration, numerous articles appeared with many plausible explanations that involve new physics, as C. Castro pointed out in [10]. However, we find in this article that no new physics is needed for the explanation of both the presence of dark matter/energy and the appearance of the Pioneer anomaly. Rather, what is needed is to cultivate the Relativistic Invariant Mass Paradox, which has recently been discovered in [1], and which is described in Section 5 below. Accordingly, the task we face in this article is to show that the Relativistic Invariant Mass Paradox of Einstein’s special relativity dictates the formation of dark matter and dark energy in the Universe and that, as a result, the origin of the Pioneer anomaly stems from the motions of the constituents of the Solar system relative to each other. 2 Einstein velocity addition vs. Newton velocity addition The improved way to study Einstein’s special theory of rela- tivity, offered by the author in his recently published book [1], enables the origin of the dark matter/energy in the Universe and the Pioneer anomaly to be determined. The improved study rests on analogies that Einsteinian mechanics and its underlying hyperbolic geometry share with Newtonian me- chanics and its underlying Euclidean geometry. In particu- lar, it rests on the analogies that Einsteinian velocity addition shares with Newtonian velocity addition, the latter being just the common vector addition in the Euclidean 3-space R3. Einstein addition of R3, is a binary operation in the ball R3c R3c = fv 2 R3 : kvk < cg (1) of all relativistically admissible velocities, where c is the speed of light in empty space. It is given by the equation u v = 1 1 + uv c2  u + 1 u v + 1 c2 1 u + u (u  v)u (2) where u is the gamma factor v= r 1 1 kvk2 c2 (3) k k itnhaRt t3ch,eabnadllwRh3ceirneheraintsdfrom are the inner its space R3. product and norm Counterintuitively, Einstein addition is neither commutative nor associative. Einstein gyrations gyr[u; v] 2 Aut(R3c; ) are defined by the equation gyr[u; v]w = (u v) (u (v w)) (4) 2 for all of the u; v; w Einstein groRu3cp,oaidnd(Rth3ce;y turn out to be automorphisms ). We recall that a groupoid is a non-empty space with a binary operation, and that an au- 2 fttohof(emuRgo3cyrpvroah)ntiist=oomnfisto(sofuefl)fathtghefra(oEtvuirn)pesofstoipedreinca(tlRgsl r3cuto;h;uevp)obiiidnsRa(ra3cRy.3coTo;nopee-e)rtmoaat-priooehnnaae,suitmtzhoeaamtpthoiarsf-t, phisms of the groupoid, gyrations are also called gyroauto- morphisms. Thus, gyr[u; v] of the definition in (4) is the gyroautomor- 2 puwh;(iviusnmR(3cvoRifn3c ,twotthh)te)hateiEntraieRnklsae3ctts.eivitnhisetigrceraollaluytpivoaidisdmtici(asRslli3cyb;laedv)me, liosgsceiibtnyleeravt(eeuldocivbtyy) The gyrations, which possess their own rich structure, measure the extent to which Einstein addition deviates from commutativity and associativity as we see from the following identities [1, 11, 12]: u v = gyr[u; v](v u) u (v w) = (u v) gyr[u; v]w (u v) w = u (v gyr[u; v]w) gyr[u; v] = gyr[u v; v] gyr[u; v] = gyr[u; v u] Gyrocommutative Law Left Gyroassociative Right Gyroassociative Left Loop Property Right Loop Property Einstein addition is thus regulated by its gyrations so that Einstein addition and its gyrations are inextricably linked. In- deed, the Einstein groupoid (R3c ; ) forms a group-like math- ematical object called a gyrocommutative gyrogroup [13], which was discovered by the author in 1988 [14]. Interestingly, Einstein gyrations are just the mathematical abstraction of the relativistic Thomas precession [1, Sec. 10.3]. The rich structure of Einstein addition is not limited to its gyrocommutative gyrogroup structure. Einstein addition admits scalar multiplication, giving rise to the Einstein gyrovector space. The latter, in turn, forms the setting for the Beltrami-Klein ball model of hyperbolic geometry just as vector spaces form the setting for the standard model of Euclidean geometry, as shown in [1]. Guided by the resulting analogies that relativistic mechanics and its underlying hyperbolic geometry share with classical mechanics and its underlying Euclidean geometry, we Abraham A. Ungar. On the Origin of the Dark Matter/Energy in the Universe and the Pioneer Anomaly 25 Volume 3 PROGRESS IN PHYSICS July, 2008 are able to present analogies that Newtonian systems of particles share with Einsteinian systems of particles in Sections 3 and 4. These analogies, in turn, uncover the Relativistic Invariant Mass Paradox in Section 5, the physical significance of which is illustrated in Section 6 in the frame of Einstein’s special theory of relativity. Finally, in Sections 7 and 8 the Paradox reveals the origin of the dark matter/energy in the Universe as well as the origin of the Pioneer anomaly. 