Einstein for Everyone Lecture 3: Special Relativity

Transcription

1 Einstein for Everyone Lecture 3: Special Relativity Dr. Erik Curiel Munich Center For Mathematical Philosophy Ludwig-Maximilians-Universität

2 1 Summary of Historical Background 2 Emission Theories Introduction Einstein s Objections 3 Einstein s Motivation 4 Relativity Einstein s Breakthrough Absolute Time and Simultaneity Light Signals 5 Signaling and Synchronization 6 Einstein s Definition 7 Light Clock Time Dilation 8 Relativity of Simultaneity 9 FAQs 10 Velocity Addition

3 Conventional Wisdom ca 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 motion through ether - Lorentz: not a question of limited precision; theoretical prediction... so the principle holds observationally but not theoretically

4 Conventional Wisdom ca 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 motion through ether - Lorentz: not a question of limited precision; theoretical prediction... so the principle holds observationally but not theoretically

5 Conventional Wisdom ca 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 motion through ether - Lorentz: not a question of limited precision; theoretical prediction... so the principle holds observationally but not theoretically

6 1 Summary of Historical Background 2 Emission Theories Introduction Einstein s Objections 3 Einstein s Motivation 4 Relativity Einstein s Breakthrough Absolute Time and Simultaneity Light Signals 5 Signaling and Synchronization 6 Einstein s Definition 7 Light Clock Time Dilation 8 Relativity of Simultaneity 9 FAQs 10 Velocity Addition

7 Introduction Emission Theories 101 Basic Idea Light propagates at c with respect to source Image from Norton, Einstein for Everyone

8 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 principle of the constancy of c, as was demanded by Maxwell s equations, or the constancy of c, exclusively for an observer sitting at the light source. I decided in favor of the first, since I was convinced that each light [ray] should be defined by frequency and intensity alone, quite independently of whether it comes from a moving or a resting light source. (Einstein to Ehrenfest, 1912)

9 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 principle of the constancy of c, as was demanded by Maxwell s equations, or the constancy of c, exclusively for an observer sitting at the light source. I decided in favor of the first, since I was convinced that each light [ray] should be defined by frequency and intensity alone, quite independently of whether it comes from a moving or a resting light source. (Einstein to Ehrenfest, 1912)

10 1 Summary of Historical Background 2 Emission Theories Introduction Einstein s Objections 3 Einstein s Motivation 4 Relativity Einstein s Breakthrough Absolute Time and Simultaneity Light Signals 5 Signaling and Synchronization 6 Einstein s Definition 7 Light Clock Time Dilation 8 Relativity of Simultaneity 9 FAQs 10 Velocity Addition

11 Chasing A Beam of Light If I pursue a beam of light with the velocity c (velocity of light in a vacuum), I should observe such a beam of light as an electromagnetic field at rest though spatially oscillating. There seems to be no such thing, however, neither on the basis of experience 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 would have to happen according to the same laws as for an observer who, relative to the earth, was at rest. For how should the first observer know or be able to determine, that he is in a state of fast uniform motion? (Einstein 1946)

12 On the Electrodynamics of Moving Bodies Einstein (1905) Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise assuming equality of relative motion in the two cases discussed to electric currents of the same path and intensity as those produced by the electric forces in the former case.

13 Einstein (1905) continued Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the light medium, suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.

14 Einstein vs. Conventional Wisdom Lorentz (and others) Immobile ether, definite rest frame Laws apply in ether s rest frame Null results: consequence of dynamical effects of motion through ether... length contraction, time dilation, etc. Einstein Ether is superfluous! Laws apply in all inertial frames Relativity (observer-dependence) of spatial distance, temporal intervals

15 Einstein vs. Conventional Wisdom Lorentz (and others) Immobile ether, definite rest frame Laws apply in ether s rest frame Null results: consequence of dynamical effects of motion through ether... length contraction, time dilation, etc. Einstein Ether is superfluous! Laws apply in all inertial frames Relativity (observer-dependence) of spatial distance, temporal intervals

16 1 Summary of Historical Background 2 Emission Theories Introduction Einstein s Objections 3 Einstein s Motivation 4 Relativity Einstein s Breakthrough Absolute Time and Simultaneity Light Signals 5 Signaling and Synchronization 6 Einstein s Definition 7 Light Clock Time Dilation 8 Relativity of Simultaneity 9 FAQs 10 Velocity Addition

