Gravity and the quantum vacuum inertia hypothesis

Transcription

1 Ann. Phys. (Leipzig) 14, No. 8, (2005) / DOI /andp Gravity and the quantum vauum inertia hypothesis Alfonso Rueda 1, and Bernard Haish 2, 1 Department of Eletrial Engineering, California State University, 1250 Bellflower Blvd., Long Beah, CA 90840, USA 2 Chief Siene Offier, ManyOne Networks, 100 Enterprise Way, Bldg. G-370, Sotts Valley, CA 95066, USA Reeived 27 January 2005, revised 3 April 2005, aepted 14 April 2005 by F.W. Hehl Published online 15 July 2005 Key words Quantum vauum, mass, zero-point field, inertia, gravitation, stohasti eletrodynamis, priniple of equivalene. PACS Sq, Cv, a, i In previous work it has been shown that the eletromagneti quantum vauum, or eletromagneti zero-point field, makes a ontribution to the inertial reation fore on an aelerated objet. We show that the result for inertial mass an be extended to passive gravitational mass. As a onsequene the weak equivalene priniple, whih equates inertial to passive gravitational mass, appears to be explainable. This in turn leads to a straightforward derivation of the lassial Newtonian gravitational fore. We all the inertia and gravitation onnetion with the vauum fields the quantum vauum inertia hypothesis. To date only the eletromagneti field has been onsidered. It remains to extend the hypothesis to the effets of the vauum fields of the other interations. We propose an idealized experiment involving a avity resonator whih, in priniple, would test the hypothesis for the simple ase in whih only eletromagneti interations are involved. This test also suggests a basis for the free parameter η(ν) whih we have previously defined to parametrize the interation between harge and the eletromagneti zero-point field ontributing to the inertial mass of a partile or objet. 1 Introdution It appears that the eletromagneti quantum vauum should make a ontribution to the inertial mass, m i, of a material objet in the sense that at least part of the inertial fore of opposition to aeleration, or inertia reation fore, springs from the eletromagneti quantum vauum [1 3]. The relevant properties of the eletromagneti quantum vauum were alulated in a Rindler frame (a name we proposed for a rigid frame that performs uniformly aelerated motion) to find a fore of opposition exerted by the quantum vauum radiation to oppose the aeleration of an eletromagnetially interating objet. We all this fore assoiated with the quantum vauum radiation in aelerating referene frames the Rindler frame fore. It originates in an event horizon asymmetry for an aelerated referene frame. Owing to the symmetry and Lorentz invariane of quantum vauum radiation in unaelerated referene frames, the Rindler frame fore is zero in onstant veloity (inertial) frames. Not all radiation frequenies are equally effetive in exerting this opposition. The effetiveness of the various frequenies may be parametrized by a funtion of the form η(ω) whose postulation is later justified by means of a ompelling illustration. This suggests an experimental approah to measure the eletromagneti quantum vauum ontribution to inertial mass Corresponding author arueda@sulb.edu haish@alphysis.org

2 480 A. Rueda and B. Haish: Gravity and the quantum vauum inertia hypothesis in a simple alulable situation. The motivation for our interpretation omes from the disovery that the resulting fore proves to be proportional to aeleration, thus suggesting a basis for inertia of matter [1 3]. It thus appears that Newton s equation of motion ould be derived in this fashion from eletrodynamis, and it has been shown that the relativisti version of the equation of motion also naturally follows. The energy of a quantum harmoni osillator is allowed to take on only disrete values, E n =(n+ 1 2 ) ω, implying a minimum energy of ω/2. This an be viewed as a onsequene of the Heisenberg unertainty priniple. The radiation field is quantized by assoiating eah mode with a harmoni osillator [4]. This implies that there should exist a ground state of eletromagneti quantum vauum, or zero-point, energy. Even though this appears to be an immediate and inesapable onsequene of quantum theory, it is usually argued that suh a field must be virtual, sine the energy density of the field would be expeted to have osmologial effets grossly inonsistent with the observed properties of the Universe, sometimes ited as a 120 order of magnitude disrepany. We set this objetion temporarily aside to explore an intriguing onnetion between the properties of this radiation field and one of the fundamental properties of matter, i.e. mass. (This will, in fat, suggest an approah to resolving the disrepany.) We do so using the tehniques of stohasti eletrodynamis (SED). Alternatively, we reently have used the tehniques of quantum eletrodynamis (QED) for obtaining exatly the same result [5]. Even this, though, may still be labelled a semilassial result in the sense that we only onsider the quantum struture of the field in its general form in the Rindler frame regardless of the details of the partiles it may interat with. If one assumes, as in SED, that the zero-point radiation field arries energy and momentum in the usual way, and if that radiation interats with the partiles omprising matter in the usual way, it an be shown that a law of inertia an be derived for matter omprised of eletromagnetially interating partiles that are a priori devoid of any property of mass. In other words, the f = ma law of mehanis as well as its relativisti ounterpart an be traed bak not to the existene in matter of mass (either innate or due to a Higgs mehanism), but to a purely eletromagneti effet (and possibly analogous ontributions from other vauum fields). It an be shown that the mass-like properties of matter reflet the energy-momentum inherent in the quantum vauum radiation field. We all this the quantum vauum inertia hypothesis. There are additional onsequenes that make this approah of assuming real interations between the eletromagneti quantum vauum and matter appear promising. It an be shown that the weak priniple of equivalene the equality of inertial and gravitational mass 1 naturally follows. In the quantum vauum inertia hypothesis, inertial and gravitational mass are not merely equal, they prove to be the idential thing. Inertial mass arises upon aeleration through the eletromagneti quantum vauum, whereas gravitational mass as manifest in weight results from what may in a limited sense be viewed as aeleration of the eletromagneti quantum vauum past a fixed objet. The latter ase ours when an objet is held fixed in a gravitational field and the quantum vauum radiation assoiated with the freely-falling frame instantaneously omoving with the objet follows urved geodesis as presribed by general relativity. Finally, the interations of the quantum vauum radiation field with massless partiles results in Shrödinger s zitterbewegung, onsisting of random speed-of-light flutuations for a partile suh as the eletron. These flutuations an be shown to ause a point-like partile to appear spread out in volume over a region all it the Compton sphere reminisent of that predited by the orresponding wave funtion. When this is viewed from a moving referene frame, the Doppler shift of these flutuations results in an observed superimposed envelope that has properties like the de Broglie wave [7]. We thus tentatively suggest that it may be worth onsidering the so-alled rest mass of partiles to be a manifestation of the energy assoiated with zitterbewegung, espeially sine simulations are showing that in the presene of an external field, massless partiles undergoing zitterbewegung will aquire helial, spin-like properties (L. J. Nikish, priv. omm.). 1 When we write gravitational mass in this paper we mean passive gravitational mass unless otherwise speified. Only for point partiles or for spherially symmetri stationary matter distributions an passive and ative gravitational mass densities be deemed idential. In the general ase they are not the same. This differene mainly appears in osmology, where passive and ative gravitional mass densities in general differ [6].

