Engineering the Zero-Point Field and PolarizabJlBeI SV,a Vcuoul.m 5 5fo, rp Ipn.t1e3rs7t-e1l4la4r, F2l0ig0h2t Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flight H.E. PUTHOFF*, S.R. LITTLE AND M. IBISON Institute for Advanced Studies at Austin, 4030 West Braker Lane, Suite 300, Austin, Texas 78759-5329, USA. *Email:[email protected] A theme that has come to the fore in advanced planning for long-range space exploration is the concept of (cid:147)propellantless propulsion(cid:148) or (cid:147)field propulsion(cid:148). 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 flight 1. Introduction The concept of (cid:147)engineering the vacuum(cid:148) found its regarding global thermodynamic and energy con- first expression in the mainstream physics litera- straints. Furthermore, the energetic components ture 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: (cid:147)The experimental method to neering solutions. With regard to perturbation of alter 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 values alter the vacuum, then we may encounter some achievable with existing engineering techniques. new phenomena, totally unexpected.(cid:148) This legitimi- Nonetheless, we can examine the constraints, pos- zation of the vacuum engineering concept was sibilities and implications under the expectation that based on the recognition that the vacuum is char- as technology matures, felicitous means may be acterized 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 (cid:147)empty its own right. Foremost among these are its proper- space(cid:148). ties that (1) within the context of quantum theory the vacuum is the seat of energetic particle and 2. Propellantless Propulsion field fluctuations, and (2) within the context of gen- eral relativity the vacuum is the seat of a space- 2.1 Global Constraint time 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 (cid:147)propellantless(cid:148) or (cid:147)field(cid:148) pro- properties of the vacuum we find the statement pulsion of a spaceship, there exist certain con- (cid:147)The vacuum is fast emerging as the central struc- straints that can be easily overlooked but must be ture of modern physics(cid:148) [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 if immediately obvious how to catalyze robust inter- not acted upon by outside forces. This means that actions of the type sought for space-flight applica- propellantless or field propulsion, whatever form it tions. 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 H.E. Puthoff, S.R. Little and M. Ibison of the CM of the spaceship is matched by a coun- ExH fields do carry momentum as in the angular teracting effect in the universe to which it is cou- case, the symmetry conditions for the linear case pled, so as not to violate the global CM constraint. are such that there exists a cancelling mechanical Therefore, before one launches into a detailed in- momentum contained in the structures even though vestigation 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 grounds constraint to determine whether the principle is vio- that, contrary to the case for angular momentum lated. Surprising subtleties may be involved in such (e.g., the Feynman disk), the total linear momentum an assessment, as illustrated in the following exam- of any stationary distribution of matter, charge and ple. their currents, and their associated fields, must van- ish. In other words, barring a new discovery that 2.2 An Example: (cid:147)ExH(cid:148) 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 propulsive force, what we might call ExH propulsion. The idea 3. The Quantum Vacuum is based on the fact that propagating electromag- netic fields (photons) possess momentum carried 3.1 Zero-Point Energy (ZPE) Background by the crossed (orthogonal) E and H fields (Poynting vector). This raises the issue as to whether static Quantum theory tells us that so-called (cid:147)empty space(cid:148) (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 field particular whether changes in static fields could theory tells us that, even in empty space, fields result 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 energy illustrated 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 such of a non-rotating disk generates a static electric activity remains even at a temperature of absolute field 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 we the plane of the disk generates a static dipole mag- will now discuss. netic field. The two fields result in a static ExH distribution that encircles the disk. Even though noth- When a hypothetical ZPE-powered spaceship ing is apparently in motion, if we take the ExH mo- strains against gravity and inertia, there are three mentum concept seriously it would appear that there elements of the equation that the ZPE technology is angular momentum (cid:147)circulating(cid:148) about the disk in could in principle address: (1) a decoupling from the 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 Gravity disk begins to rotate. This behaviour supports