Einstein correctly described the equivalence of mass and energy as“the most important upshot of the special theory ofrelativity” (Einstein 1919), for this result lies at the core ofmodern physics. Many commentators have observed that inEinstein’s first derivation of this famous result, he did notexpress it with the equation \(E = mc^2\). Instead, Einstein concludedthat if an object, which is at rest relative to an inertial frame,either absorbs or emits an amount of energy \(L\), its inertial masswill correspondingly either increase or decrease by an amount\(L/c^2\). In Newtonian physics, inertial mass is construed as anintrinsic property of an object that measures the extent to which anobject resists changes to its state of motion. So, Einstein’sconclusion that the inertial mass of an object changes if the objectabsorbs or emits energy was revolutionary and transformative. For asEinstein concluded “If the theory agrees with the facts, thenradiation transmits inertia between emitting and absorbingbodies” (Einstein, 1905b). Yet, in Newtonian physics, inertia isnot the kind of thing that can be transmitted between bodies.

Over a century after Einstein’s first derivation of mass-energyequivalence, as his famous result is called because one can selectunits in which one can express it with an equation of the form \(E =m\), the result continues to receive outstanding empirical support.Furthermore, as the physicist Wolfgang Rindler has pointed out, theresult “has been found applicable and valid in many branches ofphysics, from electromagnetism to general relativity” (Rindler1991, p. 74). Thus, from Rindler’s perspective, which is sharedby many physicists, mass-energy equivalence “… is truly a newfundamental principle of physics” (Rindler 1991, p. 74).

The two main philosophical questions surrounding Einstein’sequation, which are the focus of this entry, concern how we ought tounderstand the assertion that mass and energy are in some senseequivalent and how we ought to understand assertionsconcerning the convertibility of mass into energy (or vice versa).

The equation \(E = mc^2\) is, arguably, the most famous equation in20^th century physics. To appreciate what Einstein’sfamous result is about, and what it is not about, we begin in Section1 with a description of the physics of mass-energy equivalence. InSection 2, we survey six distinct, though related, philosophicalinterpretations of mass-energy equivalence. We then discuss, inSection 3, the history of derivations of mass-energy equivalence andits philosophical importance. Section 4 is a brief and selectiveaccount of empirical confirmation of Einstein’s result thatfocuses on Cockcroft and Walton’s (1932) first confirmation ofmass-energy equivalence and a more recent, and very accurate,confirmation by Rainville et al. (2005).

• 1. The Physics of Mass-Energy Equivalence
  • 1.1 Review of Mass, Momentum, and Kinetic Energy in Newtonian Physics
  • 1.2 Mass and Energy in Relativity: Preliminaries and Notation
  • 1.3 The Physical Significance of Einstein’s Equation
  • 1.4 Mass and Energy in Atomic Physics
  • 1.5 WhyDoes \(E_o\) Equal \(mc^2\)?
  • 1.6 Einstein’s Equation and the Iconic Equation
• 2. Philosophical interpretations of \(E_o = mc^2\)
  • 2.1 Misconceptions about \(E_o = mc^2\)
  • 2.2 Same-property interpretations of \(E_o = mc^2\)
  • 2.3 Different-properties interpretations of \(E_o = mc^2\)
    • 2.3.1 Bondi and Spurgin’s Different-Properties, No-Conversion Interpretation
    • 2.3.2 Rindler’s Different-Properties, Conversion Interpretation
    • 2.3.3 Lange’s One-Property, No-Conversion Interpretation
  • 2.4 Interpretations of \(E_o = mc^2\) and hypotheses concerning the nature of matter
  • 2.5 Ontological interpretations of \(E_o = mc^2\)
• 3. History of Derivations of Mass-Energy Equivalence
  • 3.1 Derivations of \(E_o = mc^2\) that Use Maxwell’s Theory
  • 3.2 Purely Dynamical Derivations of \(E_o = mc^2\)
• 4. Experimental Verification of Mass-Energy Equivalence
• 5. Conclusion
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. The Physics of Mass-Energy Equivalence

A lot of philosophical attention has been paid to how the meaning ofstatements concerning lengths and times changes in the context ofspecial relativity. For example, we have learned from Stein (1968)that a statement such as “the length of the table is 2m,”while perfectly meaningful in the context of Newtonian physics, iseither elliptical at best or meaningless at worst in the context ofspecial relativity. However, comparatively little philosophicalattention has been paid to how special relativity brings aboutcorresponding changes to the third fundamental dimension in basicphysics: mass. Yet, assertions concerning mass, and other dynamicalconcepts in mechanics into which it figures centrally, notablymomentum and kinetic energy, also have their meanings changed inspecial relativity.

To illustrate these changes, Section 1.1 reviews the concepts of mass,momentum and kinetic energy in Newtonian physics. In Section 1.2, webegin exploring the concepts of mass and energy in special relativityand state the notation we use in all further discussions, a step wemust make explicit as discussions of the equivalence of mass andenergy have historically used different notations, which can lead toconceptual confusion. In Section 1.3, we discuss the physicalsignificance of mass-energy equivalence as it applies to the analysisof bodies (construed as indivisible wholes) and idealized versions ofcomposite systems, while in Section 1.4 we discuss mass and energy inatomic physics. In Section 1.5, we present an elementary derivation ofEinstein’s famous result with the hope that our readers will geta sense of “why” mass and energy are equivalent accordingto special relativity. Finally, in Section 1.6, we consider therelationship between the equivalence of mass and energy codified inthe equation \(E_o = mc^2\) and the nearly identical, thoughconceptually quite different, ubiquitous equation \(E = mc^2\).

1.1 Review of Mass, Momentum, and Kinetic Energy in Newtonian Physics

In Newtonian physics, a typical physical object, such as a billiardball, has associated with it a positive, real number called itsmass, for which the symbol \(m\) is commonly used. When weare focusing on describing mathematically the motion of this objectunder the action of contact forces, such as when one tries to predictwhere a billiard ball will go after a collision, the mass in questionis also called theinertial mass. In the special case inwhich one considers how the billiard ball moves in response togravity, say as it falls toward the earth, the measure of how thebilliard ball “responds” to the gravitational field iscalled thegravitational mass of the object. Usingexperiments in which he filled wooden boxes with different materialsand suspended them from strings to construct pendulums, Newtondiscovered that inertial mass and gravitational mass are directlyproportional. Physicists have since customarily treated inertial andgravitational mass as numerically equal.

The inertial mass in Newtonian physics (and even the gravitationalmass) is routinely interpreted as an intrinsic property of an object.The inertial mass of an object is a measure of the body’sinertia, i.e., its tendency to resist changes to its state of motionin response to the action ofany kind of force. Since atleast the late nineteenth century and Mach’s criticisms ofNewton’s physics, physicists have deprecated thinking of mass asthe “quantity of matter.”

The physical intuition behind the Newtonian concept of inertial massis basic if we can avail ourselves of the notion of a Newtonian force.Given two bodies \(B_1\) and \(B_2\) in suitably idealized conditions,if it takes twice the force for \(B_2\) to attain the same finalvelocity as \(B_1\), \(B_2\) has twice the mass of \(B_1\), i.e.,\(B_2\) has twice the inertia of \(B_1\). In Newtonian physics, theinertial mass of an object, and even its gravitational mass, can onlychange by either physically removing a part of the object or attachinga part to the object to make a bigger whole.

Two important quantities for describing the motion of objects inNewtonian physics are momentum and kinetic energy. Unlike mass, eachof these quantities is what we might call a relational, or extrinsic,quantity. Although all physical objects have some value (or amount) ofmomentum and kinetic energy, the value of each of these quantitiesdepends on the inertial reference frame relative to which one ismeasuring these quantities.

Newton called momentum the “quantity of motion,” which isan apt label, because, very roughly, it is a measure of the extent towhich an object is moving relative to an inertial reference frame. Ourcolloquial use of the word “momentum” is related to thisintuition and to its formal definition as the product of the mass ofan object times its velocity relative to some inertial referenceframe. If object \(B_2\) has twice the mass as \(B_1\) but moves withthe same velocity relative to a reference frame, \(B_2\) has twice themomentum.

Velocity, however, is a directed quantity, because it codifies notjust the speed \(v\) with which the object moves, but also itsdirection. Velocity is formally represented by a vector\(\mathbf{v}\), the magnitude of which is the speed of the object\(v\), which we sometimes also call the “velocity” of theobject while allowing the context to indicate that this is ellipticalfor “the magnitude of the velocity.” Momentum is thereforealso a directed quantity, represented formally by a vector\(\mathbf{p}\). To paraphrase Taylor and Wheeler (1992, p. 191), thedirection of momentum matters: A glancing blow is never as damaging asone that is head on. By analogy with velocity, the magnitude of themomentum is represented by the letter \(p\).

Finally, in Newtonian mechanics, an object in motion relative to areference frame also has kinetic energy, or energy of motion. Kineticenergy, like all forms of energy, can be transformed into another kindof energy. So, for example, one could measure the kinetic energy of amoving billiard ball by having it collide with a ball of soft putty ora spring that brings the ball to rest and measuring the energyabsorbed by the putty or spring. There is no single standard symbolfor kinetic energy, though most elementary physics textbooks use\(KE\), more advanced books tend to use \(T\).

