EINSTEIN’S SPECIAL RELATIVITY FOR THE LAYMANBY

V. E. “BILL” HALOULAKOS (vehaloulakos@att.net)

Aerospace Science Consultant

AIAA National Distinguished Lecturer

Everything in life is relative, including success, and the more the success the more the relatives!

SUMMARY

This paper is a brief discussion of Einstein’s Theory of Special Relativity with a specific aim of explaining it for the layman, without compromising any of the scientific content. If you find the mathematics intimidating, skip it but accept it, and read the narrative. The historical background and the special considerations, that allowed Einstein to arrive at the momentous conclusions that truly revolutionized the entire world, not just the world of physics, are examined and evaluated. Special emphasis is placed on the “simplicity” of the basic concepts once erroneous preconceived notions are removed from the thinking process.

PROLOGUE

Here we are 101 years later and the world is still talking and pondering about Albert Einstein and his Theory of Relativity. This theory, which rendered Einstein as a mythical person, even while still living, has literally and figuratively revolutionized our world. It has made possible technological things such as nuclear energy, lasers and the future possibilities appear endless. Known mostly as being shy and recluse in his latter years, Einstein actually is also known to have taken part in many public events and forums and on numerous occasions was quite influential. It was his famous letter to President Roosevelt in 1939 that set in motion the Manhattan Project that resulted in the development of the Atomic Bomb. On another event, the dedication of the Albert Einstein Medical Center (now known as memorial) in New York while he was still living he was asked and had agreed to be the principal speaker. This was indeed unusual for he was not known to give many or any public addresses. The publication of the event and the speaker resulted in the onslaught of the entire New York social elites to be there. As the time for the main speaker event approached, every one present was “sitting” on needles waiting to hear his famous voice and also note his remarks. He stepped onto the podium, looked around and said, “I just decided I have nothing to say” and promptly sat down! As odd or unusual he might have appeared to be on occasions, his contributions to the field of physics and scientific thinking process, are monumental and of everlasting effect. It is said that he proved Newton wrong but that is not really true. The true statement is that Einstein’s physics cover the areas of the “very large” and the “very small”. In the middle, Newton’s physics is very much alive and well. While it is Newton’s physics that helped us develop our space ships and fly to the other planets, it cannot take us down into very small world of the atom or to the very large distances to the outer galaxies. For example Newton’s physics cannot help us in the

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development of the laser, or the design of a nuclear power generating plant. Let us now take a look as to how some one, like Einstein, happened to develop such revolutionary ideas.

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HISTORICAL BACKGROUND

Let us now go back to the latter years of the 1800s.The Victorian Era still reigns supreme.PAX Britannica Prevails throughout the world.Vienna waltzes to Strauss’ music.The United States is preparing to deploy troops to the Philippines.The general belief was that science and technology are fully developed.This may have been what caused Charles H. Duell, Director of the US Patent Office, to advise President McKinley to close down the Patent Office, because it no longer serves a purpose since: “Everything that could ever be invented had already been invented”!

In the field of physics in the 1860s, James Clerk Maxwell building and expanding the knowledge in the field of electromagnetism, primarily the works of Michael Faraday, had developed his famous equations, named after himself, which predicted the existence of electro-magnetic (EM) waves in the universe and specifically that light was a special class of an EM wave distinguished by the fact that was simply visible to the human eye. Thus, he united the fields of electricity, magnetism and optics into a single field. About 20 years later Heinrich Hertz produced EM or “Maxwellian” waves in his laboratory, which we know today as radio waves. It took Marconi and a few others to fully develop and exploit these new discoveries that brought us radio and telecommunications.Light being the “visible” EM wave had been extensively studied throughout the centuries. Specifically its phenomena of reflection (mirrors) and refraction (lenses) had been studied and utilized commercially. Also its speed of propagation in space had been measured and was determined to be 300,000 km/s (186,000 mi/sec). Maxwell in his EM wave theory proved that all EM waves travel at that speed. Furthermore, experiments with waves in physics laboratories had been mostly done with sound waves. It had been proven conclusively that sound waves travel through a medium (air most commonly and solid objects as well) via elastic collisions between its molecules. The question immediately arose that if light is also a wave then it needs a medium through which it needs to travel. Therefore, the fact that we can see the Sun, the stars and other space objects, it follows that space is NOT a vacuum but it must be filled with something! This “something” was named ETHER through which the Sun’s rays travel, via molecular elastic collisions, as the sound waves do travel through the air.Although the Earth’s orbital velocity around the Sun had been established via Newton’s Orbital Mechanics to be about 30 km/s, it was suggested that now with this “new knowledge” of light waves traveling through the ether it should be possible to conduct experiments with light waves via which the Earth’s speed “through” the ether could be measured. In other words as the Earth speeds through the ether one should be able to measure the speed of this ether wind caused by such motion.Hence, the idea of the famous, or rather infamous, Michelson-Morley Experiment was conceived and plans for its implementation proceeded “at the speed of light”!

