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 conventional computer chips, containing the IIPCS circuit elements about 18 micrometers or less apart, synchronizing clock, input/output ports, and sub-millimeter coils on 50 to 100 micrometer centers. The chips are about 6 mm to 9 mm square and are obtained from silicon wafers. These chips are sewn into a circuit-board roll with an approximately 25-micrometer-diameter gold thread. Several layers of this roll (for example, 25) are connected in a fixed location or band adjacent to the moving or non-moving spindle’s rim and form the IIPCS in the spindle rim’s magnetic field. Such rolls are routinely fabricated by French-owned Oberthur Card Systems (plant in Rancho Dominguez, California), French-based Gemplus, Schlumberger (Paris and New York), and California-based Frost & Sullivan.

In the proposed miniaturized integrated circuit devices, as exhibited in FIG. 9 A, there will be a very large number of ultra-small, sub-millimeter or microscopic coil sets or elements, 56, embedded in or imprinted upon a silicon chip, 57, in multiple layers. Ultra-fast switches or transistors of the IIPCS, 58, will launch a long a series of current pulses, 59, of approximately nanosecond to picosecond or less duration moving at the electron’s mobility speed (approximately light speed, c) that will be timed to reach the individual coil sets or elements almost simultaneously (with the same rise time as discussed in Y. Acremann et al ¹⁸) along several individual conductors, as in FIG. 9A, or one single conductor per line, as in FIG. 9B, and thereby interact with the magnetic field, 60, in concert. This interaction will result in a third-time-derivative motion or jerk of the uncharged magnetic mass to generate a train of gravitational waves. The ultra-fast switches are preferably semiconductor-based, such as the semiconductor optical amplifier (SOA), a semiconductor nonlinear interferometer such as a nonlinear Sagnac interferometer on a phosphide semiconductor chip, etc. (see, for example, D. Cotter, et al ⁵, pp. 1523-1528). In FIG. 9B, the IIPCS and its array of ultra-fast switches is programmed to launch a train of current pulses of approximately a picosecond duration, 59, such that each member of the pulse train will reach each of the coils or coil sets at the same time. The duration of the pulses will be such to completely energize any given coil set as it passes through it in order to produce a magnetic field interaction. The interaction will result in a third-time-derivative lateral motion or jerk of a cylindrical, central magnetic core, 63, shown in FIG. 10 and, as is discussed, generate a long GW train of successive GW pulses having axis, 29. This core, piston, or barrel is surrounded by and immediately adjacent to a sheath of IIPCS-controlled coil sets, 64. In the case of the current-pulse train on a single conductor interconnecting a line of coil sets, there will be a build up of impulses to full value as the current-impulse train progresses down the line of coil sets. Use of a single conductor wire for each line of coil sets reduces the resistive power loss. In each line of coils set in series 61 there will be time delays, 62, between coil sets to ensure simultaneity of the current pulses reaching any given coil set (U.S. Patent No. 6,417,597).

In FIG. 11, ultra-fast switches or transistors of the IIPCS, 58, will launch a long series of direct-current pulses acting in either direction, 59, of approximately nanosecond or picosecond or less duration moving at the electron’s mobility speed (approximately light speed, c) along individual conductors or single interconnecting computer wires in order to produce current pulses, 59, acting in concert to generate modulated jerks and resulting HF GW (GHz to THz and higher) with axis, 29. The current pulses will be timed to reach parallel-plate conductors, 66, which may have different masses or may have ballast, 67, attached and/or carry different current, and/or have different modulus of elasticity and/or are constructed differently in their mountings for the purpose of exhibiting asymmetrical mass displacements, jerks or “hammer blows”. As a GW front passes by the energizable, in this case parallel-plate, elements (schematically shown in FIG. 14E and 14F as 80, 84, 86, and 88) they are energized in sequence thereby increasing the wave’s amplitude. In FIG. 14F such an effect is schematically illustrated as GW 83 ,85, 87, and 89 build up to accumulate the GW, 82, wave front shown also in FIG. 14E. Thus a device having a much longer effective length (or radius of gyration), r, than that of the individual energizable elements is emulated. Again, subject to experimental verification (Test Objectives (1), (9), and (11) of Section VII). It is to be emphasized that any unwanted EM radiation can be screened out.

In FIG. 12, ultra-fast switches or transistors of the IIPCS, 58, will launch a long series of current pulses, 59, of approximately picosecond duration moving at the electron’s mobility speed (approximately light speed, c) along individual conductors or single, interconnecting conductor wires that will be timed to reach individual, sub-millimeter electromechanical elements, 65, in sequence to reinforce and cause a build up of the amplitude of a coherent GW beam (as in FIG. 14F) having axis, 29. The ensemble of electromechanical elements (including other kinds of energizable elements such as nanomachines) will be embedded in or imprinted on a silicon chip, 57, in multiple layers (U. S. Patent No. 6,417,597).

Summary

The problem, which all of the devices discussed in this paper solve, is to cause a mass composed of individual molecules or submicroscopic particles, termed energizable elements, to move in concert (with a jerk or a harmonic – possibly dipole --oscillation) in order to build up (generate) a GW with either planar or cylindrical wave propagation to produce a very long sequence of GW pulses having significant average power without causing disruptive g loads or generating overpowering EM radiation. This is accomplished in several alternative ways by utilizing an array of energizable elements (e. g., magnets, coils, parallel plates, piezoelectric crystals, dielectrics, capacitors, nanomachines, high-temperature superconductors, electrons, nuclear particles, etc.) to be activated by energizing elements (e.g., coils, submicroscopic particles, etc.) under computer control. These energizable elements are activated or energized in the correct sequence by the IIPCS computer to accumulate a GW (moving at local GW speed in the energizable mass, which may or may not be near to the vacuum light speed) as the GW front moves in the mass. Essentially, the IIPCS causes the entire mass (or collection of masses or molecules) to jerk effectively in unison or in step with the GW wave front and generate coherent HF GW. That is, the jerk (third time derivative of mass movement) is caused to progress in step with the GW front and build the GW amplitude up – somewhat similar to a cyclotron pulsing a charged particle as it circles around in its magnetic field, or, possibly, like a traveling-wave amplifier. (Also similar to the coherent GW generation suggested by Romero and Dehnen⁵² .) Energizable elements (that jerk when energized) are energized in sequence as the GW front passes. These elements taken together emulate a much larger, more extensive mass. That is, the entire mass “appears” to the GW (as it passes) to be a single larger mass being jerked cohesively. Experiments suggested in this paper (see Test Objective (3)) would not only shed light on such high-frequency GW characteristics, but also, as suggested by Y. Acremann, et al ¹⁸ in their discussion of the processional motion of the magnetization vector “… form the basis for realistic models of magnetization dynamics in a largely unexplored but technologically increasingly relevant (picosecond) time scale.” (Please see Test Objective (11) of Section VII.)

V. Quadrupole Moment and Alternative Configurations of HFGW Generators

We now turn to the quantitative estimate of the GW power generated by the various devices described in this paper. Although the specific relationship for GW generation from jerks or harmonic, multipole motion will be an outcome of the experimental use of the devices described in this paper; as an example of that relationship consider the standard (that is, originating with Einstein) GW quadrupole Eq. (110.16), p. 355 of L. D. Landau and E. M. Lifshitz ¹ or Eq. (1), p. 463 of J. P. Ostriker,⁶ which gives an approximation to the magnitude of the GW radiated power [watts] as

P = │ - dE/dt │ =κ (G/45c⁵)(d³D_a_b/dt³)² [watts] (3)

where (as in Eq. (1.1))

E = energy [joules],

t = time [s],

G= 6.67423x10^-11 [m³/kg-s²] (universal gravitational constant – not the Einstein tensor),

c= 3x10⁸[m/s] (speed of light; approximately the electron’s mobility speed of 2.3x10⁸[m/s] in metal),

D_a_b [kg-m²] = I is the quadrupole moment-of-inertia tensor of the mass (mass-energy quadrupole tensor of the source),

κ = a dimensionless coefficient or factor of the kernel of the quadrupole approximation equation,

I = the classical moment of inertia [kg-m² ],

ω = an angular rate [1/s],

and the

a and b subscripts signify the tensor components and directions.

We define the kernel of the quadrupole approximation equation to be (Iω³)².

Note especially the third time derivative in the squared term of Eq. (3). Such a time-rate-of-change of the second derivative ("acceleration") is herein referred to as a "jerk". Because the factor of the kernel is so small (as we will see, = 1.76x10^-52) the kernel and hence the jerk must be very large. In the following and foregoing examples of GW generation by various devices, we often cite astrophysical analyses of the same GW formulation. It should be recognized, however, that although kernels are analogous in the terrestrial devices and in the celestial astrophysical systems (or events) their operation is quite different. In most cases the celestially generated GW rely on rather slow-moving, low-frequency (sometimes single) events (a small fraction of a Hertz to possibly a MHz – perhaps higher, MHz to THz, for important relic and primeval background GW)³³ and weak gravitational attraction. On the other hand, the terrestrial devices discussed herein (and potentially useful for spacecraft propulsion and communication) rely on a long sequence of very fast-moving, high-frequency events (e. g., up to THz or QHz down to a GHz) and relatively strong magnetic, electric or electromechanical forces.

There is extensive literature in the general-relativity community that estimates the amount of gravitational radiation to be expected from just about any celestial source imaginable (please see, for example, the references^6,9,11 to the proceedings volume on “Sources of Gravitational Radiation” edited by Larry L. Smarr and the paper by Flanagan and Scott⁴¹). In order to generate GW by gravitational attraction one needs to move stellar masses around very rapidly. Gravity is extremely weak. For example, the electrical force between two electrons is on the order of 10⁴⁰larger than the force of their gravitational attraction. Thus the Earth-based generation of GW by gravitational means is not feasible by many orders of magnitude. Therefore one cannot utilize the gravitational attraction of matter on Earth to create GW in the Earth’s weak gravitational field and one must turn instead to electric, magnetic, electromechanical, or nuclear forces! Furthermore, it is not necessary to use gravitational attraction to generate gravitational waves! In fact, any force can be utilized as well as the gravitational attraction of matter. (Please see, for example, Weber^2,7.) GW are related directly to an inertial mass in motion, a jerk, or a harmonic oscillation and not directly due to the gravitational field of the mass. It is interesting to note Abraham Pais¹⁷ quote (p. 242) of Einstein and Grossman¹⁰ that “… we may consider a ‘centrifugal field’ to be a gravitational field.” and the remark of Joseph Weber⁷ (p. 97) “… elastic (springs, or even thrust, drag, etc.) forces … are electromagnetic in origin.” With regard to the quadrupole approximation, Bonner and Piper⁵⁰ suggest the emission of quadrupole and octupole GW (even dipole if non-linearity is considered), but the multipole emission is due to mass motion internal to a GW propulsion system and the force mechanism for the achievement of this motion is not discussed by them.

Spinning-Rod GW

Although the derivation of the quadrupole equation results from relativistic mechanics, one can utilize ordinary classical mechanics to obtain many useful results. From Eq. (1), p. 90 of Joseph Weber⁷, one has for Einstein's formulation⁸ of the gravitational-wave (GW) radiated power of a rod spinning about an axis through its midpoint having a moment of inertia, I [kg-m²], and an angular rate, w [radians/s] (please also see, for example, pp. 979 and 980 of Misner, Thorne, and Wheeler²⁰ in which I in the kernel of the quadrupole equation also takes on its classical-physics meaning of an ordinary moment of inertia):

P = 32GI²w⁶/5c⁵ = G(Iw³)²/5(c/2)⁵ [watts] (4.1)

(so that κ = 288 in Eq. (3) subject, of course, to the results of Test Objective (10) found in Section VII) or

P = 1.76x10^-52(Iw³)² = 1.76x10^-52(r[rmw²]w)² [watts] (4.2)

where [rmw²]² can be associated with the square of the magnitude of the rod’s centrifugal-force vector, f_cf, for a rod of mass, m, and radius of gyration, r. This vector reverses every half period at twice the angular rate of the rod (and a component’s magnitude squared completes one complete period in half the rod’s period). Thus the GW frequency is 2w and the time-rate-of-change of the magnitude of, say, the x-component of the centrifugal force, f_cfx is

Df_cfx/Dt µ 2f_cfxw. (4.3)

(Note that frequency, u = w/2p.) The change in the centrifugal-force vector itself (which we call a “jerk” when divided by a time interval) is a differential vector at right angles to f_cf and directed tangentially along the arc that the dumbbell or rod moves through. Equation (3), like Eqs. (4.1) and (4.2), are approximations and only hold accurately for r << l_GWand for speeds of the GW generator components far less than c. Please see, for example, Pais¹⁷, p. 280 and Thorne⁵⁷, p. 357 (but still roughly analyzable by Linearized theory such as found in Gertsenshtein³⁷ and Grishchuk and Sazhin³² ).

