Engineering the Zero-Point Field and Polarizable Vacuum forpp.137-144, Flight                                             ...
H.E. Puthoff, S.R. Little and M. Ibisonof the CM of the spaceship is matched by a coun-           ExH fields do carry mome...
Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flightthors to flesh out the details of the Sakha...
H.E. Puthoff, S.R. Little and M. Ibisonwhich a patent has been issued) is based on the          approach treats metric cha...
Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flightess when closer to the body where         T...
H.E. Puthoff, S.R. Little and M. Ibisonit is premature to even guess at an optimum strat-             where the sense is f...
Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flighttion that plays the role of the Einstein eq...
H.E. Puthoff, S.R. Little and M. Ibison    point-fluctuation force”, Phys. Rev. A, 47, p.3454, 1993.               energy ...
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Harold puthoff, little, ibizon engineering the zero-point field and polarizable vacuum for interstellar flight, 2002

  1. 1. Engineering the Zero-Point Field and Polarizable Vacuum forpp.137-144, Flight JBIS, Vol. 55, Interstellar 2002Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar FlightH.E. PUTHOFF*, S.R. LITTLE AND M. IBISONInstitute for Advanced Studies at Austin, 4030 West Braker Lane, Suite 300, Austin, Texas 78759-5329, USA.*Email: A theme that has come to the fore in advanced planning for long-range space exploration is the concept of “propellantless propulsion” or “field propulsion”. One version of this concept involves the projected possibility that empty space itself (the quantum vacuum, or space-time metric) might be manipulated so as to provide energy/thrust for future space vehicles [1]. Although far reaching, such a proposal is solidly grounded in modern theory that describes the vacuum as a polarizable medium that sustains energetic quantum fluctuations. Thus the possibility that matter/vacuum interactions might be engineered for space-flight applications is not a priori ruled out, although certain constraints need to be acknowledged. The structure and implications of such a far- reaching hypothesis are considered herein. Keywords: Zero-point energy, warp drive, propellantless propulsion, metric engineering, interstellar flight1. IntroductionThe concept of “engineering the vacuum” found its regarding global thermodynamic and energy con-first expression in the mainstream physics litera- straints. Furthermore, the energetic componentsture when it was introduced by T. D. Lee in his of potential utility involve very small-wavelength,textbook Particle Physics and Introduction to Field Theory high-frequency fields and thus resist facile engi-[2]. There he stated: “The experimental method to neering solutions. With regard to perturbation ofalter the properties of the vacuum may be called the space-time metric, the required energy densi-vacuum engineering.... If indeed we are able to ties exceed by many orders of magnitude valuesalter the vacuum, then we may encounter some achievable with existing engineering phenomena, totally unexpected.” This legitimi- Nonetheless, we can examine the constraints, pos-zation of the vacuum engineering concept was sibilities and implications under the expectation thatbased on the recognition that the vacuum is char- as technology matures, felicitous means may beacterized by parameters and structure that leave found that permit the exploitation of the enormous,no doubt that it constitutes an energetic medium in as-yet-untapped potential of so-called “emptyits own right. Foremost among these are its proper- space”.ties that (1) within the context of quantum theorythe vacuum is the seat of energetic particle and 2. Propellantless Propulsionfield fluctuations, and (2) within the context of gen-eral relativity the vacuum is the seat of a space- 2.1 Global Constrainttime structure (metric) that encodes the distribu-tion of matter and energy. Indeed, on the flyleaf of Regardless of the mechanisms that might be enter-a book of essays by Einstein and others on the tained with regard to “propellantless” or “field” pro-properties of the vacuum we find the statement pulsion of a spaceship, there exist certain con-“The vacuum is fast emerging as the central struc- straints that can be easily overlooked but must beture of modern physics” [3]. taken into consideration. A central one is that, be- cause of the law of conservation of momentum, the Given the known characteristics of the vacuum, center of mass-energy (CM) of an initially station-one might reasonably inquire as to why it is not ary isolated system cannot change its position ifimmediately obvious how to catalyze robust inter- not acted upon by outside forces. This means thatactions of the type sought for space-flight applica- propellantless or field propulsion, whatever form ittions. To begin, in the case of quantum fluctuations takes, is constrained to involve coupling to the ex-there are uncertainties that remain to be clarified ternal universe in such a way that the displacement 137
