Linerman-Peter: Primer

Peter A. Lindemann's "cold electricity" thesis about Edwin V. Gray's capacitor-discharge motor and "conversion switching tube", and how to keep an energy ledger for a pulsed circuit.

Source tags: [read] = from a document in the project folder (p.N is the PDF page of free_energy_secrets.pdf, which runs 8 ahead of the book's printed page numbers, so printed p.59 is PDF p.67; "ebook p.N" is the 2001 essay; some earlier review files cite printed pages); [web] = Internet source, URL given; [memory] = recalled from training, with a confidence level; [calc] = arithmetic done for this primer and reproduced by example.py. Verdicts come from arbiter.md; this primer adds no new verdicts.

Limits of the record (repeated from arbiter.md; not a source gap). Both works in the folder have usable text: Lindemann's book The Free Energy Secrets of Cold Electricity (Clear Tech, Inc., preface dated October 2000, 156 pp.) and his 2001 web essay The World of Free Energy [read]. But: - Neither document is a test report. Every device figure in the book and the essay is Lindemann's assertion. - After the QE round, every part of the book except the reference and back-cover pages (pp.153-156) has a deep read: pp.7-21, 22-38, 39-70, 74-81, 110-125, 125-139, 141-152. Gray's motor patent (pp.74-81) and the circuit-patent drawing (p.115) were checked on page images. - The only numerical test claim (Crosby Research Institute / J. A. Maize, 1973) and the only witnessed failure (Donald Cronin, about 1974) are known only from reprints on rexresearch.com [web: https://www.rexresearch.com/evgray/1gray.htm]. The Crosby "203 hours" is a projection: the one text that gives a run length gives 21 minutes. No Crosby report, court file or death certificate has been obtained. - The 1973 press demonstrations, the most-cited evidence for Gray, were reported by a sympathetic journalist; the only instrument mentioned is a voltmeter. - Some verdict inputs on cold fusion, Stanley Meyer and Bearden's MEG rest on [web] secondary summaries. Jean-Louis Naudin's MEG page could not be opened, so what Naudin actually measured is unknown to this review.

Who has a stake. Lindemann (honorary D.Sc. in complementary medicine; Vice President of Borderland Sciences 1987-99; co-founder of Clear Tech, Inc.; sells the book and runs free-energy.cc) is an interested party [web: https://wiki.naturalphilosophy.org/index.php?title=Peter_A_Lindemann]. Gray's patents are the inventor's own, attorney-drafted descriptions. The National Tattler (reporter Tom Valentine) and Rex Research were advocates. Pan World Enterprises (Japan) was an investor in Gray; EvGray Enterprises and Gray's Vice President of Engineering, Richard Hackenberger, were Gray's own company. Gerry Vassilatos, whose chapter the book reprints, is the author-publisher of a book arguing the same thesis. The Los Angeles District Attorney was the prosecutor. Each account below is that party's account.

Safety. Lindemann urges readers to build these devices. The motor patent's capacitor bank is rated 3,500 V [read: page image p.85]. Lindemann writes that "if you try to build this motor according to the principles outlined in this patent ... If you happened to get into the discharge path of those capacitors, you'd be thrown clear across the room" [read: p.16]. Vassilatos's claim, repeated in the book, that impulses under 100 microseconds are "completely safe" is backed by no measurement in the folder [read: p.30; arbiter.md QE revision (h)]. A charged high-voltage capacitor bank can kill. Do not build one without training.

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1. The idea in one paragraph

In 1973 Edwin Gray, a self-taught California inventor, demonstrated to a newspaper reporter and invited visitors (among them the president of an oil company) electromagnets, lamps and a motor that he said ran on batteries that never ran down. His patent (US 3,890,548, 1975) describes a motor driven by charging capacitors to high voltage from a battery, dumping them through spark gaps into pairs of electromagnets that kick apart, and catching the coils' collapsing-field "overshoot" in a second battery. Gray's company sold shares, the Los Angeles District Attorney raided it in 1974 and filed charges in 1975, and Gray later patented a spark-gap "conversion switching tube" (1986-87). Peter Lindemann's book argues that Gray had rediscovered what Nikola Tesla called "radiant energy": a very short, one-way spark releases an "Electro-Radiant Event" that copper grids can collect, giving a cool, non-shocking "cold electricity" that comes from the ether, ignores Ohm's law and thermodynamics, and yields about 100 times more work than the spark consumes. Lindemann also argues that Gray's own explanation was wrong, that many other "free energy" machines are proven, that the laws of thermodynamics are not universal, and that such technology is kept from the public. The useful lesson is that almost every ingredient here (capacitor discharge, back e.m.f., energy recovery, short high-voltage pulses, heat pumps) is real physics, and the claims stand or fall on one thing that was never done: a complete energy ledger with every input metered.

2. What you need to know first

Energy, power and their units. Energy E is measured in joules (J); power P is the rate of energy flow, in watts (W = J/s). A watt-hour (Wh) is energy: 1 Wh = 3,600 J. "Watts per hour", which appears in the 1973 test report, is neither power nor energy, and spotting this kind of unit slip is half of this project. When voltage v(t) and current i(t) change with time, the energy delivered over a time T is E = ∫₀ᵀ v(t) i(t) dt, the area under the instantaneous power curve. Read: Halliday, Resnick and Walker, Fundamentals of Physics, the chapters on current, resistance and circuits.

