Einstein's Mass-Energy Equation, Volume I

Francisco Fernflores

• Publisher: Momentum Press
• Year: 2017
• ISBN:
  9781606508589
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

In volume one of Einstein's Mass-Energy Equation, we examine the history and philosophical significance of several demonstrations Einstein published for his mass-energy relation, which is often expressed by the iconic equation E = mc2. Our goal is to illustrate how these demonstrations display a clear shift away from a reliance on electromagnetic phenomena culminating in Einstein's 1934 purely ÒdynamicÓ demonstration. Philosophically, this trend signals the importance of recognizing special relativity as what Einstein called a Òprinciple theory.Ó Volume two of this work examines the role that Einstein's mass-energy relation played in the development of quantum mechanics and general relativity. We also discuss the first empirical confirmation of E = mc2 and some contemporary debates concerning the philosophical interpretation of this important result.

CHAPTER 1

INTRODUCTION AND CONCEPTUAL CONTEXT

1.1 EINSTEIN’S 1905 DISCOVERY OF THE MASS–ENERGY RELATION

Einstein first expressed the relativistic relationship between mass and energy, which we now routinely denote E = mc², in a 1905 article titled “Does the Inertia of a Body Depend upon Its Energy-Content?” The implicit answer to the question in the title already says a lot. Einstein is claiming that the extent to which an object resists a change in its state of motion, be it by an impulsive force or a continuous force, depends on the “energy-content” of the body. The energy-content of a body is today sometimes called its internal energy. Given the general trend among physicists in the early 20th century to regard energy as a property of physical systems and not as a material entity, it would no doubt have seemed quite surprising in 1905 that, e.g., upon losing thermal energy a body would offer less resistance to changes in its state of motion.

The first conclusion Einstein states in his 1905 article is this: “If a body releases the energy L in the form of radiation, its mass decreases by L/V²” [1, p. 174], where Einstein is using V to designate the speed of light in vacuo, for which we will hereafter consistently use the more familiar c. Given the physical configuration Einstein uses to arrive at this result, the term “radiation” in this conclusion refers to the electromagnetic radiation. However, Einstein quickly points out that the form of energy the body gives off seems to be irrelevant to the result, assuming, as he seems to do, that any one kind of energy can be transformed into another. Thus, he says, “we are led to the more general conclusion” that

the mass of a body is a measure of its energy content; if the energy changes by L, the mass changes in the same sense by L/9 · 10²⁰, if the energy is measured in ergs and the mass in grams [1, p. 174].

It has become commonplace to note that in this first demonstration of the relativistic relationship between inertial mass and energy, Einstein nowhere writes down any version of his iconic equation E = mc².

For Einstein, a good candidate for testing his second, more general conclusion empirically would be to measure the changing mass in bodies “whose energy content is variable to a high degree (e.g., salts of radium)” [1, p. 174]. The more energy is released by a sample of radium, the more its mass should diminish. Given that the change in mass is equal to the change in energy divided by the speed of light squared, the change in mass would be very small. Still, Einstein’s claim is that a sample of radium should resist changes to its state of motion less after it has given off energy in the form of radiation.

Finally, Einstein states a closely related third conclusion: “If the theory agrees with the facts, then radiation transmits inertia between the emitting and absorbing bodies” [1, p. 174]. The change in mass that correlates with a change in the internal energy of a body occurs, as Einstein says, “in the same sense” as the change in energy. If a body absorbs energy, e.g., in the form of radiation, its inertial mass will increase; if it radiates energy, its inertial mass will decrease. So, if we consider a pair of bodies such that the first emits energy and the second absorbs it, some of the inertia from the first body will be “transmitted” to the second body.

Einstein’s third conclusion must have seemed rather surprising to Newtonian eyes. For example, suppose a sample of radium is situated inside a box of finite mass with perfectly absorbing walls. If in a given period of time the radium sample emits energy E, the mass of the radium sample decreases by an amount E/c². However, since the walls of the box are absorbing an amount of energy E, the mass of the box increases by an amount E/c². Consequently, if after this given period of time we magically remove the radium sample from the box, we shall find that the radium sample offers less resistance to changes in its state of motion, while the box offers more resistance to changes in its state of motion. The surprising conclusion, from a Newtonian perspective, is that inertia has been “transmitted” from the emitting to the absorbing body.

Attending carefully to the evolving and increasingly rigorous meaning of assertions made by physical theory will be one of our primary tasks in this study. For example, Einstein nowhere states that matter is converted into energy. Similarly, Einstein remains entirely silent about whether the entire inertial mass of an object could be “radiated” away. Details of this sort will be important to us as we try to bridge the sometimes yawning gap between popular presentations of Einstein’s famous equation and its rigorous application by physicists and engineers.

