On the Pioneer acceleration as light acceleration

Transcription

1 On the Pioneer acceleration as light acceleration arxiv:gr-qc/ v1 2 Mar 2004 Antonio F. Rañada Facultad de Física, Universidad Complutense, E Madrid, Spain 10 February 2004 Abstract Einstein showed in 1907 that the light speed must be larger at spacetime points with higher gravitational potential. This idea is used here to present a relativistic version of a recent model which explains the acceleration of the Pioneer 10 spaceship as the effect of a universal adiabatic acceleration of light. Such acceleration would be due to a cosmological acceleration of the proper time of bodies with respect to the coordinate time and would have the same observational signature as an acceleration of the ship. PACS numbers: Cc, Eg, Pe 1. Introduction. In several previous papers [1, 2, 3], a model was presented (from now on, this model or the model ) that gives an explanation of an observed but as yet unexplained phenomenon, potentially very important: the anomalous acceleration found in the Pioneer 10/11, Galileo and Ulysses spacecrafts by Anderson et al [4, 5]. What they saw is an unmodelled acceleration a P m/s 2, constant and directed towards the Sun, observed as a Doppler blue shift in the radio signals from these ships, which increases linearly in time, so that ν/ν = a P /c (= const). Obviously, its simplest interpretation is that the ships were not following the predicted orbit but had an extra unmodelled acceleration towards the Sun, as if our star pulled a bit too much from them with a force independent of the distance. Intriguingly enough, this effect does not show up in the planets. 1

2 A.F. Rañada, Pioneer acceleration as light acceleration 2 The version of the model discussed here is based on an argument which shows that the combined effect of the fourth Heisenberg relation and the expansion of the universe produces a progressive decrease of the optical density of the quantum vacuum. As a consequence the light speed must increase adiabatically, its acceleration being of the order of H 0 c m/s 2, very close to the acceleration of the Pioneer (H 0 is the present value of the Hubble parameter). It turns out (and this must be stressed) that an adiabatic acceleration of light a l has the same observational signature as a Doppler blue shift, as was shown in [2] (see also [3]). This can be understood easily, just by taking the wave equation with variable speed for the field, for instance [ 2 c 2 (t) t 2]E = 0, with c(t) = c 0 + a l t (the same result is obtained by using H instead). At first order in a l the plane wave solutions are E = E 0 exp{ 2πi[z/λ (ν 0 + νt/2)t]} with constant λ and ν/ν 0 = a l /c (note that the frequency is the time derivative of the phase, i.e. ν 0 + νt = ν 0 [1 + a l t/c]). In other words, all the increase of the speed is used to increase the frequency, the wavelength remaining constant. This has two consequences: (i) an acceleration of light a l = a P would produce an extra blue shift in radio signals, linearly increasing in time and having therefore the same observational effect as the unmodelled acceleration towards the Sun of the Pioneer and other spacecrafts, and (ii) the atomic clocks would be accelerating, since their periods would decrease. The blue shift would be due to the acceleration of light, not to the motion of the ships which would be following then and now the standard laws of gravitation without any extra pull from the Sun. In other words, the Pioneer effect would be explained by finding a good reason for the light to accelerate with a l close to a P. This is what this model does. In the previous version of the model presented in [1, 2, 3], the light acceleration arises in a Newtonian scheme because the permittivity and permeability of the quantum vacuum do depend on the gravitational potential Φ. The purpose of this paper is to present a simpler and relativistic version of that model which explains neatly the Pioneer effect. 2. On Einstein and the speed of light. In many people s mind, Einstein s name is associated to the second postulate of his special relativity, frequently misinterpreted as the statement that the observed speed of light is a universal constant. This was not his view, in fact he discovered that it depends on space and time, during his approach to general relativity between 1907 and His papers of that period, often considered just as matter

