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.