3 Newtonian systems of particles In this section we set the stage for revealing analogies that a Newtonian system of N particles and an Einsteinian system of N particles share. In this section, accordingly, as opposed Rto3S, eacntdiomn 04,isvtkh,ekN=ew0t;o1n;i:a:n:r;eNsu,latarnetNmeawstsoonfiathnevceoloncsittiiteuseinnt masses mk, k = 1; : : : ; N of a Newtonian particle system S. Accordingly, let us consider the following well known classical results, (6) – (8) below, which are involved in the calculation of the Newtonian resultant mass m0 and the clas- sical center of momentum (CM) of a Newtonian system of particles, and to which we will seek Einsteinian analogs in Section 4. Thus, let S = S(mk; vk; 0; N) ; vk 2 R3 (5) be an isolated Newtonian system of N noninteracting material particles the k-th particle of which has mass mk and Newtonian uniform velocity vk relative to an inertial frame 0, k = 1; : : : ; N. Furthermore, let m0 be the resultant mass of S, considered as the mass of a virtual particle located at the center of mass of S, and let v0 be the Newtonian velocity relative to 0 of the Newtonian CM frame of S. Then, 1 = 1 m0 X N k=1 mk (6) and v0 = 1 m0 X N mkvk k=1 u + v0 = 1 m0 X N mk(u k=1 + vk) 9 >>>>>= ; >>>>>; (7) u; vk2R3, mk > 0, k = 0; 1; : : : ; N. Here m0 is the Newto- nian mass of the Newtonian system S, supposed concentrated at the center of mass of S, and v0 is the Newtonian velocity relative to 0 of the Newtonian CM frame of the Newtonian system S in (5). It follows from (6) that m0 in (6) – (7) is given by the Newtonian resultant mass equation X N m0 = mk : (8) k=1 The derivation of the second equation in (7) from the first equation in (7) is immediate, following (i) the distributive law of scalar-vector multiplication, and (ii) the simple relation- ship (8) between the Newtonian resultant mass m0 and its constituent masses mk, k = 1; : : : ; N. 4 Einsteinian systems of particles In this section we present the Einsteinian analogs of the New- tonian expressions (5) – (8) listed in Section 3. The presented analogs are obtained in [1] by means of analogies that result from those presented in Section 2. k In this section, accordingly, as opposed = 0; 1; : : : ; N, are Einsteinian velocities to in Section R3c , and 3, vk, m0 is the Einsteinian resultant mass, yet to be determined, of the masses mk, k = 1; : : : ; N, of an Einsteinian particle system S. In analogy with (5), let S = S(mk; vk; 0; N); vk 2 R3c (9) be an isolated Einsteinian system of N noninteracting material particles the k-th particle of which has invariant mass mk and Einsteinian uniform velocity vk relative to an inertial frame 0, k = 1; : : : ; N. Furthermore, let m0 be the resultant mass of S, considered as the mass of a virtual particle located at the center of mass of S (calculated in [1, Chap. 11]), and let v0 be the Einsteinian velocity relative to 0 of the Einsteinian center of momentum (CM) frame of the Einsteinian system S in (9). Then, as shown in [1, p. 484], the relativistic analogs of the Newtonian expressions in (6) – (8) are, respec- tively, the following Einsteinian expressions in (10) – (12), v0 = 1 m0 X N mk k=1 vk 9 >>>>>= u v0 = 1 m0 X N mk k=1 u vk >>>>>; (10) and v0 v0 = 1 m0 X N k=1 mk vk vk u v0 (u v0) = 1 m0 X N mk k=1 u vk (u vk) 9 >>>>>= ; >>>>>; (11) u; vk2R3c, mk > 0, k = 0; 1; : : : ; N. Here m0, m0 = v u u u u t X N !2 X N mk + 2 mjmk( k=1 j;k=1 j mk (13) k=1 so that, paradoxically, the invariant resultant mass of a system may exceed the sum of the invariant masses of its constituent particles. The paradoxical invariant resultant mass equation (12) for m0 is the relativistic analog of the non-paradoxical Newtonian resultant mass equation (8) for m0, to which it reduces in each of the following two special cases: (i ) The Einsteinian resultant mass m0 in (12) reduces to ! 