  2. 2. H.E. Puthoff, S.R. Little and M. Ibisonof the CM of the spaceship is matched by a coun- ExH fields do carry momentum as in the angularteracting effect in the universe to which it is cou- case, the symmetry conditions for the linear casepled, so as not to violate the global CM constraint. are such that there exists a cancelling mechanicalTherefore, before one launches into a detailed in- momentum contained in the structures even thoughvestigation of a proposed propulsion mechanism it a structure’s CM itself is stationary (see Appendix A).is instructive to apply this principle as an overall Specifically, it can be shown on very general groundsconstraint to determine whether the principle is vio- that, contrary to the case for angular momentumlated. Surprising subtleties may be involved in such (e.g., the Feynman disk), the total linear momentuman assessment, as illustrated in the following exam- of any stationary distribution of matter, charge andple. their currents, and their associated fields, must van- ish. In other words, barring a new discovery that2.2 An Example: “ExH” modifies the present laws of physics, any such dis- Electromagnetic Field Propulsion tribution cannot generate a propulsive force with- out emitting some form of reaction mass or energy,A recurring theme in electromagnetic propulsion or otherwise imparting momentum to another sys-considerations is that one might employ crossed tem [7].electric and magnetic fields to generate propulsiveforce, what we might call ExH propulsion. The idea 3. The Quantum Vacuumis based on the fact that propagating electromag-netic fields (photons) possess momentum carried 3.1 Zero-Point Energy (ZPE) Backgroundby the crossed (orthogonal) E and H fields (Poyntingvector). This raises the issue as to whether static Quantum theory tells us that so-called “empty space”(i.e., non-propagating) ExH fields also constitute mo- is not truly empty, but is the seat of myriad ener-mentum (as the mathematics would imply), and in getic quantum processes. Specifically, quantum fieldparticular whether changes in static fields could theory tells us that, even in empty space, fieldsresult in the transfer of momentum to an attached (e.g., the electromagnetic field) continuously fluctu-structure. As it turns out, the answer can be yes, as ate about their zero baseline values. The energyillustrated in the example of the Feynman disk para- associated with these fluctuations is called zero-dox [4]. Electric charge distributed around the rim point energy (ZPE), reflecting the fact that suchof a non-rotating disk generates a static electric activity remains even at a temperature of absolutefield that extends outward from the rim, and a cur- zero. Such a concept is almost certain to have pro-rent-carrying coil of wire mounted perpendicular to found implications for future space travel, as wethe plane of the disk generates a static dipole mag- will now discuss.netic field. The two fields result in a static ExHdistribution that encircles the disk. Even though noth- When a hypothetical ZPE-powered spaceshiping is apparently in motion, if we take the ExH mo- strains against gravity and inertia, there are threementum concept seriously it would appear that there elements of the equation that the ZPE technologyis angular momentum “circulating” about the disk in could in principle address: (1) a decoupling fromthe static fields. That this is in fact the case is dem- gravity, (2) a reduction of inertia, or (3) the genera-onstrated by the fact that when the current in the tion of energy to overcome both.coil is interrupted, thereby extinguishing the mag-netic field component of the ExH distribution, the 3.2 Gravitydisk begins to rotate. This behaviour supports thenotion that, indeed, the static fields do contain an- With regard to a ZPE basis for gravity, the