Unlike the momentum, kinetic energy is not a directed quantity. It is,like momentum, a function of the speed \(v\) of the object relative toan inertial frame. The faster an object moves relative to someinertial frame, the more momentum and kinetic energy it has. However,for an object that is not accelerating, i.e., an object that coversequal spaces in equal times, one can always find an inertial referenceframe in which both the momentum and the kinetic energy of the objectis zero.

Following Griffiths’ approach, we might say that so far we havemerely defined some quantities (Griffiths 1999, p. 509 ff.). Thephysics really lies in the three corresponding conservation principlesassociated with these quantities: the principle of conservation ofmass, the principle of conservation of energy, the principle ofconservation of (linear) momentum. These principles contain thephysics, because each one states that a certain quantity, mass,energy, or momentum, is conserved in all interactions. So, forexample, if we consider the collision of two billiard balls, one canuse the conservation of momentum to predict, given the initial motionsand masses of the billiard balls, how they will move after thecollision.

1.2 Mass and Energy in Relativity: Preliminaries and Notation

As Hecht has emphasized, Einstein never wrote down his famous resultusing the symbols “\(E\)” and “\(m\)” as theyappear in the famous equation attributed to him (Hecht 2012). Althoughpart of the reason is merely that Einstein used different letters forenergy, mass, and the speed of light in his early papers discussingmass-energy equivalence (in 1905 and 1906), deeper reasons related tohow we should understand the result quickly emerged.

In his review article on special relativity from 1907, Einstein showsthat a body of mass \(\mu\) that has absorbed an amount of energy\(E_o\) as measured in its rest frame executes motion, in an inertialframe relative to which it moves with some velocity, as if its mass\(M\) was given by the expression (Einstein 1907b, p. 286):

In a footnote, Einstein explains the convention, which he had alreadyadopted in an earlier paper (Einstein 1907a, p. 250), of using“the subscript ‘\(o\)’ to indicate that the quantity inquestion refers to a reference system that is at rest relative to thephysical system considered” (Einstein 1907b, p. 286).

After 1907, Einstein’s notation crystalizes so that by 1921, inhis Princeton Lectures, Einstein expresses his famous result bywriting (Einstein 1922, p. 46):

Our main task in the next section is to explain the physicalsignificance of the equation \(E_o = mc^2\), which we will henceforthcall “Einstein’s equation”, and its relationship toits iconic variant without the subscript “\(o\)” appearingbeside the letter “\(E\)”.

However, it is important before going much further to note that inEinstein’s equation \(E_o = mc^2\), the symbol \(m\) is the massof an object as measuredin the inertial frame in which thatobject is at rest. Physicists also call this mass \(m\) the“rest-mass” of the object. The rest-mass of an object isnumerically equal to its Newtonian inertial mass, though arguably thesymbol \(m\) (or the corresponding term “mass”) hasdifferent meanings in Newtonian and relativistic physics (see, e.g.,Kuhn 1962, p. 101 ff. and Torretti 1990, p. 65 ff.).

Furthermore, there are conceptual reasons why both Einstein and manycontemporary physicists do not add a subscript “\(o\)” to\(m\) when denoting rest-mass. For example, Taylor and Wheeler arguethat the phrase “rest-mass” engenders potential confusionas it might lead the reader to ask: What happens to the rest-mass whenthe object is moving? Answer: nothing (Taylor and Wheeler 1992, p.251). The rest-mass of an object is an invariant quantity in specialrelativity; it has the same value for all inertial observers. Taylorand Wheeler quip: “In reality mass is mass is mass”(Taylor and Wheeler 1992, p. 251). From their perspective, which isquite standard now in physics textbooks and articles, there is no needto prefix the term “mass” with “rest,” becausethere is no other kind of mass worth speaking about in specialrelativity (seeSection 1.6).

Henceforth, we will adopt the accepted convention, though it isadmittedly not universally followed especially in older sources, ofusing the following symbols with their stated meanings (see Table 1):

We will first focus on the mechanics of idealized point particles inspecial relativity. Regardless of the macroscopic objects theyrepresent, we shall treat such particles, and by extension thecorresponding physical objects, as un-analyzable wholes. We will thenseparately discuss “composite systems” or systems composedof such particles. Physicists use composite systems to approximatephysical objects when they are interested in examining the“inner workings” of such objects, such as when they treata gas as a collection of idealized particles.

1.3 The Physical Significance of Einstein’s Equation

Following Geroch, we can begin to explain the physical significance ofEinstein’s equation \(E_o = mc^2\) by considering a very simplephysical system. Imagine, as Geroch suggests, a brick being heated ora battery being charged (Geroch 2005, p. 198). Suppose we considerthese objects in the inertial reference frame in which they are atrest. As the brick is heated, or the battery charged, it absorbs anamount of energy \(E_o\) as measured in its rest frame.Einstein’s equation tells us that the mass of the brick, orbattery, after it absorbs an amount of energy \(E_o\) is increasedexactly by an amount \(E_o /c^2\). The value of the mass of the hotbrick or charged battery is greater than it was before it absorbedenergy. So, for example, it takes justa little bit moreforce to move the charged battery than it did to move the unchargedbattery. How much is “a little bit more”?

Geroch has a clever way to answer this question. Suppose we use theamount of energy \(E_o\), as measured in the rest-frame of thebattery, to accelerate the battery (instead of charging it) so that iteventually reaches a final speed of 670 mph. Since 670 mph seems likesomething moving pretty fast, at least by the non-relativisticstandards of human travel, one might think that one is using quite abit of energy to accelerate the battery to that speed. Now supposethat instead of using the energy \(E_o\) to accelerate the battery, weuse that same amount of energy \(E_o\) to charge the battery. Theincrease in the mass of the charged battery is \(E_o /c^2\), butbecause the speed of light is such a large number, approximately 670million mph, “the mass of the battery would beincreased by about 1 part in a million-million (i.e., by a fraction\(10^{-12}\)…)” (Geroch 2005, p. 199).

There is nothing unique about energy absorption in these examples. Asthe battery loses energy, say by powering a device, or the brick emitsthermal energy as it cools, its mass decreases. Consequently, imaginea suitably idealized closed system in which two objects \(B_1\) and\(B_2\) are in a state of relative rest. If \(B_1\) radiates an amountof energy \(E_o\) and \(B_2\) fully absorbs that same amount ofenergy, the mass of \(B_1\) decreases by \(E_o /c^2\) while the massof \(B_2\) increases by the same amount. This physical change in themasses of \(B_1\) and \(B_2\) is a novel prediction in specialrelativity. For in Newtonian physics, there is no relationship at allbetween the inertial mass of a body and the amount of energy itradiates or absorbs. This is why Einstein was led to conclude that“If the theory agrees with the facts, then radiation transmitsinertia between emitting and absorbing bodies” (Einstein 1905b).If special relativity is supported by empirical evidence, the inertialmass of an object can change, not because we have chopped off a pieceof the object or attached more stuff to it, but merely because theobject has radiated or absorbed energy. To physicists and philosopherstrained exclusively in the Newtonian tradition, this result would haveseemed perhaps extraordinary but certainly revolutionary.

So far, we have been focusing on what physicists such as Baierlein(2007) call the incremental version of mass-energy equivalence,because we have focused on the strict correlation in specialrelativity between a change in the rest energy of a body \(E_o\) and achange in its mass \(m\). However, Einstein also emphasized two subtlydifferent “readings” of his equation. First, as early as1906, Einstein argued that when one considers physical systems inwhich there are electromagnetic processes, such as a“complex” of light being emitted from the inside wall of afreely-floating box and absorbed by the opposite wall, one could avoida fundamental conflict with the laws of mechanics “if oneascribes the inertial mass \(E/V^2\) to any energy \(E\)”(Einstein 1906, p. 206. Note that \(V\) represents the speed of light. See also Taylor and Wheeler 1992, p. 254, fora detailed discussion of this example). This insight, sometimesexpressed with talk about the “inertia of energy,” was animportant step in the development of general relativity, because thattheory uses the principle of equivalence, which states very roughlythat inertial mass and gravitational mass are directly proportional,as a foundational principle.

At a time when physicists were moving toward regarding energy-carryingfields, such as the electromagnetic field, as entities in their ownright, combining the principle of equivalence with the “inertiaof energy” led Einstein to the insight that fields themselvescould gravitate. So, for example, Einstein claims in 1907 to show that“radiation enclosed in a cavity possesses not only inertia butalso weight” (Einstein 1907b, p. 288). A more contemporary andfairly common way of making the same point is to say that a mirror boxwith perfectly reflecting walls filled with light (conceived here forconvenience as an energy-carrying disturbance in the electromagneticfield) is attracted by gravity by an amount that is greater than therest mass \(M\) of the box itself. Specifically, if the energy of thelight in the box is \(E_L\), gravity acts on the box not as if the boxhas a mass \(M\) but as if the box has a mass \(M + E_L /c^2\).Although the difference is numerically tiny, strictly speaking abalance with an empty mirror box on one side and an identical mirrorbox filled with light on the other would not be level.

Conversely, Einstein also emphasized reading his famous equation“in the other direction,” as it were. For example, in hisreview article from 1907, he says, “with respect to inertia, amass \(\mu\) is equivalent to an energy content of magnitude \(\muc^2\)” (Einstein 1907b, p. 287). Einstein uses the phrase“energy content” (translated from the original German“Energieinhalt”) to convey a notion that is new in specialrelativity: A physical object contains within it energy that, at leastin principle, could be transformed into other forms of energy such askinetic energy (as we now know only too well). Because this restenergy is contained within the boundaries of a body when we treat theobject as a whole, physicists, such as Rindler, also call it“internal energy” (Rindler 1991, p. 71).

