The Exact Solution of the Pioneer Anomaly and Flyby Anomaly and the Interpretation of Inertia from an asymmetric Casimir effect

Transcription

1 The Exat Solution of the Pioneer Anomaly and Flyby Anomaly and the Interpretation of Inertia from an asymmetri Casimir effet Abstrat Azzam Almosallami Zurih, Switzerland In this paper, I ll propose an exat solution for the Pioneer anomaly and the flyby anomaly. My solution interprets inertia from an asymmetri Casimir effet, and it interprets why inertial is related to Unruh effet and a Hubble-sale Casimir effet. My solution of the Pioneer anomaly and the flyby anomaly is depending on my new transformation and my equivalene priniple. Aording to my transformation, the reality is observer dependent as a result of translating the retardation aording to the invariane by the entanglement whih is leading to the wave-partile duality and the unertainty priniple by the vauum flutuation. In this ase by translating the retardation aording to my transformation, it is resulted the relativisti ether as observer dependent, and thus in this ase the reality in my transformation is observer dependent. This relativisti ether is not lassial, but it is depending on the energy of the vauum by translating the retardation aording to my transformation. My transformation expresses about field by the vauum flutuation, and that explains also why field is observer dependent, and that explains also why Unruh effet is observer dependent. Introdution Radio metri data from Pioneer indiate an apparent anomalous, onstant, aeleration ating on the spaeraft with a magnitude ~ m/s 2, direted towards the Sun [11,12]. Turyshev [13] examined the onstany and diretion of the Pioneer anomaly, and onluded that the data a temporally deaying anomalous aeleration m s 2. yr with an over 10% improvement in the residuals ompared to a onstant aeleration model. Anderson, who is retired from NASA s Jet Propulsion Laboratory (JPL), is that study s first author. He finds, so it s either new physis or old physis we haven t disovered yet. New physis ould be a variation on Newton s laws, whereas an example of as-yet-to-be-disovered old physis would be a loud of dark matter trapped around the sun. Now I introdue an exat solution for the Pioneer anomaly depending on my transformation and my equivalene priniple. and the Hubble s law. Aording to my solution, there are two terms of deelerations that ontrols the Pioneer anomaly. The first is produed by moving the Pioneer spaeraft through the gravitational field of the Sun, and this deeleration is responsible for varying behaviour of the Pioneer anomaly in Turyshev [13]. And aording to the priniple of quantum superposition and Heisenberg unertainty priniple by the vauum flutuation [1-5], we find that the seond term is depending on the Hubble s law whih is equal to the Hubble s onstant multiplied by the speed of light in vauum.

2 This solution of the Pioneer anomaly and the flyby anomaly aording to my transformation and my equivalene priniple will give us the origin of the problem of dark matter and dark energy and thus the osmologial onstant problem. Sonnleitner [10] showed that how a simple alulation leads to the surprising result that an exited two-level atom moving through a vauum sees a tiny frition fore of first order in v/. That is explained in my transformation as a result of translating the retardation in my transformation aording to the invariane by entanglement whih is leading to the wave-partile duality and the unertainty priniple by the vauum flutuation [1]. Also Shrodinger [18] showed that the emission of a light quantum by a (flying) atom is regulated by the onservation laws of energy and linear momentum. Therefore, the Doppler effet for photons is the onsequene of the energy and momentum exhange between the atom and the photon: a entral role is played by the quantum energy jump E of the transition (a relativisti invariant). That is explained ompletely aording to my transformation speially in my solution of Sagna effet [1]. An empirial equation for the anomalous flyby veloity hange was proposed by J. D. Anderson et al. dv V = 2ω ER E (osφ i osφ o ) (1) where ωe is the angular frequeny of the Earth, RE is the Earth radius, and φi and φo are the inbound and outbound equatorial angles of the spaeraft [7,9]. One of the proposed solutions of the flyby anomaly is proposing a dark matter halo around Earth [8], whih is the same as proposed in ase of the Pioneer anomaly, but dark matter aording to my transformation and my equivalene priniple is explained and thus no need to propose dark matter. Theory In my paper [1], I have reahed to my new transformation x = γ 2 (x vt ) t = γ 2 (t vx 2 ) y = γy z = γz Aording to my transformation, spae is invariant, and thus the speed of light of light is onstant in the loal lassial vauum and equals to. Globally as a result of the retardation, the speed of light is not onstant but flutuates depending on the energy of the vauum resulted by the retardation. Thus, globally we have the measured speed of light aording to the phase given as = γ 1 And the group veloity by the vauum flutuation

