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Special Relativity

by

James Adrian


September 10, 2020

Introduction

Light (and all other electromagnetic radiation) travels at 299,792,458 meters per second in a vacuum, but even a mere 458 meters per second is over 1024 miles per hour. Our daily experience with time and distance, and even the vast majority of our manufacturing and tool-making pursuits do not involve speeds that are an appreciable fraction of the speed of light or the speed of radio waves. As a result, time dilation, length contraction, and twins aging at different rates due to high relative speeds are effects that are absent from our daily lives. These effects are, however, becoming increasingly relevant to our wellbeing and happiness.

      This article explains what Einstein's Theory of Special Relativity is, with virtually no comment on its history of development or the personalities of those involved. All of special relativity is derived with no more trigonometry than the Pythagorean Theorem and no more algebra than the simplest parts of Elementary Algebra. The next section will explain our first clue:


The Michelson-Morley Experiment


Strange as it may seem, the speed of light in a vacuum, as measured by the receiver of that light, is the same, regardless of the speed or direction of travel of the source of that light. The first evidence leading to this conclusion was the result of an experiment conducted by Albert A. Michelson and Edward W. Morley that took place during April and July of 1887. Thereafter, this experiment has been known as the Michelson-Morley Experiment.

      This experimental result does not depend on the accuracy of our standard for time or our standard for distance. A speed equals a distance traveled divided by the time elapsed. If we arbitrarily choose a distance to be a meter, and arbitrarily choose a duration to be a second, and we use those meanings for distance and time, the receiver of light will always observe the same speed, regardless of the frame of motion of the source of that light, and regardless of the frame of motion of the receiver.

      You can change the length that you are calling a meter and do another thousand calculations of the speed of light. In that event, you will find that each of those thousand calculations gives you the same result, whatever it may be. You can change what a second is and calculate a different speed result, but if you measure the speed of light another thousand times with this new unit of time, all one thousand of those results will be the same.

      As long as we define speed as being distance divided by time, the speed of light is constant. Here is how that remarkable result was discovered:


The Interferometer

Michelson and Morley made their measurements by means of the interferometer. An understanding of the workings of this instrument is crucial to the understanding of the experiment and its results.

      In this experiment, an interferometer was used to detect constructive or destructive interference between two beams of light traveling on coincident paths. Prior to these two beams being made coincident, their paths were set ninety degrees apart to see whether the speed of light would depend on the direction of travel.

      In each of the six pictures below, there are two mirrors that are labeled A and B. The diagonal line in the middle of each picture represents a half-mirrored sheet of glass serving as a beam splitter. This beam splitter is drawn with a red line representing the mirrored side and a black line representing the vast majority of the thickness of the glass. A light source marked S and a light detector marked D are also present in each of the pictures.

      In the first time interval of interest, the light from source S is reflected by the beam splitter to mirror A and simultaneously transmitted through the beam splitter to mirror B, because the glass of the beam splitter is half mirrored. This is illustrated in pictures A - t1 and B - t1.

      In the second time interval of interest, the light reflected from mirror A is transmitted through the beam splitter to detector D, while at the same time, the light reflected from mirror B is reflected by the beam splitter to detector D. This is illustrated in pictures A - t2 and B - t2.


                A - t1                                       A - t2                                         A - t2a



       



                  B - t1                                       B - t2                                         B - t2a


      The second time interval of interest also includes two light paths that do not affect the operation of the interferometer, and do not affect the results of the experiment. Pictures A - t2a and B - t2a each show a beam of light returning to source S. The beam splitter reflects light from mirror A to source S, and also transmits light form mirror B to source S. Neither has any influence on source S or the experiment.


 



 Refraction A - t2             Refraction B - t1


      The two pictures above illustrate refraction as light is being transmitted through the beam splitter. This does not alter the direction of the light beam, but it slightly offsets its path to one that is parallel to what it would follow if the beam splitter did not refract the light.

