Section 15–1 The principle of relativity

(Einstein’s formula of mass / Relativity postulate / Light postulate)

In this section, Feynman discusses Einstein’s corrected formula of mass, the postulate of relativity, and the postulate of constant speed of light.

1. Einstein’s formula of mass:

“…we now know that this is not true, and that the mass of a body increases with velocity. In Einstein’s corrected formula m has the value m = m₀/√1−v²/c²… (Feynman et al., 1963, section 15–1 The principle of relativity).”

Feynman says that Newton’s Second Law, which is expressed by the equation F = d(mv)/dt, was stated with the assumption that the mass of an object is constant. He proposes that this is not true because the mass of a body increases with velocity in accordance with Einstein’s corrected formula, m = m₀/√(1−v²/c²). Feynman even claims that Einstein’s formula has been amply confirmed by the observation of many kinds of particles that move at speeds ranging up to practically the speed of light. Curiously, Einstein was not consistent in his definition of mass. In his seminal paper on the electrodynamics of moving bodies, Einstein (1905) defines the longitudinal mass of an object as m₀/(1 – v²/c²)^3/2 and transverse mass as m₀/(1 – v²/c²). Conversely, particle physicists tend to prefer the concept of invariant mass that does not increase with velocity (Rindler, 1990).

To support one’s position that velocity-dependent mass is not a good concept, Okun (1989) quotes Einstein’s letter to Barnett in 1948, “It is not good to introduce the concept of the mass, m_r = m₀/Ö(1 – v²/c²) of a moving body for which no clear definition can be given (p. 32).” On the other hand, in his autobiography that was published in the following year, Einstein (1949) explains the idea of how kinetic energy may contribute to mass: In his own words, “…the theory had to combine the following things: 1. From general considerations of special relativity theory it was clear that the inert mass of a physical system increases with the total energy (therefore, e.g., with the kinetic energy) (p. 61).” However, the increase in mass of a moving object cannot be directly confirmed by experiment within one’s own frame of reference.

2. Relativity postulate:

“…the laws of Newton are of the same form in a moving system as in a stationary system, and therefore it is impossible to tell, by making mechanical experiments, whether the system is moving or not (Feynman et al., 1963, section 15–1 The principle of relativity).”

Feynman discusses the first postulate of special relativity (principle of relativity) without using the word postulate. According to Feynman, the principle of relativity means that all of the physical phenomena in a space ship (moving at constant speed) will appear the same as if the ship were not moving. This principle is closer to Einstein’s formulation that is subtly different from Poincaré’s principle of relativity: “the laws of physical phenomena must be the same for a fixed observer as for an observer who has a uniform motion of translation relative to him, so that we have not, nor can we possibly have, any means of discerning whether or not we are carried along in such a motion (Feynman et al., 1963, section 16–1).” In essence, Feynman’s statement of the principle of relativity does not explicitly refer to an observer. However, he discusses Poincaré’s principle of relativity in chapter 16 with greater details.

Some physicists may disagree with Feynman for not using the word postulate in this chapter. In his paper, On the electrodynamics of moving bodies, Einstein (1905) writes that “[w]e will raise this conjecture (the purport of which will hereafter be called the ‘Principle of Relativity’) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.” Importantly, a postulate is not subject to either direct confirmation or disconfirmation by experiment (Goldberg, 1984). This is a subtle point that is less commonly discussed in the theory of special relativity.

3. Light postulate:

“Another consequence of the equations is that if the source of the disturbance is moving, the light emitted goes through space at the same speed c (Feynman et al., 1963, section 15–1 The principle of relativity).”

Feynman explains that light rays emitted from a moving source travel through space at the same speed c (a consequence of Maxwell’s equations). He adds that this is analogous to the case of sound because the speed of sound waves is independent of the motion of the source. More important, the phrase speed of light can be distinguished as one-way speed and two-way speed (Bridgman, 1962). The speed of light that can be directly measured experimentally is the “two-way speed” (or round trip speed) of light from a source to a detector and back to the source. On the other hand, “one-way speed” (or two-clock speed) of light from a source to a detector cannot be directly measured by synchronizing the clock at the source and the clock at the detector without specifying a convention.

In the paper On the electrodynamics of moving bodies, Einstein (1905) explains the apparent incompatibility of the relativity postulate and the postulate of constant speed of light. Furthermore, he defines the light postulate as “[e]very light ray moves in the ‘rest’ coordinate system with a fixed velocity V, independently of whether this ray of light is emitted by a body at rest or in motion.” One weakness of this statement is that the medium (vacuum) in which the light ray travels through is omitted. Another possible confusion of the light postulate is to write that light rays always travels in any inertial frame at the same speed no matter how fast the light source and the observer are moving toward or away from each other. In a sense, this statement is deduced by combining the relativity postulate with the light postulate.

Questions for discussion:

1. Would you explain the mass of an object is dependent on its kinetic energy?

2. How would you state the relativity postulate?

3. How would you state the postulate of constant speed of light?

The moral of the lesson: Einstein’s formula of mass, the principle of relativity, and the constancy of the speed of light are indirectly confirmed by experiment.

References:

1. Bridgman, P. W. (1962/1983). A Sophisticate’s Primer of Relativity (2nd ed.). Mineola, NY: Dover.

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

3. Einstein, A. (1949). Autographical notes (Translated by Schilpp). La Salle, Illinois: Open court.

4. 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.

5. Goldberg, S. (1984). Understanding Relativity. Origin and Impact of a Scientific Revolution. Oxford: Clarendon.

6. Okun, L. B. (1989). The concept of mass. Physics today, 42(6), 31-36.

7. Rindler, W. (1990). Putting to Rest Mass Misconceptions. Physics Today, 43(5), 13.