3 = γ 2 Aording to my equivalene priniple [1] γ 1 = (1 GM 2 r ) whih is depending on the gravitational potential. Aording to that, in the gravitational field we have the phase veloity globally = (1 GM 2 r ) And the group veloity globally aording to unertainty priniple by the vauum flutuation = (1 GM 2 r )2 By my equivalene priniple by translating the retardation in my transformation aording to the invariane by the entanglement, I found the relativisti esape veloity of the free fall objet under the gravitational field is given loally as V esape loally = 2GM r G2 M 2 2 r 2 (2) Whih is approximated in ase of weak gravitational field to V esape loally = 2GM r (3) The esape veloity of the free fall objet as observed globally is given aording to two veloities, the phase veloity when we make a loalization and in this ase the phase veloity and the group veloity are equal whih is equivalent to a motion in linear dispersion, and this in ase we have V esape global phase = (1 GM 2 r ) 2GM G2 M 2 r 2 r 2 (4) Now during the free fall, we have here a vauum flutuation, whih is equivalent to motion in nonlinear dispersion, and in this ase the unertainty priniple by the vauum flutuation plays the rule, where in this ase even if we start with a fairly loalized partile, it will soon loose this loalization. Aording to that the group veloity of the free fall objet under the gravitational field is not equal to the phase veloity, and in this ase the group veloity is given aording to

4 V esape global group = (1 GM 2 r )2 2GM G2 M 2 r 2 r 2 (5) Whih is approximated in ase of weak gravitational field to V esape global group = (1 2GM 2 r ) 2GM r (6) Whih is the same equation derived from the Shwarzshild Geometry in ase of weak gravitational for the free fall, but aording to the Shwarzshild geometry this equation has no any physial meaning, beause in reality it is in violation with the equivalene priniple of Einstein, and also it is in violation with reality is observer independent aording to Minkowski Geometry of spae-time. Sagna effet an be explained aording to my transformations by onsidering the t- term in my transformation. t = γ 2 (t vx 2 ) If we onsidered t = γ 2 (t vx ) and 2 t+ = γ 2 (t + vx 2 ), in this ase we get t = γ 2 ( 2x v 2 ) And sine L is invariant and by onsidering x = L, then we get t = γ 2 ( 2Lv 2 ) This result is exatly the same result whih derived by Engelhardt [19] in explaining Sagna effet in the framework of the ether theory and Galilean transformation, but instead aording to my transformation, it is the relativisti ether from the point of view of quantum vauum. My transformation is transformation of field, quantized field, and thus the relativisti ether appears aording to my transformation as observer dependent by the retardation same as in ase of Unruh effet. That explains why Unruh effet is observer dependent, and thus aording to my transformation and equivalene priniple [1], that gives a full interpretation of inertia from an asymmetri Casimir effet. The referene to the Doppler effet was only indiret (the experiments by Stark to the first order of v/) [18], and now it an be explained aording to my transformation and my equivalene priniple [1]. Aording to that for low veloities omparing to the speed of light whih is equivalent to motion in weak gravitational field, the differene between the predited frequeny and the referene frequeny ν 0 as the result of the red shift is ν model given as

5 ν model = V esape loally ν 0 (7) Now by onsidering the observed frequeny differene globally ν obs is depending on Eq. (6) in ase of weak gravitational field aording to my transformation and my equivalene priniple, in this ase we get ν obs (1 2GM = 2 r ) V esape loally ν 0 (8) Thus from Eqs. (7)&(8), and by substituting from Eq. (3) V esape loally = 2GM, we get r [ ν obs ν model ] ν 0 = ( 2GM 2GM 2 r ) r (9) From eq. (9) we get the observed differene frequeny is less than the predited. That means there is a slight blue shift. Aording to the Pioneer team alulations, the observed, two-way anomalous effet by a DSN antenna an be expressed to first order in V/C as in [11] [ ν obs ν model ] DSN = 2a p t (10) ν 0 By DSN onvention [ ν obs ν model ] usual = [ ν obs ν model ] DSN Thus from that and from eq. (9) we get ( 2GM 2GM 2 r ) r = 2a p t (11) By onsidering in Eq. (11) t = r And from that we get we get ( 2GM 2GM 2 r ) r = 2a p r 2 Whih is equal to a p = GM 2GM r r 2