3 A.F. Rañada, Pioneer acceleration as light acceleration 3 for historians, are however of great interest for the physicists, in particular because of his discussions on the variation of the light speed in a gravitational field [6, 11]. In the last section of a review paper in 1907 [7], Einstein introduces his principle of equivalence, derives from it his first formula for the bending of a light ray and concludes that the light speed must depend on the gravitational potential. According to Pais, the study of Maxwell equations in accelerated frames had taught him that the velocity of light is no longer a universal constant in the presence of gravitational fields [6]. In 1911 he takes anew the question in a paper entitled On the influence of gravitation on the propagation of light [8], where he uses again his principle of equivalence. After a discussion on the synchronization of clocks, he concludes that if we call the velocity of light at the origin of coordinates c 0, where we take Φ = 0, then the velocity of light at a place with gravitational potential Φ will be given as c = c 0 ( 1 + Φ/c 2 ). (1) He thus confirms, with a more detailed analysis, the conclusion of this 1907 paper, stating also that eq. (1) is a first order approximation. It can be written a bit more explicitly as c(r, t) = c 0 { 1 + [Φ(r, t) ΦR ]/c 2 0}, (2) where Φ R is a reference potential (at present time in a terrestrial laboratory R, for instance). From now on in this paper, c 0 will be the light speed observed at R, i. e. the constant that appears in the tables. Equations (1)-(2) state that c must vary, so that the deeper (more negative) is the potential, the smaller is the light speed and conversely (at the surface of the Sun, c must be about 2 ppm lower than here at Earth). Note that they are contrary to the frequent assumptions that it is a universal constant of nature and that relativity precludes absolutely any variation of the light speed (see also [8]). In two papers in 1912 [9], he states that the non constancy of light speed in the presence of gravitational fields, excludes the general applicability of Lorentz transformations. Going even further, he considers the light speed as a field in spacetime c(r, t) and says a clock runs faster the greater the c of the location to which we bring it, a statement quite similar to consequence (ii) at the introduction of this work that the acceleration of light must cause an acceleration of atomic clocks, to be discussed in section 5. Einstein proposes

4 A.F. Rañada, Pioneer acceleration as light acceleration 4 then for the variable light speed in a static situation the equation c = kcρ, k being a constant and ρ the density of matter. In the second paper, he reaches the conclusion that the variation of the light speed affects Maxwell s equations, since c appears in the Lorentz transformations, what establishes a coupling between the electromagnetic and the gravitational fields which is not easy to analyze since the first is generally time dependent. He proposes then the equation c = k[cσ + ( c) 2 /2kc], as a dynamical equation for the field c = c(r, t), where σ is the sum of the mechanical and electromagnetic densities [6, 11]. In 1912, in a reply to a critical paper on relativity by M. Abraham, Einstein states clearly the constancy of the velocity of light can be maintained only insofar as one restricts oneself to spatio-temporal regions with constant gravitational potential. This is where, in my opinion, the limit of the principle of the constancy of the velocity of light thought not of the principle of relativity and therewith the limit of the validity of our current theory of relativity, lies [10]. Note that what Einstein says here is that the principle of relativity is not the same thing as the principle of constancy of light speed: the latter must not be taken as a necessary consequence of the former. Furthermore, he admits that the light velocity can depend on Φ, as was the case with his eq. (1). Indeed, in the Schwarzschild metric obtained four years later, one has for a null geodesic dl/dt = c(1 2GM/c 2 r) 1/2 which, for r much larger than the Schwarzschild radius 2GM/c 2, reduces to dl/dt = c(1 GM/c 2 r), i. e. to Einstein formula (1), c being here the light speed at infinite distance. It seems clear, therefore, that, according to (1)-(2), the light speed must be a function c(r, t), with both a space and a time variation. The first depends on the distribution of matter and energy. The second however must be dominated by a secular progressive increase, since the potential Φ produced by all the matter and energy must be an increasing function of time because of the universal expansion. Note that all this refers to the observed speed of light in empty space. However, c has several other meanings, for instance the constant in the Lorentz transformations, the speed of gravitational waves in empty space or the constant that appears in the Einstein Lagrangian of General Relativity [12]. 3. The potential of all the universe and the acceleration of light. Einstein s variable light speed approach to General Relativity fell into oblivion, superseded as it was by general relativity that gives a deeper insight on Gravitation. However his arguments leading to eq. (1) are still