17 Einstein s Breakthrough: Revising Time But a friend of mine living in living in Bern (Switzerland) [Michele Besso] helped me by chance. One beautiful day, I visited him and said to him: I presently have a problem that I have been totally unable to solve. Today I have brought this struggle with me. We then had extensive discussions, and suddenly I realized the solution. The very next day, I visited him again and immediately said to him: Thanks to you, I have completely solved my problem. My solution actually concerned the concept of time. Namely, time cannot be absolutely defined by itself, and there is an unbreakable connection between time and signal velocity. Using this idea, I could now resolve the great difficulty that I previously felt. After I had this inspiration, it took only five weeks to complete what is now known as the special theory of relativity. (Einstein, 1922 lecture)

18 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 time interval between any two events Absolute: time interval does not depend on state of motion

19 Absolute Time and Simultaneity Question How to establish simultaneity of distant events? - Synchronized Clocks,... but how are they synchronized? there is an unbreakable connection between time and signal velocity. Signal Speed - Instantaneous Signal Assumed in earlier theories - Signal with Finite Speed (e.g. light)

20 Absolute Time and Simultaneity Question How to establish simultaneity of distant events? - Synchronized Clocks,... but how are they synchronized? there is an unbreakable connection between time and signal velocity. Signal Speed - Instantaneous Signal Assumed in earlier theories - Signal with Finite Speed (e.g. light)

21 Absolute Time and Simultaneity Question How to establish simultaneity of distant events? - Synchronized Clocks,... but how are they synchronized? there is an unbreakable connection between time and signal velocity. Signal Speed - Instantaneous Signal Assumed in earlier theories - Signal with Finite Speed (e.g. light)

22 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 (L 1 and L 2 ) - Flashes reach Bob at the moment he passes Al - Train moving at velocity v past Al

23 Light Signals Signaling and Simultaneity 2 Bob s Description Light moves at velocity c with respect to me from each light Distance traveled by pulse from L 1 = distance from L 2 Therefore, lights flashed simultaneously

24 Light Signals Signaling and Simultaneity 3 Al s Description Light moves at velocity c with respect to me from each light Bob moves away from L 1 and towards L 2 Distance traveled by pulse from L 1 > distance from L 2 Pulses reach Bob at same time Therefore, L 1 occurred before L 2

25 Light Signals What s Relative? Bob s Description Speed of light = c Light pulses reach Bob simultaneously Distance traveled by pulse from L 1 = Distance traveled by pulse from L 2 Flash L 1 simultaneous with L 2 Al s Description Speed of light = c Light pulses reach Bob simultaneously Distance traveled by pulse from L 1 > Distance traveled by pulse from L 2 Flash L 1 earlier than L 2 Relative Quantities Time elapsed and spatial distance between events relative to inertial observers

26 Light Signals What s Relative? Bob s Description Speed of light = c Light pulses reach Bob simultaneously Distance traveled by pulse from L 1 = Distance traveled by pulse from L 2 Flash L 1 simultaneous with L 2 Al s Description Speed of light = c Light pulses reach Bob simultaneously Distance traveled by pulse from L 1 > Distance traveled by pulse from L 2 Flash L 1 earlier than L 2 Relative Quantities Time elapsed and spatial distance between events relative to inertial observers

27 Light Signals What s Relative? Bob s Description Speed of light = c Light pulses reach Bob simultaneously Distance traveled by pulse from L 1 = Distance traveled by pulse from L 2 Flash L 1 simultaneous with L 2 Al s Description Speed of light = c Light pulses reach Bob simultaneously Distance traveled by pulse from L 1 > Distance traveled by pulse from L 2 Flash L 1 earlier than L 2 Relative Quantities Time elapsed and spatial distance between events relative to inertial observers

28 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 electrodynamics 2. Revise concepts of space and time! (critical analysis of earlier assumptions regarding space and time)

29 combining Einstein s two postulates in a slightly principle is compatible with light consisting of space. Light Signals But then the velocity of light depends m which the light particles are emitted, which Velocity Addition we insist on having both the relativity and the n neither be a particle nor a wave. Summary of Historical Background Emission Theories Einstein s Motivation Relativity Signaling and Synchronization Images by Laurent Taudin, from Janssen ostulates and adding velocities. (forthcoming)

30 respect to the latter. Yet experiment suggeste principle Summary of Historical is compatible Background Emission withtheories lighteinstein s consisting MotivationofRelativity Signaling and Synchronization for light as well. The relativity principle is space. Light Signals But then the velocity of light depends particles moving through empty space. But m which the light particles are emitted, which Velocity Addition on the velocity of the source from which the we insist on having both the relativity and the contradicts the light postulate. If we insist on n neither be a particle nor a wave. light postulate, light, it seems, can neither be Images by Laurent Taudin, from Janssen ostulates and adding velocities. (forthcoming) combining Einstein s two postulates in a slightly Figure 2. The postulates an

