Inertia and Gravitation in the Zero-Point Field Model

Final Report, NASA Contract NASW-5050.

Bernhard Haisch, Principal Investigator. Lockheed Martin Solar and Astrophysics Laboratory, Dept. L9-41, Bldg. 252, 3251 Hanover St., Palo Alto, CA 94304.

Alfonso Rueda, Co-Investigator. Dept. Electrical Engineering, Calif. State Univ., Long Beach, CA 90840.

Submitted March 31, 2000.

The results of this four-year research program are documented in the following published and as yet unpublished papers.

• Inertia: Mach's Principle or Quantum Vacuum?, B. Haisch, A. Rueda and Y. Dobyns. copy attached - intended for Physics Today

• On the Relation Between Inertial Mass and Quantum Vacua, B. Haisch and A. Rueda. copy attached - intended for Annalen der Physik

• The Case for Inertia as a Vacuum Effect: A Reply to Woodward and Mahood, Y. Dobyns, A. Rueda and B. Haisch, Foundations of Physics, in press (2000). (arXiv: gr-qc/0002069) copy attached - to appear in Foundations of Physics

• Toward an Interstellar Mission: Zeroing in on the Zero-Point-Field Inertia Resonance, B. Haisch and A. Rueda, Space Technology and Applications International Forum (STAIF-2000), Conference on Enabling Technology and Required Developments for Interstellar Missions, Amer. Inst. Phys. Conf. Publ. 504, p. 1047 (2000). (arXiv: physics/9909043)

• On the relation between a zero-point-field-induced inertial effect and the Einstein-de Broglie formula, B. Haisch and A. Rueda, Physics Letters A, in press, (2000). (arXiv: gr-qc/9906084)

• Electromagnetic Zero Point Field as Active Energy Source in the Intergalactic Medium, A. Rueda, H. Sunahata and B. Haisch, 35th AIAA/ASME/SAE/ASEE AIAA Joint Propulsion Conference, AIAA paper 99-2145, (1999). (arXiv: gr-qc/9906067)

• Progress in Establishing a Connection Between the Electromagnetic Zero-Point Field and Inertia, B. Haisch and A. Rueda, Space Technology and Applications International Forum-99, American Institute of Physics Conference Proceedings 458, Mohammed S. El-Genk, ed., p. 988 (1999). (arXiv: gr-qc/9906069)

• The Zero-Point Field and the NASA Challenge to Create the Space Drive, B. Haisch and A. Rueda, Proc. NASA Breakthrough Propulsion Physics Workshop, NASA/CP-1999-208694, p. 55 (1999).

• Advances in the Proposed Electromagnetic Zero-Point Field Theory of Inertia, B. Haisch, A. Rueda and H. E. Puthoff, 34th AIAA/ASME/SAE/ASEE AIAA Joint Propulsion Conference, AIAA paper 98-3143, (1998).

• Contribution to inertial mass by reaction of the vacuum to accelerated motion, A. Rueda and B. Haisch, Foundations of Physics, Vol. 28, No. 7, pp. 1057-1108 (1998). (arXiv: physics/9802030)

• Inertial mass as reaction of the vacuum to accelerated motion, A. Rueda and B. Haisch, Phys. Letters A, vol. 240, No. 3, pp. 115-126, (1998). (arXiv: physics/9802031)

• An Electromagnetic Basis for Inertia and Gravitation: What are the Implications for 21st Century Physics and Technology?, B. Haisch and A. Rueda, CP-420, Space Technology and Applications International Forum (M. S. El-Genk, ed), DOE Conf. 960103, American Inst. of Physics, p. 1443 (1998).

• The Zero-Point Field and Inertia, B. Haisch and A. Rueda, in "Causality and Locality in Modern Physics." G. Hunter, S. Jeffers and J.-P. Vigier (eds.),Kluwer Acad. Publ., pp. 171-178, (1998). (arXiv: gr-qc/9908057)

• Physics of the Zero-Point-Field: Implications for Inertia, Gravitation and Mass, B. Haisch, A. Rueda and H.E. Puthoff, Speculations in Science and Technology, Vol. 20, pp. 99-114, (1997).

• Reply to Michel's "Comment on Zero-Point Fluctuations and the Cosmological Constant", B. Haisch and A. Rueda, Astrophys. J., 488, 563, (1997).

Inertia: Mach’s Principle or Quantum Vacuum?

Bernhard Haisch (Solar & Astrophysics Laboratory, Lockheed Martin, L9-41, B252, 3251 Hanover St., Palo Alto, CA 94304), Alfonso Rueda (Dept. of Electrical Engineering, California State University, Long Beach, CA 90840) and York Dobyns (C-131 Engineering Quad, Princeton University, Princeton, NJ 08544-5263).

Abstract

Two competing theories are tackling the foundational question of whether inertia may have an extrinsic origin. One based on Mach's principle makes the startling prediction that transient mass fluctuations may be created to yield propellant-free propulsion. One based on quantum vacuum fluctuations may revise the conventional understanding of why moving particles have wavelike properties.

Background

Perhaps the most basic equation of physics is f = ma, Newton's equation of motion, in which m is the inertial mass of any object. Hereafter we specifically designate inertial mass as 'm_ to differentiate it from other aspects of mass. such as gravitational mass, mu, and the rest mass of special relativity based on the energy content, of an object in its rest frame, m0 = E/c 2. It is usually assumed that mi is an intrinsic property of matter. In that case any deeper understanding of the nature of inertial mass must be sought in the standard model of particle physics and experiments attempting to elucidate the interconnections among the fundamental forces and the many apparently fundamental properties of matter, such as charge, spin, parity, etc. But there is the possibility that mi is extrinsic to matter, arising from interactions between the innermost fundamental entities, such as leptons and quarks, constituting matter and some inherently external field. Such an idea was proposed by Mach in the 19th century: he proposed that a given object acquires its inertial mass via interaction with all other matter in the Universe. This concept was dubbed "Mach's principle" by Einstein, but for decades it remained more a matter of philosophy than science. Indeed, there was the nagging problem that general relativity (GR) appeared to be inconsistent with Mach's principle since solutions of the field equations of GR allowed for both an empty Universe in which a test particle could still possess mass, and a rotating Universe which would make no sense from the Machian perspective since the matter in the Universe must define the rotational frame of reference.

A significant development was the publication in 1953 by Sciama [1] of a simplified but nonetheless quantitative link between a hypothesized gravitational vector potential and inertia. A scalar potential for the Universe may be defined as

where as usual p is the local density corresponding to a source point inside the volume, dV, and r is the distance of the point of observation, or test point, from the source point. The integration extends over the Universe presumably out to the limit of causal connection which would be the cosmological event horizon. If one moves "relative to the smoothed out universe" (as Sciama wrote prior to the discovery of the cosmic microwave background and its role as a reference frame) with velocity v. then one may define a gravitational vector potential A = _v/c. The gravitational force on a small object (the smallness becomes important later on) having (passive) gravitational mass m_ would then be

1 0t (2) fg = -mgV_ - m_ c Ot

In any region of the Universe in which the scalar potential is constant, we find that

4, 0v (3) f. = -my cO - Ot

which now becomes relevant for an object undergoing acceleration.

What has been accomplished with this? This equation tells us that a reaction force proportional to and opposed to acceleration would arise as a result of what might be termed an inductive interaction between an object and the gravitational vector potential. To maintain the acceleration, one thus would have to apply a compensating motive force, f = -fq, and therefore we arrive at

4, f = m_2a . (4)

If cI_= c2 then this looks identical to Newton's equation of motion with mg standing in for mi. In other words, inertial mass in this view becomes a manifestation of the (passive) gravitational mass, and the property of inertia itself as a resistance to acceleration is merely a reaction force generated by the vector gravitational potential of the entire Universe: inertia would be a gravitational induction effect.

As intriguing as this is, there are several problems. First of all. we have simply substituted one mass for another. If inertial mass is really (passive) gravitational mass reacting to acceleration via a gravitational induction effect, then what is gravitational mass? We do not dwell on this though, because it would still be a major advance in our understanding to know that inertia is really an induction effect of gravitation, not something separate. A more serious problem is the requirement that _I_= c2. If this is not satisfied exactly the principle of equivalence is lost. Equally serious is the problem of causality. For a Universe of uniform density on average, 4_ is dominated by the most distant matter (as is evident in eqn. (1) by letting dV = 4rrrZdr). The shell of matter at distances of billions of parsecs thus dominates in producing the inertia-induction effect. But how can all of that cosmic matter in the most remote galaxies react collectively and instantaneously to any local acceleration, such as lifting a paperclip or pushing a pencil?

