Special Relativity Basics · 2019. 12. 5. · William Shaw Special Relativity Basics reaches Earth...

Special Relativity Basics

William Shaw

December 29, 2014

Abstract

An introduction to the basics of Special Theory of Relativity (timedilation and length contraction) through thought experiments and (rel-atively) simple math.

1 Introduction

The Special Theory of Relativity was born in 1905 with Albert Einstein’spaper On the Electrodynamics of Moving Bodies [Einstein (1905)]1

Since that first paper the theory has been expanded and explored and mis-understood. Mostly, it is simply not understood. Time dilation, the phe-nomenon that “moving clocks run slow” and length contraction are unknownto most people because they have no real relevance to their daily lives. How-ever, there are a lot of people who are curious about Relativity and the mys-tique associated with theoretical physics. Albert Einstein is the archetypalgenius and his renown is founded on the success of the Special and Generaltheories of relativity.

By the standards of modern day theoretical physics the Special Theory ofRelativity is not mathematically sophisticated. In fact, only algebra andnot calculus is needed to describe the theory, whereas the mathematics ofthe General Theory of Relativity is very advanced. The mystique of SpecialRelativity arises out of the difficulty of understanding how Time and Spaceappear to change in objects that are moving close to the speed of light. Whyshould the speed of an object affect how time and distance are observed tobehave?

1In German, Zur Elektrodynamik bewegter Korper.

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William Shaw Special Relativity Basics

The leap that Albert Einstein made was to accept that for some as yetunexplained reason the speed of light is the same everywhere: if you measurethe speed of a light pulse that is sent by a spaceship moving away fromyou, it will have the same speed as measured by someone on the spaceship.While that doesn’t sound too shocking, the behaviour of the light pulseis completely different than a bullet that is emitted by the spaceship: themeasured speed of the bullet as it passes you is the muzzle speed when it wasfired from the spaceship minus the speed of the spaceship. The differencebetween the two may sound minor, but it has very significant implications.

Einstein started from two simple principles:

1. The laws of physics are invariant in all inertial (non-accelerated) framesof reference.

2. The speed of light in a vacuum is the same for all observers.

The first principle seems pretty obvious: why would the laws of physics everbe different than what we experience in everyday life? The clue comes fromthe restriction that the laws are always the same (“are invariant”) in inertialframes of reference. This means that the laws of physics are the same forsomeone standing on a sidewalk watching a car go by and for a passengerin the moving car as long as the car is moving at a constant speed.

There is no guarantee that the laws of physics are the same inside the carwhen it accelerates or decelerates, or when it hits a bump and suddenlyjumps upward. So the first principle actually puts a pretty strong restrictionon where the laws of physics remain the same. Up to the point where this“new” principle was incorporated into a theory of physics everyone assumedthat the laws of physics were always the same.2

The second principle is not at all obvious. In fact, it was assumed up to thelate nineteenth century that light needed a medium to carry it in the sameway as water waves propagate over the surface of a pond and sound wavespropagate through air. If light were waves of electromagnetic fields then thewaves must be a distortion of some medium: scientists called this unknownmedium the aether (which we now spell as ether). The belief turned outto be wrong and was proven to be wrong in a very famous experiment, theMichaelson-Morley experiment, that showed that light from distant stars

2In the General Theory of Relativity the restriction is different but stronger: the lawsof physics are the same at all events on a spacetime geodesic, or “in free fall”. But that’sanother story.

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reaches Earth at the same speed no matter whether the Earth is movingtoward or away from the stars.

Interpreting the results of the Michaelson-Morley experiment was very dif-ficult. How could the speed of light in a vacuum possibly be the same nomatter what speed and direction the observer is travelling?3

As often happens in physics, the answer was to turn the question into anassumption (or “principle”) and then to figure out a theory that is based onthe assumption.4

The Special theory of relativity5 is special because it is limited to framesof reference that are either stationary or moving at a constant speed withrespect to each other.