Equation (4.2) is the same equation as that given for two bodies on a circular orbit on p. 356 of Landau and Lifshitz¹ (I=mr² in their notation) where w = n, the orbital mean motion.

Equation (4.3) substituted into Eq. (4.2) with rmw²associated with Df_cf yields

P = 1.76x10^-52(2rDf_cf /Dt)², (4.4)

where (2rDf_cf /Dt)² is the kernel of the quadrupole approximation equation.

As a validation of Eq. (4.4), that is a validation of the use of a jerk to estimate gravitational-wave power, let us utilize the approach for computing the gravitational-radiation power of PSR1913+16. From section 3, Eq. (2) we computed that each of the components of force change, Df_cfx,y = 5.77x10³² [N] (multiplied by two since the centrifugal force reverses its direction each half period) and Dt = (1/2)(7.75hrx60minx60sec) = 1.395x10⁴ [s]. Thus using the jerk approach:

P = 1.76x10^-52{(2rDf_cfx/Dt)² + (2rDf_cfy/Dt)²} = 1.76x10^-52(2x2.05x10⁹x5.77x10³²/1.395x10⁴)²x2

= 10.1x10²⁴ [watts] (4.5)

versus 9.296x10²⁴ [watts] using Landau and Lifshitz’s more exact two-body-orbit formulation given by Eqs. (1.1) and (1.2) integrated using the mean anomaly not the true anomaly. The most stunning closeness of the agreement is, of course, fortuitous since due to orbital eccentricity there is no symmetry among the Df_cfx,y components around the orbit and, as will be shown, there are errors inherent in the approximations of Eqs. (18) and (20) leading to Eq. (4.4). Nevertheless, since the results for GW power are so close, orbital-mechanic formulation compared to the utilization of a jerk, the correctness of the jerk formulation is well demonstrated!

Spin-up and Spin-down GW of a large Terrestrial Spindle Device (U. S. Patent No. 6,160,336)

As discussed by Rizzi²⁵ , the spin up or spin down of a system of masses results in GW. It is reasonable (that is, by Ockham’s Razor) to suggest that for the spin up/down of a terrestrial spindle device (referred to herein as the “(Id²w/dt²)² formulation or component”):

P = G k_I_w2dot(Id²w/dt²)²/5(c/2)⁵ [watts] , (5)

where k_I_w2dot = a dimensionless constant or function to be established by the experiment (please see Test Objective (10)) and d²w/dt² = second time derivative of the spindle’s angular velocity, w, or third time derivative of it’s angle, termed, an angular “jerk”. In fact, as noted by M. S. Turner and R. V. Wagoner ⁹(p. 383.) “If the angular velocity w … is non-uniform, octupole (post-Newtonian) radiation is generated (in addition to the quadrupole (Newtonian) radiation…” (emphasis added) and on p. 385 they state “This radiation is generated not by non-spherical distribution of matter…, but by internal motions.”

This third derivative of v, or second derivative of w, d²w/dt² , is computed by introducing the equation of motion for a rotating body

Idw/dt = rf_t (6)

where

r = radius of the spindle’s rim or radius of gyration [m] and

f_t = force tangential to the rim [N].

The derivative is approximated by

Id²w/dt² @ 2 D(Idw/dt)/Dt =2 D(rf_t)/Dt =2 rDf_t/Dt; (7)

in which Df_t is the nearly instantaneous increase in the force tangential to the rim, or jerk, caused by the magnetic field when it is sequentially turned on or turned off or pulsed by the transistors or ultra-fast switches of the Individual Independently Programmable Coil System (IIPCS), that is, a tangential jerk. Thus

P= 1.76x10^-52 (k_I_w2dotrDf_t/Dt)²[watts]. (8)

Equation (8) is essentially identical in form to Eq. (4.4), but arrived at by a different path. Since the quadrupole equation involving the jerk (Δf/Δt) is the same as the more conventional formulation, the jerk approach or formulation is again confirmed.

As a numerical example, for a spindle GW-generation device described herein that need not be rotating itself, we theoretically set k_I_w2dot = 2 (subject to experimental determination later), r = 1000 [m], Df_t = 1.8x10⁷ [N], and Dt = 10^-12 [s]. These numbers arise as follows: The rim of the spindle embodiment of the device is a thin (approximately one cm thick) band of Alnico 5 permanent magnets (or electromagnets) facing radially outward. In general, permanent magnets exhibit irregular magnetic fields and associated forces. As a rule of thumb such a band of juxtaposed magnets will produce in excess of 30 pounds per 1.75 inches (or 206 pounds per foot) of tangential rim force. Each 1.75-inch permanent magnet has a flux density, B, of about 2,600 gauss or 0.26 [Tesla] developed every 4.4 cm. The kilometer-radius rim is a large hoop connected to a central spindle/hub as described in Section VII of this paper. The IIPCS coil sets at the rim’s periphery, when switched on generate a 0.26 [Tesla] flux density every 0.044 [m] and produce in excess of a 200 pound per foot or 3000 [N/m], which is defined as Df/Dl, or impulse of tangential force every meter on the rim (that is, a force built-up almost to full value during spin up in approximately a picosecond and a similar build up of retarding force during spin down; with magnetic field rise times as discussed on p. 494 of Y. Acremann, et al ¹⁸). Since the rim's circumference is 2p (1000) (3.28 feet per meter) = 20,600 feet, the tangential rim force produced when the coils are fully energized is 4.1x10⁶ pounds or 1.8x10⁷[N]. The 10^-12 [s] intervals, with the coils being turned on and then off, will generate a train of direct-current, approximately one-picosecond pulses (or shorter or longer). In each line of coils there will be an ultra-fast switch (such switches could be located near to the coils and each one energizing a large number of coil sets or else co-located with a central control computer). For HF GW the wave length, l_GW ,can be on the order of 10^-4 [m] and for Eq. (3) to be a good approximation, r, for any given energizable element should be far less than 10^-4 [m] and if not, then k_I_w2dot will have some value or be some function that will be determined experimentally. As discussed on pp. 348 and 349 of Landau and Lifshitz¹, high-frequency gravitational waves can be modeled for regions in space with “…dimensions large compared to l_GW (e. g., > 10^-4 [m]), but small compared to L (distance)”, but the intent of this paper is to rely on experiment (Test Objectives (9) through (12)) to establish the model.

Inserting the numbers in Eq. (8) for the non-rotating spindle’s gravitational-wave (GW) power for a tangential jerk of uncharged mass yields (for each member of the train of jerks)

P = 1.76x10^-52 (2x10³ x 1.8x10⁷/10^-12)² = 2.3x10^-7 [watts]. (9)

The reference area of the 1 cm thick rim is (0.01) 2p(1000) = 63 [m²], so that the magnitude of the GW energy flux near the device is 3x10^-9 [watts/m²]. For a long (multi-second) sequence of jerks, the average flux of the GW pulse train would be about 2x10^-9 [watts/m²] compared to 4x10^-11 [watts/m²] for the LF, long-wavelength GW from a binary black hole or BBH at a distance of 500 mega parsecs [or Mpc] having a 100-BH radius osculating circular orbit. There may be a somewhat less simple, k_I_wdot_w([dw/dt]w)², formulation or component of the GW power for spin up/down, but as will be seen, for most applications it is expected to result in a smaller power than the larger of the k_I_w2dot(Id²w/dt²)² or (Iw³)² formulations or components. It is to be emphasized again that the rim need not be rotating. It is like a gigantic ratchet gear or wheel in a mechanical watch or the jitter of a servo motor that causes the jerk and resulting HF GW.

Note that the coil sets must be very close together. In order for the coils fields to interact with the whole rim's magnetic field and impart the mechanical impulse or jerk, they must be spaced no more than 0.3 mm or 300 [mm] apart (the distance light and, hence, the magnetic field and resulting impulse on the permanent magnets, travels in a picosecond – here it is assumed for convenience that GW propagates at light speed in the magnetic mass; in a superconductor GW may, however, propagate much more slowly according to Li and Torr²⁶). As noted in Acremann, et al ¹⁸, p.494, “The applied magnetic field is only immediately present on the ‘skin’ of the magnetic sample,…” or in the present case the permanent magnets if they were very rapidly magnetized (which the magnetic mass(es) are not). Of course, this statement by Y. Acremann et al applies primarily to rapidly recording on a magnetic memory device not to the present situation of a quasi-static magnetic field’s interaction with a rapidly built up coil-generated field. If all coil sets in a line of coil sets are connected in series by the same conductor as shown in FIG. (9B), then each member of the pulse train traverses a 300-micrometer-length coil set, separated from the next coil set by a time delay circuit. Such a time-delay circuit could be simply a 300-micrometer-long jumper (see 62 in FIG. 9B) between coil sets. In this connection it is noted that if each coil set is connected by its own unique conductor as in FIG. 9A, instead of one single conductor wire along each line of coils, then the communications lines or conductors to all of the coil switches from the logic circuits of the control computer must be equal to better than 0.01 mm or 10 [micrometers] in order to ensure near simultaneity. That is, the electrons must reach all of the coils sets along the rim at the appropriate times in less than approximately a fraction of a picosecond of time difference.

Magnetic Field Build Up and Heat Loss

Although of little concern in most applications, the length of time to "build-up" the magnetic field of the coils is important here as it is in the experimental work of Y. Acremann^18. The electrons must complete sufficient coil turns (moving at the electron’s mobility speed – about 2.3x10⁸[m/s]) in approximately a picosecond to "launch" most of the magnetic field that produces the impulsive force (like a “hammer blow”) or jerk when it interacts with the static magnetic field of the permanent or electromagnets carried around by the rim. Thus, they must be very tightly wound with each coil "set" having a total length of less than 0.3 mm (0.0003[m] or 300 micrometers). If each of the ultra-small, sub-millimeter coil sets consist of two coils or turns, as exhibited in FIGS. 7A, 7B, 7C, 7D, 7E, 7F, 7G, and 7F, then their diameters are on the order of d = 0.3/2 p = 0.05[mm] = 50 [μm] or less. (Please note that the single-turn coil of Al, utilized by Y. Acremann, et al ¹⁸, was about 6 [μm] in diameter.) The coil wire could be made of gold having about a 0.015-mm or 15-micrometer diameter. The resistance for such wire at room temperature is about 135 [ohms/m] -- high-temperature superconducting material would be useful here. In the spin-up mode the IIPCS will need to build up 0.26 [Tesla] flux density, at the appropriate polarity, every 0.044 [m] (the requirements for the spin-down mode are essentially the same, but reversed). Thus, subject to the results of Test Objective (11),

B = m_oni/l [Tesla] (10)

where m_o = 4px10^-7 (permeability of free space), n is the number of coil turns, i is the current through the coils [amps], and l is the length of the coil conductors [m]. The double coil sets will be placed on 50 to 100 [micrometer] centers, so that there will be about 2x100x100 = 2x10⁴ coil turns on each square-centimeter level of the stack of 25 coil levels or layers. With l = 0.044 [m] and B = 0.26 [Tesla], ni = 9.1x10³ [amp turns]. For n = 25x2x10⁴= 5x10⁵, i = 9.1x10³/5x10⁵ = 0.018 [amps] or 18 milliamperes. The total length of 15-micrometer- diameter gold wire across any given layer or level is 100(rows) x100(coil & jumper/time-delays)x(600 micrometers) = 6 [m]. For the 25 layers or levels there will 150 [m] of wire with a resistance of 150[m]x135[ohms/m] = 2.025x10⁴ [ohms]. Since on average every other pulse interval across a conductor wire will carry no current (resulting of course, in a lower average GW power), the heat loss per centimeter of chip stack or semi-conductor layers is (subject to the results of Test Objective (12)):

(1/2)i²R = 3.28 [watts]. (11)

This heat loss can be reduced by 32% by using 25-micrometer-diameter wires for the time-delay jumpers, but high-temperature superconductors for this purpose are contemplated. In addition there may be some energy loss or resistance occasioned by EM radiation generated during the GW generation process – reduced or eliminated since the jerked masses are uncharged. Such a loss can be reduced by the design of the energizing, for example coil, elements and controlling the direction of current pulses by the IIPCS. Concerns of the influence of magneto resistance (MR) of both the conductors and the semiconductor circuits and the dynamics of the impulsive magnetic field buildup should be addressed during the experiment as would be the aforementioned EM radiation, which could be significantly reduce the efficiency of the HFGW generator (please see Test Objective (12)). Note that alternating currents are not utilized (only direct-current, positive pulses) in order not to drive the electrons to the conductor’s skin and thereby increase resistance. (Not a problem when superconductors are utilized.)