One might think that geometrodynamics could solve the causality problem, but it does not. According to GR, the gravitational potential at any given point in space is really a spacetime curvature. The most distant matter has already left its (retarded) local signature in the spacetime geometry of any point. This is true, but what this accomplishes is simply to specify the geodesic path for a freely moving object. Curved spacetime is no more capable of generating a force in and of itself than is flat spacetime. If an object is forced to move along some other path, i.e. to accelerate, geometrodynamics itself cannot be the source of a force. One is merely back to the square-one argument that one has to overcome the inertia of an object to make it deviate from the local geodesic; but that of course takes us full circle: one has to assume inertia to explain inertia in the context of geometrodynamics. Whether one accelerates an object in curved spacetime or in flat spacetime amounts to the same thing, viz. forced deviation from the local geodesic path. But this tells us immediately that the spacetime curvature itself does not generate forces anymore than does ordinary space. The point is that geometrodynamics does not offer any way out of the problem of instantaneous gravitational induction of a reaction force over billions of light years that appears locally as inertia in the Machian view.

Gravitomagnetism and Transient Mass Terms

A report by the National Academy of Sciences in 1986 [2] declared that "At present there is no experimental evidence arguing for or against the existence of the gravitomagnetic effects predicted by, general relativity." This report led to the publication in 1988 by Nordvedt [3] of arguments in favor of the existence of gravitomagnetism which appear to be irrefutable unless one discards both special and general relativity. One case involves the classical GR effect of light deflection by the Sun. How would the light deflection measurement be modified for an observer moving radially away from the Sun at a sufficiently large distance. This is easily calculated by a Lorentz transformation from a stationary to a moving frame with respect to the Sun. According to relativity, one can just as well assume, though, that the moving observer is stationary and the Sun is moving away: the calculated deflection had better be the same. Nordvedt show that it is not.., unless one assumes the existence of a gravitational vector potential. The effects of a gravitational potential make the two calculations agree.

But Nordvedt did more than show that gravitomagnetic effects are real: he also showed that they can be surprisingly large. If one regards the entire Universe as being in motion relative to a test particle, one can couch Mach's principle in terms of his linear-order relativistic gravitational development. Curiously though, the requirement for the Nordvedt formulation to yield them, = m 9 identity aspect of Mach's principle is 4(I_ = c2. Compare this to Eq. (4) where _ = c 2 is required to make the connection between gravitation and inertia. Given the inherent uncertainty in how to properly judge the gravitational potential of the entire Universe, a factor of four should perhaps not be worrisome.

In the discussion above, m_g was assumed to represent the gravitational mass of a small object. This is an important limitation: an ordinary object of matter will possess gravitational self-energy. Would the identity of mi with mu still hold if in addition to the summation of masses of atoms or molecules in an object one adds the mass equivalent of the interaction energy? If it is assumed that mi = m u when mu includes the self-energy term, then there results an acceleration-dependent correction to the inertial reaction of a body, or to mi in this Machian perspective. This is called the Nordvedt effect. A nice discussion of it has been given in the book by Ohanian and Ruffini and an article by Will. [4] It appears to be a necessary correction to properly account for the highly precise observations of the orbit of the moon, for example.

The Nordvedt effect and Machian inertia are very similar effects but on different scales. In Machian inertia, acceleration of an object with respect to the gravitational potential of the entire Universe generates a reaction force which we interpret as inertia and we thus attribute inertial mass mi to an object on this basis. In the Nordvedt effect, acceleration of an object with respect to the potential of its own self-interaction generates a much smaller but not necessarily negligible reaction force which we may interpret as a mass shift, 6m_. For the case of the earth, the Nordvedt effect results in a mass shift 6mi = 3.5 x 10-_mi which must be taken into account for the most precise celestial dynamics. The self-energy potential of the Earth and its acceleration are essentially unchanging in magnitude, so that _r_z, is a constant. But if rapid changes in the self-energy potentials of objects could be induced, significant changes in 6mi might result.

The Nordvedt effect was the inspiration for a series of papers by Woodward, beginning in 1990 [5], which have resulted in further development of the gravitomagnetic version of Mach's principle leading even to a patent (No. 5,280,864) for a "Method for Transiently Altering the Mass of Objects to Facilitate their Transport or Change their Stationary Apparent Weights." One application of this would allow a science-fiction sort of propellantless propulsion which Woodward has indeed likened to a Star Trek-like impulse engine.

In Box 1 we follow Woodward's arguments leading to prediction of possible transient changes in the proper mass density of any object attributable to the Nordvedt effect resulting in the relation:

_p = ( 4rrGpc2 ) c02E_Ot 2 (5)

Woodward claims that rapid changes in energy, in this case electrical energy, on the order of 101° to 1012 erg cm -3 s -1 can be induced by charging and discharging capacitors. This would result in milligram-level fluctuations in _mi, where 6mi is the integral of 6p over the device.

While minute changes in _m,/m, would be of considerable theoretical significance, it would take values near unity to be of any practical use as a means to effectively modify weight of an object. However the real potential would lie in the ability to phase the ejection and retraction of an object with changes in 5mi. This would result in creation of a net unidirectional force: throw out an object when it is heavy, retract it when it is light, and one has a seemingly miraculous means of propulsion without the use of expendible propellant. This would indeed constitute a violation of momentum conservation at the level of the device. It is difficult to say whether this does or does not violate momentum conservation at the Machian level of the entire Universe since there is no definable reference of motion for the Universe itself.

The Quantum Vacuum Approach

While the Machian approach to inertia depends on an instantaneous reaction from the most distant matter in the Universe, the alternative is a theory, which involves local interaction between the quarks and leptons in matter and the electromagnetic component of the quantum vacuum, i.e. the zero-point fluctuations. Quantum field theory predicts an enormous electromagnetic zero-point energy density for these fluctuations which can be understood from the Heisenberg uncertainty relation. The uncertainty relation states that the ground state of a harmonic oscillator has a non-zero minimum energy of haJ/2 because an oscillator cannot simultaneously be exactly at the bottom of its potential well and have exactly zero momentum. The same logic applies to the electromagnetic field, which is quantized"by the association of a quantum mechanical harmonic oscillator with each mode k of the radiation field." [6] Summing up the energy over the modes for all frequencies, directions, and polarization states, one arrives at a zero-point energy density for the electromagnetic fluctuations of

'x'..... --d.., (6) W" = _0"_...... p=p(_')d,_' = frO0 h,..,¢ 21T, 3 2C3

where, z, n,_ is a postulated cutoff in frequency.

There is an obvious problem: Beyond what frequency do the zero-point fluctuations cease and why? One plausible cut-off is the Planck frequency which originates from the following considerations. The minimum quantum size of an object is roughly a sphere whose Compton radius is h/mc. The Schwarzschild radius for the same object is Grn/c 2. Any object so dense that the two radii become the same would put the two conflicting requirements of quantum physics and GR in direct opposition: a further compression should lead to collapse to a mini-black hole, yet the uncertainty relation should forbid any further collapse. This density corresponds to a Planck mass (2.2 × 10⁻⁵ g) in a sphere whose radius is the Planck length (1.6 × 10⁻³³ cm). The Planck length is thus usually interpreted as the smallest allowable physical interval of space. The Planck time is the time it would take light to traverse one Planck length: the Planck frequency is the inverse of that, _'p = (4"_2c5/Gh) U2 = 1.2 × 10⁴⁴ rad s⁻¹.

Assuming that w'...... = a.,e results in a zero-point energy density of ~10¹¹⁵ ergs cm⁻³. Adler, Casey and Jacob [7] have dubbed this the vacuum catastrophe to parallel the ultraviolet catastrophe that Planck and other physicists faced in 1900: the problem being that if one naively assumes that the energy density of the electromagnetic fluctuations gravitates, the Universe should be microscopic in size, yet the arguments leading to the existence of zero-point fluctuations are quite fundamental and so these fluctuations cannot just be dismissed out of hand. The enormity of this energy density is certainly worrisome, yet the useful concept of the Dirac sea, for example, suffers a similar problem.

As summarized some years ago by Sir William McCrea [8] there are numerous phenomena which point to the reality of zero-point fluctuations. One is spontaneous emission: it can almost (there is a nagging factor of two) be attributed to stimulation by the zero-point fluctuations. This would neatly account for the inhibition of spontaneous emission in suitable cavities. Writing on cavity quantum electrodynamics involving suppression of spontaneous emission Haroche and Raimond [9] raise a paradox:

These experiments indicate a counterintuitive phenomenon that might be called "no-photon interference." In short, the cavity prevents an atom from emitting a photon because that photon would have interfered destructively with itself had it ever existed. But this begs a philosophical question: How can the photon "know," even before being emitted, whether the cavity is the right or wrong size?

There is no such paradox if the inhibition of spontaneous emission reflects merely a reduction by the cavity of the zero-point fluctuations which are actually doing the stimulating which only appears to be spontaneous.