the notion that, indeed, the static fields do contain an- With regard to a ZPE basis for gravity, the Russian gular 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 present properly 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 (cid:147)hidden momentum(cid:148) that, perhaps surprisingly, de- Sakharov in 1967, this hypothesis has led to a rich nies this possibility [6]. The (cid:147)hidden momentum(cid:148) 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 Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flight thors to flesh out the details of the Sakharov pro- practical application in the field of cavity quantum posal [9], though difficulties remain [10]). Given the electrodynamics, where the spontaneous emission possibility 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 the would 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 Extraction 3.3 Inertia With regard to the extraction of energy from the Closely related to the ZPE basis for gravity is the vacuum fluctuation energy reservoir, there are no possibility 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 in rate; one is associated with the gravitational attrac- Nature in the (cid:147)powering up(cid:148) of cosmic rays [19], or tion between bodies, while the other is a measure as the source of energy release from supernovas of resistance to acceleration, even far from a gravi- [20] and gamma-ray bursts [21]. tational field. Addressing this issue, the author and his colleagues evolved a ZPE model for inertia which For our purposes, the question is whether the developed the concept that although a uniformly ZPE can be (cid:147)mined(cid:148) at a level practical for use in moving 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 order celerated body meets a resistive force proportional of nuclear energy densities or greater [22], it would to the acceleration [11], an approach that has had a constitute a seemingly ubiquitous energy supply, a favourable reception in the scientific community veritable (cid:147)Holy Grail(cid:148) energy source. [12]. Again, as in the gravity case, it would therefore appear that a potential route to the reduction of One of the first researchers to call attention to inertial mass would be via control of vacuum fluc- the principle of the use of the Casimir effect as a tuations. 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 (cid:147)proof-of-principle,(cid:148) unlike the Force(cid:146)s Advanced Concepts Office at Edwards Air astrophysical implications cited above the amount Force Base resulted in the generation of a report of energy release for mechanical structures under entitled Mass Modification Experiment Definition Study laboratory conditions is minuscule. (The collapse of that 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 appear tween closely spaced metal or dielectric plates (or to prevent recycling, though there have been some other 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, such point fluctuations of the vacuum electromagnetic as plasma pinches [25] or bubble collapse in field, which results in unbalanced ZPE radiation sonoluminescence [26], have been investigated in pressures [14]. Since issuance of the report, such our laboratory and elsewhere, but as yet without precision 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 other tention 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(cid:146)s tutes experimental evidence that vacuum fluctuations Mass Modification… study [13] was the suggestion can be altered by technological means. This suggests that the ZPE-driven cosmic ray model be explored the possibility that, given the models discussed, under laboratory conditions to determine whether gravitational 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 H.E. Puthoff, S.R. Little and M. Ibison which a patent has been issued) is based on the approach treats metric changes in terms of the concept of beat-frequency downshifting of the more permittivity and permeability constants of the energetic high-frequency components of the ZPE, vacuum, e and m , essentially along the lines of the o o by use of slightly detuned dielectric-sphere anten- (cid:147)THem (cid:148) methodology used in comparative studies of nas [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 (cid:147)metric engineering(cid:148) approach. approach based on perturbation of atomic or mo- lecular ground states, hypothesized to be equilib- In brief, Maxwell(cid:146)s equations in curved space are rium states involving dynamic radiation/absorption treated in the isomorphism of a polarizable medium exchange with the vacuum fluctuations [28]. In this of variable refractive index in flat space [32]; the model 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 the would alter the spectroscopic signatures of refractive index of the vacuum near the body; the excitations involving the ground state. We have initi- reduction in the velocity of light in a gravitational ated experiments at a synchrotron facility to ex- potential is represented by