Precession Of Mercury

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A phenomenological model for the precession of planets and deflection of light Arbab I. Arbab Department of Physics, Faculty of Science, University of Khartoum, P.O. 321, Khartoum 11115, Sudan Department of Physics and Applied Mathematics, Faculty of Applied Science and Computer, Omdurman Ahlia University, P.O. Box 786, Omdurman, Sudan E-mail: [email protected]

We presented a phenomenological mode that attributes the precession of perihelion of planets to dipole distribution of matter and relativistic correction . This modifies Newton’s equation by adding an inversely cube term with distance. The total energy of the new system is found to be the same as the Newtonian one. Moreover, we have deduced the deflection of light formula from Rutherford scattering. The relativistic term can be accounted for quantum correction of the gravitational potential on electron orbit in hydrogen atom.

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Introduction

Though Newton’s theory was successful in describing gravitational interactions related to the orbital motion of planets, it fell short to account for the anomalous precession of the perihelion of Mercury and the bending of light by the Sun. In this regard, Einstein theory of relativity predicts such extra terms. Hence, Einstein theory of general relativity became the theory of gravitation. Objects with strong gravity like binary pulsars are well treated by the general theory of relativity (GTR). According to GTR a gravitational wave is librated from these highly spinning objects. The experimental results confirm the theoretical finding of the GTR [1,2]. Careful observations of Mercury showed that the actual value of the precession disagreed with that calculated from Newton’s theory by 43 seconds of arc per century. A number of ad hoc solutions had been proposed, but they tended to introduce more problems. In general relativity, this remaining precession is explained by gravitation being mediated by the curvature of space-time. The GTR predicts exactly the observed amount of perihelion shift. The precession of Mercury was the result of many interactions with the planets of the solar System. However, in actual physical situations, the gravitating body may not be exactly spherical. For example, if the central body is spinning about its axis, it will be slightly oblate. In such a case, the Newtonian gravitational field is not spherically symmetrical, and the force exerted on a test particle at a distance r is not exactly proportional to r−2 . As a result, the actual orbit

of a test particle in such a case will not be exactly Keplerian. In order to mimic the relativistic prediction it would be necessary to hypothesize a gravitational potential that is dependent on the angular velocity of the test particle, not just on its position. A relativistic kinetic energy correction to the Newtonian orbit will also have a similar contribution (i.e., ∝ r−2 ). We consider in this letter a potential that varies inversely with the cube of the radial distance. This is also equivalent to a relativistic correction to the Newtonian potential (or force). Calculations show that this term has a contribution exactly equals to the GTR prediction. The cubic term can arise in the Newton law of gravitation due to tidal force existing between any two extendable objects. This term is responsible of slowing down the spin rotation of the Earth. Notice that a cubic term can be added to the Schwartzchild metric of general relativity. Hence, Newton law can still be hold for celestial objects and becomes indistinguishable from GRT. The effect of this modified potential is to reduce the angular momentum of the planet by a relativistic correction. 2

The model

Consider here the potential energy produced by a mass distribution M arising from monopole and dipole radial contribution. These terms can be written as U(r) = −

GmM m A − 2 , r r

(1)

