Section 18–1 The center of mass

(Motion of center of mass / Center of mass vector / Stationary center of mass)

In this section, Feynman discusses the motion of center of mass, a definition of center of mass vector, and an example whereby the center of mass is stationary. This section only gives a short introduction to the concept of center of mass. The next chapter elaborates on this concept further with two sections: 19–1 Properties of the center of mass and 19–2 Locating the center of mass. We may understand this section from the perspectives of idealization, approximation, and exception.

1. Motion of center of mass:

“That is called the theorem of the center of the mass, and the proof of it is as follows… (Feynman et al., 1963, section 18–1 The center of mass).”

According to Feynman, the first theorem concerning the motion of complicated objects can be observed if we throw an object made of some blocks held together by strings, into the air. It demonstrates that there is a mean position (or effective center) which moves parabolically, but it is not definitely a point of the material itself. The theorem of the center of mass’s motion can be developed from a simpler model involving two particles first. The simplified theorem may be stated as “[t]he motion of the center of mass of a system of two particles is the same as the motion of a single particle of mass equal to the total mass of the system acted on by the resultant of all the external forces which act on the individual particles (French, 1971, p. 353).” Then, we can let students deduce a theorem involving a system of many particles.

It is not straightforward to develop an equation to describe the motion of complicated objects such as “flowing water” or whirling galaxies.” The equation describing the center of mass of an extended body involves idealizations and simplifications. Firstly, we idealize a rigid body as “a collection of particles whose relative distances are constrained to remain absolutely fixed (Thornton & Marion, 2004, p. 411).” The rigid body is an idealization that does not exist in nature because the atoms within the body are always in motion. In addition, we simplify the motion using the equation F = MR that describes only the translation of the body (or the motion of center of mass). This idealized equation does not provide information on the body’s orientation in space as well as the location of every atom within the body.

2. Center of mass vector:

“… M is the sum of all the masses, i.e., the total mass. Then if we define a certain vector R to be R = S_im_ir_i/M, Eq. (18.3) will be simply F = d²(MR)/dt² = M(d²R/dt²), since M is a constant (Feynman et al., 1963, section 18–1 The center of mass).”

Feynman defines the center of mass vector using R = S_im_ir_i/M and expresses the external force F on the total mass M of a body as F = d²(MR)/dt² = M(d²R/dt²). The point at R is called the center of mass of the body and it is a kind of average r in which the different r_i’s have weights that are proportional to the masses. In a sense, one may assume the weights of the masses are under a parallel gravitational field. Importantly, the center of gravity for an extended object that is relatively large does not necessarily coincide with the center of mass of the same object. By assuming the “same” gravity, we may say that the center of gravity of a body is approximately located at the same point as the body’s center of mass.

Another approximation is based on the assumption in which the parts of a body are moving at speeds very much slower than the speed of light. Using the nonrelativistic approximation for all quantities, we may deduce that the mass of the moving body is constant such that we can write F_i = d²(m_ir_i)/dt². We could include internal forces within the body using Newton’s second law by writing S_iF_i^ext + S_iF_i^int = M d²R/dt² and stating S_iF_i^int = 0 due to Newton’s third law. However, Newton’s third law is not always valid for moving charged particles because electromagnetic forces are velocity dependent. To be precise, magnetic forces exerted on a moving charge q in a magnetic field B obey the weak form of the third law (F_ab = -F_ba), but not the strong form (action and reaction must act along the line joining the particles).

3. Stationary center of mass:

“… then this little blob of gas goes one way as the rocket ship goes the other, but the center of mass is still exactly where it was before (Feynman et al., 1963, section 18–1 The center of mass).”

Feynman discusses a special case (or exception) of a center of mass’s motion. He asks whether it is absolutely impossible to have rocket propulsion because one cannot move the center of mass. Although the center of mass is stationary, he clarifies that we can propel an interesting part of the rocket by throwing away the uninteresting part (exhaust gases). On the other hand, there could be a discussion about the relativistic limitation to the concept of an absolutely rigid body. For instance, if we strike a blow at one end of the rocket, it is impossible for the effect to be felt instantaneously at the opposite end. This is related to the transmission of a signal with an infinite velocity that is forbidden by the theory of relativity.

In the real world, the lift-off of a rocket has to overcome the Earth’s gravitational forces while it accelerates to the outer space. Similarly, a system of (colliding) particles cannot be completely isolated from gravity. In the words of French (1971), “[t]he conservation of momentum in a collision process holds good only to the extent that the effect of any external forces can be ignored. If external forces are indeed present, the duration Dt of the collision must be so short that the product F Dt is negligible. A different way of stating this same condition is that the forces of interaction between the colliding particles must be much greater than any external forces which may be acting (p. 353).”

Questions for discussion:

1. What are the idealizations or simplifications in conceptualizing the center of mass’s motion?

2. What are the approximations needed in writing the equation of the center of mass’s motion?

3. What are the exceptions that are related to the center of mass’s motion?

The moral of the lesson: we may define the center of mass vector using R = S_im_ir_i/M and express the external force F on the total mass M of an idealized object as F = M(d²R/dt²).

References:

1. 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.

2. French, A. (1971). Newtonian Mechanics. New York: W. W. Norton.

3. Thornton, S. T. & Marion, J. B. (2004). Classical Dynamics of Particles and Systems (5th Edition). Belmont, CA: Thomson Learning-Brooks/Cole.

Section 17–5 Four-vector algebra

(Relativistic invariance / Scalar product / Momentum of a photon)

In this section, Feynman discusses relativistic invariance, scalar product of four-vectors, and momentum of a photon (an exception of four-vector).