Notice also that Einstein’s 1905 result about the relationship between mass and energy is conceptually distinct from considerations in special relativity about how measurements of inertial mass are related for a pair of observers who are in a state of relative inertial motion. The issue here is not about what has sometimes been called “relativistic mass.” Also, because some contemporary treatments can leave the reader with the impression that the relationship between energy and mass in special relativity issues directly from the definition of the four-momentum, it is also important to note that Einstein’s result, as we shall see in later chapters, depends on accepting at least the principle of relativity, the light principle (which roughly states that light travels at the same velocity regardless of the state of motion of the observer or emitter), and the principle of conservation of energy. It is from these physical hypotheses, and not merely from definitions of physical quantities, that Einstein derives an empirically testable prediction. This prediction, when verified, shows that one property of physical objects that Newton believed to be unchangeable except by physically separating a part of the object, viz., its inertial mass, can vary depending on the “energy-content” of that body. For example, having radiated some of its thermal energy, a rock that has cooled resists changes to its state of motion a tiny bit less. Quantitatively, of course, the change to the inertial mass for a typical rock is minute. Still, from a Newtonian perspective, the temperature of the rock, or the amount of thermal energy it has released, is entirely irrelevant to any consideration of its inertial mass.

With this rough understanding of Einstein’s 1905 mass–energy result, we shall devote the remainder of this chapter to setting Einstein’s result in its historical and philosophical context. Our focus will be to step gingerly across some of the stones in the river that separates Newtonian physics and Einstein’s special relativity. Specifically, we wish to illustrate the accepted and common usage of the terms “mass” and “energy” in physics near the beginning of the 20th century so that we may better understand how Einstein might have thought about his result and how others would have understood it.¹

1.2 MASS: FROM NEWTON TO EINSTEIN

It is sometimes erroneously said that Einstein’s famous equation entails that matter can be converted into energy. Curiously, though perhaps understandably given the history of nuclear energy, seldom is it said that energy can be converted into matter. Yet, the correlation between an object’s energy and its inertial mass in special relativity is clearly symmetric. Furthermore, there is ample evidence that, at least around 1905 and in the following years that led to the development of general relativity in 1916, Einstein did not interpret his result as best expressed by talk of converting matter into energy. One of the main reasons for this is that Einstein focused on how a change to an object’s inertial mass correlates to a change in that object’s “internal” energy, where what is meant by “inertial mass” is not, as Newton would have said, a measure of the “quantity of matter.” We can further appreciate this point by visiting both Newton’s original definition of mass and Mach’s famous criticism of it.

1.2.1 NEWTON AND MASS

In Definition 1 of Newton’s Principia, he states, “Quantity of matter is a measure of matter that arises from its density and volume jointly” [2, p. 403]. In the explanation of the definition, Newton states, “I mean this quantity whenever I use the term ‘body’or ‘mass’in the following pages” [2, p. 404].² If we interpret this definition using our contemporary understanding of the mentioned quantities and without attending to the context of the definition, it is clear that Newton is defining the mass m as the product of the density ρ times the volume V. It is as if Newton is defining mass using the following equation:

(1.1)

The circularity of this definition is patent, since density is routinely defined as “mass per unit volume,” i.e., ρ = m/V. One wonders how Newton himself could have missed this circularity and how it could continue to be missed in reformulations of Newtonian physics such as Euler’s [3] and Maxwell’s [4].

It was Mach, in 1883, who famously pointed out the circularity when he said:

With regard to the concept of “mass,” it is to be observed that the formulation of Newton, which defines mass to be the quantity of matter of a body as measured by the product of its volume and density, is unfortunate. As we can only define density as the mass of unit volume, the circle is manifest [5, p. 194].

There are, then, two questions we wish to address: (1) Why would Newton define mass in an apparently circular way? (2) How do Mach’s criticism and his own attempts to define mass help us clarify the meaning of the concept of mass? Let us address question (1) first.

In A Guide to Newton’s Principia, I. B. Cohen gives a compelling account of why the charge of circularity against Newton’s definition of mass is misplaced [6]. The first important lesson to glean from Cohen’s work is that in an earlier draft of Book I of Principia, Newton clearly illustrates the need to introduce a term other than “weight” to discuss a body’s resistance to changes in its state of motion. According to Cohen, Newton was aware that “weight” varies at different locations on earth. For Newton, “weight” is due to the force of gravity acting on an object, and it is this weight that Newton claims to have found, by experiment, to be proportional to a body’s “quantity of matter” [6, p. 87]. This is significant because it signals that prior to Newton there was not a single, widely used term with a prescribed meaning to refer to what we now call inertial mass. In the scholium to the definitions, Newton states “Thus far it has seemed best to explain the senses in which the less familiar words are to be taken in this treatise” [2, p. 408]. At the time Newton wrote Principia, the term “mass” was in this category of “less familiar words” whose meaning is in need of explanation.

When we turn to the explanation that directly follows Definition 1 in Principia, we find three important aspects of the explication of the term “mass” that merit our attention. First, Newton lists a variety of examples of materials who...

Table of contents

• Cover
• Title
• Copyright
• Abstract
• Contents
• Preface
• Acknowledgments
• 1. Introduction and Conceptual Context
• 2. Early Days of the Mass–Energy Relation
• 3. Einstein and Dynamic Demonstrations of E = mc2
• Bibliography
• Index