Chaos, Solirons

& Fractals Vol. 7, No. 8, pp. 1261-1303, 19% Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 096&0779/96 $15.00 + 0.00

0960-0779(95)00101-8

Fluctuons - III. Gravity MICHAEL

CONRAD

Department of Computer Science, Wayne State University, Detroit, MI 48202, USA (Accepted 29 November 1995)

Abstract -The fluctuon principle provides a model of gravity. Mass and field are interpreted in terms

of density depression in the full sea of vacuum fermions. Gravitons are interpreted as self-perpetuating chains of transient fermion-antifermion pairs (propagating gravitational fluctuons) and gravitational waves as compression-expansion waves of vacuum density. The direct gravitational interaction between absorbing particles is repulsive (whether the absorbers are permanent positive energy particles or trapped fluctuons). The attractive gravitational interaction arises from the ‘pushing’ effect of propagating gravitational fluctuons originating in the low mass regions of space, where the density of trapped fluctuons is highest. The model implies the principle of equivalence and through this the general relativistic equations of gravitation. Copyright 0 1996 Elsevier Science Ltd

1. INTRODUCTION

Fluctuons, as described in Parts I and II of this series [l, 21, are chains of short-lived pair creation-annihilation events that propagate in a sea of unmanifest vacuum fermions. Virtual photons, for example, are interpreted as chains of transient electron-positron pairs that carry a quantity of momentum between electrically charged absorbers that depends specifically on the density of vacuum electrons (i.e. on the density of potentialities for electron-positron pair production). Real photons are chains that carry real energy. Such chains are, in some respects, reminiscent of solitons, since they serve to transmit solitary excitations that carry momentum and energy (either virtual or real) without dissipation. The mechanism of origin and persistence is different from that of conventional solitons however. The fluctuon originates as an uncertainty fluctuation in the neighborhood of an absorbing particle. It undergoes a persistent skipping (or hopping) motion in free space since it cannot decay and simultaneously satisfy conservation of energy and conservation of momentum. The term ‘fluctuon’ captures the feature that the excitation is a fluctuation that must regenerate itself in a frustrated manner until it transmits its momentum to a second absorber. The absorber may be another fluctuon, subject to the restriction that mutual annihilation of propagating fluctuons is forbidden. The fluctuon-fluctuon interaction is thus a further feature that distinguishes fluctuons from conventional solitons. The electromagnetic interaction is mediated by chains that propagate in the subsea of vacuum electrons, while the gravitational interaction is mediated by the full sea of vacuum fermions. Gravitons are interpreted as chains of transient fermion-antifermion pairs propagating in this full sea. Particle masses are associated with lacunae in the sea of vacuum particles, either because they represent holes occupied by positive energy objects or holes per se. Field is associated with depression of vacuum density induced by mass or charge. As previewed in Part II, the direct gravitational interaction between absorbers, like the electromagnetic interaction, is most naturally treated as repulsive. The net attractive interaction between two masses is then due to their being pushed together by gravitons emanating from the high vacuum density (low mass) regions of space. The Newtonian law 1261

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of gravitational attraction follows from this global picture, provided that the effect of mass on distance as measured by fluctuation lengths is ignored. The general relativistic description of cavitation also follows. The analysis comprises three subarguments. The first is that the density structure of the vacuum is stable in the sense that it changes only in response to motions of positive energy particles-in contrast to systems of positive energy particles there is no tendency for depressions in the vacuum density to disappear. The second subargument is that inertia may be interpreted in terms of the same chain propagation interactions as gravitation, with the asymmetry of fluctuon bombar~ents arising from depressions in vacuum density induced by the motions of positive energy particles. The third step is to show, through a perpetual mobile argument, that the gravitational and inertial mass must be equal in the fluctuon model (i.e. the principle of equivalence is an inherent feature). From this it follows that the law of gravitation implicit in the fluctuon model must correspond to the metric tensor description of general relativity. However, the geometrical inte~retation of gravity in terms of space curvature is replaced by an isomorphic interpretation in terms of vacuum density structure. Mass has a different dependence on vacuum density when defined by the fluctuation energy required for pair production that when defined by the gravitational interaction. This feature, which corresponds to the effect of space curvature, imposes severe self-corrective constraints on the relation between particle mass and vacuum density. These constraints are complicated by the fact that the subsea of electrically charged fermions (electrons) must behave in a manner consistent with the much more dense sea of uncharged fermions (to be called ‘massons’) in which they reside. Strictly speaking the model requires the local properties of the vacuum and the properties of the universe in the large to evolve to mutual consistency. This aspect of the model will be treated, so far as details are concerned, in Part IV of this series. For the present purposes it is sufficient to picture a sea of vacuum massons with properties analogous to the Dirac sea of vacuum electrons, but with the provision that the mass assigned to the vacuum particles is a formal property inherited from their trapped fluctuon mode of existence.

2, GRADATIONS

MODELS

Let us begin by noting some of the considerations that suggest that gravity emerges from a direct repulsive interaction. The main one ,is that as mass increases, the density of vacuum particles in the surrounding region decreases [cf. equation (2), Part II]. In the direct repulsion model this is due to the fact that ordinary positive energy particles are continual repellers, whereas vacuum particles act as repellers only in their trapped fluctuon mode of existence. Ordinary particles here include nonfluctuonic (‘permanent’) fermions, bosons constructed from nonfluctuonic fermions (such as the electron pairs in a superconductor) and fluctuonic bosons (such as real and virtual photons). The decreases in vacuum density in the regions surrounding such particles must be compensated by elevations in vacuum density in distant regions of space. The attractive gravitational interaction between them is due to their being pushed together by propagating ~a~tational fluctuons emanating from the high vacuum density (low mass) regions of space. Consider two ordinary masses. Screening of each mass by the other plays an important role. Two screening effects are operative (see Fig. 1). First, each mass screens the repulsive effect of the other on trapped fluctuons in the space external to both of them, thereby distorting the density depression structure in these regions. The density structure will be relatively less depressed along the line passing through both masses and external to both of them. Second, each mass screens the other from bombardments by propagating fluctuons

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lower density Fig. 1. Double screening effect. Propagating fluctuons are screened, but the movement of trapped fluctuons is not. Thus the latter are pumped out of the region between the two masses, yielding a relatively higher vacuum density in external regions of space. Propagating gravitational fluctuons (gravitons) impinging on the masses from these regions push them toward each other, thereby turning the direct repulsive gravitational interaction into a net attractive interaction. Gravitons initiated by trapped fluctuons repelled into the external regions of space would also have a density decreasing effect on these regions, but since this effect is delayed the whole system is always out of equilibrium.

emanating from trapped fluctuons residing in this external direction of space. The pushing effect is stronger than the repulsion because the density of trapped fluctuons is always less in the region between the two masses than in the external regions, due to the fact that the effect of propagating fluctuons on trapped fluctuons in the external regions is always delayed relative to their effect on the trapped fluctuons in the space between the masses. It is also possible to construct models with an inverse square attractive interaction, but these require mixing the attractive interaction with a repulsive interaction. Thus suppose that the trapped fluctuons exert an attractive inverse square force on each other. Positive energy masses, since they are associated with or correspond to holes in the vacuum density, would then appear to repel trapped fluctuons. Actually this apparent repulsion would be due to the trapped fluctuons in the neighborhood of the mass being attracted by trapped fluctuons in the low mass regions of space, where the density of trapped fluctuons is higher. The attractive interaction between positive energy masses would then be due to their being pushed together by propagating gravitational fluctuons emanating from the high vacuum density regions of space, as in the direct repulsion model. The repulsive effect of trapped gravitational fluctuons on positive energy mass is equivalent to the attraction of a vacuum particle, since the positive energy mass is associated with a hole in the vacuum density. However, the repulsive effect is still necessary, since vacuum particles are not absorbers. The assumption that positive energy masses directly attract each other similarly requires a mixed attraction-repulsion model, since positive energy masses would then be required to repel trapped fluctuons in order to induce a surrounding vacuum depression. The pure repulsion model is internally consistent, whereas the models involving attraction are inconsistent (in particular mixing attraction and repulsion entails inconsistent spin requirements). It is the only model that need be kept in mind in what follows. However, the thermodynamic version of the argument will actually depend in the main on two features shared by both the pure repulsion model and the trapped fluctuon attraction model: the depressing effect of mass on surrounding vacuum density and the pushing together of positive energy masses by propagating gravitational fluctuons emanating from low mass regions of space. Propagating gravitational fluctuons play the role of gravitons in the fluctuon model. Gravitons are ordinarily supposed to be spin 2 particles, in accordance with the tensor character of the gravitational field in general relativity. In the fluctuon model they are more naturally viewed as spin 1 particles, since it is awkward to build spin 2 bosons out of fermion-antifermion pairs (e.g. the fluctuon would have to be a more complex or asymmetrical composite). Spin 1 gravitons built from spin l/2 vacuum fermions (‘massonantimasson’ pairs) fit most naturally to the tensor character of the gravitational field in the fluctuon model since this field arises indirectly from direct inverse square repulsions between all absorbing particles in the universe. Spin 0 bosons could also be built from vacuum fermions, but would more naturally be associated with attractive interactions.

M. CONRAD

1264 3. VIRTUAL

WORK POSTULATE

Three types of interactions are pertinent to the direct repulsion model: direct repulsive interactions between ordinary masses, direct repulsive interactions between ordinary masses and trapped fluctuons, and direct repulsive interactions between trapped fluctuons. Screening effects are also important. The force expressions required to describe these interactions have been developed in Parts I and II [e.g. equation (17), Part I and equation (l), Part II]. The purpose of this and the following section is to rederive the relevant formulas using a postulational approach [3] and, more importantly, to put them into a form that is pertinent to the analysis of gravity (i.e. that accommodates the relativity of distance, time, and mass). The postulate, to be called the virtual work assumption is: the uncertainty energy GE, required of a virtual momentum carrier in order for its position uncertainty to be ox increases linearly with ox. This linear dependence is expressed by o&q = Kox

(1)

where K is a constant with dimensions of force and, for the moment, we choose our coordinate system so that the direction of motion is along the x-axis. The term virtual work is motivated by the analogy to macroscopic work, since both are dimensionally the product of force and distance. The analogy becomes clearer when oEreq is represented by the energy range AE,,, and OX by the spatial range Ax. The assumption then reads: the amount of fluctuation energy required to move a virtual momentum carrier a distance x increases linearly with X. The choice of a linear dependence is motivated by the linear relationship E = pc for a zero rest mass particle (or more generally by the relativistic energy momentum relation, E2 = p2c2 + m$z4), since this suggests that the momentum transfer per unit time should depend on the fluctuation energy. Since we are dealing with a virtual momentum carrier we should be able to write equation (1) as (TE,,~ = Kcat. But according to the uncertainty principle aE = h/at. Setting aE = (TE,,~, we obtain Of2 z - h KC

(2)

where for now we ignore any dependence of c on mass. If K is indeed a constant, characteristic of the particular kind of virtual momentum carrier, then at must be a sharply restricted (quantized) time interval. The notation rK = (h/Kc)‘i2 is used to represent this feature, where the subscript expresses the parametric dependence on K. If the fluctuation time is quantized the uncertainty (TX (or spatial range Ax) must be as well. The notation XK = CTK, or more generally rK = crK) is used to express this quantization. The fluctuation energy AE and the fluctuation momentum in the direction of propagation must similarly be highly restricted quantities. The virtual work postulate implies that virtual particle exchanges are mediated by a chain of excitations, since clearly the distance between interacting absorbers can be arbitrary. The chain propagation model in turn suggests a vacuum density model in which the chains comprise sequences of transient particle-antiparticle excitations. If the fluctuation length separating vacuum particles varies

‘This generalization is possible as long as it is recognized that the momentum transfer between the absorber and the vacuum particle is along the direction of subsequent chain propagation, even though the particle must exhibit positional uncertainty in all spatial directions. This restriction is necessary because of the commutation relations among xK, yK, and TV (see Part I, Section 5).

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as a function of distance, the sequence of excitations is accompanied by a compression-expansion wave that allows fluctuation lengths and times to dynamically fulfill the restriction imposed by equation (2). The virtual work postulate opposes the uncertainty principle. This is why it yields the chain propagation picture. Fluctuation energy increases with distance (or time) according to virtual work, but decreases as time increases according to the time-energy uncertainty principle (in conjunction with the requirement of conservation of energy). Obviously the fluctuation energy cannot both increase and decrease. The virtual work postulate and the uncertainty principle are therefore consistent for only one particular value of the fluctuation energy, with that value depending on the parameter K. Virtual work requires the spatial range and hence the momentum to increase with fluctuation energy, whereas the uncertainty principle requires the time interval to decrease with fluctuation energy. If the time interval and spatial range are connected in a definite way by the velocity of light, the time and space intervals must be quantized in a way that depends on K. The postulate therefore captures the competition between energy and momentum fluctuations that underlies chain propagation. For the purposes of developing the model it is convenient to identify the fluctuation energy with the magnitude of the energy fluctuation, and to identify the fluctuation lengths and times with spatial and temporal intervals. Strictly speaking the fluctuation quantities represent statistical distributions that would be obtained in an ensemble of measurements, if indeed such individual measurements could be made. The distribution of vacuum particles can then be initially conceived of as having well defined average properties. Whether two vacuum particles are neighbors can thus be defined in two ways: they are neighbors if they have physical proximity; or they are neighbors if a transfer of momentum occurs from one to the other in a single step. These two definitions should be compatible in a statistical sense, since the real requirement of the fluctuon model is that the fluctuation energy and momentum be transferred on the average over time and space intervals compatible with the uncertainty principles. In a true quantum mechanical treatment it would be preferable to think in terms of superpositions of possible energies, momenta, and positions (time is a more ambiguous quantity in this respect, since it remains a variable rather than an operator in most quantum theories).