      This link fully explains the effects of this refraction.

      When the light from mirror B is reflected by the beam splitter, the direction and placement of the light moving toward the detector is identical to that in picture named Refraction A - t2. Both refractive delays are also identical. (The speed of light in glass is less that the speed of light in air or in a vacuum.) When the light from mirror B is reflected downward on the drawing to the detector D, the light must pass, once again, through the thickness of the glass to reach the half-mirrored side of the beam splitter (that side is shown in red). This adds a very slight delay that is not present in the light coming from mirror A. Adjustments to what would otherwise be an equal distance to the two mirrors from the center of the beam splitter can be made to compensate for this additional delay.

      Michelson and Morley expected to find that the motion of the Earth moving around the sun would alter the speed of light in the direction of travel of the Earth. This would have created phase differences in the light coming from the two directions causing some destructive interference rather than an entirely constructive adding of amplitude at detector D. This did not happen.

      To fully understand destructive and constructive interference, I recommend this link to an article on wave interference.


Albert Einstein's Theory of Special Relativity


Albert Einstein (originally Albrecht Einstein) deduced Special Relativity from these two features of our perceived reality:

1. The laws of physics are identical in all non-accelerating frames of motion.

2. The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or the observer.

      Michelson and Morley did not expect the results they got, and in fact, their experiment was motivated by a theory that could not be confirmed. As indicated by statement 2 above, Einstein accepted their well-confirmed experimental result.

      Einstein used mathematical tools and insights concerning relative motion developed before Einstein's 1905 paper and stemming from the work of Hippolyte Fizeau, Woldemar Voigt, Henri Poincaré, Hermann Minkowski, George FitzGerald, and Hendrik Lorentz; however, the understandings of relative motion that prompted these developments could not be supported by all of the empirical facts. Einstein fully acknowledged the work of these physicists. He importantly made use of the Lorentz Transformation in developing Special Relativity, which was the first theory to correctly and fully explain the effects observable when objects travel at an appreciable fraction of the speed of light. For physicists, Einstein's Special Theory of Relativity, together with his General Theory of Relativity, constitute the established laws of motion.

      Gamma is the name of a Greek letter famously set equal to an expression specified by Dutch physicist Hendrik Antoon Lorentz, creating and equation thereafter known as the Lorentz Transformation:

              1
        ________ 
γ =     _______  
        √ 1 - v2/c2

      On the basis of the starting point selected by Albert Einstein (statements 1 and 2 above), all of special relativity can be derived with no more trigonometry than the Pythagorean Theorem and no more algebra than the simplest parts of Elementary Algebra.


The Invariant Interval

      Consider the following picture of three right triangles and a vertical line:

             I                  II                           III                               IV

      Actually, the vertical line on the left is being analyzed as a degenerate triangle (triangle I) where side A0 has a length equal to zero.

      Because they are right triangles, A2 + B2 = C2, whether the labels have subscripts or not. Notice that there is no subscript associated with the B side of any of the triangles. This is because the B side of each triangle is equal in length to the B side of any other triangle in the picture.

      We have the following equations:

A02 + B2 = C02

B2 = C02 - A02


A12 + B2 = C12

B2 = C12 - A12


A22 + B2 = C22

B2 = C22 - A22


A32 + B2 = C32

B2 = C32 - A32


B2 = C02 - A02 = C12 - A12 = C22 - A22 = C32 - A32

      Notice that the degenerate triangle labeled I (the vertical line) has the same relationship as the others.

      When any two right triangles, p and q, have in common a side, like B, that is the same length for each of them, the remaining sides, in this case C and A, have the following relationship:

Cp2 - Ap2 = Cq2 - Aq2

      Appraised of this bit of trigonometry, the constancy of the speed of light (and the constancy of the speed of of all other electromagnetic radiations) can be shown to have some unfamiliar consequences whenever objects travel at an appreciable fraction of the speed of light.