1 the Newtonian resultant mass m0 in (8) in the limit as c ; and (ii ) The Einsteinian resultant mass m0 in (12) reduces to the Newtonian resultant mass m0 in (8) in the special case when the system S is rigid, that is, all the internal motions in S of the constituent particles of S relative to each other vanish. In that case vj vk = 0 so that vj vk = 1 for all j; k = 1; N . This identity, in turn, generates the reduction of (12) to (8). The second equation in (11) follows from the first equation in (11) in full analogy with the second equation in (7), which follows from the first equation in (7) by the distributivity of scalar multiplication and by the simplicity of (8). However, while the proof of the latter is simple and well known, the proof of the former, presented in [1, Chap. 11], is lengthy owing to the lack of a distributive law for the Einsteinian scalar multiplication (see [1, Chap. 6]) and the lack of a simple relation for m0 like (8), which is replaced by (12). Indeed, the proof of the former, that the second equation in (11) follows from the first equation in (11), is lengthy, but accessible to undergraduates who are familiar with the vector space approach to Euclidean geometry. However, in order to follow the proof one must familiarize himself with a large part of the author’s book [1] and with its “gyrolanguage”, as indicated in Section 2. It is therefore suggested that interested readers may corroborate numerically (using a computer software like In this section we present two classically physical significant results that remain valid relativistically owing to the Relativistic Invariant Mass Paradox, according to which the rel- ativistic analog of the classical resultant mass m0 in (8) is, paradoxically, the relativistic resultant mass m0 in (12). To gain confidence in the physical significance that results from the analogy between (i ) the Newtonian resultant mass m0 in (8) of the Newtonian system S in (5) and (ii) the Einsteinian invariant resultant mass m0 in (12) of the Einsteinian system S in (9) we present below two physically significant resulting analogies. These are: (1) The Kinetic Energy Theorem [1, p. 487]: According to this theorem, where K = K0 + K1 ; (14) (i) K0 is the relativistic kinetic energy, relative to a given observer, of a virtual particle located at the relativistic center of mass of the system S in (9), with the Einsteinian resultant mass m0 in (12); and (ii) K1 is the relativistic kinetic energy of the constituent particles of S relative to its CM; and (iii) K is the relativistic kinetic energy of S relative to the observer. The Newtonian counterpart of (14) is well known; see, for instance, [15, Eq. (1.55)]. The Einsteinian analog in (14) was, however, unknown in the literature since the Einsteinian resultant mass m0 in (12) was unknown in the literature as well till its first appearance in [1]. Accordingly, Oliver D. Johns had to admit for his chagrin that “The reader (of his book; see [15, p. 392]) will be disappointed to learn that relativistic mechanics does not have a theory of collective motion that is as elegant and complete as the one presented in Chapter 1 for Newtonian mechanics.” The proof that m0 of (12) is compatible with the va- lidity of (14) in Einstein’s special theory of relativity is presented in [1, Theorem 11.8, p. 487]. (2) Additivity of Energy and Momentum: Classically, en- ergy and momentum are additive, that is, the total en- ergy and the total momentum of a system S of particles is, respectively, the sum of the energy and the sum of momenta of its constituent particles. Consequently, Abraham A. Ungar. On the Origin of the Dark Matter/Energy in the Universe and the Pioneer Anomaly 27 Volume 3 PROGRESS IN PHYSICS July, 2008 also the resultant mass m0 of S is additive, as shown in (8). Relativistically, energy and momentum remain additive but, consequently, the resultant mass m0 of S is no longer additive. Rather, it is given by (12), which is the