Section 15–6 Simultaneity

(Concepts of simultaneity / Synchronization method / The term ux/c²)

In this section, Feynman discusses concepts of simultaneity, a synchronization method, and the term ux/c².

1. Concepts of simultaneity:

“…when a man in a space ship thinks the times at two locations are simultaneous, equal values of t′ in his coordinate system must correspond to different values of t in the other coordinate system! (Feynman et al., 1963, section 15–6 Simultaneity).”

The concept of simultaneity can be distinguished as local simultaneity and distant simultaneity. In his seminal paper, Einstein (1905) first defines local simultaneity as “… for example, I say that ‘the train arrives here at seven o’clock,’ that means, more or less, ‘The pointing of the small hand of my clock to 7 and the arrival of the train are simultaneous events.” On the other hand, Feynman explains the concept of distant simultaneity in which an observer (in an inertial frame of reference) thinks the times at two locations are the same, but it must correspond to different values of t in another inertial frame. In other words, the impossibility of distant simultaneity is illustrated by a disagreement among observers from different inertial frames of reference on the order of events.

It is worthwhile to discuss the concept of an event in the context of simultaneity. Specifically, an event occurs at a point in space-time whereby it has a definite place and a definite time (Mermin, 2009). One may expect Feynman to elaborate that the concept of an event is an idealization just like the concept of a point in space. It is different from a real event where a lightning strike that has a finite extent. Furthermore, physicists idealize Einstein’s clocks as infinitely small objects. Importantly, distant simultaneity is related to two spatially separated events and it can be shown to be a logically contradictory concept using Einstein’s postulate of constant speed of light.

2. Synchronization method:

“Let us then suppose that the man in S′ synchronizes his clocks by this particular method. Let us see whether an observer in system S would agree that the two clocks are synchronous (Feynman et al., 1963, section 15–6 Simultaneity).”

According to Feynman, one way of synchronizing two clocks in a moving space ship (system S′) is to place a clock each at the front end and rear end of the ship. Light signals are sent out from the midpoint of the ship in opposite directions and they arrive at both clocks at the same time because they move at the same speed. A man in an inertial frame of reference S would reason that more time is needed for the light signal to reach the front clock because the ship was moving forward (or moving away from the light signal). This is in contrast to the rear clock that was moving toward the light signal and so the distance between them appears shorter. In summary, the light signal would reach the rear clock earlier as compared to the front clock and it means that the two observers would disagree with which event would occur first.

One may hope Feynman to discuss why the synchronization of clocks is based on the speed of light instead of the speed of sound. There could be further clarifications whether this method of synchronization is a matter of convention and the possibility of using sound waves to synchronize clocks. However, it is advantageous to use light signals because they are essentially electromagnetic waves that do not require a material medium for transmission and the speed of light in vacuum is independent of its wavelength, amplitude, or direction of propagation (Resnick, 1968). It is worth mentioning that Einstein’s invention of the theory of special relativity helps to solve problems of Maxwell’s equations. We should not overlook the obvious fact that Maxwell’s equations describe not only space and time, but electromagnetic waves.

3. The term ux/c²:

“The most interesting term in that equation is the ux/c² in the numerator, because that is quite new and unexpected (Feynman et al., 1963, section 15–6 Simultaneity).”

Feynman mentions that the most interesting term in the equation is ux/c². The term means that two events that occur at two separated places at the same time, as seen by Moe in S′ frame, do not happen at the same time as seen by Joe in S frame. Feynman calls this circumstance as “failure of simultaneity at a distance,” but he did not provide a derivation of the term. Interestingly, Morin (2003) provides derivations of this term in eight different contexts. However, the impossibility of distant simultaneity can be explained using the term ux/c². That is, two events that appear simultaneously to an observer may appear to have a time difference depending on the relative velocity of another observer (u) and the distance between the two events (x).

