Einstein for Everyone Lecture 3: Special Relativitystrangebeautiful.com/lmu/lectures/einstein-lecture-03-sr.pdf · Einstein for Everyone Lecture 3: Special Relativity ... Problems

Einstein for EveryoneLecture 3: Special Relativity

Dr. Erik Curiel

Munich Center For Mathematical PhilosophyLudwig-Maximilians-Universitat

Summary of Historical Background Emission Theories Einstein’s Motivation Relativity Signaling and Synchronization Einstein’s Definition Light Clock Relativity of Simultaneity FAQs Velocity Addition

1 Summary of Historical Background2 Emission Theories

IntroductionEinstein’s Objections

3 Einstein’s Motivation4 Relativity

Einstein’s BreakthroughAbsolute Time and SimultaneityLight Signals

5 Signaling and Synchronization6 Einstein’s Definition7 Light Clock

Time Dilation8 Relativity of Simultaneity9 FAQs

10 Velocity Addition


Conventional Wisdom ca. 1905

Principle of Relativity fails . . .

- True rest state of the ether- Motion through the ether has real dynamical effects

. . . but we cannot directly establish failure observationally

- Dynamical effects block all attempts to measure motionthrough ether

- Lorentz: not a question of limited precision; theoreticalprediction

. . . so the principle holds observationally but not theoretically


























Introduction

Emission Theories 101

Basic Idea

Light propagates at c with respect to source

Image from Norton, Einstein for Everyone


Einstein’s Objections

Problems with Emission Theory?

Requires Modification of Electrodynamic Theory . . .

- Einstein pre-1905: attempts, all unsatisfactory

- Ritz (1908), Ehrenfest (1912)

Einstein’s Objections (cf. Norton 2004 )

I considered what would be more probable, the principleof the constancy of c, as was demanded by Maxwell’sequations, or the constancy of c, exclusively for anobserver sitting at the light source. I decided in favor ofthe first, since I was convinced that each light [ray]should be defined by frequency and intensity alone, quiteindependently of whether it comes from a moving or aresting light source. (Einstein to Ehrenfest, 1912)


Einstein’s Objections

Problems with Emission Theory?

Requires Modification of Electrodynamic Theory . . .

- Einstein pre-1905: attempts, all unsatisfactory

- Ritz (1908), Ehrenfest (1912)

Einstein’s Objections (cf. Norton 2004 )

I considered what would be more probable, the principleof the constancy of c, as was demanded by Maxwell’sequations, or the constancy of c, exclusively for anobserver sitting at the light source. I decided in favor ofthe first, since I was convinced that each light [ray]should be defined by frequency and intensity alone, quiteindependently of whether it comes from a moving or aresting light source. (Einstein to Ehrenfest, 1912)










Chasing A Beam of Light

If I pursue a beam of light with the velocity c (velocity of light in avacuum), I should observe such a beam of light as anelectromagnetic field at rest though spatially oscillating.

There seems to be no such thing, however, neither on the basis ofexperience nor according to Maxwell’s equations.

From the very beginning it appeared to me intuitively clear that,judged from the standpoint of such an observer, everything wouldhave to happen according to the same laws as for an observer who,relative to the earth, was at rest. For how should the first observerknow or be able to determine, that he is in a state of fast uniformmotion? (Einstein 1946)

“On the Electrodynamics of Moving Bodies”Einstein (1905)

Take, for example, the reciprocal electrodynamic action of amagnet and a conductor. The observable phenomenon heredepends only on the relative motion of the conductor and themagnet, whereas the customary view draws a sharp distinctionbetween the two cases in which either the one or the other of thesebodies is in motion. For if the magnet is in motion and theconductor at rest, there arises in the neighbourhood of the magnetan electric field with a certain definite energy, producing a currentat the places where parts of the conductor are situated. But if themagnet is stationary and the conductor in motion, no electric fieldarises in the neighbourhood of the magnet. In the conductor,however, we find an electromotive force, to which in itself there isno corresponding energy, but which gives rise — assuming equalityof relative motion in the two cases discussed — to electric currentsof the same path and intensity as those produced by the electricforces in the former case.

Einstein (1905) continued

Examples of this sort, together with the unsuccessful attempts todiscover any motion of the earth relatively to the “light medium,”suggest that the phenomena of electrodynamics as well as ofmechanics possess no properties corresponding to the idea ofabsolute rest. They suggest rather that, as has already been shownto the first order of small quantities, the same laws ofelectrodynamics and optics will be valid for all frames of referencefor which the equations of mechanics hold good.


Einstein vs. Conventional Wisdom

Lorentz (and others)

Immobile ether, definiterest frame

Laws apply in ether’s restframe

Null results: consequenceof dynamical effects ofmotion through ether. . . length contraction,time dilation, etc.

Einstein

Ether is superfluous!