Capacitors. A capacitor of capacitance C (farads, F; 1 µF = 10⁻⁶ F) charged to voltage V holds charge Q = CV (coulombs, C) and energy E = ½CV². Example: 2.5 µF at 3,500 V stores ½ × 2.5×10⁻⁶ × 3500² = 15.3 J, about the energy needed to lift a 1.5 kg book one metre. High voltage does not by itself mean much energy. Read: Griffiths, Introduction to Electrodynamics, the sections on energy in electrostatics and capacitors (Chapter 2).

Inductors and back e.m.f. A coil of inductance L (henries, H) carrying current I stores energy ½LI² in its magnetic field. The coil resists any change in current with a voltage v = L di/dt, called back e.m.f. (e.m.f. = electromotive force). If you suddenly interrupt the current, di/dt is huge and the coil produces a large voltage spike (the "inductive kick" or flyback), which is how a car's ignition coil makes a spark from 12 V. If a diode gives that current a path into a battery, part of the coil's stored energy is recovered. This is ordinary engineering (regenerative braking in electric cars works on the same accounting). Recovery can return part of what you put in, never more. Read: Griffiths, Chapter 7 (Faraday's law, inductance, energy in magnetic fields).

The LC discharge. Connect a charged capacitor to a coil and the charge sloshes back and forth: the capacitor voltage falls as a cosine and the current rises as a sine, with angular frequency ω = 1/√(LC). The current peaks a quarter cycle later at I₀ = V₀√(C/L). With C = 2.5 µF and L = 10 mH, the quarter period is (π/2)√(LC) = 248 µs and I₀ = 55 A. Resistance R in the loop turns the oscillation into a decaying one (a damped harmonic oscillator, the same ODE as a mass on a spring with friction). Read: any ODE or circuits text on the series RLC circuit; Horowitz and Hill, The Art of Electronics, early chapters.

Spark gaps and arcs. Two electrodes in air do not conduct until the voltage exceeds the breakdown voltage (very roughly 3 kV per mm in dry air at small gaps [memory, medium]). Once an arc forms it is a good conductor with a fairly constant voltage drop across it, "typically tens to hundreds of volts" [web: https://en.wikipedia.org/wiki/Spark_gap, via search summary in arbiter.md], regardless of the current. A spark gap is therefore a voltage-triggered switch. It is not an ohmic resistor (Ohm's law, V = IR, does not describe it), which is a fact about arcs, not a breakdown of physics.

Peak power versus energy. A pulse can have an enormous instantaneous power and still carry little energy: one megawatt for one microsecond is one joule. Also, the product of the peak voltage and the peak current is not the peak power unless the two peaks happen at the same instant, in the same circuit element.

Conservation of energy and the First Law. Over a complete cycle, the energy a device delivers (work plus heat plus stored energy gained) equals the energy it takes in from all sources. For electromagnetic fields this is Poynting's theorem: ∂u/∂t + ∇·S = −J·E, where u is the field energy density, S = (1/μ₀) E × B is the energy flow per unit area (the Poynting vector), and J·E is the rate at which the field does work on charges. It follows from Maxwell's equations, so a circuit that made energy would mean Maxwell's equations fail. Read: Griffiths, Chapter 8 (Conservation Laws).

The Second Law, engines and heat pumps. A heat engine working between a hot reservoir at absolute temperature T_h and a cold one at T_c can turn at most a fraction 1 − T_c/T_h of the heat it takes in into work (the Carnot efficiency). Run backwards, the same machine is a heat pump: it uses work W to move heat, and the heat delivered per unit of work, the coefficient of performance (COP), can be well above 1: ideally COP = T_h/(T_h − T_c). A kitchen refrigerator with COP about 3 moves three joules of heat per joule of electricity. No machine can run continuously on heat from a single reservoir (the Kelvin-Planck statement). Read: Schroeder, An Introduction to Thermal Physics, Chapter 4 (Engines and Refrigerators).

Electromagnetic waves are transverse in vacuum. In empty space Maxwell's equations include ∇·E = 0. For a plane wave E = E₀ exp[i(k·r − ωt)], where k is the wave vector (pointing along the direction of travel), this gives k·E₀ = 0: the field is perpendicular to the direction of travel. Longitudinal electric oscillations do exist, but in plasmas and in the non-radiating near field of an antenna, not as free waves in vacuum [memory, high]. All such waves travel at or below c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s. Read: Griffiths, Chapter 9 (Electromagnetic Waves).

Resonance and Q. A resonant circuit with quality factor Q (roughly, the number of cycles its energy takes to die away, times 2π) builds up voltages and currents far larger than the source's while storing energy, not making it. Apparent power (rms voltage times rms current) in a resonant circuit is mostly reactive power that sloshes back and forth; the real power, the time average of v(t) i(t), is what the source supplies. This is why a Tesla coil can show 400 kV and amperes in its ground lead while drawing 2 kW from the wall.

3. The textbook view

Here is how standard physics describes each piece of Gray's machines, with the equations a reader needs to keep the books.

3.1 Gray's motor (US 3,890,548), as circuit physics. The patent's chain is: battery 10 → a vibrator and automotive ignition coils step the 12 V up → a rectifier charges capacitors to high voltage → when a rotor contact lines up, a spark gap breaks down and the capacitor dumps through a rotor coil and a stator coil, which repel → when the arc goes out, the coils' collapsing field sends an "overshoot" pulse into a second battery 10a [read: pp.75-78, deep read section 1-2]. In energy terms, each firing goes like this:

• Charging: battery 10 supplies E_drawn = E_cap / η, where E_cap = ½CV₀² and η < 1 is the efficiency of the vibrator, ignition coil and rectifier.
• Discharge (phase 1): the capacitor drives current through the arc (voltage drop V_a) and the coils (inductance L, resistance R). Kirchhoff's voltage law gives C dv/dt = −i and L di/dt = v − V_a − Ri, where v is the capacitor voltage.
• Recovery (phase 2): once the capacitor is empty the arc goes out, and the coil current can only flow through a diode into battery 10a, at voltage V_b: L di/dt = −V_b − V_d − Ri, with V_d the diode drop. The battery receives ∫ V_b i dt.
• Work: if the rotor moves while the coils carry current, the coils' changing mutual inductance takes energy from the circuit and turns it into shaft work W (this appears in the circuit equation as an extra "motional" back e.m.f.).