      Keep in mind that we normally do not see these consequences. They are immeasurably small when objects travel at only thousands of miles per hour. The speed of light, c, being 299,792,458 meters per second in a vacuum, is over 95,800 times the speed of an object traveling at 7000 miles per hour (3129.28 meters per second).

      The Light Clock enables us to see important effects associated with speeds which are an appreciable fraction of the speed of light.

      The performance of an ideal light clock would be very difficult to approximate in a physical device, but it is instructive to imagine an ideal light clock in order to appreciate the real consequences of the constancy of the speed of light.

      A vertical line having a fixed length is the path of a pulse of light being reflected by two mirrors, one mirror at the bottom of the path and one mirror at the top. This light pulse is reflected downward toward the bottom mirror by the top mirror and is reflected upward toward the top mirror by the bottom mirror. We may presume that the reflectivity of the two mirrors is 100% and that this path is retraced indefinitely while being observed. A single complete cycle is observed or counted when the light pulse moves from the bottom mirror to the top mirror and then back to the bottom mirror. This cycle is considered one tick of the light clock.

      The picture below contains four diagrams labeled D0 through D3.

                D0                 D1                            D2                          D3

      In physics literature, the letter c famously represents the speed of light. The distance between the two mirrors of the light clock (the length of the light path) is seen by an observer in the frame of motion of the light clock as cΔt0. This is the product of the speed of light, c, multiplied by the time the light pulse takes to travel the distance, H, once. That time interval is Δt0. Thus, in the frame of motion of the light clock, cΔt0 = H, as illustrated in diagram D0.

      As in the triangles of the first picture above, the vertical line in D0 will be analyzed as a degenerate triangle. In this case, the side that has a length equal to zero is labeled ΔX0.

      Let us assume the point of view of an observer, Dale, standing still on a relatively elevated area of the moon. This observer is the owner and operator of the light clock shown in diagram D0. Looking low in the sky, a vehicle, piloted by Major Tola passes by three different times. Thinking in terms of a two-dimensional Cartesian Coordinated System, Dale measures the observed horizontal travel as length along the X axis.

      The vehicle has a light clock that is clearly seen by Dale through a transparent section of the vehicle. This light clock has the same height, H, as Dale's light clock. Each time the vehicle passes, it is traveling at a sizable fraction of the speed of light.

      The light clock on this vehicle has an indicator that flash a light that Dale can see when the light-clock pulse is reflected by the top mirror. Another indicator, of a different color, flashes when the beam in the clock is reflected by the bottom mirror. These two indicators are duplicated on the dash board observed Major Tola.

      Distances, ΔX1, ΔX2, and ΔX3 are the distances along the X axis traveled during a light-pulse rise from the bottom mirror to the top mirror. This is half a tick (or half a cycle) for the purpose of counting time intervals.

      From diagram D1, we see that the distance that the light beam must travel from the bottom mirror to the top mirror is greater in length than the height, H. This is because the vehicle is moving to the right and the light beam must travel farther before reaching the top mirror. Because the speed of light is constant, the speed with which this beam travels is c. The length of this path is cΔt1.

      This is how it is measured by Dale, but to Major Tola, who sees the light clock on board behind him through a mirror, the light clock works just as Dale observes the light clock on the ground (in Dales's frame of motion).

      The light clock on the moving vehicle is observed by dale as producing fewer ticks per unit of time than are counted by those in the frame of motion of their own light clock.

      Because the measured speed of light is constant regardless of the state of motion of the observer or the source of the light, Dale measures cΔt1 as being a greater distance than cΔt0.

      Because they are right triangles, (ΔX)2 + H2 = (cΔt)2, whether the labels have subscripts or not. Notice that there is no subscript associated with the H side of any of the triangles. This is because the H side of each triangle is equal in length to the H side of any other triangle in the picture of four diagrams above.

