Explanation of Gravity Probe B Experimental Results using Heaviside-Maxwellian (Vector) Gravity in Flat Space-time
aa r X i v : . [ phy s i c s . g e n - ph ] M a r Explanation of Gravity Probe B Experimental Results using Heaviside-Maxwellian(Vector) Gravity in flat Space-time
Harihar Behera ∗ BIET Degree College and BIET Higher Secondary School, Govindpur, Dhenkanal-759001, Odisha, India.
Niranjan Barik † Department of Physics, Utkal University, Vani Vihar, Bhubaneswar-751004, Odisha, India (Dated: March 6, 2020)The Gravity Probe B (GP-B) Experiment of NASA was aimed to test the theoretical predictionsof Einstein’s 1916 relativistic tensor theory of gravity in curved space-time (General Relativity(GR)) concerning the spin axis precession of a gyroscope moving in the field of a slowly rotatingmassive body, like the Earth. In 2011, GP-B mission reported its measured data on the precession(displacement) angles of the spin axes of the four spherical gyroscopes housed in a satellite orbiting642 km (400 mi) above the Earth in polar orbit. The reported results are in agreement with thepredictions of GR. For the first time, here we report an undergraduate level explanation of the GP-Bexperimental results using Heaviside-Maxwellian (vector) Gravity (HMG) in flat space-time, firstformulated by Heaviside in 1893, and later considered/re-discovered by many authors. Our newexplanation of the GP-B results provides a new test of HMG apart from the existing ones, whichdeserves the attention of researchers in the field for its simplicity and new perspective.
I. INTRODUCTION
Gravity Probe B (GP-B), launched on 20 April 2004,was a NASA physics mission to experimentally investi-gate Einstein’s 1916 theory of general relativity (GR) -his relativistic theory of gravity in curved space-time.GP-B used four spherical gyroscopes and a telescope,housed in a satellite orbiting 642 km (400 mi) above theEarth in a polar orbit, to measure two of its significantpredictions (with unprecedented accuracy) on the pre-cession of the spin axis of a gyroscope moving in thefield of the Earth predicted by Pugh [1] in 1959 and bySchiff [2–4], in 1960, using GR. This is called Schiff ef-fect/precession, which was measured in a satellite (a co-moving frame) in which gyroscopes are at rest. In thecontext of GR, the working formula for the GP-B ex-periment in the polar orbit (Schematic in figure 1) wasworked out by Schiff [2–4] as d s dτ = (cid:0) Ω GR gd + Ω GR fd (cid:1) × s (1)where s is the spin angular momentum vector of a gyro-scope in its rest frame, τ is the time measured in gyro’srest frame (i.e. the satellite), called proper time, vectors Ω GR gd representing the Geodetic Effect and Ω GR fd Frame-dragging effect, as they are called in the GR parlance,are given by Ω GR gd = 3 GM c r ( r × v ) (2) Ω GR fd = Gc r (cid:20) r r ( r · S E ) − S E (cid:21) (3) ∗ [email protected] † [email protected] G being the gravitational constant, c is the velocity oflight in vacuum, S E = Iω E is the spin angular momen-tum of the Earth ( I and ω E are Earth’s moment of inertiaand angular velocity respectively), r is the position vectorof a gyroscope from Earth’s center and v is the orbitalvelocity of a gyroscope. The predicted quantities Ω GR gd in eq. (2) and Ω GR fd in eq. (3) could be computed sinceeach quantity in the equation eqs. (2)-(3) was known:the instantaneous orbital velocity and radius from GP-Bs on-board GPS detector; the mass, moment of inertia,and angular velocity of the Earth from geophysical andastrophysical data. As we know, GP-B experiment hasconfirmed both significant predictions of GR: the geode-tic effect (2) to a precision of 0 .
3% and frame-dragging(3) to 20% [5, 6]. However, in this communication weshow how one can obtain the eqs. (1)-(3) within theframework of Heaviside-Maxwellian Gravity (HMG) inflat space-time briefly described below.