The consequences of these two principles are very difficult to understand,since the behaviour of objects that are travelling close the the speed of lightis very different than our daily experience of the world. The only recourseis to (i) rigorously build up a theory based on the principles and (ii) runexperiments to check that the theory works.

But clearly something very strange is happening here.

2 Non-special relativity: moving frames of refer-ence

As a first step let’s review relative motion in everyday life (non-special rel-ativity). We’ll start with a thought experiment that does not involve light.

See Figure 1. A bullet is fired from a gun mounted on a tripod on railwaytracks toward a train car that is moving away from the gun. Just in frontof the gun’s muzzle there is an apparatus that can measure the speed of abullet fired from the gun. The same type of apparatus is mounted on the

3The speed of light is different in substances like diamond, water and air and is alwaysslower when travelling through some substance than the speed of light in a vacuum. Wealways use the speed of light in a vacuum as the standard because there is no room formisinterpretation.

4Two of the best known principles being Heisenberg’s Uncertainty Principle and thePauli Exclusion Principle. No fundamental explanation of any of the principles has everbeen found but experiments show that they work.

5At the time the phrase “Principle of Relativity” was used and only later when theGeneral Theory of Relativity was published was there a need to distinguish between thetwo theories.

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train car to measure the speed of the bullet as it passes the car.

There are two frames of reference that are of interest:6

1. The gun, tripod, train track and stationary measuring apparatus arein one frame of reference. They are not moving relative to each other.

2. The train car and the moving measuring apparatus are in anotherframe of reference. They are not moving relative to each other.

The train car reference frame is moving at constant speed relative to thegun-tripod reference frame.

Gun StationaryMeasuringApparatus

MovingMeasuringApparatus

Train car moving at 120 km per hour

Bullet moving at 4,320 km per hour(1.2 km per second)

Figure 1: Relative motion of a bullet and a train car

The train car is moving away from the gun at a speed of 120 kilometres perhour. When the gun is fired the bullet passes by the stationary measuringapparatus and continues on to pass by the moving measuring apparatus onthe train car. What bullet speed does each measuring apparatus measure?

A typical muzzle speed for the bullet is 1,200 metres per second, which is1.2 kilometres per second, which is 4,320 kilometres per hour. That is thespeed that the stationary measuring apparatus measures.

When the bullet catches up to the train car and passes the measuringapparatus mounted on it, what speed is measured? The answer is easy:4, 320− 120 = 4, 200 kilometres per hour. The bullet is moving slower rela-tive to the moving train car (4,200 km per hour) than it is to the gun (4,320km per hour).

6The bullet has its own frame of reference, moving with respect to both the gun andthe train car. We won’t discuss the bullet’s frame of reference in this thought experiment,however.

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The difference between the speed that is measured by the stationary appa-ratus and the speed that is measured by the moving apparatus is what wenormally experience.

Here’s a much simpler thought experiment that you could try out withoutany apparatus. It illustrates adding speeds instead of subtracting them asin the example above.

If I’m walking toward you at 4 kilometres per hour and you’re walking towardme at 6 kilometres per hour, our relative speed is 4 + 6 = 10 kilometres perhour. To calculate how long it will take for us to meet you just need to knowthe distance between us when we start walking and divide by the relativespeed: if we’re 5 kilometres apart when we start walking we will meet in (5kilometres) ÷ (10 kilometres per hour) = 1/2 hour. This is exactly the sameas if I were standing still and you were walking toward me at 10 kilometresper hour (probably running rather than walking).

We can also subtract walking speeds. If I’m walking at 6 kilometres perhour along a straight street and you‘re walking in the same direction (thatis, you‘re walking away from me along the same street) at 4 kilometres perhour, your speed relative to me is 6−4 = 2 kilometres per hour. EventuallyI will catch up to you even though you’re walking away from me becauseI’m walking faster than you. If we both start walking at the same time andwe’re 4 kilometres apart when we start walking, how long will it take beforeI catch up to you? The time is the distance divided by the relative speed,so I’ll catch up to you in 4÷ 2 = 2 hours.

Even though most us of don’t normally do these calculations, they are com-mon sense.

