A design equation fits some body of data. Otherwise it is useless, either for engineering or for producing insights for further experimentation.
E = MCsquare is a design equation. It helped make Einstein justly famous. However, the existence of a design equation does not prove that any particular theory is "true". Some assume that because Einstein based E = MCsquare on Relativity, Relativity has physical meaning. A derivation of this design equation based on Relativity does NOT prove that Relativity has physical meaning. This point can be made clear by showing that E = MCsquare can be derived without invoking Relativity:
(calculation using Doppler shift formula)
* Momentum of particle = mv
* momentum of light-pressure = 0.5 E/c
* Energy of light taken as being proportional to its frequency
* Newton's law of the conservation of momentum
* the Doppler shift formula, f'/f = (c-v)/c
A particle travels forwards at v m/s inside a box. When it hits the front wall of the box, it gives the box a forwards momentum of mv. We'll assume that the mass of the particle is negligible compared to the mass of the box, so that the box velocity hardly changes after the impact, so that we can ignore relativistic acceleration effects.
The same particle is traveling forwards at the same speed, but this time it emits a burst of energy E as two plane waves each with energy E/2, one traveling "forwards" and one traveling "backwards" along the particle's path. The speed of the particle is unchanged, because the reaction forces of both plane-waves cancels out.
However, in the frame of the box, the two waves do not have the same energy. The forward one is blueshifted by
f'/f = (c-v)/c
and the rearward one is redshifted by
f'/f = (c-(-v))/c
and the energies of the waves hitting the box are therefore altered by the same amount. The combined momentum of the two waves striking the opposite sides of the box is therefore:
E(c-v)/2cc - E(c+v)/2cc = E(c-v-c-v)/2c² = Ev/c²
so the effect of emitting those two pulses of light is to give the box Ev/c² of momentum.
If the particle and its light have the same total momentum at the end of the experiment as in [#1], then the momentum of the particle after giving off the light must be reduced by Ev/c².
If the box is large enough compared to the amount of radiation, then the amount of recoil can be negligible, and the relative velocity of the particle and the box after the radiation has been absorbed can still be taken as being v, to as many decimal places as you like. We can now express the reduced remaining momentum of the particle in terms of a reduced mass, since
mass = momentum/velocity
mass1 - mass2 = momentum1/v1 - momentum2/v2
Since both velocities are indistinguishable,
mass1 - mass2 = (mom1 - mom2) /v
masslost = (Ev/c²) /v
masslost = E/c²
In order to lose an amount of inertial mass m, a particle with a fixed velocity has to lose an amount of energy equal to
E = mc²
This calculation gives us the energy associated with a particular mass without worrying whether this "mass" value depends on speed - the relative speed of box and particle is supposed to be held (effectively) constant throughout the experiment.
At this point, it should be fairly obvious that the E=mc² result isn't unique to SR - it actually turns up under a range of different theories, via at least two different shift equations.
Einstein published the equation as a consequence of his 1905 "... Electrodynamics ..." paper in his followup, "Does the inertia of a body depend upon its on its energy-content?", but it's entirely possible that he had already spotted the relationship from his earlier work on emitter-theory, and had held the result back until was satisfied that he had a consistent-looking theory that could use it."
For Further Information:
Eric Baird: E = m c²
maintained by Eric Baird, who also gives the following information:
"* A number of Einstein's contemporaries got extremely close to stating the E=mc² relationship, but held back from presenting it as a serious result (and/or got the proportionality slightly wrong). For anyone who is interested in the historical nitty-gritty, there's a paper by W.L.Fadner ("Did Einstein really discover ''E=mc²''?", Am.J.Phys. , 2, 114-122 (February 1988) ) that sifts through some of the claims and counter-claims."
Another design equation for charged particles moving near the speed of light is:
KE = mv^2/2 + q^2v^2k/3r
derived by Dr. Paul M. Brown,
"An Alternate Interpretation Of Mass-Gain
At Near Light Velocities," Infinite Energy,
Vol. 3, No. 13 and No. 14, 1997, pages 52-53.
where r = radius of the charged particle, q = the charge, and v = the velocity of the particle. k = 3.336 x 10^-4. m is the gravitational or rest mass of the particle. This design equation also fits the data very well. Dr. Brown writes that a 1 GeV electron has a rest mass of 9.107 x 10^-31 kg and a velocity of 2.9999994 x 10^8 m/sec.
KE = mc^2 = 1.294 x 10^-10 Joule.
KE = q^2v^2k/3r = 1.364 x 10^-10 Joule.
Dr. Brown's equation apparently contributed to the invention of his resonant nuclear generator (US patent # 4,835,433), an apparatus for the direct conversion of radioactive decay energy to electrical energy; so this is a demonstrably useful design equation.
