Section 15–8 Relativistic dynamics

(Relativistic momentum / Relativistic mass / An approximate formula)

In this section, Feynman discusses relativistic momentum, relativistic mass, and an approximate formula.

1. Relativistic momentum:

“Momentum is still given by mv, but when we use the new m this becomes p = mv = m₀v/√1−v²/c²… (Feynman et al., 1963, section 15–8 Relativistic dynamics).”

According to Feynman, the relativistic momentum of an object is given by the same mathematical expression mv. In addition, under Einstein’s modification of Newton’s laws, the modified mass (m) of the relativistic momentum can be expressed as m₀/√(1−v²/c²) such that conservation of momentum holds. This is not quite correct because Einstein’s definitions of transverse mass and longitudinal mass are not equal to m₀/√(1−v²/c²). Historically, Lewis and Tolman (1909) first derived the relativistic momentum using conservation of momentum in a paper titled The Principle of Relativity and non-Newtonian mechanics. Feynman held the Richard Chace Tolman professorship in theoretical physics at the California Institute of Technology, but he could have acknowledged Tolman’s contribution to relativistic momentum.

Textbook authors may define relativistic momentum in terms of p = gmv in which g is equal to 1/√(1−v²/c²). For example, Halliday et al. (2014) state that p = mDx/Dt₀ = gmv and explain that “Dt₀ is the time required to travel that distance, measured not by the observer watching the moving particle but by an observer moving with the particle (p. 1138).” This should not be viewed as a rigorous derivation or complete explanation of relativistic momentum. Although observers in all inertial frames of reference observe different values of dx/dt of a particle, they agree with the term dx/dt₀ where the proper time t₀ is measured by the moving observer. In a sense, the gamma factor may mix with rest mass to become relativistic mass or proper length (instead of only proper time), however, there could be only a modification of mass, time, or length in p = gmv.

2. Relativistic mass:

“Of course, whenever a force produces very little change in the velocity of a body, we say that the body has a great deal of inertia, and that is exactly what our formula for relativistic mass says… (Feynman et al., 1963, section 15–8 Relativistic dynamics).”

Feynman uses the relativistic mass formula m₀/√(1−v²/c²) to explain that the inertia is comparatively higher when the velocity of an object (v) is nearly as fast as c. He relates the effect of relativistic mass to the deflection of high-speed electrons in the synchrotron of Caltech. For instance, the relativistic mass of an electron in the synchrotron is explained to be about 2000 times of the rest mass that is comparable to the mass of a proton. Thus, there is a need for a much stronger magnetic field to deflect the high-speed electrons. Some physics teachers may prefer to use the concept of relativistic energy instead of relativistic mass that may cause a distraction in understanding the relativistic effect.

Lewis and Tolman (1909) derive a mathematical expression of relativistic momentum that needs the concept of relativistic mass using a thought experiment involving an elastic collision. We can also derive the relativistic momentum formula p = gmv without the concept of relativistic mass (Adkins, 2008). This can be achieved using a Lewis-Tolman inelastic collision of an object (m) with another identical object (initially at rest) that coalesce and form a particle of mass M. The relativistic momentum is conserved in this collision. This is in contrast to the Newtonian momentum that is not conserved. Some physicists may argue that Feynman could have avoided using the obsolete concept of relativistic mass to discuss relativistic momentum.

3. An approximate formula:

“An approximate formula to express the increase of mass, for the case when the velocity is small, can be found by expanding m₀/√(1−v²/c²)−in a power series, using the binomial theorem (Feynman et al., 1963, section 15–8 Relativistic dynamics).”

Feynman discusses the motion of molecules in a small tank of gas using the concept of relativistic mass. When we heat the gas in the small tank, it increases the average speed of the molecules and therefore, the total mass of the gas. Using the binomial theorem, Feynman suggests expanding m₀/√(1−v²/c²) in a power series to obtain an approximate formula for the increase of mass. Because the average speed of the molecules is relatively slow in comparison to the speed of light, the terms after the first two are negligible and we can get m = m₀+ ½m₀(v²/c²). In other words, the increase in the mass of the gas is approximately equal to the total increase in kinetic energy of molecules divided by c², or Δm = Δ(K.E.)/c².