The spin-up/down of the entire rim is not instantaneous and is anticipated to progress at the local speed of light in the rim from the juxtaposed permanent-magnet sites on the rim acted upon by the coil- magnetic fields. Spin up/down jerks do not progress at the speed of sound, but rather it is expected to progress at the speed of light like a signal being transmitted by pushing a frictionless rod. This conclusion is subject to experimental verification (Test Objective (11), all of the ferromagnetic atoms comprising the magnets on the periphery of the rim move in concert impelled by their magnetic fields. On the other hand, impulsive stresses in the spindle or dumbbell device are expected to propagate inwardly at the speed of sound in the material of the device rather than at the speed of light. By the way, as already noted the GW may propagate at significantly less than light speed in the magnetic inertial mass (especially if it involves superconductors). Thus, for example, at a one picosecond cycle or switching rate and in-rim light speed of 3x10⁸ [m/s] some (10^-12)(3x10⁸) = 0.0003 [m] or 0.3 [mm] or 300 micrometers of the rim on each side of the juxtaposed coil/magnet activity sites will respond (spin up or down) during each picosecond after coil activation. This process, in the case of a dumbbell-shaped rim, will generate and ever widening fan of gravitational waves (which propagate in both directions from the jerk since there is a square associated with the kernel of the quadrupole equation—so there is no preferred direction along the axis of the jerk), 51, as exhibited in FIG. 8B. Such a concept will be subject to experimental test (Objective (2))and the propagation direction may be dependent upon whether or not there is circular polarization or “+” polarization or a combination.. Many more than the 4,290 coil sites of the exemplar device could be distributed around the rim (perhaps over only a single sector or selected sectors adjacent to the rim; thereby greatly reducing the required number of coils and “focusing” the GW). A large number of ultra-fast switches, preferably semiconductor based, would be activated simultaneously along the progressing gravitational wave front by the coil-control computer, with communication lines of nearly equal length to all switches. The coil-control computer of the IIPCS can also activate coil sets above and below the rim or near the faces of a wedge-shaped sector or sectors of the rim at the appropriate times. These coils are separated closely in the radial direction to produce a series of very brief, approximately picosecond-duration radial pulsation displacements or vibration of the rim or of a juxtaposed wedge-shaped rim sector or sectors to generate radially directed gravitational waves. In this case the coils, which are closely adjacent to magnets or magnetic sites in the rim or rim sectors, are to be sequentially activated. Starting from the innermost coil sets and moving radially out to the outermost coil sets they will be activated with time delays to follow the rim or sector-material’s local GW-crest speed (probably near light speed) displacement in the radial direction and thereby build up the gravitational waves radially. A section later in this report discusses radial-jerk generated GW in more detail. In this connection it should be noted that, depending upon the impressed HF magnetic field or EM (microwave) field, a HTSC could generate either tangential or radial HF GW (US Pat. 6,417,597; 6,160,336; and patents pending).

Depending upon the proximity of the coils and the duration of the current pulses, there may be currents induced in one coil juxtaposed to another. Any induced currents will produce deterministic reverberations for which the IIPCS that can be programmed to account for. Thus the GW can generate definitive signals. In any event the reverberations would subside as the current-produced magnetic pulses either collapse or clear the ensemble of chip layers at light speed.

Rim Material Accelerations

A random series of positive and negative jerks (like reciprocating “hammer blows”) tend to build up a positive acceleration over time by random walk if not countered by the IIPCS. Consider an extreme case, however, in which the maximum jerk builds up rim acceleration (spin up, or deceleration, spin down) continuously over a length of time, dt = 10^-7 [s] or 100 nanoseconds. The mass per unit length of the magnetic mass of the rim for the exemplar device is approximately Dmass/Dl = 3.83 [kg/m]. The jerk is

da/dt = d³s/dt³ = ([Df_t/Dl]/[Dmass/Dl]/Dt

= (3000[N/m]/3.8[kg/m]/10^-12 = 7.89x10¹⁴[m/s³], (12)

where s is the displacement. Therefore, in dt = 100 nanoseconds (10^-7 [s]) of continuous and constant jerk, da/dt, the acceleration, a, would build up to

a(t) = ò_o^dt(da/dt)dt = (7.89x10¹⁴)(10^-7) = 7.89x10⁷ [m/s²], (13)

the speed to

ds/dt = ò_o^dt(da/dt)dt = (da/dt)dt²/2 = (7.89/2)x10¹⁴x(10^-7)² = 3.9 [m/s], (14)

and the displacement to

s = ò_o^dt(ds/dt)dt = (da/dt)dt³/6 = (7.86/6)x10¹⁴x(10^-7)³

= 1.315x10^-7 [m]. (15)

Of course, there would be no need to reach an acceleration as large as 7.89x10⁷ [m/s²], which might affect structural stability. In fact, if the IIPCS reverses the jerk in two picoseconds (2x10^-12 [s] ) or less (e. g., an alternating jerk or series of reciprocating “hammer blows” – not harmonic oscillations), then the acceleration would be (7.89x10¹⁴)(2x10^-12) =1.6x10³ [m/s²] or less. The jerk experienced by a projectile in a gun tube would generate similar acceleration. The jerk-produced acceleration is less than two hundred g’s and not structurally damaging and the internal-lattice structural stability will not be lost! In fact, the acceleration after two picoseconds would be about the same as the 112 g’s of the double star PSR 1913+16 at periastron computed in Section III. Since it is validated by the observations of PSR 1913+16, the weak-gravitational-field approximation is preserved for relatively large g accelerations! (Please see Test Objective (6).) Nevertheless, the IIPCS should be programmed to avoid the build up of large internal accelerations. If^‑ the angular rate build up was over 100 nanoseconds (very unlikely), then the angular rate is 3.9[m/s]/1000[m] = 3.9x10^-3[radians/s] versus, for example, the mean motion of double star PSR 1913+16 of 2.25x10^-4[radians/s]. Thus there could be considerable “motion” in the magnetic mass, but essentially the mass goes only a very short distance. In this regard, the IIPCS can be programmed to ensure that there is not a secular increase or accumulation of magnetic-mass acceleration, speed, or displacement beyond certain prescribed limiting values. The stress moves in and away from the magnetic mass at sound speed (for example, 5000 [m/s] in metal) or about 500 micrometers in 100 nanoseconds. They represent microscopic shock waves that will dissipate by internal friction (possibly with anelastic aftereffects).

Radial-Jerk GW (U. S. Patent No. 6,417,597)

In the case of the radially directed pulses or displacements of the rim or rim sector or sectors, they also result in a time-variable value of the moment of inertia, I. These displacements are built up in the radial direction by the sequential activation of radial arcs of coils in a given, single wedge-shaped sector or juxtaposed sectors of the rim or dumbbell as exhibited in FIG. (8B). The concept is that as a GW crest reaches each magnetic site (or energizable element), such as 80, 84, 86, and 88 of FIGS. 14E and 14F, the site (which could be a myriad of energizable elements abreast and parallel to the crest line) is energized or jerked in a timed sequence in order to add amplitude (83, 85, 87, and 89 of FIG. 14F) to the progressing wave front (82 of FIGS. 14E and 14F). Thus a coherent HF GW is generated in the “forward” direction (non-coherent HF GW is generated in the opposite direction since the gravitational wave front is out of step with the jerks so that it does not accumulate). Under the control of the IIPCS, radially oriented strips of the aforementioned circuit-board or computer-chip rolls are sequentially activated to build up or generate a (d³I/dt³)² formulation or GW component as the GW moves outward at light speed (or at the local GW speed). The radial displacements should be asymmetrical as controlled by the IIPCS (please see, for example, Pais¹⁷ p. 280), such that there will not be cross-rim destructive GW interference, albeit for short GW wave lengths relative to the rim radius this may not be a great concern. If moving entirely in a HTSC , then wavelengths internal to the HTSC will be much shorter than those outside. It makes no difference to the material in the dumbbell or sector (or an “observer” there; assuming speeds far less than c, a quasi-inertial framework and a weak, static gravitational field) whether the change in force acting upon it (jerk) is centrifugal in nature (as in Eq. (4.2)) or due to magnetic or other influences (as already noted, we may consider a centrifugal field to be a gravitational field according to Einstein and Grossmann¹⁰). The celestial analogy here is a vibrating white dwarf star emitting GW (see, for example, D. H. Douglas, p.491, of L. L. Smarr¹¹).

As is well known and noted specifically by Dr. Geoff Burdge¹², Deputy Director for Technology and Systems of the National Security Agency “Because of symmetry, the quadrupole moment can be related to a principal moment of inertia, I, of a three-dimensional tensor of the system and … can be approximated by

-dE/dt » G/5c⁵ (d³I/dt³)² = 5.5x10^-54 (d³I/dt³)².” (16)

In which k in Burdge’s notation is G (not, however, the Einstein tensor) and the units are in the MKS system [watts] not the cgs. In this case the magnitude of the GW power is given by

P = Gk_I3dot(d³I/dt³)²/5c⁵ [watts], (17)

where k_I3dot = 32 (subject to the results of Test Objective (10))

I = dm r² [kg-m²],

dm = mass of an individual rim sector or a number of sectors (or magnetic sites) [kg], and

r = the distance from the pivot out to the single dm [m] (or more exactly, the radius of gyration). Thus

d³I/dt³ = dm d³r²/dt³ = 2rdmd³r/dt³ +…@ 2rdmd³r/dt³ (18)

and d³r/dt³ is computed by noting that by Newton’s second law of motion

2rdm d²r/dt² = 2rf_r [N-m] (19)

where f_r = radial force on a single rim sector, rim sectors, or dumbbell. The derivative is approximated by

d³I/dt³ @ 2r Df_r/Dt , (20)

in which Df_r is the nearly instantaneous increase in the radial force on magnetic sites, dm, on rim caused by the magnetic field when it is turned on and off or pulsed by the transistors or ultra-fast switches of the IIPCS, that is, a radial jerk. This radial jerk, unlike the tangential jerk, acts perpendicular to the direction of the rim’s rotation and may be more stressful to the device. In this regard, as already discussed and illustrated schematically in FIGS. 14E and 14F, the coils are sequenced radially outward by the IIPCS (at the local GW speed, say the speed of light) in order to generate or build up the train of coherent HF gravitational waves as they move through the energizable magnetic sites. Thus

P = 5.5x10^-54 k_I3dot(2rDf_r/Dt)² [watts] . (21.1)

Or with k_I3dot equal to its theoretical value of 32 (please see Test Objective (10))

P = 1.76x10^-52 (2rDf_r/Dt)² [watts] . (21.2)

It is important to emphasize that Eq. (21.2) is essentially identical to both Eqs. (4.4) and (8) yet arrived at by still another path. It is apparent that for “jerk-generated” GW (as opposed to “harmonic-oscillation-generated” GW as analyzed first by Einstein and Rosen¹⁹) Eq. (4.4) is a most useful form of the quadrupole approximation.