EINSTEIN AFTER SEVEN DECADES
Why after almost eighty years do we still need to test Einstein's theory of general relativity? The answer is that although it is among the most brilliant creations of the human mind, weaving together space, time, and gravitation, and bringing an understanding of such bizarre phenomena as black holes and the expanding Universe, it remains one of the least tested of scientific theories. General relativity is hard to reconcile with the rest of physics, and even within its own structure has weaknesses. Einstein himself was dissatisfied, and spent many years trying to broaden his theory and unify it with just one other branch of physics, electromagnetism . Modern physicists seeking wider unification meet worse perplexities. Above all, essential areas of general relativity have never been checked experimentally.
Gravity, Relativity, and the Speed of Light
But surely Einstein's ideas must have been checked by now. Does not everyone believe that E=mc2? Yes, indeed; but Einstein advanced two distinct theories of relativity, the special and the general theory.
Special relativity weaves space and time together but does not touch gravitation. It states that no signal can be propagated faster than the speed of light. It leads to E=mc2 and to phenomena such as changes in the mass and shape of a body with velocity, and changes in clock-rates seen by different moving observers. These predictions are verified every day in particle accelerators and nuclear power stations.
General relativity stands otherwise. This is Einstein's theory of gravitation. Once given special relativity Einstein faced an awkward problem. No signal can travel faster than light, but in Newton's time-honored theory of the Universe, gravity is a force transmitted instantaneously over vast distances. Something must be wrong. After ten years, Einstein produced in 1916 a new theory of gravity, interpreted not as a force but as a "field" distorting space and time. The planets in their courses, which seem to us to be moving in elliptic orbits around the Sun, are in reality following straight lines ("geodesics") through curved space-time.
Einstein's Two-and-a-Half Tests
Different as Einstein's and Newton's theories are, within the solar system their results are almost identical. Only on a cosmic scale, or near extremely dense objects such as black holes, does general relativity bring large changes. Einstein in 1916 could only think of three potential manifestations of general relativity, all minuscule.
* perihelion precession: Mercury's orbit around the Sun should gradually turn in its plane through an angle minutely different from Newtonian prediction -- an effect called perihelion precession.
* starlight deflection: Stars observed near the edge of the Sun should appear slightly displaced outward from their normal positions.
* gravitational redshift: Light leaving a star should change color slightly, shifting toward the red.
For over forty years, these three effects -- weak both in what they tested and in how well they tested it -- were all there was. Starlight deflection proved frustratingly difficult to measure. Mercury's orbit, though better, was subject to Newtonian disturbances. Least satisfactory was the redshift, which was observationally messy and hinged on the assumption (the "Einstein equivalence principle") far short of general relativity. This was at most a half-test.
Worse, competing theories soon appeared giving the same predictions for Einstein's tests of general relativity.
New technologies and Negative Experiments
The 1960s began a new era in experimental relativity, exploiting new technologies -- radar, lasers, inertial instrumentation, hydrogen maser clocks, space. Einstein's tests have been tightened, other tests proposed, and, unexpectedly, a special circumstance has produced experiments of a new kind that do discriminate between general relativity and some of its rivals.
General relativity is a minimalist theory. Its assumptions are few, and (more remarkably) often where other theories predict a non-Newtonian effect, it yields nothing. The theoretical log-jam can be broken by negative experiments -- searches for phenomena that are absent from general relativity and Newtonian gravitation but present in competing theories. An example is the Nordtvedt effect, a hypothetical 28-day non-Newtonian oscillation in the Earth-Moon distance as the two bodies orbit each other in the Sun's gravitational field. Limits on this effect have been set by bouncing laser beams from the Earth off retroreflectors planted on the Moon by the Apollo astronauts. The resulting measurements have demolished several theories.
The Problem of General Relativity and the Need for Further Tests
The demolition work from the negative experiments, valuable as it is, does not prove general relativity. If one asks for positive evidence, the story is in one view much better than it was, in another distinctly unsatisfactory.
The Einstein tests seem secure. The redshift has been confirmed -- notably in the elegant NASA program Gravity Probe A. The perihelion data have been strengthened, and supplemented by evidence from an astrophysical object, the Taylor-Hulse binary pulsar (though other astrophysical data from eclipsing binary stars conflict). Starlight deflection is established, while a closely related new test, the Shapiro time delay experiment, based on radar ranging measurements to planets and spacecraft, has been executed very precisely.
All of this indicates (what few physicists doubted) that Einstein was on the right track. Other more profound phenomena, however, remain untested. Save for some indirect evidence from the binary pulsar, no data exist on gravitational radiation. Even less is known about a vitally important relativistic effect -- "frame-dragging."