We may not be able to measure the increase in mass because the gain in kinetic energy is relatively small compared to the term c². Proponents of relativistic mass would conceptualize every gas molecule has a mass that can be expressed in terms of m₀/√(1−v²/c²) and explain that the total mass of the gas may be expressed as M = Sm_i. To support their position, they may cite the following words of Einstein (1922), “Mass and energy are therefore essentially alike; they are only different expressions for the same thing (p. 47).” However, proponents of invariant mass may argue that the definition of mass in terms of energy (m = E/c²) is a redundant concept. In a sense, physicists that disagree with the concept of relativistic mass prefer the equation E² = (pc)² + (m₀c²)² instead of E = mc².

Questions for discussion:

1. Would you agree with Feynman in using the mathematical expression of relativistic momentum in terms of p = mv instead of p = gmv?

2. Would you explain the relativistic mass of an electron in a synchrotron to be about 2000 times of its rest mass that is comparable to the mass of a proton?

3. How would you explain that the increase in the mass of a gas is proportional to the increase in temperature?

The moral of the lesson: when a gas is heated, the average speed of molecules is increased and the approximate increase in mass of the gas can be estimated using the formula Δm = Δ(K.E.)/c².

References:

1. Adkins, G. S. (2008). Energy and momentum in special relativity. American Journal of Physics, 76(11), 1045-1047.

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. Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics (10th ed.). Hoboken, NJ: Wiley.

4. Lewis, G. N., & Tolman, R. C. (1909). The Principle of Relativity, and non-Newtonian mechanics. Proceedings of the American Academy of Arts and Sciences. 44(25), 711-724.

5. Einstein, A. (1922). The meaning of relativity. Princeton: Princeton University Press.

Section 15–7 Four-vectors

(Four-position / Rotation of space-time / Four-momentum)

In this section, Feynman briefly discusses the position four-vectors, rotation of space-time, and four-momentum.

1. Four-position:

“… we expect that there will be vectors with four components, three of which are like the components of an ordinary vector, and with these will be associated with a fourth component, which is the analog of the time part (Feynman et al., 1963, section 15–7 Four-vectors).”

According to Feynman, we can extend the concept of vectors that have only space components by including a time component. The new concept may be known as position four-vectors that have four components: three of which are the components of a position vector bounded in a three-dimensional space and a fourth component that is associated with time. It means that space and time are inter-related to the extent that space should not be independently defined without time, whereas time should not be independently defined without space. However, it is debatable whether the definition of space-time in terms of four-vector is merely a convention. In other words, one may argue that space-time is arbitrarily defined as a matter of convenience depending on how a physicist formulates a theory of space and time.

A four-vector is a set of four real (or complex) numbers that can be transformed by the Lorentz transformation equations. In his seminal paper titled space and time, Minkowski (1907) writes that “I will call a point in space at a given time, i.e. a system of values x, y, z, t a worldpoint. The manifold of all possible systems of values x, y, z, t will be called the world (p. 112).” Currently, physicists prefer the term event instead of worldpoint. This idealized point in Minkowski spacetime has a time and spatial position that can be represented by a position four-vectors: R = (ct, r) where c is the speed of light and r is a three-dimensional position vector. Alternatively, some authors may adopt Poincare’s (1906) imaginary time dimension ict that is used as a matter of convention.

2. Rotation in space-time:

“… Lorentz transformation is analogous to a rotation, only it is a “rotation” in space and time, which appears to be a strange concept (Feynman et al., 1963, section 15–7 Four-vectors).”

Feynman explains that Lorentz transformation equations help to determine a new x′ which is a mixture of x and t (or a new t′ is a mixture of t and x). Mathematically, the Lorentz transformation of x and t can be visualized as a rotation of space and time in a space-time diagram. It can be further shown that the mathematical equation x′²+y′²+z′²−c²t′² = x²+y²+z²−c²t² is invariant for different values x, y, z, t in all inertial frames of reference. Physics teachers may clarify that Minkowski did not unify space and time such that they are equivalent to each other just like the equation E = mc² that means mass-energy equivalence. Although space and time can be transformed into different values, the invariant quantity x²+y²+z²−c²t² can be related to the postulate of the absolute world (Minkowski, 1907).