II. HEAVISIDE-MAXWELLIAN GRAVITY(HMG)
The fundamental equations of HMG are analogous tothe Maxwell-Lorentz equations of electromagnetism andare called gravito-Maxwell-Lorentz equations (g-MLEs).Recently Behera [7], using Galileo Newtonian Relativity(GNR) and Behera and Barik [8], using the U(1) localphage (or gauge) in-variance of Dirac’s massive field the-ory (of charged as well as neutral particles), have foundtwo equivalent mathematical representations of HMG:(1) Heaviside Gravity (HG) as originally written downby Heaviside in 1893 [9–11] and (2) Maxwellian Gravity(MG) [7, 8, 12, 13]. The g-MLEs of HG and MG, whichrepresent the same physical theory HGM, are listed inTable 1. It is to be noted that, Heaviside speculated a
FIG. 1. Schematic of the orbit of the Gravity Probe B satel-lite. The Geodetic and Frame-dragging effects. NorthSouth,EastWest relativistic precessions with respect to the guidestar IM Pegasi for an ideal gyroscope in polar orbit aroundthe Earth. [Figure Courtesy: C. W. F. Everitt. gravito-Lorentz force with a wrong sign in the velocitydependent term by electromagnetic analogy, i.e. of MG-type in Table 1. But the correct form is given in Table 1,as derived Behera [7] using Schwinger’s formalism withinGNR and Behera and Barik [8] using the principle of localphase (or gauge) invariance of Dirac’s massive quantumfield theory .In Table 1, in analogy with Maxwell-Lorentz equations gravito-MLEs of MG gravito MLEs of HG ∇ · g = − πGρ g = − ρ g ǫ g ∇ · g = − πGρ g = − ρ g ǫ g ∇ · b = 0 ∇ · b = 0 ∇ × b = − µ g j g + c g ∂ g ∂t ∇ × b = + µ g j g − c g ∂ g ∂t ∇ × g = − ∂ b ∂t ∇ × g = + ∂ b ∂td p dt = m g [ g + u × b ] d p dt = m g [ g − u × b ] b = + ∇ × A g b = −∇ × A g g = − ∇ φ g − ∂ A g ∂t g = − ∇ φ g − ∂ A g ∂t TABLE I. Physically Equivalent Sets of gravito-Maxwell-Lorentz Equations (g-MLEs) representing Heaviside-Maxwellian Gravity (HMG). of classical electromagnetism in SI units, we have intro-duced two new universal constants for vacuum: ǫ g = 14 πG , µ g = 4 πGc g ⇒ c g = 1 √ ǫ g µ g (4)where the speed of gravitational waves in vacuum c g = c ,as theoretically shown in ref. [8, 12, 13] - an impor-tant prediction of HMG which agree well with the re-cent (almost) simultaneous detection of the GravitationalWaves and Gamma-Rays from a Binary Neutron StarMerger events: GW170817 and GRB 170817A [14], grav-itatinal mass density ρ g = ρ , the positive (rest) massdensity (as theoretically proved in [8, 12, 13] withoutsacrificing the time-tested empirical law of universalityof free fall of Galileo), j g = ρ v stands for the mass current density ( v is velocity of ρ ) and by electromag-netic analogy, b may be called the gravito-magnetic field,the Newtonian gravitational field g may be called asgravito-electric field, ǫ g and µ g may be named respec-tively as the gravito-electric (or gravitic) permitivity andthe gravito-magnetic permeability of vacuum. The g-MLEs of MG-type have also been derived from other ap-proaches to gravito-electromagnetic (GEM) theory using(a) Galileo-Newtonian Relativity [13], (b) special relativ-ity [12, 13, 15, 16], (c) principle of causality [17, 18] (d)common axiomatic methods applicabe to electricity andgravity [19, 20] and (e) a specific approach to linearizeGR in weak field and slow motion approximation [21].These variant of approaches that lead to HMG in its ei-ther form, establish g-MLEs as self-consistent. It is tobe noted that the explanations for the (a) perihelion ad-vance of Mercuty (b) gravitational bending of light and(c) the Shapiro time delay within the vector theory ofgravity exist in the literature [22–25]. Recently Hilborn[26] following an electromagnetic