In the next section we will use frames of reference that are moving at con-stant speed with respect to each other, but at what we call relativistic speeds.Relativistic speeds are speeds at which the behaviours associated with Spe-cial Relativity become measurable: speeds close to the speed of light.

3 Moving frames of reference in Special Relativity

In this section we discuss measurements on frames of reference that aremoving at close to the speed of light relative to each other. The relativespeed is constant (neither frame of reference is accelerating), so SpecialRelativity applies.

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The simplest two phenomena associated with relativistic speeds are timedilation and length contraction.

In order for time dilation and length contraction to be observable, the mov-ing frame of reference must be moving toward or away from the observer.If the moving frame is moving perpendicularly to the line-of-sight from theobserver to the moving frame, no time dilation will be observed (See Figure2).

Case 1: Line-of-sight (directly toward or directly away from)

· Time dilation in the car is observed by the person

· Time dilation of the person is observed by the car

Case 2: Perpendicular (neither receding nor approaching)

· Time dilation in the car is NOT observed by the person

· Time dilation of the person is NOT observed by the car

Case 3: Oblique (approaching or receding but not along the line-of-sight)

· Some time dilation in the car is observed by the person

· Some time dilation of the person is observed by the car

Time dilation from the point of view of a person observing a moving car

Figure 2: Time dilation and relative direction of motion

In the real world, objects usually move toward or away from each other onan oblique path, neither directly toward or away nor perpendicularly. Ifthe path that the moving frame follows is not exactly perpendicular to theline-of-sight, some degree of time dilation and length contraction will beobservable.

For introductory thought experiments we assume that the observer and themoving frame (the spaceship) are moving directly toward or away from eachother to keep things as simple as possible. In mathematical terms, we haveboth the observer and the spaceship move along the X-axis of a graph.

A typical definition of time dilation is:

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Time Dilation is the difference in elapsed time between twoevents as measured by observers moving relative to each other.7

The usual catch phrase is “moving clocks run slow”.

But what does that mean?

It means that if you measure the length of time some standard event takesin your own frame of reference (five minutes according to a clock sitting onyour night table, say, as the minute hand moves from the 11 to the 12) andsomehow measure the length of time the same event takes on a spaceshipthat is moving directly toward or away from you at close to the speed oflight, it will look like it took more than five minutes for the moving clock tomove from the 11 to the 12 according to the spaceship’s clock. That is, thespaceship’s alarm clock will appear to be running slow compared to yourown alarm clock.

To someone on board the spaceship the clock appears to be running correctly,however. And if the person on board the spaceship observed your clockit would appear to be running slow. This is another weird thing aboutRelativity: to each observer it looks like the other person’s clock is runningslow. But you can see that this is what has to happen. If the spaceshipis moving at a constant speed, it looks to a person on board the spaceshipthat you are moving away at constant speed and they are stationary. Thesituation is symmetric: observers in both frames of reference measure thesame effects in the other frame of reference.

3.1 A Thought Experiment to Illustrate Time Dilation

We’re going to work through a typical thought experiment that explainstime dilation, the phenomenon that leads to “moving clocks run slow”.

To do the experiment we have to use mathematics (no calculus), but if youcan follow the logic you can see that we get the answer that Special Relativityspecifies. You should be warned that it will still be mind-stretching.

The thought experiment uses an idealized clock which is illustrated in Figure3.

7There is also a time dilation caused by gravitational fields but that is outside thescope of Special Relativity.

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L

Lightclock

Light pulse emitted at right angle to the lightclock when ER emits or reflects a light pulse

ER (Emitter-Receiver)

ER (Emitter-Receiver)

Figure 3: Idealized clock (“lightclock”)

The clock consists of two electrical devices that can both emit and receivepulses of light. We’ll call the devices emitter-receivers or ERs for short. Thetwo ERs are held at a fixed distance away from each other by a carbon fibreframe inside a vacuum chamber with very thin but very strong transparentwalls that allow light to pass through without changing speed.