The k_I3dot will be a constant or a function determined experimentally (Test Objective (10)) to account for the fact that r may not be less than l_GW for most HF GW of interest. Again all roads lead to the jerk formulation of the quadrupole approximation!

As a numerical example, for a non-spinning spindle similar to the one mentioned in the prior numerical example, but with a one-meter wide apron of peripheral magnets and IIPCS coil sets both top and bottom (thus 2x100 cm/m = 200 times more force per meter along the rim’s periphery), k_I3dot = 32, Df_r = 3.6x10⁹ [N], r = 1000 [m], and Dt = 10^-12 [s], so that

P = 1.76x10^-52 (2x1000x3.6x10⁹/10^-12)² = 9.12x10^-3 [watts]. (22)

Again the reference area is 63 [m²], so that the magnitude of the GW energy flux near the device is about 1.45x10^-4 [watts/m²]. For a train or sequence of jerks, for example comprising a message packet, the average flux or message-packet HF GW signal would be about 1x10^-4 [watts/m²] near the device. As a point of reference, the LF GW flux near LIGO for the detection of a 500 Mpc distant BBH having a semimajor axis of six black-hole (BH) radii for the osculating circular orbit (just prior to the inspiral becoming dynamically unstable and the BHs merge) is 5x10^-5[watts/m²].

Linear-Motor, Linear-Jerk GW (U. S. Patent No. 6,417,597)

The preferred linear-motor design of the device (sometimes referred to as a linear induction motor or LIM) is visualized to involve a single sector of the rim with the impulsive forces being longitudinal, Df_l_, rather than radial. Alternatively, it can be conceptualized as the rim magnets and adjacent coils being peeled off from the rim and laid out flat (linear motor). Please see FIGS. 14A – 14D for a schematic of the progression of such a “peeling”. In this very hypothetical case an exemplar device would be 2πr = 2πx2000 = 6283 [m] in length and, since for this linear mass distribution I = (1/3)mr² and d³I/dt³ @ (2/3)r Df_l /Dt , the effective radius is 6283/3 = 2094 [m] ≈ 2000 [m] (a measure of the mass distribution) and 3 [m] in diameter. The approximately one-centimeter-wide chip rolls would be placed longitudinally along the sides of central, cylindrical, permanent- (or electro-) magnetic core, piston, or barrel consisting of an array of magnetic (energizable element) sites, 57, as shown in FIG. 10. Each meter-long, square-centimeter segment of the roll would produce about 3000 [N] of longitudinal force, f_l, and all together they form a sheath of sub-millimeter coils (energizing elements) surrounding the central magnetic core, piston, or barrel. The impulsive force per unit volume, Δf_l/ΔV = 3000[N/m]/(0.01[m])² = 3x10⁷ [N/m³]. As mentioned already, lines of such uncharged elements, set abreast and parallel to the passing GW crest, will be energized as the GW crest passes and add to its amplitude so as to generate coherent HF GW. Note that in this case the motion of the magnetic mass is asymmetrical (either “in” or “out”) so that there is a quadrupole moment without GW cancellation. Pinto and Rotoli⁵⁶ (p. 567) indicate that “... the quadrupole formula is only valid provided a suitable surface integral (vanishes), which is the case for a series of point sources” such as the energizable elements of the subject device. As already noted and subject to verification (Test Objective (2)), since the kernel of Eq. (21.2) involves a square, bi-directional GW radiation may result because there is no preferred direction along the axis of the jerk; subject, of course, to validation by Test Objective (2). (Since r is so large relative to GW wavelength the usual form of the quadrupole equation probably does not hold exactly: please see Test Objective (10) of Section VII.) In practice the sheath of energizing elements would be on both sides of the uncharged energizable-element, thin-wall cylinder core shown in FIG. 10. Because of the small λ_GW ( = cΔt = 3x10⁸ x 10¹² = 3x10^-4 [m] or 300 [μm] and the small movement of the electrons (2.38x10⁸ x 10^-12 = 238 [μm]) in the coils, the device’s wall would, no doubt, be composed of interleaved energizing and energizable-element layers – a laminate – in order to minimize the “reaction” distance between, for example, the magnetic sites and the coils. Also, whatever the form of the energizing or energizable elements (quadrupoles or some other kind of element) they could be contained in a superconducting medium and the λ_GW may be approximately 100 times shorter within the medium. (Please see Li and Torr²⁶.) Thus the geometry and energizing timing of the jerked energizable elements could be such that any accompanying EM radiation would not build up coherently and be much weaker than the GW. Also one might arrange the geometry and the IIPCS control of the energizable elements to produce destructive interference of any EM radiation generated and constructive interference (build up) of the GW radiation.

Numerical Example

As a numerical example, there would be about one roll or 25-layer strip of chips spaced around and adjacent to the cylindrical barrel of the linear motor (64, shown in FIG. 10) in a longitudinal direction (parallel to the barrel axis) every two centimeters forming the sheath. Thus there would be px3[m]x100[cm/m]/2[cm] = 471 strips around the barrel’s circumference, each one having a length of 2πx1000 = 6283 [m] so that

Df_l=(471)(6283[m])(3000[N/m]) =9x10⁹ [N] (23)

and with k_mr3dot = 32 (to be established experimentally in Test Objective (10) since r may not be less than l_GW),

P= 1.76x10^-52 (⅔x6283x9x10⁹/10^-12)² = 0.25 [watts]. (24)

Thus, with the reference area being the two 3 [m] diameter ends (GW propagating in both directions so the area is doubled) with a thickness of one centimeter, area = 2 (3π) (0.01) = 0.19 [m²], the generated HF GW flux is about 1.3 [watts/m²] near the hypothetical device. The average HF GW flux or signal would be about 1 [watt/m²]. As a point of reference we again compare our terrestrial HF GW generator to celestial LF GW generation. Thus the 1 [watt/m²] is compared to the 4x10^-16 [watts/m²] maximum signal from a 500 mega parsec [Mpc] distant, 1000 black-hole (BH) radius semimajor-axis binary black hole (BBH) osculating orbit and 5x10^-5 [watts/m²] from a 6-BH radius osculating orbit –or over ten-thousand times stronger than the LF GW signal from a 6 BH-radii BBH osculating orbit just before merger! For cylindrical GW, in case the barrel magnets participated in harmonic oscillation (each end’s uncharged magnetic sites moved in and out harmonically relative to the other like a dipole), the reference area would be (6283)(3p) = 6x10⁴ [m²] and the GW flux would be 0.25/6x10⁴ = 4x10^-6 [watts/m²]. Although the jerked masses are uncharged, the high-frequency electromagnetic fields may generate significant EM radiation that will be studied in Test Objective (12).

Material Accelerations

In this case the jerk is obtained from

(da/dt)_{per unit area} = (Df/Dt)/(Dmass/DA) (25)

where Df = Df₁ [N]/(2000[m]x3[m]p) = 2.83x10⁹/1.885x10⁴ = 1.5x10⁵ [N/m²], so that

Df/Dt = 1.5x10⁵/10^-12 = 1.5x10¹⁷ [N/m²-s] (26)

and Dmass/DA = mass per area (3.8 [kg/m] of strip)(471 strips per meter) = 1.79x10³ [kg/m²],

so that da/dt = 1.5x10¹⁷/1.79x10³ = 8.38x10¹³ [m/s³]. Therefore, in the extreme case 100 nanoseconds of continuous jerk the acceleration would build up to

a = d²S/dt² = (da/dt)dt = (8.38x10¹³)(10^-7) (27)

= 8.38x10⁶ [m/s²]. As already noted, the IIPCS would be programmed so that accelerations would never approach this value! As an example, for a one THz alternating jerk the acceleration would only build up to (8.38x10¹³)(10^-12) = 83.8 [m/s²] = 8.6 [g’s] (alternating or reciprocating “hammer blows” acting on a single mass or masses {such as magnetic sites}; not oscillation of two masses). In the extreme case of 100 nanoseconds of continuous jerk in the same direction, the speed would build up to

ds/dt = (da/dt)dt²/2 = (8.38x10¹³)x10^-14 = 0.42 [m/s] (28)

and the displacement of the magnetic mass (composed of many magnetic surface sites, 57, of the linear motor, piston, or barrel shown in FIG. 10) is

s = (da/dt)dt³/6 = (8.38x10¹³/6)x10^-21 = 1.40x10^-8 [m]. (29)

Again there could be considerable “motion” of the magnetic mass, but even in the most unlikely case of an extremely long series of jerks in the same direction it goes a very small distance before the IIPCS reverses the built-up acceleration, speed, and displacement and the stresses in the material of the device would be minimal.

Infinite-radius Coil GW (U. S. Patent No. 6,417,597)

For comparison with the foregoing designs of the device, consider the evolution of a given coil pair into a flattened-out pair of parallel wires (that is, infinite-radius coils) situated very close to each other and carrying a large current in the same direction (and, therefore, attracting each other). This current, which can go either way, is to be pulsed by a large number of ultra-fast switches or transistors about every picosecond by the IIPCS to produce pulses of electrical direct current through the wire. Again it is noted that we are only introducing direct-current pulses not alternating current in order to avoid conductor surface resistance. (The situation is moot if superconductors are introduced.) For simplicity, consider the wires to be flat, one-meter square plates (therefore a one-square-meter GW reference area or smaller down to a current-pulse wavelength across, or larger by constructing a mosaic of individual plate pairs) as exhibited schematically in FIG. 11. As a numerical example, we select a 1 [m] length device, let each plate carry a one-thousand ampere current and the plates are situated one-micrometer (10^-6 [m]) apart (thus one could achieve r << l_GW in keeping with the assumptions underlying the quadrupole approximation for those individual energizable elements). In order to exhibit asymmetrical mass displacement (so that the GW do not cancel and become null) one plate could be considerably more massive than the other, or joined to a ballast, or be constrained differently in their mountings, have a different modulus of elasticity, or carry more current, etc. If the IIPCS pulsed these conducting plates with picosecond or longer duration pulses (sequenced in order to follow and build up the GW at local GW speed as exhibited in FIGS. 14E and 14F), then during each cycle the attractive, impulsive force (lateral jerk) would be (with m_o = 4πx10^-7 )

Df= (m_o/2p)(1000[amps]x[1000[amps])/10^-6[m] = 2x10⁵ [N]. (30)

As usual let GW-radiated power be given by

P= 1.76x10^-52(2rDf/Dt)² [watts], (31)

where Df/Dt = 2x10⁵/10^-12 = 2x10¹⁷ [N/s] and r = 1 [m].

Thus

Pµ1.76x10^-52(4x10¹⁷)²=2.8x10^-17 [watts] (32)

and since the reference area is two square meters ( again it is recognized that the GW propagates in both directions) the GW flux near the device = 1.4x10^-17 [watts/m²]. The kernel, i.e., the product of the amperage of the two plates multiplied by the length, r, divided by the distance between the plates would need to go up by a factor of at least 10⁸ or consist of over a thousand plate pairs (or a mosaic of plate pairs per level [and multiple levels] ) or some combination thereof in order to approach the GW-flux values for the other devices already discussed in this paper (i.e., 2x10^-9 to 1 [watt/m² ] ). Such a current is, however, exceeded by the eighteen-million-ampere current passed through Sandia Laboratories Z-pinch machine (A. Wilson¹³, p. 2059). The current-produced jerk of this machine would be expected to generate a GW pulse as its tungsten wires collapse on each other.