      Now, we have the following equations:

(ΔX0)2 + H2 = (cΔt0)2

H2 = (cΔt0)2 - (ΔX0)2


(ΔX1)2 + H2 = (cΔt1)2

H2 = (cΔt1)2 - (ΔX1)2


(ΔX2)2 + H2 = (cΔt2)2

H2 = (cΔt2)2 - (ΔX2)2


(ΔX3)2 + H2 = (cΔt3)2

H2 = (cΔt3)2 - (ΔX3)2

H2 = (cΔt0)2 - (ΔX0)2 = (cΔt1)2 - (ΔX1)2 = (cΔt2)2 - (ΔX2)2 = (cΔt3)2 - (ΔX3)2

      Notice that the degenerate triangle labeled D0 (the vertical line) has the same relationship as the others.

      We now have the invariant interval:

(cΔt)2 - (ΔX)2

      This means that regardless of the speed of an object or a vehicle, there is a quantitative relationship that is true of each frame of motion (a fact that will be very useful later).

      Notice also that Dale measures the height, H, as being unaffected by the speed of the vehicle in the X direction. Neither would any measurement in the Z direction be affected. Relativistic effects apply only to motion in the direction of travel. The Y and Z measurements in Dale's coordinates system are invariant with respect to the movement in the X direction.


Time Dilation

      From Dale's perspective, a light pulse traveling from the bottom mirror to the top mirror of a moving light clock must travel a distance greater than the height, H, of the light clock. This is because the top mirror is in a different place along the X axis by the time that the light pulse reaches it. Because the speed of light is constant, the time it takes to reach the top mirror is greater than it is in Dale's light clock, which is not moving relative to Dale.

      Dale will always observe moving clocks ticking less often and measuring time slower than clocks that are in Dale's frame of motion. To quantify this, let us again examine diagrams D0 and D1 above.

      The time it takes for light to travel from the bottom mirror to the top mirror in Dale's frame of motion is Δt0. The ascending light pulse in diagram D1 travels a greater distance, as observed by Dale. It does so during a time interval that Dale measures as Δt1. The time interval Δt1 is greater than the time interval Δt0.

      In the pass described by diagram D1, the vehicle is traveling at a velocity, v, that is equal to ΔX1 divided by Δt1 (distance divided by time).

v = ΔX1/Δt1

ΔX1 = v(Δt1)

      The time interval Δt0 is the distance traveled by the light pulse, H, divided by the speed of light, c.

Δt0 = H/c

H = cΔt0

      Pythagoras tells us that, in Diagram D1, we have the following:

(cΔt1)2 = (H1)2 + (ΔX1)2

c2(Δt1)2 = (H1)2 + (ΔX1)2

c2(Δt1)2 = (cΔt0)2 + (ΔX1)2

c2(Δt1)2 = c2(Δt0)2 + (ΔX1)2

c2(Δt1)2 = c2(Δt0)2 + (ΔX1)2

c2(Δt1)2 = c2(Δt0)2 + v2(Δt1)2

c2(Δt1)2 - v2(Δt1)2 = c2(Δt0)2

(Δt1)2(c2 - v2) = c2(Δt0)2

(Δt1)2 = c2(Δt0)2/(c2 - v2)

(Δt1)2 = (c2/(c2 - v2))(Δt0)2

(Δt1)2 = (1/(1 - v2/c2))(Δt0)2

Δt1 = (1/(1 - v2/c2))1/2(Δt0)

      Recall that gamma is the following:

              1
        ________ 
γ =     _______  
        √ 1 - v2/c2

      The time interval Δt1 observed by Dale is indeed larger that Δt0 in Dales's frame of motion since γ is larger than 1 for all values of v greater than zero and less than c.

Δt1 = γΔt0

      Therefore, the lapse time being counted by the moving light clock as observed by Dale is the following:

Δt0 = (1/γ)Δt1

      While this effect is famously known as Time Dilation, that name was prompted by the larger time between ticks of the clock. A more appropriate name would be Time Abbreviation because the clock on board of a vehicle moving an appreciable fraction of the speed of light relative to the observer, is observed as measuring less time passage relative to measurements of time in the observer's frame of motion.