A simple derivation of the term ux/c² is to compare the time difference when two light signals reach clock A and clock B in a moving train that is x distance in length and moving at a speed of u from a ground observer’s (S) perspective (Resnick, 1968). When a light signal moving to left meets clock A, at t = t_A, we have ct_A = (x/2)Ö(1 – u²/c²) – ut_A. The term ut_A has a minus sign because the light signal and the clock A move in the opposite directions. When a light signal moving to the right meets clock B at t = t_B, we have ct_B = (x/2)Ö(1 – u²/c²) + ut_B. The term ut_B has a plus sign because the light signal and the clock A move in the same direction.) Comparing the time of two clocks (S frame), the time difference observed is Dt = (x/2)Ö(1–u²/c²)/(c–u)–(x/2)Ö(1–u²/c²)/(c+u) = (ux)[Ö(1–u²/c²)]/(c²–u²). Lastly, taking into account of time dilation, Dt' = DtÖ(1–u²/c²), it can be simplified as Dt' = (ux)(1–u²/c²)/(c²–v²) = ux/c².

Questions for discussion:

1. How would you define local simultaneity and distant simultaneity?

2. Why is the synchronization of clocks based on the speed of light instead of the speed of sound?

3. Does the term ux/c² mean that it is impossible for a guy and a gal living separately in two different countries to fall in love simultaneously?

The moral of the lesson: the term ux/c² implies that the clocks at two locations that appear the same in an inertial frame of reference must correspond to different values of time in another inertial frame.

References:

1. Einstein, A. (1905). On the electrodynamics of moving bodies. Annalen der Physik, 322(10), 891-921.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Mermin, N. D. (2009). It’s about time: understanding Einstein’s relativity. Princeton: Princeton University Press.

4. Morin, D. (2003). Introductory Classical Mechanics. Cambridge: Cambridge University Press.

5. Resnick, R. (1968). Introduction to Special Relativity. New York: Wiley.

Section 15–5 The Lorentz contraction

(Real distance / Apparent length / Experimental verification)

In this section, Feynman briefly discusses the real distance, apparent length, and experimental verification of Lorentz contraction.

1. Lorentz’s suggestion:

“… Moe is using a foreshortened ruler, so the “real” distance measured is x′√(1−u²/c²) meters (Feynman et al., 1963, section 15–5 The Lorentz contraction).”

Lorentz’s first equation is based on his suggestion of contraction along the x-direction. (Feynman mentions that the real distance measured is x′√(1−u²/c²) meters without explaining the meaning of real.) Interestingly, Lorentz explains that length contraction is a dynamical effect that is related to molecular forces. In Lorentz’s (1927) words, “I should like to emphasize the fact that the variations of length caused by a translation (i.e., a change of place or motion) are real phenomena, no less than, for instance, the variations that are produced by changes of temperature (p. 95).” Feynman could have clarified the length contraction is a kinematical effect (a problem of simultaneity) instead of simply mentioned real distance.

Varićak (1911) distinguished Lorentz’s view of length contraction as an objective occurrence that is in contrast to Einstein’s subjective phenomenon (Miller, 1981, p. 236). Einstein (1911) disagreed with Varićak’s interpretation and clarified that “[t]he author unjustifiably stated a difference of Lorentz’s view and that of mine concerning the physical facts. The question as to whether length contraction really exists or not is misleading. It doesn’t ‘really’ exist, in so far as it doesn’t exist for a comoving observer; though it ‘really’ exists, i.e. in such a way that it could be demonstrated in principle by physical means by a non-comoving observer.” Physics teachers could explain that the effective length of a ruler is really shorter due to a “synchronization error” (the front end and rear end of the ruler are located at different times).

2. Apparent length:

“So, in order for the experiment to give a null result, the longitudinal arm BE must appear shorter, by the square root √1−u²/c² (Feynman et al., 1963, section 15–5 The Lorentz contraction).”

The Michelson-Morley experiment’s null results can be related to the longitudinal arm BE that must appear shorter by the square root √(1−u²/c²). Importantly, Feynman asks for the meaning of the contraction in terms of measurements made by Joe and Moe. His answer is based on Joe’s frame of reference (S system): Moe is using a foreshortened ruler such that the real distance measured is x′√(1−u²/c²) meters. However, there could be a clarification of whether the length contraction is related to an optical effect or dependent on the time the light beam needs to reach an observer’s eye. Better still, the length contraction can be explained as a rotation in space-time and it can be visualized with the help of a Minkowski’s space-time diagram.