Every joule is accounted for: E_cap = (arc loss) + (resistive loss) + (diode loss) + W + (energy returned to battery 10a) + (a little radiation). So the energy returned is always less than E_cap, and the net drain on the batteries per firing is at least W plus the losses. This is exactly what the patent says in words: the engine "can regenerate a part of the energy consumed", it runs "as long as d-c power is applied", and it is "believed to operate on the principle of conservation of energy" [read: pp.75, 79; claim 1, pp.82-83]. The patent nowhere claims self-running or output above input, and gives no efficiency, power, torque, capacitance or inductance figure at all [read: deep read section 7].

3.2 The motor's geometry checks out. Nine stator pairs sit 40° apart; each rotor and stator pair has its two magnets 13⅓° apart. The rotor advances 40°/3 = 13⅓° per firing, so one revolution takes 27 firing steps, and 27 × 13⅓° = 360° [read: page images pp.76, 79-81; calc]. The OCR text reads "134°", which is an artifact of the printed fraction 13⅓°. Table I of the patent agrees with this pattern. There are real drafting slips ("450°" printed where 240° is meant; two passages naming different firing pairs at step 3; capacitor counts not reconciled) but none affects any energy question [read: deep read section 5]. One physical point the patent does not discuss: two like poles face to face push straight apart and give no sideways force, so a coil fired exactly at alignment gives no torque; the patent's ±6⅔° "faster/slower" offsets fire slightly off alignment, which is where torque must come from [read: p.76; deep read 5.5].

3.3 Gray's circuit patents (US 4,595,975 and 4,661,747). These describe a battery-fed charger raising a capacitor to high voltage, discharged through a spark gap inside a "conversion switching element tube" with "charge receiving grids", into an inductive load. The 1986 patent says the system will, "on theory", deliver "a working force at the inductive output of the device that is more efficient than that which is capable of being developed directly from the energy source", and that efficiency is "further enhanced" by returning "that portion of the initial energy developed, and not used by the inductive load ... to the same or second energy reservoir" [read: p.111]. The return is regeneration, not gain. The "more efficient" phrase is vaguer: it names no input or output to compare, and the same page says the device works by "harnessing the 'electrostatic' or 'impulse' energy created by a high-intensity spark" [read: p.111], which is the hook in Gray's own words on which Lindemann builds his radiant-energy reading. Gray's 10 April 1980 application, "Regenerative Energy Recovery System, e.g., for Electromagnetic Propulsion", which the book does not mention, is saved in the project's research files. A 24 V battery and a DC-to-DC converter charge a capacitor "to a high voltage of about 2000 volts"; the discharge passes "through the inductive load ... and through the energy recovery battery number 2"; then a diode lets the "inductively maintained current" keep flowing "to charge a battery"; the inventor "believes this to be a more efficient use of energy" and speaks of recapturing "a part of the EMF, that is normally lost" [web: saved copy of https://www.rexresearch.com/evgray/1gray.htm]. That is the textbook mechanism of 3.1, with the recovery battery sitting in the discharge path itself. Its only numbers are 24 V and about 2000 V: no measurement, efficiency, run time or witness. The circuit patents likewise contain no figure for volts, amperes, farads, watts, rpm or efficiency, and every efficiency statement is hedged ("on theory", "anticipated") [read: deep read pp.110-125; arbiter.md QE revision (c)]. The Caltech endorser Norm Chalfin (affiliation as stated by the Tattler) described it the same way: "energy is used only during a small fraction of a millisecond. Energy not used is returned to an accessory battery for reuse" [web: rexresearch].

Lindemann objects that the 1986 patent's FIG.1 cannot be built as drawn: part #42 is drawn shorted, the grid of tube #28 is unattached, the load #36 has no coil symbol and #38 has no discharge path [read: pp.58-59]. Checked on the page image, three of the four points hold and the fourth half holds (no return loop through the load is drawn, but #38 is connected to the batteries through a switch) [read: page image p.115; arbiter.md QE revision (c)]. The charging chain is conventional; the discharge, control and recovery loops are not closed, and no component values are given. That means nobody can build a verified copy from the drawing, which weakens a "failed copy" reading as much as a "working copy" reading. Lindemann's own "probable schematic" (Fig. 34, "DRAWING BY: Peter Lindemann D.Sc., August, 2000") is his conjecture, not Gray's, and swaps the roles of the two batteries between p.55 and pp.64-67 [read: pp.55, 64-67].

3.4 Short pulses and "cold" sparks. Tesla's 1890s high-voltage, high-frequency experiments really did produce startling effects: lamps lit with no wires, and currents passed through the body without the pain of low-frequency mains [memory, high]. Mainstream explanations are pulse energy (a short pulse carries little energy), the strength-duration behaviour of nerves (shorter pulses need much more current to stimulate a nerve) and, at high frequency, skin effect and capacitive coupling [memory, medium]. None of these needs a new energy source, and none makes a high-voltage capacitor discharge safe: burns and arcs still injure.