Electromechanical-force Produced GW (U. S. Patent No. 6,417,597)

The concern that a GW generator must involve very massive, rapidly moving objects is countered by Joseph Weber’s remarks², p. 313, which although related to electromechanical-force-produced GW of piezoelectric crystals, apply more generally: “Waves one meter long could be radiated by a crystal with dimensions about fifty centimeters on a side. If it is driven just below the breaking point, each crystal would radiate »10^-20 [watts], assuming P_max to be its static published value.” At one THz the GW wavelength, l_GW, is about 3x10^-4 [m] or 300 micrometers so that the half wavelength and crystal dimension is 150 micrometers and, of course, even smaller for the approximate quadrupole equation to hold in its usual form, as in Eq. (3). If the ensemble of electromechanical elements, for example, piezoelectric crystals, nanomachines, nanomechanical systems etc. were controlled by the IIPCS and replaced the coils, and were on 160 micrometer centers in the chips, then there would be about 60x60 = 3.6x10³ per square centimeter. If there were 25 chip levels or layers, then there would be about 25x3.6x10³ = 9x10⁴crystalsper square centimeter or 9x10⁹ per [m²] as shown schematically in FIG. 12 (these numbers are many orders of magnitude more effective then those discussed by Romero and Dehnen⁵²). According to Weber² , as just quoted, the energy could be as much as 10^-20 [watts] per crystal (if each driven just below its breaking point as enhanced by low-temperature and high-frequency operation) multiplied by 9x10⁹ (crystals) @ 10^-11 [watts] without significant attendant EM radiation. With the crystals properly oriented and programmed by the IIPCS to accumulate and propagate coherent GW radiation (please see Test Objective (3)) out of the side of the centimeter-thick, one-meter-square crystal array (whose area is about one centimeter by one meter, that is a reference area of 2x10^-2 [m] since GW goes each way) the GW flux near the device would be 10^-11/2x10^-2 = 5x10^-10 [watts/m²]. As noted by Joseph Weber², such a system could be employed “… to generate and detect gravitational radiation.” (Emphasis added.) With regard to detection, the crystals would represent very small resonators whose natural frequencies were approximately in the gigahertz to terahertz range. Please see Addendum A. Alternatives to the preferred design of the device using as energizable elements: piezoelectric crystals (or piezoelectric polycrystalline ceramics), include, but are not limited to either P or N processed strain-gauge silicon semiconductors, thin-film piezoelectric resonators, nanomachines, nanoelectromechanical systems, dielectric resonators, solenoids and piezoelectric polymers. For specific design details G. L. Wojcik, et al¹⁴, pp. 1107-1112 or Jan Kocback¹⁵ can be consulted.

In the case of solenoids (or nanosolenoids), some nanomachines, nanoelectromechanical systems, current-carrying plates, etc. the energizing and energizable elements can be collocated, for example the energizing coil around the energizable central magnetic core in the case of the nanosolenoids. By the way, if nano-solenoids, with superconductors passing 180 milliamper current were used, then 3000 to 4000 [N/m] would result; see pp. 15-77 to 15-79 of Mark’s Handbook, Eighth Edition. Superconductors will, no doubt, also improve performance in general.

Advances in nanotechnology and ultra-fast science and HF GW generation go hand in hand. The smaller the energizing/energizable elements are, the more powerful is the generated GW and the higher the frequency (and wider the bandwidth; please see Addendum A). This is because, according to Eqs. (4.4), (8), and (21.2) the GW power is proportional to the square of the quantity, rΔf/Δt. Whereas r and Δf depend upon the overall length and volume of the collection of energizing/energizable elements (and, therefore, are invariant for a given size GW generator), Δt is inversely proportional to the gravitational wavelength. Since the dimensions of the energizable element is proportional to the gravitational wavelength, the smaller the element size, the smaller the Δt, and the more powerful the GW! The ultra-fast, femtosecond to attosecond (10^-18 [s]) pulse lengths involve the increased power in, for example, Eqs. (24), (40A), and (43A) and would be an outcome of ultra-fast science. The improvement is especially striking as discussed in connection with Eq. (43A) in Addendum A in which communication improves in proportion to the sixth power of Δt!

As discussed in detail in the 1960 and 1964 Joseph Weber articles referred to above, the passage of a gravitational wave deforms an object or set of objects as it passes through them. For example, a piezoelectric polymer, a silicon semiconductor, a thin-film piezoelectric resonator, a piezoelectric-crystal functioning as a collector element is deformed by a HF GW and produces a small electrical current. Likewise, the plates of a capacitor functioning as a collector element are slightly moved relative to each other and thereby produce a signal. In fact, these elements are both energizable and generate HF GW and also are collectors and detect HF GW through the same conductors. (Please see Addendum A for a discussion HF GW communication.) The nanomachine collectors operate in a similar fashion. A nanomachine is a microscopic or molecular sized machine, for example, a microscopic version of the dumbbell motor/generator discussed in sections IV and VII of this paper. As a GW passes through the collector, the dumbbell moves slightly and submicroscopic coils respond to this motion and generate a small current. Likewise, energizing the coils in the motor mode will generate HF GW due to dumbbell motion. Electrical transducers, including parametric transducers or micro strain gauge nanomachines respond to the deformation occasioned by the passage of a HF GW in exactly the same fashion as it does to a mechanically induced strain and thereby function as a HF GW collector. The nanomachine pressure transducer collector element responds to a slight change in pressure of a set of particles comprising a fluid as the HF GW passes through it. The location of the collector elements and their connection with ultra-fast switches or transistors is identical to the location of the energizer elements, shown in FIG. 12 and, as already noted, may be one and the same element acting as either an energizer or a collector. The collector elements would be sequentially connected by the IIPCS or control computer to an information-processing device in order to follow the expected incoming HF GW-frequency pulses as they pass through the ensemble of such elements. That is, the collection elements will be interrogated sequentially or dynamically tuned (US Pat. 6,417,597; 6,160,336; and patents pending) to the expected HF GW frequency. Such a process is subject to experimental verification as Test Objective (5) of the next Section. As was noted by Gianluca Gemme and Andrea Chincarini of INFN Genoa, Italy, the HFGW detectors are very much smaller than the LFGW detectors. In consequence they are less expensive and less subject to environmental noise than LFGW detectors especially since they can be tuned to the frequency of the HFGW generator. As of the date of this writing there are at least two functioning HFGW detectors or collection elements (Bernard, et al³¹ and Cruise⁴⁰). To be sure the sensitivity of spacetime strain-measuring detectors are often less for high frequencies, but even in the case of burst and periodic celestial GW sources there is significant high-characteristic-frequency sensitivity (Thorne⁵⁷ Figs. 9.4 and 9.6, pp. 375 and 387).

There is a difference in the orientation of energizable and collection elements to the direction of the HF GW. For example, the lateral motion of the central magnet core, piston or barrel of the single-sector or linear-motor design moves the magnetic central mass or masses longitudinally like a microscopic-scale star collapsing. Thus as Burdge¹² states "… the gravitational vector will propagate along the axis of the (star’s) mass collapse." Or in this case, along the axes of the core, piston or barrel (the HF GW vector propagates in both directions). If, of course, the effect of energizing the elements results in a harmonic oscillatory motion of mass pairs rather than a jerk of a mass or masses (magnetic sites), then according to Einstein and Rosen¹⁹ a cylindrical gravitational wave results. The distortion occasioned by the passage of HF GW is somewhat different due in part to polarization. According to Weber⁷, p. 95, it affects the space transverse to the GW and so, although the elements and connections may be the same, the IIPCS or control computer will need to be programmed differently for transmission and for reception of GW.

VI. Spacecraft-Propulsion Concepts

No doubt HF GW experiments will reveal many applications of GW to propel spacecraft by means of remote GW generators, which change the gravitational field near a spacecraft, and to observe such a spacecraft (or other celestial object) by means of a HF GW (US Pat. 6,417,597; 6,160,336; and patents pending and please see Addendum B). In the former regard, on p. 349 of Landau and Lifshitz¹, they comment: “Since it has a definite energy, the GW is itself the source of some additional gravitational field. Like the energy producing it, this field is a second-order effect in the h_ik (tensor describing a weak perturbation of the galilean metric). But in the case of high-frequency gravitational waves the effect is significantly strengthened…” (Emphasis added.) Please see Test Objective (7) of Section VII. In the latter regard, communications back to Earth from the spacecraft can be integrated into the propulsion system if that system involves the generation of a HF GW beam onboard (as discussed in Addendum B) by modulating the HF GW propulsion beam (as discussed in Addendum A) and receiving it on Earth by means of the HF GW Telescope during interstellar flight.

The axis of jerk rotation of a spindle GW-generation device defines a preferred, single, unique direction in space and also a preferred, single, unique plane. The axis of the single-sector, linear-motor HF GW generator device defines a preferred, unique direction in space as well. Thus there is an asphericity or pattern to the gravitational radiation, an anisotropy or focusing (as in a HF GW Telescope), including refraction by a superconductor (please see Li and Torr²⁶and Test Objective (14)) that is analogous to a radio-antenna pattern of field strength. (Patents pending.) The concept that, as a part of this pattern, the gravitational waves are constrained to the “preferred” axis of the linear-motor’s “preferred” line in space, possibly without diffraction, will also be tested (Test Objective (2)). GW polarization may play a role here as it does in the case of coalescing binary black holes. These concepts have potential application to spacecraft propulsion either by remote “gravitational force field” generation or by placing GW generators on board a spacecraft as discussed in Addendum B – a “Relativistic Rocket” (also considered by W. B. Bonnor and M. S. Piper in their “Gravitational-Wave Rocket” paper⁵⁰and by Giorgio Fontana⁵¹ in his paper concerning colliding GW beams as applied “… to space travel” ).

As a rule of thumb the GW flux near the emitting, coherent-radiation end of the preferred, linear-motor device design for each of its concentric cylindrical layers is proportional to

{2rΔf/Δt}²/ΔA = {(⅔l)(Volume of sheaths)(Δf_l/ΔV)/Δt}²/ΔA = {(⅔)(2πrδrl)(Δf_l/ΔV)/Δt}²/(2πrδr)

µ {l²r²δ(Δf_l/ΔV)/Δt}²/{r²δ} = l⁴r²δ(Δf_l /ΔV)²/(Δt)² (33)

(note that this is simply the kernel of the quadrupole equation, such as in Eq. (4.4)) where

l = length of the magnetic core, piston, or barrel (or cylindrical layer) [m],

r = outside radius of the energizable magnetic cores, pistons, or barrels and inside radius of the energizing sheaths (or cylindrical layers) [m],

r = radius of gyration or effective radius of the magnetic core, piston, or barrel, e.g., = ⅓l [m],

Df_l/DV = longitudinal impulsive force per unit volume of energizing sheath(s) acting on the energizable magnetic core(s), piston(s), or barrel(s) [N/m³],

δ = thickness of the energizable barrel wall (e.g., magnet sites) as a fraction of barrel radius, r,

ΔA = area of the emitting end of the energizable barrel(s) [m²], and

Dt = impulse time [s].

In order to estimate the potential of GW for space propulsion it is useful to predict future improvement due both to technological advancements that are anticipated and the potential advantages of space-based operation due to high vacuum, low to negligible gravitational stresses on large space structures, etc. Let us, therefore, predict some possible improvements in the capability of HF GW generation. We will utilize Eq. (33) with the following parameters:

The diameter of the preferred device increased by a factor of 10 to 30 [m], that is, r = 10 multiplied by the nominal length of 3 [m], Dt=10^-2 multiplied by the nominal = 10^-14 [s] (occasioned by the possible design of a 10 femtosecond ultra-fast switch and pulse duration), and Df_l/DV = 100 multiplied by the nominal 3x10⁷ [N/m³] value (assume increased magnet efficiencies due to, for example, use of electromagnets rather than permanent magnets and high-temperature superconductors that would yield magnetic field strengths far in excess of 0.26 [T]). Thus Eq. (33) yields (with no change in length, l) an increase in the kernel of the quadrupole equation amounting to

(1)²(10²)(2)(100)²/(10^-2)² = 1x10¹⁰ . (34)

So that from Eq. (24), P = 0.25 [watts] over a 0.19 [m²] area, yielding a 1 [watt/m²] average HF GW flux, the potential average continuous HF GW flux near the emitting (coherent GW) end of the device available for space propulsion is

(1 [watt/m²)(1x10¹⁰) = 10¹⁰ [watts/m²] (35)

or 10 gigawatts per square meter! Such an energetic wave would no doubt necessitate a nuclear power supply or the use of nuclear reactions (utilizing, for example, antiprotons as discussed in Addendum B, and also discussed by Kammash²⁷ and by Schmidt et al²⁸) the size of which would depend upon the radius and length of the magnetic core and the efficiency of the GW generation. It should be noted that the relationship (33) can be utilized to scale the devices to significantly slower switching speeds and lower GW frequency (longer wavelength) depending upon the use of strong electromagnets instead of weak permanent magnets, increased length, etc. and, of course, there is a k factor (function) to be experimentally determined (Test Objective (10)) since in this extreme case r >> l_GW not << l_GW.