Feynman’s phrase “appear shorter” may cause confusion in understanding length contraction. For example, Penrose (1959) proves that a sphere appears to have a circular outline by all observers that have a different relative velocity. In addition, Terrell (1959) shows that a meter ruler moving in high speed will appear to have undergone a rotation instead of contraction. This visual phenomenon is commonly known as Penrose-Terrell effect (or Lampa effect) due to the finite speed of light. In essence, a snapshot of a fast-moving ruler would appear rotated if we take into account the location of an observer and the distance between the different parts of the moving ruler to an observer. One may prefer to use the phrase “really shorter” and explain the meaning of real instead of “appear shorter.”

3. Experimental verification:

“In the Michelson-Morley experiment, we now appreciate that the transverse arm BC cannot change length, by the principle of relativity; yet the null result of the experiment demands that the times must be equal (Feynman et al., 1963, section 15–5 The Lorentz contraction).”

Feynman discusses the length contraction using the null results of Michelson-Morley experiment and the principle of relativity. Although the null results of Michelson-Morley experiment can be explained by the theory of special relativity, the length contraction phenomenon is not directly observable. In general, a direct length measurement refers to the proper length of a stationary object that is directly measured by an observer in one’s own frame of reference. Besides, it is difficult to accelerate a macroscopic object such that its length can be directly observable; it is also challenging to observe the length of a sub-microscopic particle that is considerably small. Thus, one may prefer to say that the Michelson-Morley experiment supports the theory of special relativity instead of validates it.

Some physicists may feel disappointed that Feynman did not discuss the barn-and-pole paradox (or ladder paradox) in this section. However, Feynman also explains that the effect of length contraction in Volume II. In Feynman’s words, “the charge q on a particle is an invariant scalar quantity, independent of the frame of reference. That means that in any frame the charge density of a distribution of electrons is just proportional to the number of electrons per unit volume. We need only worry about the fact that the volume can change because of the relativistic contraction of distances… (Feynman et al., 1964, section 13–6 The relativity of magnetic and electric fields).” In a sense, this is possibly the closest we can have in directly confirming the so-called effect of length contraction (Rindler, 2006).

Questions for discussion:

1. Is length contraction a dynamical effect or kinematical effect?

2. Is the measured length of a moving ruler really shorter or apparently shorter (or an optical effect)?

3. Is the length contraction directly verifiable by the Michelson-Morley experiment?

The moral of the lesson: the length contraction is a kinematical effect due to a problem of simultaneity (the front end and rear end of a moving ruler are located at different times).

References:

1. Einstein, A. (1911). Zum Ehrenfestschen Paradoxon. Eine Bemerkung zu V. Variĉaks Aufsatz. Physikalische Zeitschrift, 12, 509–510.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

4. Lorentz, H. A. (1927). Problems of Modern Physics. Boston: Ginn.

5. Miller, A. I. (1981). Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911). Reading: Addison–Wesley.

6. Penrose, R. (1959). The apparent shape of a relativistically moving sphere. Mathematical Proceedings of the Cambridge Philosophical Society, 55(1), 137–139.

7. Rindler, W. (2003). Relativity: special, general, and cosmological. Oxford: Oxford University Press.

8. Terrell, J. (1959). Invisibility of the Lorentz contraction. Physical Review, 116(4), 1041–1045.

Section 15–4 Transformation of time

(Light clocks / Biological clocks / Experimental verification)

In this section, Feynman discusses light clocks, biological clocks, and experimental verification of length contraction.

1. Light clocks:

“… it is a rod (meter stick) with a mirror at each end, and when we start a light signal between the mirrors, the light keeps going up and down, making a click every time it comes down, like a standard ticking clock (Feynman et al., 1963, section 15–4 Transformation of time).”