Russiangular momentum that is then transferred to the disk physicist Andrei Sakharov was the first to propose(to conserve angular momentum) when the field that in a certain sense gravitation is not a funda-momentum is extinguished [5]. This leads one to mental interaction at all, but rather an induced ef-wonder if the same principle could be applied to fect brought about by changes in the quantum-fluc-generate linear thrust by changes in static ExH fields, tuation energy of the vacuum when matter is presentproperly arrayed. [8]. In this view, the attractive gravitational force is more akin to the induced van der Waals and Casimir Pursuit of the linear thrust possibility, however, forces, than to the fundamental Coulomb force. Al-leads one to a rich literature concerning so-called though quite speculative when first introduced by“hidden momentum” that, perhaps surprisingly, de- Sakharov in 1967, this hypothesis has led to a richnies this possibility [6]. The “hidden momentum” literature on quantum-fluctuation-induced gravity.phrase refers to the fact that although the linear (The latter includes an attempt by one of the au-138
  3. 3. Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flightthors to flesh out the details of the Sakharov pro- practical application in the field of cavity quantumposal [9], though difficulties remain [10]). Given the electrodynamics, where the spontaneous emissionpossibility of a deep connection between gravity rates of atoms are subject to manipulation [17].and the zero-point fluctuations of the vacuum, it Therefore, it is not unreasonable to contemplate thewould therefore appear that a potential route to possibility of such control in the field of space pro-gravity decoupling would be via control of vacuum pulsion.fluctuations. 3.4 Energy Extraction3.3 Inertia With regard to the extraction of energy from theClosely related to the ZPE basis for gravity is the vacuum fluctuation energy reservoir, there are nopossibility of a ZPE basis for inertia. This is not energetic or thermodynamic constraints prevent-surprising, given the empirical fact that gravitational ing such release under certain conditions [18]. And,and inertial masses have the same value, even in fact, there are analyses in the literature that sug-though the underlying phenomena are quite dispa- gest that such mechanisms are already operative inrate; one is associated with the gravitational attrac- Nature in the “powering up” of cosmic rays [19], ortion between bodies, while the other is a measure as the source of energy release from supernovasof resistance to acceleration, even far from a gravi- [20] and gamma-ray bursts [21].tational field. Addressing this issue, the author andhis colleagues evolved a ZPE model for inertia which For our purposes, the question is whether thedeveloped the concept that although a uniformly ZPE can be “mined” at a level practical for use inmoving body does not experience a drag force from space propulsion. Given that the ZPE energy den-the (Lorentz-invariant) vacuum fluctuations, an ac- sity is conservatively estimated to be on the ordercelerated body meets a resistive force proportional of nuclear energy densities or greater [22], it wouldto the acceleration [11], an approach that has had a constitute a seemingly ubiquitous energy supply, afavourable reception in the scientific community veritable “Holy Grail” energy source.[12]. Again, as in the gravity case, it would thereforeappear that a potential route to the reduction of One of the first researchers to call attention toinertial mass would be via control of vacuum fluc- the principle of the use of the Casimir effect as atuations. potential energy source was Robert Forward at Hughes Research Laboratories in Malibu, CA [23]. Investigation into this possibility by the U.S. Air Though providing “proof-of-principle,” unlike theForce’s Advanced Concepts Office at Edwards Air astrophysical implications cited above the amountForce Base resulted in the generation of a report of energy release for mechanical structures underentitled Mass Modification Experiment Definition Study laboratory conditions is minuscule. (The collapse ofthat addressed just this issue [13]. Included in its a pair of one-centimeter-square Casimir plates from,recommendations was a call for precision meas- say, 2 microns to 1 micron in 1 microsecond, gener-urement of what is called the Casimir force. The ates around 1/10 microwatt.) In addition, the con-Casimir force is an attractive