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THE MICHELSON–MORLEY EXPERIMENT

A. Preliminary Considerations

As is almost always the case, each new step in scientific discovery makes use of all pertinent knowledge from previous works and discoveries. Light had been extensively studied and its wave nature well documented via its interference and diffraction wave phenomena. The idea of an ether-filled space through which light waves would travel gave impetus to the idea that some well known facts about speed boats moving up, down, and across river streams might be applicable to the problem of light moving “up, down, and across” the ether wind. Consider two boats, as shown in Figure 1, traveling in a river of width D and with a uniformly flowing stream of velocity v. Boat A travels across the river distance D and returns to the same point. Boat B travels down the river, along the flow, the same distance D and then it turns back to its starting point. We proceed to calculate the times of travel for each boat as follows.Boat A, being hit by the cross-stream, will be carried downstream unless it turns its velocity vector into the upstream direction to counter this cross current effect. From the velocity vector diagrams of Figure 1, via the famous Pythagorean Theorem for right triangles, we have the boat velocity V combining with the river velocity v giving us the net crossing velocity in the direction across the river.This then is:

(1)

So, that the actual velocity (speed) with which the boat crosses the river is

(2)

Hence the time for the initial crossing is the distance D divided by the velocity . Since the reverse crossing is exactly the same length of time the total round-trip time tA is twice D/ , or

(3)

The case of boat B is somewhat different. As it travels the same distance D down stream its net velocity is the direct sum of its velocity plus the river velocity, V+v, and while traveling upstream, also distance D, is the simple difference, V-v.

The total time of travel tB is then given by:

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(4)

Using the common denominator (V+v)(V-v) for both terms,

(5)

Thus, the ratio between the two times is

(6)

v

V

v

V

BOAT

V v

V

CROSSING

BOAT B DOWNSTREAM

V

v

V

BOAT B UPSTREAM

Figure 1. River Boat Velocity Diagrams

D

A

B

B

A RIVER VELOCITY

Thus, knowing the common boat speed V of both boats and by measuring the time ratio of equation (6), we can determine the speed of the river v.

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It should be noted that equation (6) tells us that it takes longer to go down and back than across

and back, since . The validity of this had been verified conclusively and it was known to

be a well-established fact.

Therefore, it was postulated, if one could sent light beams, starting from the same point, along and across the Earth’s velocity vectors similar time differences would be observed and upon return the beam that went down and back would arrive a bit later at the point of origin. Therefore, if the nature of these two beams were identical then a precisely known wave interference pattern would be observed. This then, as in the case of the boat example above, would allow the calculation of the “river” velocity which in this case is the velocity of the ether wind, or the velocity of the Earth around the Sun!

B. The Experiment

The first step was to design the equipment to be used. The main instrument was what is now known as The Michelson Interferometer that would direct the light beams in their proper directions and also conduct the measurements. A schematic of this interferometer is shown in Figure 2. The elements of this device are: (a) a half-silvered mirror which can split an incident light beam into two identical half beams, and direct them into two perpendicular directions one across the Earth’s velocity vector, Path A, and one down and up parallel to the Earth’s velocity vector, Path B. (b) Two reflectors A and B which reflect the half beams back to the half-silvered mirror, and (c) an image viewing screen where the light beams return and the image collected and viewed.As the returning half beams return to the half-silvered mirror, they combine into a single beam as the original one emanating from the light source. This total beam is directed to the image-viewing screen where a precisely “known” wave interference pattern appears. So, the expectation was that light beam B would return to the screen later, as per the boat model described above, and thus give this “precisely known” pattern.