The theory of special relativity is not only about the relative space and time that can be calculated using Lorentz transformation. In addition, the interval x²+y²+z²−c²t² is invariant for all inertial frames of reference. Thus, Minkowski (1907) explains that “I think the word relativity postulate used for the requirement of invariance under the group G_c is very feeble. Since the meaning of the postulate is that through the phenomena only the four-dimensional world in space and time is given, but the projection in space and in time can still be made with certain freedom, I want to give this affirmation rather the name the postulate of the absolute world (p. 117).” Perhaps special relativity should be named as “theory of variance (relativity) and invariance.” Although Einstein has destroyed absolute space and absolute time, Minkowski has helped to develop the concept of invariant (absolute) space-time interval.

3. Four-momentum:

“… the transformation gives three space parts that are like ordinary momentum components, and a fourth component, the time part, which is the energy (Feynman et al., 1963, section 15–7 Four-vectors).”

Feynman briefly mentions that four-momentum will be analyzed further in the next chapters. However, we can simply apply the idea of four-vectors to momentum such that the Lorentz transformation gives three components of linear momentum (space parts) and a fourth component (time part) which is the energy. Mathematically, we can multiply the four-velocity (or velocity 4-vector) by the rest (invariant) mass to obtain the four-momentum, P ≡ mV = (E, p). Using this approach, the law of conservation of energy and momentum appears as a single law (or conservation of four-momentum). In a sense, momentum is not completely defined without the concept of energy, whereas energy is not completely defined without the concept of momentum.

One may prefer Feynman to explain how direct experimental observations support the special theory of relativity. This can be achieved by many convincing experiments that are related to the law of conservation of energy and momentum. On the other hand, it is necessary for the law of conservation of momentum to include the time component (energy) in order to have the property of Lorentz invariance. That is, we need the law of conservation of energy to merge with the law of conservation of momentum to form invariant four-vectors in the geometry of space-time. Essentially, the energy and momentum are conserved in an interaction from the perspective of observers in all inertial frames of reference.

Questions for discussion:

1. How would you define the position four-vectors?

2. How would you explain that the Lorentz transformation is analogous to a “rotation” in space and time?

3. How would you define the four-momentum?

The moral of the lesson: the Lorentz transformation of x and t of a moving object as viewed by an inertial observer can be visualized as a rotation of space and time in a Minkowski space-time diagram.

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. Minkowski, H. (1907). Space and Time. In Petkov, V., Ed. Minkowski’s Papers on Relativity. Moscu: Minkowski Institute Press.

3. Poincaré, H. (1906). Sur la dynamique de l’electron. Rendiconti del Circolo Matematico di Palermo, 21, 129-176.

Section 15–6 Simultaneity

(Concepts of simultaneity / Synchronization method / The term ux/c²)

In this section, Feynman discusses concepts of simultaneity, a synchronization method, and the term ux/c².

1. Concepts of simultaneity:

“…when a man in a space ship thinks the times at two locations are simultaneous, equal values of t′ in his coordinate system must correspond to different values of t in the other coordinate system! (Feynman et al., 1963, section 15–6 Simultaneity).”

The concept of simultaneity can be distinguished as local simultaneity and distant simultaneity. In his seminal paper, Einstein (1905) first defines local simultaneity as “… for example, I say that ‘the train arrives here at seven o’clock,’ that means, more or less, ‘The pointing of the small hand of my clock to 7 and the arrival of the train are simultaneous events.” On the other hand, Feynman explains the concept of distant simultaneity in which an observer (in an inertial frame of reference) thinks the times at two locations are the same, but it must correspond to different values of t in another inertial frame. In other words, the impossibility of distant simultaneity is illustrated by a disagreement among observers from different inertial frames of reference on the order of events.

It is worthwhile to discuss the concept of an event in the context of simultaneity. Specifically, an event occurs at a point in space-time whereby it has a definite place and a definite time (Mermin, 2009). One may expect Feynman to elaborate that the concept of an event is an idealization just like the concept of a point in space. It is different from a real event where a lightning strike that has a finite extent. Furthermore, physicists idealize Einstein’s clocks as infinitely small objects. Importantly, distant simultaneity is related to two spatially separated events and it can be shown to be a logically contradictory concept using Einstein’s postulate of constant speed of light.