According to Feynman, a light clock will work in principle as a rod (meter stick) with a mirror at each end in which we can send a light signal between the mirrors. Physics teachers should clarify that the idealized light clock relies on the principle of constant speed of light. Simply phrased, the light clock is an imaginary time-keeping instrument that is based on the motion of a light beam. Feynman could have explained that the units of space and time are dependent on the speed of the observer’s inertial frame of reference. In essence, the speed of light must appear the same to the observers (whether at rest or in motion) and the units of length and time can be defined by the motion of the light beam.

Feynman held the Richard Chace Tolman professorship in theoretical physics at the California Institute of Technology. In an article titled The Principle of Relativity, and non-Newtonian mechanics, Lewis and Tolman (1909) write that “… the velocity of light will seem the same to two different observers, even though one may be moving towards and the other away from the source of light, constitutes the really remarkable feature of the principle of relativity, and forces us to the strange conclusions which we are about to deduce. Let us consider two systems moving past one another with a constant relative velocity, provided with plane mirrors aa and bb parallel to one another and to the line of motion (Figure 1). An observer, A, on the first system sends a beam of light across to the opposite mirror, which is reflected back to the starting point…” The light clock is also known as Langevin clock, but Feynman could have coined the term Lewis-Tolman clock to recognize Lewis and Tolman for their contribution to the concept of light clock.

Note: Langevin’s paper on the evolution of space and time was published in 1911, two years after Lewis and Tolman’s (1909) paper.

2. Biological clocks:

“…the man’s pulse rate, his thought processes, the time he takes to light a cigar, how long it takes to grow up and get old—all these things must be slowed down in the same proportion… (Feynman et al., 1963, section 15–4 Transformation of time).”

Feynman explains that all phenomena such as a man’s pulse rate, his thought processes, and the time he takes to light a cigar must be slowed down in the same rate because he cannot tell whether he is at rest or moving. In one of his Messenger Lectures, he elaborates that “…if the clock is ticking and I look at the clock in the space ship, then I can see that it is going slow. No, your brain is going slow too! (Feynman, 1965, p. 92).” However, the physics of physical clocks is the same as the physics of biological clocks. Physics teachers should elaborate that daily phenomena such as a man’s pulse rate and his thought processes are mediated by electromagnetic waves that travel at the speed of light. Similarly, gravitational waves travel at the speed of light because gravitons have no mass (just like photons).

Feynman asks whether all moving clocks run slower and what if there is no way of measuring time that ticks at a slower rate. In a sense, it is potentially misleading for Feynman to say that all moving clocks run slower. Physics students may interpret this as a possibility to enjoy a longer lifespan. Specifically, a stationary biological clock tick at the same rate in the same stationary frame of reference. One may emphasize that the biological clock appears slower depending on the relative speed. (Feynman could have used the term relative speed instead of simply speed.) Therefore, physics students should not expect to live almost forever by developing a spacecraft that can move at close to the speed of light.

3. Experimental verifications:

“A very interesting example of the slowing of time with motion is furnished by mu-mesons (muons), which are particles that disintegrate spontaneously after an average lifetime of 2.2×10⁻⁶ sec. (Feynman et al., 1963, section 15–4 Transformation of time).”

Feynman cites an example of the slowing of time with motion using muons that disintegrate spontaneously after an average lifetime of 2.2 µs. To be precise, muons are produced during interactions of cosmic rays with particles near the top of the Earth’s atmosphere. (It is potentially confusing to say that “they come to the earth in cosmic rays” because muons are the products of cosmic rays.) Historically, Rossi and Hall (1941) compared the number of muons at Echo Lake (3240 m) and Denver (1616 m) in Colorado and determined the average lifetime of muons to be 2.4 µs. In a more precise experiment conducted at Mount Washington (using a difference in height of 1907 m), Frisch and Smith (1963) determined the average lifetime of muons (moving between 0.995 c and 0.9954 c) to be 2.2 µs.