The effect most often attributed to the zero-point fluctuations is the Casimir force which has recently been well measured [10]. One physical interpretation of the Casimir force is that it is a radiation pressure from the zero-point fluctuations [11]; however the Casimir force, and other effects such as the Lamb Shift and van der Waals forces, can equally be attributed to either radiation-reaction fields (due to the quantum motions of particles) or to the vacuum zero-point fluctuations; and most characteristically to combinations of both, in several possible proportions, according to the various possible equivalent orderings of the creation and annihilation quantum operators. [12]

The ontological status of the electromagnetic zero-point fluctuations thus remains an outstanding problem. However the discipline of stochastic electrodynamics (SED) has demonstrated the usefulness of treating the zero-point fluctuations as if they constituted real electromagnetic fields with average energy hw'/2 in each mode and using the techniques of classical electrodynamics to solve quantum problems. [13] The random electromagnetic fluctuations provide a physical mechanism for the spread in particle position, momentum, energy etc. that quantum wave functions normally represent. It is possible, for example, to derive the blackbody spectrum without the assumption of quantization using SED. [14] Using SED a local origin for inertia can be attributed, at least in the sense of its electromagnetic aspect, to the interactions between the quarks and leptons in matter and the electromagnetic zero-point fluctuations. This is interesting as it indicates that a more advanced theory should produce all inertia reaction force coming from the vacua of its quantized fields. A corollary of this SED analysis also results in an electromagnetic basis for interpreting the de Broglie wavelength of a moving object.

Another major objection to a real ZPF has to do with its presumed gravitational effect. According to general relativity theory, the energy density of the ZPF would generate an enormous spacetime curvature, akin to a huge cosmological constant. This is, of course, true in the standard interpretation of mass-energy. However one has to be careful to maintain self-consistency when comparing theoretical models: the ZPF-inertia concept implies, via the principle of equivalence, that gravitation must also have a connection to the ZPF (along lines conjectured by Sakharov in 1968). If that is the case, then the ZPF cannot gravitate, because gravitation would involve the interaction of the ZPF with fundamental particles, not with itself. The energy density of the ZPF could then no longer be naively equated to a source of gravitation. Such an electromagnetically-based theory of gravitation has only undergone a preliminary development, but it does appear that the general relativistic curvature of spacetime can be mimicked by a vacuum having variable dielectric properties in the presence of matter. This raises the question of whether spacetime is actually physically non-Euclidean or whether our measurements of curvature merely reflect light propagation through a polarizable medium (the vacuum itself). Since the assumed curvature of spacetime is measured (by definition) via light propagation, there may be no way to distinguish one from the other: curved spacetime vs. light propagation with a dielectrically-modified speed-of-light. (We note that Einstein himself spent many years looking for an electromagnetic basis for gravitation, albeit unsuccessfully.)

An Electromagnetic Basis for Mass and the Wave Nature of Matter

In 1994 a first attempt was made, using SED, to find a connection between inertia and the zero-point fluctuations. [15] This was successful in that it demonstrated that the magnetic component of the zero-point fluctuations acting on a classical Planck oscillator would generate a reaction force proportional to the acceleration of the oscillator. (The acceleration of the oscillator was in the direction perpendicular to the oscillation.) In this representation then, inertia is actually the electromagnetic Lorentz force provided by the zero point fluctuations. There were several limitations to this approach: (1) the analysis was dependent on a very specific interaction between the zero-point fluctuations and the fundamental particles constituting matter, namely that of a classical Planck oscillator; (2) the requisite mathematical development was sufficiently complex so as to make it difficult to assess the validity; and (3) the interaction was assumed to take place at a presumed very high frequency (_.,p) cutoff of the zero-point fluctuations.

Thanks to a NASA research contract a completely new approach was carried through which proved to be analytically simpler and yet at the same time yielded the proper relativistic equation of motion, Y = dP/dr, from electrodynamics as applied to the zero-point fluctuations. [16] The analysis hinged on finding the Poynting vector of the zero-point fluctuations in an accelerating frame of reference. Due to the perfect randomicity of the fluctuations, no net energy flux accompanies the huge energy density of eqn. (6). That is why, in principle at least, it is possible to conceive of this vast sea of zero-point energy filling the universe without apparent electromagnetic consequences: it is perfectly uniform and isotropic, inside and outside all matter. All other electromagnetic radiation that we see and measure is over and above this apparently vast electromagnetic ground state.

Once again using SED, but this time concentrating solely on the electromagnetic fields of the zero-point fluctuations it was possible to show that the Poynting vector becomes non-zero when viewed from an accelerating frame, and that in the subrelativistic regime the strength of the Poynting vector increases linearly with the acceleration. A non-zero Poynting vector implies a non-zero momentum flux, the two being related by simply a factor of c. If we assume that the quarks and electrons in atoms of matter scatter this radiation in the same way that ordinary electromagnetic radiation would be scattered, then a net reaction force on matter results from the scattering of the momentum flux of the zero-point fluctuations. This reaction force is proportional to acceleration, and indeed owing to the fact that the transformation of the electromagnetic zero-point fluctuations from a stationary to an accelerating frame can be carried through exactly, the resulting equation of motion proves to have the relativistically correct form:.T = d'P/dr.

The resulting expression for the electromagnetic parameter that behaves like inertial mass is

,,,, = _ ,j(_lo_(_,)d_, (r) C"

where,/(w,) is a frequency-dependent fraction ranging from zero to, perhaps, unit3,. This "mass", mi, is actually a manifestation of an electromagnetic reaction force. It is assumed that momentum is carried by the electromagnetic fields of the zero-point fluctuations, and that this momentum is transferred to massless scattering centers throughout any object (the quarks and electrons in atoms of matter) resulting in a reaction force that is identical to what would ordinarily be called the inertia of the object. The physical interpretation of eqn. (7) is that some fraction rl(,; ) of the energy of the zero-point fluctuations at frequency _., instantaneously contained in the volume, V0, of an object is scattered, i.e. is the part of the total ZPF energy that actually interacts with the object.

It was speculated that the scattering parameter, 7l(,a), would be found to be a resonance at some frequency, rather than be associated with the cutoff frequency of the zero-point fluctuations as in the 1994 approach. A very interesting corollary follows from this assumption. It was proposed by de Broglie that an elementary particle is associated with a localized wave whose frequency is the Compton frequency, yielding the Einstein-de Broglie equation: t_oc = rnoc2. (8)

As summarized by Hunter [17]: '... what we regard as the (inertial) mass of the particle is, according to de Broglie's proposal, simply the vibrational energy (divided by c 2) of a localized oscillating field (most likely the electromagnetic field). From this standpoint inertial mass is not an elementary property of a particle, but rather a property derived from the localized oscillation of the (electromagnetic) field. De Broglie described this equivalence between mass and the energy of oscillational motion.., as 'une grande loi de la Nature' (a great law of nature)." The rest mass r_0 is simply mi in its rest frame. What de Broglie was proposing is that the left-hand side of eqn. (8) corresponds to physical reality; the right-hand side is in a sense bookkeeping, defining the useful but not truly ontological concept of rest mass.

This perspective is consistent with the proposition that inertial mass, m,, is also not a fundamental entity, but rather a coupling parameter between particles and the zero-point fluctuations, i.e. the vacuum fields if we contemplate prospective generalizations of our approach. De Broglie assumed that his wave at the Compton frequency originates in the particle itself. An alternative interpretation is that a particle "is tuned to a wave originating in the high-frequency modes of the zero-point background field." [12][18] The de Brogiie oscillation would thus be due to a resonant interaction with the zero-point fluctuations, presumably the same resonance that is responsible for creating inertial mass as in eqn. (7). In other words, the zero-point fluctuations would be driving this _'c oscillation of a fundamental particle, such as the electron. These particle oscillations were named zitterbewegung by Schrödinger.

We therefore suggest that an elementary charge driven to oscillate at the Compton frequency by the zeropoint fluctuations may be the physical basis of the r/(w) scattering parameter in eqn. (7). For the case of the electron, this would imply that q(_.') is a sharply-peaked resonance at the frequency, expressed in terms of energy, h_ = 512 keV. The inertial mass of the electron would physically be the reaction force due to scattering of the zero-point fluctuations at that resonance.

This leads to a surprising corollary. It can be shown that as viewed from a laboratory frame, the standing wave at the Compton frequency in the electron's own rest frame transforms into a traveling wave having the de Broglie wavelength,,kB = hip, for a moving electron, as first measured by Davisson and Germer in 1927. The wave nature of the moving electron appears to be basically due to Doppler shifts associated with its Einstein-de Broglie resonance frequency. This has been shown in detail in the monograph of de la Peña and Cetto [12] (see also Kracklauer [18]). The approach described above thus suggests very intriguing connections between electrodynamics, inertia and the quantum wave nature of matter.

Mach’s Principle or Quantum Vacuum?

The Machian approach to inertia as developed by Woodward has led to a remarkable prediction, viz. that transient changes in mass may be achieved via the inflow and outflow of electrical energy to a device. Such transient mass changes could even result in the generation of a net unidirectional force which could serve for propulsion. The NASA Breakthrough Propulsion Physics program has selected an investigation by John Cramer of the University of Washington to attempt to experimentally verify this prediction. It is not yet known whether the quantum vacuum approach to inertia will make the same or an analogous prediction. Since the quantum vacuum approach finds mass to be, in part at least, an electromagnetic phenomenon it would not be surprising to find some way to electromagnetically vary inertial mass.