analogy, calculated thewave forms of gravitational radiation emitted by orbit-ing binary objects that are very similar to those observedby the Laser Interferometer Gravitational-Wave Observa-tory (LIGO-VIRGO) gravitational wave collaboration in2015 up to the point at which the binary merger occurs.Hilborn’s calculation produces results that have the samedependence on the masses of the orbiting objects, the or-bital frequency, and the mass separation as do the resultsfrom the linear version of general relativity (GR). But thepolarization, angular distributions, and overall power re-sults of Hilborn differ from those of GR. Recognizing thisand out of scientific curiosity, we ventured to adopt theMG form of HMG in Table 1 to calculate the spin axisprecession in the gravito-electromagnetic (GEM) field ofthe spinning Earth according to MG in flat space-timeby electromagnetic analogy as under. III. SPIN GRAVITOMAGNETIC MOMENTAND ASSOCIATED GRAVITOMAGENIC FIELD:
For the purpose of the present paper, the relevant elec-tromagnetic analogy is the motion of a small sphericaluniformly charged sphere (corresponding to a gyroscopein GP-B Experiment) in an orbit around another hugespherical spinning charged sphere (coresponding to thehuge Earth in GP-B Experiment). In electromagnetism,a spinning charged sphere creates a di-polar magneticfield B around it and this field is related to its spin mag-netic moment which is proportional to the spin angularmomentum of the spinning object. Kramers [27], Corben[28] and Bohm [29] have provided us classical derivationsof spin magnetic moment of a spinning charged particle,which precisely matches with Dirac’s quantum mechani-cal finding the of spin magnetic moment of the electron: µ s = ( q/m ) s = ( g s q/ m ) (in SI units) , (5)where q = −| e | is the charge and s is the spin angu-lar momentum the electron, g s is the so-called g -factorassociated with the spin magnetic moment of a chargedparticle - called the gyromagnetic ratio; g s = 2 for anelectron as found by Dirac quantum mechanically and byothers [27–29] classicaly. In classical electromagnetism,equation (5) also holds good for a uniformly magnetizedsphere [30]. The spin magnetic moment produces a di-polar magnetic field B ( r ) at a field point r = in vacuum[30]: B ( r ) = µ π (cid:20) ˆr ( ˆr · µ s ) − µ s r (cid:21) (in SI units) , (6)where µ is the magnetic permeability of vacuum and ˆr = r /r . Similarly, in MG, a spinning particle/objectcreates a di-polar gravitomagnetic field b determined byits spin gravitomagnetic moment µ gs propertional to itsspin angular momentum s . Using the classical methodsof Kramers [27], Corben [28] and Bohm [29] for obtainingspin magnetic moment, in the case of MG, we found µ gs = s , (7)which is true for a uniformly gravito-magnetized sphere.It is to be noted that without doing detailed calcula-tions as in refs. [27–29], one can quickly get the result ineq. (7) from eq. (5), by replacing the electric charge q with m , which represents the gravitational charge in theframe-work of MG as proved in ref. [12, 13]. Further, wenote that for a spin 1 / µ gs = s = ( ~ / σ . The gravitationalanalogue of eq. (6), in the case of MG is b ( r ) = − µ g π (cid:20) ˆr ( ˆr · µ s ) − µ s r (cid:21) (8)where µ g = 4 πG/c is the gravito-magnetic permeabil-ity of vacuum. The “minus” sign on the right hand sideof eq.(8) is due to the fact that the source term ( j g ) in thegrvito-Amp´ere of MG, viz., ∇ × b = − µ g j g has a minussign before it, in contrast with the positive sign before thecorresponding term ( j e ) in Amp´ere’s law ( ∇× B = + µ j e )in electromagnetism. For the spinning Earth, µ gs = S E ,so the gravitomagnetic field of the Earth in the case ofMG follows from eq. (8) as b ( r ) = − Gc (cid:20) ˆr ( ˆr · S E ) − S E r (cid:21) . (9) IV. SPIN PRECESSION IN MAGNETIC FIELDAND GRAVITOMAGNETIC FIELD(NON-RELATIVISTIC THEORY):
