## 1. The Physics of Mass-Energy Equivalence



A lot of philosophical attention has been paid to how the meaning of
statements concerning lengths and times changes in the context of
special 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, is
either elliptical at best or meaningless at worst in the context of
special relativity. However, comparatively little philosophical
attention has been paid to how special relativity brings about
corresponding changes to the third fundamental dimension in basic
physics: mass. Yet, assertions concerning mass, and other dynamical
concepts in mechanics into which it figures centrally, notably
momentum and kinetic energy, also have their meanings changed in
special relativity. 



To illustrate these changes, Section 1.1 reviews the concepts of mass,
momentum and kinetic energy in Newtonian physics. In Section 1.2, we
begin exploring the concepts of mass and energy in special relativity
and state the notation we use in all further discussions, a step we
must make explicit as discussions of the equivalence of mass and
energy have historically used different notations, which can lead to
conceptual confusion. In Section 1.3, we discuss the physical
significance of mass-energy equivalence as it applies to the analysis
of bodies (construed as indivisible wholes) and idealized versions of
composite systems, while in Section 1.4 we discuss mass and energy in
atomic physics. In Section 1.5, we present an elementary derivation of
Einstein's famous result with the hope that our readers will get
a sense of "why" mass and energy are equivalent according
to special relativity. Finally, in Section 1.6, we consider the
relationship between the equivalence of mass and energy codified in
the equation \(E\_o = mc^2\) and the nearly identical, though
conceptually 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 billiard
ball, has associated with it a positive, real number called its
*mass*, for which the symbol \(m\) is commonly used. When we
are focusing on describing mathematically the motion of this object
under the action of contact forces, such as when one tries to predict
where a billiard ball will go after a collision, the mass in question
is also called the *inertial mass*. In the special case in
which one considers how the billiard ball moves in response to
gravity, say as it falls toward the earth, the measure of how the
billiard ball "responds" to the gravitational field is
called the *gravitational mass* of the object. Using
experiments in which he filled wooden boxes with different materials
and suspended them from strings to construct pendulums, Newton
discovered that inertial mass and gravitational mass are directly
proportional. Physicists have since customarily treated inertial and
gravitational mass as numerically equal. 



The inertial mass in Newtonian physics (and even the gravitational
mass) is routinely interpreted as an intrinsic property of an object.
The inertial mass of an object is a measure of the body's
inertia, i.e., its tendency to resist changes to its state of motion
in response to the action of *any* kind of force. Since at
least the late nineteenth century and Mach's criticisms of
Newton's physics, physicists have deprecated thinking of mass as
the "quantity of matter." 



The physical intuition behind the Newtonian concept of inertial mass
is 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 final
velocity 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, the
inertial mass of an object, and even its gravitational mass, can only
change by either physically removing a part of the object or attaching
a part to the object to make a bigger whole. 



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



Newton called momentum the "quantity of motion," which is
an apt label, because, very roughly, it is a measure of the extent to
which an object is moving relative to an inertial reference frame. Our
colloquial use of the word "momentum" is related to this
intuition and to its formal definition as the product of the mass of
an object times its velocity relative to some inertial reference
frame. If object \(B\_2\) has twice the mass as \(B\_1\) but moves with
the same velocity relative to a reference frame, \(B\_2\) has twice the
momentum. 



Velocity, however, is a directed quantity, because it codifies not
just the speed \(v\) with which the object moves, but also its
direction. 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 the
object while allowing the context to indicate that this is elliptical
for "the magnitude of the velocity." Momentum is therefore
also a directed quantity, represented formally by a vector
\(\mathbf{p}\). To paraphrase Taylor and Wheeler (1992, p. 191), the
direction of momentum matters: A glancing blow is never as damaging as
one that is head on. By analogy with velocity, the magnitude of the
momentum is represented by the letter \(p\). 



Finally, in Newtonian mechanics, an object in motion relative to a
reference frame also has kinetic energy, or energy of motion. Kinetic
energy, like all forms of energy, can be transformed into another kind
of energy. So, for example, one could measure the kinetic energy of a
moving billiard ball by having it collide with a ball of soft putty or
a spring that brings the ball to rest and measuring the energy
absorbed by the putty or spring. There is no single standard symbol
for 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 to
an inertial frame. The faster an object moves relative to some
inertial frame, the more momentum and kinetic energy it has. However,
for an object that is not accelerating, i.e., an object that covers
equal spaces in equal times, one can always find an inertial reference
frame in which both the momentum and the kinetic energy of the object
is zero. 



Following Griffiths' approach, we might say that so far we have
merely defined some quantities (Griffiths 1999, p. 509 ff.). The
physics really lies in the three corresponding conservation principles
associated with these quantities: the principle of conservation of
mass, the principle of conservation of energy, the principle of
conservation of (linear) momentum. These principles contain the
physics, because each one states that a certain quantity, mass,
energy, or momentum, is conserved in all interactions. So, for
example, if we consider the collision of two billiard balls, one can
use the conservation of momentum to predict, given the initial motions
and masses of the billiard balls, how they will move after the
collision. 