The Machian approach states that inertial mass is the very same thing as gravitational mass, the latter being the interaction of matter with the scalar gravitational potential, the former with an additional vector gravitational potential. Nordvedt has shown why such a vector potential must exist. The Machian approach simplifies things by reducing the types of mass -- by having inertial mass and gravitational mass be the same thing -- but it does not offer any new explanation of mass itself. Moreover there is the problem that for deviations from geodesic motion there is no explanation for why a reaction force arises which must be overcome by a motive force to bring about the acceleration. Geometrodynamics can only specify which path a free particle will take: it cannot generate forces to oppose motion on a non-geodesic path. To some extent one could argue that the Machian approach must therefore really assume inertial mass as the fundamental entity, and that gravitational mass must be a form of inertial mass, rather than vice versa. The bottom line is that it may be an accomplishment to link inertia and gravitational mass via a gravitational vector potential, the concept of mass as an intrinsic feature of matter of one sort (gravitational) or the other (inertial) still lies at the root of Machian inertia.

The major weakness of the Machian approach is that it would appear to call for an instantaneous and collective reaction of cosmically remote matter to any local acceleration. The quantum vacuum approach, by contrast, is based on local interaction, but one can argue that it too has its own major weakness: that one must accept the existence of a zero-point ground state of electromagnetic fluctuations of enormous energy density in the first place. However if one does this, one can arrive at a purely local explanation of inertia which does do away with the concept of inertial mass itself, interpreting it as simply a background vacuum fields force. If one also assumes that the interactions between the quarks and electrons in matter takes place at a resonance frequency identified with the Compton frequency, then one can also provide a new physical interpretation for the wave nature of matter as described by the de Broglie wavelength of a moving object. One has therefore arguably suggested the path for a true reduction in fundamental concepts from the quantum vacuum approach.

The issue of binding energies and fundamental particle masses is an area where the quantum vacuum approach to inertia may have an opportunity to make predictions that a Machian approach might not. If the scattering of zero-point radiation takes place at specific resonances, then there may be the opportunity to discover why, for example, a muon appears to be just a heavy electron via arguments based on resonance frequencies. A muon might just be an electron excited to a higher resonance. Similarly, the resonance of an ensemble of bound quarks would not be expected to be simply a linear function of the number of quarks. The 12 quarks bound together in a He nucleus would not be expected to have the same resonance as the sum of the four triplets of quarks in two protons and two neutrons. Changes in resonance thus afford a potential explanation for binding energies. Moreover in the quantum vacuum approach to inertia there is no need to postulate that one thing, mass, can be converted into something else, energy (and vice versa) via the E = mc 2 relationship. All forms of mass really trace back to the energy of the zero-point fluctuations and their association with zitterbewegung of and scattering by fundamental particles.

A massive neutrino poses no known problem for the Machian perspective, but the quantum vacuum approach in its restricted electromagnetic zero-point field formulation could not explain the mass of a truly charge-free particle. However it is important to bear in mind that the mass determination of the neutrino is not a direct measurement of inertial mass: it is all indirect inference based on a measurement of muon-to-electron neutrino populations resulting from cosmic rays. The existence of mass is then inferred from application of the current standard model. Since the quantum vacuum approach offers a completely new interpretation of mass itself, this indirect inference based on the current standard model may prove to be inappropriate.

It is also important to bear in mind that no particle is truly charge-free. The purely electromagnetic derivation of inertia from ZPF [14][15], as a necessary simplifying measure, glosses over the existence of other fields which must have their own zero-point oscillations, and with which particles must interact. It is known that electromagnetism is merely one aspect of a more general electroweak interaction. Neutrinos, while electrically neutral, have a nonzero coupling to the "weak" aspects of the electroweak force and so must interact with their quantum vacuum oscillations. A fully rigorous theory of ZPF-based inertia must deal with the quantum vacua, not only of electromagnetism, but of the full electroweak force and of quantum chromodynamics as well. The current, purely electromagnetic theory is known to be incomplete, and we should not be surprised that it omits such features as possible neutrino masses.

Acknowledgments

We acknowledge support of NASA contract NASW-5050 for this work.

(Box 1 — Derivation of the Woodward Effect, Box 2 — The Zero-Point Field in Quantum Physics, and the reference list are omitted for length; the complete text is at the source.)

On the relation between inertial mass and quantum vacua

Bernhard Haisch (Solar and Astrophysics Laboratory, Dept. H1-12, Bldg. 252, Lockheed Martin, 3251 Hanover Street, Palo Alto, California 94304) and Alfonso Rueda (Department of Electrical Engineering & Department of Physics, ECS Building, California State University, Long Beach, California 90840).

1. Introduction

In his 1905 paper "On the Electrodynamics of Moving Bodies" Einstein eliminated the notions of a mechanical ether and of an absolute frame of rest [1]. A consequence of his resulting principle of relativity was the abandonment of the concepts of absolute space and of absolute time. The new mechanics of relativity replaced that of Newton through an epistemological change in foundation: relativity is founded upon physically measureable quantities rather than abstract concepts such as absolute space and absolute time. It is the observation of light signals that defines the lengths of rulers and durations of time intervals. A similar emphasis on measureable quantities is the basis of the standard interpretation of quantum mechanics. We propose that such an epistemology of observables is also appropriate for the interpretation of the concept of a mass.

The existence of matter is self-evident and fundamental: we are made of matter. Mass however -- like absolute space and time -- is an abstraction. Though it is usually regarded as an innate property of matter, mass is not in fact directly observable. The mass we habitually attribute to matter manifests in two ways: as a force and as energy. In classical mechanics, one applies a force, f, to an object and measures its resultant acceleration, a. The force and the acceleration are the observables. We relate these two observables by assuming the existence of an innate property of matter known as inertial mass and thus we write f = ma. The existence of an innate inertial mass, m, is an inference and an abstraction. Using the methodology of stochastic electrodynamics [3] it has been shown that it may be possible to view Newton's equation of motion, f = ma, as well as its relativistic generalization, 7- = dT)/dr, as a consequence of the electromagnetic zero-point field (ZPF) or more generally of the quantum vacuum fields [4] [5]. Of course the situation is more complex in that quantum vacua other than the electromagnetic ZPF must presumably also be involved. The resistance to acceleration attributed to the existence of inertial mass in matter appears to be logically and quantitatively attributable instead to a resistance on accelerated matter due to the zero-point vacuum fields. In other words, inertia would appear to be a kind of reaction force that springs into existence out of the quantum vacuum whenever acceleration of an object takes place, for reasons given below. The rn in f = ma thus would become a coupling parameter that quantifies a more fundamental relationship between the elementary charged particles (quarks and electrons) in matter and the surrounding vacuum. This is not inconsistent with the ordinary concepts of momentum and kinetic energy which are calculated using the same m. Momentum and kinetic energy of a moving object can take on any value depending on the relative motion of the observer. It is only changes in momentum or kinetic energy that manifest as real measureable effects when a collision or a mechanical interaction takes place. Momentum, kinetic energy and mass itself are useful bookkeeping tools that become manifest only upon acceleration. Our attempts to link inertia to the actions of the quantum vacuum have been limited to the electromagnetic zero-point field. We have not considered the zero-point fields of the weak or strong interactions.

A lucid discussion concerning the epistemology of observables is found in Phillip Frank's "Einstein, Mach and Logical Positivism" [2]. The influence on the early work of Einstein (up to approximately 1920) by Mach and his Logical Positivistic viewpoint is widely known. The emphasis on observables as the essence of scientific verification was widely promoted by the thinkers of the Vienna Circle and by Auguste Compte.

