 # Illustrated Special Relativity Through Its Paradoxes

John de Pillis
José Wudka
Illustrations and animations by José Wudka
Series: Spectrum
Edition: 1
Pages: 481
https://www.jstor.org/stable/10.4169/j.ctt13x0nbx

1. Front Matter
(pp. i-viii)
(pp. ix-xvi)
3. I. A FIRST PASS
• Preface
(pp. 2-10)
• 1 Introduction to the Paradoxes
(pp. 11-42)

Aristotle (384–322 B.C.) believed that a stone fell toward the earth because the stone and the earth were both in the “Earth” category among the four basic elements, Earth, Air, Fire, and Water. According to this reasoning, smoke, which consists of air and fire, wants to be closer to the sky (air) and further from an unlike element (earth). Hence, earthly objects move naturally toward the earth while “airy” objects move naturally upward. The assumed fifth element, the heavenly substance he called the Quintessence along with an assumed Prime Mover accounted for the “perfect” circular and uniform motion of...

• 2 Clocks and Rods in Motion
(pp. 43-53)

(2.1.1a) Starting the Clock. The clock is started when an initial photon passes the clock and triggers the clock’s first tick or ping as required in Definition (2.1.1). Although 0.00 is a natural starting time, the initial starting time can be set arbitrarily.

(2.1.1b) Setting the Tick Rate. A second passing photon defines the length of the time interval that will be repeated for all subsequent ticks or pings—the clock “remembers” the time interval between the first and second photons and it waits for this interval to pass before it...

• 3 The Algebra of Frames
(pp. 54-65)

We shall see how a physical idea—the inertial frame of reference—is modeled as a mathematical concept—a set of ordered [x, t]pairs or vectors. The following is a repeat of Definitions (1.2.1), pg. 13 and (1.5.1), pg. 16.

For example, in three-space each point is described by its space coordinates [ x, y, z] and its time coordinate t. The quartet [ x, y, z, t] therefore describes the state of each point in the frame of reference F.

As seen in Figure (3.1.2), each point x on the ruler of the frame of reference corresponds to...

• 4 The Graphing of Frames
(pp. 66-82)

Our objective is to provide an intuitive way of graphing particles in motion.

In 3-dimensional space, we may represent a particle by the quartet [x, y, z, t]. When the particle moves, it will change its location with time, that is, the space coordinates become functions of the time coordinatet, that is, x = x(t), y = y(t), and z = z(t). The set of these coordinates gives us a picture of themotionof that point as well as its position.

We shall concentrate on motion in 1-dimensional space only so that points are represented by coordinates of...

4. II. GALILEAN TRANSFORMATIONS OF FRAMES
• 5 Galilean Transformations
(pp. 84-95)

We assume frame$fB$moves with constant speedvrelative to frame${f^A}$.

The set of all spacetime ordered pairs$\{ [x,t]\}$for inertial frameFform a mathematical vector space (A.2.1).

The line-of-sight function,${G_v}:{F^A} \to {F^B}$sending vector space${f^A}$to vector space${f^B}$, which takes straight lines to straight lines, where${G_v}:() = (),$, is necessarily linear (A.4.1b).

If the spacetime group showing overlapping frames${f^A}$and${f^B}$is 2-dimensional, then each output vector of the function${G_v}{F^A} \to {F^B}$is the 2 × 1 column${G_v}$(X) which Theorem (A.4.6), pg. 372, tells us can always be computed by multiplying the input vectorXon the...

5. III. The SPEED OF LIGHT IS CONSTANT
• 6 Constant c in Spacetime
(pp. 97-102)

In the definition following, we extend the notion of a spacetime diagram (4.1.2) to a Minkowski diagram by adding the time-paths (worldlines) of leftward traveling photons and rightward traveling photons ! (e.g., Figure (6.1.2)).

Diagram (6.1.2) shows perpendicular XAand TAaxes for Ashley’s frame. There are four synchronized worldlines (see Def. (3.4.5a)) namely,

The two worldlines of the synchronized leftward and rightward traveling photons, respectively.

Ashley’s vertical worldline in frame FAwhich coincides with TA, the FAtime axis,

Bernie’s slanted worldline in frame FBwhich coincides with TB, the FBtime axis that forms angle with the vertical...