Each ER can emit a pulse of light and each ER can “reflect” a pulse of lightthat it receives back along the pulse’s incoming path. There is no time delay

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between the time a pulse is received and the time it is reflected by the ER.

The time for one “tick” of the lightclock is the time it takes for a pulse totravel from one ER to the other and for the reflected pulse to travel back tothe first ER.

In addition, each time an ER emits or reflects a pulse of light it emits anotherpulse at right angles at exactly the same time. These light pulses can beused by a distant observer to measure when the light pulses were receivedby each ER. Without these pulses there would be no way for an observer toknow when pulses were emitted or reflected within the lightclock.

The distance between the two ERs is L as shown in Figure 3. In the light-clock’s frame of reference the time for a light pulse to travel from the lowerER to the upper ER is L/c, where c is the speed of light in a vacuum. Thetime is the same for the reflected pulse to travel from the upper ER to thelower ER. Therefore one tick of the lightclock is 2× L/c.8

Before describing the experiment we should explain why we’re using suchan odd apparatus instead of using a more conventional clock. The reasonis very straightforward: using light pulses makes the explanation simplerthan using a more conventional clock. We can use the speed of light directlyin the calculations without having to translate measurements made with aclock mechanism. And a lightclock as described could actually be built.

For our thought experiment we’re going to have an observer and a movinglightclock. The lightclock is moving away from the observer at speed V .

To the observer, he or she is at rest, feet planted firmly on the ground, andthe lightclock is moving. The observer can measure time with his or herown clock. We can start the clocks running whenever we want, so as thelightclock moves past the observer its time is set to zero and the observer’sclock is also set to zero.

It will be easier to discuss what’s happening if we start to use names.

We’ll call the observer O and the observer’s frame of reference S. S includesall the things (like the ground) that are stationary from O’s point of view.

We’ll call the lightclock’s frame of reference S′. In this case, all that S′

contains is the lightclock. In the S′ frame of reference, time is representedby t′ and distance is represented by x′.

8If L = 30 metres then it takes 2 × (30 ÷ (3 × 108)) = 2 × 10−7 = 120,000,000

seconds(twenty millionths of a second) for a light pulse to go from the lower ER to the upper ERand back to the lower ER.

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Since the lightclock’s frame of reference, S′, is moving along a straight lineat a constant speed V , we know that S′’s position relative to the observer,O, is V × t, that is x = V × t. If the lightclock is moving at 1000 metres persecond, then after 5 seconds it is 1000 × 5 = 5000 metres away from O (attime t = 5 the distance x = 5000).

The starting point for the experiment is shown graphically in Figure 4. Thetwo frames of reference are shown separately for clarity’s sake, but in theexperiment they coincide: the lightclock and the observer are at the samelocation and both of their clocks are set to zero; the lightclock is movingaway from the observer O at speed V .

S frame

S’ frame

t = 0 x = 0 v = 0

V

t’ = 0 x’ = 0 v’ = V

L

Figure 4: S and S′ frames of reference at time t = t′ = 0

At time t = t′ = 0 a light pulse is emitted from the lower ER of the lightclockand starts to move toward the upper ER.

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At the same time as the light pulse is moving toward the upper ER thelightclock is moving away from the observer O.

The time for the light pulse to move from the lower ER to the upper ER inthe S′ frame is L/c (distance divided by speed). We’ll call this time intervalt′1 = L/c.

From O’s point of view, the light pulse is moving in two directions at thesame time: it’s moving from the lower to the upper ER and at the sametime it’s moving away from O with speed V . The path for the light pulsethat O sees is the diagonal line AB in Figure 5. So from O’s point of viewthe pulse travels from A to B.

S frame

S’ frame

x = 0

v = 0 x’ = 0

x = V(L/c)

S’ frame

x’ = 0

x = 2V(L/c)

L

at time t’ = L/c at time t’ = 2L/c

A

B

C

V

Observer “O”

Figure 5: Lightclock experiment

To calculate the distance AB all we need is Pythagoras’ theorem that saysthe square of the length of the hypotenuse of a right-angled triangle is thesum of the squares of the other two sides. If we let AB, BC and CArepresent the lengths, then AB2 = BC2 + CA2.