3 Ann. Phys. (Leipzig) 14, No. 8 (2005) / It is thus suggestive that both mass and the wave nature of matter an be traed bak to speifi interations with the eletromagneti zero-point field and possibly the other bosoni vauum fields. Given this possible reinterpretation of fundamental properties, we suggest that it is premature to take a firm stand against the reality of the zero-point radiation field and its assoiated energy on the basis of osmologial arguments, espeially given the possible relation between quantum vauum, or zero-point, radiation and dark energy. SED is a theory that inludes the effets of the eletromagneti quantum vauum in physis by adding to ordinary Lorentzian lassial eletrodynamis a random flutuating eletromagneti bakground onstrained to be homogeneous and isotropi and to look exatly the same in every Lorentz inertial frame of referene [8 10]. This replaes the zero homogeneous bakground of ordinary lassial eletrodynamis. It is essential that this bakground not hange the laws of physis when exhanging one inertial referene system for another. This translates into the requirement that this random eletromagneti bakground must have a Lorentz invariant energy density spetrum. The only random eletromagneti bakground with this property is one whose spetral energy density, ρ(ω), is proportional to the ube of the frequeny, ρ(ω)dω ω 3 dω. This is the ase if the energy per mode is ω/2 where ω is the angular frequeny, the other fator of ω 2 oming from the density of modes [4]. (The ω/2 energy per mode is of ourse also the minimum energy of the analog of an eletromagneti field mode: a harmoni osillator.) The spetral energy density required for Lorentz invariane is thus idential to the spetral energy density of the zero-point field of ordinary quantum theory. For most purposes, inluding the present one, the zero-point field of SED may be identified with the eletromagneti quantum vauum [10]. However SED is essentially a lassial theory sine it presupposes only ordinary lassial eletrodynamis and hene SED also presupposes speial relativity (SR). Our reliane on SED [1 3] has been due to the fat that SED allows a very lear intuitive representation. Nevertheless, the SED-based results of [1, 2] have been readily rederived within a QED formalism [5]. In Set. 2 we briefly review a previously-derived development of the quantum vauum inertia hypothesis. In Set. 3 we introdue an example that allows a very lear visualization of the oupling of an eletromagnetially interating objet with the surrounding vauum fields and the evaluation of the funtion η(ω) mentioned above [1, 2]. We turn next to the priniple of equivalene. Aording to the weak equivalene priniple (WEP) of Newton and Galileo, inertial mass is equal to gravitational mass, m i = m g. If the quantum vauum inertia hypothesis is orret, a very similar mehanism involving the quantum vauum should also aount for gravitational mass. This novel result, restrited for the time being to the eletromagneti vauum omponent, is preisely what we show in Set. 4 by means of formal but simple and straightforward arguments requiring physial assumptions that are unontroversial and widely aepted in theoretial physis. In Set. 5 the onsisteny of this argument with so-alled metri theories of gravity (i.e. those theories haraterized by spaetime urvature) is exhibited. In addition to the metri theory, par exellane, Einstein s GR, there is the Brans-Dike theory and other less well known ones, briefly disussed by Will [11]. A non-spaetime urvature theory is briefly disussed in Set. 6. Nothing in our approah points to any new disriminants among the various metri theories. Nevertheless, our quantum vauum approah to gravitational mass will be shown to be entirely onsistent with the standard version of GR. Next, in Set. 7, we take advantage of geometrial symmetries and present a short argument supported by standard potential theory to show that in the weak field limit a Newtonian inverse square fore must result. A new perspetive on the origin of weight is presented in Set. 8. In Set. 9 we disuss an energetis aspet, related to the derivation presented herein, and resolve an apparent paradox. A brief disussion on the nature of the gravitational field follows in Set. 10, but we infer that the present development of our approah does not provide any deeper or more fundamental insight than GR itself into the ability of matter to bend spaetime. We present onlusions in Set. 11.

4 482 A. Rueda and B. Haish: Gravity and the quantum vauum inertia hypothesis 2 The Rindler frame fore and inertia It has been shown [1 4] that the eletromagneti quantum vauum makes a ontribution to the inertial mass, m i, of a material objet in the sense that at least part of the inertial fore of opposition to aeleration, or inertia reation fore, springs from the eletromagneti quantum vauum. Speifially, in the previously ited work, the properties of the eletromagneti quantum vauum as experiened in a Rindler, or onstant proper aeleration, frame were investigated, and we alulated a resulting fore of opposition on material objets fixed in suh a Rindler non-inertial frame and interating with the eletromagneti zero-point radiation field. It originates in an event horizon asymmetry for an aelerated referene frame. Owing to the symmetry and Lorentz invariane of zero-point radiation in unaelerated referene frames, the Rindler frame fore vanishes at onstant veloity, i.e. in inertial frames. We further hypothesized that if V 0 is the proper volume of the material objet, some frequeny-dependent fration, η(ω), of this radiative momentum will, via interations, be onferred upon the partiles omprising the matter. The motivation for this interpretation omes from the disovery that the resulting Rindler frame fore proves to be proportional to aeleration. It thus appears that Newton s equation of motion an be derived in this fashion from eletrodynamis, and it has been shown that the relativisti version of the equation of motion also naturally follows. We have alled the notion that at least part of the inertia of an objet should be due to the individual and olletive interation of its quarks and eletrons with the quantum vauum as the quantum vauum inertia hypothesis with the proviso that analogous ontributions are expeted from the other bosoni vauum fields. The eletromagneti field inside a avity of perfetly refleting walls an be shown to be expandable in ountably infinite different modes where eah mode orresponds to an independent osillation whih always, even in the ase of free spae, an be represented as a harmoni osillator [12]. When quantized, the harmoni osillator gives an energy of the form ω(n +1/2), where ω is the harateristi frequeny of the mode (ω =2πν), and when the integer n is set to zero, there is still a remnant energy ω/2. Alternatively we also have that for the harmoni osillator, the Heisenberg unertainty relation tells us that the minimum (zero-point) energy is ω/2. The eletromagneti field therefore has a minimum quantum energy state onsisting of zero-point flutuations having an average energy per mode of ω/2. The density of modes is, see e.g. [4], N(ω)dω = ω2 dω π 2 3, (1) and this ombined with the average energy per mode of ω/2 yields a spetral energy density for the zero-point flutuations of ρ(ω)dω = ω3 2π 2 3dω. (2) In the semi-lassial approah known as stohasti eletrodynamis (SED) the quantum flutuations of the eletri and magneti fields are treated as random plane waves summed over all possible modes with eah mode having this ω/2 energy. The eletri and magneti zero-point flutuations in the SED approximation are thus E zp (r,t)= B zp (r,t)= λ=1,2 λ=1,2 d 3 k d 3 k ( ) 1/2 ω 2π 2 ˆɛ(k,λ) os [k r ωt θ(k,λ)], (3a) ( ) 1/2 ω ] [ˆk ˆɛ(k,λ) 2π 2 os [k r ωt θ(k,λ)], where the sum is over the two polarization states, ˆɛ is a unit vetor, and θ(k,λ) is a random variable uniformly distributed in the interval (0, 2π). For simpliity and to failitate omparison with muh previous SED work, we omit the Ibison and Haish modifiation of Eqs. (3) in whih the amplitudes are also randomized in suh a way as to bring the quantum and lassial statistis of the eletromagneti zero-point field into exat agreement [13]. (3b)

5 Ann. Phys. (Leipzig) 14, No. 8 (2005) / Sine the E zp and B zp field flutuations are entirely random, there is no net energy-momentum flow aross any surfae, or in other words, the value of the stohastially averaged Poynting vetor must be zero: N zp = 4π Ezp B zp =0. (4) It is nowadays well known how to transform the E zp and B zp field vetors from a stationary frame to one undergoing uniform aeleration, i.e. having onstant proper aeleration, that we have labelled a Rindler frame. Suh a frame will experiene an asymmetri event horizon leading to a non-zero eletromagneti energy and momentum flux as alulated by a stationary observer. The veloity and the Lorentz fator for suh a frame are, v ( ατ ) = tanh, (5a) ( ατ ) γ τ = osh, (5b) where α is the objet s proper aeleration and τ its proper time. The general, ompat form of the Lorentz transformation of eletromagneti fields is (f. Eq in [14]) E = γ(e + v ( ) γ 2 v B) γ +1 (v E), (6a) B = γ(b v ( ) γ 2 v E) γ +1 (v B). (6b) Partiularizing v to be along the x-diretion and using omponent form of Eqs. (6), one transforms E zp and B zp to the Rindler frame (in a way explained in onsiderably more detail in [1]) in order to obtain: E zp (0,τ)= λ=1,2 +ẑ osh [ os B zp (0,τ)= λ=1,2 +ẑ osh [ os d 3 k ( ω 2π 2 ( ατ )[ ˆɛ z + tanh k x 2 a osh ( ατ d 3 k ) 1/2 {ˆxˆɛ x +ŷ osh ( ατ ) ]} (ˆk ˆɛ) y ) ω ( ατ a sinh ( ) 1/2 ω {ˆx(ˆk 2π 2 ˆɛ) x +ŷ osh ( ατ )[ (ˆk ˆɛ) z tanh k x 2 a osh ( ατ ) ω ( ατ a sinh ( ατ )[ ( ατ ) ] ˆɛ y tanh (ˆk ˆɛ) z ) ] θ(k,λ), (7a) ( ατ ) ]} ˆɛ y ( ατ )[ (ˆk ˆɛ) y + tanh ( ατ ) ] ˆɛ z ) ] θ(k,λ), (7b) where ˆɛ x signifies the salar projetion of the ˆɛ unit vetor along the x diretion, and similarly for ˆɛ y and ˆɛ z. Our goal here has been to alulate the rate of hange of the momentum applied by the zero-point field on the eletromagnetially interating, aelerating objet. Eah individual inertial frame has assoiated with itself its own (or proper) random eletromagneti zero-point field bakground and thus its proper zero-point field vauum spetral energy density distribution. For eah inertial frame its proper eletromagneti vauum is homogeneously and isotropially distributed. This means that the omponents of the net Poynting vetor of the zero-point field of a given frame when observed in that same frame all should vanish. Moreover, of the omponents of the 4 4 eletromagneti energy-momentum stress tensor only the diagonal omponents

