Section 15–2 The Lorentz transformation

(Lorentz’s mathematics / Poincaré’s philosophy / Einstein’s physics)

In this section, Feynman discusses the contributions of Lorentz, Poincaré, and Einstein with regard to the special theory of relativity.

1. Lorentz’s mathematics:

“H. A. Lorentz noticed a remarkable and curious thing when he made the following substitutions in the Maxwell equations: x′ = (x − ut)/√(1−u²/c²), y′ = y, z′ = z, t′ = (t − ux/c²)/√(1−u²/c²)…(Feynman et al., 1963, section 15–2 The Lorentz transformation).”

Feynman says that Hendrik Antoon Lorentz noticed a remarkable fact after the following substitutions in the Maxwell equations: x′ = (x − ut)/√(1 − u²/c²), y′ = y, z′ = z, t′ = (t − ux/c²)/√(1 − u²/c²). Essentially, Maxwell’s equations remain invariant when the Lorentz transformation of coordinates is applied. However, Lorentz failed to grasp the notion of relative time and simultaneity. In 1915, Lorentz writes that, “[t]he chief cause of my failure was my clinging to the idea that only the variable t can be considered as the true time and that my local time t’ must be regarded as no more than an auxiliary mathematical quantity (Pais, 1982, p. 167).” Thus, Lorentz is commonly recognized for his contribution in the mathematics (instead of physics) of special relativity.

FitzGerald-Lorentz contraction hypothesis was an attempt to reconcile the null result of the 1887 Michelson-Morley experiment. In a letter titled The Ether and the Earth’s Atmosphere, George Francis FitzGerald (1889) writes that “I would suggest that almost the only hypothesis that could reconcile this opposition is that the lengths of material bodies changes, according as they are moving through the ether or across it, by an amount depending on the square of the ratio of their velocities to that of light (p. 390).” He proposes that electrical forces are affected by the motion of the electrified bodies relative to the ether and thus, the size of a body and the molecular forces are affected by the motion. Three years later, Lorentz (1892) conceives the same idea and introduces a second-order contraction effect (v²/2c²) to explain the null result.

2. Poincaré’s philosophy:

“…following a suggestion originally made by Poincaré, then proposed that all the physical laws should be of such a kind that they remain unchanged under a Lorentz transformation (Feynman et al., 1963, section 15–2 The Lorentz transformation).”

According to Feynman, Einstein follows a suggestion originally made by Poincaré and proposes that all the physical laws should be of such a kind that they remain unchanged under a Lorentz transformation. (However, all the physical laws should include mechanical laws, optical laws, and the laws of electrodynamics, but not the laws of quantum mechanics.) Historically, there is no direct evidence that Einstein was aware of Poincaré’s works during his development of special relativity. In a letter to Seelig, Einstein writes that “Concerning myself, I knew only Lorentz’s important work of 1895… but not Lorentz’s later work, nor the consecutive investigations by Poincaré (Born, 1956, p. 194).” More important, Poincaré (1906) should be credited for his realization that the Lorentz transformation is a rotation in the four-dimensional space-time and the invariance of the expression: x² + y² + z² − c²t².

It is unclear whether Poincaré accepted Einstein’s theory of special relativity (Pais, 1982). In a speech delivered in St. Louis, Poincaré (1904) says that “[t]he principle of relativity, according to which the laws of physical phenomena should be the same, whether for an observer fixed, or for an observer carried along in a uniform movement of translation; so that we have not and could not have any means of discerning whether or not we are carried along in such a motion.” This paragraph provides an important philosophical insight into the relativity principle: impossibility to determine absolute motion. However, subsequent paragraphs within the same speech indicate that Poincaré failed to grasp the concept of simultaneity (Yang, 2006). Thus, Poincaré is recognized for his contribution to the philosophy (instead of physics) of special relativity.

3. Einstein’s physics:

“…as Einstein did, we too must analyze our ideas of space and time in order to understand this transformation (Feynman et al., 1963, section 15–2 The Lorentz transformation).”

Feynman suggests that we must analyze our ideas of space and time in order to understand Lorentz transformation just like Einstein. Similarly, Yang (2006) explains that “Lorentz had the mathematics, but not the physics, and Poincaré had the philosophy, but also not the physics. It was the 26-year-old Einstein who dared to question mankind’s primordial concept about time, and insisted that simultaneity is relative (p. 3033).” In essence, Einstein resolves the apparent incompatibility between Newton’s mechanics and Maxwell’s electrodynamics by destroying Newton’s notion of absolute space and absolute time. During a speech in Kyoto University on 14 Dec 1922, Einstein elaborates that “[a]n analysis of the concept of time was my solution. Time cannot be absolutely defined, and there is an inseparable relation between time and signal velocity. With this new concept, I could resolve all the difficulties completely for the first time (Einstein, 1982, p. 46).”

