Abstract

Albert Einstein in his General Theory of Relativity predicted gravitational waves (GW). Such waves have never been detected, but an extra-terrestrial source of low-frequency GW (10 KHz and below); namely, a neutron double star, has been observed to coalesce at a rate exactly as predicted if it radiated GW. The extra-terrestrial GW are generated by relatively weak gravitational attraction among large celestial masses. In this paper several devices are proposed that allow for the production of high-frequency GW (MHz to THz and above), by means of a sequence of pulses having a significant average power, that are generated by relatively strong magnetic, electric, electromechanical, and nuclear forces acting on relatively small masses with a “jerk” or “shake”. The process that actually generates GW is the rapid change in acceleration not necessarily the acceleration itself. The essence of the jerk approach is that although it introduces no new theory it does make the engineering applications more apparent. A conductor would screen off any accompanying electromagnetic signal. The analyses rely on conventional, classical physics and no new physical principles, concepts or theories need to be introduced. Preliminary tests of such devices are suggested, which involve existing high-frequency GW detectors, including those devices that produce coherent high-frequency GW as well as a superconductor in an alternating magnetic field, which may generate high-frequency GW that in turn may change the gravitational field local to it. High-Frequency (HF) GW is chosen since it exhibits higher energy for propulsion, wider bandwidth for communication, and less diffraction for imaging than does Low-Frequency (LF) GW. The results of the proposed tests will lead to insights concerning HF GW spacecraft propulsion, HF GW communications, and imaging leading to the design of a HF GW Telescope. It is also the purpose of this paper to challenge the imagination of the reader concerning applications of HF GW.

Nomenclature

A = area

a = semi-major axis of a two-body orbit

a = acceleration

B = bandwidth

c = speed of light or, alternatively, approximately the electron mobility speed

D_a_b = quadrupole moment-of-inertia tensor

d = diameter

E = energy

e = eccentricity of a two-body orbit

f = force

f_cf = centrifugal-force vector

G = universal gravitational constant

I = moment of inertia

i = current

l = length

M = mean anomaly for a two-body orbit

m = mass of an object on orbit in characteristic units

m = sum of the masses of a pair of binary stars or mass of a rod in kilograms

N = noise or index of GW refraction

n = mean motion for a two-body orbit

n = number of coil turns

P = the magnitude of the power of a gravitational-radiation source

p = parameter or semilatus rectum = a(1-e² )

Q = p times the number of oscillations a free oscillator undergoes before its amplitude decays by a

factor of e

R = resistance or lens surface radius

r = radial distance to an object on orbit; alternately, the effective radius of gyration

r = radius of a magnetic core, piston or barrel

S = signal or GW flux

s = distance or displacement

V = volume or speed

v = true anomaly of a two-body orbit

v = speed

t = time

x = axis of orthogonal coordinate system

y = axis of orthogonal coordinate system

z = axis of rotation orthogonal to x and y axis

a = attenuation or diffraction angle

b = propagation constant

D = small increment

Df_cfx= incremental x component of centrifugal force

Df_cfy = incremental y component of centrifugal force

Dt = time increment

δ = fraction of a linear-motor, GW generator’s barrel radius that is an energizing-element sheath

and/or energizable-element core

dm = differential mass

dt = differential time

k_I_w2dot = coefficient (constant or function) of the kernel in the Iw² formulation of the quadrupole

k_I3dot = coefficient (constant or function) of the kernel in the d³I/dt³ formulation of the quadrupole

l = wavelength

m = m₁ +m₂ = sum of masses on a two-body orbit in characteristic units

m₀ = permeability of free space

n = frequency

s = absorption cross section

t = characteristic time; for heliocentric unit systems 5.022x10⁵ seconds

w = Angular rotational rate

Subscripts

1 = refers to mass one or front lens surface (one)

