Well, Shenma R, here is a sample of the types of answers i use to give here before this category was taken over by the UFO and Moon landing hoax nuts and I stopped coming by here. It is getting better, so I am slowly coming back.
Anyway, he is an answer I gave a few months ago. . .
Why does E=mc²?
I once wrote a series of post on this and in that series, I started with Aristotle, this time I will be a bit shorter.
We will start with the Michelson-Morley experiment.
The laws of motion, set down by Newton depended on the assumption that absolute motion existed, but, everything is in motion, the sun, the earth, heck we can even measure continental drift.
In the late 1800 the big thing is scientific circles was the “Luminiferous Ether” Light, at that time, was considered a wave, (although the debate was still on and the particle theory still had it’s supporters) and if there is one thing everyone agreed on, it was that a wave need something to travel through. Water waves traveled through water, sound waves traveled to almost any medium. If light was a wave, it had to travel through something. Ether filled every bill.
Ether had been around in theory since Aristotle, and while by the 1700 most of Aristotle’s theories had fallen to pieces, ether was still around.
You see, everyone (by then) knew that the Earth was bound to the sun by gravity, but come to think of it, How? If you wish to exert a force on something, you have to push it, pull it, lift it, throw a stick at it, or tie a rope to it, but there must be some physical connection between you and the object. How does gravity reach out across 93,000,000 miles of vacuum and still exert a pull on earth. Ether was the answer, gravity traveled through the ether and pulled on all the planets. Ether has no true substance, no mass, no matter, it just existed as a conduit for gravity.
Exactly what ether’s properties were could not be shown through direct observation, for it could not be directly observed, it was not matter or energy, for when ever only ether was present, all we could detect was a vacuum. At the same time, ether (whatever it was) was to be found not only in empty space, but permeating all matter. During an eclipse of the sun, when there was 2000 miles of moon between the earth and the sun, there was no lessening of gravity, so the moon, and all matter must be saturated with ether.
Furthermore, ether didn’t interfere with the movements of the planets. Planets moved through ether like it wasn’t there. It seemed that matter and ether didn’t interact at all. Ether could conduct forces, but was not subject to them at all.
This meant that ether was not moving. How could it move unless some force was acting on it?
In 1801 an Englishman, Thomas Young, showed that it was possible to combine two rays of light in such a way as to get alternating bands of light and darkness. This was the first observation that showed light doing something that a particle could never do (How can 2 particles combine to make no particles?), but which waves could easily do.
You see, at this time, there was an argument in scientific circles around the nature of light, was it a wave, or was it a particle?
With Young, the particle theory of light was doomed. It’s advocates put up argument after argument, but, there was little they could do, when it was shown that everything the “light particle” could do, the light wave could also do. (Remember, this was a century before quantum theory and the duality of light.)
Ultimately, the wave theory won, and then began the argument of what was waving (see last post about the property of waves, they need an agency to wave through). Light went through a vacuum with the greatest of east, and a vacuum consisted of nothing but ether, but if ether was the medium that light traveled through then how could you explain the differences between light and gravity. You couldn’t bend, split, reflect, or absorb gravity, while you could do it with light very easily. Was it possible that there was two ethers, one to conduct light and one to conduct gravity?
The question was never answered, but in the 19th century, light was far more important to the development of theoretical physics than gravity was, and it was the light carrying ether that was under constant discussion. Physicist referred to it as “luminiferous ether”.
Albert Michelson was obsessed! I mean we all have cravings and desires, but this guy was obsessed. His obsession? The Speed of Light. Ole Christensen Roemer has shown, way back in 1676 that the speed of light was finite. He proved this by observing the eclipses of the moons of Jupiter, and even came up with a figure of 140,000 miles per second (not bad considering the instruments and measurement he had at the time).
Since then the speed of light had be refined, redefined, measured, re-measured and included in a cookbook. The figures kept getting better and better as instruments got better and better, but Michelson wanted to know exactly what it was. His attempts to measure the speed of light led him to develop the interferometer, an exceptionally useful device.