6 484 A. Rueda and B. Haish: Gravity and the quantum vauum inertia hypothesis survive as all other omponents should be zero. Extreme are should be exerised when performing suh Lorentz transformations. Eah of the omponents of the tensor is represented by an improper integral over k spae, whih is the k spae representation assoiated with the partiular inertial frame that we are dealing with. When transforming to another inertial frame it is not enough to transform the interior of the integral, i.e. its integrand. One must also take are of what in Appendix C of [1] we alled the k sphere. This is done by means of a utoff onvergene fator e k/k that effetively uts off the zero-point radiation beyond some maximal magnitude of the wave vetor in the k spae of the original inertial frame, thus the name k sphere. The utoff parameter k learly makes the zero-point energy as well as all other omponents of the tensor, like the Poynting vetor omponents and the Maxwell 3 3 stress tensor omponents, all finite, whih is a desirable onsequene. Nevertheless, the spetral Lorentz invariane holds even when at the end we make k anyway. In suh ases e k/k beomes just a form fator to regularize the integration [1]. Observe however that in the many integrations we performed in [1] we really did not need to use suh a utoff fator. The reason is lear: the final integrations were performed at aelerating objet proper time τ =0, when the laboratory frame and the aelerated-partile frame instantaneously omoved. Indeed, we observe that all final integrations for the inertia reation fore in [1] are performed at τ =0, whih is the time instant when the uniformly-aelerated objet omoves with the inertial laboratory frame. This means that both the momentum density, and thereby the Poynting vetor, of the zero-point field are for the objet, those of the laboratory frame. But there is a more subtle hange that needs to be aptured. Both the momentum density and the total momentum of the zero-point field inside the volume V 0 of the body display a time rate of hange and have a non-vanishing time derivative. Thus although both the Poynting vetor and the momentum of the zero-point field inside the body instantaneously vanish, their time derivatives at that oinidene time do not vanish. This an also be visualized as a manifestation of a redution in symmetry of the event horizon that passes from the perfet three-dimensional symmetry of a sphere to a lower two-dimensional symmetry that is merely axial. This results in a speial situation that may be visualized as a stress or tension in the vauum field whih is manifested on the aelerated partile as the Rindler frame fore. From the eletromagneti zero-point field in the Rindler frame of the aelerated objet we alulate the Poynting vetor (see [1] for details). For uniform aeleration in the x-diretion, the Poynting vetor is N zp (τ) = ˆx 2 ( ) ( ) 2ατ ω 3 2ατ 3 sinh 2π 2 3dω = ˆx2 3 sinh ρ(ω)dω, (8) where ρ(ω) is the spetral energy density of the zero-point flutuations. The amount of radiative momentum arried by the zero-point flutuations that are passing through and instantaneously ontained in the aelerated objet whih is undergoing uniform proper aeleration α = αˆx and that has proper volume V 0,is p zp = ˆx V ( ) 0 2 2ατ γ τ 3 sinh ρ(ω)dω = ˆx 4 [ 1 3 V 0v τ γ τ 2 ] ρ(ω)dω. (9) Let us assume that at frequeny ω some small fration η(ω) of this energy interats with the partiles of matter ontained in V 0. We an then write the interating fration of the zero-point momentum instantaneously ontained in and passing through the objet as p zp = ˆx 4 [ ] 1 3 V 0v τ γ τ 2 η(ω)ρ(ω)dω. (10) If we take the time derivative of this quantity, we find that [1] dp zp [ ] 4 = f zp V 0 = dt 3 2 η(ω)ρ(ω)dω a. (11)

7 Ann. Phys. (Leipzig) 14, No. 8 (2005) / A fully ovariant analysis eliminates the fator of 4/3 and also yields a proper relativisti four-vetor fore expression (see Appendix D of [1]). Eqn. (11) is telling us that in order to maintain the aeleration of suh an objet, a motive fore, f, must ontinuously be applied to balane the vauum eletromagneti ounterating reation fore, that we may all the Rindler frame fore, f zp. The motive fore is, [ ] f = f zp V0 = η(ω)ρ(ω)dω a. (12) 2 where we have suppressed the fator of 4/3. This strongly suggests Newton s f = ma, and that the zero-point field ontribution to the inertial mass ontained in V 0 is the mass-like oeffiient [ ] V0 m i = η(ω)ρ(ω)dω. (13) 2 In other words, some of the apparent inertial mass of an objet originates in the interating fration of the zero-point energy instantaneously ontained in an objet. In this view, the apparent momentum of the objet an be traed bak to the momentum of the zero-point radiation field. We suspet that η(ω) involves some kind of resonane at the Compton frequeny of individual partiles, (ω C = m 2 / ), sine this suggests a lose onnetion between the origin of mass and the de Broglie wavelength, both stemming from interations of matter with the quantum vauum [7]. We speulatively digress for a moment in order to try to gain some physial insight into how the interrelationships of minutely different oordinate systems an translate into real measurable effets. Sine momentum is not absolute but relative to the observer, and indeed always zero in one s own frame, one wonders how the time derivative of this arbitrary quantity an yield something real and absolute, i.e. a fore. We suggest that the answer is deeply rooted in the basis of speial relativity that all interations are limited by the speed of light. Hene if one pushes on an objet at point A, it will take a finite time for that push to be transferred to point B in the same objet. The aeleration of any objet of finite extent involves a o-mingling of an infinite number of minutely different referene frames. Aeleration neessarily mixes infinitely many adjaent referene frames as a result of the propagation limitations of speial relativity. This analogously suggest a physial meaning for the otherwise seemingly purely mathematial operation of alulating how E zp and B zp would look to an observer in one (inertial) referene frame from the basis of another (aelerating). In Appendix B of [1] we also showed an alternative but omplementary approah. Instead of onsidering the opposition to aeleration due to the eletromagneti zero-point field bakground through whih the objet is being aelerated (of the previous approah in the body of [1]) we also looked at the amount of eletromagneti energy ontained within the objet aording to the viewpoint of an inertial stationary observer (Appendix B of [1]). The very natural result that ensues for that stationary observer is that as the objet moves faster and faster the amount of enlosed energy of zero-point radiation grows in exat proportion to γ τ, whih is exatly the same way as the mass of a moving and aelerating objet of rest mass m i behaves aording to an inertial stationary observer. In this omplementary approah the rest mass turns out to be exatly the same as in the zero-point field drag approah of Eq. (13) above. Moreover, whereas in Eq. (10) for the zero-point field drag approah we obtained the fore f zp that the zero-point field exerts on the objet when opposing its aeleration, in the new approah we diretly get f, the fore that the external agent has to apply to the objet to aelerate it, whih we already notied in Eq. (12). The formal development, i.e. the equations in the derivation, of this omplementary approah omes out to be supefiially similar to the formal development of the zero-point field drag approah, exept that all equations sine the very first ones [1] have the opposite sign than in the previous ase. Coneptually though, the two approahes are not similar. Instead they are very omplementary. We ould say that they are like the two sides of the same oin. One annot exist without the other, but they are distintly different. The rest mass in Eq. (13) exatly orresponds to the amount of zero-point field mass equivalent enlosed within the volume of the body and that, thanks to the η(ω) spetral fator interats with the body. If the mass