2 = refers to mass two or back lens surface (two)

cf = centrifugal

d = diffraction

GW = gravitational wave

l = longitudinal

r = radial

t = tangential

x = x component

y = y component

I. Introduction

In the last four decades there has been considerable progress developing instruments that detect low-frequency (LF) gravitational waves (GW) 10 KHz and below. Up to this time no one has proposed a practical device to generate GW artificially on the Earth. Heretofore it has been assumed that artificially produced GW are of far too small of an intensity to be of any useful value. They are considered to be ultra-weak disturbances and, in most cases, masses and accelerations of man-made terrestrial devices have never produced useful or even measurable GW. Thus many physicists in the general-relativity community have been absolutely certain that it is impossible to generate GW on Earth! Such a reaction was not surprising (and another reason for the proposed experiment) since as Abraham Pais¹⁷ has stated: “Physicists – good physicists … are conservative revolutionaries, resisting innovation as long as possible and at all intellectual cost….” In fact, there is no experimental evidence on Earth that validates the generation of GW based upon accepted physical principles using gravitational attraction of masses; but no such bar exists for electric, magnetic, and nuclear forces. As a matter of fact, it is NOT necessary to use gravitational attraction to generate gravitational waves! In fact, Joseph Weber⁷ (1964, page 97) writes that "The non- gravitational forces play a decisive role in methods for detection and generation of gravitational waves …" and his comments have never been disputed. In fact, in Warsaw, Poland Leopold Infeld among others (such as Halpern and Jouvet ⁵⁴ p. 25) believed that objects moving under purely gravitational forces will not radiate gravitational waves (GW), but believed that gravitational waves can only be generated by non-gravitational forces. Infeld was proved wrong -- any force will generate gravitational waves: electrical, magnetic, electromagnetic, nuclear, centrifugal, gravitational, etc. Indirect confirmation of the existence of GW generated extra terrestrially has come from observations of the binary pulsar PSR 1913+16. Similar to the operation of one of the devices described herein it is spinning down. Different from the described devices, however, it relies on the relatively weak gravitational attraction between a pair of very massive neutron stars to produce strong, 10²⁵ watt, LF GW (frequency about 0.00007 Hz = 0.07mHz) rather than relatively strong magnetic, electric, electromechanical, and nuclear forces to produce HF GW (MHz to THz and above). The observational evidence and the recognition of the importance of GW was the basis of the 1993 Nobel Prize in Physics awarded to R. A. Hulse and J. H. Taylor. Data regarding PSR 1913+16 will be utilized to validate numerically the algorithms derived in this paper.

The rapid movement, or “jerk” or “shake” of an uncharged mass or the rapid change, or “jerk” in angular momentum with time, over a nanosecond to less than a picosecond time span, caused by the operation of the contemplated devices (US Pat. 6,417,597; 6,160,336; and patents pending), will produce a quadrupole moment and could generate useful HF GW without attendant overpowering electromagnetic radiation (please see Addendum A). The devices discussed will accomplish this GW generation in several alternative ways based upon the terrestrial device’s rotating and non-rotating, symmetrical and non-symmetrical masses, sometimes in a superconducting state, acted upon by means of relatively strong magnetic, electric, electromechanical, and nuclear forces. Such forces are produced by an ensemble of very small, sub-millimeter (often much smaller than a GW wavelength), energizing or stimulating elements (including particle beams, microwaves, lasers, alternating magnetic fields, etc. -- please see Addendum B) operating in concert under the control of the device’s computer on energizable or stimulated elements (including submicroscopic particles). Successive energization pulses generate a long train of short (e. g., picosecond duration) GW pulses having a significant average power. The energization can be timed to follow a GW crest and accumulate coherent GW (similar to the suggestion of F. Romero and H. Dehnen⁵² who consider a chain of thin piezoelectric crystals and Pinto and Rotoli ⁵⁶who use a laser beam to excite an array of Germanium crystals). It is also noted in this paper that GW can be refracted and focused in order to increase its intensity for reception or for use in a HF GW Telescope (Patents pending; please see end of Addendum B). As noted, the terrestrial process is different from the extra-terrestrial generation of LF GW by very large rotating and non-rotating celestial masses acted upon by relatively weak gravitational attraction often producing a sequence of very long-wave GW pulses or bursts. It is to be noted that it is not possible in a terrestrial laboratory to produce large mass-times-velocity or momentum values compared to those produced by celestial events. The process that actually generates the GW is the rapid change in momentum not the momentum itself—it is not acceleration, but its change, a jerk or a shake, that generates GW! Proof of this concept is an objective of this paper (Test Objective (1) of Section VII).