Let the spin magnetic moment of a particle with in-trinsic spin s , say an electron, be denoted by µ s = α s ,where α is proportionality constant (for electron α = q/m = −| e | /m ). In non-relativistic classical theory of electromagnetism, if such a particle is placed in a mag-netic field, its spin angular momentum s and hence spinaxis changes with time as d s dt = µ s × B = ( − α B ) × s = Ω f × s (10)where B is the magnetic field at the position of the par-ticle and the spin axis rotates with angular velocity Ω f = − α B . (11)Similarly, in the non-relativistic (i.e., slow motion) for-mulation of gravito-electromagnetic (GEM) theory [7,13] within domain of Newtonian physics, the gravito-magnetic analogues of eqs. (10)-(11) are d s dt = µ gs × b (0) = ( − b ) × s = Ω GEM fd × s (12) Ω GEM fd = − b = Gc (cid:20) ˆr ( ˆr · S E ) − S E r (cid:21) = Ω GR fd (13)which exactly matches with eq. (3) predicted by GR, ifwe consider µ gs = s as the gravitomagnetic moment of agyroscope in GP-B experiment and the gravitomagneticfield b of the Earth as given by eq. (9). V. SPIN PRECESSION IN ELECTRIC FIELDAND GRAVITO-ELECTRIC FIELD(NON-RELATIVISTIC THEORY):
Suppose an electron with charge q = − e , rest mass m , spin magnetic moment µ es = ( q/m ) s = − ( e/m ) s moves with velocity v in an external electric field E .Then in the instantaneous rest frame of the electron itnot only experiences the electric field E but also a mag-netic field given by (in SI units) B = − v × E c = E × v c (14)Then the equation of motion for its spin angular momen-tum s in its rest frame is (cid:18) d s dt (cid:19) rest = µ es × B = − (cid:18) q E × v m c (cid:19) × s = Ω Ie × s (15)and the spin axis of the electron precesses with an angularvelocity Ω Ie = − (cid:18) q E × v m c (cid:19) = (cid:18) e E × v m c (cid:19) (16)In the case of an atomic electron moving around a nucleusof charge Ze ( Z being the atomic number), the field E =( Ze r ) / (4 πǫ r ) and in this case eq. (16) becomes Ω Ie = Ze πǫ m c r ( r × v ) (17)In the context of Maxwellain Gravity an analogous situ-ation arises in the case of a spinning body moving in aNewtonian gravitoelectric field, just as a gyroscope mov-ing around the Earth as in GP-B experiment. The quick-est way to find the gravitational analogue of eq. (16), isto replace q E by m g = − ( GM m r ) /r , where m nowrepresents the mass of the gyroscope and g is the New-tonian gravito-electric field of the Earth of mass M; theresult is Ω Ig = − b = GMc r ( r × v ) (18)which is short of Ω IIg = 12
GMc r ( r × v ) (19)to get the correct GR value in eq. (2). This deficiencyis corrected from special relativistic contribution tothe spin axis precession of an electron in its rest frameas calculated by Thomas [31] in 1927, which we shallconsider now. VI. SPIN PRECESSION IN ELECTRIC FIELDAND GRAVITO-ELECTRIC FIELD(RELATIVISTIC THEORY OF THOMAS):
For an electron with charge q = − e , rest mass m ,spin angular momentum s , spin magnetic moment µ es ,Thomas [31] first calculated the relativistic change in thedirection of its spin axis in electron’s rest frame, whichfor the simple case of µ es = ( q/m ) s , is given by his eq.4.122 in [31] (written here in SI units and in differentnotations): d s dτ = Ω × s , (20)where Ω = − qm (cid:18) B + 1 c γ γ ( E × v ) (cid:19) , (21) τ is the proper time, γ = (1 − v /c ) − / is the Lorentzfactor, the fields E and B respectively refers to the elec-tric