4. MODELING THE DIRECT FLUCTUONIC INTERACTION

4.1.

Choice of coordinates

The chain interaction picture elicited from the virtual work postulate can serve as a starting point for obtaining macroscopic expressions for the forces between particles. The analysis must be more detailed for gravity than was required for electromagnetism, since the source of force (mass as opposed to charge) is no longer quantized and since variations in vacuum density, and therefore in vacuum lengths and times, must be considered. Additionally, the indirect nature of the gravitational interaction means that screening plays a dominant rather than a modifying role. To accommodate variations in density it is necessary to consider more carefully the operations involved in defining distance and how these relate to the conceptual requirements of the fluctuon model. Consider two absorbers and suppose that the propagating fluctuon that mediates the interaction between them comprises N links. The distance between the two absorbers should then be the sum of N fluctuation lengths. This can be defined in two ways in the fluctuon model. The first is based on the number N of vacuum

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particles between the two absorbers and on the average distance between these vacuum particles. The method of ascertaining this average defines a coordinate system. The second definition is based on the length of the initial fluctuation interval, since the conservation laws require that all N fluctuation intervals in the chain be of the same size. The size of the initial fluctuation interval depends on the relative state of motion of the observer, since it depends on the mass-energy of the initiating absorber, and consequently it also requires a procedure for defining a coordinate system. The actual value of N, and therefore the actual fluctuation lengths, need not be known. It is only necessary that N be the same for both definitions. Let ?, represent the average value of the fluctuation length in the region of space between the, two particles. (Most of what follows is only relevant when rK = rg, where rg is short for rgrav, but for later reference we will develop the formulas without at this point specializing to particular kinds of forces.) This is a purely notational representation, since this average could not be known without specifying a procedure for choosing coordinates. Even before doing this we can use this notation to express the distance between two absorbers as r = N fK, that is, continue to express it in terms of chain propagation steps. We could also assign a notational average value to the fluctuation times by writing Y = NcZ, = NT,. But some caution is required, since c varies with distance from mass. The dependence of c on gravitational field will eventually play an important role in the theory, and therefore for future reference we will from now on use the symbol c0 to denote light velocity at an infinite distance from mass (i.e. its value in special relativity) and write C(T) to the express functional dependence of light velocity on mass and distance from mass. The simplest choice of coordinates is one that allows the force law derivations developed in Part I for a homogeneous vacuum to carry over with minimal alteration to the inhomogeneous case. For this purpose suppose that one of the two absorbers can be treated as a test particle that has negligible influence on the vacuum density. Consider a spherical surface defined by all rotationally symmetric positions of the test absorber. The distance r can be identified with a coordinate distance, as measured by an observer always at rest relative to a measuring instrument used to ascertain the surface of this sphere (i.e. used to measure its proper area). This procedure for defining TK corresponds to the procedure for defining Schwarzchild coordinates. Now let us turn to the second definition, based on the initial fluctuation interval. Let rK(ur, N) denote the fluctuation length at N excitation steps from absorber ai. Then rK(Ul7 1) denotes the fluctuation length separating an initiating absorber from the immediate neighboring vacuum particle and rK(ai, 1) denotes the corresponding fluctuation time. The vacuum length denoted by the absorber per se will be denoted by rK(ui, 0). Later we can replace ai by m, to represent the mass of the absorber or with qi to represent its charge. Thus rg(mi, 1) would be the vacuum length associated with an absorber of mass mi and r,,(qi, 1) would be the vacuum length associated with an absorber of charge 9. The initial fluctuation length re,(qi, 1) is with high precision proportional to rem(qi, 0), due to the requirement for overall charge neutrality (cf. Part II, Section 8). The initial fluctuation length r!(mi, 0) could be affected by massesother than mi in a very high mass, high velocity domain; but in the low mass, low velocity domain, or in the case of a single large mass, we can always write rK(ui, 1) = CKrK(uI, 0), where CK is a proportionality constant specific to the particular kind of force (i.e. for gravity CK = C, and for electromagnetism CK = C,,). If the vacuum is inhomogeneous N FK < NrK(ai, 1). But as noted above, the intervals rK and rK must dynamically stay constant throughout the whole exchange process in order to satisfy conservation principles (i.e. chain propagation entails compression-expansion waves). By itself this cannot change N, the number of vacuum particles between the absorbers, since a change in N would mean jumping over vacuum particles. The distance

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between the two absorbers may therefore also be written as L = NrK(ai, l), where L > r. The distance L is the distance traveled by gravitons or photons transmitting an influence from the source absorber to the test absorber. (It is unrealistic to think in terms of a single graviton or photon as transmitting the influence if the source and test particles are separated by a significant distance, since in general the chain initiated by the source particle would be terminated by an intermediating trapped fluctuon before reaching the test particle). That L is larger than r is consistent with the fact that the proper distance between two absorbers (i.e. the distance measured by an observer at rest relative to his ruler) is larger than would be expected on the basis of the proper area of the sphere defined by rotational symmetry, due to the shrinking effect of gravity on the ruler. This shrinking effect is consistent with the fluctuation lengths being larger in the neighborhood of absorbers, since for given N the distance from an absorber would always be estimated high if the immediate neighboring length is taken as the standard. However, in fact L would not be the proper distance between two arbitrary absorbers, since over any measurable distance trapped fluctuons between the two absorbers will terminate propagating fluctuons with different fluctuation lengths; furthermore, in the case of gravity the indirect interactions emanating from high density regions of the universe play a major role. The interval rK(ai, 1) and the number of excitation steps N are not directly measurable. But the fluctuon model imposes requirements on these quantities. The interval rg(mi, 1) must increase as mass appears to increase (we single out rg here since the effect would be negligible for rem except under the most extreme conditions). So right at this stage it can be seen that mass, length, time and signal velocity must all be tied together relativistically in the fluctuon model. A change in the mass of one of the absorbers would, ceteris paribus, alter N as well. In the Newtonian (low mass, low velocity) limit the consequent effects become negligible and the difference between r and L becomes insignificant. For the purposes of eventually dealing with high mass, high velocity phenomena it is important to distinguish these two quantities, however. The analysis of fluctuon exchanges will be based on I, and not on L, even if the force expressions are strictly valid only in the Newtonian limit. As noted above, this is because the probability of an interaction between two absorbers is most easily described in terms of its dependence on the proper area of the spherical surfaces that they define around each other. The distinction between r and L will become important in Section 15, where we use the fluctuon exchange mechanism to model the Schwarzschild metric. Finally we should note that if c varies with distance from an absorber it is its value at the locus of chain initiation, where r = rK(ai, l), that is controlling so far as the probability of chain initiation is concerned. For mnemonic convenience we will refer to this particular value of c(r) by c(i). 4.2.

Exchange probability

Consider two particles separated by a distance r and exchanging momentum though a virtual process. At least one of the particles is an absorber that initiates the virtual exchange. The second is any absorber or a vacuum particle located a distance r from the first particle. According to first order, time dependent perturbation theory, the probability that two particles and surrounding vacuum undergo a transition from state a to state b in a time interval t in the presence of a perturbing potential constant over that time interval is given by

4lblVlb)l ‘sin2{ $AE/h)r} P(a, b) =

AE2

(3)

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where AE = E, - Eb [cf. equation (lo), Part I]. The perturbation V is viewed, as in the earlier development, as a self-perturbation allowed by a virtual fluctuation of energy compatible with AEr = A. The term self-perturbation is used because the transition is triggered by a virtual (spontaneous) fluctuation of energy, not by an imposed potential. Equation (3) can also be interpreted as describing a two-particle scattering process. Equation (3) does not presuppose the vacuum density background. At first sight the fluctuation energy, AE, could have any value. But according to the virtual work postulate z is restricted to tK(ui, 1). In accordance with our choice of coordinates the distance between the two absorbers is expressed as r = Nc(r)fK = NP,. The quantities rK and c may be taken as constant with a high degree of accuracy for the subsea of vacuum electrons, except in the immediate neighborhood of a charged absorber. This is never an accurate assumption for the full sea of vacuum fermions. But it is a good approximation, except in the immediate neighborhood of absorbers, when the mass and velocity of the absorbers are low (this corresponds to the approximation of curved space by flat space in general relativity). Unless the two particles whose states are connected by equation (3) are immediate neighbors (i.e. unless N = 1) the interaction between them must be mediated by a chain of virtual excitations. Let us first consider the N = 1 case, to be denoted by P(a,, b2), where the subscript refers to the value of N. The relevant fluctuation energy AE then has the value h/rK(ai, 1). This also follows directly from the virtual work postulate, since evaluating equation (2) for K and working in fluctuation lengths yields

K=

h(i) TK(ai,

l)’

where c(i) is the appropriate value of c. Substituting for K in equation (1) and noting that r is now restricted to the value IK(ai, 1) gives AE,, = hc(i)/rK(ai, 1). Thus AE,, = h/zK(Ui, l), which as noted above is the same as the fluctuation energy obtained directly from the vacuum density picture. Setting AE = AE,,, in both the numerator and denominator of equation (3) then yields

P(q, b2) =

0.92((b~V~~)~~r,(u,, l)* c(i)2h2

(5)

This is equivalent to P(u, b), the probability that a vacuum particle that is undergoing a fluctuation of energy AE = h/~rc(Ui, 1) interacts with a neighboring absorber to initiate a propagating fluctuon [equation (ll), Part I]. If the neighbors are both absorbing particles, at least one of them must undergo an initiating fluctuation. This could be a spontaneous fluctuation, or it could itself be initiated by a neighboring vacuum particle. Now suppose that N > 1. The receiving particle may be initially pictured as residing somewhere on a spherical surface centered on the initiating absorber and of radius r = NYK, where N > 1. The probability of the two particles undergoing a transition that represents a delivery of momentum from the absorber to some receiver on this surface may, in accordance with the above notation, be denoted by P(ar, bN). According to the chain propagation model P(u,, bN) = P(u,, b2), independent of N. This is because the chain propagates in a deterministic manner, since it cannot disappear in the absence of a terminating absorber without violating conservation of momentum. Thus, once the chain is initiated each transiently excited pair must transmit its fluctuation energy to a neighboring vacuum particle with unit probability. If an absorbing particle acts as the neighbor the chain is terminated. (A neighboring particle is always available, since the absence of an unmanifest neighboring vacuum particle defines a neighboring absorber.) Strictly speaking the imagined sphere is not operationally tenable. Both the initiating and

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receiving particles are in superpositions of possible states that interfere with each other; they can only be associated with particular position and momentum states with probabilities consistent with the uncertainty principle. We are free to stipulate that the most probable location of the initiating particle was a posteriori the center of an imagined sphere. But we are not then free to say that a receiving particle is actually located at any particular locus on the surface defined by this center and by the radius r = NFK (since this is a measurement question that ties the geometry to the physics). This has no effect on P(ar , b,,,) since the momentum has to be transferred to some receiver with unit probability in any case and since any variations in rK between absorbing particles must to a good approximation be compensated by compression-expansion waves. We can now be more precise about what P(ar , bN) is and is not. It is the probability that an exchange of momentum occurs between two particles, at least one of which must be an absorber, subject to the requirement that the receiving particle could be any particle on [or less than TK(ai, N)/2 distant from] the surface of a sphere of radius r = N fK. This is different from the probability, to be denoted by P(i, f), that an energy fluctuation leads to an exchange between two absorbers potentially at particular locations or between an absorber and a vacuum particle potentially at particular locations. P(i, f), depends on the direction in which the momentum vector is pointing and on the probability amplitude that the receiving particle is actually at the particular location. This is the case even if the initiating and receiving particles are immediate neighbors. It is also the case if the distance separating two absorbers is arbitrary. Chain propagation then provides the means for the interaction. The transfer of momentum is of course only observable if the receiving particle is an absorber. To estimate P(i, f) think of the potential receiving particles as small circles of diameter TK(ai, N) residing on the spherical surface. But since we are interested in the force between absorbers we can take the potential receivers to be potential absorbers and write their diameter as rK(af, 0). The probability that a chain initiated at the center hits one of the small circles is rK(af, 0)2/16N2& (cf. Part I, Section 9). This is roughly the probability that it comes into the neighborhood of any particular particle located a distance r from the initiating particle. The probability P(i, f) of an exchange between two particular particles separated by a distance r is thus given by P(u, b)rK(uf, 0)2/16N2&, yielding fyi, f) =

0.921 (b(Vlu) 12rK(Qi, 1)2r~(uf, 0)2

(6)

16N27&(i)2h2 where N2& = r2. Suppose (taking an overly classical view) that it were possible to locate an absorber with complete certainty in the small circle that is hit. The chain would then be terminated with certainty, since it could not bypass the absorber without an unacceptable violation of energy conservation. In reality, as noted above, an absorber is in a superposition of possible locations; consequently it is impossible for an absorber to be situated between an initiator and a potential receiver in such a way that it completely screens the potential receiver from the initiator. This has no effect on P(i, f), however, since the initiating absorber sends momentum carrying chains in all different directions with equal probability. Any chain that a receiving absorber fails to terminate by virtue of its ‘quantum mechanical indefiniteness’ will be compensated by another chain that it succeeds to terminate because of this indefiniteness. Any chain that an unmanifest vacuum particle fails to be excited by due to its indefiniteness will be similarly compensated. This N2 factor in the denominator, which enters because of the uncertainty in the propagation direction, is what allows the fluctuon to propagate for an indefinite amount of time with constant fluctuation energy, and nevertheless to comply with the uncertainty principle. If the fluctuation energy A E lasts for a time interval NT,(u,, 1) it should,

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according to the uncertainty principle, decrease in accordance with AE - fi/NrK(ai, 1). This is in apparent contradiction to the requirement that the fluctuation energy of the fluctuon remain constant, and equal to ~/~x(~i, l), regardless of how large N is. The total energy of all the excitations that comprise the fluctuon should therefore grow as A &xal = Nh/rK(Ui, 1). However, the chance of actually detecting the fluctuation with a measuring instrument placed on the spherical surface decreases by the factor 16N2 as N increases. Dividing hEtotal by this factor yields AE 2 hjl6N rK(ai, l), which easily satisfies the uncertainty principle. 4.3.