Certainly, gravitational waves are real, can be generated on Earth, do transmit momentum, and may change a gravitational field. Hence the potential for a new form of communication and spacecraft propulsion (please see the Addenda A and B) and the concurrent desirability to test the concept. Two alternative spacecraft propulsion means or concepts are described: first, the use of HF GW “… as a source of some additional gravitational field…” at a distance, as suggested by L. D. Landau and E. M. Lifshitz¹ (p. 349) possibly near to and acting on a spacecraft (Fontana⁵¹) and, second, anisotropic gravitational-wave radiation from GW generators onboard the spacecraft as described in Addendum B. Preliminary tests or experiments that would validate one or both of these propulsion concepts are discussed. Such tests involve determination of the intensity, propagation characteristics (including possible lack of diffraction and/or dispersion: Test Objectives (2) and (13)), absorption, and influence on a gravitational field of HF GW (Test Objective (7)) and the characteristics of magnetic-field build up on nanosecond to picosecond or less time scales (Test Objective (11)).

II. History of HFGW Generation Devices, Characteristics and Concept of Gravitational Waves

History of HFGW Generation Devices

Although not well known generally, the concept of gravitational waves (GW) is extremely well known and documented in the scientific community. Albert Einstein in his General Theory of Relativity predicted gravitational waves. In many ways they are similar to electromagnetic (EM) waves e.g., light, radio, microwave, X-rays, etc., that are produced when a charged particle is accelerated. In fact, GW occur when mass is accelerated or decelerated with a jerk (or shake) or subjected to harmonic motion and generate second-rank tensors (not EM vectors). The predicted effect is usually quite small, but as previously noted has been confirmed observationally in the gradual slowing of the rotation of binary pulsar neutron stars. One of the earliest pioneers in GW research was Joseph Weber². One of his associates, Robert L. Forward, worked at the Hughes Aircraft Company Research Laboratories in Malibu, California. Dr. Forward’s Ph.D. thesis (in 1965) involved the construction of the first bar antenna designed for the detection of low-frequency (LF) GW. This GW detector is often referred to as the “Weber Bar”. Such a bar is essentially a large cylinder of aluminum that is super cooled, isolated and under a vacuum in order to minimize any thermal noise. When a LF GW impacts it, the bar begins to vibrate or “ring” very slightly at its fundamental frequency. This ring manifests itself in a very small change in length of the bar (less than the diameter of a proton) that can be sensed, for example, by a laser or by piezoelectric crystals. By having two bars separated thousands of kilometers away one can subtract out the earth-based vibrations (e.g., a hurricane coming to rest, micro seismically vibrating Earth’s crust, etc.) and record only the extra-terrestrial waves coming from celestial sources. In this rudimentary GW antenna only the total energy of the waves can be determined and the time that they occurred. Such resonance LF GW detectors are now supplemented by interferometric detectors that measure minute changes in length occasioned by a LF GW.

Weber’s instrument observed what were presumed to be high energies of GW and prompted other groups to develop similar instruments. Over the years considerable progress has been made in the development