6. IV. LORENTZ TRANSFORMATIONS OF FRAMES
• 7 Lorentz Transformations
(pp. 104-117)

In the Minkowski diagram (7.1.2) following, the worldlines are plotted for four players:

Ashley at the origin in inertial frame${f^A}$whose worldline is vertical,

Bernie at the origin in synchronized frame${f^B}$(3.4.5a) moving with relative speedv.Bernie’s worldline coincides with${T^B}$, the time axis of${T^B}$.

Leftward and a rightward traveling photons whose worldlines contain the line segments from$\boxed1$to$\boxed3$and from$\boxed2$to$\boxed4$, respectively.

The values$P = \left( {1 + \nu /c} \right)$and$q = \left( {1 + \nu /c} \right)$in Figure (7.1.2) will be established in (7.1.3f).

Line-of-Sight Coordinates of Line-of-Sight Coordinates of Point$\boxed0$, Fig. (7.1.2). Ashley and...

• 8 The Hyperbola of Time-Stamped Origins
(pp. 118-125)

As noted on page 83, non-relativistic Galilean physics assumes that time is universal, which implies there is one universal clock whose readings are valid for all observers, regardless of their relative motion. This universal clock physical property is interpreted mathematically by saying that time is invariant under the Galilean line-of-sight transformation (5.4.2), pg. 90,${G_v}:{[x,t]_A} \to {[x',t]_B}$. That is to say, the time entries of the input coordinate pair${[x,t]_A}$and the output coordinate pair${[x',t]_B}$are always equal.

Similarly, we may ask what quantity might be invariant under the Lorentz transformation${L_v}:{[x,t]_A} \to [x',t']B$

A first glance might not...

7. V. GRAPHIC RESOLUTION OF THE PARADOXES
• 9 The Accommodating Universe Paradox
(pp. 127-131)

Summary of the Accommodating Universe, Section (1.9). If a skateboarder in frame${f^S}$races on a straight-line track in${f^E}$towards the finish line, then the faster he runs, the shorter the track becomes from his point of view in frame${f^S}$—the Universe isaccommodatingin the sense that the faster he goes, the shorter the track becomes from his point of view and, hence, the closer the finish line is to the runner.

The paradox lies in the statement that if the length of the track can be made arbitrarily small as viewed from fame${f^S}$, then the skateboarder...

• 10 The Length-Time Comparison Paradoxes
(pp. 132-140)

We have two inertial reference frames,${f^A}$and${f^B}$, with respective observersAandBthat are at rest. Observer A sees ObserverBmove with speedvandBsees A move with speed −v.BothAandBhave identical clocks and rulers. Then the first results, (1.7.2a) and (1.7.2b), along with the illustrations of (1.10.4a) and (1.10.4b) lead to the following observations:

Both moving clocks run slow. Each observer (on the platform, or in the railway car) sees the other’s moving clock run slower than his/her own clock.

Both moving rulers shrink. Each observer sees the other’s...

(pp. 141-152)

A graphical analysis of Figure (11.3.1) completely resolves the twin paradox (Section(1.11 )) which we restate as follows:

At time 0 years, one twin leaves Earth traveling at an outbound speed of 0.8c . After 25 years by the Earth clock, the traveling twin instantly boards a returning spaceship which travels in the opposite direction, at an inbound speed of 0.8c. After 25 more years, or at time 50 years by the Earth clock, the traveling twin returns to Earth to greet his 50 year-old twin while he, himself, is only 30 years old.

(pp. 153-162)

Static Train and Tunnel are of Equal Length.

The common at-rest length of the tunnel in frame${f^A}$and the train in frame

${f^B}$is d₀.

In${f^A}$, the origin x = 0 is at the left entrance of the tunnel — x = d₀ marks the right entrance.

In${f^B}$, x = 0 is at the right end (the front) of the train — x = −d₀ marks the left end (the rear) of the train.

Train in Motion Shrinks, Tunnel is Stationary.

The frame${f^B}$of the train has speed v relative to the stationary frame${f^A}$of the tunnel....