8 486 A. Rueda and B. Haish: Gravity and the quantum vauum inertia hypothesis is moving with respet to a stationary observer, with speed v τ, the assoiated momentum is p and in the seond approah omes as given by eqn, (11) but with the positive sign. (Reall that the 4/3 fator omes from not using a ovariant approah. The ovariant approah of Appendix D [1] removes the spurious fator by onsidering all omponents in the eletromagneti stress-energy momentum tensor.) This is analogous to the situation of relativisti mass inrease. A stationary observer would ome to the onlusion that the mass of an aelerating objet is steadily inreasing as γ τ m, but in the aelerating frame no mass hange is evident. One s momentum in one s own referene frame is always zero. Physial onsequenes only ensue when there is a hange in momentum, akin to a physial hange from one referene frame into another. In the approah of the quantum vauum inertia hypothesis, it beomes physially evident how the Lorentz fator γ τ, whih haraterizes a spae-time geometry relationship, aquires the physial role of relativisti mass inrease parameter. It also beomes lear why the relativisti mass inrease must beome infinite at : One annot propagate the effet of any fores due to the zero-point field at speeds faster than. 3 On the eletromagneti model of the aelerated objet The seond or omplementary approah of [1], Appendix B, that we mentioned above, very learly shows that m i 2 is just the amount of zero-point energy loated inside the aelerated objet and that instantaneously omoves with the objet as viewed by an inertial observer. Here we would like to visualize the meaning of the parameter V 0, the volume of the objet in an eletromagneti sense, and the effiieny or interation funtion η(ω). For this we just need to think of an eletromagneti avity omprised of surrounding onduting walls. In suh a ase V 0 is just the internal volume of the avity. The most familiar shape is a retagular parallelepiped of sides a, b and, with a<b<. We onsider here only the volume inside the avity and as far as m i goes, the mass is just 1/ 2 times the total amount of zero-point field energy that an be aumulated within the struture of the avity. The avity walls are onduting and as suh serve to onfine the eletromagneti osillating radiation within its boundaries. Internally, the avity an sustain a ountable and finite number of harateristi or proper modes of osillation. It is well known that beyond a threshold frequeny ω p that orresponds to the plasma frequeny of the eletrons in the onduting walls, mirowave avities beome essentially transparent and annot sustain any more modes internally. The lowest frequeny modes are omparatively widely distributed in frequeny but as the harateristi frequenies of the modes inrease, they beome more losely spaed in frequeny. For wavelengths λ of the modes muh smaller than the dimensions of the avity, λ a<b<, their frequeny number density starts to grow roughly in proportion to ω 2 /π 2 3 whih, not surprisingly, is the N(ω) of Eq. (1). Reall that N(ω) is obtained when the avity that enloses the radiation grows in size by letting >b>a, whih yields what we all the limit of the ontinuum. The walls, beause they are onduting, serve to indiretly onnet the inside of the avity to the outside. At zero temperature, on the outside we have the zero-point eletromagneti radiation of Eqs. (1) and (2) and in the inside only the frequenies orresponding to the modes whih an be exited. The modes of the avity weakly ouple through the onduting walls to the random eletromagneti radiation outside. At zero temperature, the avity l th mode, being a harmoni osillator, aquires the osillation of energy ω l /2 where ω l represents the l th mode harateristi frequeny. Let us first assume that the avity is an ideal one in whih dissipation may be negleted. This by itself does not prelude some broadening around the harateristi frequeny beause there are, in the walls, quantum flutuations of the eletrons and of the plasma of eletrons at zero temperature whih indue some small albeit nonvanishing blurring of the exat mode harateristi frequeny due to the Doppler shifts and flutuations in the avity geometry (dimensions a, b and are not exat anymore). Hene there is always some broadening. For the time being, however, let us neglet suh broadening and onsider instead an ideal avity with a finite number of disrete modes up to ω p, the plasma frequeny of the eletrons in the walls, and at stritly

9 Ann. Phys. (Leipzig) 14, No. 8 (2005) / zero temperature, T =0. Eah mode then osillates at its exat harateristi frequeny, ω k, and with an energy orresponding to that of the harmoni osillator zero-point osillation, ω k /2, k =1, 2,..., N, where N is the maximal mode frequeny suh that ω N ω p. The total energy is then E = N k=1 But this energy must be exatly equal to the energy given by Eq. (13): m i 2 = ω k 2, ω 1 ω 2 ω 3... ω N ω p. (14) ω=0 V 0 η(ω)ρ(ω)dω = E = N k=1 ω k 2. (15) As ρ(ω) is given by Eq. (2), it is easy to see that in order to satisfy (15) we need an η(ω) of the form: N 1 η(ω) = π2 3 N V 0 ω 2 δ(ω ω π k)= V 0 ω 2δ(ω ω k). (16) k=1 The idealized avity that we have onsidered gives rise to an η(ω) that essentially onsists of a finite sum of idealized line-shaped funtions of the usual form present when dissipation is negleted. A better model is obtained when we allow for some dissipation in the interation of the modes with the walls, or if no dissipation is onsidered, some broadening must still be present beause of the reasons indiated above. It is well-known that in many ases, if the dissipation is small (see, e.g. [15 17]), that the lineshape funtion is the so-alled Lorentzian lineshape funtion ω g(ω)dω = [ 2π (ω ω 0 ) 2 + k=1 ( ) ] 2 dω. (17) ω 2 The ubiquity of the Lorentzian lineshape omes from the fat that it originates in a dissipative exponential time-dependent deay (see [18]). The funtion g(ω) is alled the Lorentzian lineshape funtion and it is suh that g(ω)dω =1 (18) 0 and that lim g(ω) =δ(ω ω 0). (19) ω 0 The lineshape broadening parameter ω is due to various forms of dissipation and other broadening effets suh as those mentioned above and 4π/ ω is the exponential time onstant of deay for the relevant dissipative part of the proess. In the more realisti ase when line broadening around the modes harateristi frequenies is onsidered, ombining the inputs from (16) and (17) yields K(ω) V 0 η(ω) = π2 3 ω 2 N ω [ k ( ) ] 2 (20) 2π (ω ω k ) 2 ωk + 2 k=1 where, as explained above, the ω k is the harateristi frequeny of the k th mode and ω k, with ω k > 0, its orresponding line broadening. Only at the highest frequenies in the mirowave avity are the modes

10 488 A. Rueda and B. Haish: Gravity and the quantum vauum inertia hypothesis expeted to overlap beause the ω k broadenings beome omparable to the frequeny separations between the modes. All in all the orresponding ontribution of Eq. (20) to the inertial mass of a physial avity resonator when the walls are inluded is fairly small, but what is in priniple interesting is that it is not negligible! The inertial mass assoiated with the avity itself (without the mass of the walls) of Eq. (20) beomes m i = 1 2 = ω=0 ω=0 = 1 2 N k=1 π 2 ω 2 V 0 η(ω)ρ(ω)dω N ω [ k ( ) ] 2 2π (ω ω k ) 2 ωk + 2 k=1 ω=0 ω 3 2π 2 3dω ω [ k ( ) ] 2 ω dω (21) 2π (ω ω k ) 2 ωk whih is a rather interesting expression. We learly just add up the zero-point energies of all the resonant avity modes broadened by their orresponding Lorentzian broadening fators. Observe that in Eq. (21) the volume V 0 of the avity seemingly has disappeared. Of ourse that is just a superfiial remark as the volume as well as the geometry are determining fators in the spetrum of frequenies {ω k } distribution up to the plasma frequeny of the avity, ω p, with ω N ω p. The advantage for the general ase of a more general objet (or partile) is that V 0 is a parameter that in Eq. (21) has disappeared and been replaed by eletromagneti quantities like the ω k and ω k that, among other things, depend on the geometry. In the ase of a mirowave avity resonator mode with good onduting walls, relatively, the deay times of the modes are very long and onsequently the ω k parameters are very small with ω k ω k. This allows us to propose a simple experiment to measure the eletromagneti quantum vauum ontribution to inertial mass. If the ω k are indeed negligible for a given ondutor for all modes ω k up to the maximal one ω N, where ω N ω p, and where ω p is the plasma frequeny of the ondution-band eletrons in the metal of the walls, then the ontribution to inertial mass given by the avity is as given by Eq. (15), a quantity whih in priniple is readily alulable. This is partiularly so for a avity of retangular shape with nonommensurable a<b<in order to avoid mode degeneraies. Observe that the walls are made of a ondutor and their rystal struture. In turn we expet that several vauum fields beside the eletromagneti are going to make ontributions to the mass of the walls, but in the avity by itself the only ontribution an be that of Eq. (15) whih is purely eletromagneti. This suggests a simple experimental proedure in priniple. First, weigh in a preision balane the avity with its walls in ryogeni onditions, i.e. at T as low as possible. Then disassemble the avity and weigh its omponents. The first measurement under areful experimental onditions should yield a slightly larger mass than the seond one. This exess mass must be the one assoiated with the energy E in Eq. (15). This is of ourse an extremely idealized experiment. The experiment, if properly aomplished, would give interesting onfirmations. First, some onfirmation would be given to the eletromagneti vauum ontribution to inertial mass onept, or quantum vauum inertia hypothesis. An important onfirmation would be the reality or virtuality of the zero-point field. The reality of the zero-point field or quantum vauum would be unquestionably established. The diffiulty in the experiment is that even though there are many modes inside a avity resonator, the mass assoiated with Eq. (15) should be very small even for a large avity.