In a sense, Einstein did not fully understand the implications of his theory of special relativity when he called Minkowski’s approach to space-time diagram “superfluous learnedness (Pais, 1982, p. 152).” Einstein was also negative towards his contribution and said that “[s]ince the mathematicians have invaded the relativity theory, I do not understand it myself anymore (Sommerfeld, 1949, p. 102).” Shortly after, Einstein realized that his theory of gravity would be impossible without Minkowski’s space-time. In his autobiography, Einstein (1949) recognizes Minkowski’s contribution in special relativity and writes that “[h]e also showed that the Lorentz transformation (apart from a different algebraic sign due to the special character of time) is nothing but a rotation of the coordinate system in the four-dimensional space (p. 55).”

Questions for discussion:

1. What are Lorentz’s contributions to the theory of special relativity?

2. What are Poincaré’s contributions to the theory of special relativity?

3. What are Einstein’s contributions to the theory of special relativity?

The moral of the lesson: Lorentz contributed to the mathematical transformation of coordinates, Poincaré contributed to the philosophy of the principle of relativity, but Einstein revolutionized the concept of space and time in special relativity.

References:

1. Born, M. (1956). Physics and Relativity. In Physics in my generation: A selection of papers. New York: Pergamon Press.

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

3. Einstein, A. (1982/1922). How I Created the Theory of Relativity (translated by Yoshimasa A. Ono). Physics Today, 35(8), 45-47.

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. FitzGerald, G. F. (1889). The Ether and the Earth’s Atmosphere. Science, 13, 390.

6. Lorentz, H. A. (1892). De relatieve beweging van de aarde en den aether. Koninklijke Akademie van Wetenschappen te Amsterdam. Wis- en Natuurkundige Afdeeling. Verslagen der Zittingen. 1, 1892-93, pp. 74-79.

7. Pais, A. (1982). Subtle is the Lord: The Science and the Life of Albert Einstein. Oxford: Clarendon Press.

8. Poincaré, H. (1904). L'état actuel et l'avenir de la physique mathématique. Bulletin des sciences mathématiques, 28(2), 302-324.

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

10. Sommerfeld, A. (1949). To Albert Einstein’s Seventieth Birthday. In P. A. Schilpp (ed.), Albert Einstein: Philosopher-Scientist. La Salle, IL: Open Court.

11. Yang, C. N. (2006). Albert Einstein: Opportunity and perception. International Journal of Modern Physics A, 21(15), 3031-3038.

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.

Section 14–5 Potentials and fields

(Potential = (−field).(ds) / F_x = −∂U/∂x / Field = grad potential)

In this section, Feynman discusses the integral of (−field).(ds), partial derivative of force, and gradient of potential.

1. Potential = (−field).(ds):

“Since the potential energy, the integral of (−force).(ds) can be written as m times the integral of (−field).(ds), a mere change of scale… (Feynman et al., 1963, section 14–5 Potentials and fields).”

The potential energy function U(x, y, z) is equal to the integral of (−force).(ds) and it can be written as m times the integral of (−field).(ds). (When the mass of an object is equal to 1 kg, the force is numerically equal to the field and the potential energy is numerically equal to the potential.) Strictly speaking, whether the work done is positive or negative is dependent on the convention chosen. One explanation is that an external force on an object within a gravitational field results in positive work and store “more” potential energy. On the other hand, the work done by the gravitational field may release the stored energy and thus, the potential energy is reduced through the negative work. In other words, the work done by an internal (gravitational) force results in a decrease in the potential energy of an object-Earth system.

In the last chapter, Feynman has already explained that there is no work done in moving an object from one place to another where the potential energy is constant because there is no (conservative) force. In this section, he suggests the convenience of giving a scalar function Ψ instead of writing three complicated components of a vector function C. Some may criticize Feynman for the use of the symbol C instead of g, which may refer to gravitational field strength or acceleration vector that is due to gravity. However, one may guess the symbol C is chosen to represent a conservative field. More important, a definition of conservative field could be provided and specifically, it may refer to a gravitational field or electrostatic field.

2. F_x = −∂U/∂x:

“Therefore, we find that the force in the x-direction is minus the partial derivative of U with respect to x: F_x = −∂U/∂x (Feynman et al., 1963, section 14–5 Potentials and fields).”