The interferometer is a device which split a light beam in two, sent the parts along different paths at right angles to each other and then brought them back together again. The two beams were made to travel the same distance, and, presumably, take the same time in their travels, it this was the case, then the reunited beam would have the same properties as the original beam.
In 1887, Michelson and Edward Morley carried out a very careful and difficult experiment at the Case School of Applied Science (now Case Western Reserve University) in Cleveland. They realized that because the earth orbits the sun at a speed of nearly twenty miles per second, their lab itself must be moving at a relatively high rate of speed through the ether. Of course, no one knew in which direction or how fast the ether might be moving with respect to the sun, or whether it was moving at all. But by repeating an experiment at different times of the year, when the earth was in different positions along its orbit, they could hope to account for this unknown factor. So Michelson and Morley set up an experiment to compare the speed of light measured in the direction of the earth’s motion through the ether (when we were moving toward the source of the light) to the speed of light at right angles to that motion (when we were not moving toward the source). To their great surprise, they found the speed in both directions was exactly the same!
You see, if you measured the speed of light in the direction that earth traveled, then you would assume that the measured speed would be the speed of light + the speed of earth’s travel, or c+v.
On the return trip however, the opposite is true, the light is now bucking a head wind that the new velocity is c-v. The time for the return trips is d divided by (c-v).
The total time for the round trip is:
(d/(c+v)) + (d/(c-v))
Combining the terms we get,
(d(c-v)+d(c+v))/(c+v)(c-v) = (dc-dv+dc+dv)/c²-v² = 2dc/c²-v²
Now, suppose that using the interferometer we send a light beam to a mirror in a direction at right angles to the earths motion.
The beam of light is aimed from S (the source) to M (the mirror) over the distance d. However, during the time it takes the light to reach the mirror, the earths motion has carried the mirror from M to M’, so the actual path traveled by the light is from S to M’. This distance, x, and the distance from M to M’ we will call y.
(Unfortunately, I can’t include a really neat drawing here that would show you exactly what I am talking about, but you should get the idea.)
While the light is moving the distance x at it’s velocity c, the mirror is moving the distance y at the velocity of the earths motion. Since both the light and the mirror arrive at M’ simultaneously, the distances traveled must be proportional to the velocities, therefore:
y/x =v/c
or,
y=xv/c
We can solve for the value of x by using the Pythagorean theorem (no, I’m not going into Pythagoras). In the triangle SMM’, substituting vx/c for y:
x² = d² + (vx/c)²
x² - (vx/c)² = d²
x² - (v²x²/c²) = d²
(c²x² - v²x²)/c² = d²
(c² - v²)x² = d²c²
x² = d²c²/c²-v²
x = dc/sqr(c²-v²)
(sqr = square root. There may be better ways to show this, but that is what I am going to use)
The light is reflected from the mirror at M’ to the source, which has moved to S”. Since the distance S’S” is equal to SS’, the distance M’S” is equal to x. The total path traveled by the light beam is 2x, or
2dc/sqr(c²-v².)
The time taken by the light beam to cover this distance at it’s velocity c is:
2dc/sqr(c²-v²) ÷ c = 2d/sqr(c²-v²).
How does this compare with the time it takes light to travel for the round trip in the direction of earth’s motion? Let us divide the time in the parallel case (2dc/(c²-v²)) by the time in the perpendicular case (2d/sqr(c²-v²)):
2dc/c²-v² ÷ 2d/sqr(c²-v²) = (2dc/c²-v²)(sqr(c²-v²) /2d = c sqr(c²-v²) /c²-v² .
Now, any number divided by it’s square root is equal gives the same square root as the quotient that is x/sqr(x )=sqr(x). Conversely, sqr(x) /x = 1/(sqr(x) . So the last equation simplifies to: c/sqr(c²-v²).
This expression can be further simplified if we multiply but the numerator and the denominator by sqr(1/c²) (which is equal to 1/c).
c (sqr(1/c²) /sqr(c²-v²) sqr(1/c²) = c/c / sqr(c²/c²) - v²/c² = 1/sqr(1 - v²/c²) .