EXCEPT THAT NO SUCH INTERFERNCE PATTERN WAS OBSERVED!

The immediate conclusion was that there were mistakes in the experiment and therefore they needed to repeat it and, this time, more carefully. This “more careful” repetition was done over many times, in different locations, sometimes six months later (when the Earth would be traveling in the opposite direction) and by different researchers. But lo and behold, the results were the same! NO INTERFERENCE PATTERN SHIFT!

So, what did happen? How does one explain the apparently erroneous results suggesting that the ether wind velocity, caused by the Earth’s motion through the ether, had absolutely no effect on the light speed as the river velocity affected the boat speed?

OF COURSE, THE EARTH MUST NOT BE MOVING!

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LIGHT SOURCE

Figure 2. Michelson Interferometer

REFLECTOR B

REFLECTOR

IMAGE VIEWING SCREEN

HALF LIGHT BEAM DOWN& UP: PATH B

HALF LIGHT BEAM ACROSS & BACK: PATH A

TOTAL LIGHT BEAM

A

HALF-SILVERED MIRROR

Ve

ETHER WIND

Perhaps, those ancients, proponents of the Ptolemaic Geocentric (read EGO-centric) system, were probably correct, after all! How nice of an explanation that would be! But, of course this easy way out did not fly! Might it be possible that this “ether” may have undetectable properties so its effects could not be measured? That thought of course has no place in science. How can one postulate about something whose very existence cannot be verified? So, the problem persisted. If only those guys Michelson and Morley had left well enough alone and not done this “troublesome” experiment!The biggest problems seemed to be a set of preconceived notions about stationary vs. moving coordinate systems and their applicability to the study of the newly developed field of wave mechanics as a result of Maxwell’s Equations of EM waves. There is a set of mathematical relations termed Galilean Transformations that were used to handle all problems of Newtonian

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Physics but seemed to have difficulty in handling Maxwell’s EM equations. This can be illustrated by considering the basic relationships between mathematical equations and geometric figures.A straight line is described by a first degree, or order, equation of the type ax +by + c =o, and for this reason this equation is called linear. Likewise second degree geometric curves, e.g., parabolas, circles, etc., are represented by second degree, or quadratic, equations of the type ax2 + by2 + cx + dy + e = 0. There are physical wave phenomena some of which are described by the basic trigonometric function of y = sinx. These wave phenomena are described in mathematical functional terms as follows:

For a function φ (x, t) to be a wave function it must satisfy or be a solution of the partial differential equation given by

(w-1)

where c is the wave velocity. So, it is true that a physical phenomenon that is not described by the above wave equation is not a wave.

The problem arises when the wave equation is transformed from a fixed coordinate system (x , t) to a moving system ( , ) using the Galilean transformations of x = + vt and t = , the resulting equation is of the form (details are given in Ref. 3):

(w-2)

which no longer looks like the wave equation. This means that if one could picture a long vibrating string across a room, as viewed while standing next to it, it would not look like a wave if one chose to observe it while walking along it and that difference would continue to change as the speed of the walk changes. This, of course, fails the simplest experiment. The vibrating string wave looks the same whether one stands next to it, walking or running forward, backward, or sideways!As simple as these concepts may appear in retrospect, at that time the greatest scientific minds seemed hesitant to apply them to the results of the Michelson-Morley experiment because they carried some rather revolutionary consequences which no one had the courage to face. For it was thought those results were counter to some pronouncements and conclusions proposed by none other than Sir Isaac Newton! There was great risk that an occupant of some prestigious university physics chair might damage his reputation.

To put it simply: No one had the courage to state out loud what these troublesome Michelson-Morley experiment results really meant!