2. Synchronization method:

“Let us then suppose that the man in S′ synchronizes his clocks by this particular method. Let us see whether an observer in system S would agree that the two clocks are synchronous (Feynman et al., 1963, section 15–6 Simultaneity).”

According to Feynman, one way of synchronizing two clocks in a moving space ship (system S′) is to place a clock each at the front end and rear end of the ship. Light signals are sent out from the midpoint of the ship in opposite directions and they arrive at both clocks at the same time because they move at the same speed. A man in an inertial frame of reference S would reason that more time is needed for the light signal to reach the front clock because the ship was moving forward (or moving away from the light signal). This is in contrast to the rear clock that was moving toward the light signal and so the distance between them appears shorter. In summary, the light signal would reach the rear clock earlier as compared to the front clock and it means that the two observers would disagree with which event would occur first.

One may hope Feynman to discuss why the synchronization of clocks is based on the speed of light instead of the speed of sound. There could be further clarifications whether this method of synchronization is a matter of convention and the possibility of using sound waves to synchronize clocks. However, it is advantageous to use light signals because they are essentially electromagnetic waves that do not require a material medium for transmission and the speed of light in vacuum is independent of its wavelength, amplitude, or direction of propagation (Resnick, 1968). It is worth mentioning that Einstein’s invention of the theory of special relativity helps to solve problems of Maxwell’s equations. We should not overlook the obvious fact that Maxwell’s equations describe not only space and time, but electromagnetic waves.

3. The term ux/c²:

“The most interesting term in that equation is the ux/c² in the numerator, because that is quite new and unexpected (Feynman et al., 1963, section 15–6 Simultaneity).”

Feynman mentions that the most interesting term in the equation is ux/c². The term means that two events that occur at two separated places at the same time, as seen by Moe in S′ frame, do not happen at the same time as seen by Joe in S frame. Feynman calls this circumstance as “failure of simultaneity at a distance,” but he did not provide a derivation of the term. Interestingly, Morin (2003) provides derivations of this term in eight different contexts. However, the impossibility of distant simultaneity can be explained using the term ux/c². That is, two events that appear simultaneously to an observer may appear to have a time difference depending on the relative velocity of another observer (u) and the distance between the two events (x).

A simple derivation of the term ux/c² is to compare the time difference when two light signals reach clock A and clock B in a moving train that is x distance in length and moving at a speed of u from a ground observer’s (S) perspective (Resnick, 1968). When a light signal moving to left meets clock A, at t = t_A, we have ct_A = (x/2)Ö(1 – u²/c²) – ut_A. The term ut_A has a minus sign because the light signal and the clock A move in the opposite directions. When a light signal moving to the right meets clock B at t = t_B, we have ct_B = (x/2)Ö(1 – u²/c²) + ut_B. The term ut_B has a plus sign because the light signal and the clock A move in the same direction.) Comparing the time of two clocks (S frame), the time difference observed is Dt = (x/2)Ö(1–u²/c²)/(c–u)–(x/2)Ö(1–u²/c²)/(c+u) = (ux)[Ö(1–u²/c²)]/(c²–u²). Lastly, taking into account of time dilation, Dt' = DtÖ(1–u²/c²), it can be simplified as Dt' = (ux)(1–u²/c²)/(c²–v²) = ux/c².

Questions for discussion:

1. How would you define local simultaneity and distant simultaneity?

2. Why is the synchronization of clocks based on the speed of light instead of the speed of sound?

3. Does the term ux/c² mean that it is impossible for a guy and a gal living separately in two different countries to fall in love simultaneously?

The moral of the lesson: the term ux/c² implies that the clocks at two locations that appear the same in an inertial frame of reference must correspond to different values of time in another inertial frame.

References:

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

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. Mermin, N. D. (2009). It’s about time: understanding Einstein’s relativity. Princeton: Princeton University Press.

4. Morin, D. (2003). Introductory Classical Mechanics. Cambridge: Cambridge University Press.

5. Resnick, R. (1968). Introduction to Special Relativity. New York: Wiley.