11 Ann. Phys. (Leipzig) 14, No. 8 (2005) / The physial basis of the priniple of equivalene We intend to show not only that the quantum vauum inertia hypothesis is onsistent with GR, but that it answers an outstanding question regarding a possible physial origin of the fore manifesting as weight [19]. We also intend to show that just as it beomes possible to identify a physial proess underlying the f=ma postulate of Newtonian mehanis (as well as its extension to SR [1]), it is possible to identify a parallel physial proess underlying the weak equivalene priniple, m i = m g. Within the standard theoretial framework of GR and related theories, the equality (or proportionality) of inertial mass to gravitational mass has to be assumed. It remains unexplained. As orretly stated by Rindler [20], the proportionality of inertial and gravitational mass for different materials is really a very mysterious fat. However here we show that at least within the present restrition to eletromagnetism the quantum vauum inertia hypothesis leads naturally and inevitably to this equality. The interation between the eletromagneti quantum vauum and the eletromagnetially-interating partiles onstituting any physial objet (quarks and eletrons) is idential for the two situations of (a) aeleration with respet to onstant veloity inertial frames or (b) remaining fixed above some gravitating body with respet to freely-falling loal inertial frames. A related theoretial launa involves the origin of the fore whih manifests itself as weight. Within GR theory one an only state that deviation from geodesi motion results in a fore whih must be an inertia reation fore. We propose that it is possible in priniple to identify a mehanism whih generates suh an inertia reation fore, and that in urved spaetime it ats in the same way as aeleration does in flat spaetime. Begin by onsidering a marosopi, massive, gravitating objet, W, whih is fixed in spae and whih for simpliity we assume to be solid, of onstant density, and spherial with a radius R, e.g. a planet-like objet. At a distane r R from the enter of W there is a small objet, w, that for our purposes we may regard as a point-like test partile. A onstant fore f is exerted by an external agent that prevents the small body w from falling into the gravitational potential of W and thereby maintains w at a fixed point in spae above the surfae of W. Experiene tells us that when the fore f is removed, w will instantaneously start to move toward W with an aeleration g and then ontinue freely falling toward W. Next we onsider a freely falling loal inertial frame I (in the ustomary sense given to suh a loal frame [21]) that is instantaneously at rest with respet to w.atw proper time τ, that we selet to be τ =0, objet w is instantaneously at rest at the point ( 2 /g, 0, 0) of the I frame. The x-axis of that frame goes in the diretion from W to w and, sine the frame is freely falling toward W,atτ =0objet w appears aelerated in I in the x-diretion and with an aeleration g w =ˆxg. As argued below, w is performing a uniformly-aelerated motion, i.e. a motion with a onstant proper aeleration g w as observed from any neighboring instantaneously omoving (loal) freely-falling inertial frame. In this respet we introdue an infinite olletion of loal inertial frames I τ, with axes parallel to those of I and with a ommon x-axis whih is that of I. Let w be instantaneously at rest and o-moving with the frame I τ at w proper time τ. So the τ parameter representing the w proper time also serves to parametrize this infinite olletion of (loal) inertial frames. Clearly then, I is the member of the olletion with τ =0, so that I = I τ=0.at the point in time of oinidene with a given I τ, w is found momentarly at rest at the ( 2 /g, 0, 0) point of the I τ frame. We selet also the times in the (loal) inertial frames to be t τ and suh that t τ =0at the moment of oinidene when w is instantaneously at rest in I τ and at the aforementioned ( 2 /g, 0, 0) point of I τ. Clearly as I τ=0 = I then t =0when τ =0. All the frames in the olletion are freely falling toward W, and when any one of them is instantaneously at rest with w it is instantaneously falling with aeleration g = g w = ˆxg with respet to w in the diretion of W. It is not diffiult to realize that w appears in those frames as uniformly aelerated and hene performing hyperboli motion with onstant proper aeleration g w. This situation is equivalent to that of an objet w aelerating with respet to an ensemble of I τ referene frames in the absene of gravity. In that situation, the onept of the ensemble of inertial frames, I τ, eah with an infinitesimally greater veloity (for the ase of positive aeleration) than the last, and eah oiniding

12 490 A. Rueda and B. Haish: Gravity and the quantum vauum inertia hypothesis instantaneously with an aelerating w is not diffiult to piture. But how does one piture the analogous ensemble for w held stationary with respet to a gravitating body? We are free to bring referene frames into existene at will. Imagine bringing a referene frame into existene at time τ =0diretly adjaent to w, but whereas w is fixed at a speifi point above W,we let the newly reated referene frame immediately begin free-falling toward W. We immediately reate a replaement referene frame diretly adjaent to w and let it drop, and so on. The ensemble of freely-falling loal inertial frames bear the same relation to w and to eah other as do the extended I τ inertial frames used in the ase of true aeleration of w [1,2]. For onveniene we introdue a speial frame of referene, S, whose x-axis oinides with those of the I τ frames, inluding of ourse I, and whose y-axis and z-axis are parallel to those of I τ and I. This frame S stays olloated with w whih is positioned at the ( 2 /g, 0, 0) point of the S frame. For I (and for the I τ frames) the frame S appears as aelerated with the uniform aeleration g w of its point ( 2 /g, 0, 0). We will assume that the frame S is rigid. If so, the aelerations of points of S suffiiently separated from the w point ( 2 /g, 0, 0) are not going to be the same as that of ( 2 /g, 0, 0). This is not a onern however sine we will only need in all frames (I τ, I and S) to onsider points in a suffiiently small neighborhood of the ( 2 /g, 0, 0) point of eah frame. The olletion of frames, I τ, as well as I and S, orrespond exatly to the set of frames introdued in [1]. The only differenes are, first, that now they are all loal, in the sense that they are only well defined for regions in the neighborhood of their respetive ( 2 /g, 0, 0) spae points; and seond, that now I and the I τ frames are all onsidered to be freely falling toward W and the S frame is fixed with respet to W. Similarly to [1], the S frame may again be onsidered to be, relative to the viewpoint of I,aRindler noninertial frame. The laboratory frame I we now all the Einstein laboratory frame, sine now the laboratory is loal and freely falling. We all the olletion of inertial frames I τ the Boyer family of frames as he was the first to introdue them in SED analysis of the Unruh-Davies effet [21]. The relativity priniple as formulated by Einstein when proposing SR states that all inertial frames are totally equivalent for the performane of all physial experiments. [11] Before applying this priniple to the freely-falling frames I and I τ that we have defined above it is neessary to draw a distintion between these frames and inertial frames that are far away from any gravitating body, suh as W. The free-fall trajetories, i.e. the geodesis, in the viinity of any gravitating body, W, annot be parallel over any arbitrary distane owing to the fat that W must be of finite size. This means that the priniple of relativity an only be applied loally. This was preisely the limitation that Einstein had to put on his infinitely-extended Lorentz inertial frames of SR when starting to onstrut GR [11,20,22]. We adopt the priniple of loal Lorentz invariane (LLI) whih an be stated, following Will [11], as the outome of a loal nongravitational test experiment is independent of the veloity of the freely falling apparatus. A non-gravitational test experiment is one for whih self-gravitating effets an be negleted. We also adopt the assumption of spae and time uniformity, whih we all the uniformity assumption (UA) and whih states that the laws of physis are the same at any time or plae within the Universe. Again, following Will [11] this an be stated as the outome of any loal nongravitational test experiment is independent of where and when in the universe it is performed. We do not onern ourselves with physial or osmologial theories that in one way or another violate UA, e.g. beause they involve spatial or temporal hanges in fundamental onstants [23]. Loally, the freely falling loal Lorentz frames whih we now designate with a subsript L i.e. I,L and I τ,l are entirely equivalent to the I and I τ extended frames of [1]. The free-falling Lorentz frame I τ,l loally is exatly the same as the extended I τ. Invoking the LLI priniple we an then immediately onlude that the eletromagneti zero-point field, or eletromagneti quantum vauum, that an be assoiated with I τ,l must be the same as that assoiated with I τ. From the viewpoint of the loal Lorentz frames I,L and I τ,l the body w is undergoing uniform aeleration and therefore for the same reasons as presented in [1] a peuliar aeleration-dependent fore arises, that for onreteness we shall all a drag fore. These formal arguments demonstrate that the analyses of [1] whih found the existene of a Rindler frame fore in an aelerating referene frame translate and orrespond exatly to a referene frame fixed