      Keep in mind that observers on a moving vehicle do not observe any time dilation measured by the clocks in their own frame of motion.

      This time dilation is not a mathematical artifact or some kind of trick or misunderstanding. Despite the fact that, during a high-speed pass, Major Tola measures no difference between the rate of ticks on the vehicle's clock when moving fast and when moving slowly, the on-board clock shows less time passed than Dale's clock when Major Tola lands. Upon landing, this is observed by both Major Tola and Dale.

      This is not due to the construction of the light clocks. Atomic clocks show the same thing. When two carefully made atomic clocks are tested, one kept on Earth while the other is flown at high speed with astronauts traveling in space, even modest speeds (several thousand miles per hour) produce time dilation that can be measured and has been measured. Astronauts have returned from their journeys a fraction of a second younger than they would have been relative to the people they have returned to. Whenever relative speeds approaching the speed of light become involved (some time in the future), the time difference will be months or years. This is the basis of the so-called Twin Paradox, which is no longer a true logical paradox. We now know why.


Length Contraction

      Length contraction is a decrease in length in the direction of travel. This is currently difficult to measure except at very high speeds relative to the observer; nonetheless, this length contraction is present (usually to a minuscule extent) at any non-zero speed. Length contraction arises due to the fact that the speed of light in a vacuum is constant in any frame of motion.

      Consider the vehicle and light clock described by diagram D2. Dale measures its speed as approximately 75% of the speed of light, giving it a γ of about 1.5. It's time dilation (time abbreviation) in the D2 pass by Major Tola is this:

Dale's lapse time = (1/γ)Major Tola's lapse time

Dale's lapse time = (1/1.5)Major Tola's lapse time

Dale's lapse time = (0.66)Major Tola's lapse time

      The ever cooperative Major Tola had installed a flat reflecting strip on the underside of the vehicle which also detects being struck by blue laser light. For this D2 pass, Dale installed a laser to produce a coherent vertical beam, and a laser-light detector surrounding it on the ground:



      Traveling to the right, Major Tola's careful navigation causes the flat reflective strip on the underside of the vehicle to reflect the blue laser beam down to the source which detects the reflection for as long as the reflective strip is above it.

      The time it takes for the laser beam to reach the reflective strip and travel back down to the detector is both minuscule and identical to the delay in detecting the absence of reflection after the vehicle has passed. The duration of this reflection is recorded by both Dale and Major Tola.

      Major Tola's speed, as measure by Dale, is about 75% of the speed of light. This is very nearly 225 million meters per second.

      The reflector on the vehicle, as measured by Major Tola, is 40 meters long.

      Because the lapse time measured by Dale is .66 times the lapse time measured by Major Tola, Dale calculates the length of the reflective stip to be 40 meters multiplied by .66 which equals 26.4 meters.

      If the length measured by Dale is LDale and the length measured by Major Tola is LTola then we have the following:

LDale = 1/γ LTola

      This is called Length Contraction.


Deriving the Lorentz Transformation

      Recall that the invariant interval is (cΔt)2 - (ΔX)2. This means that for any two frames of motion, p and q, we have the following:

(cΔtp)2 - (ΔXp)2 = (cΔtq)2 - (ΔXq)2

or

c2Δtp2 - ΔXp2 = c2Δtq2 - ΔXq2

      First (because the math is easier), the transformation will be derived in the case where ΔXq2 is zero.

c2Δtp2 - ΔXp2 = c2Δtq2

Since distance equals velocity times time we have the following:

c2Δtp2 - (vtp)2 = c2Δtq2

and to simplify matters a bit further

c2tp2 - (vtp)2 = c2tq2

tp2(c2 - v2) = c2tq2

tp2 = c2tq2/(c2 - v2)

tp2 = tq2/(1 - v2/c2)

tp = tq/((1 - v2/c2)1/2)

tp = γtq

      Further, since Xp = vtp we have the following:

Xp = γ(vtq)

      The Lorentz transformation is derived from the invariant interval in this special case where the distance value, Xq is zero.