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where m is the mass of the test body in this potential energy. where h = r2 dϕ dt . The first term the ordinary Newtonian potential energy, the The additional term will induce and extra acceleration on second term arises due to a dipole distribution of the mass orbiting bodies. producing the potential energy. This term also mimics the relThe solution of Eq.(6) is given by [4] ativistic correction of the Newtonian term. If we believe this p , (7) r= term embodies also the relativistic correction, using dimen(1 + e cos γ ϕ) sional argument, only M, G and c can be considered. Hence, where one can write γ 2 h2 2 GmM 2mA 2Eh2 γ2  GM 2 , p = ,E = − 1 − γ2 = , e − 1 = 2 2 GM 2a Mh mG A=β , where β = comst. . (2) c (8) and E is the energy at the perihelion (ϕ = 0), so that p = dU This will lead to a force, F = − dr , on a body of mass m of 2 ). The radial period P is given by P = 2π . The angular a(1 − e ω the form  GM 2 1 ∆ϕ of the perihelion precession during one period is mGM F = − 2 − 2β m . (3) c r r3 (1 − γ) ∆ϕ = 2π ' 2π(1 − γ), (9) The second term in Eq.(3) can be written as γ  GM 2 1  v 2 and its mean precession rate per period is given by (for γ ' 1) GM GmM  v 2 GmM m = × = = F , (4) N c c c r3 rc2 r2 r2 β 2π GM ∆ϕ = 2 . (10) GmM GM 2 c a(1 − e2 ) for a circular orbit, where v = r and F N = r2 is the Newtonian gravitational force. Therefore, we may treat the Comparing this equation with the Einstein formula, one obsecond term in Eq.(3) as a relativistic correction to the New- tains tonian force. β = 3. (11) Notice that in electromagnetism a dipole contribution comes Hence, Eq.(3) becomes from the fact that we have positive and negative charges. How GM 2 1 mGM ever, we see here even a negative mass doesn’t exist, the secF = − 2 − 6m , (12) c ond term gives the same contribution, and has always an atr r3 tractive nature. Such an additional attractive force will have and the corresponding potential energy∗ in Eq.(5) its effects on the orbital motion of Earth’s satellites. This term  GM 2 1 GmM might also arise due to tidal force which is inversely related to U(r) = − − 3m . (13) r c r2 the cube of the distance of the two bodies. This force eventuThis implies that the inclusion of a relativistic correction (and/or ally leads to tidal locking of the two masses. The potential energy in Eq.(1) is that of a Keplerian mo- a dipole distribution) results in making the orbit precess with tion perturbed by an inverse cube force, hence Eq.(1) can be an angle coincides with the GTR prediction [1]. This is evident if we use Eq.(4) in Eq.(9) so that one gets written as  v 2  GM 2 1 GmM ∆ϕ = 6π (14) U(r) = − − mβ . (5) c r c r2 Once again, from the relativistic kinetic energy correction one which is the precession angle per period. 4 We remark here that if one included a force term, F = 2 = GM , one obtains a term of the form 83 mv 2 . Hence, for v r c 3GmMh2  2 , besides Newton force, one would obtain a value close c2 r 4 1 finds a contribution ∝ GM to the potential energy. Using 2 c to GTR precession. r the Binet formula [3], Eq.(3) can be written as ∗ metric would become ds2 =  A modified  Schwartzchild  ! −1 2u 2GM 6G2 M 2 2GM 6G2 M 2 2 2 2 2 2 c 1 − − dt − 1 − − dr − r dΩ . Moreover, d dU 1 c2 r c4 r2 c2 r c4 r2 F = −h2 u2 +u =− , u= , (6) g = − (1 + 2U/mc2 ) 00 dr r dϕ2 2

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where K = 21 mv2 is the kinetic energy of the mass m. In polar coordinates (r , θ) this can be written as

Relativistic acceleration

The relativistic force in Eq.(12) (using Eq.(4)) produces an  GM 2  GmM 1  2 additional acceleration on all planets of the solar system, given 2 ˙2 E = m r˙ + r θ − − 3m , L = mr2 θ˙ , 2 r c by (22) !  GM 2 1 where L is the angular momentum. This can be written as an 1 ac = 6 , ac = 3.55 × 10−10 3 , (15) equation of the mass m in an effective potential (U ) 3 E c r r 1 L02 GmM where ac is measured in m/s2 and r in AU. E = m˙r2 + U E , U E = − , (23) 2 2 r 2mr The centripetal force on a mass moving in a circular orbit under this force is given by where  GmM 2 02 2   2 2 L = L − 6 , (24) mv GM 1 GmM c + 6m . (16) = r c r2 r3 is the reduce angular momentum of the mass. Thus L0 should Hence, the orbital velocity will be be the conserved angular momentum and not L as defined in Eq.(22). r ! GM 3GM 1 Using Eqs.(13) and (16) this yields v= 1+ 2 , (17) r c r GmM E=− . (25) 6GM 3GM 2r for an orbit r > c2 . Defining r0 = c2 = r0 = 4.4 km for all planets, Eq.(17) is transformed into This is the same as the Newtonian total energy. Thus, the r potential we have considered is unique. This may suggest that r0  GM  1+ . (18) gravity is not exactly inverse square law. What happens here v= r r is that due to the extra force term, the velocity increases while the distance decreases in such a way the total energy remains Hence, the orbital period will be constant. Notice that if we had included any other terms in the ! 2 2r0 4π potential energy, we wouldn’t have obtained this conclusion. T ' TN 1 − , T N2 = r3 . (19) r GM 4 Deflection of light and Rutherford deflection Therefore, the period will be shorter than the Keplerian period. A significant period decay has been observed in binary Gravitational deflection of light by matter is one of the definpulsars, which is attributed to the emission of gravitational ing predictions of Einstein’s General Relativity (GR) [1, 2]. waves from these pulsars [5]. According to GR, the deflection of a light ray just grazing the The rate of energy loss due to gravitational radiation (by Sun is 1.75 seconds of arc. The deflection angle at a Sun’s rim binary pulsars) is given by [5] b (impact parameter) is given by 48π 4π GM dP =− 5 dt P 5c

!5/3 ,

(20)

where P is the orbital period of the binary pulsar. The total energy of a mass m orbiting in a potential given in Eq.(13) is E = K+U, (21)

∆θ =

4GM . bc2

(26)

The light is deflected towards the Sun. GR interprets this bending of light to the curvature of space-time the Sun makes when light passed by the Sun. However, Newton obtained a value halved this value, using the Newtonian laws. One of the important events to test the bending of light was done in 3