As Rindler suggests, it is not at all an idle or childish question toask: Where does all that “internal energy” reside (Rindler1991, p. 75)? Rindler’s answer to this question assumes that wewish to analyze a body all the way down to its subatomic components.He explains:

A very small part of this energy resides in the thermal motions of themolecules constituting the particle, and can be given up as heat; apart resides in the intermolecular and interatomic cohesion forces,and some of that can be given up in chemical explosions; another partmay reside in excited atoms and escape in the form of radiation; muchmore resides in nuclear bonds and can also sometimes be set free, asin the atomic bomb. But by far the largest part of the energy (about99 per cent) resides simply in the mass of the ultimate particles andcannot be further explained. Nevertheless, it too can be liberatedunder suitable conditions, e.g., when matter and antimatter annihilateeach other (Rindler 1991, p.75).

However, when developing a theoretical description of a macroscopicobject, it is seldom practical to construct that description bytreating subatomic quantum objects as the fundamental constituents, asRindler suggests.

Instead, when one analyzes a physical object theoretically by lookingat its component parts, one has to decide the level of granularity atwhich one wishes to analyze the object. For example, if the object weare analyzing is a chunk of iron, and we are studying magnetism, itmay be sufficient to analyze the iron by treating its magnetic domainsas fundamental constituents. However, a magnetic domain is a chunk ofiron that contains many, many atoms. For other purposes, we may electto analyze the sample of iron all the way down to the atomic level. Ofcourse, the atoms themselves have parts. So, for yet differentpurposes, we may elect to analyze the sample of iron at the sub-atomiclevel of quantum objects.

As Rindler’s description of where the rest energy resides makesclear, at each level of analysis, there are two main contributors tothe rest energy of the “macroscopic” sample we areanalyzing: (1) the energy equivalent of the sum of the rest-masses ofthe constituent elements considered as fundamental and (2) the sum ofthe energy “stored” or “carried” by theconstituent elements. To perform this kind of analysis for a concreteobject is a rather subtle affair. So physicists and philosopherswriting about mass-energy equivalence tend to focus on the highlyidealized notion of an ideal gas.

For the purposes of this kind of discussion, an ideal gas is composedof molecules that are treated as idealized point particles to each ofwhich we assign a mass (i.e., a rest-mass). The molecules are treatedas moving uniformly, i.e., with constant velocity, and as interactingonly in perfectly elastic collisions. Consequently, if we consider asample of gas contained in a massless vessel, the only energy“carried” by the constituent elements is the kineticenergy of the molecules.

When viewed from a Newtonian perspective, and assuming that the vesselcontaining the gas is itself massless, the mass of the vessel of gasis simply equal to the sum of the masses of the molecules. From arelativistic point of view, this last assertion statesincorrectly that the rest-mass of the vessel of gas is equalto the sum of the rest-masses of the molecules. Yet, from arelativistic point of view, the rest-mass of the vessel of gas isequal to the sum of the rest-masses of the moleculesplus thekinetic energy of the molecules divided by \(c^2\). Since, accordingto the Kinetic Theory of Gases, the temperature of the gas isproportional to the average kinetic energy of its molecules, if thegas temperature increases or decreases, the rest-mass of the vessel ofgas increases or decreases accordingly by a tiny amount.

Assuming the principle of equivalence, which entails that we canmeasure inertial mass by measuring gravitational mass with a balance,we can illustrate the difference between the Newtonian andrelativistic understanding of the ideal gas as follows. Imagine thatwe have two otherwise identical massless vessels filled with exactlythe same amount and type of gas. In one vessel, the gas is at atemperature very near absolute zero, so its molecules have very littlekinetic energy. In the other vessel, the gas is at a temperature of500° C. Place these two vessels of gas on the ends of a balance.According to Newtonian physics, the balance will be level, becauseboth gas samples have exactly the same mass. According to relativity,the balance will not be level and will be tipped on the side of thehot gas, because the high kinetic energy of the molecules contributesto the rest energy of the gas, which contributes, throughEinstein’s equation, to the rest-mass of the vessel of gas.

1.4 Mass and Energy in Atomic Physics

Perhaps the most common examples used to illustrate Einstein’sequation concern collisions among sub-atomic objects. For ourpurposes, it is safe to treat atomic and sub-atomic objects asparticles involved in collisions where the total number of particlesmay or may not be conserved.

The bombardment of a Lithium nucleus by protons is a historicallysignificant and useful example for discussing mass-energy equivalencein collisions where the number of particles is conserved. Cockcroftand Walton (1932) were the first to observe the release of two\(\alpha\)-particles when a proton \(p\) collides with a \({^7}\Li\)nucleus. The reaction is symbolized as follows:

That the number of particles is conserved in this reaction becomesclear when we recognize that the \({^7}\Li\) nucleus consists of threeprotons and four neutrons and that each \(\alpha\)-particle consistsof two protons and two neutrons.

In the bombardment of Lithium reaction above, the sum of therest-masses of the reactants (the proton and the \({^7}\Li\) nucleus) isgreater than the sum of the rest-masses of the products (thetwo \(\alpha\)-particles). However, the total kinetic energy of thereactants isless than the total kinetic energy of theproducts. Cockcroft and Walton’s experiment is routinelyinterpreted as demonstrating that the difference in the rest-masses ofthe products and reactants (times \(c^2)\) is equal to the differencein the kinetic energies of the products and reactants (but seeSection 4 for further discussion of this experiment as a confirmation ofmass-energy equivalence).

Descriptions of collisions among sub-atomic particles such as thebombardment of Lithium make it seem as though one must admit that massisconverted into energy. However, influenced perhaps by thewidely-known discussion of mass-energy equivalence by Bondi andSpurgin (1987) (seeSection 2.3.1), physicists now explain such reactions not as cases of mass beingconverted into energy, but merely as cases where energy has changedforms. Typically, in these types of reactions, the potential energythat “contributes” to the rest-mass of one (or possibly)more of the reactants istransformed in a non-controversialway to the kinetic energy of the products. As Baierlein (2007, p. 322)explains, in the case of the bombardment of \({^7}\Li\) with protons andits subsequent decomposition into two \(\alpha\)-particles, theapparently “excess” kinetic energy of the\(\alpha\)-particles did not simply “appear” out ofnowhere. Instead, that energy was there all along as the potential andkinetic energy of the nucleons. In other words, one can explain thechange in mass and energy in the bombardment reaction by saying (i)that the potential and kinetic energies of the nucleons that make upthe \({^7}\Li\) nucleus contribute to its rest-mass and (ii) that thevast amount of energy of the \(\alpha\)-particles was not“created” in the reaction, or “converted” frommass, but was simply transformed from the various forms of energy thenucleons possess.

Collisions among sub-atomic particles and their correspondinganti-particle are not quite so easily explained as merely involvingthe re-arrangement of particles and re-distribution of energy. Themost extreme example of this sort, and one that is often used in thephysics literature, is pair annihilation. Consequently, let usconsider a collision between an electron \(e^-\) and a positron\(e^+\), which yields two photons \(\gamma\). Symbolically, thisannihilation reaction is written as follows:

According to the currently accepted Standard Model of particlephysics, electrons and photons are both “fundamentalparticles,” by which physicists mean that such particles have nostructure, i.e., such particles are not composed of other, smallerparticles. Furthermore, the photons that are the products in theannihilation reaction have zero rest-mass. Thus, the rest-masses ofthe incoming electron and positron seems to “disappear”and an equivalent amount of energy “appears” as the energyof the outgoing photons. Of course, Einstein’s famous equationmakes all of the correct predictions concerning the relevant massesand energies involved in this reaction. So, for example, the totalenergy of the two photons is equal to the sum of the kinetic energiesof the electron and positronplus the sum of the rest-massesof the electron and positron multiplied by \(c^2\).

Finally, although mass and energy seem to “disappear” and“appear” respectively when we focus on the individualconstituents of the physical system containing the incomingelectron-positron pairand the outgoing photons, the mass andenergyof the entire system remains the same throughout theinteraction. Before the collision, the rest-mass of the system issimply the sum of the rest-masses of the electron and positron plusthe mass-equivalent of the total kinetic energy of the particles.Consequently, the entire system (if we draw the boundary of the systemaround the reactants and products—which is, of course, a spatialand temporal boundary), has a non-zero rest-mass prior to thecollision. However, after the collision, the system, which nowconsists of two photons moving in non-parallel directions, also has anon-zero rest-mass (for a detailed discussion concerning the rest-massof systems of photons, see Taylor and Wheeler, 1992, p. 232).

1.5 WhyDoes \(E_o\) Equal \(mc^2\)?

A common way in which the question “Whydoes \(E_o\)equal \(mc^2\)?” is interpreted by philosophers and physicistsis that it is a request for a derivation that shows that given certainphysical principles, such as the principle of relativity and theprinciple of conservation of energy, Einstein’s equation is alogical consequence of those assumptions. We will discuss briefly thehistory of derivation of Einstein’s equation inSection 3.