As you can see by the formulas, the light traveling perpendicular to the earth motion should take longer then the light traveling with the earths motion to cross the same distance.
Michelson and Morley made herculean efforts to free their equipment from vibrations. They used every means at their disposal to insure that nothing would interfere with their observations. Finally, they sent a beam of light out, split it, rejoined it, and saw...Nothing.
Of course, it might be that the rays of light weren’t heading exactly upwind and downwind, but in such a direction that the ether wind had no effect. However, the instrument could be rotated. They took measurements at all angles— surely the ether wind had to be blowing in some direction. They kept talking measurements all year while the earth itself changed direction as it moved around the sun.
They made thousands of observations, and in July of 1887 they were ready to make their report. The results were negative. They had tried to measure the Earth’s absolute velocity and they failed.
There had to be an explanation of this failure and no less then five of them can be considered for the moment.
1.) The experiment can be dismissed. Perhaps something was wrong with the equipment or the procedure or the reasoning behind it. Lord Kelvin and Oliver Lodge took that point of view.
However, this point of view is not tenable. Since 1887, numerous physicists have repeated the experiment, in 1960, masers were used for this purpose and an accuracy of one part in a trillion was achieved. But, always, the failure was repeated.
2.) Well, the experiment is valid and there is no ether wind for the following reasons.
a) The Earth is not moving. It is the center of the universe and everything revolves around it.
Let’s get real, ok. This would throw everything we ever learned about astronomy out the window.
However, in the interest of proving beyond any doubt that this is wrong, there are plans to duplicate the experiment on the moon as soon as it is feasible.
b) The Earth does move, but in doing so it drags the neighboring ether with it so that it seems motionless compared with the ether at the earths surface.
British physicist George Stokes suggested this, but, this implies that there is friction between the Earth and the ether, and this would raise the question as to why the motions of heavenly bodies weren’t slowing down due to “ether drag”. Stokes’ notion died a quick and painless death.
There are, however two suggestions that survived.
c) The Irish physicist George FitzGerald suggested that all object (and therefore all measuring apparatus) grew shorter in the direction of motion in accordance to a formula which was easily derived.
We’ll get back to that in a few minutes.
d) The Austrian physicist Ernst Mach went right for the heart of the matter. He said there was no ether wind because there was no ether. What could be simpler?
This still doesn’t explain how light could cross a vacuum. (I could say more about Mach, he denied any theory or suggestion the could not be proved by sensory experience (observation) and I have very little good to say about a man who believed that atoms were a convenient fiction.)
OK, so lets look at what FitzGerald had to say:
The FitzGerald contraction (derived from the Michelson-Morley experiment) was one of the formulas that Albert Whatzname used in formulating his general theory of relativity. He suggested that all objects decrease in length in the direction in which they are moving by an amount equal to: sqr(1-v²/c²).
Thus: L’ = L sqr(1-v²/c²) where L’ is the length of a moving object in the direction of it’s motion and L is what the length would be at rest.
The foreshortening fraction √1-v²/c² is just enough to cancel the ratio 1/√1-v²/c², which related the maximum and minimum velocities of light in the Michelson-Morley experiment. The ratio would become 1, and the velocity would of light would seem to out foreshortened instruments and senses to be equal in all directions, regardless of movement of the source of light through the ether.
Lets take a close look at this ratio for a minute. As you can see, in order to have any noticeable foreshortening you have to traveling at an appreciable fraction of the speed of light. Let us work this out for an object traveling at .1c (one tenth of the speed of light or 186,282 miles per second).
L’ = L sqr(1 - 0.1² /1²)
L’ = L sqr(1-/0.01)
L’ = L sqr0.99
L’ = .995L
So, as you can see, even moving at 0.1c, you only have a foreshortening of about half of 1 percent. For moving bodies, velocities such as this only exist in the realm of subatomic particles. The foreshortening of an airplane traveling at 2000 miles per hour would be inconsequential, as you can calculate yourself.