6 a p = 2 r (GM r)3/2 (12) In Eq. (12), we find r represents the distane between the spaeraft and the Sun, and thus we find the deeleration of the spaeraft is depending on the distane of the spaeraft from the Sun aording to my equivalene priniple and my transformation by translating the retardation aording to the invariane by the entanglement whih is leading to the wave-partile duality and the unertainty priniple by the vauum flutuation. That s why Eq. (6) whih an be derived from Shwarzshild Geometry in ase of weak gravitational field has no any physial meaning in general relativity of Einstein GR, where in this ase it is in violation with the equivalene priniple of Einstein. Also GR an t even desribe the gravitational field in ase of strong gravitational field beause in ase of strong gravitational the esape veloity must be defined as relativisti as defined in my equivalene priniple and my transformation, not lassial as defined in ase of Lorentz transformation and the equivalene priniple of Einstein in GR. Now by onsidering G = m 3 kg. s 2, M = kg are respetively the gravitational onstant and the mass of the Sun. Nasa data [13] show that in the very middle part ( ) of the whole observation period of Pioneer 10, its radial distane from the Sun hanges from r 28.8 AU = m to r 48.1 AU = m. Thus by omputing a p from Eq. (12), we get a 10 = m s 2 and a 10 = m s 2. We have seen that the deeleration of the pioneer 10 anomalies is dereased depending on the distane from the Sun as from Eq. (12) aording to my equivalene priniple and my transformation [1], and that what is ausing the varying behavior of the Pioneer anomalies aording to Turyshev [13]. Aording to the period of observation 7.5 years from ( ) as noted by Anderson [11], we find for the Pioneer 10 a 10 is given as a 10 = = m s 2. yr 7.5 Markwardt [14] obtained an improved fit of Pioneer 10 data when estimating a jerk of a p 10 = m s 2. yr whih is exatly same as in my alulations. Also Toth [15] obtained a p 10 = m s 2. yr whih is in full agreement with my alulations. We find Eq. (6) whih an be derived also from Shwarzshild geometry in ase of weak gravitational field in GR an aount exatly for varying behavior of the Pioneer anomaly depending on the distane of the spaeraft from the Sun depending on the gravitational potential of the Sun, but this equation has no any physial meaning in GR, beause in reality if we onsider that in GR, then that will be in violation with the equivalene priniple of Einstein and the independent reality resulted from Minkowski geometry of the spae-time in relativity of Einstein. Now there is another term must be added to the Pioneer anomaly in Eq. (12) aording to the priniple of the quantum superposition in my equivalene priniple and my transformation by translating the retardation depending on the Heisenberg unertainty

7 priniple by the vauum flutuation. This term is related to the Hubble s law. We have from Hubble s law this aeleration is given aording to the equation a H = H (13) Where a H is the deeleration is aused by the Hubble, where is this ase sine the spaeraft is going far away from the Sun, in this ase it is observed for an observer on ground, there is a slight blue-shift given aording to the Eqs. (12)&(13). If the spaeraft is in a free fall toward the Sun, in this ase, it will be observed a slight redshift whih is given also aording to Eqs. (12)&(13), and that in reality explains the Hubble s law whih is leading to solve the osmologial onstant problem by solving the problem time in physis aording to my transformation by translating the retardation aording to invariane by the entanglement. Aording to that we get the full Pioneer anomaly is given aording to a p = H 2 r (GM r)3/2 (14) An estimate of the Hubble onstant, whih used a new infrared amera on the Hubble Spae Telesope (HST) to measure the distane and redshift for a olletion of astronomial objets, gives a value of H= 73.8 ± 2.4 (km/s)/mp or about H = 73.8 ± 2.4 (km s)/mp [16,17]. Thus from Eq. (40) we get for the Pioneer 10 at distane r = 28.8 AU or after 11 years of lunh a 10 = a H a 10 = = m s 2 This quantity is in omplete agreement with the observed Pioneer 10 aeleration ( at t=11 years of lunh), in Fig. (1) taken from Turyshev [13]. At a distane r = 48.1 AU at t = 18 years of lunh, we get a 10 = = m s 2 This quantity is in omplete agreement with the observed Pioneer 10 aeleration ( at t=18 years of lunh), in Fig. (1) taken from Turyshev [13]. We find from my transformation by translating the retardation aording to the invariane by the entanglement whih is leading to the wave-partile duality and the unertainty priniple by the vauum flutuation, the gravitational field is expressed aording to the energy flutuation, the vauum energy flutuation effetively gives a orret explanation of dark energy and dark matter, where in this ase dark matter and dark energy are explained, and that will provide a solution to the osmologial onstant problem. Figure (2) illustrates the predited Pioneer 10 anomaly aording to Eq. (14) whih gives an exat solution of the Pioneer anomaly.

8 Fig. (1): Comparison of the thermally-indued and anomalous aelerations for Pioneer 10. The estimated thermal aeleration is shown with error bars [13].

9 a 10 ( m s 2 ), H = 73.8 ± 2.4 (km s ) Mp, a p = H + 2 r (GM r)3/2 Fig. (3), the predited Pioneer 10 anomaly versus distane from the Sun aording to my solution.

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