an effective increase in plore this ZPE/ground-state relationship, though so the refractive index of the vacuum, and so forth. As far 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 a perimental 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 match our website, www.earthtech.org. to the classical experimental tests of those equa- tions. Whether tapping the ZPE as an energy source or manipulating the ZPE for gravity/inertia control are Specifically, the PV approach treats such meas- but gleams in a spaceship designer(cid:146)s eye, or a Royal ures as the velocity of light, the length of rulers Road 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 variable scribed here will the answer emerge. In the interim vacuum dielectric constant K in which vacuum per- a quote by the Russian science historian Roman mittivity e transforms to e fi Ke , vacuum perme- o o o Podolny would seem to apply: (cid:147)It would be just as ability to m fi Km . In a planetary or solar gravita- o o presumptuous to deny the feasibility of useful ap- tional potential K » 1+2GM /rc2 >1, and the results plication as it would be irresponsible to guarantee are as shown in Table 1. Thus, the velocity of light is such application(cid:148) [29]. reduced, light emitted from an atom is redshifted as compared with an atom at infinity (K = 1), rulers 4. The Space-Time Metric shrink, etc. ((cid:147)Metric Engineering(cid:148) Approach) As one example of the significance of the tabu- lated values, the dependence of fundamental length Despite 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 adjust cant degree, we examine the structure that such in accordance with local changes in vacuum perturbations would take under conditions useful polarizability - thus there is no such thing as a per- for space-flight application, a (cid:147)Blue Sky(cid:148) 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 (cid:147)curved space-time, we shall find it convenient for our pur- space.(cid:148) In the vicinity of, say, a planet or star, where poses to utilize one of the alternative methodolo- K(cid:160)>(cid:160)1, if one were to take a ruler and measure along gies for treating metric changes that has emerged a radius vector R to some circular orbit, and then over the years in studies of gravitational theories. measure the circumference C of that orbit, one The approach, known as the polarizable vacuum would obtain C < 2p R (as for a concave curved sur- (PV) representation of general relativity (GR), treats face). This is a consequence of the ruler being the vacuum as a polarizable medium [30]. The PV relatively shorter during the radial measuring proc- 140 Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flight ess 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 v(K) v = c/K velocity of light < c ence on the measuring process due L L to induced polarizability changes in mass m(K) m = m o K3/2 effective mass increases the vacuum near the body leads to frequency w (K) w = w / K redshift toward lower frequencies o the GR concept that the presence time interval D t(K) D t = D t clocks run slower of the body (cid:147)influences the metric,(cid:148) o K and correctly so. energy E(K) E = E / K lower energy states o length dim. L(K) L = L / K objects shrink o We are now in a position to con- sider application of this (cid:147)metric en- TABLE 2: Engineered Metric Effects in the Polarizable Vacuum (PV) Representation of gineering(cid:148) 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 v(K) v = c/K velocity of light > c can in principle be modified to re- L L duce the value of the vacuum di- mass m(K) m = m K3/2 effective mass decreases o electric constant K to below unity. frequency w (K ) w = w / K blueshift toward higher frequencies o Returning to Table 1, we see that a time interval D t(K) D t = D t clocks run faster K < 1 solution permits the addition o K of another column for which the energy E(K) E = E / K higher energy states o descriptors are reversed, as shown length dim. L(K) L = L / K objects expand o in 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 energy perturbation, the local velocity of light (as seen from a requirement to somewhat below a solar mass, an reference 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 Krasnikov presumably 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 breakthrough generated 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 that by distorting the local space-time metric in the re- Clearly then, calculations for the proposed gion of a spaceship in a certain prescribed way, it geometries are by no means directly applicable to would 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 of ers outside the disturbed region, without violating potentially useful trends derivable on the basis of the local velocity-of-light constraint within the re- the application of GR principles as embodied in a gion. 