Laws apply in all inertialframes

Relativity(observer-dependence) ofspatial distance, temporalintervals


Einstein vs. Conventional Wisdom

Lorentz (and others)

Immobile ether, definiterest frame

Laws apply in ether’s restframe

Null results: consequenceof dynamical effects ofmotion through ether. . . length contraction,time dilation, etc.

Einstein

Ether is superfluous!

Laws apply in all inertialframes

Relativity(observer-dependence) ofspatial distance, temporalintervals









Einstein’s Breakthrough: Revising Time

But a friend of mine living in living in Bern (Switzerland) [MicheleBesso] helped me by chance. One beautiful day, I visited him andsaid to him: ‘I presently have a problem that I have been totallyunable to solve. Today I have brought this “struggle” with me.’We then had extensive discussions, and suddenly I realized thesolution. The very next day, I visited him again and immediatelysaid to him: ‘Thanks to you, I have completely solved my problem.’My solution actually concerned the concept of time. Namely, timecannot be absolutely defined by itself, and there is an unbreakableconnection between time and signal velocity.Using this idea, I could now resolve the great difficulty that Ipreviously felt. After I had this inspiration, it took only five weeksto complete what is now known as the special theory of relativity.(Einstein, 1922 lecture)


Absolute Time and Simultaneity

Absolute Time (Classical, Newtonian Picture)

From Roger Penrose, The Road to Reality

Absolute Time

Spacetime composed of“events”

Global: well-defined timeinterval between any twoevents

Absolute: time intervaldoes not depend on stateof motion



Question

How to establish simultaneity of distant events?

- Synchronized Clocks, . . . but how are they synchronized?. . .“. . . there is an unbreakable connection between time andsignal velocity.”

Signal Speed

- Instantaneous SignalAssumed in earlier theories

- Signal with Finite Speed (e.g. light)



Question



Signal Speed





Question



Signal Speed




Light Signals

Signaling and Simultaneity 1

Setup (see Janssen, “Appendix on SR” §1.2)

- Bob: in the middle of rail car, lights flash once at each end(L1 and L2)

- Flashes reach Bob at the moment he passes Al- Train moving at velocity v past Al


Light Signals


Bob’s Description

Light moves at velocity c with respect to me from each light

Distance traveled by pulse from L1 = distance from L2

Therefore, lights flashed simultaneously


Light Signals


Al’s Description

Light moves at velocity cwith respect to me fromeach light

Bob moves away from L1

and towards L2

Distance traveled by pulsefrom L1 > distance fromL2

Pulses reach Bob at sametime

Therefore, L1 occurredbefore L2


Light Signals

What’s Relative?

Bob’s Description

Speed of light = c

Light pulses reach Bobsimultaneously

Distance traveled by pulsefrom L1 = Distancetraveled by pulse from L2

Flash L1 simultaneouswith L2

Al’s Description

Speed of light = c


Distance traveled by pulsefrom L1 > Distancetraveled by pulse from L2

Flash L1 earlier than L2

Relative Quantities

Time elapsed and spatial distance between events relative toinertial observers


Light Signals

What’s Relative?

Bob’s Description

Speed of light = c




Al’s Description

Speed of light = c




Relative Quantities



Light Signals

What’s Relative?

Bob’s Description

Speed of light = c




Al’s Description

Speed of light = c




Relative Quantities



Light Signals

Einstein’s Insight

Einstein’s Two Postulates (1905):

1 Principle of Relativity

2 Light Postulate

How can these two postulates be reconciled?

(agreement about speed of light, disagreement about spatial and

temporal intervals)Possibilities:1. Revise electrodynamics2. Revise concepts of space and time! (critical analysis of earlierassumptions regarding space and time)


Light Signals

Velocity Addition

4 Michel Janssen

expected the relativity principle to break down. An observer in uniform motionthrough the ether is not equivalent to an observer at rest in the ether. A wavein the ether only moves with the same constant velocity in all directions withrespect to the latter. Yet experiment suggested that the relativity principle holdsfor light as well. The relativity principle is compatible with light consisting ofparticles moving through empty space. But then the velocity of light dependson the velocity of the source from which the light particles are emitted, whichcontradicts the light postulate. If we insist on having both the relativity and thelight postulate, light, it seems, can neither be a particle nor a wave.

Figure 2. The postulates and adding velocities.

Fig. 2 illustrates the problem of combining Einstein’s two postulates in a slightlydifferent way. We have two observers, Al and Bob, moving with respect to oneanother (like the drivers of the SUVs in Fig. 1). They examine a flash of lightemitted by a light bulb, a cork shooting out of a champagne bottle, and a bulletfired from a gun. Common sense tells us that the light, the cork, and the bulletwill have different velocities for Al and Bob. Special relativity confirms this in thecases of the cork and the bullet, even though, as we shall see, it calls for correctionsof the common-sense values v + vcork and v + vbullet shown in Fig. 2. In the case oflight, however, it is a direct consequence of Einstein’s two postulates that Al andBob, despite being in motion with respect to one another, register the exact samevelocity! How can this possibly be? It is clear that something will have to give.