Lindemann gives one first-hand account of such an effect. In 1986, on the day the photograph of Eric Dollard's Magnifying Transmitter discharge was taken (Fig. 24, by Alison Davidson, supplied by Tom Brown), he held the screw base of an ordinary filament bulb while Brown touched its bottom contact, the two men standing about 6 and 8 ft from the transmitter, and the filament lit with no sensation; no instrument or bulb rating is given [read: pp.50-51]. The mathematician's reading is near-field coupling: two bodies standing in the strong field of a large coil act as a capacitor plate and lead [mathematician.md]. That is a physics explanation of the account, not a refutation of it. Brian Desborough's back-cover testimonial of "incoming spikes over 1000 amps" names no instrument and is an endorsement of the book [read: p.156].

3.5 Tesla's "radiant energy" patents. Tesla's US 685,957 (apparatus) and 685,958 (method), both filed 21 March 1901, describe radiation (sunlight, X-rays, ultraviolet) falling on an insulated, elevated plate and charging a capacitor connected between the plate and ground [web: https://patents.google.com/patent/US685957A/en ; https://patents.google.com/patent/US685958A/en]. Today we would explain that by the photoelectric effect and by ionisation of the air near the plate, both real and both small in power: sunlight brings about 1 kW per square metre at best, and photoemission converts a tiny fraction of it [memory, high]. This is a different thing from the spark-gap "Electro-Radiant Event" in Lindemann's book, which is worth noticing: even Tesla's own patented "radiant energy" is not the effect Lindemann attributes to it. Nor is it an ether claim: US 685,958 explains the charging by "minute particles of matter" thrown off by the sun, with the ground as "a vast reservoir of negative electricity", and mentions the ether only as a view it does not adopt [read: p.131].

3.6 Thermodynamics. Lindemann's essay starts from a true point: "they use a small amount of one form of energy to control or release a large amount of a different kind" [read: ebook p.1]. That describes a transistor, a relay, a heat pump or a match, all of which obey the First and Second Laws because the large amount of energy comes from somewhere (a power supply, the outdoor air, the chemical energy of the wood). He is also right that a refrigerator moves about three units of heat per unit of electricity [read: ebook p.2] and that heat pumps across a small temperature lift can reach COP 8-10: the ideal heating COP for a 20 K lift at 313 K is 313/20 = 15.6, and a real machine at about half of ideal gives about 8 [calc]. The textbook adds the constraint he leaves out: the heat moved is not work, and an engine running on that heat is again limited by Carnot.

4. Where the inventor departs

There are two inventors here. Gray's patents stay close to the textbook; Lindemann's book departs from it, and also departs from Gray.

The exact point of departure is the passage on pp.67-68. Lindemann accepts that the capacitor's energy comes from the battery, then says it is entirely "lost" in the spark gap's voltage drop, and then counts the output against that lost energy alone. Two things go wrong there, and both are checkable with the tools in section 2:

1. The arc does not take 99%. The energy dissipated in an arc with drop V_a while charge Q passes is V_a Q. If the capacitor empties, Q = CV₀, so the fraction lost in the arc is V_a C V₀ / (½CV₀²) = 2V_a/V₀. At V₀ = 3,000 V and V_a = 50 V, that is 3.3%. A small voltage drop means a small loss, the opposite of what the text says. (To lose 99% the arc would need a drop of about 1,500 V.) Where the energy really goes depends on the rest of the circuit: in Lindemann's own "probable schematic" there is a resistor (#30) between the gap and the battery [read: p.67], which would turn much of it into heat.
2. The 100:1 is not derived. It compares two energies, neither of which is measured, and the denominator is only part of the input. Even if the arc did take all of E_cap, the battery had to supply E_cap first. A ratio of output to part of the input is not a gain.

Where Lindemann is right. He is right, and the mainstream agrees, that Gray's "peak apparent power" explanation is no explanation: multiplying one source's voltage by another's current does not give power, and a high peak power in a sub-millisecond pulse does not mean a lot of energy (section 5, Part C). He also writes, of the 1975 motor patent he received in 1978, that "if you try to build this motor according to the principles outlined in this patent, it doesn't perform anything like what was described in the Valentine articles", that touching the discharge path would throw you "clear across the room", and that "the amount of energy that could be recycled from this arrangement is negligible" [read: p.16]. He never says he built one, and gives no build record, date, values or instruments. It reads as first-hand experience and is evidence against the motor as the 1973 press described it, coming from the inventor's most committed advocate; but it is his statement, not a documented replication, and it says nothing direct about Gray's own hardware [arbiter.md QE revision (a)].

Lindemann on fraud, for balance. Lindemann is not uncritical. His essay describes, without naming him, a con man who in the previous 15 years had raised "more than $100,000,000", was "barred from doing business in the State of Washington" and "jailed in California" (a claim checkable in public records), and lists Keely, Hubbard, Coler and Hendershot as cases he cannot decide [read: ebook p.5]. He writes that "the patent office is crammed with hundreds of 'permanent magnet motors', none of which work, to my knowledge" [read: p.150], and he calls Gray's own explanation "clearly not true" [read: p.67].

Where he departs from Gray. Gray's patents and his 1980 application describe recovery of inductive energy. Lindemann sets that aside ("nothing to do with recharging") and substitutes an ether mechanism. So the "radiant energy" reading is Lindemann's reading of Gray, not Gray's own patented description.