In the electromagnetic case, this comes about through the Poynting vector of the ZPF: in an accelerating reference frame it becomes non-zero and proves to be proportional to acceleration, b A non-zero Poynting vector implies a non-zero radiative momentum flux transiting any accelerating object. If one assumes that the quarks and electrons in such an object scatter this radiation, stochastic electrodynamics shows that there will result a reaction force on that accelerating object having the form fr = -aa, where the a parameter quantifies the strength of the scattering process. In order to maintain the state of acceleration, a motive force f must continuously be applied to balance this reaction force L. Applying Newton's third law to the region of contact between the agent and the object, f = -f_, we thus immediately arrive at f = ca, which is identical to Newton's equation of motion. However now a parameter originating in the zero-point field scattering, a, accomplishes the very thing that inertial mass, m, is assumed to do: resist acceleration. One can conceptually replace inertial mass. m, by a ZPF-based parameter representing a scattering process, _. We discuss this relationship in § 4. This is not merely a trivial substitution of nomenclature: Taking this approach one may be able to eliminate a postulate of physics. Newton's second law. f = ma, may cease to be fundamental as it can be derived from the vacuum fields plus the third law. Newton's third law of action and reaction would be axiomatic; Newton's second law would not. For practical purposes one can retain the concept of inertial mass, m, while realizing that it is not physically fundamental. One might regard mass in the same way, one makes use of a classical thermodynamic parameter, such as heat capacity, for example. The measurable heat capacity of a given substance is a useful concept, but we know that it really represents an ensemble of atomic processes at a more fundamental level. So it appears to be with inertial mass as well: it represents a more fundamental vacuum process involving interactions like that between the ZPF and the particle and anti-particle pairs in the Dirac sea. Inertial mass would be due to interactions between the ZPF and the quarks and electrons constituting the matter of an accelerating object, c In conventional QCD the proton and neutron masses are explained as being primarily the energies associated with quark motions and gluon fields. That sort of reasoning is considered sufficient explanation of nucleon masses, but the quantum vacuum-inertia hypothesis addresses the possibility that there is a deeper level to the nature of mass by asking where inertia itself comes from. We are proposing that there is a physical basis underlying the reaction force that characterizes inertia. If this is true, that would certainly be a deeper explanation than simply saying that there is so much energy (mass) in the quark motions and gluon fields and by definition that such energy (mass) simply resists acceleration. Where does the specific reaction force that opposes acceleration come from? Why does mass resist acceleration? Even more puzzling, why does the energy equivalent of mass resist acceleration? One possibility is that this will never be solved and forever remain a mystery. Another possibility is that this can be explained and that the present approach offers a truly new insight. Inertial mass is only one of several manifestations of the concept of mass. If a ZPF-scattering process can account, at least in part, for inertial mass is there an analogous basis for the E = mc 2 relation? This equation is Universally regarded as a statement that one kind of thing (energy) can be transformed into a totally different kind of thing (mass), and vice versa. Following an epistemology of observables, we propose that this is not the case, and that just as the physical reality of inertial mass is force, the physical reality of rest mass is energy. In a preliminary attempt to develop the Sakharov [8] conjecture of a vacuum-fluctuation model for gravity, Hestenes and Krüger [9] proposed that the E = mc 2 relationship reflected the internal energy associated with Zitterbewegung of fundamental particles (see also Puthoff [10] for a similar suggestion). Zitterbewegung, so named by Schrödinger [11], can be understood as the oscillatory motion associated with the center of charge operator in the electron with respect to the center of mass operator. It can be interpreted as a motion of the center of charge around the averaged center of mass point. It is attributed in stochastic electrodynamics to the fluctuations induced by the ZPF. In the Dirac theory of the electron the eigenvalues of the Zitterbewegung velocity are +c (see [12]), and the amplitude of these oscillations are on the order of the Compton wavelength. In the view proposed by Puthoff, the rest mass of a particle is actually the field energy associated with point charge particle oscillations driven by the ZPF.If that is the case, there is no problematic conversion of mass into energy or enigmatic creation of mass from energy, but rather simply a concentration or liberation of ZPF-associated energy. Here too mass may become a useful but no longer fundamental concept. This approach appears to allow yet another reduction in physical postulates. Just as the laws of electrodynamics applied to the ZPF appear to explain and support a former postulate of physics (f = ma) via a new interpretation of inertial mass, a postulate of quantum mechanics can be derived via a new interpretation of rest mass as the energy of ZPF-driven Zitterbewegung: The de Broglie relation for the wavelength of a moving particle, A = h/p, may be derived from straightforward application of relativity theory. This is discussed in § 5. There is one final mass concept: gravitational mass. Einstein's principle of equivalence dictates that inertial and gravitational mass must be the same. Therefore if inertial mass is a placeholder for vacuum field forces that arise in accelerating reference frames, then there must be an analogous connection between gravitation and vacuum fields. The attempt of Puthoff a decade ago to develop the Sakharov conjecture along the lines of a stochastic electrodynamics approach seemed promising, but now we know that it needs considerable further development. [13] We limit our discussion on gravitation to some comments on this and on the associated problem of the cosmological constant in § 6. To summarize the view that emerges, all energy and momentum that we normally associate with matter appears to actually reflect some part of the energy and momentum of the underlying vacuum. The classical kinetic energy, T = my2/2, or momentum, /Y = rag, that we ascribe to an object depend entirely on the relative motion of the object and the observer. Both T and/Y are necessarily calculated quantities; a real observation only arises when object and observer are made to closely interact, e.g. when brought together into the same frame, which is to say when a collision occurs. But to achieve that requires a change in velocity, and it is precisely upon deceleration that the vacuum generates a reaction force that is called the inertial reaction force which Newton took to be an irreducible property of the so-called inertial mass, m. Again, we may retain the concept of inertial mass as a convenient bookkeeping tool for kinetic energy, momentum and other calculations, but the actual observable measurement of forces can be traced back to the vacuum reaction force on the most elementary components of matter (e.g., in the electromagnetic case, quarks and electrons) that accompanies acceleration.

b In this respect, the fact that here we deal with a vector field that has a Poynting vector and not with a scalar field may be critical. For simple scalar fields such a resistance opposing acceleration is not present. This has been reviewed and studied by, e.g., Jaekel and Raynaud [6]. Here however [5] we are dealing with a vector field with a well defined Poynting vector and associated momentum density c j. p. Vigier [7] has proposed that there is a contribution to inertia due to the interaction of the accelerated particle with the surrounding virtual particles of the Dirac vacuum.

(Section 2, ‘Historical remarks on the zero-point field of Stochastic Electrodynamics’, is omitted for length; the complete text is at the source.)

3. The zero-point field in accelerating reference frames

The ZPF spectral energy density

4_rh_ 3 [)Zp(ll) -- C3 (3)

would indeed be analogous to a spatially uniform constant offset that cancels out when considering net energy fluxes. However an important discovery was made in the mid-1970s that showed that the ZPF acquires special characteristics when viewed from an accelerating frame. In connection with radiation from evaporating black holes as proposed in 1974 by Hawking [27],Davies [28] and Unruh [29], working independently, determined that a Planck-like component of the ZPF will arise in a uniformly-accelerated coordinate system, namely one having a constant proper acceleration a (where a = la[) with what amounts to an effective "temperature"

fia To - (4) 2Trek"

This "temperature" does not originate in emission from particles undergoing thermal motions, d As discussed by Davies, Dray and Manogue [30]:

One of the most curious properties to be discussed in recent years is the prediction that an observer who accelerates in the conventional quantum vacuum of Minkowski space will perceive a bath of radiation, while an inertial observer of course perceives nothing. In the case of linear acceleration, for which there exists an extensive literature, the response of a model particle detector mimics the effect of its being immersed in a bath of thermal radiation (the so-called Unruh effect). This "heat bath"is a quantum phenomenon. The "temperature" is negligible for most accelerations. Only in the extremely large gravitational fields of black holes or in high-energy particle collisions can this become significant. This effect has been studied using both QED[28][29] and in the SED formalism[31].For the classical SED case it is found that the spectrum is quasi-Planckian in To. Thus for the case of zero true external thermal radiation (T = 0) but including this acceleration effect (To), eqn. (3) becomes [31] ¢

d One suspects of course that there is a deep connection between the fact that the ZPF spectrum that arises in this fashion due to acceleration and the ordinary blackbody spectrum have identical form.

p(u, To)- ca 1+ _ +e_"/_r,,-1 '

where the acceleration-dependent pseudo-Planckian component is placed after the h_,/2 term to indicate that except for extreme accelerations (e.g. particle collisions at high energies) this term is negligibly small. While these additional acceleration-dependent terms do not show any spatial asymmetry in the expression for the ZPF spectral energy density, certain asymmetries do appear when the (vector) electromagnetic field interactions with charged particles are analyzed, or when the momentum flux of the ZPF is calculated. The ordinary plus a 2 radiation reaction terms in eqn. (12) of HRP mirror the two leading terms in eqn. (5). An analysis was carried through by HRP and this resulted in the apparent derivation of at least part of Newton's equation of motion, f = ma, from Maxwell's equations as applied to the ZPF. In that analysis it appeared that the resistance to acceleration known as inertia was in reality the electromagnetic Lorentz force stemming from interactions between a charged particle (such as an electron or a quark) treated as a classical Planck oscillator and the ZPF, i.e. it was found that the stochastically-averaged expression < Vo.,_ x B ZP > was exactly proportional to and in the opposite direction to the acceleration a. The velocity Vo._ represented the internal velocity of oscillation induced by the electric component of the ZPF, E zP, on the harmonic oscillator. This internal motion was restricted to a plane orthogonal to the external direction of motion (acceleration) of the particle as a whole. The Lorentz force was found using a perturbation technique due to Einstein and Hopf [33]. Owing to its linear dependence on acceleration we interpreted this resulting force as a contribution to Newton's inertia reaction force on the particle. The HRP analysis can be summarized as follows. The simplest possible model of a particle (which, following Feynman's terminology, we referred to as a parton) is that of a harmonically-oscillating point charge ("Planck oscillator"). Such a model would apply to electrons or to the quarks constituting protons and neutrons for example. Given the peculiar character of the strong interaction that it increases in strength with distance, to a first approximation it is reasonable in such an exploratory attempt to treat the three quarks in a proton or neutron as independent oscillators. This Planck oscillator is driven by the electric component of the ZPF, E zP, to harmonic motion, Vo_, assumed for simplicity to be in a plane. The oscillator is then given a constant proper acceleration, a, by an independent, external agent. This acceleration is in a direction perpendicular to that plane of oscillation, i.e. perpendicular to the Vo,_ motions. New components of the ZPF will appear in the frame of the accelerating particle having the spectral energy density given in eqn. (5). The leading term of the acceleration-dependent terms is taken; the electric and magnetic fields are transformed into a constant proper acceleration frame using well-known relations. The Lorentz force arising from the acceleration-dependent part of the B zv acting upon the Planck oscillator is calculated. This is found to be proportional to acceleration. The constant of proportionality is interpreted as the inertial mass, 77_/,of the Planck oscillator and thus at least as a contribution to the total mass of the particle. This inertial mass, m,, is a function of the Abraham-Lorentz radiation damping constant, F, of the oscillator and of the interaction frequency with the ZPF,