BC is just L, the distance between the lower and upper ERs. We know thatthe distance AB must be the speed of light times the time, so AB = c× tAB

where tAB is the time the light pulse takes to travel from A to B. CA isthe distance the lightclock has moved away from the observer in the time ittook for the pulse to travel between the two ERs, which is x = V × tAB.

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From

AB2 = BC2 + CA2 (1)

we get

(c× tAB)2 = L2 + (V × tAB)2 (2)

by replacing AB, BC and CA as above. This can be rearranged to

(c× tAB)2 − (V × tAB)2 = L2. (3)

Since tAB appears in both terms on the left hand side of the equation,equation (3) becomes:

t2AB · (c2 − V 2) = L2. (4)

Taking positive square roots of both sides an dividing by√c2 − V 2 gives

tAB = L · 1√c2 − V 2

(5)

which can be written as

tAB =L

c· 1√

1− V 2

c2

. (6)

But Lc = t′1, so

tAB = t′11√

1− V 2

c2

. (7)

That is, the time observed by O is larger than the locally measured S′ time,t′1, by the factor 1√

1−V 2

c2

. This is always greater than 1 if V is less than c.

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This shows that O “sees” the S′ clock as running slow because O knowsthat the time for the light pulse to get from the lower ER to the upper ERshould just be L

c .9.

Get it? You’re forgiven if it doesn’t seem completely clear.

The key idea is that the speed of light is the same in both the stationaryframe and in the moving frame but travels different distances in the twoframes. Since we know that the speed of light is the same relative to theframe in which it is observed and we know the distance the light pulse isobserved to travel in each frame, time in the moving frame has to be observedto run slower to an observer in the stationary frame.

But as we saw above, an observer in the S′ frame has to see time in theS frame running slow in exactly the same way. Figure 6 shows the reversesituation where the lightclock is in the stationary frame, S, and the observeris in the moving frame.

This time the light pulse is emitted at time t = t′ = 0 as before, but nowit moves from the lower ER to the upper ER in the S frame. To observerO′ in the S′ frame the light pulse moves on the path from A to B as thelightclock appears to move away at speed V . The same calculation as aboveshows that O′ measures the S frame time to be running slow.

9This thought experiment is derived from one described in Relativity for Scientists andEngineers by Ray Skinner [Skinner (1982), p. 42 ff.]

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S frameS’ frame

L

A

B

C

V

Observer O’

at time t = L/c

Figure 6: Lightclock experiment as seen from the moving frame

This probably doesn’t seem to make sense at first. How can both observersmeasure time in the “other” reference frame as running slow?

As noted above, the moving clock running slow is an effect due to relativemotion and the fact that the speed of light is always measured to be thesame relative to the observer. The formula tells us how time in a localstationary frame is related to observed time in a moving frame.

3.2 Length Contraction in Special Relativity

Here’s another thought experiment. As before there is a stationary frameof reference we’ll call the S frame and a moving frame of reference we’ll callthe S′ frame. The S′ frame is moving relative to the S frame with speedV in the positive X direction (see Figure 7). The clocks in both frames areset so that at time t = 0 in the S frame, t′ = 0 in the S′ frame. Again asbefore, x = x′ = 0 when t = t′ = 0. What we are saying is that at the startof the experiment the two frames coincide in both space and time, althoughthe S′ frame is moving at speed V .

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S frame

S’ frame

V

at time t = 0

at time t’ = 0

Light pulse emitted at x = x’ = 0

X axis and X’ axis coincide

Y axis andY’ axiscoincide

Figure 7: S frame and S′ frame at t = t′ = 0

At time t = t′ = 0 a light pulse is emitted at the location x = x′ = 0 (that’sthe star shape in Figure 7).

The light pulse creates a spherical wave front that propagates at the speedof light, c, away from the point at which it was emitted (see Figure 8). We’relimited to two dimensions in the diagram, so the wave front appears as anarc.