quantum force be- servative nature of the Casimir effect would appeartween closely spaced metal or dielectric plates (or to prevent recycling, though there have been someother structures) that derives from partial shielding suggestions for getting around this barrier [24]. Al-of the interior region from the background zero- ternatives involving non-recycling behaviour, suchpoint fluctuations of the vacuum electromagnetic as plasma pinches [25] or bubble collapse infield, which results in unbalanced ZPE radiation sonoluminescence [26], have been investigated inpressures [14]. Since issuance of the report, such our laboratory and elsewhere, but as yet withoutprecision measurements have been made which real promise for energy applications.confirm the Casimir effect to high accuracy [15],measurements which even attracted high-profile at- Vacuum energy extraction approaches by othertention in the media [16]. The relevance of the than the Casimir effect are also being considered.Casimir effect to our considerations is that it consti- One approach that emerged from the Air Force’stutes experimental evidence that vacuum fluctuations Mass Modification… study [13] was the suggestioncan be altered by technological means. This suggests that the ZPE-driven cosmic ray model be exploredthe possibility that, given the models discussed, under laboratory conditions to determine whethergravitational and inertial masses might also be amenable protons could be accelerated by the proposed cos-to modification. The control of vacuum fluctuations mic ray mechanism in a cryogenically-cooled, colli-by the use of cavity structures has already found sion-free vacuum trap. Yet another proposal (for 139
  4. 4. H.E. Puthoff, S.R. Little and M. Ibisonwhich a patent has been issued) is based on the approach treats metric changes in terms of theconcept of beat-frequency downshifting of the more permittivity and permeability constants of theenergetic high-frequency components of the ZPE, vacuum, εo and µo, essentially along the lines of theby use of slightly detuned dielectric-sphere anten- “THεµ” methodology used in comparative studies ofnas [27]. gravitational theories [31]. Such an approach, rely- ing as it does on parameters familiar to engineers, In our own laboratory we have considered an can be considered a “metric engineering” approach.approach based on perturbation of atomic or mo-lecular ground states, hypothesized to be equilib- In brief, Maxwell’s equations in curved space arerium states involving dynamic radiation/absorption treated in the isomorphism of a polarizable mediumexchange with the vacuum fluctuations [28]. In this of variable refractive index in flat space [32]; themodel atoms or molecules in a ZPE-limiting Casimir bending of a light ray near a massive body is mod-cavity are expected to undergo energy shifts that elled as due to an induced spatial variation in thewould alter the spectroscopic signatures of refractive index of the vacuum near the body; theexcitations involving the ground state. We have initi- reduction in the velocity of light in a gravitationalated experiments at a synchrotron facility to ex- potential is represented by an effective increase inplore this ZPE/ground-state relationship, though so the refractive index of the vacuum, and so forth. Asfar without success. In addition to carrying out ex- elaborated in Ref. 30 and the references therein,periments based on our own ideas, our laboratory though differing in some aspects from GR, PV mod-also acts as a clearing-house to evaluate the ex- elling can be carried out for cases of interest in aperimental concepts and devices of others who are self-consistent way so as to reproduce to appropri-working along similar lines. Details can be found on ate order both the equations of GR, and the matchour website, to the classical experimental tests of those equa- tions. Whether tapping the ZPE as an energy source ormanipulating the ZPE for gravity/inertia control are Specifically, the PV approach treats such meas-but gleams in a spaceship designer’s eye, or a Royal ures as the velocity of light, the length of rulersRoad to practical space propulsion, is yet to be (atomic bond lengths), the frequency of clocks, par-determined. Only by explorations of the type de- ticle masses, and so forth, in terms of a variablescribed here will the answer emerge. In the interim vacuum dielectric constant Κ in which vacuum per-a quote by the Russian science historian Roman mittivity εo