However, further analysis by Boyer [32] showed that although the spectrum of the fields in an accelerated frame is correctly given by eqn. (5), a dipole oscillator attached to the frame will have an additional radiation reaction term that exactly compensates for the additional factor 1 + (a/Jrc_,) in eqn. (5). As a result the detector will still detect only a Planckian spectrum insofar as the scalar detector-ZPF interaction is concerned.

rt .o (G)

where we have written uo to indicate that this may be a resonance rather than the cutoff assumed by HRP. Since both F and uo are unknown (but see § 5) we can make no absolute prediction of mass values in this simple model. Nevertheless, if correct and considering only the electromagnetic interaction, the HRP concept substitutes for Mach's principle a very specific electromagnetic effect acting between the ZPF and the charge inherent in matter. Inertia appears as an acceleration-dependent electromagnetic (Lorentz) force. Newtonian mechanics would then be derivable in principle from the ZPF via Maxwell's equations and in the more general case from the other vacuum fields also. Note that this coupling of the electric and magnetic components of the ZPF via the technique of Einstein and Hopf is very similar to that found in ordinary electromagnetic radiation pressure. A similar observation, we conjecture, should hold for the other vacuum fields. So we conclude that inertia appears as a radiation pressure exerted by the fields in the vacuum opposing the acceleration of material elementary particles.

4. The relativistic formulation of inertia from the ZPF Poynting vector

The oversimplification of an idealized oscillator interacting with the ZPF as well as the mathematical complexity of the HRP analysis are understandable sources of skepticism, as is the limitation to Newtonian mechanics. A relativistic form of the equation of motion having standard covariant properties has been obtained [5], which is independent of any particle model, relying solely on the standard Lorentz-transformation properties of the electromagnetic fields. Newton's third law states that if an agent applies a force to a point on an object, at that point there arises an equal and opposite force back upon the agent. Were this not the case, the agent would not experience the process of exerting a force and we would have no basis for mechanics. The mechanical law of equal and opposite contact forces is thus fundamental both conceptually and perceptually, but it is legitimate to seek further underlying connections. In the case of a stationary object (fixed to the earth, say), the equal and opposite force can be said to arise in interatomic forces in the neighborhood of the point of contact which act to resist compression. This can be traced more deeply still to electromagnetic interactions involving orbital electrons of adjacent atoms or molecules, etc. A similar experience of equal and opposite forces arises in the process of accelerating pushing on) an object that is free to move. It is an experimental fact that to accelerate an object a force must be applied by an agent and that the agent will thus experience an equal and opposite reaction force so long as the acceleration continues. It appears that this equal and opposite reaction force also has a deeper physical cause, which turns out to also be electromagnetic and is specifically due to the scattering of ZPF radiation. Rueda g: Haisch [5] demonstrated that from the point of view of the pushing agent there exists a net flux (Poynting vector) of ZPF radiation transiting the accelerating object in a direction opposite to the acceleration. The scattering opacity of the object to the transiting flux creates the back reaction force called inertia. The new approach is less complex and model-dependent than the HRP analysis in that it assumes simply that the elementary particles in any material object interact with the ZPF in some way that is analogous to ordinary scattering of radiation. It is well known that treating the ZPF-particle interaction as dipole scattering is a successful representation in that the dipole-scattered field exactly reproduces the original unscattered field radiation pattern in unaccelerated reference frames.j14] It is thus likely that dipole scattering is an appropriate way -- at least to first order -- to describe the ZPF-particle interaction, but in fact for the more general RH analysis one simply needs to assume that there is some dimensionless efficiency factor, r/, that describes whatever the process is (be it dipole scattering or not). We suspect that r/(_o) contains one or more resonances -- and in the following section discuss why this resonance likely involves the Compton frequency -- but again this is not a necessary assumption. The new approach relies on making standard transformations of the E "p and B -'v from a stationary to an accelerated coordinate system.j34] In a stationary or uniformly-moving frame the E -'v and B =p constitute an isotropic radiation pattern. In an accelerated frame the radiation pattern acquires asymmetries. There is thus a non-zero Poynting vector in any accelerated frame carrying a non-zero net flux of electromagnetic momentum. The scattering of this momentum flux generates a reaction force, f_. Moreover since any physical object will undergo a Lorentz contraction in the direction of motion the reaction force. L, can be shown to depend on %, the Lorentz factor (which is a function of proper time, r, since the object is accelerating).[5] Rueda and Haisch find [5] that

71'l i = ,(_')pzP(_') &', (7)

where Pzp is the well known spectral energy density of the ZPF:

h.j 3 (*zr(._)- 2,.r.c 3 • (S)

The momentum of the object is of the form

p = nL%v_. (9)

Thus, one can also obtain the relativistic equation of motion [5]

d7) d .7- dr - dr (%'mic' p ) (10)

The origin of inertia, in this picture, becomes remarkably intuitive. Any material object resists acceleration because the acceleration produces a perceived flux of radiation in the opposite direction that scatters within the object and thereby pushes against the accelerating agent. Inertia in the present model appears as a kind of acceleration-dependent electromagnetic vacuum-fields drag force acting upon elementary charged particles.

5. Inertial mass and the de Broglie relation for a moving particle

De Broglie proposed that an elementary particle is associated with a localized wave whose frequency is the Compton frequency, yielding the Einstein-de Broglie equation:

hwc = moc 2. (11)

As summarized by Hunter [35]: "... what we regard as the (inertial) mass of the particle is. according to de Broglie's proposal, simply the vibrational energy (divided by c 2) of a localized oscillating field (most likely the electromagnetic field). From this standpoint inertial mass is not an elementary property of a particle, but rather a property derived from the localized oscillation of the (electromagnetic) field. De Broglie described this equivalence between mass and the energy of oscillational motion.., as 'une grande loi de la Nature' (a great law of nature)." The rest mass m0 is simply mi in its rest frame. What de Broglie was proposing is that the left-hand side of eqn. (11) corresponds to physical reality; the right-hand side is in a sense bookkeeping, defining the concept of rest mass. This perspective is consistent with the proposition that inertial mass, mi, is also not a fundamental entity, but rather a coupling parameter between electromagnetically interacting particles and the ZPF as discussed above. De Broglie assumed that his wave at the Compton frequency originates in the particle itself. An alternative interpretation is that a particle "is tuned to a wave originating in the high-frequency modes of the zero-point background field."[36] The de Broglie oscillation would thus be due to a resonant interaction with the ZPF, presumably the same resonance that is responsible for creating a contribution to inertial mass as in eqn. (7). In other words, the ZPF would be driving this a,'c oscillation. We therefore suggest that an elementary charge driven to oscillate at the Compton frequency, :oc, by the ZPF may be the physical basis of the r](a_) scattering parameter in eqn. (7). For the case of the electron, this would imply that r/(_,) is a sharply-peaked resonance at the frequency, expressed in terms of ener_,, h,_'c = 512 keV. The inertial mass of the electron would physically be the reaction force due to resonance scattering of the ZPF at that frequency. This leads to a surprising corollary. It can be shown that as viewed from a laboratory frame, the standing wave at the Compton frequency in the electron frame transforms into a traveling wave having the de Broglie wavelength, AB = h/p, for a moving electron. The wave nature of the moving electron appears

to be basically due to Doppler shifts associated with its Einstein-de Broglie resonance frequency. This has been shown in detail in [36]. Assume an electron is moving with velocity v in the +x-direction. For simplicity consider only the components of the ZPF in the 4-x directions. The ZPF-wave responsible for driving the resonant oscillation impinging on the electron from the front will be the ZPF-wave seen in the laboratory frame to have frequency 4'_ = "ywc(1 -v/c), i.e. it is the wave below the Compton frequency in the laboratory that for the electron is Doppler shifted up to the wc resonance. Similarly the ZPF-wave responsible for driving the electron resonant oscillation impinging on the electron from the rear will have a laboratory frequency _'+ = "y: vc(1 + v/c) which is Doppler shifted down to wc for the electron. The same transformations apply to the wave numbers, k+ and k_. The Lorentz invariance of the ZPF spectrum ensures that regardless of the electron's (unaccelerated) motion the up- and down-shifting of the laboratory-frame ZPF will always yield a standing wave in the electron's frame. It has been proposed [36] that in the laboratory frame the superposition of these two waves results in an apparent traveling wave whose wavelength is

cAc _ , (i2)

which is simply the de Broglie wavelength, AB = h/p, for a particle of momentum p = nl0")'v. This is evident from looking at the summation of two oppositely moving wave trains of equal amplitude, _b+ and __, in the particle and laboratory frames. In the rest frame of the particle the two wave trains combine to yield a single standing wave. In the laboratory frame we have for the sum,