However, in this section, we wish to present a rather simplifiedversion of just one of Einstein’s derivations, published in 1946(Einstein 1946). We will follow closely the simplified version ofEinstein’s 1946 derivation developed by Ralph Baierlein (1991),who has used his derivation to teach Einstein’s equation toundergraduates who are not science majors. John Norton has usedessentially the same simplified derivation in theCambridgeCompanion to Einstein (Norton 2014).

As we will see, if Baierlein’s assessment of his own derivationis correct, the price to pay for this particular simplification isthat the derivation ceases to be relativistic, in the sense that noneof the core principles at the heart of special relativity seem to berequired to derive Einstein’s equation. Nevertheless, it mayhelp readers get a “feel” for why \(E_o = mc^2\) and itdoes familiarize those not already conversant in the methods ofrelativistic physics with a style of reasoning that is common in thatfield. Those interested in a presentation of Einstein’s own 1946derivation, which explicitly shows all of the relativistic assumptionsand key approximation steps Einstein takes, may be interested inconsulting the exposition by Fernflores (Fernflores 2018, vol. II,§3.3).

Like Einstein, many physicists and philosophers who wish to deriveEinstein’s equation do so by considering an idealized physicalconfiguration. Typically, one considers a physical object \(B\)involved in a symmetric physical interaction. For example, inEinstein’s original derivation from 1905, \(B\) emits twoequally energetic pulses of light in opposite directions. Mermin andFeigenbaum have shown how one can also derive Einstein’sequation by considering the case where \(B\) emits two physicalbodies, instead of pulses of light (Mermin and Feigenbaum 1990).

In Einstein’s 1946 derivation, instead of emitting light, \(B\)absorbs two equally energetic pulses of light symmetrically. The goalin all of these approaches is to perform a “before” and“after” comparison and to show that “after”the body \(B\) absorbs or emits energy, its mass (i.e., rest-mass)increases or decreases according to Einstein’s equation.

Using a common heuristic in relativistic physics, one considers thephysical interaction first in the inertial reference frame in which\(B\) is at rest, which is sometimes called the “restframe.” One then compares the mathematical description of thisinteraction to the corresponding description from the perspective of adifferent inertial reference frame that moves with a constant velocityrelative to the rest frame. Finally, using dynamical principles, suchas the conservation of energy or conservation of momentum, and the twoprinciples at the core of special relativity (i.e., the relativityprinciple and the light principle), one shows that the dynamicalprinciples require that after \(B\) suffers the interaction in whichit absorbs or emits energy, its mass (i.e., rest-mass) must change byan amount given by Einstein’s equation.

From a very general point of view, the reasoning in these types ofderivations can be displayed schematically, if somewhat roughly, likethis:

For a particular idealized physical interaction, if:

• certain conservation principles are true, such as the principle ofconservation of energy and the principle of conservation ofmomentum,
• the two principles at the core of special relativity are true,i.e., the principle of relativity and the light principle, and
• a body, treated as an un-analyzed whole, absorbs or emits an amountof energy \(E_o\) as measured in its rest frame,

then:

• its mass (i.e., rest-mass) increases or decreases by an amount \(E_o /c^2\).

Even at this very general level, one can see the limitations of theseapproaches. For example, since the analysis is based on a particularphysical interaction \(\boldsymbol{I}\), one cannot immediatelyconclude that a body’s mass will change according toEinstein’s equation in a different type of physical interactionI’, for instance one in whichelectromagnetism plays no role.

Reasoning along these lines, Ohanian (2009, 2012) has argued thatEinstein should not be credited with proving his famous equation. Onthe other hand, the physicist N. David Mermin (2011, 2012) claims thatOhanian’s demands for what counts as a “proof” inphysics are too stringent. Even though we are considering a veryspecific interaction \(\boldsymbol{I}\), Mermin might say, we canunderstand that it is such a generic situation that we can confidentlyexpect the result in all circumstances. Einstein himself believed thatif \(\boldsymbol{I}\) involved a physical process in which there wasan interaction between an electromagnetic field (say in the form oflight) and an ordinary physical object, then \(\boldsymbol{I}\) wastoo specific to support the general conclusion that inallcases, a change in the rest energy of a body is accompanied by achange to its mass (i.e., rest-mass).

With all of these caveats in place, we are almost ready to understandBaierlein’s derivation of Einstein’s equation. BecauseBaierlein’s derivation involves analyzing a physical interactionin which an object emits light, i.e., electromagnetic radiation, weneed first to state two dynamical properties of electromagneticradiation. First, considered as an electromagnetic wave, like allwaves, light carries energy. This is quite familiar to us today at aneveryday level, especially because of all the “solar”devices we use. Second, light also carries momentum. As Baierleinreports, in the late nineteenth century, James Clerk Maxwell hadalready determined that a burst of light of energy \(E\) had momentum\(E/c\). So, for example, if a laser beam strikes a freely-floatingmirror in outer space, the collision of light against mirror willimpart a finite, non-zero momentum to the mirror. One can calculateexactly how the mirror will have its state of motion changed as aresult of this interaction by using the principle of conservation ofmomentum.

Baierlein asks us to consider an atom that emits two photons of equalenergy “back-to-back” in opposite directions. AlthoughBaierlein uses the Planck-Einstein expression \(E = hf\) for theenergy of a photon, where \(f\) is its frequency, and \(h\) isPlanck’s constant, as he himself points out the right-hand sideof that expression does not figure significantly in his derivation.Baierlein’s derivation, he tells us, works in exactly the sameway if we just think of an object that emits two “bursts”of light construed as classical electromagnetic radiation. We willadopt this latter approach and consider a body \(B\) that emits twobursts of light in opposite directions, each with an energy \(E/2\)(this is the approach Norton 2014 uses). The total energy emitted by\(B\) is therefore simply \(E\).

We wish now to examine this emission of light by \(B\) from twodifferent inertial frames. The first inertial frame we consider issimply the inertial frame in which \(B\) is at rest. Since we will notbe performing many calculations using quantities defined relative tothis inertial frame, we will label it \(K'\). The direction ofmotion of the second inertial frame, which we will label \(K\), iscarefully chosen to simplify the calculations.

First, we choose \(K'\) so that \(B\) emits the two bursts oflight along \(z'\)-axis in opposite directions. We then choose\(K\) so that \(K\) moves with velocity \(v\) in the negativedirection along the \(x'\)-axis. Using standard conventions,this means that an observer at rest in \(K'\) (Alice) judgesthat the light emitted by \(B\) travels up and down the page. For anobserver at rest in \(K\) (Bob), the atom \(B\) moves to the rightwith velocity \(v\) and the light is emitted toward the right makingan angle \(\theta\) with the \(x\)-axis.

According to Baierlein, “symmetry alone requires that the atom\([B]\) remain at rest in Alice’s frame” (1991, p. 170).It also follows directly from this that since \(B\) remains at rest in\(K'\), the velocity of \(B\) does not change in \(K\) after theemission of light. However, as we will see shortly, the momentum of\(B\) does change in both \(K\) and \(K'\) because the light itemits carries away momentum. If we assume the classical definition forthe momentum of \(B\) as the product of its mass \(m\) and itsvelocity \(v\), and \(v\) does not change, it follows that in orderfor the law of conservation of momentum to be satisfied, the mass(i.e., rest-mass) of \(B\) must change.

Let us now examine the light emission from the perspective of theinertial frame \(K\). Relative to \(K, B\) moves (to the right) withvelocity \(v\). Because relative to \(K'\) the bursts of lightare collinear, the \(x\)-component of the velocity of the light asmeasured in \(K\) must be \(v\). However, the velocity of the lightalso has a vertical component, i.e., a \(z\)-component. This issignificant, because we wish to calculate the change in the momentumof \(B\).

Relative to \(K\), the momentum of \(B\) only changes along the\(x\)-direction, because the changes to the momentum along the\(z\)-direction are equal and opposite. Using elementary trigonometry,for one of the bursts of light, the momentum along the \(x\)-directionis:

So, the total momentum of the emitted photons, relative to \(K\), issimply:

which, by the principle of conservation of momentum, must be equal tothe amount of momentum lost by \(B\).

Now, since the velocity of \(B\) does not change (in either \(K\) or\(K'\)), and if we assume the classical expression for the momentum of\(B\), we have,

or simply,

Finally, since \(E\) is the amount of energy lost by \(B\) as measuredin \(B'\)s rest-frame, we could more perspicuously write:

We have thus shown that when \(B\) emits an amount of energy \(E_o\)(while remaining in its current state of inertial motion, which is astate of rest relative to \(K'\)), the mass (i.e., rest-mass) of \(B\)decreases by an amount \(E_o /c^2\).

Regardless of whether Ohanian (2009, 2011) is correct thatEinstein’s own derivations do not constitute a“proof” of mass-energy equivalence because they considerphysical configurations that are too specific, a new concern ariseswhen one considers “simplified” versions of derivations of\(E_o = mc^2\) such as the one we have just reviewed. Baierleinconcludes his derivation by praising, as one of its merits, that it“requires only the ratio of momentum to energy forelectromagnetic radiation” (Baierlein 1991, p. 172). We havealready used this aspect of the derivation in our presentation above.

However, Baierlein then goes on to state:

Moreover, this derivation—unlike Einstein’s 1905derivation—makes no use of Lorentz transformations or otherresults from the special theory of relativity. In short, by 1873,Maxwell knew everything necessary to derive the equation \(\Delta E =(\Delta m)c^2\). All that was missing was a context of inquiry thatwould have led him to search for a connection between energy andinertia (Baierlein 1991, p. 172).