of GW detectors or antenna. Today there are well over a dozen such instruments operating or under construction throughout the world. One example is the Laser Interferometer Gravitational-wave Observatory (LIGO) being developed at the California Institute of Technology. The purpose of all these instruments has been to detect low-frequency, extra-terrestrial sources of gravitational radiation. To this date there has been no known development or construction of a functioning HF GW generator whose “signal” can be sensed by HF GW detectors, however many designs have been proposed. As noted by Portilla and Lapiedra³⁶, in 1962 Gertsenshtein³⁷ discussed (in a very brief, two-page paper) the resonance of light and gravitational waves (termed by them as Gertsenshtein waves). In 1964 Halpern and Laurent⁵³ (pp. 747-750) and in 1968 Halpern and Jouvet⁵⁴ defined a gravitational counterpart of the laser called a “gaser” an acronym for Gravitational-wave Amplification by Stimulated Emission of Radiation that does not produce significant EM; but indicated ( p. 41) that “... the effects are ... below the threshold of observability...”. Halpern and Laurent⁵³ suggest that “... the maximum of the gravitational radiation occurs in a direction from which the corresponding electromagnetic radiation is excluded.” In U. S. Patent No. 3,722,288 (filed January 31, 1969), Weber alluded to a GW generator in this GW-detection patent; but never fully described or claimed it. The interaction of a small dielectric sphere (an “energizable element” in the context of the present paper) and a plane electromagnetic wave (an “energizing element” in the context of the present paper) “… seem promising for the generation of detectable high-frequency gravitational waves in the laboratory.” (Portilla and Lapiedra³⁶, p. 044014 -1; emphasis added). Essentially, they suggest that an electric charge shaken (or “jerked” in the context of the present paper) in a homogeneous stationary magnetic field produces both electromagnetic and gravitational waves. However, they have not reported the design (as is done in this paper), fabrication or operation of any such device to date. In 1974 Grishchuk and Sazhin³² proposed a device that was according to Weiss³³ “… only a factor of one hundred thousand (10⁵) from being feasible …” at 10^-10 [watts/m²]. On the other hand, Vinet³⁴ stated “… terrestrial generation of gravitational waves has been addressed by several authors in the seventies who proposed … pure electromagnetic effects (you can move packets of energy at high speed in wave guides) … All these attempts failed due to the very low masses or equivalent masses involved.” Also Klimenko³⁵ advises that “… people made this kind of experiment at the Integrated Nuclear Research Institute, Dubna, Russia, 15 years ago … the experiment was a mistake and never was completed … it took several years for the Dubna physicists to realize that their calculations were wrong.” In 1981 Romero and Dehnen⁵² proposed a row of ten thousand, 10 [cm] long, 0.5 [cm] wide piezoelectric crystals 0.5 [cm] apart to generate coherent GW (with two polarized states perpendicular to the direction of propagation along the row) the stronger, coherent, forward component having an intensity of only 10^-29 [watts] in a 20 degree “needle radiation” beam and the attendant EM from the crystals was small. This intensity they felt was not high enough for a successful laboratory experiment; but the size of their energizable or stimulated elements was many orders of magnitude larger and their number many orders of magnitude smaller than the submicroscopic elements considered herein. For example, if THz frequencies and 10⁹ closely spaced nanopiezoelectric crystals were utilized, then the intensity climbs to more than 10^-9 [watts] according to their Eq. (A.11). A theoretical EM-GW converter suggested by Pinto and Rotolli⁵⁶ in 1988 could generate 10^-17 [watts] of GW power (with a small resulting EM component) – still too small and “... at the limit of (the) state of the art.” In 1991 Astone, et al⁵⁵ operated a spinning (30,000 rpm) rotor GW generator (actually a gravitational-field oscillator producing waves of gravity not gravitational waves) near a resonance GW detector at CERN, but they had “... difficulties in controlling the detector frequencies...” and their results were inconclusive. High-temperature superconductors (HTSCs) under the influence of a high-frequency magnetic field may have serendipitously generated HF GW ^44,45,46by means of a jerk. Fontana ⁴⁷ has, in fact, suggested that an HF GW flux of 10⁵ [watts/m²] could be generated by such a device without significant attendant EM, but proof of that at the time of this writing is incomplete. (If such an experiment involving currently operating HFGW detectors were implemented, then the jerk approach could be proved experimentally.) Thus this paper and patents; and patents pending break new ground.

Characteristics

Gravitational waves (GW) are absorbed differently and propagate differently through matter and space than are electromagnetic (EM) waves. By the way, the term “gravity waves,” strictly speaking, refers to water waves in which buoyancy acts as a restoring force, as opposed to relativistic “gravitational waves” (GW) as referred to in this paper. Gravitational waves also differ from oscillating or periodic “waves of gravity,” which evoke, say, a tidal response or perturbation in masses in its vicinity. For example, a spinning neutron star generates gravitational waves in the spacetime continuum, but not significant waves of gravity evoking a tidal response. Contrary wise, a mass dipole generates no gravitational waves (Weber [1964]), but could evoke a nearby tidal response. GW may offer advantages over EM in that like the gravitational field itself GW are transmitted through material opaque to EM and their intensity may fall off less rapidly with distance than EM.

Concept

The general concept of the devices discussed herein is to simulate scientifically accepted GW generation by energizable celestial systems (rotating binary stars, star explosions, star collapse, binary black holes, etc.) by the use of small macro- and micro-, terrestrial energizable systems. Such terrestrial systems generate well over 40 orders of magnitude more force intensity (nuclear or electromagnetic compared to gravitational) and well over 12 orders of magnitude greater frequency (THz or QHz compared to KHz or very small fractions of a Hz) than the celestial systems. Terrestrial energizable systems produce significant and useful GW according to the various designs of the devices described herein, even though they are orders of magnitude smaller than the extraterrestrial celestial systems. In the various designs of the devices the large numbers of small energizable elements are energized in the sequence or in concert by energizing or stimulating elements emulating the motion of a much larger and extended body in order to enhance the generation of GW.