1. Relativistic invariance:

“...complete the law of conservation of momentum by extending it to include the time component. This is absolutely necessary to go with the other three, or there cannot be relativistic invariance (Feynman et al., 1963, section 17–5 Four-vector algebra).”

According to Feynman, we must complete the law of conservation of momentum by extending it to include the conservation of energy (time component) to obtain a valid four-vector relationship in the space-time geometry. That is, it is absolutely necessary to have four components of momentum such that there is relativistic invariance. We can define relativistic invariance (or Lorentz invariance) in terms of a four-vector a_μ under a Lorentz transformation whereby a_t′²−a_x′²−a_y′²−a_z′² = a_t²−a_x²−a_y²−a_z². In chapter 16, Feynman explains that “the conservation of mass which we have deduced above is equivalent to the conservation of energy” in the context of collisions within the special theory of relativity. In a sense, this implies that we have a single principle involving the conservation of momentum, mass, and energy.

In Volume II, Feynman adds that “the quantity which is analogous to r² for three dimensions, in four dimensions is t²−x²−y²−z². It is an invariant under what is called the “complete Lorentz group”—which means for transformation of both translations at constant velocity and rotations (Feynman et al., 1964, section 25–2 The scalar product).” He also discusses the four-velocity vector with components v_x = dx/dt, v_y = dy/dt, v_z = dz/dt, and mentions that an incorrect guess of the time component is v_t = dt/dt = 1. It turns out that the four “velocity” components that we have written down also have an invariant quantity (the speed of light) if we include the proper time t in all denominators (cdt/dt, dx/dt, dy/dt, dz/dt). The invariant quantity of the four-velocity can be calculated easily using (cdt/dt)² – (dv/dt)² = g²(c² – v²) = c².

2. Scalar product:

“if a_μ is one four-vector and b_μ is another four-vector, then the scalar product is S′a_μb_μ = a_tb_t−a_xb_x−a_yb_y−a_zb_z. It is the same in all coordinate systems (Feynman et al., 1963, section 17–5 Four-vector algebra).”

Feynman states the notations that are used for a scalar product (or inner product) in terms of S′_μA_μA_μ = A_t²−A_x²−A_y²−A_z². The prime on S′ means that the first term, the “time” term, is positive, but the other three terms have minus signs. This invariant quantity is the same in any coordinate system (or inertial frame), and we may call it the square of the length (or norm) of a four-vector. In addition, Feynman (1964) mentions that “the only real complication is the notation.” However, some authors complicate the situation by changing the sign of all the terms and state the square of the length of the four-vector as +a_x²+a_y²+a_z²−a_t². Alternatively, one may state Sa_μ² = +a_x²+a_y²+a_z²+a_t² in which all have the same signs, and define a_t in terms of ict instead of ct (i is an imaginary number).

Feynman elaborates that the square of the length of a four-vector momentum of a single particle is equal to p_t²−p_x²−p_y²−p_z², or in short, E²−p². He says that the invariant quantity must be the same in every coordinate system (or inertial frame) and it is purely its energy, which is the same as its rest mass. In other words, the norm of the four-momentum vector is equal to m₀²c⁴ (i.e., E²−p²c² =m₀²c⁴) and the invariant quantity can be stated as rest energy instead of rest mass. In essence, we may save time in problem-solving if we use the four-momentum vector with a scalar product (relativistic invariant) that is the same in all inertial frames of reference. Simply put, the knowledge of an invariant quantity allows us to choose an inertial frame in which a problem can be solved more easily.

3. Momentum of a photon:

“Such a photon also carries a momentum, and the momentum of a photon (or of any other particle, in fact) is h divided by the wavelength: p = h/λ (Feynman et al., 1963, section 17–5 Four-vector algebra).”

Feynman ends the chapter by discussing the momentum of a photon. This is an interesting example because the rest mass of a photon is zero. Thus, the equation E²−p²c² = m₀²c⁴ can be simplified as E²−p²c² = 0 or E = pc. However, Feynman could mention that the momentum of any (massless) particle is equal to its total energy times its velocity (p = vE/c²) can be reduced to E = p if c = 1. More importantly, physics teachers should explain that the equations E = gm₀c² and p = gm₀v are not really useful for massless particles because m₀ = 0 and g approaches infinity for objects moving at the speed of light (i.e., E and p are not equal to zero). The energy of a photon would change and this is related to the relativistic Doppler effect that is elaborated later in the section 34–6 The Doppler effect (Feynman et al., 1963).

Feynman mentions that the rest mass of a photon is zero and ask whether the photon’s energy is zero using the formula m₀/√(1−v²) instead of E = gm₀c². He adds that the photon really can have energy even though it has no rest mass, but it possesses energy by always moving at the speed of light. In Volume II, Feynman states that “the energy E₀ has the relativistic mass E₀/c² the photon has a mass (not rest mass) ℏω₀/c², and is ‘attracted’ by the earth (Feynman et al., 1964, section 42–6 The speed of clocks in a gravitational field).” On the other hand, from an interpretation of Meissner effect, “[t]he photon mass inside a conventional superconductor is 10^–11 proton masses, or less (Wilczek, 2008, p. 213).” It is good to end the chapter by discussing a special case (or exception) of four-vector using the momentum of a photon.

Questions for discussion:

1. How would you define the concept of relativistic invariance?

2. Would you define the scalar product of a four-vector that has the same sign or opposite signs?

3. How would you explain the mass and momentum of a photon?

The moral of the lesson: the relativistic invariance of a four-vector a_μ is related to the principle of the conservation of momentum, mass, and energy.

References:

1. 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.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

3. Wilczek, F. (2008). The lightness of being: Mass, ether, and the unification of forces. New York: Basic Books.

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.