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Special Relativity

by

September 10, 2020

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:

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:

Michelson and Morley made their measurements by means of the

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 - t

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 - t

A - t_{1}
A - t_{2}
A - t_{2a}

B - t_{1}
B - t_{2}
B - t_{2a}

B - t

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 - t

Refraction A - t

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

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 (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.

1

________

γ = _______

√ 1 - v^{2}/c^{2}

√ 1 - v

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.

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 A

Because they are right triangles, A

We have the following equations:

A

B

A

B

A

B

A

B

B

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:

C

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

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 D

D

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 cΔt

As in the triangles of the first picture above, the vertical line in D

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 D

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, ΔX

From diagram D

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 cΔt

Because they are right triangles, (ΔX)

Now, we have the following equations:

(ΔX

H

(ΔX

H

(ΔX

H

(ΔX

H

H^{2} = (cΔt_{0})^{2} - (ΔX_{0})^{2} = (cΔt_{1})^{2} - (ΔX_{1})^{2} = (cΔt_{2})^{2} - (ΔX_{2})^{2} = (cΔt_{3})^{2} - (ΔX_{3})^{2}

Notice that the degenerate triangle labeled D

We now have the invariant interval:

(cΔt)

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.

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 D

The time it takes for light to travel from the bottom mirror to the top mirror in Dale's frame of motion is Δt

In the pass described by diagram D

v = ΔX

ΔX

The time interval Δt

Δt

H = cΔt

Pythagoras tells us that, in Diagram D

(cΔt

c

c

c

c

c

c

(Δt

(Δt

(Δt

(Δt

Δt

Recall that gamma is the following:

1

________

γ = _______

√ 1 - v^{2}/c^{2}

√ 1 - v

The time interval Δt

Δt

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

Δt

While this effect is famously known as

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 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 D

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 D

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 L

L

This is called Length Contraction.

Recall that the invariant interval is (cΔt)

(cΔt

or

c

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

c

Since distance equals velocity times time we have the following:

c

and to simplify matters a bit further

c

t

t

t

t

t

Further, since X

X

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

In the general case, X

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:

t

X

These equations must produce the results obtained from the special case above. Namely, if X

t

X

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

t

X

We begin again with the ever-helpful invariant equation:

c

c

c

c

(c

t

t

Because there is a t

c

γ

γ

γ

γ

γ = 1/(1 - v

1

________

γ = _______

√ 1 - v^{2}/c^{2}

√ 1 - v

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:

c

2γ(c

Recall that γ is never zero.

c

G = Mv/c

Because we have an X

c

- c

M

M

M

M

M

M

M = 1/(1 - (v

M = γ

G = γv/c

t

X

t

X

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.

Lieber, Lillian R.

Lagerstrom, Larry R.

https://www.coursera.org/learn/einstein-relativity