The specific concept, which will be expanded upon in this paper, is that by applying a long series of rapid “jerks” or third-time-derivative motion to a mass or masses, using relatively strong magnetic, electric, or nuclear forces, the devices described in the present paper will be shown to generate a significant HF GW without disruptive g loads. The effect will be measurable in the laboratory since it affects or warps the spacetime geodesic over very small distances (due to high frequency and short GW wavelength) and thereby will produce detectable GW. If the energizable elements are uncharged, then there is little or no EM radiation.

III. Analysis of PSR 1913 +16 and Binary Black Holes

Since the observation of the binary pulsar PSR 1913+16 (identifies right ascension of 19 degrees 13 minutes and declination of 16 degrees North) represents the only experimental confirmation of GW, insight into the jerk approach can be found in the analyses of such a double-star system. Thus please bear with the rather laborious arithmetic. The pair of stars will coalesce in 3.5x10⁸ years due to GW radiation and produce a rather continuous GW until that time. It is the pair’s slowing that exactly agrees with GW-generation theory (utilizing orbital mechanics) that indirectly confirms the existence of GW. According to J. H. Taylor, Jr.³, the period of their mutual rotation is 7.75 hours (or 2.79x10⁴ [s]), periastron is 1.1 solar radii (one solar radius is 6.965x10⁸ [m]), and apastron is 4.8 solar radii. It’s radius of gyration is essentially the semi-major axis = (1.1 + 4.8)/2 = 2.95 solar radii = (2.95)(6.965x10⁸) = 2.05x10⁹ [m]. Each star exhibits a mass of about 1.4 solar masses (one solar mass is 1.987x10³⁰ [kg]) so that together their mass is m = (2)(1.4)(1.987x10³⁰) = 5.56x10³⁰ [kg]. According to a perusal of binary-star catalogs by John Mosley of the Griffith Observatory, the binary pulsar PSR 1913+16 is at a distance from our Sun of 23,300 light years (one light year is 9.5x10¹⁵ [m]). If there was complete diffraction, then the reference area over which the GW would spread at the Sun’s distance would be a sphere having an area of (4p)(2.33x10⁴ x 9.5x10¹⁵)² = 6.2x10⁴¹ [m²].

In the case of a binary star pair such as PSR 1913+16 the magnitude of the GW power, P, is computed from the quadrupole moment, which for two masses on orbit about one another is given, for example, by an equation on p. 356 of L. D. Landau and E. M. Lifshitz¹or Peters and Mathews (1963)⁴². The time-constant factor in the equation for P is

8G⁴m₁²m₂²μ/(15c⁵). (1.1)

They give the time-variable factor in P as a function of the true anomaly, v, and orbital eccentricity, e, as

(1+ecosv)⁴([1+{e/12}cosv]²+e²sin²v)/(a[1-e²])⁵. (1.2)

In conventional astrodynamic/celestial-mechanics notation (please see Samuel Herrick³⁰ ) this factor is

p/r⁶+(dr/dt)²/12mr⁴, (1.3)

where p is the orbital “parameter” or semilatus rectum (= a{1 – e²}) in [AU], r is the radial distance between the two masses [AU], t is the characteristic time measured in k_sdays or in units of 5.022x10⁶[s] for a heliocentric-unit system (utilized by Taylor³ and others for PSR 1913+16), m is the sum of the two masses, m₁+ m₂ [solar masses], G = 6.67423x10^-11 [m³/kg-s], and c is the speed of light = 3x10⁸ [m/s]. Note that one AU (astronomical unit) = 1.496x10¹¹ [m] and one solar mass = 1.987x10³⁰ [kg]. The dr/dt term is related to dI/dt (=-2mr[dr/dt]), d³I/dt³ (=-2m²[dr/dt]/r²), d²v/dt² (=-Ömp[dr/dt]/r³), and d³v/dt³ (=-2mÖmp[1/r-1/a-4{dr/dt}²/m]/r⁴), where a = the semi-major axis of the orbit [AU] and for a circular orbit dr/dt = 0. These time derivatives are related to some of the devices discussed in this paper.