13 Ann. Phys. (Leipzig) 14, No. 8 (2005) / above a gravitating body. In the same manner that light rays are deviated from straight-line propagation by a massive gravitating body W, the other forms of eletromagneti radiation, inluding the eletromagneti zero-point field rays (in the SED approximation), are also deviated from straight-line propagation. Not surprisingly this reates an anisotropy in the otherwise isotropi eletromagneti quantum vauum. In [1] we interpreted the drag fore exerted by the eletromagneti quantum vauum radiation as the inertia reation fore of an objet that is being fored to aelerate through the eletromagneti quantum vauum field. Aordingly, in the present situation, the assoiated nonrelativisti form of the inertia reation fore should be f zp = m i g w (22) where g w is the aeleration with whih w appears in the loal inertial frame I. As shown in [1] the oeffiient m i is [ ] V0 m i = 2 η(ω) ω3 2π 2 3dω (23) where V 0 is the proper volume of the objet, is the speed of light, is Plank s onstant divided by 2π and η(ω), where 0 η(ω) 1, is a funtion that spetralwise represents the relative strength of the interation between the zero-point field and the massive objet, interation whih ats to oppose the aeleration. If the objet is just a single partile, the spetral profile of η(ω) will haraterize the eletromagnetiallyinterating partile. It an also haraterize a muh more extended objet, i.e. a marosopi objet, but then the η(ω) will have muh more struture (in frequeny). We should expet different shapes for the eletron, a given quark, a omposite partile like the proton, a moleule, a homogeneous dust grain or a homogeneous marosopi body. In the last ase the η(ω) beomes a ompliated spetral opaity funtion that must extend to extremely high frequenies suh as those haraterizing the Compton frequeny of the eletron and even beyond. Now, however, what appears as inertial mass, m i, to the observer in the loal I,L frame is of ourse what orresponds to passive gravitational mass, m g, and it must therefore be the ase that [ ] V0 m g = 2 η(ω) ω3 2π 2 3dω. (24) As done in [1], Appendix B, it an be shown that the right hand side indeed represents the energy of the eletromagneti quantum vauum enlosed within the objet s volume and able to interat with the objet as manifested by the η(ω) oupling funtion. A more thorough, fully ovariant development an also be implemented to show that the fore expression of Eq. (1) an be extended to the relativisti form of the inertia reation fore as in [1], Appendix D. (This development also served to obtain the final form of m i given above in Eq. (23) eliminating a spurious 4/3 fator.) 2 Summarizing what we have shown in this setion is that if a fore f is applied to the w body just large enough to prevent it from falling toward the body W, then in the non-relativisti ase that fore is given by f = mg (25) where we have dropped the nonessential subsripts i and g and supersripts, beause it is now lear that m i = m g = m follows from the quantum vauum inertia hypothesis. The physial basis for the priniple of equivalene is the fat that aelerating through the eletromagneti quantum vauum is idential to remaining fixed in a gravitational field and having the eletromagneti 2 We use this opportunity to orret a minor transription error that appeared in the printed version of [1]. In Appendix D, p. 1100, the minus sign in Eq. (D8) is wrong. It should read η ν η ν =1 (D8) and the orresponding signature signs in the line just above Eq. (D8) are the opposite of what was written and should instead read (+ ).

14 492 A. Rueda and B. Haish: Gravity and the quantum vauum inertia hypothesis quantum vauum fall past on urved geodesis. In both situations the observer will experiene an asymmetry in the radiation pattern of the eletromagneti quantum vauum whih results in a fore either the inertia reation fore or weight whih beome then within this more general Einsteinian perspetive the same thing. 5 Consisteny with Einstein s general relativity The statement that m i = m g onstitutes the weak equivalene priniple (WEP). Its origin goes bak to Galileo and Newton, but it now appears, as shown in the previous setion, that this priniple is a natural onsequene of the quantum vauum inertia hypothesis. The strong equivalene priniple (SEP) of Einstein onsists of the WEP together with LLI and the UA. Sine the quantum vauum inertia hypothesis and its extension to gravity allow us to obtain the WEP assuming LLI and the UA, this approah is onsistent with all theories that are derived from the SEP. In addition to GR, the Brans-Dike theory is derived from SEP as are other lesser known theories [11], all distinguished from eah other by various partiular assumptions. All theories that assume the SEP are alled metri theories. They are haraterized by the fat that they ontemplate a bending of spaetime assoiated with the presene of matter. Two important onsequenes of the LLI-WEP-UA ombination are that light bends in the presene of matter and that there is a gravitational Doppler shift. Sine the quantum vauum inertia hypothesis is onsistent with this same ombination, it would also require that light bends in the presene of gravitational fields. This an, of ourse, be interpreted as a hange in spaetime geometry, the standard interpretation of GR. However this does not mean that we have explained the mehanism for the atual bending of spae-time in the viinity of a material objet. This is the origin of so-alled ative gravitational mass that still requires an explanation within the viewpoint of the quantum vauum inertia hypothesis. 6 A reent alternative vauum approah to gravity The idea that the vauum is ultimately responsible for gravitation is not new. It goes bak to a proposal of Sakharov [24] based on the work of Zeldovih [25] in whih a onnetion is drawn between Hilbert-Einstein ation and the quantum vauum. This leads to a view of gravity as a metri elastiity of spae (see Misner, Thorne and Wheeler [26] for a suint review of this onept). Following Sakharov s idea and using the tehniques of SED, Puthoff proposed that gravity ould be onstrued as a form of van der Waals fore [27]. Although interesting and stimulating in some respets, Puthoff s attempt to derive a Newtonian inverse square fore of gravity proves to be unsuessful [28 31]. An alternative approah has reently been developed by Puthoff [32] that is based on earlier work of Dike [33] and of Wilson [34]: a polarizable vauum model of gravitation. In this representation, gravitation omes from an effet by massive bodies on both the permittivity, ɛ 0, and the permeability, µ 0, of the vauum and thus on the veloity of light in the presene of matter. That is learly an alternative theory to GR sine it does not involve urvature of spaetime. On the other hand, sine spaetime urvature is by definition inferred from light propagation in relativity theory, the polarizable vauum gravitation model may be labelled a pseudo-metri theory of gravitation sine the effet of variation in the dieletri properties of the vauum by massive objets on light propagation are approximately equivalent to GR spaetime urvature as long as the fields are suffiiently weak. In the weak-field limit, the polarizable vauum model of gravitation dupliates the results of GR, inluding the lassi tests (gravitational redshift, bending of light near the Sun, advane of the perihelion of Merury). Differenes appear in the strong-field regime, whih should lead to interesting tests.

