Section 16–3 Transformation of velocities

(1-D Velocity addition / Constant speed of light / 2-D Velocity addition)

In this section, Feynman discusses velocities addition in one dimension and two dimensions, as well as verifies the constant speed of light using the velocities addition formula.

1. 1-D Velocity addition:

“Next we discuss the interesting problem of the addition of velocities in relativity (Feynman et al., 1963, section 16–3 Transformation of velocities).”

Feynman presents a general problem as follows: “an object is moving with velocity v inside a space ship and the space ship itself is moving with velocity u with respect to the ground. What is the apparent velocity v_x of this object from the point of view of a man on the ground?” Feynman derives the one-dimensional velocity addition formula (or longitudinal velocity addition) of v_x using the ratio of x to t instead of the formula v_x = Dx/Dt. He clarifies that the velocity of a moving object as seen by the outside observer is equal to the observer’s distance divided by the observer’s time and it should not be the moving man’s time. In general, we can substitute Dx = g(Dx¢+uDt¢) and Dt = g(Dt¢+uDx¢/c²) into v_x = Dx/Dt = (Dx¢+uDt¢)/(Dt¢+uDx¢/c²) and simplify it into (u+v_x¢)/(1+uv_x¢/c²).

The symbols used in velocities addition may be revised as follows: let u be the velocity of an object measured by an observer in S frame and u¢ be the velocity measured by another observer in S¢ frame. We may also use the symbol v to represent the relative velocity between the S frame and S¢ frame. Alternatively, we can use u_A and u_B to represent the velocities that are measured by A and B respectively. Thus, the formula of relative velocity of an object (or observer) A with respect to another object B can be revised from u_A – u_B to (u_A – u_B)/(1 – u_Au_B/c²) that has a minus sign. If the two objects are moving in an opposite direction, the formula can be expressed as (|u_A| + |u_B|)/(1+|u_A||u_B|/c²) that has a plus sign.

2. Constant speed of light:

“… suppose that inside the space ship the man was observing light itself. In other words, v = c, and yet the space ship is moving. How will it look to the man on the ground? (Feynman et al., 1963, section 16–3 Transformation of velocities).”

Feynman’s question may be rephrased as follows: “suppose that a light beam is in a space ship that is moving at a constant velocity u with respect to the ground. What is the apparent speed of light observed by a man on the ground?” By using the longitudinal velocity addition formula, we obtain v = (u+c)/(1+uc/c²) = c. Feynman explains that this result is good because Einstein’s theory of relativity was designed to do this in the first place. In a sense, it is unnecessary to verify the velocity addition formula because Lorentz’s equations are derived from the light postulate. Essentially, the transformation of velocities is based on the “invariant” speed of light in vacuum that is independent of the motion of an observer in any inertial frame of reference.

One may relate Einstein’s beam of light thought experiment to the velocities addition formula. In his Autobiographical Notes, Einstein (1949) writes that “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... (pp. 49-51)” One may also introduce this problem: “if an observer moves at the speed of light, what will be the speed of light that is observed in the mirror and the speed of light that is moving in the same direction as observed by the observer?” Furthermore, physics teachers could let students verify the velocities addition formula using two different velocities of light: v = (u±c)/(1±uc/c²) = c.

3. 2-D Velocity addition:

“… since the speed c_y is less than the speed of light, the speed v_y of the particle must be slower than the corresponding speed by the same square-root ratio (Feynman et al., 1963, section 16–3 Transformation of velocities).”

To illustrate the addition of velocities in two dimensions (or transverse velocity addition), Feynman gives an example of an object inside a ship which is just moving “upward” with the velocity v_y′ with respect to a horizontally moving ship. He shows that using the relevant Lorentz transformations and the results, y = y¢ = v_y¢t′, we can deduce that if v_x′ = 0, the vertical velocity of the object as measured by an observer on the ground, v_y = v_y′Ö(1−u²/c²). However, Feynman did not explain how the factor Ö(1−u²/c²) is related to time dilation. To be more comprehensive, we can explain that: if the object’s velocity has a horizontal component, then Dx¢ is not zero and Dt = g(Dt¢ + uDx¢/c²). Thus, the velocity of the object moving in two dimensions measured by an observer on the ground is v_y = Dy/Dt =Dy¢/g(Dt¢+uDx¢/c²) = v_y′/g(1+uv_x¢/c²).

According to Feynman, each click of the “particle clock” will coincide with each n-th “click” of the light clock because the physical phenomenon of coincidence will be a coincidence in any frame. He concludes that the speed c_y is less than the speed of light c and the speed v_y of the particle must be slower than the corresponding speed by the same square-root ratio. Importantly, the square root ratio may be explained by Pythagoras theorem: a² + b² = c². The apparent vertical speed of light (v_y¢) in the moving frame, the horizontal speed of the moving system (u) and the speed of the light (c) are related by the equation v_y¢² + u² = c². We may deduce that v_y¢² = c² - u² and thus, the ratio of the two speeds: v_y¢/c = Ö(1 – u²/c²). This is based on the light postulate that is applicable to a light beam that is moving horizontally or any direction.

Questions for discussion:

1. How would you derive the relative velocity of two objects in the context of special theory of relativity?

2. Do we need to verify that the speed of light remains constant using the velocities addition formula?

3. How would you explain that the speed v_y of the particle in a moving space ship must be slower than the corresponding speed v_y′ by the same square-root ratio?

The moral of the lesson: the velocity of a moving object as observed by the outside observer is equal to the observer’s distance divided by the observer’s time and it should not be the moving man’s time.

References:

1. Einstein, A. (1949/1979). Autographical notes (Translated by Schilpp). La Salle, Illinois: Open court.

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.