The GW power radiated, P, which causes a perturbation in the semi-major axis, a, (and an attendant secular decrease in the orbital period) is obtained by integrating the time-variable factor, Eq. (1.3), over the orbital period using the mean anomaly, M, which is directly proportional to the time (that is, M = n [t-T], where n is the mean motion [w in Landau and Lifshitz’s¹ notation, p. 357] and T is the time of periastron passage). The value of the average GW power, P, is computed from observations that define the eccentricity (based primarily upon Doppler-shift determination of the range rate at periastron and apastron), semi-major axis, and orbital orientation angles of PSR1913+16. The error in the computed value of P is related to the observational error leading to the determination of the orbital elements as well as the determination of the masses of the pair of neutron stars, m = m₁ + m₂ = 1.4 + 1.4 = 2.8 [solar masses]. For example, a 0.1 percent change in the measurement of range rate at periastron results in a 0.28 percent change in GW power, P, and a 0.1 percent change in the mass of the stars results in a 0.33 percent change in GW power. The average centrifugal force component, Df_cfx,y (which will later be utilized to validate the fundamental jerk equation) is

man²= (5.56x10³⁰)(2.05x10⁹)(2.25x10^-4)² = 5.77x10³² [N] (2)

divided by m yields the average centrifugal acceleration = 103.78 [m/s²] = 10.6 [g’s]. At periastron, r = q = a(1-e) = (2.05x10⁹)(1-0.641) = 7.36x10⁸ [m] (e = 0.641), the centrifugal acceleration is q(dv/dt)² where dv/dt = Ö(mp)/r² (please see Baker⁴, p. 13). In this latter case m = 2.8 [solar masses], a = 2.95 [solar radii] = (2.95)(6.965x10⁸ [m/solar radii]/1.496x10¹¹ [m/AU] = 0.01373 [AU], p = a{1-e²} = 0.01373{1- 0.4109} = 0.00809 [AU], and q = r = 7.36x10⁸ [m]/1.496x10¹¹[m/AU] = 0.00495 [AU]. After inserting these numbers we have dv/dt = (Ö[2.8x0.00809]/[0.00495]²)/5.022x10⁶[s/k_sday] = 1. 223x10^-3 [radians/s]. Thus the centrifugal acceleration at periastron of the star pair is q(dv/dt)² = (7.36x10⁸ [m])(1.223x10^-3 [radians/s])² = 1.101x10³ [m/s²] = 112 [g’s] – apparently still within the weak-field approximation of Einstein’s GW equations.

The observed accumulated shift in the times of periastron passage, T, caused by the secular shortening of the orbital period of PSR 1913+16, compares closely, within observational error, to that predicted by General Relativity and confirms the existence of GW radiation. Likewise is confirmed the existence of a dr/dt component, which is related to d²w/dt² (»d³v/dt³) and d³I/dt³ , that are involved in the GW-generator aspect of this paper. The average magnitude of the GW power, P, established by Landau and Lifshitz¹, p. 357, by analytical integration and given as a function of eccentricity, e, is for e = 0.641, 9.293x10²⁴ [watts]. By numerically integrating (see, for example, Baker⁴, pp. 263-272) over the mean anomaly the average GW power, P, is 9.296x10²⁴ [watts] and exhibits low frequency (0.00007 Hz = 0.07mHz) associated with the orbital period of the star pair. The peak GW power, 1.73x10²⁶ [watts], occurs at the time of periastron passage (every 7.75 hours; when the neutron stars rapidly jerk around each other) and at the Sun’s distance would result in GW bursts having a GW-flux magnitude of 1.73x10²⁶/(4p[2.33x10⁴x9.5x10¹⁵]²) = 2.81x10^-16 [watts/m²] if totally diffracted over the spherical reference area and the GW propagation was approximately spherically isotropic.

