Section 17–4 More about four-vectors

(Energy is mass / Composite velocity / Transformation of energy and momentum)

 

In this section, Feynman discusses four-momentum from the perspectives of mass-energy equivalence, composite velocity, and the transformation of energy and momentum. The section could be titled “four-momentum” instead of “more about four-vectors” because it is also closely related to energy, momentum, and invariant mass.

 

1. Energy is mass:

“Energy and mass, for example, differ only by a factor c² which is merely a question of units, so we can say energy is the mass (Feynman et al., 1963, section 17–4 More about four-vectors).”

 

There are many examples of four-vectors in special relativity, including the four-velocity and four-force. Another example is the four-momentum, which has three spatial components (linear momentum) and a temporal component (energy). To simplify the notation and avoid repeatedly writing factors of c, physicists often adopt natural units where c = 1, allowing them to write E = m. Feynman explains that energy and mass differ only by a factor c² which is merely a matter of unit choice, and remarks that energy is the mass. However, Okun (1989) explains that mass is not equivalent to energy and emphasize that E = m₀c² is the correct equation instead of E = mc².

 

In The Evolution of Physics, Einstein and Infeld (1938) write: “according to the theory of relativity, there is no essential distinction between mass and energy. Energy has mass and mass represents energy. Instead of two conservation laws we have only one, that of mass-energy (pp. 197-198).” Currently, physicists may argue that mass and energy are ontologically different rather than interchangeable or different forms of the same thing. However, many debates largely hinge on definitions and whether they serve a useful purpose. For instance, photons are fundamentally massless in vacuum, but in certain condensed matter systems—especially superconductors—they can acquire an effective mass through interactions with the medium. This shows how definitions of mass may vary depending on context. Notably, there are three sub-groups of physicists within the particle physics community (experimentalists, instrument developers, and theorists) that have a different specialized language (or definitions) for their internal communication (Galison, 1997).

 

2. Composite velocity:

“What is v′, the velocity as seen from the space ship? It is the composite velocity, the ‘difference’ between v and u (Feynman et al., 1963, section 17–4 More about four-vectors).”

 

To determine the momentum and energy of an object in a different inertial frame, we must know how its velocity transforms between inertial frames. If an object moves with velocity v in one frame, and an observer is in a spaceship moving at velocity u relative to that frame, we can use v′ to designate the observed velocity of the object as seen from the spaceship. Feynman adopts the term composite velocity (the “difference” between v and u in an inertial frame) and states it as v′ = (v−u)/(1−uv). This expression ensures that no observed velocity exceeds the speed of light by setting c = 1 (in natural units). Importantly, the concept of composite velocity corresponds to relativistic velocity subtraction, which is useful for switching from one inertial frame to another. It represents a special case of velocity addition formula, which can be directly derived from the Lorentz transformations of space and time.

 

Feynman’s method to obtain the 4-momentum may seem unnatural by using a trick, v′² = (v²−2uv+u²)/(1−2uv+u²v²). However, we can directly substitute the composite velocity v′ = (v−u)/(1−uv) into the Lorentz factor g_v′ = 1/√(1−v′²). Thus, we can get g_v′ = 1/√(1−[(v−u)/(1−uv)]²) = (1−uv)/√([1−uv]²–[v−u]²). The expression g_v′ could be simplified as (1−uv)/√(1−u²)√(1–v²) and it is equal to g_ug_v(1−uv). Alternatively, Feynman could use the relativistic velocity addition formula to get the same expression. The relativistic velocity addition formula applies in two distinct contexts:

1.   Switching Between Inertial Frames — Transforming the velocity of an object from one inertial frame (S′) to another (S) that is moving at a constant relative velocity.

2.  Relative Velocity Between Two Objects — Determining the relative velocity between two moving objects (in the same or opposite directions), as measured from a third inertial frame.

The distinction between the two contexts is a matter of perspectives — one emphasizes switching inertial frames, while the other focuses on comparing two moving objects — both are physically and mathematically equivalent.

 

3. Transformation of energy and momentum:

“… transformations for the new energy and momentum in terms of the old energy and momentum are exactly the same as the transformations for t′ in terms of t and x, and x′ in terms of x and t… (Feynman et al., 1963, section 17–4 More about four-vectors).”