C. EINSTEIN’S PRONOUNCEMENTS

Einstein was no ordinary young man. He recalled later in his autobiography that at the age of four the sight of a compass needle that his father showed him fascinated him. His next

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fascinating experience was at the age of 12 when he was exposed to Euclidean plane geometry, which he mastered effortlessly. Between the ages of 12-16 he mastered differential and integral calculus. By 1905 he was a young (26 years old) physicist with all the requisite credentials except a good job or a prestigious professorship. He was a clerk in the Swiss Patent Office, a job not the most exciting and challenging for a genius in physics. As he read the results of the Michelson-Morley experiment and the confusion that had followed it, he proceeded to give his thoughts and opinions on them. Note that, he had no reputation to protect or anything to lose even if he were proved to be wrong. But, he stood to gain a lot if he turned out to be correct!So, he dared tell the world that:

1. They did not find any effects of the ether wind BECAUSE THERE IS NO ETHER!2. The speedboat model upon which the basic hypothesis was based was totally

inapplicable for two reasons. (a) The boat example used the riverbank as a fixed frame of reference. NO SUCH THING EXISTS OUT IN SPACE, (b) Extrapolating from the velocities of the boats (a few miles per hour) to the speed of light (300,000 km/s or 186,000 mi./sec) is a mighty big jump and there is nothing there to suggest that such an extrapolation is valid.

It should be noted here that extrapolations always run the risk of being wrong especially when done over a very large range. E.g.. It would almost be similar to one using the beneficial results from taking one or two aspirin tablets to taking 1000! Why not? After all this is also an ”extrapolation” from the very small to the very large.He then followed with his monumental explanation of the experiment results incorporated into what have now become known as the two postulates of special relativity.

Postulate 1. The Principle of Invariance: The laws of physics are the same on all frames of reference. This simply says that nature is what it is and is not going to change according to the way one chooses to look at it. This is similar to an axiom of Euclid’s plane geometry.

Postulate 2. The Principle of the Constancy of the Speed of Light: the speed of light in vacuum is always a constant equal to c, and is independent of the relative motion of the frame of reference, the source and the observer.This then says that the Michelson-Morley experiment simply measured this fact when it noted that the light beams spent the same length of time traveling across the Earth’s velocity vector and back as they did traveling down and back. That is the reason that the expected interference pattern shift was not there.

Let us examine the consequences of these very simplistic postulates. We should mention that the conventional way had been to use the Galilean Transformations in combining motion, displacement, velocities and accelerations. According to these transformations when two cars move in the same direction with a constant velocity v, then their relative velocity is zero, i.e. they move together side by side. If they were traveling in opposing directions their relative velocity is 2v. This, however, is not the case when objects move at the speed of light as per Postulate 2! That is to say that if the cars are moving at a velocity c, then their relative velocity will be c no matter if they move in the same or opposing directions!

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Thus, if we have two coordinate systems (frames of reference) one stationary S and one moving at some velocity v relative to S, then by above postulates the displacement in both frames is of the same form.

Therefore, we have

(7)

(8)

We should note here that in the old Galilean transformations these equations would be

(9)

which is in direct contradiction to Postulate 2, a firm experimental fact.

The mathematical problem here is to find the relationships of and in terms of x and t. This is done in the Appendix. The results are the, now well known, Lorentz Transformations named after Hedrik A, Lorentz of Leyden University who had almost developed them while studying Maxwell’s equations of wave mechanics, although Einstein did derive these equations independently:

(10)

(11)

We may also obtain the inverse transformations (from system to S) by replacing v by –v and simply interchanging primed and unprimed coordinates. This gives,

(12)

(13)

So, this indeed is “the bombshell” that Einstein threw into the world of physics! These equations say that time t and displacement x are NOT fixed invariants, as stated in the classical Newtonian Physics, but they depend on the velocity v of the coordinate frame relative to S.

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By way of a quick test these new equations are tested in the transformation of the wave equation, Equa. (w-1), from the fixed frame S to the moving frame . The details of this can be found in Reference 3, the resulting equation is

(w-3)

which is the wave equation hence these transformations are indeed invariant.