Momentum transfer and dimensional analysis

Suppose that the coordinate system is chosen so that the two absorbers of interest lie on the x-axis. Strictly speaking the fluctuon should be described in terms of a superposition of its possible directions of motion. The chance that a fluctuon initiated by either of the absorbers propagates in the direction of the other along this axis can only be known after the fact. All measured motions of the absorbers that are oppositely directed to each other along the x-axis will be due to the exchange of propagating fluctuons whose direction of propagation can be aligned with this axis. With this choice of coordinate system P(i, f) represents the probability that one of the absorbing particles absorbs momentum h/x,(ai, 1) from a neighbo~ng vacuum particle that undergoes a proper ~uctuation [that is, a fluctuation of energy AE = h,hK(ai, l)] and that the propagating fluctuon thereby initiated is absorbed by the second absorbing particle, P(i, f) increases as xK(ai, 1) increases and consequently it increases as h/XK(ai, l), the momentum transferred, decreases. It does not by itself represent the quantity of momentum transferred. It also does not represent the probability that the neighboring vacuum particle undergoes the required ~uctuation. This along with the actual ~uctuation energy contributes to the pe~urbing potential V; but V does not represent this probabiiity and cannot, since (as will be elaborated shortly) it should approximate a perturbation constant over the time interval rK(ai, 1). Either absorber could of course be the inhibitor. The momentum transferred from the one arbitrarily identified as being at the center of the sphere to the one identified as being on the surface is equal and opposite to the momentum transferred from the one on the surface to the one at the center, assuming a constant vacuum density. The force between the two particles should be proportional to the amount of momentum carried by the propagating fluctuon [i.e. to h/x&al, l)] and to the probability that the fluctuon is initiated and that it hits the target absorber. The latter probability is given by f,P(AE,)P(i, f), where P(AE,) is the probability that a neighboring particle undergoes a proper fluctuation of energy AE,, fi, is the frequency of proper ~uctuations that last long enough to permit chain initiation, and P(i, f) includes the probability that initiation actually occurs in the presence of a perturbing potential concomitant to a proper fluctuation and also the probability of hitting the target absorber (at this point it is convenient to use the subscript p to distinguish a proper fluctuation). The probability of chain initiation increases quadratically with the fluctuation length rK(ai, 1) since initiation requires that the initial ~uctuation energy be completely taken up by the neighbo~ng vacuum particle, and therefore that the transition described by equation (3) be to an equal energy state (cf. Part I, Section 9). The factors h/xK(ai, l), P(AE,), and fr, effectively multiply to a constant, to be denoted by A, yielding a direct proportionality between force and P(i, f). To establish this let us look more carefully at P(AE,) and at the difference between this and the perturbing potential V. According to the Gaussian assumption [Equation (12), Part I], P(AE) is given by

P(AE)d(AE)

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Gravity

=

e-(E-@2b’(E)2d(AE)

’ +5)+7

(7)

where AE is an arbitrary fluctuation energy, E is the energy of the vacuum particle undergoing a fluctuation AE, i? is the average energy of the vacuum particle, P(AE) denotes the probability density of the fluctuation and d(AE) represents a differential element of the fluctuation magnitude [a peculiarity of notation that could be avoided in Part I by expressing equation (7) as a proportionality]+. E can be taken as zero (since energy will eventually be associated with vacuum density, and fluctuations in the direction of greater or lesser density should be symmetrical). Thus we can set A E = (E - E) = a(E). The fluctuation energy of interest is A E, = h/z,(Ui, 1). Equation (7) then reduces to P(AE,) = P(AE,)d(AE,)

=

WEpI A Epv2ne

(8)

This may be interpreted as the probability that at any given time a fluctuation of magnitude AEr occurs on a linear space interval IK(Ui, 1). It increases as tK(Ui, 1)/h increases and therefore as xK(ai, l)/hc(i) increases [since the differential element d(AE,) is independent of AE,]. Thus h

P(AE,) = --WE,) (9) c(i)j/27re xFAai, 1) is constant apart from a factor of c(i). The proportionality constant A can be written as fpd(AEp)/c(i)d2rre, where as required f,, has units of frequency. This is consistent since d(AE,)/c(i) has units of momentum and force should be equal to the momentum delivered per unit time. Furthermore, since frequency is a clock process (i.e. could be used to keep time) it should depend on vacuum density in a manner that follows the dependence of light velocity on vacuum density. Thus we can assume that fp increases as c(i)/c,,, and consequently A can in fact be taken as a constant. This can be thought of as a redshift in the frequency of proper fluctuations that last long enough to be converted to deliveries of momentum. Because of this redshift the dependence of fr on rK(ai, 1) is hidden in the constancy of A. This is not the case for P(i, f) and for macroscopic force, however. As the interval of time over which macroscopic force is defined is made smaller [approaching rK(Ui, 1) in the limit] its statistical character should become more evident, since the momentum is delivered in discrete and precisely equal packets that are themselves reduced with a probability that depends on rK(Ui, 1) and on the matrix element I(bjVla)l P.

‘The probability density in equation (12), Part I was not distinguished by boldface type. ‘The number of graviton exchanges per unit time should be smaller between two particles located in remote space (far distant from mass) than for two particles subject to a high gravitational field, due both to the decrease in the probability of fluctuations capable of initiating chain propagation and to the even faster decrease in the probability of eligible fluctuations that actually couple to a neighboring vacuum particle. But the quantity of momentum associated with each exchange should be greater under these circumstances. These effects should occur even though the gravitational interaction between any two particles is influenced by gravitons emanating from all regions of space in the fluctuon model, since most of the indirect interactions proceed through multiple chain terminations and initiations by intervening trapped fluctuons. Thus the radiation emitted by a small charged particle distant from mass would in principle reveal that the graviton impacts on it decrease in frequency, but that the probability of larger accelerations would be greater. Such fluctuation phenomena would clearly be extremely difficult to detect due to the large number of graviton exchanges that occur per unit time. Fluctuations connected with the electromagnetic interaction would in principle decrease in a sufficiently high gravitational field, but the field strengths required to produce a sufficient increase in r~m(ai, 1) for effects to occur would have to be enormous to overcome the charge neutralization feature of the electromagnetic field.

M. CONRAD

1272

Now we can consider the perturbing potential V and why it differs from both AE, and P(AE,). As the fluctuation energy A E, increases the chance that the vacuum particle acts on the absorbing particle should increase. But recall that AE, is really the standard deviation of the fluctuation energies over the time interval rK. If this interval is divided into smaller and smaller subintervals the chance of a fluctuation of the required magnitude occurring, or indeed of fluctuations of any magnitude occurring, becomes infinite. The chance of the vacuum particle acting should therefore depend not just on A E,, but also on the probability P(A EP) that it makes excursions of magnitude A Ep at any particular time and on the probability that these persist long enough to be efficacious. Thus V should be reasonably approximated by a constant, since f&A EJA E, is constant for given c(i). The autocorrelation function of AE, might alter the approximation in a second order way. The important point is that V is an energy and the contribution of [$(A EJ to it does not simply represent the probability that a neighboring vacuum particle undergoes a proper fluctuation.

4.4.

Macroscopic force

When the statistical character of the interactions is averaged out, we can sum up the momentum transfers in terms of the force expression

(10) where FK(i) is the equal and opposite force that absorber ai and af exert on each other due where CK is the to fluctuons initiated by Ui. The prefactor AK stands for CKAl(blVl~)l’, proportionality constant that translates rK(af, 0) into rK(af, l), A denotes the product fpP(AEp)h/XK(ai, I), and the factor of 2 enters because equation (10) represents only one member of a pairwise interaction. As shown above A can be taken as a constant [cf. equation (17), Part I]. In the case of gravity CK (which would be written as C,, short for C,,) is strictly speaking constant only for absorbers that are isolated from other masses, or in practice distant from very large masses (cf. Section 4.1). Similarly the equal and opposite force that Ui and af exert on each other due to fluctuons initiated by f is

AKrK(ui,1)2rK(uf,1>2 FK(fJ

=

2C(f)%%J

(11)

The total force is then AKrK(ai, l)2rK(af? 1>2[c(i)2 + c(f)2] FK

=

FK(i)

-I- FK(f)

5

2c(i)2c(f)2h2r2

(12)

If c(i) = c(f) we have FK =

AKrK(ai,

1)2rK(ufy

c(i)‘fi2r”

1j2

(13)

But as noted earlier the interaction between two arbitrary particles would in reality be mediated by trapped fluctuons in the intervening space. Consequently FK as calculated using rx(ai, 1) and rK(af, 1) would in general not correspond precisely to the measured force, except in the case of a direct transfer. The measured value could only be obtained by correcting for all these intervening interactions, but this point is somewhat moot since the picture of two isolated permanent absorbers is an idealization in any case.

Fluctuons-III.

Gravity

1213

The product TK(Ui, l)*rK(af, l)* in the numerator of equation (10) explains why pairwise products of charges (qlq2) or masses (m1171*)occur in classical force laws. The probability of an interaction is given by the product of the probability of chain initiation [controlled by rK(Ui, l)* and r,(af, l)*] and by the probability that the chain hits the terminating absorber [controlled by rK(Ui, 1)*/C: and rK(af, 1)*/C:]. A product of charges occurs when rK(Ui, 1) and rk(uf, 1) are lengths in the subsea of vacuum electrons and a product of masseswhen Ix(Ui, 1) and rK(uf, 1) are lengths in the full sea of vacuum fermions. If rK(“i, l) = rK(uf7 1) = rK we regain the simplified form FK =

AK

(14)

C*h*p~3r2

where pK = l/r; serves to characterize the density of vacuum particles without explicit representation of the spatial variation. For the electromagnetic field (i.e. when FK = Fern) this is an adequate picture, apart from slight variations of light velocity in high gravitational fields, since the vacuum density structure could only be altered by the most extreme conditions. For the gravitational field (FK = FD,, to be denoted by Fg for short) the more general equation is conceptually necessary, though detectable effects only occur in the high mass, high velocity region. The inclusion of these effects does not by itself put enough structure into the equation to describe the direct interaction between two particles in an arbitrary coordinate system, let alone the net attractive interaction. But the more general equation will provide a tool for constructing locally valid descriptions of multiple interactions that can be pasted together to provide a microscopic interpretation of a relativistically correct global description that can be independently deduced from the fluctuon principle.