15 Ann. Phys. (Leipzig) 14, No. 8 (2005) / Derivation of Newton s law of gravitation Our approah allows us to derive a Newtonian form of gravitation in the weak-field limit based on the quantum vauum inertia hypothesis, loal Lorentz invariane and geometrial onsiderations. In [1] we showed how an asymmetry that appears in the radiation pattern of the eletromagneti quantum vauum, when viewed from an aelerating referene frame, leads to the appearane of an apparent non-zero momentum flux. Individual and olletive interation between the eletromagnetially-interating partiles (quarks and eletrons) omprising a material objet and the eletromagneti quantum vauum generates a reation fore that may be identified with the eletromagneti ontribution to the inertia reation fore as it has the right form for all veloity regimes. In partiular in the low veloity limit it is exatly proportional to the aeleration of the objet. In Set. 4 we showed from formal arguments based on the priniple of loal Lorentz invariane (LLI) that an exatly equivalent fore must originate when an objet is effetively aelerated with respet to free-falling Lorentz frames by virtue of being held fixed above a gravitating body, W. We infer that the presene of a gravitating body must distort the eletromagneti quantum vauum in exatly the same way at any given point as would the proess of aeleration suh that a = g. Simple geometrial arguments [35] now suffie to show that the gravitational fore an only be the Newton inverse-square law with distane (in the weak field limit). It has been shown above that (outside W ) the g field of Eq. (25) generated by W is entral, i.e., entrally distributed with spherial symmetry around W. It has to be radial, with its vetorial diretion parallel to the orresponding radius vetor r originating at the enter of mass of W where we loate the origin of oordinates. The spherial symmetry implies that g is radial in the diretion ˆr, and depends only on the r-oordinate, g = ˆrg(r). The field is learly generated by mass. A simple symmetry argument based on Newton s Third Law, here omitted for brevity, shows that indeed if we sale the mass M of W by a fator α to beome αm then the resulting g must be of the form αg. IfM goes to zero, g disappears. The mass M must be the soure of field lines of g, and these field lines an be disontinuous only where mass is present. The field lines an be neither generated nor destroyed in free spae. Sine g is the fore on a unit mass, we must expet that g behaves as a vetor, and speifially that g follows the laws of vetor addition. Namely, if two masses M 1 and M 2 in the viinity of eah other generate fields g 1 and g 2 respetively, the resulting g at any given point in spae should be the linear superposition of the g 1 and g 2 fields at that given point, namely their vetor sum g = g 1 + g 2. (26) Finally, from the argument that leads to Eq. (25) we an see that the field g must be unbounded, extending essentially to infinity. With all of these onsiderations, learly the lines of g must obey the ontinuity property outside W.If there is no mass present inside a volume, V, enlosed by a surfae S, we expet, using Gauss divergene theorem, that 0= g n ds = g dv, (27) S(V ) but sine V is arbitrary this tells us that outside the massive body W V g =0, r > R. (28) In the presene of our single, spherially symmetri massive body, W, but outside that body, g is radial with respet to the enter of W. Hene we must have g =ˆrΦ(r), and therefore from Eq. (28), beause Φ is only a funtion of r, it neessarily follows that g is of the form g ˆr 1 r2, (29)

16 494 A. Rueda and B. Haish: Gravity and the quantum vauum inertia hypothesis where the restrition r>ris hereafter understood. Sine the field g is also proportional to the mass M whih is its origin, we onlude that g must be of the form g = ˆr GM r 2, (30) where G is a proportionality onstant and from Eq. (17) we have that F = ˆr GMm r 2, (31) whih is Newton s law of gravitation. It is remarkable that after finding the entral and radial harater of g by means of the vauum approah of our quantum vauum inertia hypothesis, one an immediately obtain Newtonian gravitation, an endeavour keenly but unsuessfully pursued for quite some time from the viewpoint of the vauum fields [24 26] and in partiular of SED [27 31]. In this last ase (SED), it was proposed that gravity was a fore of the van der Waals form, a view whih has been shown to be unsuessful [31]. As a final point here we observe that when the two bodies of masses M and m are redued to point partiles, a simple argument based on the symmetry of the situation shows that the two roles of the masses m and M (passive and ative) an be interhanged and that similarly the fore on M due to m should be equal in magnitude but opposite in diretion to that of Eq. (22) so that Newton s third law of mehanis is satisfied. 8 On the origin of weight We have established that the quantum vauum inertia hypothesis leads to a fore in Eq. (22) whih adequately explains the origin of weight in a Newtonian view of gravitation. How is this onsistent with the geometrodynami view of GR? Geometrodynamis speifies the effet of matter and energy on an assumed pliable spaetime metri. That defines the geodesis whih light rays and freely-falling objets will follow. However there is nothing in geometrodynamis that points to the origin of the inertia reation fore when geodesi motion is prevented, whih manifests in speial irumstanes as weight. Geometrodynamis merely assumes that deviation from geodesi motion results in inertial fores. That is, in fat, true, but as it stands it is devoid of any physial insight. What we have shown above is that an idential asymmetry in the quantum vauum radiation pattern will arise due to either true aeleration or to effets on light propagation by the presene of gravitating matter. In the ase of true aeleration, the resulting fore is the inertia reation fore. In the ase of being held stationary in a non-minkowski metri, the resulting fore is the weight, whih is also the enforer of geodesi motion for freely-falling objets. 9 A phantom anomaly Following the reasoning of Set. 7 one would onlude that at every point of fixed r above W there is an inflowing energy-momentum flux from the falling eletromagneti quantum vauum. This would seem to imply a ontinuous energy flux toward and into W, an apparently paradoxial situation onsisting of quantum vauum-originating energy streaming toward W from all diretions in spae. But this paradox is only apparent, as we show. It is true that for a freely falling observer attahed to a loal inertial frame I τ,l whih is falling past the surfae of the Earth and towards its enter ( 4), the objet w, that we take now to be loated on the surfae of the Earth, appears to be uniformly aelerated with onstant proper aeleration g w = g. Thus this observer onludes as he freely falls that the mass of w must grow as γ τ m, where m is the rest mass of w and γ τ is the Lorentz fator, (1 β 2 τ ) 1/2. Conommitantly, this observer of I τ,l, in light of our disussion in App. B of [1] and beause of the SEP, asserts that the energy ontent of w, whih for simpliity we take here as purely eletromagneti and

17 Ann. Phys. (Leipzig) 14, No. 8 (2005) / oming from the eletromagneti vauum, is steadily inreasing. Eqn. (33) of [1] gives an energy growth rate de/dt = f v, where f in the present ase orresponds to the fore of support against the Earth s gravity exerted on w by the surfae of the Earth. Clearly then this freely falling observer of I τ onludes that objets on the surfae of the Earth are steadily inreasing their internal energy at the expense of the falling vauum, namely of the eletromagneti zero-point field of I τ, that freely falls with I τ. However when exatly the same onsideration is made but now from the point of view of an observer fixed on the surfae of the Earth and in the neighborhood of w, the onlusion is quite different. In this ase the observer is attahed to S, the loal Rindler non-inertial frame where w is fixed and that oinides with the surfae of the Earth around the spot where w is loated. To alulate de/dt as seen from S we use the loal instantaneously omoving, freely-falling inertial frame I,L = I τ=0,l whih at that point in time o-moves with w. In that frame I,L the partile veloity is of ourse zero, and therefore the rate of energy growth is de/dt =0. So the observer in S onludes that w does not hange its energy ontent. The above shows that for an observer on the surfae of the Earth there is no real transfer of energy from the vauum to objets on the surfae of the Earth. In this respet the energy growth above, that a freely falling observer sees as time progresses, is purely a kinematial effet that omes form the fat that in the freely-falling frame the veloity of the objet w is growing with time and does not vanish exept at the instant of oinidene of I,L with S, atτ =0. In speial relativity, a onstantly aelerating spaeship is said by an external observer to be inreasing its mass steadily as γ(τ)m, but of ourse an observer onboard that raft will not pereive any relativisti mass inrease. On the other hand as seen in S, for the frame at the surfae of the Earth, even though de/dt vanishes, the fore f does not vanish, as shown in [1]. The fore f is needed to ompensate the fore f zp, i.e. f zp = f, exerted on w by the freely-falling vauum of I τ, that indeed orresponds to what we alled above the Rindler frame fore. 10 Disussion In light of what has been proposed herein, what an we eluidate about the nature of the gravitational field? In the low fields and low veloities version, or the Newtonian limit, we have seen above that gravity manifests itself as the attrative fore per unit mass, g, of Eq. (30) that pulls any massive test body present at a given point in spae towards the body W that originates the field. Sine we assumed the Einstein LLI priniple and from this derived the WEP, this, together with the very natural UA of invariane in the laws of physis throughout universal spaetime, leads us to the Einstein SEP whih neessarily implies the spaetime bending representation of the generalized gravitational field [11,20]. A simple thought experiment (Einstein s lift) immediately shows that light rays propagate along geodesis, and more speifially along null geodesis [20,26,36]. The spaetime bending is dramatially evident when a light ray goes from one side to the other of the freely falling elevator. For the observer attahed to the elevator s frame that indeed ats as a LLI frame, the light ray propagates in a straight line from one side to the other of the elevator. But for the stationary observer who sees the elevator falling with aeleration g, the light ray bends along a path that loally is seen as a paraboli urve. Undoubtedly the most natural explanation for the stationary observer is that spaetime bends and therefore the assoiation that this bending is a manifestation of the gravitational field of W, or rather that this bending of spaetime is the gravitational field itself [36]. Starting from the above fat taken as a given, the various metri theories proeed from there to formulate their equations. In the version of Brans and Dike a salar-tensor field is assumed. In Einstein s GR only a tensor field is proposed. Following this maximally simplisti proposal and guided by general onsiderations of general ovariane (the need of arbitrary oordinates and tensor laws) Einstein was led to his field equations in the presene of matter... and then GR naturally unfolded [11,20,26,36].