relativistic analog of (8). The proof that m0 of (12) is compatible with the ad- ditivity of energy and momentum in Einstein’s special theory of relativity is presented in [1, pp. 488–491]. Thus, the Einsteinian resultant mass m0 in (12) of the Einsteinian system S in (9) is the relativistic analog of the Newtonian resultant mass m0 in (8) of the Newtonian system S in (5). As such, it is the Einsteinian resultant mass m0 in (12) that is responsible for the extension of the validity of (14) and of the additivity of energy and momentum from classical to relativistic mechanics. However, classically, mass is additive. Indeed, the New- ttohneicaonnrsetistuuletannt tpamrtaiscslems, 0me0qu=alsPtNkh=e 1smumk, of as the we masses of see in (8). Relativistically, in contrast, mass is not additive. Indeed, the mEiansssteesinoiafnthreescuolntasntittumenatsspamrt0iclmesa,yme0xceedPtNkh=e1smumk, of as the we see from (12). Accordingly, from the relativistic viewpoint, the resultant mass m0 in (12) of a galaxy that consists of stars that move relative to each other exceeds the sum of the masses of its constituent stars. This excess of mass re- veals its presence only gravitationally and, hence, we iden- tify it as the mass of dark matter. Dark matter is thus vir- tual matter with positive mass, which reveals its presence only gravitationally. In particular, the dark mass mdark of the Einsteinian system S in (9), given by (16) below, is the mass of virtual matter called the dark matter of S. To con- trast the real matter of S with its virtual, dark matter, we call the former bright (or, luminous, or, baryonic) matter. The total mass m0 of S, which can be detected gravitationally, is the composition of the bright mass mbright of the real, bright matter of S, and the dark mass mdark of the virtual, dark matter of S. This mass composition, presented in (15) – (17) in Section 7 below, quantifies the effects of dark matter. 7 The origin of the dark matter the equation q m0 = m2bright + m2dark (17) The mass mbright in (15) is the Newtonian resultant mass of the particles of the Einsteinian system S in (9). These par- ticles reveal their presence gravitationally, as well as by radi- ation that they may emit and by occasional collisions. In contrast, the mass mdark in (16) is the mass of virtual matter in the Einsteinian system S in (9), which reveals its presence only gravitationally. In particular, it does not emit radiation and it does not collide. As such, it is identified with the dark matter of the Universe. In our expanding universe, with accelerated expansion [16], relative velocities between some astronomical objects are significantly close to the speed of light c. Accordingly, 2 1 sloincciteiegsavmmaRf3cacatpoprrsoavchatphperosapcehed when their relative veof light, it follows from (16) that dark matter contributes an increasingly significant part of the mass of the universe. 8 The origin of the dark energy Under different circumstances dark matter may appear or disappear resulting in gravitational attraction or repulsion. Dark matter increases the gravitational attraction of the region of each stellar explosion, a supernova, since any stellar explosion creates relative speeds between objects that were at rest relative to each other prior to the explosion. The resulting generated relative speeds increase the dark mass of the region, thus increasing its gravitational attraction. Similarly, relative speeds of objects that converge into a star vanish in the process of star formation, resulting in the decrease of the dark mass of a star formation region. This, in turn, decreases the gravitational attraction or, equivalently, increases the gravitational repulsion of any star formation inflated region. The increased gravitational repulsion associated with star formation results in the accelerated expansion of the universe, first observed in 1998; see [6, p. 1764], [17] and [18, 19]. Thus, according to the present special relativistic dark matter/energy model, the universe accelerated expansion is a late time cosmic acceleration that began at the time of star formation. Let X N mbright = mk (15) k=1 and mdark = v u u u u t2 X N j;k=1 mj mk( vj vk 1) (16) j