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S frameS’ frame

Vat time t = T

at time t’ = T

d = c x T

Y axis Y’ axis

d’ = c x T

X axis and X’ axis coincide

Figure 8: S frame and S′ frame at t = t′ = T

Principle 2 says that the speed of light is the same for all observers. There-fore the spherical light wave propagates in the S frame away from the pointx = 0 and it also propagates in the S′ frame away from the point x′ = 0.

After time T the wave front has moved outward from both x = 0 and x′ = 0as shown in Figure 8. The diagram shows what the wave fronts would looklike to observers in the S and S′ frames. We notice that the two wave frontsdo not coincide since the S′ frame has moved a distance d = c × T alongthe X axis while the wave front was propagating. But it has to be the samewave front, since it started from one light pulse at time t = t′ = 0.

Length contraction resolves this discrepancy. Without going through all themath again, we’ll just give the equation:

L′ =

√1− v2

c2L. (8)

That is, distance in the direction of motion as measured by a stationaryobserver is shorter than distance measured by an observer in the movingframe.

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3.3 The Lorentz Transformation

The key issue that Special Relativity was invented to resolve is how to recon-cile observations by made by observers in reference frames moving relativeto each other. In this case, how can two observers, one at rest in the Sframe and one at rest in the S′ frame, both see the wave front propagatingas a spherical wave front in their respective frames when it’s really the samewave front?

The general answer is given by the Lorentz transformations which define therelationship between the space and time coordinates in two frames movingrelative to each other with a constant speed.

The relationship for the space (distance) coordinate is:

x′ =x− vt√1− v2/c2

(9)

and the relationship for the time coordinate is:

ct′ =ct− vx/c√1− v2/c2

. (10)

These formulas are more complicated than in equation (7) because theydescribe how time and space coordinates in one frame of reference are relatedto time and space coordinates in a different frame of reference that is inrelative motion at a constant speed v. 10

If we want to compare clocks in the two reference frames, we use equation(10). For example, there is a clock in the S frame that ticks off minutes.Let time t1 be 12:00 and time t2 be 12:01. Then t2 − t1 = 1 minute or 60seconds. Suppose that frame S′ is moving at speed V = 0.2×c ≈ 60, 000, 000kilometres per second. Then, since x = 0 because the clock is not movingin the S frame,

10See Appendix A for an explanation of how we get the non-relativistic formula formoving frames.

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t′ =t− vx/c2√1− v2/c2

=t− (v × 0)/c2√

1− v2/c2(11)

=t√

1− v2/c2(12)

=60√

1− (0.2)2(13)

=60√

1− .04(14)

=60√.96

(15)

≈ 60

.98(16)

≈ 61.2 (17)

seconds. So a minute in the S frame appears to the observer in the S′ frameto last for 61.2 seconds: “moving clocks run slow”.

4 Conclusion

What we have tried to show is that the theory of Special Relativity providesa way to coordinate measurements in frames of reference that are movingrelative to each other in a way that accommodates the second principle, thatthe speed of light is the same in every inertial reference frame.

Special Relativity is a kinematical theory, which means that it describesmotion but not the forces that cause the motion. This is important: whenthere are forces acting between or on the objects in a system the objects areaccelerating or decelerating and by definition Special Relativity does notapply to accelerated frames of reference.

It is also important to understand that all the measurements that are per-formed in Special Relativity utilize the transmission of electromagnetic ra-diation (light, radio waves, X-rays, etc.) between objects and observers. Wenever see things “immediately” because there is always a time lag betweenwhen light is emitted from an object and when it arrives at our eyes. In

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everyday life where we observe objects in the world around us through oureyes we can ignore the time lag because it’s so tiny it doesn’t make a dif-ference. The time lag is noticeable when we communicate via radio signalswith people who are far away, however, so unlike in Einstein’s time, thespeed of light is a real part of our everyday life.

There is a very rich literature that deals with Special Relativity, rangingfrom simple to sophisticated in the type of mathematics used to describethe theory.