 

Feynman explains that the transformations for energy and momentum are formally identical to the Lorentz transformations for space and time; specifically, one can obtain them by replacing time t with energy E, and spatial coordinate x with momentum component p_x​. More generally, for a particle with nonzero invariant mass m, the four-momentum (E, p_x, p_y, p_z) can be derived by multiplying the invariant mass by the four-velocity, defined as the derivative of the four-position (ct, x, y, z) with respect to proper time τ. In short, the four-velocity is (cdt/dt, dr/dt) and the four-momentum becomes (E, p) where E = γmc² and p = γmv. In practice, physicists often adopt natural units where c = 1, simplifying expressions to (E, p). However, this can lead to ambiguity: should the time-like component of four-momentum be written as E/c or just E? It depends on the unit system used, and this subtlety can be a source of confusion, especially for students encountering four-vectors for the first time.

 

“This arrow has a time component equal to the energy, and its space components represent its three-vector momentum; this arrow is more ‘real’ than either the energy or the momentum, because those just depend on how we look at the diagram (Feynman et al., 1963, section 17–4 More about four-vectors).”

 

Feynman concludes that the four-momentum transforms in the same way as spacetime coordinates under Lorentz transformations. However, Feynman’s remark that the arrow of four-vector is more real than either the energy or the momentum could be unclear. One possible interpretation is that the arrow is more real because it captures the particle’s true physical motion through spacetime. More fundamentally, what is invariant — and arguably “more real” — is not any single component of the vector, but the magnitude of the four-momentum: E² – (cp)² = (m₀c²)². This Lorentz-invariant quantity expresses the particle’s invariant mass, which remains the same in all inertial frames.

 

“Is it possible, then, to associate with some of our known ‘three-vectors’ a fourth object, that we could call the ‘time component,’ in such a manner that the four objects together would ‘rotate’ the same way as position and time in space-time? We shall now show that there is, indeed, at least one such thing (there are many of them, in fact)..." (Feynman et al., 1963, section 17–4 More about four-vectors).”

 

Feynman’s use of the term “rotate” in this context can be confusing, as it suggests the familiar notion of spatial rotation in Euclidean geometry—an idea that does not accurately capture the nature of Lorentz transformations in spacetime. There are at least two reasons why the concept of rotation is misleading when applied to these transformations:

1. Axis Tilt is Not a True Rotation: A moving clock appears to tick more slowly (Δt′ = γΔt > Δt), and a moving ruler appears shorter (Δx′ = Δx/γ < Δx). This rescaling of time and space (including the tilt of the time-axis and space-axis) violates the defining feature of rotations (See figure below).

2. Opposing Tilts result in a Shear: In a Lorentz transformation, the time axis tilts clockwise (meaning time dilation), while the space axis tilts anti-clockwise (meaning length contraction). These opposing tilts do not preserve spatial angles or distances, but form a hyperbolic shear (See figure below).

A more accurate analogy is to imagine spacetime as a rubber sheet marked with a coordinate grid: depending on the observer’s velocity, the grid is skewed and diagonally stretched. Terms like Minkowski rotation or hyperbolic rotation emphasize the secondary “side” effect (axis tilting) while neglecting the primary relativistic effect: the stretching and squeezing of spacetime—a hyperbolic shear, which is a pseudo-rotation*.

*The misconception of rotation is not new, e.g., Rindler (2012) writes: “Imparting a uniform velocity in relativity corresponds to making a pseudo-rotation in ‘spacetime’ (p. 41).”

Source: https://www.anyrgb.com/en-clipart-sylvj

 

Questions for discussion:

1. Would you explain that energy is mass in the transformation of four-momentum?

2. Would you adopt the concept of composite velocity (instead of relativistic velocity addition) in the derivation of four-momentum?

3. Why is the arrow of four-momentum more “real” than the energy or momentum?

 

The moral of the lesson: Terms like Minkowski rotation or hyperbolic rotation are sometimes used as elegant—but potentially misleading—metaphors for Lorentz transformations. Saying that energy and momentum “rotate” like space and time is similar to explaining a black hole behaves like a “dark star.” Lorentz transformations are not rotations in the usual Euclidean sense; they are hyperbolic transformations that mix space and time (or energy and momentum) in a fundamentally non-Euclidean geometry. What appears as a simple "rotation" in Minkowski diagrams is a skewed transformation of the fabric of spacetime.

 

References:

1. Einstein, A., & Infeld, L. (1938). The Evolution of Physics. New York: Simon and Schuster.

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. Galison, P. (1997). Image and Logic: A Material Culture of Microphysics. Chicago: University of Chicago Press.

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

5. Rindler, W. (2012). Essential relativity: special, general, and cosmological. Springer Science & Business Media.