VII. Preliminary Tests and Experiments

Introduction

The quintessential test of a HFGW generation device is to be able to detect and analyze its HFGW output. At present there are at least two HFGW detectors available: Bernard, et al³¹ and Cruise⁴⁰. As has been noted, a HTSC under the influence of a HF magnetic field may generate HFGW (Fontana⁴⁷) as well as will a list of HFGW generation devices described in Section II and those alternate configurations discussed in Section V (including possibly the Z-pinch device). The preferred linear-motor design detailed in Section V or the magnetically or EM actuated HTSC design (Patents pending) would be especially good choices. On the other hand, it would be desirable to test fundamental concepts such as the influence of the weak-field assumption on HFGW generation. For this reason we shall discuss the utilization of a spindle device in this paper in some detail.

Terrestrial Gravitational-Wave Generator Utilizing a Spindle (U. S. Patent No. 6,160,336)

One possible gravitational-wave test device, 10, according to the present concept is illustrated in the perspective view in FIG. 1A, the plan view in FIG. 1B and the cross-sectional view in FIG. 1C. It comprises a circular rim, 11, which is supported by streamlined struts, 12, and guy wires or streamlined spokes, 14. The streamlined struts are connected to a hub, 16, which supports a spindle, 18. The guy wires or streamlined spokes, 14, extend from the top of spindle, 18, to spaced locations on rim, 11. Likewise, streamlined struts, 12, extend from the exterior surface of the hub to the interior side of the rim. A large receptacle or basin, 20, which is filled with a liquid, 21, such as salt water, supports the hub and spindle in a floating condition thereby supporting the hub spindle and rim assembly by means of a water bearing. Such a bearing exhibits no sticksion and, therefore, if the rim is not rotating allows for almost frictionless jerks. A berm, 43, on the ground surrounds the rim in order to shield the rim from wind- and sand- storms and to reduce the hazard attendant to a possible rim break-up while rotating or jerking. The berm, together with a slope, 48, of ground surrounding the receptacle or basin forms containment channel, 46. In order to stop the rim's motion rapidly (if it is rotating), in the event of severe earthquakes, accidents, or for the purpose of generating a single gravitational-wave pulse, bearing liquid will be rapidly drained. This will then cause the rim to drop precipitously (with acceleration, g) into the containment channel that has been partially filled with a liquid such as water or Acetylene Tetrabromide and bring the rim to rest with a "jerk". The energy dissipated by the rim as it comes to rest will vaporize some of the liquid. (Please see Test Objective (8).) Since centrifugal force drives some liquid up the wall of the receptacle or basin, 49, a splashguard or lip, 40, is attached to the top, inner edge or rim of the receptacle or basin in order to retain liquid. A float valve, 39, provides liquid on demand to make up for any liquid lost from the receptacle or basin due to evaporation, splash out or leaks in order to maintain the liquid level, 50. Upper radial bearing, 45, and lower radial bearing, 44, provided at each end of the spindle are pin and jewel bearings and resist occasional wind earthquake and sandstorm as well as gyroscopic side forces that are encountered during operation.

The upper radial bearing, 45, is supported by a system of streamlined spokes or wires, 37, and support structures, 38. The lower bearing, 44, is fixed to the bottom of the receptacle or basin. Both bearings allow for moderate z-directed motion (about 12 inches) of the spindle device by allowing the pins to slip through the jewel bearings without impediment. Use of a number of radial jewel bearings in mutual alignment to a common pin is contemplated in order to counter side forces of various magnitudes including earthquakes, wind, and, especially, gyroscopic forces (about 3,320 [N] at 34 degrees latitude if the rim is rapidly rotating during test of the weak-acceleration field approximation). The center of buoyancy, 41, of the hub is above the center of mass, 42, of the spindle in order to afford hydrostatic balance. A gravitational-wave detection facility, 47, is placed at a convenient location external to and in the plane of the rim, for example, within the berm, 43. A typical gravitational-wave detector, 17, (such as the one discussed in section V and in Addendum A) is positioned in the plane of the rim and located exteriorly of the rim. As discussed in more detail in conjunction with subsequent figures of the drawing, a series of permanent magnets, 24, are imbedded in the interior or under-surface of the rim and a series of coils proximate to the rim, 26, are fixed on the ground with or without metallic cores and controlled by the IIPCS. These coils acting on the rim magnets can produce a series of tangential jerks and an attendant train of HF GW pulses. Such a continuous train is, of course, far superior and far more energetic than a single pulse achieved by rapidly braking the rim’s rotation. As indicated it is contemplated that the rim rotates only for certain tests involving the weak-field approximation.

In one design, the device is contemplated to be a circular rim having a diameter of about 455 feet. The rim is constructed of steel reinforced concrete to resist up to a 44.5-g radial acceleration and weighs approximately 2500 tons. Composites such as graphite filaments in place of steel rebar are also contemplated for hoop-tension reinforcement in the rim. Under rotation, the flywheel rotates at a speed of approximately 25-rpm, which results in a relatively high rate of speed at any given point on the rim of approximately 390 miles per hour. If the rim is rapidly stopped by means of striking the liquid in the containment channel, 46, then about 2,600 gallons of liquid is vaporized, which is a far less efficient one-shot affair than a series of jerks. This breaking process will be studied during the experiments. By the way, since centrifugal and gravitational force fields are essentially the same, part of the experiment will be to determine the effects of the force field on GW generation, that is on the HF GW pulse train (Test Objective (6) as described below). Again it is noted that a train of pulses is far superior to a single pulse and the only purpose of the rapidly rotating the rim is to study HF GW generation in various acceleration environments in accomplishing Test Objective (6).

Tests

Test of Spin Up (without IIPCS totally operational)

Short segments of the IIPCS will be tested to accelerate the rim by jerks with no initial rim spin. Alternately, a gear and motor on the hub can produce 25-rpm motion of the rim or greater if, say, jet engines are attached to the rim to test the “weak-field” approximation under various centrifugal g loads. Silicon wafer encased sub-millimeter coils will also be fabricated and tested as discussed in section V to impart a long series of tangential, reciprocating jerks to the rim. Some test objectives will be pursued during developmental testing prior to complete fabrication of the test apparatus. It is to be emphasized again that the rim need not be rotating!

Test of Rapid Spin Down

Electrically-operated sluice gates or valves will be opened in order to a rapidly drop the level of liquid in the central bearing cylinder. If the spindle is rotating, then this will cause the spindle rim to drop very quickly (at 9.8 [m/s²] acceleration) into high-density liquid, such as Acetylene Tetrabromide (density of 2,930 [kg/m³], in a peripheral channel under the rim. This, in turn, will result in a very large hydrodynamic braking force over a very brief time interval (possibly less than a microsecond) and, therefore, generate a measurable jerk-produced HF GW. Unfortunately, the brief duration of this single GW pulse embodiment of the device may not produce sufficient flux for HF GW measurement (it might have to involve the use of templates such as those utilized for LWGW detection from small-fraction-of-a-second celestial events such as BBH merger). Thus the continuous, magnetic-field induced sequence of jerks utilizing the IIPCS will, no doubt, turn out to be the only viable means for HF GW testing especially in gauging the influence of acceleration (force field), caused by the rotating spindle, on GW power.

Test Objectives

The following are some quantities or characteristics of HF GW or of the devices to be tested:

(1) Test various procedures for the fabrication and utilization of ultra-small energizing circuit elements (e.g., coils, solenoids, plates, etc.) for HF GW generation/detection by means of a jerk.

(2) Experimentally study HF GW diffraction (if any) and measurement of average HF GW power flux, polarization, and directivity.

(3) Test HF GW Amplitude/Intensity and emulation of more extensive mass by means of the build up of coherent GW as the GW moves through the target mass of energizable elements.

(4) Test HF GW absorption (if any), change in polarization, refraction, dispersion, and scattering in various materials including superconductors, super fluids, etc..

(5) Experimentally study HF GW detection sensitivity using an ensemble of piezoelectric crystals tuned by the IIPCS.

(6) Test the effect of force-field (acceleration) magnitude on HF GW generation (effect on generation caused by centrifugal g loads; that is, test the weak-gravitational-field approximation by rotating the spindle at various rates).

(7) Test the GW effect on small masses and test of hypothesis concerning HF GW modification of a gravitational field suggested by Landau and Lifshitz¹ (p. 349).

(8) Experimentally determine the hydrodynamic drag of the rim for those situations when it is rotating and rapidly impacts the containment-channel liquid including possible ablation of the rim’s under surface if the rim is rapidly rotating.

(9) Experimentally study the characteristics of the ultra-fast switches (including optical switches) , the resulting current pulses and shape.

(10) Experimentally determine the functional form of the k’s to account for the quadrupole approximation for various values of the effective system radius compared to GW wavelength.

(11) Experimentally determine realistic models of magnetization dynamics on a picosecond or less time scale and magnetic field geometry and interaction.

(12) Experimentally determine the magnetoresistance effects and alternating DC pulse resistance as well as other aspects of microcircuit resistance and heating and energy loss due to EM emissions.

(13) Establish by experiment the variation of the HF GW speed in refractive media (such as a superconductor) as a function of HF GW frequency (that is, study HF GW dispersion if any) and HF GW Telescope optics.

(14) Test the refraction of GW by media such as superconductors and the degree to which HF GW group velocity is reduced and test high-frequency, short-wavelength GW optics in general.

VIII. Historical Footnote

Although the quadrupole-equation (or approximation) for generating GW was first formulated by Einstein⁸ in 1918, the term “gravitational wave” first appeared in 1905. Jules Henri Poincaré¹⁶ concluded that Newton’s laws needed modification and that there should exist gravitational waves that propagate at the speed of light. Our research efforts will attempt to demonstrate Poincaré’s conclusion directly.

Acknowledgement: I wish to thank Paul A. Murad for his valuable contributions to the text.

ADDENDUM A

Communication Utilizing HF GW (US Pat. 6,417,597; 6,160,336; and patents pending)

As an approximate numerical example related to a possible gravitational-wave detector for high-frequency, GHz to THz or QHz, gravitational waves, consider the absorption cross section, s [m²], for such antennas as given by Joseph Weber⁷, p.99

s = 15pGIQb²N²/8wc [m²] (36A)

where G = 6.67423x10^-11 [m³/kg-s²] (universal gravitational constant),

I = moment of inertia or quadrupole moment of the detector element(s) [kg-m²],

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

b = 2p/l_GW [1/m] (propagation constant),

l_GW = c/n [m] (gravitational-wave wavelength),

N = the number of quadrupoles coupled together in the antenna (see Eq. (2.9A) of Weber et al ²⁶p.62),

c = 3x10⁸ [m/s] (the speed of light),

n = frequency of gravitational radiation [1/s or Hz], and

w = angular frequency (or mean motion) [1/s].

For Q = 10⁶ (as noted by Joseph Weber², p. 308, “A practical antenna might be expected to have Q » 10⁶.” A large Q implies that a long time is required for the collector element to reach thermal equilibrium possibly enhanced by cryogenic cooling (with the Ingley HF GW detector³⁸ such cooling implies a lower bandwidth). Also the detection devices that Weber had in mind were large isolated aluminum cylinders, suspended and well isolated from the environment. The collector elements for the present device will probably be contained on a chip with damping constraints and a much smaller Q is likely; in fact, Cruise⁴⁰suggests Q = 10³ to 10⁴),

n = 10¹² [Hz] or one [THz], and

b = 2pu/c = 2.09x10⁴ [1/m], and

w = n/2 = 5x10¹¹ [1/s]; see Weber⁷ p. 90, we have

s = 1.15x10^-15 IN²[m²].