The wider claims follow the same pattern: a real starting point, then a step that leaves out an input. - Heat pumps. A real COP of 8-10 becomes "350 hp, measured on a Dynamometer ... from just 100-sq. ft. of solar collector" with a compressor "less than 20 hp, so this system produced more than 17 times more energy than it took" [read: ebook pp.2-3]. The 17:1 leaves the solar heat out of the input; and 350 hp is 261 kW, 28 times all the sunlight that falls on 9.29 m² at 1 kW/m², while Carnot between a 100 °C collector and 55 °F ground caps the work at about 2.2 kW [calc; arbiter.md]. - Thermodynamics. His 1994 article says "The line of logic embodied in the 'Laws of Thermodynamics' is flawless" and that for Constable's etheric projectors "There is nothing over-unity about this situation ... all of the energy gain in the system occurs outside of the equipment", then says energy is "apparently appearing and disappearing from the machine ... in the local space" [read: p.145]. The first two statements describe an open system with an outside energy source, which thermodynamics allows; the third is local non-conservation, which would mean Poynting's theorem fails. The text moves between them. His rain example (100,000:1) compares the energy of falling rain with the machine's input, but condensing a million gallons of vapour releases about 2.4 GWh of latent heat, against about 1 MWh of falling-water energy from 100 m: the atmosphere is the reservoir, the machine at most a trigger [calc; arbiter.md]. - Radiant energy as longitudinal and instantaneous. "Longitudinal" contradicts k·E₀ = 0 for free waves in vacuum (section 2); "instantaneous" is a testable claim (a pulse sent 1 km should arrive 3.3 µs later, not at once), and all of radio, radar and GPS engineering measures signal delays that agree with c [memory, high]. The "instantaneous, incompressible" wording comes from Vassilatos ("Should the path be 300,000 kilometers long, the impulse at the source end will reach that point as quickly as at all other points", introduced "According to this view") and from Steiner via Lindemann, not from Tesla's patents [read: p.36; arbiter.md QE revision (g)].

A worked check: Tesla's 471,240 km/s. Tesla's own patent US 787,412 (1905), reprinted in the book, does give a speed above c. It asks for a wave train lasting "not less than one-twelfth or probably 0.08484 of a second", "taken in passing to and returning from the region diametrically opposite the pole over the earth's surface with a mean velocity of about four hundred and seventy-one thousand two hundred and forty kilometers per second" [read: pp.136-137; web: https://patents.google.com/patent/US787412A/en]. Check the numbers (example.py, Part F) [calc]: - 471,240 / 299,792 = 1.572, while π/2 = 1.5708. - 0.08484 s × 471,240 km/s = 39,980 km, about the Earth's circumference (π × 12,742 km = 40,030 km [memory, high, for the diameter]). - One reading that fits these digits: a disturbance crosses the Earth straight through the diameter, 2R, at c, taking 2R/c; the point where it emerges is half a circumference, πR, away along the surface. The surface distance divided by the through-the-Earth time is πR / (2R/c) = (π/2) c = 470,913 km/s. This is a bookkeeping speed along the surface, like the spot of a searchlight sweeping a wall, not a signal travelling faster than light. The patent states no derivation, so this is a reading (first suggested in qe.md, arithmetic confirmed in arbiter.md), not something Tesla says. - The same patent's first condition defines the quarter wavelength by "the velocity of light" divided by four times the frequency: at its lowest frequency, 6 per second, c/(4f) = 12,491 km, close to the Earth's diameter. So the patent uses c for the wave itself and gives a finite, specific time. It does not support "instantaneous" propagation. Its "thousands of times greater" describes the amplitude of the electrical movement in a resonant circuit ("directly proportionate to the inductance and frequency and inversely to the resistance", that is the quality factor Q = ωL/R of section 2), not power from nowhere; Lindemann reads it as "a gain in power" [read: pp.46, 135].

5. Worked example: one discharge, kept as an energy ledger

This example follows one firing of a Gray-type circuit from the battery to the recovery battery, then uses the result to read the one quantitative test claim on record. All of it, and the two checks of printed numbers in section 4 and section 6 (Part F: Tesla's 471,240 km/s and the 1973 EMA table), is reproduced by example.py (run command in its header). It ran cleanly; every number below is from its output.

The patent gives no capacitance, inductance or resistance (only "BATTERIES 12 V" and "18 CAPACITORS 3500 VOLTS EACH" [read: page image p.85]), so the component values are assumptions, chosen to match the mathematician's illustration. Change them in example.py and the ledger still closes; only the percentages move.

Step 1. Energy in the capacitor. E_cap = ½CV₀² = ½ × 2.5×10⁻⁶ × 3500² = 15.31 J.

Step 2. Energy taken from battery 10. E_drawn = E_cap/η = 15.31/0.80 = 19.14 J.

Step 3. Phase 1, the spark. Solve C dv/dt = −i, L di/dt = v − V_a − Ri from v = 3,500 V, i = 0, until the capacitor is empty. Without losses the quarter period would be (π/2)√(LC) = (π/2)√(10⁻² × 2.5×10⁻⁶) = 248 µs and the peak current V₀√(C/L) = 3500 × √(2.5×10⁻⁴) = 55.3 A. The simulation gives 251 µs and 54.2 A, slightly different because of the arc drop and resistance. When the capacitor reaches zero, the coil holds ½LI² = 14.69 J.

Step 4. The arc's share. Charge passed = CV₀ = 8.75 mC. Arc loss = V_a × CV₀ = 50 × 0.00875 = 0.44 J = 2.9% of E_cap, exactly 2V_a/V₀ = 100/3500. This is the check on Lindemann's "99% lost in the voltage drop": the arc takes about 3%.