transforms to εo → Kεo, vacuum perme-Podolny would seem to apply: “It would be just as ability to µo → Kµo. In a planetary or solar gravita- tional potential K ≈ 1 + 2GM / rc > 1 , and the results 2presumptuous to deny the feasibility of useful ap-plication as it would be irresponsible to guarantee are as shown in Table 1. Thus, the velocity of light issuch application” [29]. reduced, light emitted from an atom is redshifted as compared with an atom at infinity (K = 1), rulers4. The Space-Time Metric shrink, etc. (“Metric Engineering” Approach) As one example of the significance of the tabu- lated values, the dependence of fundamental lengthDespite the apparently daunting energy require- measures (ruler shrinkage) on the variable K indi-ments to perturb the space-time metric to a signifi- cates that the dimensions of material objects adjustcant degree, we examine the structure that such in accordance with local changes in vacuumperturbations would take under conditions useful polarizability - thus there is no such thing as a per-for space-flight application, a “Blue Sky” approach, fectly rigid rod. From the standpoint of the PV ap-as it were. proach this is the genesis of the variable metric that is of such significance in GR studies. It also permits Although topics in general relativity are routinely us to define, from the viewpoint of the PV approach,treated in terms of tensor formulations in curved just what precisely is meant by the label “curvedspace-time, we shall find it convenient for our pur- space.” In the vicinity of, say, a planet or star, whereposes to utilize one of the alternative methodolo- K > 1, if one were to take a ruler and measure alonggies for treating metric changes that has emerged a radius vector R to some circular orbit, and thenover the years in studies of gravitational theories. measure the circumference C of that orbit, oneThe approach, known as the polarizable vacuum would obtain C < 2πR (as for a concave curved sur-(PV) representation of general relativity (GR), treats face). This is a consequence of the ruler beingthe vacuum as a polarizable medium [30]. The PV relatively shorter during the radial measuring proc-140
  5. 5. Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flightess when closer to the body where TABLE 1: Typical Metric Effects in the Polarizable Vacuum (PV) Representation of GR.K is relatively greater, as compared (For reference frame at infinity, K = 1.)to its length during the circumfer-ential measuring process when fur- Variable Determining Equation ≥ K≥1 (typical mass distribution, M)ther from the body. Such an influ- velocity of light vL(K) vL = c /K velocity of light < cence on the measuring process due mass m(K) m = mo K3/2 effective mass increasesto induced polarizability changes inthe vacuum near the body leads to frequency ω(K) ω = ωo / K redshift toward lower frequenciesthe GR concept that the presence time interval ∆t(K) ∆t = ∆to clocks run slowerof the body “influences the metric,” K energy E(K) E = Eo / lower energy statesand correctly so. K length dim. L(K) L = Lo / K objects shrink We are now in a position to con-sider application of this “metric en- TABLE 2: Engineered Metric Effects in the Polarizable Vacuum (PV) Representation ofgineering” formalism to the type of GR. (For reference frame at infinity, K = 1.)questions relevant to space propul-sion. As we show in Appendix B, Variable Determining Equation ≤ K≤1 (engineered metric)under certain conditions the metric velocity of light vL(K) vL = c/K velocity of light > ccan in principle be modified to re-duce the value of the vacuum di- mass m(K) m = mo K3/2 effective mass decreaseselectric constant K to below unity. frequency ω(Κ) ω = ωo / K blueshift toward higher frequenciesReturning to Table 1, we see that a time interval ∆t(K) ∆t = ∆to clocks run fasterK < 1 solution permits the addition Kof another column for which the energy E(K) E = Eo / K higher energy statesdescriptors are reversed, as shown length dim. L(K) L = Lo / K objects expandin Table 2. total mass of the universe! [35] Further theoretical Under such conditions of extreme space-time effort has resulted in a reduction of the energyperturbation, the local velocity of light (as seen from a requirement to somewhat below a solar mass, anreference frame at infinity) is increased, mass de- impressive advance but still quite impractical [36].creases, energy bond strengths increase, etc., features Analysis of related alternatives such as the