This value, depending upon I and the number of quadrupoles (with masses and characteristics nearly identical) coupled together in the antenna, N, compares favorably with s = 10^-20 [m²] of the Weber Bar given on p. 102 of Weber². There are, of course, other approaches to enhance GW detection or reception capability. The small voltages and currents produced by some of the alternative GW collector elements can be measured, for example, by a superconducting quantum interference device (SQUID) using Josephson junctions (described in U. S. Patent 4,403,189) and/or by quantum non-demolition (QND) techniques utilized in optics; but applied to the problem of reducing quantum-noise limitations for high-frequency GW. The QND technique was first suggested by Vladimir B. Braginskii of the Moscow State University and published by A. M. Smith²², pp. 935-941. At the present time (2000) there are at least two functioning HFGW detectors or receivers: Bernard, et al³¹ and Cruise⁴⁰. Bernard, et al utilizes coupled microwave cavities whose resonance frequency difference can be tuned to the HFGW frequency of the generator – the dimensions of this detector are smaller than the HFGW wavelength. Cruise utilizes a microwave resonance loop and measures a polarization vector – its size is about that of a HFGW wavelength (spacetime curvature not strain is measured). By the way, any accompanying electromagnetic signal arising from HF GW generation could be screened off by a conductor or, if necessary, by a mosaic of HTSCs.

An approximate estimate of what bandwidth a HF GW transglobal communication system might achieve is obtained as follows: Suppose that the distance between the HF GW generating or transmitting device and the receiver or detector is about one Earth's radius, 7,000 [km]. We assume that due to diffraction the HF GW will fan out from the rim in a wedge shape with an apex diffraction angle, α_d of

α _d= λ_GW /edge-width-of-rim = cΔt/edge-width-of-rim [radians] (37A)

and for λ_GW = cΔt = 3x10⁸ x 10^-12 = 3x10^-4 [m] the apex angle is α _d = 3x10^-4 [m]/0.01[m] = 0.03 [radians]. Thus at a distance of 7x10⁶ [m] this angle results in a band (0.03)(7x10⁶ [m]) = 2.1x10⁵ [m] wide and (2π)(7x10⁶ [m] ) in circumference for an area of 9x10¹² [m² ]. Also, suppose that we are transmitting through the Earth's mantle and that 10 percent of the GW energy gets through (very conservative since probably all of it will get through). Thus, for the tangential-jerk situation the "signal" obtained by modulating a long train or sequence of the current pulses by the IIPCS (some pulses missing and some forming a longer-duration pulse or pulses of different amplitudes) is, using the power near the device given by Eq. (9) and the average power flux of 2x10^-9 [watts/m²] there

S = (2x10^-9)(0.1)/9x10¹² = 2x10^-23 [watts/m²] (38A)

at the receiver or detector. For the radial-jerk situation using the power near the device given by Eq. (22) and the average power flux of 1x10^-4 [watts/m²] there

S = (1x10^-4)(0.1)/9x10¹² = 1.1x10^-18 [watts/m²]. (39A)

For the preferred longitudinal-jerk, linear-motor situation (U. S. Patent No. 6,417,597) it is especially important to calculate the signal strength, S. In this device the coherent GW emanates from one end and spreads out like a cone (having an apex angle, α_d = cΔt/3[m] = (3x10⁸)(10^-12) = 1x10^-4 [radians] ) resulting in an area of π(1x10^-4x7x10⁶/2)² = 3.8x10⁵ [m²] with average power from Eq. (24) of 1[watt/m²] we have,

S = (1)(0.1)/(3.8x10⁵) =2.5x10^-7 [watts/m²]. (40A)

Let us estimate the detector’s "noise" N ≈ 10^-8 [watts/m²] in the THz band (probably not many GW sources there except for relic or primeval background and possibly HF GW generated by HF EM as suggested by Gertsenshtein³⁷, but Brownian motion, thermal and quantum fluctuations, etc. may result in much more noise than these sources). Also we have hypothesized that the GW detector exhibits sensitivity on this same order. Of course the bandwidth of the long-base-line, interferometric GW detectors now under construction, such as LIGO, are at most about a few KHz and they are not designed for THz detection. Thus it is difficult to make comparisons of HF GW detectors (receivers) with the sensitivity of long-wavelength interferometric detectors. The “signal” or GW flux from an osculating circular orbit of a BBH having between a 6 and a 100-BH-radii semimajor axis is between 5x10^-5and 4x10^-11 [watts/m²]. A ten-watt isotropically radiating radio transmitter at a distance of 7 [km] produces a signal of 10/4π(7000)² = 1.6x10^-8 [watts/m²]. Therefore a reasonable HF GW detection sensitivity is on the order of 10^‑8[watts/m²] especially since we know what HF GW frequency we are looking for and don’t need a “template”. Test Objective (5) specifically addresses the issue. Note that the sensitivity of the single-crystal detectors considered by Joseph Weber 42 years ago were on the order of about 10^-10 [watts] as given on p. 313 of Weber². In fact, Weber²³ has speculated optimistically (p. 30) that there is "… no limit to the theoretical sensitivity of a (elastic solid) gravitational radiation antenna, and perhaps no limit to the number of novel methods for improving the sensitivity of existing antennas.” More recently in an article by Bernard et aI³¹ they suggest that superconducting coupled microwave cavities could detect fractional HF GW deformations or strain amplitudes having a sensitivity of Dl/l = 10^-20/Ön_GW = 10^-26 for THz GW. Also, as previously noted, A. M. Cruise⁴⁰ and R. M. J. Ingley^{38, 39} have proposed an electromagnetic detector for HF GW. All this work is somewhat similar to that found in Weber’s 1973 U. S. Patent No. 3,722,288. In this regard, please see Test Objective (5) of Section VII. Furthermore the signal can be enhanced by the energy gathering power or grasp of a high-frequency GW focusing system or HF GW Telescope (Patents pending) – enabled by a superconductive HF GW refractive media discussed in Addendum B.

Using Shannon's classical equation (C. B. Shannon²³p. 623), the maximum rate of information transfer, C, is given by:

C = Blog₂(1+S/N) . (41A)

For the radial-jerk and tangential-jerk, non-rotating spindle GW design S < N so no transglobal communication is possible. On the other hand, for the longitudinal-jerk (linear-motor) HF GW preferred design (U. S. Patent No. 6,417,597):

C = Blog₂(1+2.5x10^-7/10^-8) @ (10¹²)log₂(26) = 4.7x10¹² [bps]. (42A)

The bandwidth, B, is taken to be the IIPCS switch on-off or "chop" rate or reciprocating “hammer blows” or jerks of about 10¹² per second (that is, one Terabit per second or Tbps and multiple HF GW generators or “transmitters” could increase the bandwidth further). Note also that here we are talking about a single “carrier” chopping frequency whereas in actuality one can spread the information over an entire band of GW frequencies! HFGW is the ultimate wireless system, even reaching submerged submarines and it has offers the potential greater than QHz point-to-multipoint communication without the need for expensive enabling infrastructure (no need for fiber optic cable, satellite transponders, microwave relays, etc.).

Let us consider potential advances in the capabilities of a HF GW communications system. For the purpose of having a specific numerical example let us suppose that the dimensions of the transmitter or GW-generation device involve an energizing-element sheath (e.g., microscopic coils) that is 6 [mm] thick surrounding a 3 [mm] radius energizable-element core (e.g., microscopic magnets) and that the device is 18 [mm] in length (the effective length or radius of gyration is 6 [mm]). At the receiver, which we assume to be 7 [km] away, we will introduce a 18 [mm] diameter superconducting lens to gather and focus the HF GW in order to concentrate or amplify the signal at the receiver. We will again consider that Δf_l /ΔV can be increased 100 fold by increased magnetic efficiencies due, for example, by the use of superconducting electromagnets(rather than rather weak permanent magnets) to 3x10⁹ [N/m³]. We will also consider a reduction in pulse time to one femtosecond or Δt = 10^-15 [s]. The longitudinal-force pulse, Δf_l = (Volume)(Δf_l/ΔV) = (π[(9x10^-3)² – (3x10^-3)²] [0.018] )(3x10⁹) = (4.07x10^-6)(3x10⁹) = 1.22x10⁴ [N]. Thus from Eq. (4.4) we find (with half the GW, the non-coherent half, going in the opposite direction)

P = ½x 1.76x10^-52{(2)(0.006)(1.22x10⁴)/10^-15}² = 1.89x10^-18 [watts]. (43A)

This power from the forward, “coherent-radiation” end is distributed over an area defined by the diffraction pattern at a distance of 7 [km]. The diffraction angle, α_d , at the apex of a cone of HF GW is, similar to Eq. (37A), given by

α_d = λ_GW /core-diameter = cΔt/(0.018) = (3x10^‑8)(10^-15)/(0.018) = 1.67x10^-5 [radians]. (44A)

The area of the conical spread of the HF GW is

A = π(α_d R/2) = π(1.67x10^-5x7x10³/2)² = 1.07x10^-2 [m²]. (45A)

The lens, which concentrates the HF GW at the receiver, has a grasp, GW gathering power, or amplification of (d/λ_GW)² = {(0.018)/(3x10⁸)(10^-15)}² = 3.6x10⁹ . Putting it all together the signal at the receiver is {(1.89x10^-18)/(1.97x10^-2)}{3.6x10⁹} = 6.3x10^-7 [watts/m²].

Note that the HF GW signal at the receiver is inversely proportional to the sixth power of the system’s pulse length, Δt, (including the lens at the receiver). The foregoing is a bit of a simplification since, like the discussion of the linear-motor design in Section V, one would turn to a concentric, cylindrical-layer construction – not to a simple sheath and core. Thus the energizing elements (e.g., coils) and energizable elements (e.g., magnetic sites) would be close enough for the GW waves (of wavelength cΔt = (3x10⁸) (10^-15) = 3x10^-7 [m] or 300 nanometers – probably much smaller in a superconductor) marching down the cylinder coherently, to build up with an electron migration distance of only (2.38x10⁸)(10^-15) = 238 nanometers.

By the way, and like the spacetime continuum through which it propagates, gravitational-wave frequencies should not be subjected to governmental regulation. Paraphrasing George Gilder²⁴ p. 162: not only can numerous HF GW transmitters and receivers operate in the same frequency band, they can also “see" other user’s HF GW signals and avoid them. The HF GW spectrum is not only abundant and virgin, but in a sense it is quite limitless -- "bandwidth wasting circuits become ideal again..." (ibid, p. 207) -- every inhabitant of planet Earth can have his or her own bandwidth -- ten or so MHz.

ADDENDUM B

Gravitational-Wave Propulsion and HF GW Telescope (US Pat. 6,417,597; 6,160,336; and patents pending)

An onboard propulsion system (Patents pending), utilizing a gravitational wave generator, is shown in the block diagram of FIG.13. As shown therein, the propulsion system provides a gravitational-wave generator 67 included within a vehicle housing, 75. The generator (Patents pending) includes a particle-beam source (or laser -, or microwave-, electromagnetic-photon source), 69, of energizing elements and the nuclear-reaction chamber 72, which includes target-mass energizable elements. Such elements could involve high-energy, nuclear-particle collisions whose products (and resulting jerks) are distributed asymmetrically in the direction of the particle-beam energizing element’s motion (such asymmetry for high-energy nuclear collisions is discussed by Charles Seife²⁹ ). Alternatively, the energizable nuclear elements could be constrained to a preferred orientation yielding a preferred direction of the collision products and, again, a nuclear jerk in a preferred -direction. As previously noted, since the kernel of the quadrupole equation (that is, Eq. (21.2) or (8) or (4.4)) involves a square, the GW is bi-directional, that is, the GW axes extends in both directions along the axis of the jerk regardless of the direction of the jerk. (To be validated by Test Objective (2).) Such GW directivity is illustrated schematically by FIGS. 8A and 8B, but as noted previously may be polarization dependent. The rearward moving gravitational waves 62 in FIG. 13 exit the rear of the vehicle propelling the vehicle in the desired direction of travel, 74. The target-mass energizable elements in the nuclear-reaction chamber, 72, buildup, by constructive interference or reinforcement, the coherent GW, 82, as exhibited in FIG.14. Due to this reinforcement the system of energizable elements comprising the target emulates a more extensive mass having a longer effective radius of gyration and, therefore, stronger HF GW and more momentum to cause the forward motion in the desired direction of travel, 74, (validated by Test Objective (3)). A refractive medium, 77, in FIG. 13 can intercept the oppositely or forward-directed HF GW and those rays can be bent or refractive to the side, 76, in order to reduce the forward components of HF GW momentum and, thereby promote forward propulsion in the desired direction of travel (Patents pending). Refraction of the HF GW, which is also the basis of a HF GW Telescope ( Patents pending), is achieved due to the fact that, according to Ning Li and Douglas G. Torr²⁶, HF GW can move more slowly than its vacuum light speed in certain media, for example a super conductor. Specifically, they state: “It should be pointed out that since nothing is known of the phase velocity of a gravitational wave …propagating within a superconductor, it is usually presumed to be equal to the velocity of light. (The phase velocity, which relates to refraction, is the velocity of propagation of uniform plane GW, i.e., the speed of individual waves– versus the group velocity, which is the speed with which the information or energy is transported.) We argue that the interaction of the coupled electromagnetic and gravitoelectromagnetic fields with the Cooper pairs in superconductors will form a superconducting condensate wave characterized by a phase velocity v_p. Since … the phase velocity can be predicted for the first time as

v_p » … 10⁶ [m/s] (1x10⁶ ± 5x10⁵ [m/s] ), (30)

which is two orders of magnitude smaller than the velocity of light.” To be validated as Test Objective (14).