Step 5. Phase 2, recovery. The arc goes out; the coil current flows through the diode into the 12 V battery: L di/dt = −(V_b + V_d) − Ri. This linear ODE has the exact solution i(t) = (I₁ + a)e^(−t/τ) − a, with a = (V_b + V_d)/R = 25.4 A and τ = L/R = 20 ms. The current reaches zero at t = τ ln((I₁ + a)/a) = 22.9 ms. The charge delivered is Q₂ = ∫₀^t i dt = τ I₁ − a t* ≈ 0.50 C, so the battery receives V_b Q₂ = 6.05 J (the SymPy closed form and the numerical integration agree).

Step 6. Close the ledger (rotor held still, so no shaft work):

Step 7. Net cost of one firing. The batteries lose 19.14 − 6.05 = 13.10 J per firing even before the rotor does any work. Any shaft work W comes out of the same 15.31 J, so it lowers the 6.05 J returned. How much is returned depends strongly on the coil resistance:

Even a resistance-free coil can return at most the coil's 14.69 J, less than the capacitor's 15.31 J, and less still than the 19.14 J the charger drew. Recovery is real and can be large; it is never above 100%. That is Gray's patent in numbers. Note that recovery into a 12 V battery is slow (tens of milliseconds) and lossy when the coil's IR drop is comparable to 12 V, a practical reason a build made to the patent might recover "negligible" energy, as Lindemann says it would [read: p.16].

Step 8. Peak power versus energy (Gray's explanation, and why Lindemann rejects it). The product of the peak voltage and peak current is V₀ × I₀ = 3,500 × 55.3 = 194 kW. But the voltage peaks at t = 0 and the current a quarter cycle later, so that product never occurs. The true peak of v(t) i(t) is V₀I₀/2 = 97 kW (simulation: 95.8 kW), and the energy of the whole pulse is still 15.3 J. "The instantaneous peak apparent power is therefore, also very high" is true and implies nothing about gain.

Step 9. Reading the Crosby/Maize report with the ledger. The Tattler (1 July 1973) reports that J. A. Maize of the Crosby Research Institute, testing for the investor Pan World Enterprises, ran the motor "into a 10-horsepower dynamometer load at 1,100 rpm, a power output of 7,460 watts", on batteries "only capable of 5,454 watts per hour", with "total battery power consumed ... only 26.8 watts per hour", so that "the system will operate continuously for 203 hours", and "measured overall system efficiency exceeding 99%" [web: https://www.rexresearch.com/evgray/1gray.htm]. The 203 hours is a projection, not a measured run: a second text in the same file, apparently from EvGray Enterprises (an interested party), gives the only run length, "a 21-minute run" consuming "only 9.75 watts" (it must mean watt-hours), "this equals 26.8 watts per hour"; it then says the system "operated" for 203 hours, which contradicts its own 21-minute run [web: same, saved copy; arbiter.md QE revision (d)]. This is the only numerical test claim on record; it comes from an advocate tabloid about a test commissioned by an investor, and no Crosby document has been found. - 9.75 Wh in 21 min = 9.75/0.35 = 27.9 W average, not 26.8 W; at 27.9 W the battery would last 196 h, so the 203 h was extrapolated from the 26.8 figure. - 10 hp = 10 × 745.7 = 7,457 W: matches 7,460 W. - 5,454 / 26.8 = 203.5 h: matches the projected 203 h, but only if "5,454 watts per hour" means 5,454 Wh and "26.8 watts per hour" means a 26.8 W average draw. - Read that way, over the projected run: out 7,460 W × 203 h = 1.51 MWh; in 26.8 W × 203 h = 5.44 kWh; ratio 278:1. That is not "efficiency exceeding 99%" (which means out ≤ in). As printed, the report is either a 278:1 over-unity claim or has garbled units. It cannot be both. - The same battery at 7,460 W would last 5,454/7,460 × 60 = 43.9 minutes.

Step 10. What 7,460 W at 1,100 rpm needs per firing. Assume the 27 firing steps per revolution of the patent's Table I (the tested motor may have differed). Firing rate = 1,100/60 × 27 = 495 per second. Shaft work per step = 7,460/495 = 15.07 J, or 5.0 J for each of the three rotor-stator pairs fired at once; a capacitor holding 5.0 J at 3,500 V is C = 2E/V² = 0.82 µF. So the hardware is not absurdly sized for that output. The problem is the input: "26.8 W" allows only 26.8/495 = 54 mJ of net battery energy per step, 0.36% of the 15 J of work each step delivers. Steps 6-7 show that the net battery cost of a firing is at least the shaft work it does (plus losses). For the report to describe a closed system, 99.6% of the shaft work would have to come from somewhere other than the batteries.

What the example shows. Everything Gray patented (charge, spark, repel, recover) works on paper within ordinary physics; recovery of part of the energy is real and can be large with low-resistance coils. The over-unity claims require either an energy input that is not being metered or an error in the numbers. Only a full-cycle measurement (battery energy integrated over a discharge-recharge cycle, including the charger) could show which. That measurement is exactly what the record lacks.

6. Evidence table

Verdicts are arbiter.md's final verdicts. "Interested party" means a source with a stake, as listed at the top.