What Einstein showed in his 1905 paper is that once we accept his two postulates—as we should given all the empirical evidence backing them up—we have to giveup some of our common-sense ideas about space and time. Now it is one thing toconcede when confronted with the relentless logic of Einstein’s 1905 paper that ourold ideas were indeed nothing but prejudices; getting comfortable with the newideas that Einstein put in their place is a different matter. Here the 1905 paper isof little help. It does not tell us how to visualize the new relativistic ideas about

Images by Laurent Taudin, from Janssen(forthcoming)


Light Signals

Velocity Addition

4 Michel Janssen





Images by Laurent Taudin, from Janssen(forthcoming)

4 Michel Janssen






Light Signals

Revising Velocity Addition, Option 1

Postulate 1: Principle ofRelativity

Light moves at c withrespect to emitter

. . . modify theory,compatible with nullresults

4 Michel Janssen






Light Signals

Revising Velocity Addition, Option 2

Both postulates → allinertial observers measuresame speed of light

. . . but velocity additionfollows from usual ideasabout space and time

. . . so these need to berevised!

4 Michel Janssen






Light Signals

Einstein’s Path to Special Relativity

Three Components of Einstein’s Discovery (following Norton)

Astute analysis of new and surprising experiments

Deeply reflective philosophical analysis of the nature of time

Solving an incongruous and overlooked problem in thefoundations of electricity and magnetism










Signaling and Simultaneity

Meaning of time for distant events

- Newtonian assumption: instantaneous signal, globalsynchronization of clocks and absolute time

- . . . But in fact use light signals. (1) Finite velocity; (2) Doesnot obey usual velocity addition rule

Consequences of using light signals

- Observers in relative motion disagree about spatial distances,temporal intervals between two events


Signaling and Simultaneity

Meaning of time for distant events

- Newtonian assumption: instantaneous signal, globalsynchronization of clocks and absolute time

- . . . But in fact use light signals. (1) Finite velocity; (2) Doesnot obey usual velocity addition rule

Consequences of using light signals

- Observers in relative motion disagree about spatial distances,temporal intervals between two events


Contrast

Newtonian View

Moving Rods: samelength

Moving Clocks: tick atsame rate

Velocities: simple additionv′b = vb + v

Relativistic View

Moving Rods: contract

Moving Clocks: tick moreslowly

Velocities: addition with“correction factor,” suchthat v′b < c


Contrast

Newtonian View

Moving Rods: samelength

Moving Clocks: tick atsame rate

Velocities: simple additionv′b = vb + v

Relativistic View

Moving Rods: contract

Moving Clocks: tick moreslowly

Velocities: addition with“correction factor,” suchthat v′b < c


Measuring Absolute Time


Measuring Time

“Ideal clock”: measurestime elapsed along itstrajectory

How to “spread time”from the trajectory todistant events?

Requires synchronization


Measuring Absolute Time


Measuring Time

“Ideal clock”: measurestime elapsed along itstrajectory

How to “spread time”from the trajectory todistant events?

Requires synchronization


Einstein’s Analysis

Newtonian View

Based on implausibleassumptions: infinitesignal velocity, global time

Incompatible with theoryof electromagnetism

Relativistic View

Theory ofelectromagnetism: lightpostulate

Revise space and timeconcepts to insurecompatibility withRelativity Principle

Recover Newtonian viewsas an approximation


Einstein’s Analysis

Newtonian View

Based on implausibleassumptions: infinitesignal velocity, global time

Incompatible with theoryof electromagnetism

Relativistic View

Theory ofelectromagnetism: lightpostulate

Revise space and timeconcepts to insurecompatibility withRelativity Principle

Recover Newtonian viewsas an approximation










Simultaneity for Distant Events

Primitives:

1 Clock: local time measurements

2 Light signals: to “spread time” to distant events

Einstein on TimeNow we must bear carefully in mind that a mathematicaldescription of this kind has no physical meaning unless we arequite clear as to what we understand by “time.” We have to takeinto account that all our judgments in which time plays a part arealways judgments of simultaneous events. If, for instance, I say,“That train arrives here at 7 o’clock,” I mean something like this:“The pointing of the small hand of my watch to 7 and the arrivalof the train are simultaneous events.”It might appear possible to overcome all the difficulties attendingthe definition of “time” by substituting “the position of the smallhand of my watch” for “time.” And in fact such a definition issatisfactory when we are concerned with defining a time exclusivelyfor the place where the watch is located; but it is no longersatisfactory when we have to connect in time series of eventsoccurring at different places, or—what comes to the samething—to evaluate the times of events occurring at places remotefrom the watch.

