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.