It should be noted that for very low velocities, compared to c, these equations reduce to the Newtonian equations.

i.e.; when v is small then and hence we have = x – vt and = t which are the Galilean transformations. In fact we conclude that for low velocities (v<<c and β ~ 0):

Limit (Lorentz transformations) = (Galilean transformations)v→0

And this is the reason that makes our world work fine as described by Newton’s Physics.

Another consequence of the Lorentz transformations is the fact that no velocity faster than that of light c is possible because the factor would be imaginary.

This, however, has not constrained physicists from theorizing that there may be particles that travel faster than the speed of light, termed tachyons or “speedies” or “swift”, which would have imaginary mass at speeds less that the speed of light. Such concepts are not totally alien to the word of physics for we have photons, which are the particles of light, that have mass when they travel, and only at the speed of light. When photons do not travel, i.e., they are at rest, their mass becomes energy E = hν, the minimum quantum energy, h being Planck’s Constant (6.63 x j-s). An analogous situation existed earlier in the field of aerodynamics where a “sound barrier” had been theorized, because drag was tending to infinity at speeds approaching the speed of sound that could never be crossed. Today, of course, we have airplanes and rockets that travel at supersonic speeds and the pertinent theories have been modified accordingly. One could, therefore, envision a similar situation with the discovery of tachyons in which case the following humorous limerick may become fashionable in social salons.

There once was a lady from Bright, who could travel faster than lightShe departed one day, in her relative way and returned on the previous night.

VELOCITY TRASFORMATIONS

As a direct consequence to these new transformations, all the other mathematical operations and physical variables follow accordingly. For example, the velocity equations though still the derivatives of the displacement assume a new form, so the Lorentz form of the velocities is:

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(14)

and the inverse is:

(15)

Again, we refer to the Appendix for the mathematical details.

LENGTH, TIME, AND MASS

Now we turn to evaluate the implications of the Lorentz transformations on the other physical variables, specifically to the concepts of length, time, and mass the three items that have always been invariant under classical Newtonian physics.

A, Length:

Consider two observers, one working on the fixed coordinate system S and the other on the moving system observing the length of a rod. If the two systems are initially at rest and coinciding then both will measure the same length L, or L0 . The observer on S will express the length as L = x2 – x1 and the one on will read it as L0 = 2 – 1. As the system begins to move with a velocity v along the X axis, the observer on will continue to read the length of the rod as L0, but the one on S will read L = x2 – x1, where x2 and x1 must be connected to 2 and 1 via the Lorentz transformations Eqs. (10) and (11). That is,

(16)

(17)

(18)

Since the observer on S will measure both ends of the rod at the same time then t2 = t1, so Equa. (18) gives

(19)

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where , , and is the length as measured by the observer

on S. Thus, we have

(20)

which simply says that length appears to shrink as observed from S, as it speeds up along its length.

B. Time:

On a similar analysis the two observers will also measure time a bit differently. From Equas. (12) and (13) we have

which says that the observer on S reads the clock on and sees ( 2 - 1) and compares it to his own time interval (t2 – t1) and sees

(21)

If we define T0 = 2 – 1 and T = t2 – t1 we have

(22)

Thus a clock appears to slow down by the factor as it speeds up relative to the fixed frame S.

C. Mass:

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Relativity effects on mass are evaluated by considering the conservation of linear momentum and connecting the variables using the Lorentz velocity transformations. From Equas. (14) and (15) we find that the relativity effects also affect the mass. Details can be found Reference 1. The result is:

(23)

(24)

Thus, mass also appears to change, it increases, and it becomes infinite as it approaches the speed of light.

So, in conclusion all the Newtonian Invariants, mass, length and time are no longer invariant but do appear to change as a function of the velocity of the frame !

ENERGY CONSIDERATIONS

This is the particular topic with which Albert Einstein is mostly identified with in the minds of the general public. Just about every one around knows that Einstein invented or developed his famous energy equation E = mc2! There have been numerous cartoons showing Einstein writing in a classroom blackboard with the following caption:

E = ma2 No! E = mb2 No! E = mc2 Yes! That’s it!

This, of course, may be humorous but it is not what happened.