For comparison with the detection sensitivity for which LIGO is designed, we turn to binary black holes (BBHs) as discussed by Flanagan and Hughes⁴¹. We choose BBHs during the inspiral phase having equal mass = 10 solar masses (if the masses are equal, then computations reveal that the GW power is independent of the BBH’s mass – their mass, m, being, however, in the range of 3< m < 2000 solar masses). Calculating from Eq. (1.1) we have for the time-constant factor, (8)(6.6743x10^-11 )⁴(10)²(10)²(20)(1.987x10³⁰)⁵/(15)(3x10⁸)⁵= 2.7x10⁷⁴ and for a 1000 black-hole (BH) radii semimajor axis (equal to both p and r for an osculating circular orbit) with the BH radius = 2.95x10⁴ [m], the variable factor, p/r⁶ = 4.476x10^-38 ; so that the power, P = 1.208x10³⁶ [watts]. For a 6 to 100 BH-radii osculating orbit the power is 1.55x10⁴⁷to 1.21x10⁴¹ [watts] (LF GW generated with frequencies of 5KHz and below). At 6 BH-radii the BBH apparently becomes dynamically unstable (see p. 4535 of Flanagan and Hughes⁴¹) and there are no more simple osculating orbits. Please note that the radial speed of contraction according to the equation on p. 356 of Landau and Lifshitz¹ is -6.3x10^-4 BH radii per period of the osculating orbit (to the spiral) at 1000 BH radii (also independent of BBH mass) and the number of cycles (half periods) for 100-BH-radii contraction at 1000 BH radii is 3.2x10⁵ so that there should be sufficient cycles for detection if the signal-to-noise ratio (using theoretical waveform templates) is proportional to the square root of the number of cycles for matched filtering (p. 4537 of Flanagan and Hughes⁴¹). Also note that the time for the BBH to turn 1% of their mass into GW energy for such an osculating 1000-BH-radii semimajor-axis orbit is about 1000 years; so the chance of a BH being present and, therefore, of LIGO observing BBHs is probably good – depends, of course, on the number density of such BBHs in the nearby universe. (The final inspiraling, merger, and ringdown of a BBH probably last but a very small fraction of a second.) The maximum BBH detection distance⁴¹for the initial LIGO interferometer is 500 mega parsecs [Mpc] or 1.6x10²⁵ [m]. Therefore, for long-wave-length LF GW, the GW-flux at the Earth = 1.208x10³⁶/(4π[1.6x10²⁵]²) = 3.7x10^-16 [watts/m²] for the 1000 BH-radii case and 5x10^-5 to 4x10^-11[watts/m²] for the 6 to 100 BH-radii cases. We take this GW flux of between 10^-16 and 10^-5 [watts/m²] to be very approximately the maximum LF GW detection sensitivity of LIGO and hypothesize that 10^-8 [watts/m²] is the background noise for HF GW (please see Addendum A where noise higher than 10^-7 [watts/m²] can be tolerated).

IV. Individual Independently Programmable Coil System (IIPCS) (U. S. Patent No. 6,160,336)

Of fundamental importance to the operation of GW generation devices discussed herein is the Individual Independently Programmable Coil System (IIPCS, U.S. Patent 6,160,336). This system is enabled by computer and associated computer software to control a system of either transistors or of ultra-fast switches. The switches rapidly turn off or on a myriad of sub-millimeter coils and/or electromechanical or other energizing elements. By means of which magnetic (or electromechanical) force produces a third time derivatives or "jerk" of a mass or of submicroscopic masses in, for example, a HTSC. As a first example, we shall consider a succession of peripheral jerks acting on a large rotating or non-rotating spindle test device.