(pp. 163-170)

Restatement of the pea-shooter paradox (1.13): You observe an airplane traveling away from you at speedwwhile the plane launches a spaceship (in the same direction of the plane) whose instantaneous speed, as measured by the pilot, isv.Your intuition and experience should tell you that you would measure the spaceship speed to bev+w.In fact, the actual spaceship speed is the theoretically accurate Lorentz sum,v$\oplus$w(7.3.1), which, due to the low speeds involved, is practically indistinguishable from the usual arithmetic sumv+wfor speedsvandw.

Our powers...

(pp. 171-183)

In frame${f^A}$, two vertical plates, each with horizontally aligned holes, have left surfaces d₀ units apart. A rivet in${f^B}$with at-rest shank length d₀ travels rightward towards the two aligned holes which are large enough so that the shank but not the head will pass through. A bug is in front of the hole in the right plate.

Viewed from the (stationary) plates, the moving rivet has length${\sigma _\nu }{d_0}\angle {d_0}$— hence, the bug is not injured when the rivet head collides with the left plate. But viewed from the (stationary) rivet, the moving plates are separated by only${\sigma _\nu }{d_0}\angle {d_0}$...

8. VI. ENERGY AND MASS
• 15 E = mc²
(pp. 185-197)

It might seem that our analyses of the paradoxes through Minkowski diagrams have little to do with$E = m{c^2},$the relation of a massmat rest and its energy equivalentE.

As it turns out, the structure underlying the pea-shooter paradox is also essential to showing that$E = m{c^2},$The structure in question is the Lorentzian addition of speeds (7.3.1 ).

Panel @ : Coordinates for the endpointsPandQare found in two frames of reference: P = p[−c, 1] and Q = q[c, 1] relative to the frame${F^A}$with perpendicular X and T axes, while P =...

9. VII. THE MATHEMATICS OF WAVES AND LIGHT
• 16 The Nature of Waves
(pp. 199-216)

A fundamental starting point of Einstein’s paper, , is that the speed of light is the same for all observers, whether or not the light source is moving. Equivalently (and non-intuitively), the speed of light is a constant 186,000 miles per second (in a vacuum) whether or not the observer is moving (see(1.5.4 )).

Why should we believe such a non-intuitive idea? The following chapters provide the answer.

The constant speed of light, independent of frame of reference (point of view) was predicted decades before Einstein’s paper, by Maxwell’s equations which connected electricity, magnetism, and optics. These equations also...

• 17 Measuring the Speed of Light
(pp. 217-237)

Why IscUsed to Represent the Speed of Light in a Vacuum? One credible reason is thatcis the first letter of Celeritas, a Latin word that means “speed.” However, from Maxwell’s writings in 1865 through Einstein’s papers in 1905, the letterV—notc—was used for the speed of light.

During this time, the symbol,c,had been waiting in the wings on the stage of history. As early as 1856, the letterc,as related to the speed of light, was used by Weber and Kohlrausch in their work on an electrodynamics force law. Eventually,...

10. VIII. MAXWELL’S EQUATIONS
• 18 Maxwell’s Mathematical Toolkit
(pp. 239-261)

We present the mathematical machinery used by Maxwell to develop his four equations (21.7.3a)–(21.7.3d). This requires a significant amount of calculus, much of which will be reviewed in this chapter. Use of calculus may seem daunting but the rewards are great. Maxwell’s equations have stood the test of time, as they describe not only the motion of charged particles, properties of magnets and electromagnetic waves at all frequencies, but they apply to phenomena in the quantum regime as well.

The language of mathematics is essential—and there does not seem to be a simpler path.

Given a set of...

• 19 Electric and Magnetic Fields
(pp. 262-275)

In 1873, the Scottish physicist James Clerk Maxwell (1831–1879) published ATreatise on Electricity and Magnetismin which he extended and unified discoveries of Charles-Augustin de Coulomb Carl Friedrich Gauss (1777–1855), André Marie Ampére (1775–1836), and Michael Faraday (1791–1867).

With only four fundamental equations (listed in Section (21.7)),Maxwell was able to integrate the theories of electricity, magnetism, and optics. Moreover, these equations enabled him to make the armchair calculation that the speed of electromagnetic waves in a vacuum wasc— 3×10³ meters/second, a number that exactly coincided with the experimentally measured speed of light.