4.5. Screening effects The crude picture is that a propagating fluctuon would be annihilated with unit probability if it could be said to enter the immediate neighborhood of .an absorber. The term ‘enter’ is misleadingly classical, however. The more correct statement is that the probability of an interaction between two absorbers is not altered by the feature that it is only possible to specify the probability amplitude that the chain and the absorber intersect. The underlying assumption is that the initiation and termination of a propagating fluctuon can be described without introducing a separate probability for termination. If the transfer of momentum is between two neighboring absorbers [i.e. if N = 1 with r = r,(Ui, l)] it is clearly a single event, described by the single probability P(u, b). If N > 1, the transfer must be mediated by a chain of vacuum fluctuations. The initiation and termination events are indistinguishable, in that it would be impossible to tell if the momentum transferred to the absorber originated in the spontaneous (or probabilistically generated) fluctuation of a neighboring vacuum particle or from a propagating fluctuon initiated elsewhere. If initiation and absorption could be treated as independent events it would be possible to ascertain the interval NtK(ai, 1) through a measurement. But if this were the case we could then write AE * h/NtK(ui, l), which contradicts the requirement that for a fluctuon A E, = h/z&, 1). The impossibility of detecting a fluctuon without annihilating it protects the chain propagation process from this contradiction. (This feature has an analog in conventional quantum field theories, where the uncertainty principle makes it impossible to ascertain the time order of emission and absorption of virtual particles. It is also connected to the fact that it is sufficient to consider a single absorber-fluctuon interaction to calculate the

1214

M. CONRAD

coupling between a charged particle and the electromagnetic field; see the discussion of the fine structure constant in Part II.) It might appear that the above argument breaks down when the vacuum density in the neighborhood of the two interacting absorbers is different, since A E would then depend on the locus of initiation. However, the perceived differences in density could always be transformed away by shifting to a coordinate system in a different state of acceleration (because of the equivalence of gravity and acceleration) and consequently no conclusion could be drawn about which absorber initiates the fluctuon that is independent of the state of motion of the observer. It is conceivable that the chain (i.e. the propagating fluctuon) could jump over a potential terminating absorber in a manner independent of the probability amplitude that their world lines intersect and nevertheless satisfy energy conservation. It could do so by inducing a compression in the vacuum density. Such compressions necessarily occur in the neighborhood of ordinary particles (particles in positive energy states excluding trapped fluctuons) since mass or charge induce variations in vacuum density. This is the reason why propagating fluctuons must be accompanied by a wave of vacuum density compression and expansion (cf. Part I, Section 5). The absorber corresponds to a discontinuity in the vacuum density structure, however, and bypassing it would therefore require a rapid distortion or discontinuity in the compression-expansion wave. Such long jumps must be possible, since virtual photons can bypass charge free masses. But in this case both the initiating and terminating absorber must possess some (masking) property that allows for the required vacuum distortion. In the absence of a bypass facilitating property common to both the initiating and terminating particles the probability of long jumps occurring should be exceedingly low, and should be zero according to the inseparability argument given above. This feature is important for the model of gravity to be developed, since it means that ordinary particles will screen each other from propagating fluctuons to the maximum extent compatible with the uncertainty principle. Fluctuons can also interact with each other since their transient constituents act as absorbers during their positive energy existence. In order for such scattering to occur the momentum of one of two intersecting fluctuons must be captured by a transient absorber in the other rather than by a vacuum particle. The terminating fluctuon may either be a trapped or propagating fluctuon. One of its constituents must enter (in the quantum mechanical sense noted above) the fluctuation volume into which a propagating fluctuon transfers its momentum. The average distance between the constituents of the terminating fluctuon is in general much larger than a fluctuation volume (cf. Part II, Section 7). Consequently, the probability amplitude that two fluctuons cross paths is much greater than the probability amplitude that a termination will occur. Accordingly, two identical fluctuons can occupy the same state with very little probability of their constituents interacting; at the same time interactions between fluctuons with non-intersecting world lines have a greater probability of occurring. Gravitons, with the above proviso, can be terminated by trapped fluctuons. But since the latter are only temporary absorbers (i.e. since vacuum particles are in a superposition of absorbing and non-absorbing states) the terminating effect is much smaller than the terminating effect exerted by permanent absorbers (i.e. ordinary particles). Furthermore, this leads to no net screening, since the probability that trapped fluctuons act as initiators balances the probability that they act as terminators. This is required for conservation of energy. If trapped fluctuons terminated more than they initiated they would act as graviton sinks, thereby producing a rectifying increase in the local vacuum density, and conversely, if the initiation probability were less than the termination probability this would result in a rectifying decrease in the local vacuum density.

Fluctuons-III.

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Gravity

Trapped fluctuons built from self-regenerating electron-positron pairs terminate photons, again with the proviso that a constituent of the trapped fluctuon enters the fluctuon volume into which the photon must transfer momentum. As with trapped fluctuons built from uncharged constituents, this leads to no net screening effect. Trapped fluctuons built from electron-positron pairs will be repelled by gravitons, but since they are electrically neutral they will not be repelled by photons so far as net effect is concerned. The photon interaction controls the average distance between the constituents of the trapped fluctuon (but again it is to be emphasized that we are dealing with a superposition of unmanifest and transient particle-antiparticle states). The direct interaction between propagating fluctuons (photons and photons, gravitons and photons, gravitons and gravitons) also requires a constituent of one of the chains to enter the fluctuation volume into which the other must transfer momentum. Otherwise the two fluctuons will pass through each other (or superpose). The net effect on the distribution of virtual photons and gravitons in the case of the photon-photon or graviton-graviton interaction will be hidden since the probability of initiation by a transient constituent is the same as the probability of termination. The average size of the fluctuon (e.g. transient positronium in the case of the photon) will be controlled by these interactions, however. Only one of the intersecting chains can be terminated in this process. The same holds for photons when viewed from a coordinate system in which they are assigned real energy. The direction of the non-terminated photon will change due to the momentum imparted to it by the terminated photon; but this will result in a photon initiated by the non-terminated photon that also carries real momentum and that hides the effect. In order for photon-photon scattering to be detectable the transient electronpositron pairs would have to exchange virtual photons. The probability is extremely small. Photons initiate gravitons and terminate them, consistent with the fact that photons gravitate. However, gravitons are not affected by photons, except through the exchange of virtual gravitons. If two gravitons interact one can be terminated and the other can be scattered. But as noted above the scattering effect is canceled out by graviton initiation, and consequently the flux of gravitons is totally determined by the distribution of ordinary and trapped absorbers. 5. DIRECT GRAVITATIONAL

INTERACTIONS

Consider two ordinary positive energy bodies, of mass mL and mR, separated by a distance r. For simplicity suppose that these bodies (to be referred to as mL and ma) are spherical, that they are isolated from all other permanent massesor other sources of force, and that the coordinate system can be chosen in such as way that their motion towards one another is not combined with any translational motion. According to the fluctuon model three types of inverse square interactions are relevant to the net interaction between mL and mR, or indeed to the net interaction among any arrangement of positive energy masses: (1) Direct repulsion between ordinary masses. This is obtained by equating the postulated repulsive interaction between mL and ma to the inverse square interaction expressed in terms of vacuum lengths [equation (12)]. Thus F

s

=

GmPR

r2

~

A,&%,

1)2rg(mRy

1)2kw2

+

cw21

(15)

2c(L)2c(R)2h2r2 G, is a constant connected with the direct repulsive interaction (later to be identified

M. CONRAD

1276

with the universal gravitational constant G). The light velocity [or factor that converts rs( m, 1) to rs( 111,l)] is denoted by c(L) in the neighborhood of mL and by c(R) in the neighborhood of m a. These velocities would be detectably different only if the disparity between the mL and ma is very great. For simplicity, and for the present purposes, we can assume that c(L) and c(R) have the common value c(i). In accordance with equation (13) we can therefore write

(2) Direct repulsion between an ordinary mass and a trapped fluctuon. In this case we can denote the mass of the positive energy body by m (which may either be mL or ma) and the mass of the trapped fluctuon by mti. Again assume that the massesare viewed from the special coordinate system that eliminates translational motions. Equating the postulated repulsive interaction between m and rntf to the inverse square interaction as expressed in terms of fluctuation lengths

F = Gmwf

g

r2

~ A,r,(m l)2rg(mti9l)‘M4’

+ 4md’l

2c(m)2c(m,)2h2r2

(17)

As before we use equation (12) for the fluctuonic force, with c(m) representing the light velocity pertinent to the initiation of fluctuons in the neighborhood of m and c(mtf) representing the velocity in the neighborhood of rntf (strictly speaking these velocities should in general differ in this case). The repulsion constant G, is the same as in equation (15), but the repulsion between m and the vacuum particle is reduced since it is active only when the latter enters the trapped fluctuonic state (m is a permanent repeller whereas the vacuum particle is a part time repeller). In effect the repulsion between ordinary and vacuum particles can be expressed by replacing G, in equation (12) by f,,G, and replacing A, by ftfA,, where ftf is the average fraction of time that the vacuum particles spend in trapped fluctuonic states. The vacuum density depressions surrounding ordinary massesare induced by the direct repulsive interaction between them and the trapped fluctuonic modes of the whole sea of vacuum particles. The net attractive interaction between ordinary masses is due to the repulsive interactions between them and the trapped fluctuons in the resulting high density regions of space, together with screening. (3) Direct repulsion between trapped fluctuons. This is obtained by equating the postulated repulsive interaction between two trapped fluctuons with the inverse square interaction as expressed in terms of fluctuation length. The rest massesof the trapped fluctuons could be different if they are formed from different types of vacuum particles. Denoting the rest massesby rnti and Mtf and assuming the same special coordinate system as above

F _ GKnf g-

r2

~ A,r,(Mti, l)2rg(mtf9 l)2[c(Md2 + 4md21 2c( M,f)2c(m,f)2h2r2

W-9

The repulsion between m tf and Mti is the same as between two ordinary particles with corresponding masses, but the effective repulsion between the vacuum particles is reduced by virtue of their both being part time repellers. The direct repulsive interaction between trapped fluctuons cannot alter the distribution of vacuum particles. This is because the density of trapped fluctuons must be proportional to the vacuum density. If this were not the case the vacuum density would not be controlled by the distribution of positive energy mass. (A trapped fluctuon in a region of relatively depressed vacuum density will exert a greater repulsive force on a trapped fluctuon in a relatively elevated region than

Flwtuons-III.

1217

Gravity

conversely; but this is compensated by the fact that the density of trapped fluctuons in the depressed region is less.) Equations (15)-(18) are valid in the specially chosen coordinate system in which the absorbers are at rest (apart from the accelerations away from one another produced by the repulsive interaction). If the repulsive interaction could be isolated the motions of the absorbers would appear to be different in different inertial coordinate systems. But unlike the electromagnetic interaction this structure could not be deduced from the special case of an inverse square interaction by transforming to different inertial coordinate systems (Part II, Section 4). The requisite assumptions on conservation of charge and other limitations of the effects of moving charges do not carry over to mass, since mass-energy increases with velocity. The gravitational force exerted by the absorbers on each other will thus increase. According to the principle of equivalence this is tantamount to a transformation to an accelerating frame. But even this is an inadequate picture, since any change in the forces exerted by the absorbers on each other must be accompanied by changes in the pair-wise interactions between them and all the trapped fluctuons in the universe. Thus the reaction of the vacuum density structure to the motions of the absorbers must also be included if a complete analysis of their motions is to be obtained. The full gravitational field equations of general relativity (to be considered in Section 14) must incorporate all these features (i.e. metric coefficients to express the density structure of the vacuum, first and second derivatives of the metric cofficients to capture the link between velocity and acceleration of the absorbing particles in response to this density structure, and non-linearity to express the back reaction of the absorbers on the density structure). The inverse square character that will eventually be deduced in the Newtonian approximation for the net attractive interaction between permanent masses is negligibly affected by the above complications, however, since the multiplicity of interactions with the sea of trapped fluctuons is essentially self-canceling and since the analysis is restricted to the low mass domain (cf. Section 12). The comparatively small number of direct exchanges between any two particular absorbers is swamped by bombarding gravitons emanating from all regions of space and consequently the motions of the absorbers are negligibly affected by the choice of inertial system. Similarly, the net repulsion of trapped fluctuons by ordinary masses is negligibly affected by the choice of inertial system, since this also arises from interactions with all other massesin the universe, including all the trapped fluctuons. In the high velocity, high mass domain the increase in relativistic mass and its effect on vacuum density structure must be taken into account. Later (in Section 14) we will see that the vacuum density surrounding an ordinary absorber appears to be topologically distorted when viewed from coordinate systems in different states of acceleration and that transformations between different frames is equivalent to transformations between different vacuum density structures. As shown immediately below, the assumption that Fg depends solely on the products mLmR, mmtf, and kftimtf is not strictly correct in the presence of ordinary masses not explicitly represented in equations (15)-(18), despite the fact that the pairwise character of the graviton exchanges still obtains. This is an intimation of the effect of accelerating frames, since the gravitational fields created by such unrepresented masses are locally equivalent to accelerating frames (see Section 11). 6. RELATION

BETWEEN

MASS AND FLUCTUATION

LENGTH

The assumption in all of the above interactions [equations (15)-(18)] is that the repulsive gravitational force is proportional to the masses of the interacting particles, just as it is in the Newtonian attractive interaction. This entails a definite relationship between mass

M. CONRAD

1278

and fluctuation length. To cover all cases let mlmz represent mrma, mmff, or Mtimti. Eliminating r2 then yields

A,r,(ml, l)‘r&m, lj2

mlm2 =

G$(l,

2)2h2

where c^(i, f) is defined by 1

= N2

+ cw2

2c(i)2c(f)2

(20)

c(i, f)” If the factors A,, G,, c(i), and h on the right hand side of equation (19) are constant or nearly so we can write m1m2 m r&ml,

1)2rg(m2, 1)2

C-21)

If the masses are equal we obtain the simple square dependence mj -

AiPr,(mi,

1)2

Gt’2c(i)h

C=)

where m, is the gravitational mass of the absorber, which may be either an ordinary particle or a trapped fluctuon. But recall (from Section 4.1) that in the low mass, low velocity region the fluctuation length associated with mass mi is proportional to rg(mi, 1). Thus Mi a Tg(mi, 0)’

(23) where rg(mi, 0) may be interpreted as the radius of the absorber. If the masses are not equal we can write mlm2 a r&ml,

O)2rg(m2, 0)”

(24) According to equations (5) and (6) the square fluctuation length r&ml, 1)2 controls the probability that an absorber interacts with a properly fluctuating vacuum particle to initiate a propagating gravitational fluctuon (i.e. a graviton). The mass of an absorbing particle, since it is proportional to this interval, can then be viewed as a measure of the probability that it absorbs gravitons. A naive mechanistic picture is of some use at this point. The absorbing particle is associated with a discontinuous change in the vacuum density, that is, with a discontinuous or sharp variation in the fluctuation length. This is required for consistency with momentum conservation, since chain initiation or termination resulting solely from induced variation in vacuum density would amount to ‘free space’ pair creation and annihilation. The fluctuation length Ts(mi, 1) is most naturally interpreted as the fluctuation length that separates mi from its immediate neighboring vacuum particles, while the length rs(mi, 0) is the length demarcated by the vacuum density discontinuity per se (shortly to be identified with the radius of the vacuum lacuna corresponding to mJ. The initial relationship between rg(mi, 0) and rs(mi, 1) is a property that we assume to be inherent in the wave function of the absorber and immediate surrounding vacuum, therefore formally a property hidden in the matrix element (b(V(a). Th’is assumption is consistent, since the N = 1 interaction between an absorber and a neighboring vacuum particle in a trapped fluctuon mode involves a direct exchange of momentum and could not be mediated by the exchange of a propagating gravitational fluctuon. The depression of vacuum density induced by ordinary particles follows from this initial setup. Recall that the fluctuation lengths at increasing distances from mi can in our notation be denoted by Tg(mi, N), where N = l., 2, 3. . . These lengths decrease with

Fluctuons-III.