Let us now really see how Einstein came up with this truly monumental conclusion.

Einstein is well known for his thought experiments. One such experiment went as follows:Consider two identical masses bound by a rubber band and compressing between them a spring of stiffness k and a compression distance x, thus having a stored potential energy U = ½(kx2), moving to the right with a constant velocity v. As these masses are passing in front of you imagine reaching in and cutting the rubber band with a pair of scissors. The compressed spring will then release its stored potential energy and the masses will be pushed in opposite directions. Let us now assume that the initial mass velocity v and the stored spring potential energy U are such that the aft mass remains stationary in front of the observer while the front mass is pushed in a forward direction. The question then arises as what is the new velocity of the mass moving to the right. “i.e. Is it 2v or not 2v? That is the question!” If we were to apply the Galilean transformations then that velocity would be 2v, i.e., v initial plus another v

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from the spring compression. If, however, we apply the new velocity addition formulas compatible with the Lorentz transformations, Equa (14), along with the conservation of energy and momentum principles, then it “appears” as if this mass that is moving away to the right has “lost” some mass. The question then is where did this mass go? Perhaps it went to energy? Strictly from the units point of view, for mass to be converted into energy it must be multiplied by the velocity squared, as in the case of the kinetic energy, K = ½(mv2). That then immediately points to some velocity that when squared and multiplied by the “missing” mass will give that extra energy increment. This, of course led to the famous formula E = mc2! The details of this development are given in the Appendix.

EPILOGUE

It is interesting to note a few things about Einstein’s difficult experiences in getting the word out on this monumental development.As it was noted earlier, he was a 26-year-old clerk in the Swiss Patent Office and had no particular standing among the elites of physics. As such, therefore, he had no reputation to risk with such revolutionary pronouncements. He prepared his paper titled “On the Electrodynamics of Moving Bodies”, actually ("Zur Elektrodynamik bewegter Körper") and proceeded to submit it for publication in the very prestigious journal Annalen der Physik, but since he was an unknown his paper was not even considered. Of course, a man of Einstein’s make would not be shut out by some small office clerk. He proceeded to write another paper on the Photoelectric Effect (he was given the 1921 Nobel Prize in Physics for this work) which discussed the particle theory of light (light was a stream of photons) and also connected it to the concepts of Quantum Mechanics, the other new science of Physics proposed by one of the Icons of the Physics World named Max Planck. Einstein sent that paper to Planck who immediately called the physics journal asking that they publish it. Well, now Einstein was “published” and therefore, when he resubmitted the previously rejected relativity paper it was published very promptly. A brief search reveals that both papers were published in the same year, Vol. 17, pp 132 and 549 respectively. Actually Einstein published five papers in the year 1905, his energy, E = mc2, paper coming out in the fall of that year. As the world of physics read these “outrageous” claims, they immediately asked, “who is this guy”? The answer, of course, was “some young Jewish guy who is a clerk in the Swiss Patent Office”! It was Hedrik A. Lorentz of Leyden University, who immediately realized the importance of this article. Lorentz, as mentioned earlier, had originated the new (now named for him in honor and recognition of his work, although it was really Einstein who had developed them independently) coordinate transformations in his attempt to explain the new theory of electromagnetism. But for some reason he did not quite come out “full force” with it. Could it be that his already great reputation might have made him cautious? We may never know for sure!In any event he saw the value of this new science and invited the 26-year-old Einstein to be the keynote speaker at the most prestigious Physics Symposium he was sponsoring at the University of Leyden, and the rest is HISTORY! Einstein’s German citizenship was officially recognized and he was promptly appointed to various prestigious professorships, ending up in 1914 at the university of Berlin where he also

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published the General Theory of Relativity. He served and flourished there until the Nazis came to power in Germany, which caused his coming to America and Princeton University. While he became “mythical” after all that, his Theory of Relativity became the talk of the town and the following saying originated.

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There once were three people named Stein, EP, GERTRUDE and EINEP’s statues were fine, GERTRUDE’s poetry was divine, and no one understood EIN!

Here we may not have done anything about Ep’s statues, or Gertrude’s poetry, but hopefully we have done something about understanding at least some part of Ein!