Krasnikovpresumably attractive for interstellar travel. Tube [37] and traversable wormholes have fared no better [38]. Thus, if success is to be achieved, it As an example, one specific approach that has must rest on some as yet unforeseen breakthroughgenerated considerable commentary in the techni- about which we can only speculate, such as a tech-cal literature is the so-called Alcubierre Warp Drive, nology to cohere otherwise random vacuum fluc-named after its creator, general relativity theorist tuation energy.Miguel Alcubierre [33, 34]. Alcubierre showed thatby distorting the local space-time metric in the re- Clearly then, calculations for the proposedgion of a spaceship in a certain prescribed way, it geometries are by no means directly applicable towould be possible in principle to achieve motion the design of a space propulsion drive. However,faster than the speed of light as judged by observ- these sample calculations indicate the direction ofers outside the disturbed region, without violating potentially useful trends derivable on the basis ofthe local velocity-of-light constraint within the re- the application of GR principles as embodied in agion. Furthermore, the Alcubierre solution showed metric engineering approach, with the results con-that the proper (experienced) acceleration along strained only by what is achievable practically in anthe spaceship’s path would be zero, and that the engineering sense. The latter is, however, a daunt-spaceship would suffer no time dilation, highly de- ing constraint.sirable features for interstellar travel. 5. Conclusions When it comes to engineering the Alcubierresolution, however, seemingly insurmountable In this paper we have touched briefly on innovativeobstacles emerge. For a 100 m warp bubble the forms of space propulsion, especially those thatbubble wall thickness approaches a Planck length might exploit properties of the quantum vacuum or(~10-35 m) and the (negative) energy required is the space-time metric in a fundamental way. At thisroughly 10 orders of magnitude greater than the point in the development of such nascent concepts 141
  6. 6. H.E. Puthoff, S.R. Little and M. Ibisonit is premature to even guess at an optimum strat- where the sense is from left to right. From this it isegy, let alone attempt to forge a critical path; in fact, concluded that there is a steady net linear momen-it remains to be determined whether such exploita- tum stored in the electromagnetic fields. We willtion is even feasible. Nonetheless, only by inquiring now show there is another momentum, equal andinto such concepts in a rigorous way can we hope opposite to this electromagnetic field arrive at a proper assessment of the possibilitiesand thereby determine the best course of action to Since the current flowing in the loop is givenpursue in our steps first to explore our solar system by I = ρav, the velocity of the fluid is everywhereenvironment, and then one day to reach the stars. v = I/ρa. Meanwhile, the external electric field E cre- ates a pressure difference between the bottom andAppendix A - Hidden Momentum the top of the fluid given by P = ρEh. Moving to the left, therefore, is a net energy flux S (energy per unitConsider a stationary current loop which consists area per unit time) given byof an incompressible fluid of positive charge den-sity ρ circulating at velocity v clockwise around a S = Pv = (ρEh)x (I/ρa ) = IEhloop of non-conductive piping of cross sectional (A4) aarea a. The loop is immersed in a constant uniformelectric field E. But since energy has mass, Eq. (A4) may be con- verted to an expression for momentum. This is w mostly easily accomplished by writing the Einstein relation E = mc2 in flux density form as S = gc2, where v g is the momentum per unit volume. It now follows that, due to the different pressures at the top and bottom of the loop, there must be a net overall a momentum - directed to the left - given by h Saw IEhw pmech = gaw = 2 = (A5) c c2 where the subscript ‘mech’ draws attention to the apparently entirely mechanical origin of this mo- mentum. E Eqs. (A5) and (A3) demonstrate that the elec-The magnetic field created by the current loop com- tromagnetic momentum is balanced by an equalbines with the electric field to produce an electro- and opposite mechanical momentum. Because ofmagnetic field momentum given by its rather obscure nature, this momentum has been referred to in the literature as “hidden momen- 1 tum”. This is a particular example