Gravity

1279

increasing N up to a point and then increase (since the propagation of a density depression entails the propagation of a density elevation). The induced elongations of vacuum density for N > 1 should be kept distinct from rg(mi, 0), the vacuum dilation directly associated with the absorber and from Ts(mi, l), the neighboring elongation directly induced by the absorber. One reason is that trapped fluctuons do not on the average induce elongations in intervals for which N > 1. Furthermore, if rg were identified as an average quantity characterizing the induced variations in vacuum structure it would have to be an average taken over all space. This is clearly untenable, since all masses would then have the same value. We can now return to the statement, made at the end of the previous section, that equations (15)-(18) require correction in the presence of additional ordinary masses. The reason is that the vacuum density depressions induced by these masses exert a stretching effect on Ts(mi, 1) in addition to that exerted by rg(mi, 0). Thus the probability that any two absorbers will exchange gravitons will depend not only on the fluctuation lengths specifically associated with them but also on all other masses in the universe (because the local vacuum density would be altered by the presence of these other masses). This effect should be extremely small unless the absorbers are close together (for example, if they border the same fluctuation interval) or if their masses are extremely large. In such situations we can write Ig(mi, 1; mj, . . ., mk), where the list mj, . . ., mk includes all the masses that exert additional influences on the fluctuation interval adjacent to mi and that therefore alter the direct proportionality between this interval and mi. The possibility of such dependency is an attractive feature, since it is consistent with the general relativistic conception of gravitating matter (here ordinary absorbers) as both controlling and being controlled by space curvature (here interpreted as vacuum density variation). But apart from extreme circumstances the deviations from a simple association between masses of particular absorbers as measured by chain initiation capacity and as measured by the Tg(mi, 0) would be small, and accordingly equations (15)-(H) would obtain to a high degree of accuracy. The relation between mass and fluctuation length is analogous to the e a r&, relation between electric charge and fluctuation length in the subsea of charged vacuum fermions (cf. Part II, Section 7). The difference is that mass can have any value, whereas electric charge is to a high degree fixed by the requirement for overall charge neutrality. This requirement suppresses variations in fluctuation length, and ensures that r,,(mi, 1) is effectively constant for electrically charged particles. 7. VACUUM DISCONTINUITIES

It is useful, again as a rough thought device, to picture vacuum fermions as having average locations, and to think of the mass of an ordinary particle as being a measure of the improbability of vacuum fermions being located in the region of space it occupies. This is consistent with the association between the mass of an absorbing particle and the fluctuation length Ts(mi, 0), since a fluctuation length represents a deficit in the capacity for pair production in a region of space. The ‘size’ of an absorber, as measured by this deficit, is not identical to its de Broglie wavelength (h = h/p for a free particle). As the mass of an absorber increases, its fluctuation length rs(mi, 0) increases quadratically; but its de Broglie wavelength decreases linearly. Setting p = miu and using equation (23) to express the de Broglie wavelength in terms of fluctuation length

1280

M. CONRAD

As il increases to infinity the absorber spreads over all space. Concomitantly rg(mi, 0) decreases to zero, corresponding in the limit to a complete absence of any reduction in pair producing capacity (strictly speaking, corresponding to the absence of any hole in the vacuum density that could exert a dilating effect on a neighboring fluctuation length). Both effects-the spreading of the particle and the decrease in the fluctuation length-imply that the mass of the absorber decreases. As ), decreases, the absorber becomes more localized. Concomitantly rg(Uli, 0) increases, corresponding to a larger region of reduced pair producing capacity. Again this is consistent, since the mass increases in both cases. The inverse relationship between de Broglie wavelength and mass does not involve charge due to the fact that the subsea of charged fermions is too dilute relative to the full sea of vacuum fermions for increases in rem to have any effect on rg. Particles in the vacuum state should also have a de Broglie wavelength, but it is entirely consistent for these not to be associated with a vacuum lacuna since they do not gravitate. In a trapped fluctuon mode they would assume mass, and would exhibit the same inverse relation between fluctuation length and de Broglie wavelength. The fluctuation length should not be identified with the diameter of the transient particle-antiparticle pair constituting the fluctuon. The radius of transient positronium, which plays a role in the derivation of the fine structure constant, is substantially greater than rem(Part II, Section 7). The relationship between A and rg(mi, 0) reflects the complementarity principle, that is, the difficulty (or impossibility) of being precise about the relationship between the particle and wave aspects of matter. As noted above, the fluctuon model requires that an absorbing particle be interpreted as being demarcated by a discontinuity in the vacuum density between the intervals Tg(mi, 0) and rg( Illi, 1). As the localization of the absorber increases (as 1 decreases) the probability of finding it in a defined region of space increases (assuming a measurement with the required degree of precision can in principle be contemplated). The discontinuity feature would be brought to the fore by such a measurement, but at the expense of enlarging the region of space defined by the discontinuity and therefore bringing back the wave aspect. If Ts(mi, 0) decreased as A decreases, it would then be possible to identify the absorber with a wave packet. In a full quantum mechanical treatment we should properly describe the absorber by a wave function whose square amplitude yields the probability of finding it at particular locations in space. If the absorber could be identified with the wave packet and the wavefunction could be taken as a wavepacket (for a one particle wavefunction) we would obtain a manifest contradiction to the interpretative postulate of quantum mechanics. That equation (25) precludes such a contradiction suggests that the fluctuon model is consistent with quantum mechanics even though we have not at this point formulated it using an apparatus that automatically builds in all the requisite quantum mechanical features. 8. INTERPRETATION

OF MASS AND FIELD

The connection between mass and deficit in pair production capacity nevertheless allows some definite statements about the relationship between particle mass and vacuum density and between field energy and vacuum density. Consider first an ordinary fermion. This can arise via the promotion of a vacuum fermion to a permanent positive energy state. The promotion removes a transiently excitable fermion from the vacuum, thereby locally reducing pg. This local reduction may be pictured as a hole and associated with the antiparticle. It is more accurate to associate it with a local reduction in pair production capacity in the region of space occupied by the particle (since the Pauli exclusion principle would not prevent the hole from being occupied by another type of particle). The promoted particle itself can cohabit the same fluctuation volume as a transiently excitable

Fluctuons-III. Gravity

1281

vacuum particle of the same type as long as this particle does not in fact undergo a

transient excitation. As soon as it does undergo such an excitation it will be repelled due to the exclusion principle, thereby again creating a local reduction in ps. Thus the appearance of two positive energy massesthrough pair production can be equated to the appearance of two depressions (or deficits) in the vacuum density. This would be the case even if the antiparticle is viewed as a positive object, since it would, by virtue of the exclusion principle, repel transiently created antiparticles. Propagating fluctuons constitute the fields produced by ordinary massesand charges, and also by trapped fluctuons. Since propagating fluctuons (gravitons and photons) comprise fermion-antifermion composites in the fluctuon model, it is possible to associate them with vacuum density depression without requiring a sea of unmanifest bosons. Gravitons or photons that carry real energy should be accompanied by a net vacuum density depression that detaches from the density depression induced by an accelerating absorber. The net depression approaches zero in the reference frame in which the fluctuon approaches pure virtuality. During its lifetime the fluctuon (the whole chain of transient excitations) is equivalent in its repelling power to an ordinary absorber, and consequently the depression structure that it is initiated with is stable whether it is virtual or real. Thus field energy is equivalent to energy in particulate form, since both are defined in terms of vacuum density depression. This is reasonable since field also has a particulate aspect. In effect trapped fluctuons also represent reductions in the availability of vacuum particles. The new feature is that during its lifetime a trapped fluctuon comprises a pair that is continually being created and annihilated. Thus rather than being associated with a fixed density depression (or a fixed elongation of rJ it is associated with a self-regenerating density fluctuation. Trapped fluctuons with a range of velocities should be possible, corresponding to a range of A E values greater than AE,. The energies of such particles could be enormous; but the particles would be rapidly terminated by the vast mesh of propagating fluctuons in their environment. Conceivably vacuum density depression could arise through direct stretching of the vacuum. Positive energy would then be created without the creation of antimatter, but only if the stretching were accompanied by the appearance of a discontinuity. The number of particles would then exceed the number of antiparticles, provided that the antiparticles are identified with the positive objects (since the particles would then be associated both with the holes produced by the promotion of antiparticles and the holes produced by stretching). In the absence of a discontinuity the stretch would be eliminated by the density controlling effects of ordinary absorbers, in violation of the principle of energy conservation. This implies that in an equilibrium universe energy is associated with vacuum depression only when it occurs in conjunction with discontinuity, i.e. in conjunction with ordinary fermions or bosons, trapped fluctuons, propagating fluctuons that carry real energy, or propagating fluctuons that carry only fluctuation energy. The stress-energy tensor that generates the gravitational field will eventually be interpreted in terms of these modes of vacuum density depression (but excluding gravitons and trapped fluctuons). 9. VACUUM

THERMODYNAMICS

Inhomogeneities of density tend to disappear in collections of manifest particles, since more particles migrate from higher density regions to lower density regions than conversely. This corresponds to the increase in entropy (or macroscopic symmetry). The statistical thermodynamics of the vacuum is entirely different. Inhomogeneities in the distribution of vacuum particles (or potentialities for pair production) appear or disappear only in response to the motions of massesand charges.

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The spontaneous disappearance of vacuum inhomogeneities would be incompatible with the identification of mass and field with vacuum density depression, since this would mean that masses and the fields associated with them could disappear without reappearing in any other guise, in manifest contradiction to conservation of energy. For example, if promotion of a vacuum particle to a positive energy state is accompanied by the appearance of an unoccupied hole in the vacuum sea, homogenization would mean the disappearance of this hole and hence the disappearance of its rest mass energy without any compensating appearance of either kinetic or potential energy. The above energy argument has a simple mechanistic correlate. Unmanifest vacuum particles in the fluctuon model inherit their properties from their transient entries into positive energy states. In order for a vacuum particle to migrate through the vacuum sea it must therefore transiently come into existence as a panicle-antiparticle (or panicle-hole) pair. Due to the Pauli exclusion principle it would also be necessary for a pathway for migration to become available at the same time. This means that a neighboring vacuum particle must simultaneously undergo a particle-hole forming fluctuation. In the absence of such a pathway of migration the exclusion principle would only allow exchange of identical particles. No detectable migration would occur, and consequently no change in vacuum density could occur. Now consider two adjacent vacuum regions, to be denoted by I and II. The number of transiently excited particles available for migrating from region I to region II is proportional to the density of region I. But the fraction of these that could actually succeed in entering region II is proportional to the density of region II. As a consequence the density structure

of the vacuum is conserved even if it is inhomogeneous. This means that the entropy of the vacuum is conserved, where entropy may be taken as a measure of the number of possible complexions of transient pair formations compatible with the density structure. The conclusion would be the same if the antiparticle were interpreted as a positive object rather than a hole, since the exclusion principle allows antiparticles to provide migration pathways for particles. The above arg~ent links the exclusion principle to conse~ation of energy. Energy could not be conserved in the fluctuon model if it were not for the exclusion principle, since it is the (unsatisfiable) requirement for migration pathways that prevents vacuum inhomogeneities from disappearing. Also note that the frictionless motion of positive energy particles through the vacuum sea is formally consistent with the exclusion principle since these particles are in different energy states than vacuum particles, This argument is somewhat misleading in the context of the fluctuon model, however, since positive energy particles are associated with deficits in pair production potentiality. Hence the world line of a positive energy particle is associated with the world line of a deficit, and consequently co-occupancy of the fluctuation volume by a positive energy particle and vacuum particle would be inconsistent with conservation of energy. Two positive energy particles (such as fermions with different spins) can occupy the same spatial state since this just means that the two associated deficits superpose without destruction of the demarcating discontinuities. If the discontinuities would be lost, and no new type of particle could be produced, a violation of energy conservation would occur. The density of trapped fluctuons should be proportional to the vacuum density, apart from disequilibrium effects induced by the motions of ordinary particles. If the density of trapped fluctuons increased relative to the local vacuum density they would repel each other into less dense regions; if the density decreased relative to the local vacuum density trapped fluctuons would be repelled into the region from more dense regions. Thus proportionality will hold even though the fluctuation energy required for the production of trapped fluctuons increases as the vacuum density increases. The relative motions of ordinary particles alter the vacuum density structure by virtue of the repelling action of