This is a remarkable conclusion, for, if correct, it suggests thatwhen we use this kind of simplified version of Einstein’s 1946derivation, we are not displaying how Einstein’s equation is aconsequence of special relativity. By contrast, Einstein’s own1946 derivation is explicitly relativistic (see Fernflores 2018, vol.II, Sec. 3.3).

1.6 Einstein’s Equation and the Iconic Equation

As we have seen, Einstein’s equation \(E_o = mc^2\) states thatwhenever there is a change in the rest energy of an object, there is acorresponding change in its mass (i.e., rest-mass). Although we havenot exactly demonstrated it, Einstein’s equation is alsointerpreted as stating that any object with a non-zero mass (i.e.,rest-mass) possesses a rest energy (also sometimes called“internal energy”).

However, in relativistic mechanics, i.e., in the study of the motionsof idealized point particles that move in accordance with the theoryof special relativity, an object’s total energy \(E\), which isdefined as the sum of its kinetic energy and its rest energy, is givenby the equation:

where \(\gamma(v)\) is to so-called “Lorentz factor.” Thistotal energy \(E\) differs from the rest energy \(E_o\) for any objectthat moves with some velocity \(v\) relative to a given inertialframe. In such an inertial frame, an object will have a non-zerorelativistic kinetic energy and itstotal energy \(E\) isgiven by the equation above. In the inertial frame in which such anobject is at rest, however, the value of \(\gamma(v)\) becomes 1 andthe total energy is equal to the rest energy \(E_o\), one might say,precisely because in that inertial frame the object’srelativistic kinetic energy is zero.

In the early and mid twentieth century, some physicists, such asRichard Feynman (1963, Vol. I, Sec. 16-4), defined a new quantity,which they labelled \(m\) and called the “relativisticmass,” as the product of the rest-mass of the object, which theylabelled \(m_o\), and the Lorentz factor \(\gamma(v)\) like this:

Usingthese notational conventions, the iconic equation \(E =mc^2\) is the equation for the total energy \(E\) of a body as afunction of itsrelativitic mass.This equation isnot really of interest to us in discussing mass-energy equivalence.Furthermore, physicists today have generally given up this notationalconvention and deprecate the notion of “relativisticmass,” which, as Griffiths quips, “has gone the way of thetwo dollar bill” (1999, p. 510 fn. 8).

2. Philosophical interpretations of \(E_o = mc^2\)

There are three main philosophical questions concerning theinterpretation of \(E_o = mc^2\) that have occupied philosophers andphysicists:

1. Are mass and energy thesame property of physical systemsand is that what is meant by asserting that they are“equivalent”?
2. Is mass “converted” into energy in some physicalinteractions, and if so, what is the relevant sense of“conversion”?
3. Does \(E_o = mc^2\) have any ontological consequences, and if so,what are they?

Interpretations of mass-energy equivalence can be organized accordingto how they answer questions (1) and (2) above (Flores 2005). As wewill see (inSection 2.5), interpretations that answer question (3) affirmatively assume thatthe answer to question (1) is yes.

The only combination of answers to questions (1) and (2) that isinconsistent is to say that mass and energy are the same property ofphysical systems but that the conversion of mass into energy (or viceversa) is a genuine physical process. All the other three combinationsof answers to questions (1) and (2) are viable options and have beenheld, at one time or another, by physicists or philosophers asindicated by the examples given inTable 2.

In this section, we will describe the merits and demerits of each ofthe interpretations inTable 2. Beyond these interpretations, we will also discuss two other types ofinterpretations of mass-energy equivalence that do not fit neatly inTable 2. First, we will discuss Lange’s (2001, 2002) interpretation,which holds that only mass is areal property of physicalsystems and thatwe convert mass into energy when we shiftthe level at which we analyze physical systems. Second, we willdiscuss two interpretations (one by Einstein and Infeld, 1938 and theother by Zahar, 1989), which we will callontologicalinterpretations, that attempt to answer question (3) aboveaffirmatively. However, we begin this section by addressing what hasformerly been a fairly common misconception concerning mass-energyequivalence.

2.1 Misconceptions about \(E_o = mc^2\)

Although it is far less common today, one still sometimes hears ofEinstein’s equation entailing that matter can be converted intoenergy. Strictly speaking, this constitutes an elementary categorymistake. In relativistic physics, as in classical physics, mass andenergy are both regarded asproperties of physical systems orproperties of the constituents of physical systems. If one wishes totalk about the physicalstuff that is the bearer of suchproperties, then one typically talks about either “matter”or “fields.” The distinction between “matter”and “fields” in modern physics is itself rather subtle inno small part because of the equivalence of mass and energy.Philosophically, to think of fields as stuff is also controversial.

Nevertheless, we can assert that whatever sense of“conversion” seems compelling between mass and energy, itwill have to be a “conversion” betweenmass andenergy, and not betweenmatter and energy. Finally, ourobservation obtains even in so-called “annihilation”reactions where the entire mass of the incoming particles seems to“disappear” (see, for example, Baierlein (2007, p. 323)). Of course, the older terminology of “matter” and“anti-matter” in the description of annihilation reactionsdoes not really help our philosophical understanding of mass-energyequivalence and is perhaps partly to blame for some of themisconceptions surrounding \(E_o = mc^2\).

2.2 Same-property interpretations of \(E_o = mc^2\)

The first interpretation we will consider asserts that mass and energyare the same property of physical systems. Consequently,there is no sense in which one of the properties is ever physicallyconverted into the other.

Philosophers such as Torretti (1996) and physicists such as Eddington(1929) have adopted the same-property interpretation. For example,Eddington states that “it seems very probable that mass andenergy are two ways of measuring what is essentially the same thing,in the same sense that the parallax and distance of a star are twoways of expressing the same property of location” (1929, p.146). According to Eddington, the distinction between mass and energyis artificial. We treat mass and energy as different properties ofphysical systems because we routinely measure them using differentunits. However, one can measure mass and energy using the same unitsby choosing units in which \(c = 1\), i.e., units in which distancesare measured in units of time (e.g., light-years). Once we do this,Eddington claims, the distinction between mass and energydisappears.

Like Eddington, Torretti points out that mass and energyseemto be different properties because they are measured in differentunits. Speaking against Bunge’s (1967) view that their numericalequivalence doesnot entail that mass and energy “arethe same thing,” Torretti explains:

If a kitchen refrigerator can extract mass from a given jug of waterand transfer it by heat radiation or convection to the kitchen wallbehind it, a trenchant metaphysical distinction between the mass andthe energy of matter does seem far fetched (1996, p. 307,fn. 13).

For Torretti, the very existence of physical processes in which theemission of energy by an object is correlated with the decrease in theobject’s mass in accordance with Einstein’s equationspeaks strongly against the view that mass and energy are somehowdistinct properties of physical systems. Torretti continues:

Of course, if lengths and times are measured with different, unrelatedunits, the ‘mass’… differs conceptually from the‘energy.’ But this difference can be understood as aconsquence of the convenient but deceitful act of the mind by which weabstract time and space from nature (1996, p. 307, fn. 13).

Thus, this footnote in his masterlyRelativity and Geometrysuggests that, for Torretti, we are misled into using different unitsfor mass and energy merely because of how we perceive space and time.As we have seen, one can use the same units for mass and energy byadopting the convention Torretti himself uses of selecting units inwhich \(c = 1\) (pp. 88–89). However, it may be useful toremember that merely using the same units for spatial and temporalintervals does not entail that space and time are treated “on apar” in special relativity; they are not, as is evident from thesignature of the Minkwoski metric.

The main merit of Torretti’s view is that it takes veryseriously the unification of space and time effected by specialrelativity and so famously announced in the opening lines of Minkowski(1908). It is also consistent with how mass and energy are treated ingeneral relativity.

Interpretations such as Torretti’s and Eddington’s draw nofurther ontological conclusions from mass-energy equivalence. Forexample, neither Eddington nor Torretti make any explicit claimconcerning whether properties are best understood as universals, orwhether one ought to be a realist about such properties. Finally, bysaying that mass and energy are the same, these thinkers aresuggesting that the denotation of the terms “mass” and“energy” is the same, though they recognize that theconnotation of these terms is clearly different.

2.3 Different-properties interpretations of \(E_o = mc^2\)

As we have displayed inTable 2, interpretations of mass-energy equivalence that hold that mass andenergy are different properties disagree concerning whether there issome physical process by which mass is converted into energy (or viceversa). Although superficially Lange’s (2001, 2002)interpretation seems to fall in this category, as he certainly treatsmass and energy as different properties, he differs from others inthis category because Lange explicitly argues that only mass is a realproperty of physical systems. Consequently, we will discussLange’s interpretation separately below (inSection 2.3.3).

We will begin with a discussion of Bondi and Spurgin’sinterpretation (inSection 2.3.1). They hold that mass and energy are distinct properties and that thereis no such thing as the conversion of mass and energy. We will thendiscuss Rindler’s interpretation (inSection 2.3.2). He maintains that mass and energy are different properties but thatgenuine conversions of mass and energy are at least permitted bymass-energy equivalence.