of the general p EM = c2 ∫ ExHdV (A1) result that a net static linear field momentum will always be balanced by an equal and oppositeHowever, in steady state situations, this is equal to hidden mechanical momentum. In practical terms,(Ref. 6) this means that the creation of linear field mo- mentum cannot give rise to motion because the field momentum is automatically neutralized by a 1 p EM = c2 ∫ JφdV (A2) mechanical momentum hidden within the struc- ture, so that the whole system remains stationary. This inability to utilize linear field momentum forWith reference to the above figure, the only non- propulsion is guaranteed by the law of momen-zero component of momentum surviving this inte- tum conservation.gration is directed horizontally across the page.Using the expression Eq. (A2), this computes to Appendix B - Metric Engineering Solutions p EM = Iw (φ top − φ bottom ) = IEhw (A3) c2 c 2 In the polarizable vacuum (PV) approach the equa-142
  7. 7. Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flighttion that plays the role of the Einstein equation(curvature driven by the mass-energy stress ten- K= ( K) 2 2  GM  = e 2GM / rc = 1 + 2 2  + ...  rc  (B3)sor) for a single massive particle at the origin is(Ref. 30) which can be shown to reproduce to appropriate order the standard GR Schwarzschild metric prop- erties as they apply to the weak-field conditions 1 ∂2 K ∇2 K − = prevailing in the solar system. (c / K ) 2 ∂t 2   −   ( K  mo c 2  1 + w 2  ) ä (r )+ 1  B  3  2 + Kε o E  2 For the case of a mass M with charge Q, the elec- tric field appropriate to a charged mass imbedded in 4λ  2 K  1  2  Kµ   (  1− w  2 2 )  o  a variable-dielectric-constant medium is given by λ  1  ∂K    2  ∫D.da = KεoE4πr2 = Q − (∇K )2 +    (B1) (B4) K  2  (c / K )  2 ∂t     which leads to (for spherical symmetry, with b2 = Q2G/4πεoc4)where   b2  2 d2 K 2d K 1  d K   −  + = w = v / (c / K ) dr 2 r dr K  dr  r4  (B5)   In this PV formulation of GR, changes in the vacuumdielectric constant K are driven by mass density which should be compared with Eq. (B2). The solu-(first term), EM energy density (second term), and tion here as a function of charge (represented by b)the vacuum polarization energy density itself (third and mass (represented by a = GM/c2) is given byterm). (The constant λ = c 4 / 32πG, where G is thegravitational constant.)  a 2 − b2   a2 − b2  K = cosh  + sinh   a  r   r  In space surrounding an uncharged spherical   a 2 − b2  mass distribution (e.g., a planet) the static solution(∂K / ∂t = 0) to the above is found by solving for 2 a 2 > b2 (B6) d2 K 2d K 1 d K    + = (B2) dr 2 r dr K  dr    For the weak-field case the above reproduces the familiar Reissner-Nordstrøm metric [39]. For b2 > a2,The solution that satisfies the Newtonian limit is however, the hyperbolic solutions turn trigonomet-given by ric, and K can take on values K < 1. References1. H.E. Puthoff, “Can the vacuum be engineered for 285, p.154, 1980. spaceflight applications? Overview of theory and 6. V. Hnizdo, “Hidden momentum of a relativistic fluid experiments”, Jour. Sci. Exploration, 12, p.295, 1998. See carrying current in an external electric field”, Am. J. also H.E. Puthoff, “Space propulsion: Can empty space Phys., 65, p.92, 1997. itself provide a solution?”, Ad Astra, 9, (National Space 7. Proposals to push directly against the space-time Society), p.42, Jan/Feb 1997. metric or quantum vacuum, i.e., use the rest of the2. T.D. Lee, “Particle Physics and Introduction to Field Theory”, Universe as a springboard by means presently Harwood Academic Press, London, 1988. unknown, fall into the latter category.3. S. Saunders and H. R. Brown, “The Philosophy of Vacuum”, 8. A.D. Sakharov, “Vacuum quantum fluctuations in curved Eds., Clarendon Press, Oxford, 1991. space and the theory of gravitation”, Dokl. Akad. Nauk4. R.P. Feynman, R.B. Leighton and M. Sands, “The Feynman SSSR, [Sov. Phys. - Dokl. 12, p.1040, 1968]. See also C. Lectures on Physics”, Addison-Wesley, Reading, MA, Vol. W. Misner, K. S. Thorne and J. A. Wheeler, “Gravitation”, II, p.17-5, 1964. Freeman, San Francisco, pp.426-428,1973.5. For experimental confirmation see, for example, G.M. 9. H.E. Puthoff, “Gravity as a zero-point-fluctuation force”, Graham and D.G. Lahoz, “Observation of static Phys. Rev. A, 39, p.2333, 1989. electromagnetic angular momentum in vacuo”, Nature, 10. H.E. Puthoff, “Reply to ‘Comment on “Gravity as a zero- 143
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