(The wave-packet derivation, Eqs. (13)–(16), is omitted for length; the complete text is at the source.)

where 01.2 are again two independent random phases 01,2 = ½(0+ +0_). Observe that for fixed z, the rapidly oscillating "carrier" of frequency _'_- is modulated by the slowly varying envelope function in frequency a.'s. And vice versa observe that at a given t the "carrier" in space appears to have a relatively large wave number k, which is modulated by the envelope of much smaller wave number kB. Hence both timewise at a fixed point in space and spacewise at a given time, there appears a carrier that is modulated by a much broader wave of dimension corresponding to the de Broglie time tB = 2rr/a.'B, or equivalently, the de Broglie wavelength AB = 2rr/kB. This result may be generalized to include ZPF radiation from all other directions, as may be found in the monograph of de la Peña and Cetto [3]. They conclude by stating: "The foregoing discussion assigns a physical meaning to de Broglie's wave: it is the modulation of the wave formed by the Lorentz-transformed, Doppler-shifted superposition of the whole set of random stationary electromagnetic waves of frequency wc with which the electron interacts selectively." Another way of looking at the spatial modulation is in terms of the wave function. Since

_.'c'_L' _ moTv _ p (17) c2 h h

this spatial modulation is exactly the e 'p_/h wave function of a freely moving particle satisfying the Schrödinger equation. The same argument has been made by Hunter [35]. In such a view the quantum wave function of a moving free particle becomes a "beat frequency" produced by the relative motion of the observer with respect to the particle and its oscillating charge. It thus appears that a simple model of a particle as a ZPF-driven oscillating charge with a resonance at its Compton frequency, may simultaneously offer insight into the nature of inertial mass, i.e. into rest inertial mass and its relativistic extension the Einstein-de Broglie formula and into its associated wave function involving the de Broglie wavelength of a moving particle. If the de Broglie oscillation is indeed driven by the ZPF, then it is a form of Schrödinger's Zitterbewegung. Moreover there is a substantial literature attempting to associate spin with Zitterbewegung tracing back to the work of Schrödinger [11]; see for example Huang [12] and Barut and Zanghi [37]. In the context of ascribing the Zitterbewegung to the fluctuations produced by the ZPF, it has been proposed that spin may be traced back to the (circular) polarization of the electromagnetic field, i.e. particle spin may derive from the spin of photons in the electromagnetic quantum vacuum [5]. It is well known, in ordinary quantum theory, that the introduction of h into the ZPF energy density spectrum Pzpof eqn. (2) is made via the harmonic-oscillators-quantization of the electromagnetic modes and that this introduction of t_ is totally independent from the simultaneous introduction of tL into the particle spin. The idea expounded herein points however towards a connection between the h in pzp (o:) and the h in the spin of the electron. In spite of a suggestive preliminary proposal, an exact detailed model of this connection remains to be developed [25]. Finally, although we amply acknowledge that other vacuum fields besides the electromagnetic do contribute to inertia, no attempt has been made within the context of the present work to explore that extension.

6. Comments on Gravitation

If inertial mass, mi, originates in ZPF-charge interactions, then, by the principle of equivalence so must gravitational mass, m_g. In this view, gravitation would be a force originating in ZPF-charge interactions analogous to the ZPF-inertia concept. Sakharov [8], presumably inspired by previous work of Zeldovich [38], was the first to conjecture this interpretation of gravity. If true, gravitation would be unified with the other forces: it would be a manifestation of the other fields. The general relativistic mathematical treatment of gravitation as a space-time curvature works extremely well. However if it could be shown that a different theoretical basis can be made analytically equivalent to space-time curvature, with its prediction of gravitational lensing, black holes, etc. this may reopen the possibility that gravitation should be viewed as a force. The following points are worth noting: (1) general relativity and quantum physics are at present irreconcilable, therefore something substantive is either wrong or missing in our understanding of one or both; (2) the propagation of gravitational waves is not rigorously consistent with space-time curvature. (The issue revolves around whether gravitational waves can be made to vanish in a properly chosen coordinate system. The discovery of apparent gravitational energy loss by the Hulse-Taylor pulsar provides indirect evidence for the existence of gravitational waves. Theoretical developments and calculations have not yet been performed to examine whether an approach based on the Sakharov [8] ideas would predict gravitational waves, but the coordinate ambiguities of GR should not appear in a ZPF-referenced theory of gravitation.) General relativity (GR) attributes gravitation to spacetime curvature. Modern attempts to reconcile quantum physics with GR take a different approach, treating gravity as an exchange of gravitons in flat spacetime (analogous to the treatment of electromagnetism as exchange of virtual photons). A non-geometric (i.e. flat spacetime) approach to gravity is legitimate in quantum gravity. Similarly another non-geometric approach would be to assume that the dielectric properties of space itself may change in the presence of matter: this can be called the polarizable vacuum (PV) approach to gravity. Propagation of light in the presence of matter would deviate from straight lines due to variable refraction of space itself, and other GR effects such as the slowing down of light (the coordinate velocity as judged by a distant observer) in a gravitational potential would also occur. But of course it is the propagation of light from which we infer that spacetime is curved in the first place. This raises the interesting possibility that GR may be successful and yet not because spacetime is really curved: rather because the point-to-point changes in the dielectric (refractive) properties of space in the presence of matter create the illusion of geometrical curvature. A PV type of model does not directly relate gravitation to the ZPF(or to the more general quantum vacuum) but it does appear to provide a theoretical framework conducive to developing the conjecture of Sakharov that it is changes in the ZPF that create gravitational forces. There were some early pioneering attempts, inspired by Sakharov's conjecture, to link gravity to the vacuum from a quantum field theoretical viewpoint (by Amati, Adler and others, see discussion and references in Misner, Thorne and Wheeler [39]) as well as within SED (see Surdin [40]). The first step in developing Sakharov's conjecture in any detail within the classical context of nonrelativistic SED was the work of Puthoff [10]. In this approach gravity is treated as a residuum force in the manner of the van der Waals forces. Expressed in the most rudimentary way this can be viewed as follows. The electric component of the ZPF causes a given charged particle to oscillate. Such oscillations give rise to secondary electromagnetic fields. An adjacent charged particle will thus experience both the ZPF driving forces causing it to oscillate, and in addition forces due to the secondary fields produced by the ZPF-driven oscillations of the first particle. Similarly, the ZPF-driven oscillations of the second particle will cause their own secondary fields acting back upon the first particle. The net effect is an attractive force between the particles. The sign of the charge does not matter: it only affects the phasing of the interactions. Unlike the Coulomb force which, classically viewed, acts directly between charged particles, this interaction is mediated by extremely minute propagating secondary fields created by the ZPF-driven oscillations, and so is enormously weaker than the Coulomb force. Gravitation, in this view, appears to be a long-range interaction akin to the van der Waals force. The ZPF-driven ultrarelativistic oscillations were named Zitterbewegung by Schrödinger. The Puthoff analysis consists of two separate parts. In the first, the energy of the Zitterbewegung motion is equated to gravitational mass, mg (after dividing by c²). This leads to a relationship between m_g and electrodynamic parameters that is identical to the HRP inertial mass, mi, apart from a factor of two. This factor of two is discussed in the appendix of HRP, in which it is concluded that the Puthoff m_g should be reduced by a factor of two, yielding m_ = m 9 precisely. The second part of Puthoff's analysis is more controversial. He quantitatively examines the van der Waals force-like interactions between two driven oscillating dipoles and derives an inverse square force of attraction. This part of the analysis has been challenged by Carlip to which Puthoff has responded [41], but, since problems remain [42], this aspect of the ZPF-gravitation concept requires further theoretical development, in particular the implementation of a fully relativistic model. Clearly the ZPF-inertia and the ZPF-gravitation concepts must stand or fall together, given the principle of equivalence. However, that being the case, the SED approach to gravity proposed by Puthoff, if correct, does legitimately refute the objection that "the ZPF cannot be a real electromagnetic field since the energy density of this field would be enormous and thereby act as a cosmological constant, A, of enormous proportions that would curve the Universe into something microscopic in size." This cannot happen in the Sakharov-Puthoff view. This situation is clearly ruled out by the fact that, in this view, the ZPF cannot act upon itself to gravitate. Gravitation is not caused by the mere presence of the ZPF, rather by secondary motions of charged particles driven by the ZPF. In this view it is impossible for the ZPF to give rise to a cosmological constant. (The possibility of non-gravitating vacuum energy has recently been investigated in quantum cosmology in the framework of the modified Born-Oppenheimer approximation by Datta [43].) The other side of this argument is of course that as electromagnetic radiation is not made of polarizable entities one might naively no longer expect deviation of light rays by massive bodies. We speculate however that such deviation will be part of a fully relativistic theory that besides the ZPF properly takes into account the polarization of the Dirac vacuum when light rays pass through the particle-antiparticle Dirac sea. It should act, in effect, as a medium with an index of refraction modified in the vicinity, of massive objects. This is very much in line with the original Sakharov [8] concept. Indeed, within a more general field-theoretical fl-amework one would expect that the role of the ZPF in the inertia and gravitation developments mentioned above will be played by, a more general quantum vacuum field, as was already suggested in the HRP appendix.