In FIG. 2, a series of permanent magnets, 24, are schematically shown that are embedded in the bottom face of a reinforced concrete rim attached to a spindle device at about 20-ft. intervals. This means that there are approximately 72 large magnets (alternatively a very large number of smaller magnets can be utilized) spaced around the lower surface of the rim. As the spindle turns about the z-axis the magnetic fields of the permanent magnets sweep over a string of individual coil sets, 26, that are attached to the ground and located in close proximity to the faces of the permanent magnets. The coils may or may not have metallic cores. In the rim acceleration or spin-up mode, a current flowing in the coils produces a magnetic field that pulls the permanent magnets in the rim around to the right. As is shown in FIG. 3A, as a rim magnet approaches a coil set, the south pole of the magnet is attracted by the north pole of the magnetic field produced by current flowing in the coil in the direction shown. Thus, the rim is urged to move more to the right as shown in the drawing and the rim speed increases. In FIG. 3B the permanent magnet has been carried around the rim so as to be directly over the coil and no current flows through the coil as there is no magnetic field in the coil and no force on the magnet. In FIG. 3C, the permanent magnet is receding from the coil and the current has been reversed in the coil, thereby reversing the coil’s magnetic field so that the north pole of the permanent magnet is repelled by the north pole of the magnetic field produced by the current in the coil now flowing in the reverse direction. Thus, the permanent magnet and the rim are urged to move even more to the right. By reversing the foregoing process the rim can decelerate or spin down. A series of rapid jerks (in either direction, i.e., urging spin up or spin down) can be imparted to the rim. This is accomplished by passing very short (e.g., picosecond duration) direct-current pulses through the coils that, in turn, impart a series of rapid jerks to the uncharged rim material. Such jerks are in addition to the continuous, very small jerks of the rim generated as the rim material’s centrifugal-force vectors jerk around if the rim is rotating (please see section V). We will discuss accelerations imparted to the rim and associated material stresses later on in this paper.

The spindle test device (only one of several possible test devices) includes an Individual Independently Programmable Coil System (IIPCS) to enable the coils to be electronically reversed as shown in FIGS. 3A and 3C and in FIGS. 4A and 4C at high frequency to generate HF GW (MHz to THz and above). Thus the flywheel-magnet/coil system can be switched between a spin-up and spin-down at will, that is, given a series of reciprocating jerks so that no net acceleration or spindle rotation is built up. Coils of various lengths can be “assembled” electronically leading or lagging a given permanent magnet or magnets by controlling the transistors or ultra-fast switches. To accomplish this dynamic assembly, shorter and shorter “strings” of coils are connected together in series as the rim slows down more and more. This is done by “breaking” or interrupting a conductor by means of a transistor or ultra-fast switch before or after a given coil set. Thus, for example, 20 coil sets can be connected in series (coil-to-coil) to a string of 20 more coil sets up the line by breaking the conductor just before the coil in the first 20 coil set string and after the first coil of the next string of 20 coils. The current will run only from the last coil of the first string to the first coil of the next string as shown in FIG. 5B.

If two conductors are used, then a sub group of coils can be assembled in parallel by connecting the ends of each coil by means of the transistors or ultra-fast switches to a different one of the two conductors. These sub groups can be connected in series by the means discussed above i.e., by using the computer-controlled transistors or ultra-fast switches on the conductors to interrupt or disconnect the conductor just before and just after the sub groups of coils so that the current will only run from the end of the last sub group to the beginning of the next sub group of coils and so on up the line on alternate conductors. Since both ends of the coil can be attached to either conductor, the current can flow in either the counter-clockwise (right-hand rule), L to R, direction, FIG. 5A, 34, or the clockwise, R to L, direction, FIG. 5B, 36, through the coil sets so that the transistors or ultra-fast switches can be "set by" the control computer almost instantaneously to reverse the coil’s magnetic field. (Subject to experiments: Test Objectives (1), (9), and (11): see Section VII.) Thus the spindle device can spin up or spin down at will and the gravitational waves can be modulated and shaped.

We will now discuss other HF GW generation devices that utilize, for example, microchip and nanotechnology. For the very large number of ultra-small, sub-millimeter coil elements utilized in some of the devices discussed, which are in addition to the spindle (e.g., linear motor, parallel current-carrying conductors, solenoids, piezoelectric crystals, nanomachines, high-temperature superconductors, etc.), a miniaturized integrated circuit can be utilized (see, for example, the coil turn of Al utilized by Y. Acremann, et al¹⁸). They will be embedded in or imprinted on, for example, a silicon chip, organic material, or in connection with polymer-based or superconductor devices. They will consist of multiple layers (with appropriate sequencing time delays to ensure near simultaneity of the magnetic fields interaction as the direct-current train of approximately one-picosecond pulses simultaneously traverse each coil set on the chip levels) and possibly integrated in the chip with the ultra-fast switches or transistors or other semi-conductors. Since the jerk is generated by an electromagnetic process, there could be significant EM radiation generated that could reduce the efficiency of the device. Test Objective (12) will address this issue. A preferred design (U. S. Patent No. 6,417,597) utilizes

Robert M. L. Baker, Jr., Ph.D.

Senior Consultant, Transportation Sciences Corporation

Abstract

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