Feynman explains that mu-mesons (muons) in their short lifetimes cannot travel much more than 600 meters even at the speed of light. (Based on current Standard Model, muons are no longer considered to be mesons.) Physics teachers should clarify that the distance from the Earth’s atmosphere to the ground appears contracted from the muon’s frame of reference. In other words, the length contraction can be explained as an effect due to the Earth moving toward the muons at a high speed. In summary, an experimenter on Earth observes the time dilation of muons, whereas another experimenter co-moving with a muon would deduce length contractions of mountains.

Questions for discussion:

1. What is the fundamental principle of a light clock?

2. Do moving clocks really run slower?

3. How would you explain that the time dilation can be verified experimentally?

The moral of the lesson: the fact that biological clocks function like a light clock can be experimentally verified using cosmic rays experiments in which muons are created in the Earth’s atmosphere.

References:

1. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Frisch, D. H., & Smith, J. H. (1963). Measurement of the relativistic time dilation using μ-mesons. American Journal of Physics, 31(5), 342–355.

4. Langevin, P. (1911). L’évolution de l'espace et du temps. Scientia, X, 31–54.

5. Lewis, G. N., & Tolman, R. C. (1909). The Principle of Relativity, and non-Newtonian mechanics. In Proceedings of the American Academy of Arts and Sciences.  44(25), 711-724. American Academy of Arts & Sciences.

6. Rossi, B., & Hall, D. B. (1941). Variation of the rate of decay of mesotrons with momentum. Physical Review, 59(3), 223–228.

Section 15–3 The Michelson-Morley experiment

(Theoretical background / Experimental setup / Null results)

In this section, Feynman discusses the theoretical background, experimental setup, and null results of the Michelson-Morley experiment.

1. Theoretical background:

“…the emerging beams D and F will be in phase and will reinforce each other, but if the two times differ slightly, the beams will be slightly out of phase and interference will result (Feynman et al., 1963, section 15–3 The Michelson-Morley experiment).”

The 1887 Michelson-Morley experiment was an attempt to detect ether by comparing the difference in time that a light beam travels parallel to the ether wind with another light beam that travels perpendicular to the ether wind. This is an interference experiment in which the interference fringes may be shifted if the light beams are slightly out of phase. To be precise, interference occurs whether the light beams are in phase or out of phase, but it may not be observable. (It is confusing for Feynman to say that the light beams will be out of phase such that interference will result.) In Michelson’s (1881) earlier experiment, the expected fringe shift was about 0.16 of the distance between the fringes, but the observed fringe shift was significantly less than the expected shift that it was deduced to be errors of measurement.

Theoretically speaking, if the two arms of the Michelson-Morley interferometer have exactly the same length, the null results of the Michelson–Morley experiment can be explained by length contraction alone. Therefore, Kennedy and Thorndike (1932) modified the Michelson–Morley interferometer by changing the length of an arm such that one side is shorter than the other. During the Kennedy–Thorndike experiment, the apparatus was held fixed in the laboratory and the interference fringes were observed over a period of months. The experiment requires extreme demands on the mechanical stability of the apparatus and it serves as a test to indirectly verify time dilation in addition to length contraction. The experimental results also help to support the claim that there are no phase shifts while the Earth is rotating around the Sun.

2. Experimental setup:

“This apparatus is essentially comprised of a light source A, a partially silvered glass plate B, and two mirrors C and E, all mounted on a rigid base (Feynman et al., 1963, section 15–3 The Michelson-Morley experiment).”

The Michelson-Morley experiment was performed using an apparatus similar to that as shown schematically in Fig. 15–2. The apparatus is essentially comprised of a light source A, a partially silvered glass plate B, and two mirrors C and E; the mirrors were placed at equal distances L from the glass plate. The function of the glass plate is to split an incoming beam of light into two outgoing beams that move in mutually perpendicular directions to the mirrors, whereby they are reflected back to the glass plate. Physics teachers may elaborate that the glass plate (or beam splitter) causes a phase shift, due to its refractive index, to the light beam that passes through it. Thus, a compensator plate of equal refractive index and thickness is used to compensate for the extra optical path length (through the glass plate) traveled by the light beam.