18 496 A. Rueda and B. Haish: Gravity and the quantum vauum inertia hypothesis We have shown here that our inertia proposal of [1] leads us, when limited by the LLI priniple, to the metri theories and therefore that it is onsistent with those theories and in partiular with Einstein s GR. In addition, there is the following interesting feature of our proposal. From our analysis in [1], and in partiular in Appendix B of [1], it was made lear that within the quantum vauum inertia hypothesis proposed therein, the mass of the objet, m, ould be viewed as the energy in the equivalent eletromagneti quantum vauum field aptured within the struture of the objet and that readily interats with the objet. This view properly and aurately mathed with the omplementary view, exposed in other parts of the paper [1], that presented inertia as the result of a vauum reation effet, a kind of drag fore exerted by the vauum field on aelerated objets. Quantitatively, both approahes lead to exatly the same inertial mass and moreover they are partly omplementary. Both viewpoints are needed. One ould not exist without the other. They were the two sides of the same oin. The question is now why massive objets, when freely falling, also follow geodesi paths. The tempting view suggested here is that, as massive bodies have a mass that is made of the vauum eletromagneti energy ontained within their struture and that readily interats with suh struture (aording to our analysis of [1], Appendix B), it is no surprise that geodesis are their natural path of motion during free fall. Eletromagneti radiation has been shown by Einstein to follow preisely geodesi paths. The only differene now is that, as the radiation stays within the aelerated body struture and is ontained within that struture and thereby its energy enter moves subrelativistially, these geodesis are just time-like and not null ones as in the ase of freely propagating light rays. We illustrate this with an example. Imagine a freely-falling eletromagneti avity with perfetly refleting walls of negligible weight, so that all the weight is due to the enlosed radiation. A simple plane wave mode deomposition shows that although individual wavetrains do still move at the speed of light, the enter of energy of the radiation inside the avity moves subrelativistially as the wavetrains reflet bak and forth. The wavetrains do indeed propagate along null geodesis, but the enter of energy propagates only along a time-like geodesi. Neither our approah nor the onventional presentations of GR for that matter, an offer a physial explanation of the mehanism of the bending of spaetime as related to energy density. Misner, Thorne and Wheeler [26] present six different proposed explanations. The sixth is the one we already mentioned due to Sakharov [24, 25] whih starts from general vauum onsiderations. As our approah starts also from vauum onsiderations, it naturally fits better the onept of the onjeture of Sakharov [24] and Zeldovih [25] than the other proposals but it is not inonsistent with any of them. In partiular the stritly formal proposal of Hilbert [26] that introdues the so-alled Einstein-Hilbert Ation, is also at the origin of the Sakharov proposal. We plan to devote more work to exploring the onnetion of our inertia [1] and gravity approah to the approah proposed by Sakharov [24]. This we leave for a future publiation. 11 Conlusions The prinipal onlusions of this paper are: (1) Identity of inertial mass with gravitational mass, m i = m g naturally follows from the quantum vauum inertia hypthesis. It has been shown that the approah of [1] ontains this peuliar feature that, so far and as we know, has never been explained. Rindler [20] alls this feature a very mysterious fat, as indeed it has been up to the present. We expet with this work to have shed some light on this peuliar feature. (2) The quantum vauum inertia hypothesis is onsistent with Einstein s GR. We have already ommented above, in partiular in Set. 8, on this interesting feature presented in Set. 5 that puts the vauum inertia approah of [1] within the mainstream thought of ontemporary gravitational theories, speifially within that of theories of the metri type and in partiular in agreement with GR. (3) Newton s gravitational law naturally follows from the quantum vauum inertia hypothesis. By means of a simple argument based on potential theory we show how to obtain in a natural way Newton s inverse

19 Ann. Phys. (Leipzig) 14, No. 8 (2005) / square fore with distane from our vauum approah to inertia of [1]. The simpliity of our approah ontrasts with previous attempts to aomplish this within the framework of SED theory. (4) An origin of weight and a physial mehanism to enfore motion along geodesi trajetories for freely-falling objets is ontained in the hypothesis. We have shown how this approah to inertia answers a fundamental question left open within GR, viz. is there a physial mehanism that generates the inertia reation fore when non-geodesi motion is imposed on an objet and whih an manifest speifially as weight. Or put another way, while geometrodynamis ditates the spaetime metri and thus speifies geodesis, is there an identifiable mehanism for enforing motion along geodesi trajetories? The quantum vauum inertia hypothesis represents a signifiant first step in providing suh a mehanism. (5) A physial basis is provided for the relativisti mass inrease. In the approah of the quantum vauum inertia hypothesis, it beomes physially evident how the Lorentz fator γ τ, whih haraterizes a spae-time geometry relationship, aquires the physial role of relativisti mass inrease parameter. It also beomes lear why the relativisti mass inrease must beome infinite at : One annot propagate the effet of any fores originating in zero-point field effets at speeds faster than. (6) The interior (exluding the walls) of a avity resonator is presented as the arhetype of a system wherein only the eletromagneti vauum ontributes to the mass. This allows an expression for a resulting inertial mass with no free parameters. This mass in priniple an be measured. (7) An experimental predition has been made that the mass of the resonant eletromagneti zero-point field modes within a avity should add to the mass of the avity struture. Aknowledgements AR reeived partial support from the California Institute for Physis and Astrophysis via a grant to Cal. State Univ., Long Beah and via release time from the Offie of Aademi Affairs, CSULB. This work was partially funded under NASA ontrat NASW Referenes [1] A. Rueda and B. Haish, Found. Phys. 28, 1057 (1998); see also A. Rueda and B. Haish, Phys. Lett. A 240, 115 (1998). [2] B. Haish, A. Rueda, and Y. Dobyns, Ann. Phys. (Leipzig) 10, 393 (2001). [3] B. Haish, A. Rueda, and H. E. Puthoff, Phys. Rev. A 48, 678 (1994). [4] R. Loudon, The Quantum Theory of Light (2nd ed.) (Clarendon Press, Oxford, 1983), p. 5. [5] H. Sunahata and A. Rueda, in preparation (2005). [6] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003) (in partiular Set. IIB). [7] B. Haish and A. Rueda, Phys. Lett. A 268, 224 (2000). [8] T. H. Boyer, Phys. Rev. D 11, 790 (1975). [9] P. W. Milonni, The Quantum Vauum: An Introdution to Quantum Eletrodynamis, Set. 8.2 (Aademi Press, New York, 1994). [10] L. de La Peña and A. M. Cetto, The Quantum Die An Introdution to Stohasti Eletrodynamis. Aad. Publ., Fundamental Theories of Physis Series (Kluwer, Dordreht, Holland, 1996) and referenes therein. [11] C. W. Will, Theory and Experiment in Gravitational Physis (Cambridge University Press, Cambridge, 1993), pp [12] H. Weyl, J. Math. 143, 177; H. Weyl, R. Cir. Mat. Palermo 39, 1 (1913). [13] M. Ibison and B. Haish, Phys. Rev. A 54, 2737 (1996). [14] J. D. Jakson, Classial Eletrodynamis (3rd ed.) (Wiley and Sons, New York, 1999), p [15] A. Yariv, Quantum Eletronis (Wiley, New York, 1967). [16] B. Saleh and M. C. Teih, Fundamentals of Photonis (Wiley, New York, 1991). [17] E.A. Hinds, Perturbative Cavity Quantum Eletrodynamis, in: Cavity Quantum Eletrodynamis, edited by P. R. Berman (Aademi Press, New York, 1994) pp [18] W. H. Louisell, Quantum Statistial Properties of Radiation (Wiley, New York, 1975). [19] Y. Dobyns, A. Rueda, and B. Haish, Found. Phys. 30 (1), 59 (2000). [20] W. Rindler, Essential Relativity Speial, General and Cosmologial (Springer Verlag, Heidelberg, 1977), p. 17.

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