7. Concluding comments on the Higgs Field as originator of mass

In the Standard Model of particle physics it is postulated that there exists a scalar field pervasive throughout the Universe and whose main function is to assign mass to the elementary particles. This is the so-called Higgs field or Higgs boson and it originated from a proposal by the British physicist Peter Higgs who introduced that kind of field as an idea for assigning masses in the Landau-Ginzburg theory of superconductivity. Recent predictions of the mass that the Higgs boson itself may have indicate a rather large mass (more than60GeV) and this may be one of the reasons why, up to the present, the Higgs boson has not been observed. There are alternative theories that give mass to elementary particles without the need to postulate a Higgs field, as, e.g., dynamical symmetry breaking where the Higgs boson is not elementary but composite. But the fact that the Higgs boson has not been detected is by no means an indication that it does not exist. Recall the 26 years which passed between the proposal by Pauli in 1930 of the existence of the neutrino and its first detection when the Reines experiment was performed. It should be clearly stated that the existence (or non-existence) of the hypothetical Higgs boson does not affect our proposal for the origin of inertia. In the Standard Model attempt to obtain, in John Wheeler's quote, "mass without mass", the issue of inertia itself does not appear. As Wilczek [44] states concerning protons and neutrons:"Most of the mass of ordinary matter, for sure, is the pure energy of moving quarks and gluons. The remainder, a quantitatively small but qualitatively crucial remainder -- it includes the mass of electrons -- is all ascribed to the confounding influence of a pervasive medium, the Higgs field condensate." An explanation of proton and neutron masses in terms of the energies of quark motions and gluon fields falls short of offering any insight on inertia itself. One is no closer to an understanding of how this energy somehow acquires the property of resistance to acceleration known as inertia. Put another way, a quantitative equivalence between energy and mass does not address the origin of inertial reaction forces. Many physicists apparently believe that our conjecture of inertia originating in the vacuum fields is at odds with the Higgs hypothesis for the origin of mass. This happens because of the pervasive, one might even say invisible, assumption that inertia can only be intrinsic to mass and thus if the Higgs mechanism creates mass one automatically has an explanation for inertia. If inertia is intrinsic to mass as postulated by Newton, then it (inertia) cannot simultaneously have an extrinsic basis deriving from either the Higgs field or from our proposed mechanism whereby real reaction forces are generated by the quantum vacua. However if one accepts that there is indeed an extrinsic origin for the inertia reaction force, be it the gravity field of the surrounding matter of the Universe (Mach's Principle) or be it the electromagnetic quantum vacuum (or more generally the quantum vacua) that we propose, then the question of how mass originates -- possibly by a Higgs mechanism -- is a separate issue from the property of inertia. This is a point that is often not properly understood. The modern Standard Model explanation of mass is satisfied if it can balance the calculated energies with the measured masses (as in the proton) but merely equating energy and mass does not explain inertia. Retuming to our epistemology of observables, it is the inertia reaction force associated with acceleration that is measureable and fundamental, not mass itself. We are proposing a specific mechanism for generation of the inertia reaction force resulting from distortions of the quantum vacua as perceived by accelerating elementary particles. We do not enter into the problems associated with attempts to explain inertia via Mach's Principle, since we have discussed this at length in a recent paper in collaboration with Y. Dobyns [45]: a detailed discussion on intrinsic vs. extrinsic inertia and on the inability of the geometrodynamics of general relativity to generate inertia reaction forces may be found therein. It had already been shown by Rindler [46] that Mach's Principle is inconsistent with general relativity, and Dobyns et al. further elaborate on a crucial point in general relativity that is not widely understood: Geometrodynamics merely defines the geodesic that a freely moving object will follow. But if an object is constrained to follow some different path, geometrodynamics has no mechanism for creating a reaction force. Geometrodynamics leaves it to inertia to generate such a force upon deviation from a geodesic path, but this becomes an obvious tautology if an explanation of inertia is sought in geometrodynamics. We acknowledge that Newton's proposal that inertia is intrinsic to mass is more economical (Occam's razor) but it is also oversimplistic as one may always continue asking for a deeper reason for the operation of physical processes or for more fundamental bases for physical laws. The question of why the mass associated with either matter or energy should possess a resistance to acceleration is a valid one that would need to be addressed even if the Higgs boson were to be found.

Acknowledgement -- We acknowledge NASA contract NASW-5050 for support of this research.

(The appendix on the quantization of the radiation field and the reference list are omitted for length; the complete text is at the source.)

The Case for Inertia as a Vacuum Effect: A Reply to Woodward and Mahood

York Dobyns (C-131 Engineering Quad, Princeton University, Princeton, NJ 08544-5263), Alfonso Rueda (Department of Electrical Engineering, ECS-561, California State University, Long Beach, CA 90840) and Bernhard Haisch (Solar & Astrophysics Laboratory, Dept. L9-41, Bldg. 252, Lockheed Martin, 3251 Hanover St., Palo Alto, CA 94304). Foundations of Physics, in press.

Abstract

The possibility of an extrinsic origin for inertial reaction forces has recently seen increased attention in the physical literature. Among theories of extrinsic inertia, the two considered by the current work are (1) the hypothesis that inertia is a result of gravitational interactions, and (2) the hypothesis that inertial reaction forces arise from the interaction of material particles with local fluctuations of the quantum vacuum. A recent article supporting the former and criticizing the latter is shown to contain substantial errors.

(Sections 1–3 — the introduction, the critique of gravitational inertia and the reply on the zero-point field — are omitted for length; the complete text is at the source.)

4. Discussion and Conclusions

In reviewing the arguments of Woodward and Mahood (1999), the following conclusions can clearly be seen:

1. Within the standard geometrical interpretation of general relativity, any attempt to identify gravity as the source of inertial reaction forces can succeed only by postulating the thesis it purports to prove. Such arguments can therefore be dismissed as circular.

2. While one can construct a gravitational theory for inertial reaction forces, as in the case of Sciama's 1953 theory, such theories are necessarily theories of explicit forces coupled to a source m 9, and therefore are quite distinct from the geometrical theory we know as general relativity.

3. The particular gravitational-inertia theory propounded by WM suffers a consistency problem in the handling of ¢0 as a quantity that (a) acts as a potential, (b) has a gradient, and (c) is a locally measured invariant. These three properties prove to be mutually incompatible.

4. The advocacy of WM for the philosophy of "radical timelessness" is, contrary to their own assertion, not a consequence of relativity but a consequence of their acceptance of nonlocal interactions in a relativistic framework.

5. The arguments of WM against the existence of quantum zero-point fluctuations are deeply flawed, being based in one case on a misunderstanding of the cosmological constant problem and in the second case on a willingness to adopt nonlocal interactions in a way which most working physicists would find unacceptable.

6. The arguments of WM against the HRP theory of extrinsic inertia arising from interactions with the ZPF make it clear that WM have misunderstood almost every important point of the argument. Their arguments are in most cases invalid, in some cases useful criticisms pointing to ways in which the theory needs to be strengthened and improved. In no case whatever do they constitute actual refutations.

Finally, we should note that among the possible theories of inertia the most plausible current contender, albeit also the least informative, remains the simplest: That inertia is inherent in mass. No theory of extrinsic inertia yet proposed has been able successfully to reproduce all of the observed phenomena which are trivial consequences of this simple premise. The alternative theories of extrinsic inertia require considerable further development before they can practically replace the standard interpretation of inertial reaction forces which has been thoroughly successful since the days of Newton.

Acknowledgements

B.H. and A.R. acknowledge support of this work by NASA contract NASW-5050.