Moreover, deep theoretical problems -- some old and some new -- remain. Einstein himself remarked that the left-hand side of his field equation (describing the curvature of space-time) was granite, but that the right-hand side (connecting space-time to matter) was sand. The mathematical structures of general relativity and quantum mechanics, the two great theoretical achievements of 20th century physics, seem utterly incompatible. Some physicists, worried by this and by our continued inability to unite the four forces of nature -- gravitation, electromagnetism , and the strong and weak nuclear forces -- suspect that general relativity needs amendment.
One obstacle to creative amendment, however, is the paucity of experimental evidence. How will Gravity Probe B contribute to meeting the need for deeper tests of Einstein's wonderful but troubling theory?
Directionality in Space-Time: Riddles of Relativity
TWO EXTRAORDINARY PREDICTIONS OF GENERAL RELATIVITY
Gravity Probe B is designed to reveal -- and check with high precision -- two extraordinary consequences of general relativity, as seen by gyroscopes.
What is a gyroscope ? The first, invented in 1852 by the French physicist J. B. L. Foucault, was an instrument for studying the Earth's rotation by means of a freely suspended flywheel. Since then gyroscopes have found many applications, especially in navigation, and many types exist. The ones for Gravity Probe B are not flywheels but electrically supported spheres, spinning in a vacuum. Others utilize the spins of atomic nuclei, circulating sound waves, even circulating laser beams. In all gyroscopes the underlying principle is that rotating systems, free from disturbing forces, should stay pointing in the same direction in space.
But what does "the same direction in space" mean? For Newton the answer was easy. Space and time were absolutes. A perfect gyroscope set spinning and pointed at a star would stay aligned forever. Not so for Einstein. Space-time is warped -- and may even be set in motion by moving matter. A gyroscope orbiting the Earth finds two distinct space-time processes -- frame-dragging and the geodetic effect -- gradually changing its direction of spin.
Frame-dragging: Measuring the Rotation of Space-time
In 1918, two years after Einstein formulated general relativity, W. Lense and H. Thirring calculated that according to the theory a rotating massive body should slowly drag space and time around with it!
Startling and far-reaching as Lense & Thirring's discovery was, any verification of frame-dragging seemed hopeless. Nothing happened until 1959 when Leonard Schiff of Stanford University (and independently George Pugh of the Defense Department) considered orbiting gyroscopes. On Schiff's calculations a gyroscope in polar orbit at 400 miles should turn with the Earth through an angle amounting after one year to 42 milliarc-seconds.
This vitally important frame-dragging effect has never been seen. Gravity Probe B will measure it to a precision of 1% or better.
The Geodetic Effect: Measuring the Curvature of Space-time
According to Einstein the Earth warps space-time. A second, much larger change in spin direction, the geodetic effect, follows from the gyroscope's motion through this space-time curvature. The phenomenon was foreshadowed in 1916 by W. de Sitter who predicted a minute relativistic correction to the complicated motions of the Earth-Moon system around the Sun -- an effect finally detected in 1988 through an elaborate combination of lunar ranging and radiointerferometry data. For a gyroscope the predicted effect is a rotation in the orbit-plane of 6,600 milliarc-seconds per year -- quite a large angle by relativistic standards.
Gravity Probe B will measure the change to 1 part in 10,000 or better, the most precise qualitative check yet of any effect predicted by general relativity.
Gravity Probe B: A Different Kind of Experiment
The Gravity Probe B experiment comprises four gyroscopes and a reference telescope sighted on HR8703 (also known as IM Pegasus), a binary star in the constellation Pegasus. In polar orbit, with the gyro spin directions also pointing toward HR8703, the frame-dragging and geodetic effects come out at right angles, each gyroscope measuring both. What do the two measurements signify, and how does Gravity Probe B differ from all previous tests of general relativity, positive or negative?
First, Gravity Probe B contrasts with earlier tests (redshift measurements apart) in being a physics experiment, not a disentangling of complex phenomena in stars or the solar system. Events are under the experimenters' control; disturbing effects are eliminated rather than calculated out; exact calibration checks can be performed on orbit to authenticate the results.
Second, Gravity Probe B supplies two new, very precise tests of relativistic effects on massive bodies. Relativity experiments form three groups, based respectively on clocks, electromagnetic waves, and massive bodies. Amazingly, except for the possible radiation drag in the binary pulsar, there is still only one secure positive result with massive bodies -- perihelion precession. Yet such tests are crucial in exploring the differences between Einstein's and Newton's dynamics. Compare, for example, starlight deflection with the geodetic precession of a gyroscope, two effects often bracketed together since both check the curvature of space-time. Starlight deflection follows from the electromagnetic theory of light plus a special limiting case of Einstein's equations. The gyroscope effects, both frame-dragging and geodetic, follow from the conservation laws for massive spinning bodies derived from Einstein's full field equations -- a critical element in the theory.