5 A.F. Rañada, Pioneer acceleration as light acceleration 5 right and could be used in the study of non local effects in weak gravitational fields, when going beyond special relativity which is only valid locally in spacetime. Indeed, special relativity is an approximation to general relativity which holds good only locally, more precisely in all the Local Inertial Frames (LIF), i. e. free-falling frames in the tangent space at each point P. However there is no reason whatsoever to require that the light speed will be the same for all points: on the contrary it can be expected that it will depend on the particular P, so that c = c(p). To illustrate the argument, let us take a team of physicists in a spaceship during a travel along regions at which the potential is weak, through the Solar System or the Galaxy for instance. It they measure locally the light speed they could find that it varies adiabatically along the trip. At any moment, special relativity would still be valid in a small region around any spacetime point but the local value of c could be different from one region to the other. This can understood by considering the element of interval in weak gravity ds 2 = e 2Φ/c2 c 2 dt 2 dl 2 (1 + 2Φ/c 2 ) c 2 dt 2 dl 2. (3) Φ is here the potential of near bodies, those that produce a non vanishing acceleration g(r, t) at the observation point. Along a null geodesic one has thus for the light speed at a generic point P, c(p) = c(r)[1 + (Φ(P) Φ(R))/c 2 ], R being here a reference point. Assume for a moment that the light speed increases adiabatically as c(t) = cf(t), c = c( t) being its value at some fixed time t in the past, f = 1 + η(t) (so that η( t) = 0), η(t) being small and increasing in time. Defining the function Π(t) = c 2 log f = c 2 log[c(t)/ c] c 2 η(t), the interval (3) can be written as ds 2 = e 2(Φ+Π)/ c 2 c 2 dt 2 dl 2, (4) at first order. It is clear therefore that Π, which is space independent but time dependent, can be interpreted as a uniform gravitational potential of global character. Since (3) and (4) give equivalent descriptions, it turns out that an adiabatic acceleration of light is equivalent to a gravitational potential due to all the universe in the following sense. At any time, we can use either (i) eq. (3) with a time dependent light speed c(t) and no potential Π, or (ii) eq. (4) with a constant value of the light speed c and with the potential Π(t). This shows that Π(t) is the increase of the gravitational potential due to all the universe since the time t, when the light speed was measured and assumed to have a fixed value. If in the second option we take t = t 0 and, consequently

6 A.F. Rañada, Pioneer acceleration as light acceleration 6 c = c 0 (i. e. the value in the tables), then Π(t 0 ) = 0 but Π(t 0 ) > 0 because the galaxies are separating in the universal expansion, this implying that light must be accelerating (overdot means time derivative). A third interpretation will be considered later. 4. Calculation of the adiabatic acceleration of light. The effect of the expansion on the potential at a spacetime point will be considered now. The potential at the terrestrial laboratory R can be written, with good approximation, as Φ = Φ inh (R) + Φ av (t). The first term Φ inh (R) is the part due to the near local inhomogeneities (the Earth, the Solar System and the Milky Way). It can be taken as constant since these objects are not expanding. The second Φ av (t) is the space averaged potential due to all the mass and energy in the universe, assuming that they are uniformly distributed. It depends on time because of the expansion. The former has a non vanishing gradient but is small, the latter is space independent but is time dependent and much larger. The value of Φ inh /c 2 0 at R is the sum of the effects of the Earth, the Sun and the Milky Way, which are about , 10 8 and , respectively, certainly with much smaller absolute values than Φ av, which of the order of 10 1 as will be seen below. Moreover Φ inh appear in the two terms of the difference of potential in eq. (2), so that, being time independent, it cancels out. In fact, that difference is [Φ av (t)+φ inh (R) (Φ av (t 0 ) + Φ inh (R)] = [Φ av (t) Φ av (t 0 )] = Π(t), so that eq. (2) takes the form c(t) = c 0 [ 1 + Π(t)/c 2 0 ]. (5) Taking the time derivative, the light acceleration a l results to be a l = a t c 0, with a t = Π/c 2 0. (6) 0 Since c(t) = c 0 (1 + a t t). (7) the inverse time a t can be interpreted as an acceleration of time or of the clocks (note that η(t) = a t t). This quantity was introduced by Anderson et al in [4] its value being deduced there from the observed a P. It can be calculated in this model as follows. Let Φ 0 be the gravitational potential produced by the critical mass density distributed up to the distance R U (i. e. Φ 0 = R U 0 Gρ cr 4πrdr 0.3c 2 if R U 3, 000 Mpc) and let Ω M, Ω Λ be the corresponding present time relative densities of matter (ordinary plus dark)