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Sudan on February 25, 1952 during a solar eclipse that had confirmed one of the valuable predictions of GTR [6]. One can compare this with Rutherford scattering by αparticle where the particle deflected away from the nucleus because of electrical repulsion. The deflecting angle is given by 4keQ ∆θ = , (27) mr0 v2

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The perihelion precession frequency for an electron at a Bohr radius is given by ω = 19.2 arcsec/s. This is a very fast precession and can be measured experimentally in hydrogen atom. The second term in the potential describe in Eq.(13) (a correction to the Newtonian potential, first order in h) can be casted in the form Vg = −

3G hg 1 , c r2

(30)

6kh 1 . c r3

(32)

where r0 is the distance of closest approach and k is the Coulomb constant.. Using our recent analogy between electrodynamics where hg = GM2 is some characteristic Planck constant [8, 9]. c and gravitomagnetic [7] entitles us to make the following re- Such a potential can give rise to some quantum mechanical placement in Rutherford formula to obtain the gravitational phenomena happening at large scale having their analogues formula, viz., at microscopic scale. We anticipate that such a term would appear in any quantum gravity theory. e (Q = Ze) → m (M) , k → G, (28) Owing to the gravito-electric analogy, the electric quantum potential (a correction to the Coulomb potential) will be and for massless particle (light) we set v → c. Hence one arrives at the formula 3kh 1 Vq = − , (31) c r2 4GM ∆θ = , (29) bc2 where h is the Planck constant. Notice that the two potentials which is the same as GR formula with b = r0 . Therefore, are independent of the charge and mass. this analogy is an interesting one, and one can use it to bridge This correction term would induce a precession of the electron safely from electromagnetic phenomena to gravitomagnetic orbit in hydrogen atom like the precession of planets. It can also be compared with the Larmor precession resulting from phenomena or vice-versa. Notice that Schwartzchild radius of a black hole could be di- the spin-orbit interaction in the hydrogen atom. We remark rectly obtained from the escape velocity of a non-relativistic here that the potential in Eq.(13) can be thought of as a first gravitating object when its escape velocity is equated to the order correction to Newton potential because it involves a term G2 . We can associate a quantum electric field with the above velocity of light in vacuum. quantum potential on the electron by the formula Eq = −∇Vq acting on all masses. 5 Gravito-Electric analogy and quantum potential corThe quantum electric field of the electron in hydrogen rection atom associated with the potential in Eq.(8) is given by Owing to the electromagnetic and gravitational analogy, we developed recently, we would like to raise the following questions:

Eq =

• Does an electron orbit (according to Bohr theory) pre- This quantum electric field amounts to cess like the precession of planets, and if so with what Eq = 8.02 × 102 V/m . frequency?

(33)

• Is that frequency is the same as the Larmor precession? The electrical quantized potential in Eq.(31) contributes to the −30 • Can we rescue Bohr model as we are doing with New- electron at Bohr radius an energy of 2.12 × 10 J. To appreciate this finding, recently Nesvizhevsky et al. have calton? culated the neutron’s ground state energy and wave function in the Earth’s gravitational field and obtained a ground-state 4

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energy of 2.25 × 10−31 J [10, 11]. Their result shows an evidence for quantized gravitational States of the Neutron. Because of the extra electric potential in Eq.(31), the electron in an atom experiences an additional energy that is a result of a quantum potential and not a manifestation of quantized gravitational state. The difference in the two energy values above could be attributed to the difference between the gravitational acceleration and the above quantum electric field. Using perturbation theory, one can find the contribution (first order correction) of the quantum potential in Eq(31) to the ground state of the electron in hydrogen atom. This yields a value of hVi0 = 4.24 × 10−30 J [12], whereas the Stark effect contribution vanishes. References 1. Will, C. M., Theory and Experiment in Gravitational Physics (CUP), (1993). 2. Hartle, J. B., An introduction to Einstein’s General Relativity, Pearson Education, Singapore, (2003). 3. Bradbury, T. C., Theoretical Mechanics, New York Wiley, (1968). 4. Jean Sivardiere, Eur. Phys. 7, 283 (1986). 5. Hulse, R.A., and Taylor, J.H., ApJ, 195, L51 (1975). 6. Eddington, A. S., Space, time and gravitation, Cambridge University Press (CUP), (1987). 7. Arbab, A. I., The analogy between electromagnetism and hydrodynamics, 2009, unpublished. 8. Arbab, A. I., Gen. Rel. Gravit., 36, No.11, 2465 (2004). 9. Arbab, A. I., Afr. J. Math. Phys., 2, No.1, 1 (2005). 10. V. Nesvizhevsky, et al., ”Quantum states of neutrons in the Earth’s gravitational field,” Nature, 415 297 (2002). 11. Bowles, T. J., ”Quantum effects of gravity,” Nature 415, 267 (2002). 12. Landau, L.D. and Lifshitz, E.M., Quantum mechanics, Pergamon Press, New York, (1965). pp.21-24. 48

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