In general, the derivative of a function U(x, y, z) with respect to x can be written as dU/dx if we ignore other variables such as y and z. Feynman explains that we can write ∂U/∂x, or include a line beside it with a subscript yz at the bottom (∂U/∂x|_yz), which means “partial derivative of U with respect to x, while keeping y and z constant.” In addition, he mentions that the mathematicians have invented a new symbol to remind us to be very careful when we are differentiating such a function, by considering only x varies, whereas y and z do not vary. Historically, in Memoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations, Adrien Marie Legendre (1786) first used the “curly d” in the form of ¶u/¶x.

Partial derivatives differ from ordinary derivatives in five ways (Roundy et al., 2015): (1) Symbolic: we need to specify varying quantities and “fixed” quantities for a ratio of small changes. (2) Graphical: a partial derivative becomes the slope of the tangent plane (or tangent space for higher dimensions) in a given direction at a specific point. (3) Algebraic procedure: the steps to obtain a partial derivative are identical to an ordinary derivative if the variables are independent of each other. (4) Verbal: we must mention the independent variables, e.g., the ratio of change of volume as pressure is changed and temperature is held constant. (5) Experimental: The representation of derivatives in experiments must include measurable (dependent) variable, manipulated (independent) variables and controlled (constant) variables.

3. Field = grad potential:

“The x-component of this “grad” is ∂/∂x the y-component is ∂/∂y, and the z-component is ∂/∂z, and then we have the fun of writing our formulas this way… (Feynman et al., 1963, section 14–5 Potentials and fields).”

According to Feynman, the mathematicians have invented a glorious new symbol, ∇, called “grad,” which is an operator that makes a vector from a scalar function. In short, we can write ∇U instead of ∂U/∂x i + ∂U/∂y j + ∂U/∂z k. The symbol ∇ (pronounced as “del”) is not a specific operator, but rather a convenient mathematical notation that simplifies many equations. (It has a different meaning when it is written as ∇².) Historically, William Rowan Hamilton (1853) invented the ∇ symbol (or “nabla”) in his Lectures on Quaternions. Writing the symbol ∇ together with a scalar function U indicate a directional derivative that always points in the direction of greatest increase of U, and it has a magnitude equal to the maximum rate of increase at the point.

Feynman ends the section by saying it is easy to show that the force on a particle due to magnetic fields is always at right angles to its velocity. He concludes that no work is done by the magnetic field on a moving charge because the motion is at right angles to the force. However, this does not mean that the magnetic field is always a conservative field. In general, if a magnetic field is changing, then it generates an electric field according to Faraday’s law of electromagnetic induction. Notably, the induced electric field can form closed paths that are non-conservative. As a result, some energy can be dissipated as internal energy in resistive materials or radiated as light energy (electromagnetic waves).

Questions for discussion:

1. How would you explain the gravitational potential energy is equal to the integral of (−force).(ds)?

2. Why does the force in the x-direction can be more precisely written as the negative partial derivative of U with respect to x: F_x = −∂U/∂x instead of F_x = −dU/dx?

3. Are magnetic forces always conservative?

The moral of the lesson: the potential energy and the integral of (−force).(ds) can be written as m times the integral of (−field).(ds), or equivalently, force is the (negative) rate of change of potential energy and field is the negative rate of change of potential.

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. Hamilton, W. R. (1853). Lectures on Quaternions. Hodges and Smith, Dublin.

3. Legendre, A. M. (1786). Mémoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations. Mémoires de l’Acad. roy. des Sciences, 1788, 7–37.

4. Roundy, D., Weber, E., Dray, T., Bajracharya, R. R., Dorko, A., Smith, E. M., & Manogue, C. A. (2015). Experts’ understanding of partial derivatives using the partial derivative machine. Physical Review Special Topics-Physics Education Research, 11(2), 020126.

Section 14–4 Nonconservative forces

(Globular star cluster / Atomic level / Energy of light)

In this section, Feynman discusses non-conservative forces from the perspectives of a globular star cluster, atomic level, and energy of light. Physics teachers may change the sequence of teaching by discussing frictional forces at the atomic level (a concept that students are likely more familiar) followed by the globular star cluster.

1. Globular star cluster:

“…we analyze a system like that great globular star cluster that we saw a picture of, with the thousands of stars all interacting… (Feynman et al., 1963, section 14–4 Nonconservative forces).”

Feynman explains that friction is apparently a non-conservative force because it has been discovered that the fundamental forces between the particles are conservative. He suggests that we can analyze a system such as a star cluster that is drifting in space. The star cluster can be conceptualized as a single object if we do not have a powerful telescope to see the details. Gravitational forces are still responsible (or “wasted”) for increasing the kinetic or potential energies of the stars inside the star cluster. Strictly speaking, the total energy of the star cluster is still conserved, but it is our inability to see the detailed motions inside that suggest the energy of the star cluster is not conserved. However, Feynman could have explained the apparent loss of kinetic energy of the star cluster using the concept of friction.