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these particles on trapped fluctuons. Readjustment of the vacuum density structure in response to such motions is associated with deviations from proportionality. Suppose that the vacuum density is defined in terms of non-absorbing vacuum particles and a separate density is defined in terms of trapped fluctuons. Consider two regions of the vacuum, I and II, and suppose that a net migration of trapped fluctuons from region I to II occurs due to the mass or charge of an ordinary particle. The non-absorbing density of regions I and II will stay the same for a short time after a net migration occurs. The non-absorbing density of region I will then decrease, since the trapped fluctuons lost to it will not decay back into non-absorbing states. Conversely, the density of region II will increase, since the new trapped fluctuons entering it will decay into non-absorbing states. 10. INTERPRETATION

OF INERTIA

In the fluctuon model (inertial) mass as defined by resistance to acceleration and mass as defined by the gravitational interaction can be given equivalent mechanistic interpretations in terms of bombardments by propagating gravitational fluctuons. The key idea is that an ordinary particle in a state of acceleration always moves in advance of the vacuum density depression induced by it, thereby increasing the fraction of bombardments by propagating fluctuons that oppose the acceleration. Consider a permanent positive energy body, of mass m, in an accelerated state of motion. The acceleration may be due to the presence of another mass M, or it may have electromagnetic or other non-gravitational origin. The mass m body will induce a depression in the surrounding vacuum, in accordance with the repulsive interaction between ordinary particles and trapped fluctuons expressed by equation (17). Since the gravitons initiated by m propagate with light velocity, the effect of m’s acceleration on surrounding vacuum density must also propagate with light velocity. More distant regions of the space will be informed about this acceleration later than nearer regions. Consequently the change in state of motion of the vacuum depression induced by m will always be retarded relative to the change in state of motion of m whenever the latter undergoes acceleration. Because of this retardation the vacuum density in the region of space into which m is moving is always greater than the vacuum density in the region of space that it is leaving. If the acceleration is due to bombardments by gravitons the inertia (resistance to the change in state of motion) of m is due to the increase in the number of counterbombardments by gravitons resulting from the greater vacuum density of the region which m is entering. If the acceleration is produced by non-gravitational means (for example, through bombardments by virtual photons) the resistance will still arise from counterbombardments by gravitons. In either case inertia is a form of gravitation. Trapped fluctuons are also subject to inertia, despite the fact that they do not induce vacuum depressions. The proportionality between the average density of trapped fluctuons and the vacuum density ensures this. Accelerations of trapped fluctuons in response to propagating fluctuons impinging on them will on the average always repel them into regions of higher density, and consequently such accelerations will always be resisted by counterbombarding gravitons. This will not happen if the trapped fluctuon is viewed from a coordinate system in which it is moving at a constant velocity, since this inherently entails a symmetry of bombardments and counterbombardments. The term ‘average’ is used here since trapped fluctuons might be surrounded by net depressions or elevations in the vacuum density due to their being particularly strong or particularly weak repellers. But these are fluctuation effects that have no net consequences when the whole sea of trapped fluctuons is considered. The principle of inertia can be stated in terms of the symmetry of bombardments and

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M. CONRAD

counterbombardments. If the momentum transferred to an absorbing particle by propagating fluctuons emanating from different directions in space is asymmetrical the particle will undergo acceleration. The acceleration always produces countervailing bombardments by gravitons that transfer equal and opposite momentum to the particle, thereby resisting the acceleration. The acceleration of a particle produced by a constant force is constant because the countervailing bombardments are always delayed relative to the bombardments that constitute the force. This is because the change in state of motion of the particle begins on arrival of the bombardments, but the counterbombardments only begin after the position of the particle relative to the vacuum density structure changes. 11. PRINCIPLE OF EQUIVALENCE

The above interpretation of inertia, complemented by a perpetual mobile argument, implies the principle of equivalence and through this the principle of general relativity. The equivalence principle asserts that the gravitational mass of a body (i.e. the mass as defined by the gravitational forces) can always be taken equal to its inertial mass (i.e. its mass as defined by its resistance to acceleration). This is exemplified by the fact that the acceleration of a body in a gravitational field is independent of its mass. The principle can also be interpreted as asserting that the behavior of a body in a gravitational field is locally equivalent to its behavior in some accelerating reference frame. This formulation, mathematically expressed as the principle of general convariance, leads to the gravitational field equations of general relativity [4]. According to the gravitational interpretation of inertia outlined above the acceleration of a body in response to a force will be independent of its mass as long as any increase in the intensity of the fluctuonic bombardments that produce the acceleration (or constitute the force) is matched by a proportional increase in the intensity of counterbombardments. Suppose, as the reductio hypothesis, that exact proportionality does not hold and that the inertial mass cannot therefore be equated to the gravitational mass. Consider first an ordinary absorber. If the reductio hypothesis were correct the ratio of gravitational to inertial mass of an ordinary absorbing body would depend on the extent to which the vacuum depression it induces is retarded relative to its motion. To produce such a retardation it is only necessary to accelerate the absorbing body. The gravitational mass, according to the reductio hypothesis, would either be larger or smaller than the inertial mass. This would mean that the weight of the body would either grow proportionally larger or smaller than its inertial mass when it is accelerated (here it is convenient to speak in terms of weight since this should be proportional to gravitational mass). Suppose the weight is larger. By lifting an absorbing body off the surface of the earth without acceleration and lowering it with acceleration we could build a device in which the body weighed more on the way down than on the way up. Such a device could serve as a perpetual motion machine of the first kind, since it would yield more work on descent than required for ascent. If the weight decreased in proportion to the inertial mass for an accelerated body we could achieve the same effect by lifting the body with acceleration and lowering it without acceleration (e.g. lowering it in small steps that approximate constant velocity). The argument is less direct for trapped fluctuons, since these can only be manipulated by acting on ordinary absorbers. For example, the distribution of trapped fluctuons could be altered by lifting a weight (to be denoted by m) comprised of ordinary particles and dropping it on the surface of the earth. Suppose that the weight can be lifted and returned to its initial position in a manner that approximates reversibility (by dropping it in small steps). The distribution of trapped fluctuons should change in response to the movement of

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the weight, and should be the same at the end of the process as at the beginning, apart from heat motion effects that are here being ignored. If this were not the case the vacuum would exhibit density depressions that are not controlled by the distribution of ordinary absorbers, in contradiction to the fluctuon model of gravity. Suppose, as the reductio hypothesis, that the mass of a trapped fluctuon, as defined by the gravitational interaction, is either larger or smaller when accelerated by the motions of m than its mass as defined in terms of resistance to acceleration by the surrounding vacuum. The vacuum will either provide too little or too much resistance to maintain a unique relationship between the position of the weight and the vacuum density depression it induces, thereby allowing for the spontaneous appearance or disappearance of energy. As an extreme case of energy disappearance, suppose that the inertial mass of trapped fluctuons is so large that the vacuum density structure is not in the least altered in response to the upward motion of m. The vacuum density structure would then be entirely independent of m’s position. Work would be required to lift m, but zero potential energy would be obtained; in fact work would be required to return M to its original position since the repulsive interaction between it and the earth would remain, but no indirect attractive interaction would be created. The opposite extreme case is that the inertial mass of the trapped fluctuons is zero. The vacuum structure would then continue to change even after the weight is no longer being raised. Gravitational potential would continue to build up with time in a spontaneous way, again in obvious violation of the conservation of energy. The above considerations are reminiscent of the perpetual mobile argument for the constancy of light velocity in the fluctuon model, and more broadly for the principle of special relativity (cf. Part I, Sections 6 and 7). The special relativity argument depended only on the fluctuon principle per se. The equivalence principle argument depends additionally on the two main features of the fluctuon model of gravity: induced vacuum depression and indirect attractive interactions. It does not, however, depend on the specific model of fluctuonic force summarized in equation (12), and therefore does constitute evidence for the details of this model. For example, it would go through even if the direct fluctuonic force turned out to depend on say, rK, rather than on &. The indistinguishability of local gravitational fields and accelerating reference frames follows from the equivalence of gravitational and inertial mass, and therefore from the fluctuon model. In terms of the vacuum densities this local indistinguishability may be formulated thus: the counterjk of gravitons that resists the acceleration produced by a gravitational force (i.e. by an ordinary mass) can always be locally reproduced by some given non-gravitational force that produces an equivalent acceleration. If it were possible for

the non-gravitational force to vary in an arbitrary way the counterflux could be globally reproduced; but this is a physical impossibility, since the non-gravitational force always originates from masses that impose their own accelerating effects. The restriction to local indistinguishability is of fundamental importance for gravity, since it provides the basis for detecting acceleration and hence for measuring vacuum density curvature (the analog of space curvature in the fluctuon model). The gravitational red shift, the slowing of a clock in a gravitational field, the reduction of light velocity in the presence of mass, and the bending of light rays in a gravitational field all follow from the above indistinguishability [5]. The fluctuon model of gravity is a fortiori compatible with all these phenomena. 12. NEWTONIAN

APPROXIMATION

The perpetual mobile approach to the equivalence principle implies that some member of the class of fluctuon-based models should yield the general relativistic description of gravity

1286

M. CONRAD

and should yield the Newtonian inverse square law in the low mass, low velocity limit. The purpose of this section is to show in a constructive way that the double screening model in fact yields an attractive interaction with the correct form in this limit. The derivation uses a specially chosen coordinate system and ignores the ‘density curvature’ concomitant to the variation of rs. These would be just the approximations that collapse the field equations of general relativity in the Newton law. The derivation provides insight into the global and very intricate vacuum particle dynamics that underlie the Newton law and that would continue to be operative in a fully covariant analysis. Consider two ordinary absorbing bodies, of mass mL and mn, separated by a distance r. For simplicity suppose that these bodies can be treated as single particles, that they are isolated from other ordinary (permanent energy) massesor other sources of force, and that the coordinate system can be chosen in such a way that their motion towards one another is not combined with any translational motion. Some terminology helps to describe the double screening that is responsible for the net attraction between mL and ma (see Fig. 2). Picture the two absorbers as being connected by an imaginary straight tube, of diameter equal to the diameter of the bodies, and picture this tube as extending infinitely into space in both directions. The two masses can thus be thought of as spheres located in a cylindrical tube. If the masses are particles the tube will be very narrow, in the limit approaching a straight line as the particles approach idealized point particles. Now piCtUrC mL and ma as being on an imaginary spherical surface, with the line between them being the diameter of this sphere. Call the region of space inside the sphere the interior region and the region outside the exterior region. The tube connecting the two masses in the interior region will be called the interior tube (along the interior axis) and the extension of this tube into the exterior regions of space will be called the exterior tube (or axis). As a convention we can designate the region of space exterior to the hemisphere on which mL is located as the left hand region of space, and we can call the segment of the exterior axis in this region of space the left hand segment of the axis. t0 ma as the right hand segment of the axis. Similarly we can designate the aXiS exterior The above construction is acceptable for the present purposes, but would break down as soon as the effect of gravity on light becomes important (since light rays would no longer define straight lines). The first screening effect is inside-outside screening. Each absorber screens the repulsive effect of the other on trapped fluctuons in the space exterior to it. Thus mL screens the repulsive effect of ma on trapped fluctuons along the left exterior axis and mR screens the repulsive effect of mL on trapped fluctuons along the right exterior axis. Screening occurs because each absorber terminates with unit probability the gravitons sent by the other along the axis joining them (cf. Section 4.5). The terminated gravitons push absorbers away

left exterior region e

4 right exterior region

Fig. 2. Double screening construction used to recover Newtonian gravitation in the low mass, low velocity domain. The axis passing through two interacting masses defines the ‘tube’ of mutual screening. The compressing graviton fluxes bombarding the two masses from exterior regions along the right exterior and left exterior segment of this tube are greater than the repelling fluxes impinging on them along the interior segment (see Fig. 1).