2.3.1 Bondi and Spurgin’s Different-Properties, No-Conversion Interpretation

Bondi and Spurgin’s (1987) interpretation of mass-energyequivalence has been influential especially among physicists concernedwith physics education. In an article where they complained about howstudents often misunderstand Einstein’s famous equation, Bondiand Spurgin argued that Einstein’s equation does not entail thatmass and energy are the same property any more than the equation \(m =\varrho V\) (where \(m\) is mass, \(V\) is volume, and \(\varrho\) isdensity) entails that mass and volume are the same. Just as in thecase of mass and volume, Bondi and Spurgin argue, mass and energy havedifferentdimensions. Ultimately, this reduces to adisagreement with philosophers such as Torretti who would argue thattime, as a dimension, is no different than any one of the spatialdimensions. Note well that this is not an issue about theunits we use for measuring mass (or energy).

Everyone agrees that according to special relativity one can measurespatial intervals in units of time. We can do this because of thepostulate of special relativity that states that the speed of lighthas the same value in all inertial frames. If we perform what amountsto a substitution of variables and take our spatial dimensions to be\(x_n^* = x_n /c\), where \(c\) is the speed of light and \(n = 1, 2,3\), then we select units in which \(c = 1\).

However, one can consistently use units in which \(c = 1\) and holdthat there is nevertheless a fundamental distinction between space andtime as dimensions. On such a view, which is the view that Bondi andSpurgin seem implicitly to be defending, while time is distinct fromany given spatial dimension, the contingent fact that \(c\) has thesame value in all inertial frames allows us to perform the relevantsubstitution of variables. However, it does not follow from this thatwe ought to treat time on a par with any spatial dimension, or that weought to treat the saptio-temporal interval as more fundamental (inthe way Torretti does).

In their influential article, Bondi and Spurgin then examine a varietyof cases of purported conversions of mass and energy. In each case,they show that the purported conversion of mass and energy is bestunderstood merely as a transformation of energy. In general, Bondi andSpurgin argue, whenever we encounter a purported conversion of massand energy, we can always explain what is taking place by looking atthe constituents of the physical system in the reaction and examininghow energy is proportioned among the constituents before and after thereaction takes place.

Explanations of purported “conversions” along the linessuggested by Bondi and Spurgin are now commonplace in the physicsliterature. These explanations have the merit of emphasizing that inmany cases the mysteries of mass-energy equivalence donotconcern one physical property magically being transfigured intoanother. However, the Bondi-Spurgin interpretation of mass-energyequivalence has the demerit that it fails to address reactions such asthe electron-positron annihilation reaction. In such reactions, notonly is the number of particles not conserved, butall of theparticles involved are, by hypothesis, indivisible wholes. Thus, theenergy liberated in such reactions cannot be explained as resultingfrom a transformation of the energy that was originally possessed bythe constituents of the reacting particles. Of course, Bondi andSpurgin may simply be hoping that physics will reveal that particlessuch as electrons and positrons are not indivisible wholes after all.Indeed, they may even use annihilation reactions combined with theirinterpretation of mass-energy equivalence to argue that it cannot bethe case that such particles are indivisible. Thus, we witness hereexplicitly just how closely related interpretations concerningmass-energy equivalence can be to views concerning the nature ofmatter.

The second demerit of the Bondi-Spurgin interpretation, which itshares with all other interpretations of mass-energy equivalence thathold that mass and energy are different properties, is that it remainssilent about a central feature of physical systems it uses inexplaining apparent conversions of mass and energy. In order toexplain purported conversions along the lines suggested byBondi-Spurgin, one must make the familiar assumption that the energyof the constituents of a system, be it potential energy or kineticenergy, “contributes” to the rest-mass of the system.Thus, for example, in the bombardment of \({^7}\Li\) reaction Bondi andSpurgin must explain the rest-mass of the \({^7}\Li\) in the familiarway, as arising fromboth the sum of the rest-masses of thenucleons, and the mass-equivalents of their energies. However, theBondi-Spurgin interpretation offers no explanation concerning why theenergy of the constituents of a physical system, be it potentialenergy or kinetic energy, manifest itself as part of theinertial mass of the system as a whole. Of course, one canalways reply that even to ask for this type of explanation is torefuse to accept relativistic thinking fully: the potential andkinetic energies of the constituents contributes to the rest energy ofthe whole, and because of Einstein’s equation, contributes tothe rest-mass of the whole.

As we shall see, Rindler’s interpretation of mass-energyequivalence attempts to address the first demerit of the Bondi-Spurgininterpretation, while Lange’s interpretation brings to theforeground that the energy of the constituents of a physical system“contributes” to that system’s inertial mass.

2.3.2 Rindler’s Different-Properties, Conversion Interpretation

Rindler’s interpretation of mass-energy equivalence is aslightly, though importantly, modified version of the Bondi-Spurgininterpretation. Rindler (for example, in 1977), agrees that there aremany purported conversions that are best understood as meretransformations of one kind of energy into a different kind of energy.

However, for Rindler, there is nothing within special relativityitself that rules out the possibility that there exists fundamental,structureless particles (i.e., particles that are “atomic”in thephilosophical sense of the term). If such particlesexist, it is possible according to Einstein’s equation that someor all of the mass of such particles “disappears” and anequivalent amount of energy “appears” within the relevantphysical system. Thus, Rindler seems to be suggesting that we shouldconfine our interpretation of mass-energy equivalence to what we candeduce from special relativity. Thus, we should hold thatEinstein’s equation at least allows for genuine conversions ofmass into energy, in the sense that theremay be cases wherea certain amount of inertial mass “disappears” fromwithin a physical system and a corresponding amount of energy“appears.” Furthermore, in such cases we cannot explainthe reaction as merely involving a transformation of one kind ofenergy into another.

The merit of Rindler’s interpretation is that it confines theinterpretation of Einstein’s equation to what we can validlyinfer from the postulates of special relativity. Unlike theinterpretation proposed by Bondi and Spurgin, Rindler’sinterpretation makes no assumptions about the constitution of matterbut leaves that for future physics to determine.

2.3.3 Lange’s One-Property, No-Conversion Interpretation

Lange (2001, 2002) has suggested a rather unique interpretation ofmass-energy equivalence. Lange begins his interpretation by arguingthat rest-mass is the onlyreal property of physical systems.This claim by itself suggests that there can be no such thing as aphysical process by which mass is converted into energy, for as Langeasks “in what sense can mass beconverted into energywhen mass and energy are not on a par in terms of theirreality?” (2002, p. 227, emphasis in original). Lange thengoes on to argue that a careful analysis of purported conversions ofmass-energy equivalence reveals that there is no physical process bywhich mass is ever converted into energy. Instead, Lange argues, theapparent conversion of mass into energy (or vice versa) is an illusionthat arises when we shift our level of analysis in examining aphysical system.

Lange seems to use a familiar argument from the Lorentz invariance ofcertain physical quantities to their “reality.” For Lange,if a physical quantity is not Lorentz invariant, then it is not realin the sense that it does not represent “the objective facts, onwhich all inertial frames agree” (2002, p. 209). Thus Langeuses Lorentz invariance as anecessary condition for thereality of a physical quantity. However, in several other places, forexample when Lange argues for the reality of the Minkowski interval(2002, p. 219) or when he argues for the reality of rest-mass(2002, p. 223), Lange implicitly uses Lorentz invariance as asufficient condition for the reality of a physical quantity.However, if Lange adopts Lorentz-invariance as both a necessary andsufficient condition for the reality of a physical quantity, then heis committed to the view that rest energy is real for the very samereasons he is committed to the view that rest-mass is real. Thus,Lange’s original suggestion that there can be no physicalprocess of conversion between mass and energy because they havedifferent ontological status seems challenged.

As it happens, Lange’s overall position is not seriouslychallenged by the ontological status of rest energy. Lange couldeasily grant that rest energy is a real property of physical systemsand still argue (i) that there is no such thing as a physical processof conversion between mass and energy and (ii) that purportedconversions result from shifting levels of analysis when we examine aphysical system. It is his observations concerning (ii) that force usto face again the question of why the energy of the constituents of aphysical system manifests itself as the mass of the system, thoughadmittedly the question itself may simply reveal a failure fully toappreciate a relativistic description of composite systems.

One of the main examples that Lange uses to present his interpretationof mass-energy equivalence is the heating of an ideal gas, which wehave already considered above (seeSection 1.3). He also considers examples involving reactions among sub-atomicparticles that, for our purposes, are very similar in the relevantrespects to the example we have discussed concerning the bombardmentand subsequent decomposition of a \({^7}\Li\) nucleus. In both cases,Lange essentially adopts the minimal interpretation we discussed inSection 1.3. In the case of the ideal gas, as we have seen, when the gas sample isheated and its inertial mass concurrently increases, this increase inrest-mass is not a result of the gas somehow being suddenly (orgradually) composed of molecules that are themselves more massive. Therest-mass of any individual molecule does not change. It is also not aresult of the gas suddenly (or gradually) containing more molecules.Instead, the increased kinetic energy of the molecules of the gasconstitutes an increase to the rest energy of the gas sample which,through Einstein’s equation, manifests as an increase in the gassample’s inertial mass. Lange summarizes this feature of theincrease in the gas sample’s inertial mass by saying:

… we have just seen that this “conversion” of energyinto mass is not a real physical process at all.We“converted” energy into mass simply bychanging ourperspective on the gas: shifting from initially treating it asmany bodies to treating it as a single body [emphasis in original](p. 236, 2002).