7 A.F. Rañada, Pioneer acceleration as light acceleration 7 and dark energy corresponding to the cosmological constant Λ. Because of the expansion of the universe, the gravitational potentials due to matter and dark energy equivalent to the cosmological constant vary in time as the inverse of the scale factor R(t) and as its square R 2 (t), respectively. Since Π(t) = Φ av (t) Φ av (t 0 ), it turns out therefore that Π(t) = Φ 0 F(t), (8) where F(t) = Ω M [1/R(t) 1] 2Ω Λ [R 2 (t) 1], with F(t 0 ) = 0, F(t 0 ) = (1 + 3Ω Λ )H 0. Let us assume a universe with flat sections t = constant (i.e. k = 0), with Ω M = 0.3, Ω Λ = 0.7 and Hubble parameter to H 0 = 71 km s 1 Mpc 1 = s 1. It follows then that the inverse time a t and the present value of the acceleration of light are a t = H 0 (1 + 3Ω Λ )Φ 0 /c 2 0 H 0, a l H 0 c 0, (9) Note that, since H 0 c 0 = m/s a P, a l is close to the Pioneer acceleration, at least, and that we will take a t = H 0. It must be emphasized that the acceleration of light is a simple consequence of Einstein formula (2), taking into account the expansion of the universe, since the potential Φ is increasing because of the expansion, so that c must increase also. This increase, in turn produces a blue shift, as shown in section 1, with the same observational signature as an extra attraction from the Sun. It could be argued, however, that the gravitational redshift would cancel the increase in c since the frequency decreases when photons climb up along an increasing potential. However, the well known expressions for the gravitational redshift ν/ν 0 = Φ/c 2 0 assume static situations, which is not the case in an expanding universe, so that their proofs fail. 5. Is time accelerating? As explained in section 2, Einstein asserted in 1912 that clocks run faster the higher is c (page 104 of the first paper of [9]). He was thinking then in clocks at different space points, but the same can be said clearly about clocks at different times: if he had known the universal expansion (unsuspected however at that time) he could have said clocks run faster as time goes on. The (cautious) claim of this and previous work is that the Pioneer acceleration is an effect of the acceleration of light. Equivalently, it can be said that it is the manifestation of the acceleration of time, as measured by atomic clocks. If the observed blue shift is the consequence of the acceleration of light, the basic units of time (i. e. the periods of electromagnetic waves) would be decreasing and atomic clocks going faster.

8 A.F. Rañada, Pioneer acceleration as light acceleration 8 To understand better this question, consider a third intriguing possibility to interpret the metric of the expanding universe (3), in addition to the two given in section 3. Let t be the last time that the light speed was measured. In our case, it could be the year 1983, which we take for a moment as the origin of time and as t in (4). If instead of introducing Π(t) when going from (3) to (4), the time is changed as t τ = t t f(t)dt = t + H 0t 2 /2 (and Π/ c 2 = H 0 t 1), we can write for the interval ds 2 = e 2Φ/ c2 c 2 dτ 2 dl 2. (10) The light speed would then be constant with respect to the time τ. Since according to (4), the proper time of a body would verify dτ 0 = g 00 dt = (1 + H 0 t)dt, it turns out that, while the variable t is the coordinate (and parametric) time, τ would be the proper (and cosmological) time (note that τ would decrease in an eventual contracting phase). In other words: the effect of the background gravitational potential of all the universe would be equivalent either to an acceleration of light with respect to coordinate time or to a constant light speed with respect to proper time. Anderson et al mention in [4] the possibility of a clock acceleration a t, so that a P = a t c (compare with eq. (6)), that would appear as a nonuniformity of time; i. e. all clocks would be changing with a constant acceleration. However, at the end of the paper they say that some results rule out the universality of the a t time-acceleration model. It seems therefore that the relation between parametric time and dynamical time is an important element in this question [15, 16]. It must be emphasized that, in this model, the quantity a t is given by eq. (7), which allow its calculation. It is the clock acceleration, what gives special interest to the fact that it is, at least, close to the Hubble parameter H The role of the quantum vacuum. We have seen that Einstein himself showed that gravitation has an influence on Maxwell equations, as is clear from his eq. (2). This means that, at least for weak gravitation, we can make use of such concepts the refractive index n(φ) = [1 + (Φ Φ R )/c 2 ] 1, equal to ǫ r µ r in terms of the relative permittivity and permeability of the quantum vacuum. In this model and as a consequence of the fourth Heisenberg relation, the average lifetime of the virtual pairs in the zero point radiation depend on Φ, so that the deeper the potential the longer the lifetime and the denser the zero point radiation [1, 2, 3]. Since Φ av increases in time, the quantum vacuum becomes thinner as time goes on, the