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from each other rather than pushing the trapped fluctuons in the exterior region further into the exterior regions (gravitons are also terminated and initiated by trapped fluctuons in the interior, but this does not alter the net effect). Consequently the density depressions along the exterior axes are each controlled by only one of the absorbers, whereas the density in all other regions of space is influenced by both absorbers. Each absorber will repel the trapped fluctuons in the interior region of the space as well. It is important that mL and mR do not screen the trapped fluctuons, since these cannot be terminated by the absorbers and their movement from the interior to the exterior tubes is consistent with the Pauli exclusion principle. Trapped fluctuons will also be repelled from the exterior tubes into the interior tube by gravitons emanating from trapped fluctuons in the higher density regions of the exterior tube. Each of the absorbers, then, will sit in a pocket of depressed density that is influenced by the other absorber except along the exterior axes. As mL and mR move relative to one another the trapped fluctuons near to them will be repelled first and the more distant ones later. The regions of relative density elevation that surround the pockets will increase until the influx of trapped fluctuons into the pockets balances the efflux from them. The net depression along the interior axis is due to the fact that the influx in response to any change in position or state of motion requires some time (determined by light velocity) to compensate the efflux. The second screening effect is outside-inside. Each absorber screens the other from the repulsive effects of gravitons emanating from trapped fluctuons that accumulate along the exterior axes. Gravitons traveling rightwards along the left exterior axes are terminated by mL, thereby screening their repulsive effect on mR. Similarly, gravitons traveling leftwards along the right exterior axes are terminated by mR, thereby screening their repulsive effect on mL. The net attraction between mL and mR is due to their being pushed together by gravitons impinging on them along the exterior axes. Let us denote the compressing effect due to the outside-inside screening by OUT-IN and the vacuum density differential produced by inside-outside screening by IN-OUT. We can then summarize the double screening model symbolically by writing F g(net)CCOUT-IN

m IN-OUT

(26)

This just says that the form of the force law should depend totally on factors that contribute to inside-outside screening, since the outside-inside effect is completely determined by this. Despite the complexity of the interaction the in-out screening must be controlled solely by the two absorbers, mR and mL, and by the distance between them, since these are the only sources of asymmetry in the system. These absorbers have only two pertinent properties: the probability that they initiate gravitons and the probability that they screen gravitons initiated by the other absorber. In symbols IN-OUT

cc P(mR initiates graviton) x P(mL screens graviton) + P( mL initiates graviton) x P( mR screens graviton)

(27) A graviton sent by ma along the interior axis will be terminated before it reaches mL if a vacuum particle along this axis enters a trapped fluctuonic state (as will almost certainly be the case except for very small r). The same will be true for gravitons sent from mL to mR. The trapped fluctuons in the interior will also be a source of gravitons, however. If ma and mL are equal, the vacuum density along the axis remains entirely symmetrical. Graviton terminations along the axis will on the average be compensated by graviton initiations, leading to no change in the propagation direction probabilities. If these masses are not equal the vacuum density along the interior axis will be more depressed near the larger

M. CONRAD

1288

mass [it will vary inhomogeneously elsewhere as well, but this does not bear on equation (27)]. Density variations along the axis, whether or not symmetrical, will modify the relationship r = A&,. In the Newtonian appro~mation r should be independent of mR and mL, however. This means that the rs values must to excellent approximation be equal to one another, except in the intervals occupied by ma and mL. According to equations (5) and (22) the initiation probabilities are given by P(mn initiates graviton) = &,&ma, 1)2 - B’mR

f2W

P(mL initiates graviton) =rcBr,(m,, 1)’ = B’mr

(=b) where B and 3’ are proportionality constants and we recall that the notation r&mi, 1) represents the fluctuation length that separates the absorber from the immediate neighboring vacuum particle. To obtain the screening probabilities suppose that ma sits at the center of an imaginary sphere of radius r and mL sits somewhere on the surface of this sphere. The probab~ity that a graviton emitted by mR hits mL is given by P(mL screen graviton) =

r&W, Oj2 -

16r2

Cm 16r2

(294

where we utilize the propo~ionality between mass and ~uctuation length and denote the p~opo~ionality constant by C [cf. equation (23) and Section 4.31. Similarly P(mR screens graviton) =

rg(mR9

16r2

O>’

-

cmR

16r2

(29b)

Gathering these probabilities together

FP(W

-RmL

(30)

cc r2

The assumption of constant (or near constant) fluctuation lengths along the interior axes is what allows the constants B’ and C to be the same for both mR and mL even if these masses are not equal. From the point of view of the mech~ism of gravity this approximation is untenable, since the variations in vacuum density are the source of gravity; but clearly they must be too small in the low mass, low velocity regions to affect the form of the net force law. The pushing effect from the exterior always dominates the direct repulsion of mL and is always attractive. That this is so is due to the asymmetry of ma, and consequently Fgcoetj the screening process: propagating ~avitational fluctuons are screened, but trapped fluctuons are not. The repelling effect of mL and mR on the trapped fluctuons is equal to their repelling effect on each other, the only difference between the two cases being the lifetime of the absorber [cf. equations (15) and (17)]. Suppose that screening could be turned off (in reality of course this would be impossible). If the repelling effect of mL and mR on each other increased, the repelling effect on the trapped fluctuons would increase by the same amount. Trapped fluctuons would continue to build up in the exterior until the net efflux of trapped fluctuons from the interior region ceased, But at this point the direct repulsion between mL and mR would also be balanced by counterbombarding gravitons. If this were not the case mL and mR could repel each other without repelling trapped fluctuons, thereby disconnecting their movements from the vacuum density structure altogether. Switching screening back on breaks the symmetry of all these repulsive interactions. If mL and mR neither attract nor repel each other in the absence of screening they must certainly attract each other in its presence, since screening increases the

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differential between the interior and exterior vacuum densities, and ensures that the graviton pressure on each side pushes only on that side. 13. DENSITY CURVATURE AND METRIC TENSOR

To analyze more complicated situations, say the behavior of a test particle centered between mL and mR, it is necessary to consider the effects of the density variations. These alter distance and time measurements, and also require gravitons to be accompanied by compressions and expansions in rg that enable them to satisfy the original conservation and uncertainty principle requirements. The fluctuon vacuum comprises a mesh of potentialities for pair creation-annihilation events whose separations provide natural units for a space-time metric. This mesh does not, however, constitute an absolute coordinate system, since observers in different states of acceleration would draw different conclusions about the separations between events. This follows from the principle of equivalence, since different gravitational fields entail different vacuum density structures. For inertial observers, however, it is possible to write

s;= c’ 2m*, corresponding to the exterior solution. When r decreases to rg(mi, 1) this assumption becomes questionable, due to the proportionality between mi and rg(mi, 1)2. At this point N = 1. The analysis breaks down totally for any r less than rs(mi, l), since no graviton exchanges occur at this scale.

16. MASS-DENSITY RELATIONS

Masses as defined by the gravitational interaction, including the mass formally assigned to vacuum particles, are controlled by the vacuum density in the fluctuon model. At the same time interactions can occur only when the mass of vacuum particles as defined by the fluctuation energy required for pair production is compatible with the vacuum density. The model is thus subject to strong self-consistency requirements. These become especially important when the density curvature becomes large (e.g. in the presence of high masses or in the immediate neighborhood of absorbing particles). When deviations from selfconsistency occur self-corrective processes come into play. These mechanisms will be described in Part IV. For the present purposes it is sufficient to note the following the key points: (1) The mass formally assigned to different vacuum particle types (vacuum electrons, vacuum massons) must be constant, or nearly so, independent of local vacuum density. This is due to the fact that the mass derives from the interactions of trapped fluctuons with propagating fluctuons emanating from all parts of the universe (the Machian character of the model). If the rest mass formally assigned to vacuum massons varied with local vacuum density, the rest mass formally assigned to vacuum electrons would have to vary as well.

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But since the density of vacuum electrons is essentially independent of the distribution of positive energy mass, this would automatically entail an inconsistency between the fluctuation energy required for electron-positron pair production and the fluctuation energy required for propagation in the form of a photon. (2) The dependence of rg on mass and distance from mass requires gravitons to be accompanied by compression-expansion waves of vacuum density, otherwise the graviton would not ‘see’ the constant density curvature required for chain propagation. Such gravitational waves correspond to the electromagnetic waves that allow photons to see an effectively constant density curvature. The difference is that pemis homogeneous except for the depressions in the immediate neighborhood of charges, whereas pg is highly inhomogeneous, and never constant except at infinite distance from mass. Also, the fluctuation intervals have a more complicated structure in the case of gravity, corresponding to the global tensor character of the gravitational field, and as a consequence the local compression-expansion waves must have a more complicated structure. The gravitational waves adduced from the field equations of general relativity [equation (33)] are built up from the local vacuum density deformations that accompany the totality of propagating gravitational fluctuons emanating from all regions of the universe. The build up of deformation energy is not as unstably large as would appear from the general relativistic analysis, though, since the distance traveled by most gravitons would be quite short, due to the presence of trapped fluctuons that serve as intermediating terminators and initiators. In the absence of such intervening absorbers the density curvatures experienced by gravitons traveling between two ordinary absorbers could become extremely large, and consequently the deformation energies would become untenably large. (3) The energy required for pair production at each step of chain propagation is determined by the energy required for pair production at the point of initiation, even though light velocity (and therefore the energy required for de IZOVOpair production) increases with distance from mass. More formally, let 111, denote the rest mass of the purely gravitational vacuum particles (which we have referred to as massons). The fluctuation energy at the point of initiation is then AEf = 2m,c(i)2, where c(i) is the light velocity at the point of initiation. This means that the energy required for each step of propagation, AE,, is also 2rn,~(i)~ regardless of distance from point of initiation and regardless of the value of c at that step. This is possible because the expansioncompression waves carry the initial density curvature along with the propagating chain, but the speed with which the chain propagates is determined by the density of the surrounding masses (since it is determined by the gravitons impinging on it from all regions of the universe). (4) The above restriction on A E, can always be satisfied in a way that allows for the initiation of gravitational chains if c(i) varies with mass in an appropriate way. The conditions that must be satisfied are suggested by the following rough calculation, based on a two body interaction in the low mass, low velocity region. Recalling [from the discussion surrounding equation (4)] that AE, = hc(i)/r,(mi, 1) and equating this to AEf h(i)

= 2m,c(i)2 (44) $,(% 1) Also recalling the relationship between absorber mass and immediate neighboring fluctuation length [equation (22)] we can write mj =

A;‘2r,(ml,

1)2

(45) Glizc(i)h where G, can now be set equal to G and c(i) can be taken as the light velocity since we

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M. CONRAD

are considering two isolated particles of equal mass. Eliminating rg( nri, 1) from equations (44) and (45) and solving for c(i) c(i) = - e lyli

If3

(46)

where Q = Af%‘/3G-1i6(2m,) -*I3. Thus c(i) decreases as mi increases, but more slowly as m becomes larger, just as if it were a one-dimensional analog of the diffusion constant for spherical molecules (with the weight of the photons or gravitons in the immediate neighborh~d of absorber mi being analogous to the molecular weight of the dif~sing molecules). The important point is that the fluctuation energy required for massonantimasson pair production can always equal the fluctuation energy required for gravitational chain propagation, as long as the light velocity falls off with mass in a manner roughly described by equation (46). The detailed manner of falloff would depend on the detailed mass dist~bution. For example, the S~hwarzs~hild (i.e. one body) dependence of Iight velocity on mass and distance from mass [equation (39)] yields a different result (which, however, may be unreliable due to problems with the coordinate representation in the region of interest). As the masses or velocities become very large the simple inverse square approximation of the gravitational field would begin to fail; the different components of the fluctuation interval would begin to play a more prominent role. The assumption buried in A, of a simple propo~ionality between rg(mi, 1) and rg(mi, 0) may also begin to fail. (5) The actual vacuum must comprise different types of vacuum particles, with different formal rest masses. This is necessary since all particles gravitate, but not all particles respond to electromagnetic or strong forces. Since [from equation (14)] the stronger forces are associated with smaller vacuum densities, they must also be mediated by vacuum particles with smaller formal rest mass assignments. The electron, for example, must have a smaller mass than the masson, corresponding to the fact that the fluctuation energy required to initiate propagation is smaller for the electron. But electrons, since they gravitate, should also be recruitable into graviton chains. In this case the transient electron-positron pair must be produced with excess kinetic energy sufficient to satisfy 2m,c(i)2 = 2m,c(i)” + T

(47) Excitations of vacuum electrons that fulfill this condition make a negligible contribution to the gravitational force, due to the virtually negligible contribution that vacuum electrons make to pp. Low energy pair production processes, of energy AEf = 2m,c(i)2, must be precluded from mediating the ~avitational interaction since this would yield a net attractive gravitational force that is just as strong as the repulsive electromagnetic force. The gravitational mass assigned to all vacuum particles would then increase, with the consequence that the energies required for pair production would become greater than the fluctuation energies defined by the vacuum densities. Chain propagation would be terminated and both the electromagnetic and gravitational fields would disappear. The gravitational masses of all vacuum particles would then fall to zero and the fields would be turned on again. Suppose that some putative precursor universe were dominated by a chaotic evolution of this type. Either it would continue in this regime indefinitely (contrary to the facts of our universe) or it would reprocess itself until it reached a point where the particles assumed properties that precluded vacuum electrons (or any vacuum fermions other than massons) from mediating the gravitationai interaction through low energy ~uctuations that allow them to make the longer jumps characteristic of their own subsea. (6) As noted above the gravitational mass should have a value that is effectively fixed by the average vacuum density of the universe (or equivalently, by the total quantity of