In astrophysics, dynamical friction (or gravitational drag) causes a loss of kinetic energy of celestial bodies through gravitational forces with matters in space (Chandrasekhar, 1943). Similarly, a rocket may accelerate and gain kinetic energy through the slingshot effect (or gravity assist). Simply put, this is accomplished via gravitational interactions and it results in a loss of kinetic energy in a large celestial body such as a planet. Importantly, the total energy of the rocket and the celestial body remain constant. Using the law of conservation of energy, we can conclude that a loss of kinetic energy of a massive body is compensated by an increase in a less massive body. In short, a star cluster loses kinetic energy because of dynamical friction or Chandrasekhar friction.

2. Atomic level:

“When we study matter in the finest detail at the atomic level, it is not always easy to separate the total energy of a thing into two parts, kinetic energy, and potential energy… (Feynman et al., 1963, section 14–4 Nonconservative forces).”

According to Feynman, it is not always easy to separate the total energy of a thing into kinetic energy and potential energy at the atomic level. He elaborates that it is not strictly possible to treat heat and chemical energy as being pure kinetic energy or pure potential energy. In a sense, he was inconsistent in using the term heat that could mean “primarily kinetic energy” or “a mixture of kinetic and potential energy.” Physicists prefer to define heat as a process (Romer, 2001) and sometimes use the term internal energy instead of heat. In general, “internal energy of a real gas” can be expressed as a sum of the potential and kinetic energies of the molecules of a system. On the other hand, “internal energy of an ideal gas” does not include potential energy because we assume there are no interactions between molecules.

From a microscopic perspective, frictional forces (or electromagnetic forces) cause a loss of kinetic energy of a large object via electromagnetic forces between charges in the atoms. For example, an atom may accelerate and gain kinetic energy through a collision. This is accomplished through electromagnetic interactions and it results in a loss of kinetic energy in the large object. Importantly, the total energy of the large object and atoms (including electrons) remain constant. Using the law of conservation of energy, we can conclude that a loss of kinetic energy of the large object is compensated by an increase in kinetic energy of atoms. This is similar to how a star cluster loses kinetic energy because of friction as mentioned earlier.

3. Energy of light:

“Light, for example, would involve a new form of energy in the classical theory, but we can also, if we want to, imagine that the energy of light is the kinetic energy of a photon… (Feynman et al., 1963, section 14–4 Nonconservative forces).”

Feynman says that the law of conservation of energy will appear to be incorrect if we neglect to take interactions into account. (In biology, the law of conservation of energy may appear incorrect because biology students study open systems instead of isolated systems). Generally speaking, the kinetic energy of an object is not really lost because the atoms inside are jiggling with a greater amount of kinetic energy. In addition, he clarifies that the formula T + U = constant (14.2) would still be right if we conceptualize the energy of light as the kinetic energy of photons. However, there is also a continuous loss of light energy from the object that is in the form of infrared radiation. Physicists may add that the object would emit even more photons if its temperature is higher.

The emission of light energy can be related to the oscillations of charged particles. In Chapter 28, Feynman explains that light was recognized as electromagnetic influences extending over long distances and generated by an almost incredibly rapid oscillation of the electrons in the atoms (light has both particle and wave properties). Essentially, the fundamental mechanism for the emission of photons is due to the acceleration of charges (e.g., electrons). Furthermore, Feynman adds that “the charge moves around, the unit vector wiggles, and the acceleration of that unit vector is what we are looking for. That is all. Thus E = (−q/4πϵ₀c²)(d²e_r′/dt²) is a statement of the laws of radiation…(Feynman et al., 1963, section 28–2 Radiation).” However, there is also an apparent loss of light energy in the universe that is known as redshift.

Questions for discussion:

1. How would a globular star cluster lose kinetic energy as a result of friction?

2. Is it always easy to separate the total energy of a thing into two parts: kinetic energy and potential energy (at the atomic level)?

3. How does the energy of light help to maintain the law of conservation of energy?

The moral of the lesson: Non-conservative forces (e.g., frictional forces) appear only because we neglect microscopic complications.

References:

1. Chandrasekhar, S. (1943). Dynamical Friction. I. General Considerations: the Coefficient of Dynamical Friction. The Astrophysical Journal, 97, 255–262.

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. Romer, R. H. (2001). Heat is not a noun. American Journal of Physics, 69(2), 107–9.