and magnetic fields at the instantaneous position ofthe electron and Ω is angular velocity of spin precession.For small velocity | v | << c , γ → Ω = − qm (cid:18) B + 12 c ( E × v ) (cid:19) . (22)If we put eq. (22) in eq. (20) we get Kramers equation(11) of 1934, in which Kramers [27] followed a simplerapproach. It should be clearly noted that the 1 / γ factorterm in eq. (21), which was absent in our non-relativisticapproach in the previous section. Now substituting thevalue of B from eq. (14) in eq. (22), we get Ω Elect.gd = − q E × v m c = 32 Ze πǫ m c r ( r × v ) , (23)for an atomic electron, which is the electrical analogue ofgeodetic precession in eq. (2) in the atomic domain.As was done in the previous section for the gravitationalsituation of GP-B experiment, we now replaced q E in eq.(23) by m g = − GM m r /r to get the correct formulafor geodetic precession according to HMG in flat space-time as Ω GEMgd = − m g × v m c = 3 GM c r ( r × v ) = Ω GRgd , (24)which is in perfect agreement with the prediction of GRin eq. (2). One can also use equation (46) of Bailey [32]to arrive at the eq. (24) in the GEM frame work of HMGin flat space-time. With our results in eqs. (13) and (24),we have thus shown that the GP-B experimental resultsmay well be explained using Heaviside-Maxwellian Grav-ity (HMG) in flat space-time without the requirement ofEinstein’s General Relativity. CONCLUSION
The GP-B experimental data are in close agreementwith the Schiff’s formula for spin axis precession of a gy-roscope in the gravitational field of the spinning Earth,which Schiff derived using Einstein’s relativistic theoryof gravity in curved space-time. However, for the firsttime, here we report a Faraday-Maxwellian field theo-retical explanation of the GP-B experimental results us-ing Heaviside-Maxwellian (vector) Gravity (HMG) in flatspace-time, first formulated by Heaviside in 1893, andlater considered/re-discovered by many authors follow-ing a variant of approaches to gravito-electromagnetism.Our new explanation of the GP-B results involves anundergraduate level derivation the electrical and gravi-tational analogues of Schiff formula in flat space-time,which provides a new test of HMG apart from the exist-ing ones noted in section 2. Well known physicist, C. M.Will concluded his viewpoint on GP-B results [Physics ,43 (2011)] by stating, “The precession of a gyroscope inthe gravitation field of a rotating body had never beenmeasured before GP-B. While the results support Ein-stein, this didn’t have to be the case. Physicists will nevercease testing their basic theories, out of curiosity that newphysics could exist beyond the “accepted” picture. [1] Pugh G. E., WSEG research memorandum No. 11 (1959). [2] Schiff, L.I. Possible new experimental test of general rel-ativity theory. Phys. Rev. Lett . , 215 (1960). [3] Schiff, L.I. On experimental tests of the general theoryof relativity. Am. J. Phys . , 340 (1960).[4] Schiff, L. I. Motion of gyroscope according to Einsteinstheory of gravitation. Proc. Natl. Acad. Sci. USA ,871 (1960). Proc. Natl. Acad. Sci. USA , 871 (1960).[5] Everitt C. W. F., et al. Gravity Probe B: Final Re-sults of a Space Experiment to Test General Relativity. Phys. Rev. Lett. http://einstein.stanford.edu [7] Behera, H. Gravitomagnetism and Gravitational Wavesin Galileo-Newtonian Physics. 2019). arXiv:1907.09910[8] Behera, H., Barik, N. A New Set of Maxwell-LorentzEquations and Rediscovery of Heaviside-Maxwellian(Vector) Gravity from Quantum Field Theory. (2019).arXiv:1810.04791[9] Heaviside, O. 1893. A Gravitational and ElectromagneticAnalogy, Part I.
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