Third, most important, Gravity Probe B investigates the gravitational action of moving matter. Matter moving through space-time can be thought of as creating a new force -- gravitomagnetism -- which John Wheeler, dean of relativists, describes as being "as different from ordinary gravity as magnetism is from electricity." The frame-dragging measurement detects this force and fixes its scale. Commenting on its unverified status, Wheeler has said "It is hard to imagine a science so exposed for lack of evidence on a force so fundamental to the scheme of physics."
Frame-dragging and Grand Unification
The frame-dragging effect, small as it is for the Earth, reaches far. It may underlie processes that generate vast amounts of power in distant quasars; it may clarify a strange physical hypothesis called Mach's principle. Above all, it may throw light on grand unification . Grand unification is the greatest challenge confronting theoretical physicists today. Gravitation, the strong nuclear forces, and the partially unified electro-weak forces must be connected, but how? Even the issues remain speculative but several clues suggest that general relativity may require amendment, and that the amendment, in the words of Nobel laureate C. N. Yang "somehow entangles spin and rotation." Says Yang: "Einstein's general relativity theory, though profoundly beautiful, is likely to be amended.... That the amendment may not disturb the usual tests is easy to imagine, since the usual tests do not relate to spin[i.e. frame-dragging]. The Stanford experiment is especially interesting in that it focuses on the spin. I would not be surprised at all if it gives a result in disagreement with Einstein's theory."
But to measure this extraordinary effect an extraordinary gyroscope is needed.
Mach's Principle and Other Mysteries
Is gravity a large or a small force and are the Sun and the Earth large or small bodies?
For the solar system -- and ordinary human experience -- gravity is all-important. Launching a spacecraft from Earth demands a huge expenditure of energy. Yet, compared to electricity, gravity is unbelievably weak. The gravitational attraction between two electrons is 10-39 (1/1,000,000,000,000,000,000,000,000,000,000,000,000,000) of their electrical repulsion. Gravity dominates the solar system only because the forces from positive and negative electrical charges balance out.
But what is large? On a galactic scale the Sun and the Earth are tiny. More profoundly, they are small bodies for general relativity. The space-time curvature they produce is minute: a part in a billion for the Earth and even for the Sun only a part in a million. That is why frame-dragging and starlight deflection are small.
Larger bodies or ones of greater density (neutron stars, black holes) offer more. Astrophysicists suspect that frame-dragging from a supermassive black hole explains why certain remote quasars eject violent opposed jets of radio power aligned over vast distances. The object NGC6251 is estimated to emit 1038 watts -- a raging inferno equivalent in power to a million million suns.
Frame-dragging is thought to influence many astrophysical processes, yet even its existence remains unverified. To observe it directly, and to check it exactly, a known source is needed -- the rotating Earth with its drag of 42 milliarc-seconds/year.
Ever since Newton's time physicists and philosophers have pondered the meaning of inertia. Why do bodies which have mass resist acceleration? Why does a gyroscope stay aligned? Newton framed answers in terms of absolute space, but Bishop Berkeley in 1710, and later Ernst Mach, argued that the local properties of matter might originate in the actions of the other distant bodies. That is to say, the stars create our inertia.
Mach's principle, as it is called, charmed Einstein. General relativity incorporates hints both for and against this hypothetical bridge from large to small. Against it is the mathematical allowability of "Godel's Universe," a solution to Einstein's equations which (to Einstein's annoyance) flouts the principle without being logically excludable. Supportive of Mach's principle (if Godel's difficulty were removed) is the phenomenon of frame-dragging. If frame-dragging from the Earth turns a gyroscope through a small angle, then perhaps a like frame-dragging from all the other matter in the Universe would keep the gyroscope locked on a star. For that speculation to work, however, the Universe must be closed -- must in effect behave as if it were a black hole with ourselves inside it.
To many physicists, Mach's principle smacks of mysticism. Other links from large to small exist, however, stretching physics beyond general relativity. One lies deep within the atomic nucleus. Physicists find that three fundamental quantities -- Newton's gravitational constant, the velocity of light, and Planck's quantum of action -- combine to form an intrinsic natural yardstick, the Planck length. The Planck length has the weirdly small value of 10-33 cm; it is small compared to an atomic nucleus as is the nucleus compared to the Earth. Physics near the Planck length must differ greatly from anything we know; with gravitational and quantum phenomena intertwined, and gravity again becoming the dominant force.
Much remains to be done before the role of gravity in connecting large with small is understood -- not least the experimental validation and calibration of frame-dragging. For Further Information:
Eric Baird: E = m c²
Riddles of Relativity
Infinite Energy Magazine Online - Cold Fusion and New Energy Technology
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E=mc^2 from ELECTRO MAGNETIC QUANTUM GRAVITY