Gray's later history, as the sources tell it. Lindemann's biography of Gray (p.21, uncited; each item documented in one published source, his book, no second source found yet) says that the third EMA prototype "successfully tested for 32 days straight before it was torn down for analysis"; that the Los Angeles Patent Attorney's Association nominated Gray "Inventor of the Year" by February 1976; that Governor Reagan gave him a "Certificate of Merit"; that he wrote to "every member of Congress ... and the President, Vice President, and every member of the Cabinet" offering the technology for the Strategic Defense Initiative (SDI) and received "not a single reply"; that he built a car with Paul M. Lewis; and that Zetech, Inc. acquired the technology in the late 1970s [read: p.21]. A letter of 11 October 1973 signed "Renate Gray" for EVGray Enterprises told Lindemann that "due to our security we are unable to give out any information about the motor or the company" [read: page image p.13]. The anonymous KeelyNet informant (2000) adds a claimed "1,000 HP" at 10,000 rpm, that "the FBI and others suppressed anything they could", that a promoter "gathered lots of investor money for Gray, with as much as 60 million being placed offshore", and that Gray "worked for NASA as a mechanic" with a Dr. Alexei Popov; each is one anonymous source [web: research/2026-10-06-keelynet-mirror.md]. The $60 million statement cuts both ways: it could describe a promoter's fraud, or money that disappeared. Sources disagree on small facts too: "one of 14 children" (Tattler, 1973) versus "one of six" (NewsReal, 1977) [read: arbiter.md QE revision (e)].

Where the record goes dark (absence-of-evidence analysis, condensed from arbiter.md). Gray had a funded company (about 800 shareholders from 1971, per AP), a Japanese backer, press attention and three patents. After the 1975-76 prosecution the press record stops, no instrumented test is ever published, and the next documents are patents in 1986-87, then a fee lapse on US 4,595,975 on 17 June 1990, two months after the VA death date [web: https://patents.google.com/patent/US4595975A/en]. Competing explanations:

Records that would decide: the Los Angeles Superior Court file for the 1975 case; SEC News Digests 1975-76; USPTO assignment records for the three patents; the Washoe County, Nevada death certificate and an April 1990 obituary; Zetech and Western States corporate records; Internet Archive copies of KeelyNet's 1990-91 Gray files; the Crosby Research Institute report; the Greek patent office (OBI) file for the 16 December 1983 priority of US 4,661,747; the Borderland Labs videos (1987-88) and the "unpublished photographs and unreleased video footage of experiments (1986-1989)" Lindemann lists [read: pp.153-154]. None has been obtained yet.

7. What a learner should take away

Solid. - Capacitor discharge through a spark gap into coils, back e.m.f., and recovery of inductive energy into a battery: real, standard, and what Gray's patents actually claim. Recovery can return a large fraction with good coils (section 5, Step 7). - Gray's motor geometry is internally consistent (13⅓° steps, 27 per revolution). - Lindemann's rejection of Gray's "peak apparent power" explanation is correct: peak power is not energy, and you cannot multiply one source's voltage by another's current. - Heat pumps really do deliver several times more heat than the work they use (COP 8-10 at small lifts is plausible). "Small input controls large output" is a real category: amplifiers, relays, heat pumps. - Short high-voltage impulses really behave differently from mains current in the body and in lamps; Tesla's observations of this are real.

Wrong, and why the mistakes are instructive. - "99% lost in the voltage drop." Confusing voltage drop with energy loss. The arc's loss is V_a Q, a few percent; the rest of the ledger has to be found elsewhere. Always ask: energy = voltage × charge, and which charge? - Ratios with part of the input in the denominator. Lindemann's 100:1 (output over the arc loss), the solar 17:1 (output over the compressor, leaving out the sun) and the Constable 100,000:1 (rain over the machine, leaving out the latent heat) all share one fault: the boundary of the ledger is drawn to leave an input outside. This is the single most useful lesson of the project. - Garbled units hide the real claim. "Watts per hour" lets one article say both "99% efficiency" and 278:1. Insist on W for power and Wh or J for energy, then check that the numbers close. - Longitudinal, instantaneous waves. ∇·E = 0 in vacuum forces transverse free waves; finite c is measured every day. The words "longitudinal" and "scalar" point to real near-field and plasma physics, which does not carry energy faster than light. Tesla's own "471,240 km/s" shows why a number needs its context: read as the speed of a point moving along the Earth's surface, (π/2)c is ordinary geometry, not a faster-than-light signal (section 4).

Genuinely open. - Whether the Crosby/Maize report exists, what units it used, and whether any run longer than 21 minutes was documented. - Whether the Canadian and Japanese family members of the motor patent (CA 1030600, JP S50-74712) contain measured data. (The saved 1980 application text has none: only 24 V and about 2000 V.) - Why Gray filed first in Greece: US 4,661,747 prints a priority of "Dec. 16, 1983 [GR] Greece 24388" [read: p.117], while Google Patents lists the Greek documents as GR73274A / GR72864B [web: https://patents.google.com/patent/US4661747A/en]. Who the applicant was and what the Greek filing says are unknown. - Who Dr. Norm Chalfin was, who took the 1973 EMA measurements, and what Wood and Hackenberger said later, if anything. - What happened to the 1975 charges, and why Lindemann's account (one charge, dropped; 1976 plea) differs from the AP's twelve counts. - When and how Gray died (1989, 1990 or 1993; no death certificate seen). - What present-day experimenters on the groups.io EVGRAY forum and Reddit r/plasma_pi measure on the simplified tube circuit (not visited; no result claimed or denied here). - What Naudin measured on the MEG, and whether DePalma's N-machine or the Dollard coil ever had an independent measurement. - How the 1989 cold-fusion replications were handled and funded (not examined here).

The deciding experiment is cheap by research standards: build the tube circuit (with trained supervision), integrate battery voltage times current over a full discharge-recharge cycle including the charger, measure the spark gap's heat with a calorimeter, measure energy into a resistive load and into the second battery, and run a dummy battery as a control. Net battery energy rising across a cycle is the only result that would matter.