According to Feynman, it is difficult to make the length of the two arms exactly equal. (His discussion is misleading because it indicates that the purpose of the rotation of the apparatus is to solve the problem of unequal lengths.) Importantly, a maximum fringe shift was expected by revolving the apparatus through 90 degrees. Furthermore, the experimental results could be more convincing when the apparatus was rotated by many different angles instead of only 90 degrees. In the words of Michelson and Morley (1887), “[t]he apparatus was revolved very slowly (one turn in six minutes) and after a few minutes, the cross wire of the micrometer was set on the clearest of the interference fringes at the instant of passing one of the marks. The motion was so slow that this could be done readily and accurately.” Note that the apparatus was mounted on a basin filled with liquid mercury such that it can be slowly rotated.

Note: One may add that this is a difficult experiment because of the need to control for vibrations. In an earlier paper titled The Relative Motion of the Earth and the Luminiferous Ether, Michelson (1881) writes: “...owing to the extreme sensitiveness of the instrument to vibrations, the work could not be carried on during the day... Here, the fringes under ordinary circumstances were sufficiently quiet to measure, but so extraordinarily sensitive was the instrument that the stamping of the pavement, about 100 meters from the observatory, made the fringes disappear entirely!”

3. Null results:

“Although the contraction hypothesis successfully accounted for the negative result of the experiment, it was open to the objection that it was invented for the express purpose of explaining away the difficulty, and was too artificial (Feynman et al., 1963, section 15–3 The Michelson-Morley experiment).”

The Michelson-Morley experiment is the most famous failed experiment because of the null results. (Perhaps Feynman should avoid the term negative results because it suggests an opposite meaning to positive results that is different from null results.) Based on the null results, Lorentz formulates a contraction hypothesis. On the other hand, Poincaré proposes that it is impossible to discover an ether wind by any experiment and there is no way to determine an absolute velocity. However, Michelson and Morley (1887) disagree with Lorentz’s hypothesis and conclude that the ether is at rest with regard to the Earth’s surface. Interestingly, Michelson is the first American to receive the Nobel Prize for Physics (precisions in optical experiments).

The experimental results that significantly influenced Einstein and form the basis of his special theory of relativity were the observations of stellar aberration and Fizeau’s experiment instead of the Michelson and Morley experiment (Shankland, 1963). In a lecture series at the University of Buenos Aires, Einstein delivered his first lecture on 25 March 1925 by saying that Fizeau’s 1851 water tube experiment was “perhaps the most fundamental to the theory of special relativity… that demonstrates the impossibility of assuming that ether is completely dragged by matter (Einstein, 1925, p. 941).” In essence, Fizeau’s experimental results can be explained using the concept of relative velocity (instead of absolute velocity) of the running water with respect to the laboratory.

Questions for discussion:

1. What is the theoretical background behind the Michelson-Morley experiment?

2. What are the experimental considerations of the Michelson-Morley experiment?

3. What are the possible conclusions of the Michelson-Morley experiment?

The moral of the lesson: based on the null results of Michelson-Morley experiment, we can deduce that it is impossible to discover an ether wind by any experiment and there is no way to determine the absolute velocity of an object.

References:

1. Einstein, A. (1925). Professor Einstein Began His Lecture Series Yesterday. In D. K. Buchwald et al. (eds.) (2015). The Collected Papers of Albert Einstein, Volume 14 (English): The Berlin Years: Writings & Correspondence. Princeton: Princeton University Press.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Kennedy, R. J., & Thorndike, E. M. (1932). Experimental establishment of the relativity of time. Physical Review, 42(3), 400–418.

4. Michelson, A. A. (1881). The Relative Motion of the Earth and the Luminiferous Ether. American Journal of Science, 22(128), 120–129.

5. Michelson, A. A., & Morley, E. W. (1887). On the Relative Motion of the Earth and of the Luminiferous Ether. American Journal of Science. 34(203), 333–345.

6. Shankland, R. S. (1963). Conversations with Albert Einstein. American journal of physics, 31(1), 47–57.