The forward-propelling portion of the HF GW generated by the jerks associated with the energization of the elements comprising the target mass (unlike the rearward moving gravitational waves) is not coherent. This as shown in FIG. 13 the HF GW portion is the result of the smaller actual radii of gyration of each individual energizable comment. Thus weaker HF GW is generated and, as previously mentioned, can be bent to a side by a HF GW refractive media and far less momentum is carried away to counter the propulsion in the desired forward direction of travel so that the forward propulsion dominates. The alternative means of HF GW propulsion involves the modification or distortion in gravity, 90, caused by the HF GW beam, 82, generated internally, or beam(s) generated externally, that results in the spacecraft moving toward, 91, or away from, 74, the distortion, 90 (U.S. Patent No. 6,417,597 and patents pending).

In FIGS. 14E and 14F the constructive interference or reinforcement or amplification of a GW by energizable elements consisting of the nuclear-reaction chamber’s target masses 80, 84, 86, and 88, HF GW is produced by micro- mass (nuclear) explosion or collapse (for example, antiproton annihilation; see, for example, p.1103 of Kammash²⁷ and also Schmidt, et al²⁸) which is emulates a macro star explosion or collapse, with the HF GW directed along its axes as predicted by Burdge¹², is illustrated. The reinforcement of HF GW is illustrated schematically by the arrows 83, 85, 87, and 89 in FIG. 14F. The HF GW builds up to a larger amplitude 82 as the energizing beam bunch and the HF GW crest or front moves with the same speed together through the particles comprising the target mass or energizable elements and generate coherent HF GW pulses. The target particles or energizable elements 80, 84, 86 and 88 are V_GW Dt apart where V_GW is the local GW speed and Dt is the time between energization. Thus an extensive mass in the propulsion system can be emulated; to be validated by Test Objective (3).

The fact that a HF GW beam can modify a gravitational field (Landau and Liftshitz¹ ) may have already been demonstrated serendipitously and independently by the experiments of Podkletnov ⁴⁴, Ning Li, et al⁴⁵, and Rounds⁴⁶. It is conjectured that a high-temperature superconductor (HTSC), when subjected to high-frequency magnetic field, generates HF GW. If it can be demonstrated experimentally that the magnitude of the change in weight of a test mass or the gravitational acceleration measured by a gravimeter above (or below) the HTSC increases in proportion to the square of the frequency of the magnetic field impressed on the HTSC, then the data would agree with the square term (kernel) of the quadrupole approximation to HF GW intensity – possibly phonons (lattice/molecule vibration)⁵⁶ or magnetic-vortex oscillations (or Brownian motion) provide the jerks. As a matter of fact, on page 10 of the Podkletnov⁴⁴ paper, the data seem to indicate just such a variation. Also if the gravitational-field effect varies radially out from the axis of rotation, r (either of the HTSC or the impressed magnetic field), then the r dependence would follow that predicted by the quadrupole kernel, (2rΔf/Δt)². At this point the conjecture would become a theory and HF GW detectors (e.g., Bernard, et al³¹ and/or Cruise ⁴⁰⁾ could be utilized to measure HF GW directly. Thus a relationship between the HF GW flux and the change in the gravitational field could be measured directly. By the way, this conjecture is supported by a 1998 paper by Giorgio Fontana⁴⁷who suggest a mechanism for the generation of HF GW by a HTSC. Fontana also suggests that one should look for additional spectral components in a local laser beam. There are also two papers that discuss the collision of GW. One by Veneziano⁴⁸ states that according to string theory the head-on GW collision focuses “... toward a point located on the (graviton) axis” and may lead to a singularity in the spacetime fabric. Another paper by Valeria Ferari⁴⁹ discusses the influence of polarization on GW collision and the relationship to BH formation. In fact, Fontana⁵¹ states “... the nonlinear behavior of spacetime may permit the generation of spacetime singularities with colliding beams of gravitational radiation; this phenomenon could become a form of propellantless propulsion.” Unless created by remotely generated HFGW beams, the onboard energy required for propulsion would not be reduced. Fontana also conjectures that “... the mutual interaction of gravitational waves would cause the appearance of a rectified wave, accompanied by a coulomb-like gravitational field. If this field is created outside a spacecraft, the spacecraft would free-fall towards the distortion ... our spacecraft would follow a depression in spacetime.” Thus we have a realistic possibility of moving inanimate objects by such remotely or onboard generated HF GW beams and a potential for interstellar travel.

With regard to the patented HF GW Telescope, it has two major components and a third component is required to test it. The first component is a one to one-hundred meter diameter multifaceted lens composed of a mosaic of several high-temperature superconductors (tiles) or other media that will refract and focus HF GW. Such a medium is state of the art or near to it. For example, a 10-inch diameter, 0.5-inch thick superconducting disk was reported built in March 1997 at the University of Alabama and a Yttrium-Barium-Copper-Oxide (YB₂C₃O₇), 22[mm] diameter, 4[mm] thick superconducting disk can be purchased for $25.00 from Futurescience Inc., P. O. Box 17179, Colorado Springs, CO 80935. Superconductor Components, Inc. in Columbus, Ohio is also fabricating a HTSC disk for NASA (for them to test the results of Podkletnov ⁴⁴).For large diameter HF GW Telescope objective lenses far less expensive (though somewhat higher temperature) HTSC such as steel-clad MgB₂ can be utilized. Note that since GW can pass through any material without attenuation, such as the detectors on the focal plane (surface) themselves, and the slope of the marginal ray through the lens at the image can exceed 90 [deg] and can be incident on the “wrong side” of the detector array. Thus focal ratios less than 0.5 might be achieved. The second component is a HF GW detector (or matrix of detector elements under computer control) placed on the focal plane (or surface) of the HF GW lens (please see the end of Section V and Addendum A). Unlike long-wave-length LF GW detectors (having dimensions of hundreds or thousands of meters) currently under construction, these detectors will make use of patented, nanoscale, sophisticated elements, already discussed in this paper but will require considerable new-technology development – albeit much of the applicable ultra-fast science, nanomachine technology, and high-temperature superconductor technology is currently under rapidly expanding development at hundreds of laboratories both here and abroad. The third component, needed for optical-bench testing of the HF GW Telescope (Test Objective (13)), is the HF GW generator device itself.

As a numerical example, for a 100[m] objective-lens-diameter, earth-based HF GW telescope, and a micrometer λ_GWand spurious disk diameter (for a point source) at the focal plane, the HF GW frequency, ν = c/λ_GW = 3x10⁸/10^-6 = 3x10¹⁴ [Hz} = 300 THz and the GW grasp, or GW gathering power, or amplification is {d/λ_GW}² = {100[m]/(10^-6)}² = 1x10¹⁴for point sources. Such HFGW point sources might include the very speculative ultra-small, nearer, relic BHs – a candidate for Dark Matter; sequences of super-nova shell material jerked from rest to a large fraction of the speed of light over a few centimeters of distance in, say, a picosecond, ... the “... extremely strong compressional shocks in matter...”, Pinto and Rotoli⁵⁶, p.568; Halpern and Laurent⁵³ , p.745, even suggest HFGW radiation from the interior of a star (Sun). As Professor John Miller of Oxford and Trieste said: “It has been the fashion to look for celestial sources of rather low-frequency GW, now my eyes are opening to the possibility of celestial sources of your high-frequency GW.”

Let us consider the case of interstellar communications with and/or observation of an interstellar spacecraft (or other celestial point source of HF GW) as mentioned at the beginning of Section VI. For a three-meter-diameter transmitter (or propulsion-generator) beam, the widening will be due to diffraction (if it exists for GW; please see Test Objective (2)) like a cone with a λ_GW/width-of-source = (10^-6)[m]/3[3] = 3x10^-7 [radian] apex angle, α_d,. Thus over a distance of 10 light years (lyr) or 9.5x10¹⁶ [m], the signal at the focal plane of the receiving HF GW Telescope will be reduced by a factor of

{1x10¹⁴}{π(3[m]/2)²}/{π(9.5x10¹⁶[m]x3x10^-7[radians]/2)²} = 9.9x10¹⁴/6.4x10²¹ = 1.5x10^-6. (46B)

From Eq. (41A) with B = 10¹⁴ [Hz] and GW flux at the transmitter (or propulsion GW generator) of 10¹⁰ [watts/m²] (from Eq. (35)) so that S = (10¹⁰)(1.5x10^-6) = 1.5x10⁴ [watts/m²], and with hypothesized noise, N = 10^-8 [watts/m²], we have

C = 1x10¹⁴ log₂{1 + (1.5x10⁴/10^-8} = 1x10¹⁴{log₂(1.5x10¹²)} ≈ 4x10¹⁵ [bps] (47B)

or 4 Qbps maximum information transfer rate.

For an extended source, such as ripples or other anisotropic features of limited angular extent in the relic or primordial cosmic background, by Ockham’s Razor the intensity a the focus, P, for an objective lens diameter, d, and a focal length, f, is

P[watts] ≈(GWflux)(ObjectiveLensArea)(1/{focal ratio}²) ≈ GWflux [watts/m²] d⁴ [m⁴]/f² [m²] . (48B)

The classical index of refraction, N, is given by

N = (velocity in a vacuum)/(velocity in the medium) = c/v_p = 3x10⁸ /(1± 0.5)x10⁶= 400 ±200.(49B)

I will rely on an old, rather standard, textbook on optics by Warren J. Smith⁴³. The standard lens equation (for example, Eq. (2.30), page 35 of Smith⁴³) is for focal length, f, and front spherical lens radius, R₁,

1/f = (N – 1)(1/R₁ – 1/R₂). (50B)

For a plane convex lens, R₂ → ∞ and with N » 1 we have

R₁=Nf, (51B)

where for f/1, i.e., unity focal ratio, f = 100 [m], so that R₁ = (400)(100) = 40000 [m], f = R₁/N and f = 40000/200 = 200 [m] to 40000/600 = 67 [m] or f/2 to f/0.67, the uncertainty being the uncertainty of the speed of GW in a superconductor as reflected in the uncertainty of the index of refraction. If a superconductor field lens or immersion lens encloses the detectors on the focal plane, then since λ_GW is greatly reduced, resolution is enhanced due to less diffraction.

If intervening matter between the HFGW generator and detector causes a change (even a very slight one) in HFGW polarization, diffraction, dispersion or results in extremely slight scattering or absorption, then it may be possible to develop a HFGW “X-ray” like system. It may, in fact, be possible to image directly through the Earth and view subterranean features, such as geological ones, to a sub-millimeter resolution for THz HFGW.

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* Copyright Ó 2000 by Robert M. L. Baker, Jr., PhD. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

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

Senior Consultant, Transportation Sciences Corporation

Abstract

Nomenclature

Subscripts

As usual let GW-radiated power be given by

References