8. Check yourself

1. A 1 µF capacitor charged to 3,000 V discharges completely through a spark gap whose arc drop is 30 V. How much energy does the capacitor hold, and how much of it is lost in the arc?
2. A battery of 5,454 Wh drives a 7,460 W load. How long does it last? If the battery instead lasts 203 hours, what average power must it be supplying, and what output-to-input ratio does that imply?
3. In an ideal LC discharge (C = 2.5 µF, V₀ = 3,500 V, L = 10 mH), why is V₀ × I₀ not the peak power, and what is the actual peak power?
4. Lindemann reports a heat-pump engine producing 350 hp from 100 sq ft of solar collector. Using 1 kW/m² for full sunlight, how many times the incident solar power is that? What is the most a Carnot engine could deliver between a 100 °C collector and 55 °F ground?
5. Show from ∇·E = 0 that a plane electromagnetic wave in vacuum cannot have its electric field along its direction of travel. Where do longitudinal electric oscillations exist?

9. Further reading

Mainstream. - D. J. Griffiths, Introduction to Electrodynamics: Chapter 2 (energy of a capacitor), Chapter 7 (inductance, energy in magnetic fields), Chapter 8 (Conservation Laws, Poynting's theorem), Chapter 9 (Electromagnetic Waves). - D. V. Schroeder, An Introduction to Thermal Physics, Chapter 4 (Engines and Refrigerators): Carnot limit and heat-pump COP. - P. Horowitz and W. Hill, The Art of Electronics: inductors, flyback, diodes, measuring pulsed signals. - E. M. Purcell and D. J. Morin, Electricity and Magnetism: a second view of fields, energy and circuits. - F. Frungel, High Speed Pulse Technology (Academic Press, 1965), cited by Gray's own patent as a reference [read: p.74]: the engineering of capacitor discharges and spark gaps. - N. Tesla, Experiments with Alternate Currents of High Potential and High Frequency (lecture, 1892) [memory, high]: the original high-frequency and impulse demonstrations behind the "radiant energy" story.

The inventors' own work and the record. - P. A. Lindemann, The Free Energy Secrets of Cold Electricity (Clear Tech, Inc., 2000), in the project folder; Chapters 3-4 (PDF pp.39-70) carry his argument. - P. A. Lindemann, The World of Free Energy (web essay, 1 March 2001), in the project folder. - P. A. Lindemann, "Thermodynamics and Free Energy" (Journal of Borderland Sciences Research, Q3 1994) and "Tesla's Self-Acting Engine" (Q3 1995), reprinted in the book, pp.141-152. - E. V. Gray, US 3,890,548 "Pulsed Capacitor Discharge Electric Engine" (1975); US 4,595,975 "Efficient Power Supply Suitable for Inductive Loads" (1986); US 4,661,747 "Efficient Electrical Conversion Switching Tube Suitable for Inductive Loads" (1987); all reprinted in the book and on Google Patents. - N. Tesla, US 685,957 and 685,958 (1901), the "radiant energy" patents [web: https://patents.google.com/patent/US685957A/en]. - Rex Research Edwin Gray dossier: Tattler articles (1973-74), the AP wire of 7 July 1975, the 1980 application text, the Crosby/Maize and EvGray "21-minute run" texts [web: https://www.rexresearch.com/evgray/1gray.htm]. Advocate compilation; read with the stakes in mind. - G. Vassilatos, Secrets of Cold War Technology (1996), Chapter 1, excerpted in the book as "The Rosetta Stone": Lindemann's secondary source on Tesla. Its one named primary source, Tesla's lecture "Dissipation of Electricity" (1892), is the document to check its experiments against. - N. Tesla, US 787,412 "Art of Transmitting Electrical Energy Through the Natural Mediums" (1905), reprinted in the book pp.135-139: the 471,240 km/s passage [web: https://patents.google.com/patent/US787412A/en].

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Answers

1. E = ½CV₀² = ½ × 10⁻⁶ × 3000² = 4.5 J. Arc loss = V_a × CV₀ = 30 × 0.003 = 0.09 J, which is 2V_a/V₀ = 2%.
2. 5,454/7,460 = 0.731 h = 43.9 min. Lasting 203 h means an average draw of 5,454/203 = 26.9 W. Output/input = 7,460/26.9 ≈ 278:1, which cannot be an efficiency "exceeding 99%"; it is a claim of 278 times more energy out than in.
3. v(t) = V₀ cos ωt peaks at t = 0, while i(t) = I₀ sin ωt peaks a quarter cycle later, so the two peaks never coincide. The power v i = V₀I₀ sin ωt cos ωt = ½V₀I₀ sin 2ωt peaks at ½V₀I₀. With I₀ = 3500 × √(2.5×10⁻⁶/10⁻²) = 55.3 A, V₀I₀ = 194 kW and the true peak is 97 kW. The energy of the pulse is still ½CV₀² = 15.3 J.
4. 100 sq ft = 9.29 m², so at most 9.29 kW of sunlight. 350 hp = 350 × 745.7 = 261 kW, 28 times the sunlight. Carnot: 1 − 286.5/373 = 0.23, so at most 0.23 × 9.29 ≈ 2.2 kW (about 2.9 hp).
5. For E = E₀ exp[i(k·r − ωt)], ∇·E = i k·E₀ exp[...]. Setting it to zero in vacuum (no charge) gives k·E₀ = 0, so E₀ is perpendicular to k, the direction of travel. Longitudinal electric oscillations exist where there is charge: in plasmas (plasma oscillations) and in conductors, and as the non-radiating near field around sources.