Einstein on Time, 2If at the point A of space there is a clock, an observer at A candetermine the time values of events in the immediate proximity ofA by finding the positions of the hands which are simultaneouswith these events. If there is at the point B of space another clockin all respects resembling the one at A, it is possible for anobserver at B to determine the time values of events in theimmediate neighbourhood of B. But it is not possible withoutfurther assumption to compare, in respect of time, an event at Awith an event at B. We have so far defined only an “A time” anda “B time.” We have not defined a common “time” for A and B,for the latter cannot be defined at all unless we establish bydefinition that the “time” required by light to travel from A to Bequals the “time” it requires to travel from B to A. Let a ray oflight start at the “A time” tA from A towards B, let it at the “Btime” tB be reflected at B in the direction of A, and arrive againat A at the “A time” t′A.In accordance with definition the two clocks synchronize if

tB − tA = t′A − tB (6.1)


Illustration: Al and Bob

Distance from P to Q = d

Send a light signal from Q to P at tQ

Clock at P synchronized if it reads tQ + d/c

. . . but Al will not agree










Time Dilation

Illustration: Light Clock

Simple clock:

Two mirrors, spaced a distanceL apart

“Tick”: light beam reflecting offbase mirror

Ticks 2Lc times per second


Time Dilation

Moving Light Clock

Verticalvelocity ismuch lower(by factor√1− v2/c2)

Result: roundtrip takeslonger, clockticks slower(2Lc√1− v2/c2

per second)


Time Dilation

Time Dilation

Direct consequence of Einstein’s postulates: moving clockruns more slowly

Analysis for simple case (light clock), but also applies to otherclocks

. . . as a consequence of the principle of relativity (Exercise:why?)


Time Dilation

Time Dilation





Time Dilation

Time Dilation





Time Dilation

Michelson-Morley Reconsidered

Michelson-Morley interferometer

- Two light clocks, perpendicular to each other- Null result: these tick at the same rate

. . . consequence of principle of relativity

- Requires ticking at same rate- Length contraction to insure that this holds


Time Dilation

Michelson-Morley Reconsidered

Michelson-Morley interferometer

- Two light clocks, perpendicular to each other- Null result: these tick at the same rate

. . . consequence of principle of relativity

- Requires ticking at same rate- Length contraction to insure that this holds


Time Dilation

Length Contraction

Length for moving light clock:(√

1− v2/c2)L (shorter,

0 <√

1− v2/c2 < 1)

Derive by requiring equal travel times for two perpendicularlight clocks (see Janssen (2013) for the details)










Common Mistake

“Appearance Simultaneity”

- Corrections for different signal speeds (sound vs. light)- Signals arriving simultaneously from events at different

distances

Contrast: relativity of simultaneity holds after these have beentaken into account. . .


Relativity of Simultaneity

Einstein’s Definition of Time

- Compatible with his basic postulates- Acknowledges importance of light signals in synchronizing

distant clocks

Time elapsed between two events relative to observer

- Common sense, pre-1905 physics → approximation, reflectsexperience with low velocities

- Time dilation, length contraction as consequences


Relativity of Simultaneity

Einstein’s Definition of Time

- Compatible with his basic postulates- Acknowledges importance of light signals in synchronizing

distant clocks

Time elapsed between two events relative to observer

- Common sense, pre-1905 physics → approximation, reflectsexperience with low velocities

- Time dilation, length contraction as consequences










Relativistic Effects: FAQs

Question 1

Do length contraction and time dilation only effect special clocksand measuring devices?

Question 2

Are length contraction and time dilation real effects? (DoesLorentz contraction hurt?)



Question 1


Question 2



Relativistic Effects: Answers

Question 1


NO! The results apply to everything . . .otherwise there would be a violation of the principle of relativity


Relativistic Effects: Answers

Question 1


NO! The results apply to everything . . .otherwise there would be a violation of the principle of relativity



Question 2


Perspectival does not imply “imaginary” or “unreal”

“Relative Quantities”, values depend on state of motion ofobserver who measures them (“perspectival”): space, time,energy, momentum, electric field, magnetic field, . . .

but underlying absolute quantities all observers agree on:spatiotemporal interval, energy-momentum, electromagneticfield, . . .



Question 2


Perspectival does not imply “imaginary” or “unreal”

“Relative Quantities”, values depend on state of motion ofobserver who measures them (“perspectival”): space, time,energy, momentum, electric field, magnetic field, . . .

but underlying absolute quantities all observers agree on:spatiotemporal interval, energy-momentum, electromagneticfield, . . .

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Einstein for Everyone Lecture 3: Special Relativitystrangebeautiful.com/lmu/lectures/einstein-lecture-03-sr.pdf · Einstein for Everyone Lecture 3: Special Relativity ... Problems