9 A.F. Rañada, Pioneer acceleration as light acceleration 9 optical density of empty space decreasing accordingly. Indeed, in its first form [1, 2, 3], the model is Newtonian and takes as first approximation that ǫ r = 1 β(φ Φ R )/c 2 0 and µ r = 1 γ(φ Φ R )/c 2 0 where β, γ are positive constants. Using instead χ = (β + γ)/2 and ξ = (3β γ)/2, it turns out that the light speed and the fine structure constant must depend on space and time as c(φ) = c 0 {1 + χ[(φ Φ R )/c 2 0 ]}, α(φ) = α 0{1 + ξ[(φ Φ R )/c 2 0 ]}. (11) Einstein s eq. (2) implies that χ = 1, a conclusion already reached in [3] on other grounds. The observations by Webb et al [13] on the cosmological evolution of α indicate that ξ 10 5 (see [1, 3]), although ξ could be smaller if they have overestimated the variation of α [14]. In that case β 0.5 and γ 1.5. If α does not vary, ξ = 0, so that γ = 3β = 1.5. It is to be noted that since α = e 2 /4πǫ 0 hc, the variations of e and c (both increasing in time) work against each other, their combined effect being nil if ξ = Summary and conclusion. The previous Newtonian version of this model [1, 2, 3] is based on an argument implying that the optical density of empty space must be decreasing adiabatically in time, as a consequence of the decrease of its permittivity and permeability, a phenomenon that was dubbed there the attenuation of the quantum vacuum. The acceleration of light is the consequence. Note that an application of the fourth Heisenberg relation is an important point in that argument. The results indicate that the model gives a description of the interaction between gravitation and Maxwell equations (by means of a permittivity and a permeability depending on Φ), which is alternative and equivalent to the one embodied in Einstein equation (2). Consequently, the integration of the optical properties of empty space in this relativistic model does not pose any problem if χ = 1 and ξ is small enough (see [17] for a detailed comparison of the model with the experiments). The conclusion of this paper is that the Pioneer phenomenon could be a consequence of the acceleration of the proper time of bodies τ (measured by atomic clocks) with respect to the coordinate time t because of the universe expansion. The light speed would still be constant with respect to τ, but would have a universal adiabatic acceleration equal to a l m/s 2 = a P H 0 c with respect to t. Since its observational signature would a blue shift increasing linearly in time as ν/ν = a P /c, this acceleration would be quite similar to the footprint of an extra acceleration of the Pioneer towards the observer. All this must be studied by the experts who know the details of the motion of the spacecrafts.

10 A.F. Rañada, Pioneer acceleration as light acceleration 10 I am indebted to Profs. A. Tiemblo for enlightening discussions on the nature of time and J.L. Sebastián for useful information on microwaves instrumentation. References [1] A. F. Rañada, astro-phys/ , Europhys. Lett. 61, 174 (2003). [2] A. F. Rañada, gr-qc/ , Europhys. Lett. 63, 653 (2003) [3] A. F. Rañada, Int J. Mod. Phys. D 12, 1755 (2003). [4] J. D. Anderson, Ph. A. Laing, E. L. Lau, A. S. Liu, M. Martin Nieto and S. G. Turyshev, Phys. Rev. Lett. 81, 2858 (1998). [5] J. D. Anderson, Ph. A. Laing, E. L. Lau, A. S. Liu, M. Martin Nieto and S. G. Turyshev, Phys. Rev. D 65, (2002). [6] A. Pais, Subtle is the Lord. The science and the life of Albert Einstein (Oxford University Press, Oxford, 1984). [7] A. Einstein, Jahrbuch Radioaktivität und Electronik 4, 411 (1907); reprinted in The collected papers of Albert Einstein, English translation, Vol 2 (Princeton University Press, 1989), p [8] A. Einstein, Annalen der Physik 35, 898 (1911); reprinted in The collected papers of Albert Einstein, English translation, Vol 3 (Princeton University Press, 1993), p [9] A. Einstein, Annalen der Physik 38, 355 (1912); 38, 443 (1912); reprinted in The collected papers of Albert Einstein, English translation, Vol 4 (Princeton University Press, 1996), pp. 95 and 107. [10] A. Einstein, Ann. Physik 38, 1059 (1912); reprinted in The collected papers of Albert Einstein (1996), English translation, Vol 4 (Princeton University Press, Princeton, 1996), p [11] J. M. Sánchez Ron, El origen y desarrollo de la relatividad (Alianza Editorial, Madrid, 1985). [12] G. F. R. Ellis and J. Ph. Uzan, gr-qc/ v2.

11 A.F. Rañada, Pioneer acceleration as light acceleration 11 [13] J.K. Webb, M.T. Murphy, V.V. Flambaum, V.A. Dzuba, J.D. Barrow, C.W. Churchill, J.X. Prochaska and A.M. Wolfe, Phys. Rev. Lett. 87, (2001). [14] R. Srianand, H. Chand, P. Petitjean and B. Aracil, astro-ph/ v1. [15] A. Tiemblo and R. Tresguerres, Gen. Rel. Grav. 34, 31 (2002). [16] J. F. Barbero, A. Tiemblo and R. Tresguerres, Phys. Rev. D 57, 6104 (1998). [17] A. F. Rañada, gr-qc/