      In the general case, Xq is not necessarily zero. Fortunately, the Lorentz Transformation can be derived from the invariant interval in this case as well.

      The derivation must assure that the units of time and distance are same on both sides of either transformation equation (for time or distance). For instance, we can't have meters on one side of the equation and meters squared on the other side. To assure this kind of linearity, here is a format that might work:

tp = (G)Xq + (H)tq

Xp = (M)Xq + (N)tq

      These equations must produce the results obtained from the special case above. Namely, if Xq = 0, tp must equal γtq; and Xp must equal γ(vtq). In that event, (G)Xq = 0, and (M)Xq = 0. We then have the following:

tp = (H)tq = γtq

Xp = (N)tq = γ(vtq)

      This reveals that H = γ and that N = γv. Now we have this:

tp = (G)Xq + γtq

Xp = (M)Xq + γ(vtq)

      We begin again with the ever-helpful invariant equation:


c2tp2 - Xp2 = c2tq2 - Xq2

c2((G)Xq + γtq)2 - Xp2 = c2tq2 - Xq2

c2((G)Xq + γtq)2 - ((M)Xq + γ(vtq))2 = c2tq2 - Xq2

c2(G2Xq2 + 2GγXqtq + γ2tq2) - (M2Xq2 + 2MγvXqtq + γ2v2tq2) = c2tq2 - Xq2

(c2γ2tq2 - γ2v2tq2) + (c22GγXqtq - 2MγvXqtq) + (c2G2Xq2 - M2Xq2) = c2tq2 - Xq2

tq2(c2γ2 - γ2v2) + Xqtq(c22Gγ - 2Mγv) + Xq2(c2G - M2) = c2tq2 - Xq2

      There is no Xqtq term on the right hand side of the equation, so the Xqtq term on the left hand side must be zero:

tq2(c2γ2 - γ2v2) + Xq2(c2G - M2) = c2tq2 - Xq2

      Because there is a tq2 term on both sides of the equation, we have the following:

c2γ2 - γ2v2 = c2

γ2(c2 - v2) = c2

γ2 = c2/(c2 - v2)

γ2 = c2/(c2(1 - v2/c2))

γ2 = 1/(1 - v2/c2)

γ = 1/(1 - v2/c2)1/2

              1
        ________ 
γ =     _______  
        √ 1 - v2/c2

      Thus the Lorentz Transformation is derived from the invariant interval for the general case. Still loving for G and M, we can make further observations:

c22Gγ - 2Mγv = 0

2γ(c2G = Mv) = 0

      Recall that γ is never zero.

c2G = Mv = 0

G = Mv/c2

      Because we have an Xq2 term on both sides, we have another equation to explore:

c2G - M2 = -1

- c2G + M2 = 1

M2 - c2G = 1

M2 - c2(Mv/c2)2 = 1

M2 - c2((M2v2)/c4) = 1

M2(1 - (c2v2/c4)) = 1

M2(1 - (v2/c2)) = 1

M2 = 1/(1 - (v2/c2))

M = 1/(1 - (v2/c2))1/2

M = γ

G = γv/c2

tp = (G)Xq + γtq

Xp = (M)Xq + γ(vtq)

tp = (γv/c2)Xq + γtq

Xp = γXq + γ(vtq)

      The invariant equation has produced the last two equations which transform the time and distance measured in one frame of motion into the time and distance of another frame of motion.


Bobliography


Lieber, Lillian R. The Einstein Theory of Relativity: A Trip to the Fourth Dimension. Paul Dry Books, 2008.

Lagerstrom, Larry R. Understanding Einstein: The Special Theory of Relativity. Stanford University online course through Cousera.org, 2020.
https://www.coursera.org/learn/einstein-relativity