Unfortunately, Lange’s characterization threatens to leavereaders with the impression that if “we” had not shiftedour perspective in the analysis of the gas, no change to the inertialmass of the gas sample would have ensued. Of course, it is unlikelythat Lange means this. Lange would likely agree that even if no humanbeings are around to analyze a gas sample, the gas sample will respondin any physical interaction differently as a wholeafter ithas absorbed some energy precisely because its inertial mass will haveincreased.

The merits of Lange’s view concerning the“conversion” of mass-energy equivalence are essentiallythe same as the merits of both the Bondi-Spurgin interpretation andRindler’s interpretation. All these interpretations agree thatthere are important cases where we have now learned enough to assertconfidently that purported “conversions” of mass andenergy are merely cases where energy of one kind is transformed intoenergy of another kind. Aside from the comparatively minor issueconcerning the “reality” of rest energy, the main demeritof Lange’s view is that it might potentially misleadunsuspecting readers.

2.4 Interpretations of \(E_o = mc^2\) and hypotheses concerning the nature of matter

The relationship between mass-energy equivalence and hypothesesconcerning the nature of matter is twofold. First, as we havesuggested, some of the interpretations of mass-energy equivalence seemto assume implicitly certain features of matter. Second, somephilosophers and physicists, notably Einstein and Infeld (1938) andZahar (1989), have argued that mass-energy equivalence hasconsequences concerning the nature of matter. In thissection, we will discuss the first of these two relationships between\(E_o = mc^2\) and hypotheses concerning the nature of matter. Wediscuss the second relationship in the next section (Section 2.5).

To explain how some interpretations of mass-energy equivalence rest onassumptions concerning the nature of matter, we need first torecognize, as several authors have pointed out, e.g., Rindler (1977),Stachel and Torretti (1982), and Mermin and Feigenbaum (1990), thatthe relation one actually derives from the special relativity is:

where \(K\) is merely an additive factor that fixes the zero-point ofenergy and is conventionally set to zero and \(q\) is also routinelyset to zero. However, unlike the convention to set \(K\) to zero,setting \(q = 0\) involves a hypothesis concerning the nature ofmatter, because it rules out the possibility that there exists matterthat has mass but which is such that some of its mass can never be“converted” into energy.

The same-property interpretation of mass-energy equivalence restssquarely on the assumption that \(q = 0\). Mass and energy cannot bethe same property if there exists matter that has mass some of whichcannot ever, under any conditions, be “converted” intoenergy. However, one could argue that although the same-propertyinterpretation makes this assumption, it is not anunjustified assumption. Currently, physicists do not have anyevidence that there exists matter for which \(q\) is not equal tozero. Nevertheless, it seems important, from a philosophical point ofview, to recognize that the same-property interpretation depends notonly on what one can derive from the postulates of special relativity,but also on evidence from “outside” this theory.

Interpretations of \(E_o = mc^2\) that hold that mass and energy aredistinct properties of physical systems need not, of course, assumethat \(q\) is different from zero. Such interpretations can simplyleave the value of \(q\) to be determined empirically, for as we haveseen such interpretations argue for treating mass and energy asdistinct properties on different grounds. Nevertheless, theBondi-Spurgin interpretation does seem to adopt implicitly ahypothesis concerning the nature of matter.

According to Bondi and Spurgin, all purported conversions of mass andenergy are cases where one type of energy is transformed into anotherkind of energy. This in turn assumes that we can, in all cases,understand a reaction by examining the constituents of physicalsystems. If we focus on reactions involving sub-atomic particles, forexample, Bondi and Spurgin seem to assume that we can always explainsuch reactions by examining the internal structure of sub-atomicparticles. However, if we ever find good evidence to support the viewthat some particles haveno internal structure, as it nowseems to be the case with electrons for example, then we either haveto give up the Bondi-Spurgin interpretation or use the interpretationitself to argue that such seemingly structureless particles actuallydo contain an internal structure. Thus, it seems that theBondi-Spurgin interpretation assumes something like the infinitedivisibility of matter, which is clearly a hypothesis that lies“outside” special relativity.

2.5 Ontological interpretations of \(E_o = mc^2\)

Einstein and Infeld (1938) and Zahar (1989) have both argued that\(E_o = mc^2\) has ontological consequences. Both of theEinstein-Infeld and Zahar interpretations begin by adopting thesame-property interpretation of \(E_o = mc^2\). Thus,according to both interpretations, mass and energy are the sameproperties of physical systems. Furthermore, both the Einstein-Infeldand Zahar interpretations use a rudimentary distinction between“matter” and “fields.” According to thissomewhat dated distinction, classical physics includes two fundamentalsubstances: matter, by which one means ponderable material stuff, andfields, by which one means physical fields such as the electromagneticfield. For both Einstein and Infeld and Zahar, matter and fields inclassical physics are distinguished by the properties they bear.Matter has both mass and energy, whereas fields only have energy.However, since the equivalence of mass and energy entails that massand energy are really the same physical property after all, sayEinstein and Infeld and Zahar, one can no longer distinguish betweenmatter and fields, as both now have both massand energy.

Although both Einstein and Infeld and Zahar use the same basicargument, they reach slightly different conclusions. Zahar argues thatmass-energy equivalence entails that the fundamental stuff of physicsis a sort of “I-know-not-what” that can manifest itself aseither matter or field. Einstein and Infeld, on the other hand, inplaces seem to argue that we can infer that the fundamental stuff ofphysics is fields. In other places, however, Einstein and Infeld seema bit more cautious and suggest only that one can construct a physicswith only fields in its ontology.

The demerits of either ontological interpretation of mass-energyequivalence are that it rests upon thesame-propertyinterpretation of \(E_o = mc^2\). As we have discussed above (seeSection 2.4), while onecan adopt thesame-propertyinterpretation, to do so one must make additional assumptionsconcerning the nature of matter. Furthermore, the ontologicalinterpretation rests on what nowadays seems like a rather crudedistinction between “matter” and “fields.” Tobe sure, mass-energy equivalence has figured prominently inphysicists’ conception of matter in no small part because itdoes open up the door to a description of what we ordinarily regard asponderable matter in terms of fields, since the energy of the field atone level can manifest itself as mass one level up. However, theinference from mass-energy equivalence to the fundamental ontology ofmodern physics seems far more subtle than either Enstein and Infeld orZahar suggest.

3. History of Derivations of Mass-Energy Equivalence

Einstein first derived mass-energy equivalence from the principles ofspecial relativity in a small article titled “Does the Inertiaof a Body Depend Upon Its Energy Content?” (1905b). Thisderivation, along with others that followed soon after (e.g., Planck(1906), Von Laue (1911)), uses Maxwell’s theory ofelectromagnetism. (SeeSection 3.1.) However, as Einstein later observed (1935), mass-energy equivalenceis a result that should be independent of any theory that describes aspecific physical interaction. This is the main reason that ledphysicists to search for “purely dynamical” derivations,i.e., derivations that invoke only mechanical concepts such as energyand momentum, and the conservation principles that govern them. (SeeSection 3.2)

3.1 Derivations of \(E_o = mc^2\) that Use Maxwell’s Theory

Einstein’s original derivation of mass-energy equivalence is thebest known in this group. Einstein begins with the followingthought-experiment: a body at rest (in some inertial frame) emits twopulses of light of equal energy in opposite directions. Einstein thenanalyzes this “act of emission” from another inertialframe, which is in a state of uniform motion relative to the first. Inthis analysis, Einstein uses Maxwell’s theory ofelectromagnetism to calculate the physical properties of the lightpulses (such as their intensity) in the second inertial frame. Bycomparing the two descriptions of the “act of emission”,Einstein arrives at his celebrated result: “the mass of a bodyis a measure of its energy-content; if the energy changes by \(L\),the mass changes in the same sense by \(L/9 \times 10^{20}\), theenergy being measured in ergs, and the mass in grammes” (1905b,p. 71). A similar derivation using the same thought experimentbut appealing to the Doppler effect was given by Langevin (1913) (seethe discussion of the inertia of energy in Fox (1965, p. 8)).

Some philosophers and historians of science claim thatEinstein’s first derivation is fallacious. For example, inThe Concept of Mass, Jammer says: “It is a curiousincident in the history of scientific thought that Einstein’sown derivation of the formula \(E = mc^2\), as published in hisarticle inAnnalen der Physik, was basically fallacious. . .  the result of apetitio principii, the conclusionbegging the question” (Jammer, 1961, p. 177). According toJammer, Einstein implicitly assumes what he is trying to prove, viz.,that if a body emits an amount of energy \(L\), its inertial mass willdecrease by an amount \(\Delta m = L/c^2\). Jammer also accusesEinstein of assuming the expression for the relativistic kineticenergy of a body. If Einstein made these assumptions, he would beguilty of begging the question. However, Stachel and Torretti (1982)have shown convincingly that Einstein’s (1905b) argument issound. They note that Einstein indeed derives the expression for thekinetic energy of an “electron” (i.e., a structurelessparticle with a net charge) in his earlier (1905a) paper. However,Einstein nowhere uses this expression in the (1905b) derivation ofmass-energy equivalence. Stachel and Torretti also show thatEinstein’s critics overlook two key moves that are sufficient tomake Einstein’s derivation sound, since one need not assume that\(\Delta m = L/c^2\).