31 Light Signals Revising Velocity Addition, Option 1 for light as well. The relativity principle is particles moving through empty space. But on the velocity of the source from which the contradicts the light postulate. If we insist on light postulate, light, it seems, can neither be Postulate 1: Principle of Relativity Light moves at c with respect to emitter... modify theory, compatible with null results Figure 2. The postulates an

32 Light Signals Revising Velocity Addition, Option 2 for light as well. The relativity principle is particles moving through empty space. But on the velocity of the source from which the contradicts the light postulate. If we insist on light postulate, light, it seems, can neither be Both postulates all inertial observers measure same speed of light... but velocity addition follows from usual ideas about space and time... so these need to be revised! Figure 2. The postulates an

33 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 the foundations of electricity and magnetism

34 1 Summary of Historical Background 2 Emission Theories Introduction Einstein s Objections 3 Einstein s Motivation 4 Relativity Einstein s Breakthrough Absolute Time and Simultaneity Light Signals 5 Signaling and Synchronization 6 Einstein s Definition 7 Light Clock Time Dilation 8 Relativity of Simultaneity 9 FAQs 10 Velocity Addition

35 Signaling and Simultaneity Meaning of time for distant events - Newtonian assumption: instantaneous signal, global synchronization of clocks and absolute time -... But in fact use light signals. (1) Finite velocity; (2) Does not obey usual velocity addition rule Consequences of using light signals - Observers in relative motion disagree about spatial distances, temporal intervals between two events

36 Signaling and Simultaneity Meaning of time for distant events - Newtonian assumption: instantaneous signal, global synchronization of clocks and absolute time -... But in fact use light signals. (1) Finite velocity; (2) Does not obey usual velocity addition rule Consequences of using light signals - Observers in relative motion disagree about spatial distances, temporal intervals between two events

37 Contrast Newtonian View Moving Rods: same length Moving Clocks: tick at same rate Velocities: simple addition v b = v b + v Relativistic View Moving Rods: contract Moving Clocks: tick more slowly Velocities: addition with correction factor, such that v b < c

38 Contrast Newtonian View Moving Rods: same length Moving Clocks: tick at same rate Velocities: simple addition v b = v b + v Relativistic View Moving Rods: contract Moving Clocks: tick more slowly Velocities: addition with correction factor, such that v b < c

39 Measuring Absolute Time From Roger Penrose, The Road to Reality Measuring Time Ideal clock : measures time elapsed along its trajectory How to spread time from the trajectory to distant events? Requires synchronization

40 Measuring Absolute Time From Roger Penrose, The Road to Reality Measuring Time Ideal clock : measures time elapsed along its trajectory How to spread time from the trajectory to distant events? Requires synchronization

41 Einstein s Analysis Newtonian View Based on implausible assumptions: infinite signal velocity, global time Incompatible with theory of electromagnetism Relativistic View Theory of electromagnetism: light postulate Revise space and time concepts to insure compatibility with Relativity Principle Recover Newtonian views as an approximation

42 Einstein s Analysis Newtonian View Based on implausible assumptions: infinite signal velocity, global time Incompatible with theory of electromagnetism Relativistic View Theory of electromagnetism: light postulate Revise space and time concepts to insure compatibility with Relativity Principle Recover Newtonian views as an approximation

43 1 Summary of Historical Background 2 Emission Theories Introduction Einstein s Objections 3 Einstein s Motivation 4 Relativity Einstein s Breakthrough Absolute Time and Simultaneity Light Signals 5 Signaling and Synchronization 6 Einstein s Definition 7 Light Clock Time Dilation 8 Relativity of Simultaneity 9 FAQs 10 Velocity Addition

44 Simultaneity for Distant Events Primitives: 1 Clock: local time measurements 2 Light signals: to spread time to distant events

45 Einstein on Time Now we must bear carefully in mind that a mathematical description of this kind has no physical meaning unless we are quite clear as to what we understand by time. We have to take into account that all our judgments in which time plays a part are always 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 arrival of the train are simultaneous events. It might appear possible to overcome all the difficulties attending the definition of time by substituting the position of the small hand of my watch for time. And in fact such a definition is satisfactory when we are concerned with defining a time exclusively for the place where the watch is located; but it is no longer satisfactory when we have to connect in time series of events occurring at different places, or what comes to the same thing to evaluate the times of events occurring at places remote from the watch.