I would not recommend Einstein’s original paper as translated into English[Einstein (1905)]. There is an ambiguity in the language and the derivationsthat makes it much harder to follow than explanations that have appearedsince. It is included in the references in case you would like to see wherethis all started in 1905.

For those with a mathematical bent there is a paper by Peter Grogono[Grogono (2007)] that presents a derivation of the Lorentz transformationthat is fairly straightforward. The paper includes additional theoreticalmaterial and examples that are interesting.

Relativity for Scientists and Engineers [Skinner (1982)] is very readable.

The book Flat and Curved Space-times ([Ellis (1988)]) gives a very clearexplanation of Special Relativity and goes beyond into the realm of basicGeneral Relativity.

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Appendix A Non-relativistic relative motion

The non-relativistic formula that describes how distance coordinates be-tween frames moving relative to each other is:

x′ = x− vt. (18)

x′ is the distance to a stationary object (like a traffic light) measured in amoving frame of reference (a car moving along a street)and x is the distanceto the traffic light measured by a person standing still on the street. v isthe speed of the car and t is the time that has passed since the car startedto drive away. See Figure A.

Jean(standing still)

Car moving at 60 km per hour = 1 km per minute

Traffic light

Jean’s distance to traffic light is x = 2 km

Car’s distance to traffic light is x’ = (2 km) – (1 km per min) × (minutes)

The car is moving along the road at 60 km per hour. There is a traffic light2 kilometres along the road from where the person (Jean) is standing, whichis the same place where the car starts to drive away. The “coordinate” ofthe traffic light according to Jean is +2.0 (in kilometres). If we use x torepresent distance according to Jean, x = 2.0.

How far away is the traffic light from the car? The distance is changingas the car moves. After 30 seconds at 60 km per hour (which is 1 km perminute) the car has moved 0.5 kilometres and the traffic light is now only1.5 kilometres away. If we use x′ to represent the distance of the traffic lightfrom the car, x′ = 1.5 after 30 seconds.

How is the distance of the traffic light in the car’s coordinate, x′ related toJean’s coordinate, x? It is given by equation (18).

Using 0.5 minutes (30 seconds) for t and 1 kilometre per minute (which is 60km per hour) for v we get the distance to the traffic light according to thecar as x′ = x− v × t = 2.0− 1× 0.5 = 2.0− 0.5 = 1.5 kilometres. Similarly,after the car has driven for one minute the distance to the traffic light fromthe car is x′ = x− v × t = 2.0− 1× 1.0 = 2.0− 1.0 = 1.0 kilometre.

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When we know t, the length of time that the car has been moving and v,the car’s speed, we can express the car’s distance from the traffic light interms of Jean’s distance from the traffic light:

(Car’s distance from traffic light) = (Jean’s distance from traffic light) -(car’s speed) × (number of minutes).

Now look back at equation (9) above. It’s the same equation except for thatcomplicated

√1− v2/c2 bit. But if the speed v is very, very small compared

to the speed of light c, v2/c2 is even smaller and√

1− v2/c2 is almost exactly1, which means that equation (9) is almost exactly x′ = x − vt. Equation(9) is the Special Relativistic form of equation (18).

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References

[Einstein (1905)] Einstein, Albert 1905, “On the Electrodynamics of Mov-ing Bodies”, translation by John Walker, http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf.

[Ellis (1988)] Ellis, George F.R. and Williams, Ruth M., Flat and CurvedSpace-Times, Oxford University Press, New York, 1988.

[Grogono (2007)] Grogono, Peter 2007, “Deriving Special Relativ-ity”, http://users.encs.concordia.ca/~grogono/Writings/

relativity.pdf.

[Skinner (1982)] Skinner, R., Relativity for Scientists and Engineers, DoverPublications Inc., 1982.

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http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf

http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf

http://users.encs.concordia.ca/~grogono/Writings/relativity.pdf

http://users.encs.concordia.ca/~grogono/Writings/relativity.pdf

Special Relativity Basics · 2019. 12. 5. · William Shaw Special Relativity Basics reaches Earth...

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