Einstein’s further conclusion that “the mass of a body isa measure of its energy content” (1905b, p. 71) does not,strictly speaking, follow from his argument. As Torretti (1996) andother philosophers and physicists have observed, Einstein’s(1905b) argument allows for the possibility that once a body’senergy store has been entirely used up (and subtracted from the massusing the mass-energy equivalence relation) the remainder is not zero.In other words, it is only an hypothesis in Einstein’s (1905b)argument, and indeed in all derivations of \(E_o = mc^2\) in specialrelativity, that no “exotic matter” exists that isnot convertible into energy (see Ehlers, Rindler, Penrose,(1965) for a discussion of this point). However, particle-antiparticleanihilation experiments in atomic physics, which were first observeddecades after 1905, strongly support “Einstein’s dauntlessextrapolation” (Torretti, 1996, p. 112).

In general, derivations in this group use the same style of reasoning.One typically begins by considering an object that either absorbs oremits electromagnetic radiation (typically light) of total energy\(E_o\) in equal and opposite directions. Because light carries bothenergy and momentum, one then uses the conservation principles forthose quantities and the standard heuristic in relativity ofconsidering the same physical process from two different inertialframes that are in a state of relative motion to show that in orderfor the conservation principles to be satisfied, the mass (i.e.,rest-mass) of the emitting or absorbing object must increase ordecrease by an amount \(E_o /c^2\). For a more detailed description ofa simplified derivation in this group, seeSection 1.5

One of the few exceptions to this approach among derivations that useMaxwell’s theory is Einstein’s 1906 derivation (Einstein1906). In this derivation, Einstein considers a freely-floating box. Aburst of electromagnetic radiation of energy \(E_o\) is emitted insidethe box from one wall toward a parallel wall. Einstein shows that theprinciple of mechanics that says that the motion of the center of massof a body cannot change merely because of changes inside the bodywould be violated if one did not attribute an inertial mass \(E_o/c^2\) to the burst of electromagnetic radiation (see Taylor andWheeler 1992, p. 254 for a detailed discussion of this example).

3.2 Purely Dynamical Derivations of \(E_o = mc^2\)

Purely dynamical derivations of \(E_o = mc^2\) typically proceed byanalyzing an inelastic collision from the point of view of twoinertial frames in a state of relative motion (the centre-of-massframe, and an inertial frame moving with a relative velocity \(v)\).One of the first papers to appear following this approach isPerrin’s (1932). According to Rindler and Penrose (1965),Perrin’s derivation was based largely on Langevin’s“elegant” lectures, which were delivered at theCollège de France in Zurich around 1922. Einstein himself gavea purely dynamical derivation (Einstein, 1935), though he nowherementions either Langevin or Perrin. The most comprehensive derivationof this sort was given by Ehlers, Rindler and Penrose (1965). Morerecently, a purely dynamical version of Einstein’s original(1905b) thought experiment, where the particles that are emitted arenot photons, has been given by Mermin and Feigenbaum (1990) and Mermin(2005).

Derivations in this group are distinctive because they demonstratethat mass-energy equivalence is a consequence of the changes to thestructure of spacetime brought about by special relativity. Therelationship between mass and energy is independent of Maxwell’stheory or any other theory that describes a specific physicalinteraction. We can get a glimpse of this by noting that to derive\(E_o = mc^2\) by analyzing a collision, one must first define thefour-momentum \(\mathbf{p}\), the “space-part” of which isrelativistic momentum \(\mathbf{p}_{\rel}\), and relativistic kineticenergy \(T_{\rel}\), since one cannot use the old Newtonian notions ofmomentum and kinetic energy.

In Einstein’s own purely dynamical derivation (1935), more thanhalf of the paper is devoted to finding the mathematical expressionsthat define \(\mathbf{p}\) and \(T_{\rel}\). This much work is requiredto arrive at these expressions for two reasons. First, the changes tothe structure of spacetime must be incorporated into the definitionsof the relativistic quantities. Second, \(\mathbf{p}\) and \(T_{\rel}\)must be defined so that they reduce to their Newtonian counterparts inthe appropriate limit. This last requirement ensures, in effect, thatspecial relativity will inherit the empirical success of Newtonianphysics. Once the definitions of \(\mathbf{p}\) and \(T_{\rel}\) areobtained, one derives mass-energy equivalence in a straight-forwardway by analyzing a collision. (For a more detailed discussion ofEinstein’s (1935), see Fernflores, 2018.)

At a very general level, purely dynamical derivations ofEinstein’s equation and derivations that appeal toMaxwell’s theory really follow the same approach. In both stylesof derivations, although it may not seem like it at first glance, weare dealing with one of the most basic dynamical interactions: acollision. So, for example, we can construe the physical configurationof Einstein’s original 1905 derivation (Einstein 1905b) as acollision in which the total number of objects is not conserved. Thisis even easier to do if one adopts a “particle”description of light. In both the purely dynamical derivations and thederivations that appeal to an interaction with electromagneticradiation, one then examines the collision in question and shows thatin order for dynamical principles to be satisfied, the relationshipamong the masses and energies of the objects involved in thecollisions must satsify Einstein’s equation.

The main difference between the two approaches to derivingEinstein’s equation is that in derivations that consider acollision with light, one must use the dynamical properties of light,which are not themselves described by special relativity. For example,as we have seen, in Einstein’s 1946 derivation (seeSection 1.5), we must appeal to the expression for the momentum of a burst oflight.

4. Experimental Verification of Mass-Energy Equivalence

Cockcroft and Walton (1932) are routinely credited with the firstexperimental verification of mass-energy equivalence. Cockcroft andWalton examined a variety of reactions where different atomic nulceiare bombarded by protons. They focussed their attention primarily onthe bombardment of \({^7}\Li\) by protons (seeSection 1.4).

In their famous paper, Cockcroft and Walton noted that the sum of therest-masses of the proton and the Lithium nucleus (i.e., thereactants) was \(1.0072 + 7.0104 = 8.0176\) amu. However, the sum ofthe rest-masses of the two \(\alpha\)-particles (i.e., the products)was 8.0022 amu. Thus, it seems as if an amount of mass of 0.0154 amuhas “disappeared” from the reactants. Cockcroft and Waltonalso observed that the total energy (in the reference frame in whichthe \({^7}\Li\) nucleus is at rest) for the reactants was 125 KeV.However, the total kinetic energy of the \(\alpha\)-particles wasobserved to be 17.2 MeV. Thus, it seems as if an amount of energy ofapproximately 17 MeV has “appeared” in the reaction.

Implicitly referring to the equivalence of mass and energy, withoutexplicitly mentioning either the result or Einstein by name, Cockcroftand Walton then simply assert that a mass 0.0154 amu “isequivalent to an energy liberation of \((14.3 \pm 2.7) \times 10^6\)Volts” (p. 236). They then implicitly suggest that thisinferred value for the kinetic energy of the two resulting\(\alpha\)-particles is consistent with theobserved valuefor the kinetic energy of the \(\alpha\)-particles. Cockcroft andWalton conclude that “the observed energies of the\(\alpha\)-particles are consistent with our hypothesis”(pp. 236–237). The hypothesis they set out to test,however, is not mass-energy equivalence, but rather than when a\({^7}\Li\) nucleus is bombarded with protons, the result is two\(\alpha\)-particles.

Stuewer (1993) has suggested that Cockcroft and Waltonusemass-energy equivalence to confirm their hypothesis about what happenswhen \({^7}\Li\) is bombarded by protons. Hence, it does not seem weought to regard this experiment as a confirmation of \(E_o = mc^2\).However, if we take some of the other evidence that Cockcroft andWalton provide concerning the identification of the products in thebombardment reaction as sufficient to establish that the products areindeed \(\alpha\)-particles, then we can interpret this experiment asa confirmation of mass-energy equivalence, which is how thisexperiment is often reported in the physics literature.

More recently, Rainville et al. (2005) have published the resultsof what they call “A direct test of \(E = mc^2\).” Theirexperiments test mass-energy equivalence “directly” bycomparing the difference in the rest-masses in a neutron capturereaction with the energy of the emitted \(\gamma\)-rays. Specifically,Rainville et al. examine two reactions, one involving neutron captureby Sulphur \((\S)\), the other involving neutron capture by Silicon\((\Si)\):

In these reactions, when the nucleus of an atom (in this case either\({^{32}}\S\) or \({^{28}}\Si)\) captures the neutron, a new isotope iscreated in an excited state. In returning to its ground state, theisotope emits a \(\gamma\)-ray. According to Einstein’sequation, the difference in the rest-masses of the neutron plusnucleus, on the one hand, and the new isotope in its ground state onthe other hand, should be equal to the energy of the emitted photon.Thus, Rainville et al. test \(\Delta E = \Delta mc^2\) by making veryaccurate measurements of the rest-mass difference and the frequency,and hence energy, of the emitted photon. Rainville et al. report thattheir measurements show that Einstein’s equation obtains to anaccuracy of at least 0.00004%.

5. Conclusion

In this entry, we have presented the physics of mass-energyequivalence as widely understood by both physicists and philosophers.We have also canvassed a variety of philosophical interpretations ofmass-energy equivalence. Along the way, we have presented the meritsand demerits of each interpretation. We have also presented a briefhistory of derivations of mass-energy equivalence to emphasize thatthe equivalence of mass and energy is a direct result to changes tothe structure of spacetime imposed by special relativity. Finally, wehave briefly and rather selectively discussed the empiricalconfirmation of mass-energy equivalence.

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