46 Einstein on Time, 2 If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an A time and a B time. We have not defined a common time for A and B, for the latter cannot be defined at all unless we establish by definition that the time required by light to travel from A to B equals the time it requires to travel from B to A. Let a ray of light start at the A time t A from A towards B, let it at the B time t B be reflected at B in the direction of A, and arrive again at A at the A time t A. In accordance with definition the two clocks synchronize if t B t A = t A t B (6.1)

47 Illustration: Al and Bob Distance from P to Q = d Send a light signal from Q to P at t Q Clock at P synchronized if it reads t Q + d/c... but Al will not agree

48 1 Summary of Historical Background 2 Emission Theories Introduction Einstein s Objections 3 Einstein s Motivation 4 Relativity Einstein s Breakthrough Absolute Time and Simultaneity Light Signals 5 Signaling and Synchronization 6 Einstein s Definition 7 Light Clock Time Dilation 8 Relativity of Simultaneity 9 FAQs 10 Velocity Addition

49 Time Dilation Illustration: Light Clock Simple clock: Two mirrors, spaced a distance L apart Tick : light beam reflecting off base mirror Ticks 2L c times per second

50 Time Dilation Moving Light Clock Vertical velocity is much lower (by factor 1 v 2 /c 2 ) Result: round trip takes longer, clock ticks slower ( 2L c 1 v 2 /c 2 per second)

51 Time Dilation Time Dilation Direct consequence of Einstein s postulates: moving clock runs more slowly Analysis for simple case (light clock), but also applies to other clocks... as a consequence of the principle of relativity (Exercise: why?)

52 Time Dilation Time Dilation Direct consequence of Einstein s postulates: moving clock runs more slowly Analysis for simple case (light clock), but also applies to other clocks... as a consequence of the principle of relativity (Exercise: why?)

53 Time Dilation Time Dilation Direct consequence of Einstein s postulates: moving clock runs more slowly Analysis for simple case (light clock), but also applies to other clocks... as a consequence of the principle of relativity (Exercise: why?)

54 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

55 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

56 Time Dilation Length Contraction Length for moving light clock: 0 < 1 v 2 /c 2 < 1) ( 1 v 2 /c 2 ) L (shorter, Derive by requiring equal travel times for two perpendicular light clocks (see Janssen (2013) for the details)

57 1 Summary of Historical Background 2 Emission Theories Introduction Einstein s Objections 3 Einstein s Motivation 4 Relativity Einstein s Breakthrough Absolute Time and Simultaneity Light Signals 5 Signaling and Synchronization 6 Einstein s Definition 7 Light Clock Time Dilation 8 Relativity of Simultaneity 9 FAQs 10 Velocity Addition

58 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 been taken into account...

59 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, reflects experience with low velocities - Time dilation, length contraction as consequences

60 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, reflects experience with low velocities - Time dilation, length contraction as consequences

61 1 Summary of Historical Background 2 Emission Theories Introduction Einstein s Objections 3 Einstein s Motivation 4 Relativity Einstein s Breakthrough Absolute Time and Simultaneity Light Signals 5 Signaling and Synchronization 6 Einstein s Definition 7 Light Clock Time Dilation 8 Relativity of Simultaneity 9 FAQs 10 Velocity Addition

62 Relativistic Effects: FAQs Question 1 Do length contraction and time dilation only effect special clocks and measuring devices? Question 2 Are length contraction and time dilation real effects? (Does Lorentz contraction hurt?)

63 Relativistic Effects: FAQs Question 1 Do length contraction and time dilation only effect special clocks and measuring devices? Question 2 Are length contraction and time dilation real effects? (Does Lorentz contraction hurt?)

64 Relativistic Effects: Answers Question 1 Do length contraction and time dilation only effect special clocks and measuring devices? NO! The results apply to everything... otherwise there would be a violation of the principle of relativity

65 Relativistic Effects: Answers Question 1 Do length contraction and time dilation only effect special clocks and measuring devices? NO! The results apply to everything... otherwise there would be a violation of the principle of relativity

66 Relativistic Effects: FAQs Question 2 Are length contraction and time dilation real effects? (Does Lorentz contraction hurt?) Perspectival does not imply imaginary or unreal Relative Quantities, values depend on state of motion of observer who measures them ( perspectival ): space, time, energy, momentum, electric field, magnetic field,... but underlying absolute quantities all observers agree on: spatiotemporal interval, energy-momentum, electromagnetic field,...

67 Relativistic Effects: FAQs Question 2 Are length contraction and time dilation real effects? (Does Lorentz contraction hurt?) Perspectival does not imply imaginary or unreal Relative Quantities, values depend on state of motion of observer who measures them ( perspectival ): space, time, energy, momentum, electric field, magnetic field,... but underlying absolute quantities all observers agree on: spatiotemporal interval, energy-momentum, electromagnetic field,...