Merely a...

• 20 Electricity and Magnetism: Gauss’ Laws
(pp. 276-288)

We consider a vector field$\overrightarrow E$(18.5.3), pg. 248, which is composed of field vectors including${\overrightarrow E _i},i = 1,2,...,k.$For example, the individual field vectors${\overrightarrow E _i}i$might be water speeds, electric field vectors, magnetic field vectors, or gravitational field vectors.

As per (18.3.3), we can represent a small element with area$\Delta$A as a vector$\overrightarrow {\Delta A}$whose length is$\Delta$A. That is,$\left\| \Delta \right.\overrightarrow A \left\| = \right.\Delta A.$

Our objective is to measure the flow of“E-stuff”or so-calledfluxpassing through a surface. To do this, we only measure the perpendicular component of${\overrightarrow E _i}$which has length where$\left\| {{{}_i}\cos (\theta )} \right\|$is the angle between vectors${{{}_i}}$

a a...

• 21 Towards Maxwell’s Equations
(pp. 289-303)

Magnetism is generated by moving electrical charges and (in a quantum setting that has nothing to do with moving electrons) by spin on particles such as the neutron and the neutrino.

Some atoms become tiny magnets because of their moving electrons and also because of the spin of the atom’s electrons, protons, and neutrons. Not all atoms have this property.

In 1819, Hans Christian Oersted noticed that a compass needle was deflected by a wire carrying a current. Then, in 1820, Jean- Baptiste Biot (1774–1862) and F’elix Savart (1791–1841) conducted experiments that measured the force exerted on a...

• 22 Electromagnetism: A Qualitative View
(pp. 304-312)

We show how Maxwell’s equations (21.7) and some accurate laboratory measurements lead to an expression forc,the speed in a vacuum of electromagnetic (EM) waves. Maxwell did not know that light was a subset of his electromagnetic waves. Other electromagnetic waves include infrared, and ultraviolet radiation, microwaves, radio waves, andXand gamma rays (see Figure(22.3.3)).

Electromagnetic waves are time-dependent. One of the simplest examples is ordinary household alternating current (AC) in a wire where electrons flow—alternately and repeatedly—in one direction, and then in the opposite direction.

Figure (22.1.1) shows a vertical wire with an alternating...

• 23 Electromagnetism: A Quantitative View
(pp. 313-333)

The proper linking of mathematics to reality is not always clear, although the power of mathematical modeling is rarely in doubt. (For an example of a model based on a false assumption, see (24.6 ), pg. 348.) A befuddled character, Miss Withers, articulates this point of view in the 1934 book,The Puzzle of the Silver persian26, by Stuart Palmer, where we read

Miss Withers . . . sat and stared at the red coals of her fire . . . they presented innumerable, fantastic pictures, but never did they suggest the clear and definite answers to this complex...

11. IX. FINAL THOUGHTS
• 24 Epilogue: Final Thoughts
(pp. 335-357)

Old Mathematics. Thousands of years of mathematics have produced more than mere lists of formulas and deduced results. One component of our intellectual heritage that has been defined and nourished by mathematical progress is theprocess of proof.

The theorem of Pythagoras of Samos (circa580–500 B.C.) generalized a property of right triangles that had been known for a few special cases. (aRight Triangleis any triangle in which two sides form a 90⁰ angle) For examle, it was Known that for the (3, 4, 5), right triangle, 3² + 4² = 5², and for the (5, 12,...

12. X. APPENDICES
• A Linear Algebra Overview
(pp. 359-375)
• B Hyperbolic Functions
(pp. 376-384)
• C Deconstructing a Moving Train
(pp. 385-387)
13. XI. SUPPLEMENTAL MATERIAL
• D Dimensional Analysis
(pp. 389-395)
• E Rings of Functions and Square Matrices
(pp. 396-405)
• F The Scientific Method
(pp. 406-434)
• G Logic of the Scientific Method
(pp. 435-446)
14. Bibliography
(pp. 447-450)
15. Index
(pp. 451-464)