The FitzGerald contraction, while explaining the lack of success in the Michelson-Morley experiment, did, at the time, seem like an excuse and not the major breakthrough it was. He seemed to be saying that nature is conspiring to keep us from measuring absolute motion by introducing an effect that just cancels out any differences we might try to use to detect that motion.
The Dutch physicist, Hendrik Lorentz carried the FitzGerald contraction a bit farther, and used it to show that the mass of an object increases with the motion.
Soon after FitzGerald advanced his equation, the electron was discovered, and scientist began to study the properties of the tiny particle.
Hendrik Lorentz worked out a theory that the mass of a particle with a given charge is inversely proportional to its radius. In other words, the smaller the volume into which a particle crowds it charge, the greater the mass.
Now if a particle is foreshortened because of its motion, its radius in the direction of motion is reduced in accordance with the FitzGerald equation. Substituting the symbols R and R’ for L and L’, we could write the equation:
R’ = R (sqr(1-v²/c²)),
or:
R’/R = sqr(1-v²/c²).
Now, the mass of a particle is inversely proportional to its radius, or: R’/R = M/M’ where M is the mass of the particle at rest and M’ is its mass in motion.
Substituting M/M’ for R’/R in the preceding equation, we get:
M/M’ = sqr(1-v²/c²)
or:
M’ = M/sqr(1-v²/c²).
This, of course, means that the faster a particle travels, the more mass it has. You have the same proportions that FitzGerald worked out or:
V=.1c
L’=-0.995
M’=+0.05
Now, lets see what happens when v=c:
M’ = M/√1-c²/c² = M/√1-1 = M/0
Now, as the denominator of any fraction with a fixed numerator becomes smaller, the value of the fraction becomes larger. In other words, from the preceding equation it would seem that mass of any object traveling at a velocity approaching that of light becomes infinitely large. Again, the speed of light would seem to be the maximum limit.
All this led Einstein to recast the laws of motion and of gravitation. He considered a universe, in other words, in which the results of the Michelson-Morley experiment would be expected.
The FitzGerald and Lorentz equations showed why the Michelson-Morley experiment produced no results, but that still left us with the problem of absolute motion. There was an examiner in the Swiss Patent Office, who, in his spare time like to think about the universe and the ramifications of the new theories which tried to explain the nature of it. He look at the problems caused by the Michelson-Morley experiment, the FitzGerald - Lorentz equations, the photo-electric effect, and the new theory by Planck called the quantum theory. (I could go on about both of them, but you will be able to understand the concepts from what I say about them.) In 1905 he published what he called The Special Theory of Relativity. His name, Albert Einstein.
Lets look at the Lorentz equation for a second. One of the consequences of the Lorentz equation was worked out by Einstein to produce the most famous equation of all time.
The Lorentz equation can be written in the form: M’ = M sqr(1-v²/c²) since in algebraic notion 1/sqr(x) can be written as x^-½. This puts the equation into a form that can be expanded by a formula discovered by (guess who) Newton. The formula is the binomial theorem.
The number of terms that the Lorentz equation can be expanded is infinite, but since each term is smaller than the one before, you can take only the first two terms you are more or less correct, the sum of the remaining terms being small enough to be neglected. The expansion becomes:
(1-v²/c²)^-½ = 1 + ½v²/c² . . . .
Substituting that in the Lorentz equation, we get:
M’ =M(1 + ½v²/c²) = M + ½Mv²/c².
Now, in classical physics the term ½Mv² represents the energy of a moving body. If we let e stand for energy, the equation becomes:
M’ = M + e/c²
or:
M’ - M = e/c² .
The increase in mass due to motion can (M’ - M) can be represented by m, so:
m = e/c²
or:
e = mc²
It was this equation that for the first time indicated mass to be a form of energy. Einstein went on to show that the equation applies to all mass, not merely to the increase in mass due to motion.
Here again, most of the mathematics involved is only at the high school level. Yet, it presented the world with the beginning of a view of the universe greater and broader even than that of Newton, and all it took was a little math, and the inspired genius of Einstein, working on base that began all the way back with Galileo.
Kinna simple, isn’t it.
2007-05-12 19:06:29
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answer #8
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answered by Walking Man 6
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