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 an the spaceship(cid:146)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 Alcubierre solution, however, seemingly insurmountable In this paper we have touched briefly on innovative obstacles emerge. For a 100 m warp bubble the forms of space propulsion, especially those that bubble 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 this roughly 10 orders of magnitude greater than the point in the development of such nascent concepts 141 H.E. Puthoff, S.R. Little and M. Ibison it is premature to even guess at an optimum strat- where the sense is from left to right. From this it is egy, 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 will tion is even feasible. Nonetheless, only by inquiring now show there is another momentum, equal and into such concepts in a rigorous way can we hope opposite to this electromagnetic field momentum. to arrive at a proper assessment of the possibilities and thereby determine the best course of action to Since the current flowing in the loop is given pursue in our steps first to explore our solar system by I = r av, the velocity of the fluid is everywhere environment, and then one day to reach the stars. v = I/r a. Meanwhile, the external electric field E cre- ates a pressure difference between the bottom and Appendix A - Hidden Momentum the top of the fluid given by P = r Eh. Moving to the left, therefore, is a net energy flux S (energy per unit Consider a stationary current loop which consists area per unit time) given by of an incompressible fluid of positive charge den- sity r circulating at velocity v clockwise around a ( ) ( ) IEh loop of non-conductive piping of cross sectional S =Pv= r Eh x I/r a = (A4) a area a. The loop is immersed in a constant uniform electric field E. But since energy has mass, Eq. (A4) may be con- verted to an expression for momentum. This is mostly easily accomplished by writing the Einstein w relation E = mc2 in flux density form as S = gc2, where g is the momentum per unit volume. It now follows v 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 p =gaw= = (A5) mech c2 c2 where the subscript (cid:145)mech(cid:146) 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 equal bines with the electric field to produce an electro- and opposite mechanical momentum. Because of magnetic field momentum given by its rather obscure nature, this momentum has been referred to in the literature as (cid:147)hidden momen- p = 1 (cid:242) ExHdV (A1) tum(cid:148). This is a particular example of the general EM c2 result that a net static linear field momentum will always be balanced by an equal and opposite However, 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 p = 1 (cid:242) Jf dV (A2) mechanical momentum hidden within the struc- EM c2 ture, so that the whole system remains stationary. This inability to utilize linear field momentum for With 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.(cid:160)(A2), this computes to Appendix B - Metric Engineering Solutions Iw( ) IEhw p = f - f = (A3) EM c2 top bottom c2 In the polarizable vacuum (PV) approach the equa- 142 Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flight tion that plays the role of the Einstein equation ( ) (curvature driven by the mass-energy stress ten- K = K 2 =e2GM/rc2 =1+2(cid:231) (cid:230) GM (cid:247) (cid:246) +... (B3) Ł rc2 ł 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 (cid:209) 2 K - ( c/K )2 ¶ t2 = prevailing in the solar system. K (cid:239)(cid:236) m c2Œ Ø ( 1+w2 ) œ ø ( ) 1(cid:231) (cid:230) B2 (cid:247) (cid:246) For the case of a mass M with charge Q, the elec- - 4l (cid:239)(cid:238) (cid:237) 2 o K Œ Œ Œ º ( 1- w2 ) 2 1 œ œ œ ß (cid:228)3 r + 2 (cid:231) Ł Km o +Ke o E2 (cid:247) ł t a r i v c a f r i i e a l b d l e a - p d p ie r l o e p c r t i r a ic te -c t o o n a s t c a h n a t r m ge e d d i m um as i s s i g m iv b e e n d d b e y d in - K l 2 Œ Œ º Ø ( (cid:209) K )2+ ( c/ 1 K )2 (cid:231) Ł (cid:230) ¶ ¶ K t (cid:247) ł (cid:246) 2 œ œ ß ø (cid:239)(cid:254) (cid:239) (cid:253) (cid:252) (B1) (cid:242)D.da = Ke o E4p r2 = Q (B4) which leads to (for spherical symmetry, with b2 = Q2G/4pe c4) o where d2 K + 2 d K = 1 Œ Ø (cid:231) (cid:230) d K (cid:247) (cid:246) 2 - b2 œ ø w=v/ ( c/K ) dr2 r dr K Œ (cid:231) Ł dr (cid:247) ł r4œ (B5) º ß In this PV formulation of GR, changes in the vacuum which should be compared with Eq. (B2). The solu- dielectric constant K are driven by mass density (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 by term). (The constant l =c4/32p G,where G is the gravitational constant.) (cid:230) (cid:246) (cid:230) (cid:246) (cid:231) a2- b2 (cid:247) a (cid:231) a2- b2 (cid:247) K =cosh + sinh (cid:231) (cid:247) (cid:231) (cid:247) In space surrounding an uncharged spherical Ł r ł a2- b2 Ł r ł mass distribution (e.g., a planet) the static solution ( ) ¶ K/¶ t=0 to the above is found by solving for d2 K + 2 d K = 1 (cid:231) (cid:230) d K (cid:247) (cid:246) 2 a2 >b2 (B6) (cid:231) (cid:247) (B2) dr2 r dr K Ł dr ł For the weak-field case the above reproduces the familiar Reissner-Nordstrłm metric [39]. 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