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APPENDIX

Here we present the derivation of the new set of equations termed, Lorentz transformations, and all the subsequent relations.

LORENTZ TRANSFORMATIONS

We consider two coordinate systems (frames of reference) one stationary S and one moving at some velocity v relative to S, then according to the two postulates of Relativity, stated in the main text, the displacement in both frames is of the same form.

Therefore, we have

(A-1)

(A-2)

We should note here that in the old Galilean transformations these equations would be

(A-3)

which is in direct contradiction to Postulate 2, a firm experimental fact.

Equations (A-1) and (A-2) can be written as

(A-4)

(A-5)

That is,

(A-6)

We are interested in finding and in terms of x and t. That is,

= (x, t) (A-7)

= (x, t) (A-8)

This is accomplished via the formation of two linear simultaneous equations as follows:

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(A-9)

(A-10)

where a11, a12, a21, and a22 are constants to be evaluated. It is required that the transformations are linear in order for one event in one system to be interpreted as one event in the other system; quadratic transformations imply more than one event in the other system.

Solution of problems involving motion begins with an assumption of their initial conditions; i.e., where does the problem begin?

The classical assumption is to set = 0 at = 0. Therefore, according to S, the system appears to be moving with a velocity v, so that x = vt. We can obtain this from Equa. (A-9) by writing it in the form = a11(x - vt) so that, when = 0, x = vt. Therefore, we conclude that a12 = -va11. We can write Equations (A-9) and (A-10) as

(A-11)

(A-12)

Substituting and into Equation (A-6) and rearranging, we get

(A-13)

Since this equation is equal to zero, all the coefficients must vanish. That is,

(A-14)

(A-15)

(A-16)

Solving these equations we obtain

(A-17)

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(A-18)

where β = v/c and .

Thus, substituting these values in Equas. (A-11) and (A-12) we obtain the famous Lorentz coordinate transformation equations connecting the fixed coordinate system S to the moving coordinate system :

(A-19)

(A-20)

We may also obtain the inverse transformations (from system to S) by replacing v by –v and simply interchanging primed and unprimed coordinates. This gives,

(A-21)

(A-22)

VELOCITY TRANSFORMATIONS

As a direct consequence to these new transformations, all the other mathematical operations and physical variables follow accordingly. For example, the velocity equations (though still the derivatives of the displacement) assume a new form, so the Lorentz form of the velocities is:

From Equas. (A-19) and (A-20) we have:

(A-23)

(A-24)

Therefore:

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(A-25)

ENERGY CONSIDERATIONS

Consider a particle of rest mass m0 being acted by a force F through a distance x in time t and that it attains a final velocity v. The kinetic energy attained by the particle is defined as the work done by the force F. The applicable equations are,

(A-26)

We note that

and that

Substituting d(γv) in Eq. (45) and integrating, we obtain

(A-27)

That is,

(A-28)

This says that K = (m – m0)c2 and finally one sees that the total energy is equal to the sum of the kinetic energy K and the rest energy E0 = m0c2.

i.e., E = K + Eo = γm0c2 = γE0, (A-29)

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where E0 = m0c2 and E = mc2.

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REFERENCES AND BIBLIOGRAPHY

1. Atam P. Arya, Elementary Modern Physics, Addison Wesley, 1974.

2. Arthur Beiser, Concepts of Modern Physics, second edition, McGraw-Hill, N. Y. 1973.

3. Ronald Gautreau and William Savin, Modern Physics, Schaum's Outline series in Science, McGraw-Hill, N. Y., 1978.

4. Dimitri Marianoff with Palma Wayne, Einstein: An Intimate Story of a Great Man, Double Day, Garden City, N. Y., 1944. This book was written from inside the Einstein household, Marianoff was his son-in-law and lived in the same house. He indeed provides a story that is truly unique.

5. Paul Arthur Schilpp, Albert Einstein, Philosopher-Scientist, Vol. VII in the Library of Living Philosophers, MJF Books, New York, 1970.

6. Herbert Goldstein, Classical Mechanics, Addison Wesley, 1959.

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