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positive energy mass in the universe). Suppose that this average density could decrease. The gravitational mass assigned to vacuum particles would increase, due to the greater strength of the gravitational interaction. At the same time the mass as defined by the fluctuation energy required for chain propagation should decrease. Pairs could not be produced with sufficiently low energy to initiate propagating chains. All interactions would be turned off, at least in regions of the universe for which 2rn,~(i)~ exceeds h/rs(mi, l), until the vacuum relaxed back to the required average value. This would happen, for example, in a universe in which chains that continue to emanate from high density regions exerted a compressing effect on the vacuum density in the overly low density region. If such self-corrective processes were incapable of re-establishing the required vacuum density, the model would have to be replaced by a broader one (e.g. one in which values of fundamental constants, such as h, could evolve in a compensating way). If the average density could suddenly increase, the mass as defined by the gravitational force would decrease, but the mass as defined by the fluctuation energy required for chain propagation would increase. Chain propagation could be initiated, but the fluctuation energy at infinite distance from mass would initially exceed 2m,c02. The net attractive gravitational interaction responsible for maintaining the vacuum density would be weaker, and consequently the vacuum would undergo a self-corrective decrease in density until the condition AEf = 2m,c02 is satisfied. The electromagnetic force would also weaken if the density were increased, but pp is controlling since the repulsive character of the electromagnetic interaction means that at equilibrium pernmust be essentially constant over the whole universe and therefore must assume the lowest possible value compatible with the volume of the universe. (7) Conceivably the above equilibration considerations could fail in a detectable way in a highly heterogeneous vacuum, or in the neighborhood of immense masses (since these would induce extremely high density curvatures). The value formally assigned to m, could then exhibit a local deviation from the fixed value assumed in the discussion above. The local density curvature might be too great or change too rapidly for light velocity or fluctuation lengths to satisfy the condition for chain initiation, namely that the fluctuation energy associated with pair production equal the fluctuation energy required for chain propagation at the point of chain initiation [cf. equation (46)]. Gravitational and in extreme instances other interactions would be shut down at a local level, leading to the same kind of self-corrective alterations in the vacuum structure outlined in paragraph (6) above and hence to the re-emergence of these interactions. The alterations in states of motion associated with this flickering out of and back into existence of force interactions should be accompanied by radiative phenomena. Under nearly all circumstances such radiation would be too negligible to detect, or too incoherent to identify in a specific way. However, under some circumstances the flickering may become coherent enough to produce macroscopically detectable effects, as noted below. (8) Black holes arise in the fluctuon model when a collection of absorbing particles becomes so massive that the initiation of photons is shut down. The production of gravitons would also be shut down. The particles belonging to the collection would no longer exert direct repulsive interactions on each other or on particles outside the collection. Initially they would still appear to exert a gravitational attraction, but this is due to particles being pushed into the collection by gravitons emanating from the high density regions of the universe. (This obviates the contradiction inherent in the conventional picture that a body of sufficient mass to prevent photons from escaping should also prevent gravitons from escaping, thereby nullifying its gravitational field altogether). This pushing effect would initially accelerate the collapse process and increase the size of the hole. However, the high density regions would immediately begin to decrease in density due to the apparent disappearance of the inducing mass. The indirect attractive interaction would then decline

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and the particles of the collection would begin to fly apart. An isolated black hole (i.e. isolated from the compressing effects of gravitons emanating from high density regions induced by other masses) would therefore be inherently unstable. Periodic radiative phenomena would occur, since the fading out and fading in of the direct and indirect gravitational interactions would change the states of motion of particles both outside and inside the collection on a massive scale. Systems on the borderline between normal and black hole regimes would produce bursts of radiation suggestive of the bursts produced by pulsars. This would be the case, for example, for a black hole continually repopulated by particles driven into it by compressing forces from high vacuum density regions induced by a companion star, but at the same time continually depopulated (in a manner that depends on its rotational motion) by screening effects of this star. (9) Expansion and contractions of the universe as a whole would be similarly selflimiting, though obviously on an altogether different time scale. From the point of view of the field equations such fading out and fading in of gravitational interactions would have the effect of dynamically introducing a cosmological constant and doing so in a way that preserves a definite correspondence to the density structure of the universe, and hence to the dist~bution of mass that controls this density structure (recall that the fluctuon model precludes a cosmological constant based on the energy inherent in vacuum fluctuations). The value and sign of the ‘constant’ would change during different phases of the expansion-contraction cycle. (10) The inclusion of such a cosmological parameter would be pertinent to the evolution of black holes as well as to the evolution of the universe. The important feature, from the microscopic point of view, is that the proportionality between absorbing particle masses and the fluctuation intervals neighboring them [equation (22)f is significantly distorted by the close proximity of other absorbing particles. The introduction of a parameter representing an extra attraction or repulsion becomes necessary when the distortion is sufficient to preclude the initiation of gravitons. This is certainly the case when c(i) = 0, since the exchange probability P(i, f) between two absorbers [equation (6)] and therefore the fluctuonic force between them would become undefined in this limit if gravitons could still be initiated. An equivalent statement is that rs = r&(i) would be undefined for c(i) = 0, even though ~ais always finite except in a vacuum that is everywhere depopulated of vacuum particles, but physically it should grow to infinity. Physically the transition probability should therefore equal zero at this point since an infinite fluctuation time means that initiations of propagating fluctuons could not occur. The general relativistic field equations accommodate departures from the propo~ionality assumption, but must become inaccurate as soon as some of the interactions that normally contribute to the gravitational field are turned off. From another point of view the fluctuation energy required for pair production goes to zero when c(i) + 0, and consequently nearly all pair production processes would lead to real particles and antiparticles rather than to virtual pairs that could initiate chain propagation processes. More accurately stated, pair production would hardly occur at all, since the vacuum would be depleted of pair producing potentialities and consequently nearly all particles would become ordinary. Photons and gravitons would only be initiated to the extent permitted by uncertainty fluctuations in the vacuum density that introduce deviations from the c(i) = 0 limit {so that in fact it is more accurate to think in terms of a nearly complete rather than a complete shutdown in chain initiation). The shift from virtual pair production to real pairs allows for the emission of particles reminiscent of Hawking radiation, but the fluctuon model suggests that it is because of the weakening of the indirect gravitational interaction that the efflux of particles from the hole is greater than the influx. Real particles escape from the hole because of the decrease in the flux of

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compressing gravitons from high density regions of space; trapped fluctuons migrate into the hole, due to the dearth of gravitons emanating from it. The inflow repopulates the hole with pair producing potentialities, thereby increasing its density and decreasing its mass- and in the process annihilating its hole-like character. The fluctuon model, as indicated by the above remarks, need not rest on a pre-established harmony among the manifest and unmanifest features of the universe. The model, as will be shown more fully in Part IV of this series, encapsulates a self-reprocessing (or recursive) dynamic that allows it to evolve to a self-consistent state, and to continually correct local deviations from self-consistency. The non-linearity of this dynamic entails departures from linear superposition, as should be required of any quantum mechanical theory capable of addressing acceleration, since gravity requires a non-linear description, and accordingly wave function collapse and irreversibility enter as objective features of the time evolution [3, 13-151. 17. THREE SEA MODEL AND THE INTERPRETATION

OF SPACE CURVATURE

Let us now consider more carefully the isomorphism between the space curvature of general relativity and the vacuum density curvature of the fluctuon model. The advantage of this reformulation is that it admits virtual particle exchange models for both gravity and electromagnetism, but at the same time allows gravity to be completely controlling so far as space-time geometry (as chosen in conjunction with physical measurement and congruence considerations [16]) is concerned. The cohabitation of these two forces in the same geometrical framework is possible because pernconstitutes an extremely sparse subsea of pg, the subsea of all vacuum particles. Furthermore, the structure of the two seas is independent, since the much greater strength of the electromagnetic interaction (concomitant to the relative sparsity of vacuum electrons) combined with the requirement for charge neutrality assures that pern is unaffected by gravity except under the most extreme high mass conditions. Thus gravity affects electromagnetic phenomena only by virtue of the fact that charged particles and photons have mass, and not at all because of any linking between pg and fern. Electromagnetic phenomena affect gravity not at all, since the contribution of pernto ps is always too small to have any detectable impact. The picture can be expanded to include the strong and weak interactions. The strong interaction requires the introduction of a second subsea, comprising the quark class of vacuum fermions (or potentialities for the creation of the various quark-antiquark combinations). The spin 1 bosons (gluons) that mediate the interquark force are interpreted as chains of transient quark-antiquark pair creation and annihilation events. This subsea must be even more sparse than pem, since the interquark force is stronger than the electromagnetic force. The virtual mesons that mediate the internuclear force are, as in the standard theory [cf. 171, also quark-antiquark pairs. Since these are spin 0 (or even spin) particles the force can be attractive, unlike the other fluctuonic forces. This is interpreted as an immediate neighbor (N = 1) fluctuonic interaction, since a massive virtual particle and an exchange of negative momentum are both possible in the immediate neighborhood of the initiating absorber. (A fluctuon carrying negative momentum would always be reabsorbed by the initiating absorber at the end of the initial fluctuation, so that chains of length N > 1 are excluded. Since the probability of reabsorption goes up with the distance between the two absorbers the force decreases rapidly with distance, as does the Yukawa potential.) The weak force, according to the fluctuon model, is the expression of the electromagnetic force in pockets of high electromagnetic vacuum density [15]. The consequent

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M. CONRAD

reduction of r,, at the point of chain initiation means higher energy fluctuations and hence a weaker force. Higher mass particles (muons) can then contribute to chain initiation. Electron-positron (or muon-antimuon) chains initiated in this region may split into separate spin l/2 chains due to the suppression of electric charge. Neutrinos are interpreted as split chains that carry with them a compression-expansion wave that continues to mask the expression of charge. The suppressed charge, which entails a disappearance of a vacuum length rem(qi, 0) associated with a previously charged particle, must appear in some other form. The suggestion is that the spin 1 vector bosom (W’ and Z*) that transmit the weak force are fluctuons that acquire mass by converting the disappearing lacunae in &, to lacunae in ps. Thus an electron-positron (or muon-antimuon) fluctuon initiated in a high pern region, including the compressed region surrounding a neutrino, will transmit charge and momentum as a W’ (massive charged spin 1) boson if charge on one of the components of the fluctuon is temporarily suppressed and will transmit momentum only as a 2’ (massive neutral spin 1) boson if charges on both components are suppressed. Because of their massive character these ‘high density electromagnetic fiuctuons’ must reconvert the extra mass back to charge in an N = 1 interaction, either by directly transmitting charge (or self-canceling charges) and momentum to an absorber or by converting back to a fermionic absorber and neutrino (i.e. split chain) within a time interval allowed by the time-energy uncertainty principle (which is possible in an N = 1 process). The accumulation of mass and transfer of charge through the interconversion of charge and mass is the fluctuon model’s version of the Higgs mechanism. The quark subsea, like the electromagnetic subsea, should contribute to the gravitational supersea through high fluctuation energy pair production processes. But as with pernit is far too small relative to pp to affect the gravitational force, and even too small relative to pem to contribute significantly to the electromagentic force. As with electromagnetism the only effect of gravity on the strong force is through the interactions of gravitons with positive energy mass and trapped quark sea fluctuons, and not with the structure of the sea itself, which because of the requirement for the various forms of quark neutrality is very little affected by gravity. The structure of the eIectroma~eti~a~ly active subsea could be slightly altered by the strong force, since this can concentrate protons in a small region, but even here the effect would be mostly canceled by the electrons attracted to this region. From the above it follows that all measurements that bear on the geometry of space and time are controlled solely by the gravitational supersea, which therefore admits corresponding geometric and virtual particle descriptions that are entirely consistent with flat space, virtual particle descriptions for the electroweak and strong forces (apart from the influence of gravity on masses). This can be reformulated in terms of the conditions for the applicability of the general and special principles of relativity. According to the p~nciple of general relativity it is possible for the observer to choose his coordinate system so that the space curvature is locally flat; in terms of the fluctuon model it means that it is possible to choose the coordinate system so that the supersea density is locally homogeneous. But in order to do so it is necessary to admit coordinate transfo~ations that hide acceleration, and that therefore hide gravity. The electromagnetic and quark subseas have equivalent density structures throughout the whole universe, apart from the most extreme curvature of the supersea, and consequently the equations for these fields need only admit transformations that hide velocity. The principle of special relativity is therefore completely adequate, as long as interactions involving gravitons are ignored. Acknowledgement-This No. ~CS-~4~7~~.

material is based uponwork supported by the National Science Foundation under Grant

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REFERENCES 1. M. Conrad, Fluctuons I. Operational analysis. Chaos, Sohtons & Fractals 3, 411-424 (1993). 2. M. Conrad, Fluctuons II. Electromagnetism. Chaos, Sohtons & Fructals 3, 563-573 (1993). 3. M. Conrad, Force, measurement, and life, in Toward A Theory of Models for Living Systems, edited by J. Casti and A. Karlqvist, pp. 121-200. Birkhauser, Boston, MA (1989). 4. P. G. Bergman, Introduction to the Theory of Relativity. Prentice-Hall, Englewood Cliffs, NJ (1951). 5. A. Einstein, On the influence of gravitation on the propagation of light, in The Principle of Relativity edited by H. A. Lorentz, A. Einstein, H. Minkowski and H. Weyl. Methuen, London (1923) (reprinted by Dover, New York). 6. W. Pauli, Theory of Relativity. Dover, New York (1981). 7. R. Penrose and W. Rindler, Spinors and Space-Time, Vol. II. Cambridge University Press, Cambridge (1986). 8. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation. Freeman, New York (1970). 9. R. M. Wald, General Relativity. The University of Chicago Press, Chicago, IL (1984). 10. T. Frankel, Gravitational Curvature. Freeman, San Francisco (1979). 11. Y. B. Zel’dovich, Cosmological constant and elementary particles. Sot. Phys. --JETP Lett. 6, 316-317 (1967). 12. A. Zichichi, V. de Sabbata and N. Sanchez (Eds), Gravitation and Modern Cosmology. Plenum Press,

New York (1991). 13. M. Conrad, Reversibility in the light of evolution. Mondes en Developpement 14, 111-121 (1986). 14. M. Conrad, Transient excitations of the Dirac vacuum as a mechanism of virtual particle exchange. Phys. Lett. A. 152, 245-250 (1991). 15. M. Conrad, The fluctuon model of force, life, and computation: a constructive analysis. Appl. Math. Computation 56, 203-259 (1993). edited by 16. H. P. Robertson, Geometry as a branch of physics, in Albert Einstein, Philosopher-Scientist, P. A. Schilpp, pp. 315-332. Harper, New York (1959). 17. G. D. Coughlan and J. E. Dodd, The Ideas of Particle Physics. Cambridge University Press, Cambridge (1991).