### 1.2 Mass and Energy in Relativity: Preliminaries and Notation



As Hecht has emphasized, Einstein never wrote down his famous result
using the symbols "\(E\)" and "\(m\)" as they
appear in the famous equation attributed to him (Hecht 2012). Although
part of the reason is merely that Einstein used different letters for
energy, mass, and the speed of light in his early papers discussing
mass-energy equivalence (in 1905 and 1906), deeper reasons related to
how we should understand the result quickly emerged. 



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



\[
M = \mu + E\_o /c^2
\]


In a footnote, Einstein explains the convention, which he had already
adopted in an earlier paper (Einstein 1907a, p. 250), of using
"the subscript '\(o\)' to indicate that the quantity in
question refers to a reference system that is at rest relative to the
physical system considered" (Einstein 1907b, p. 286). 



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



\[\tag{Einstein's Equation}
E\_o = mc^2
\]


Our main task in the next section is to explain the physical
significance of the equation \(E\_o = mc^2\), which we will henceforth
call "Einstein's equation", and its relationship to
its iconic variant without the subscript "\(o\)" appearing
beside the letter "\(E\)". 



However, it is important before going much further to note that in
Einstein's equation \(E\_o = mc^2\), the symbol \(m\) is the mass
of an object as measured *in the inertial frame in which that
object is at rest*. Physicists also call this mass \(m\) the
"rest-mass" of the object. The rest-mass of an object is
numerically equal to its Newtonian inertial mass, though arguably the
symbol \(m\) (or the corresponding term "mass") has
different 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 many
contemporary physicists do not add a subscript "\(o\)" to
\(m\) when denoting rest-mass. For example, Taylor and Wheeler argue
that the phrase "rest-mass" engenders potential confusion
as it might lead the reader to ask: What happens to the rest-mass when
the object is moving? Answer: nothing (Taylor and Wheeler 1992, p.
251). The rest-mass of an object is an invariant quantity in special
relativity; it has the same value for all inertial observers. Taylor
and Wheeler quip: "In reality mass is mass is mass"
(Taylor and Wheeler 1992, p. 251). From their perspective, which is
quite standard now in physics textbooks and articles, there is no need
to prefix the term "mass" with "rest," because
there is no other kind of mass worth speaking about in special
relativity (see
 Section 1.6).
 



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





|  |  |
| --- | --- |
| *Symbol* | *Meaning* |
| \(E\) | The total energy of a physical system, unless otherwise
indicated  |
| \(E\_o\) | The rest energy of a physical system or an amount of energy
as measured in the rest frame of an object (typically energy that is
emitted or absorbed by an object) |
| \(m\) | The mass (i.e., rest-mass) of a physical system  |


Table 1. Notation used in this entry




We will first focus on the mechanics of idealized point particles in
special relativity. Regardless of the macroscopic objects they
represent, we shall treat such particles, and by extension the
corresponding physical objects, as un-analyzable wholes. We will then
separately discuss "composite systems" or systems composed
of such particles. Physicists use composite systems to approximate
physical objects when they are interested in examining the
"inner workings" of such objects, such as when they treat
a 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 of
Einstein's equation \(E\_o = mc^2\) by considering a very simple
physical system. Imagine, as Geroch suggests, a brick being heated or
a battery being charged (Geroch 2005, p. 198). Suppose we consider
these objects in the inertial reference frame in which they are at
rest. As the brick is heated, or the battery charged, it absorbs an
amount of energy \(E\_o\) as measured in its rest frame.
Einstein's equation tells us that the mass of the brick, or
battery, after it absorbs an amount of energy \(E\_o\) is increased
exactly by an amount \(E\_o /c^2\). The value of the mass of the hot
brick or charged battery is greater than it was before it absorbed
energy. So, for example, it takes just *a little bit more*
force to move the charged battery than it did to move the uncharged
battery. How much is "*a little bit more*"? 



Geroch has a clever way to answer this question. Suppose we use the
amount of energy \(E\_o\), as measured in the rest-frame of the
battery, to accelerate the battery (instead of charging it) so that it
eventually reaches a final speed of 670 mph. Since 670 mph seems like
something moving pretty fast, at least by the non-relativistic
standards of human travel, one might think that one is using quite a
bit of energy to accelerate the battery to that speed. Now suppose
that instead of using the energy \(E\_o\) to accelerate the battery, we
use that same amount of energy \(E\_o\) to charge the battery. The
increase in the mass of the charged battery is \(E\_o /c^2\), but
because the speed of light is such a large number, approximately 670
*million* mph, "the mass of the battery would be
increased 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. As
the battery loses energy, say by powering a device, or the brick emits
thermal energy as it cools, its mass decreases. Consequently, imagine
a 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 amount
of energy \(E\_o\) and \(B\_2\) fully absorbs that same amount of
energy, the mass of \(B\_1\) decreases by \(E\_o /c^2\) while the mass
of \(B\_2\) increases by the same amount. This physical change in the
masses of \(B\_1\) and \(B\_2\) is a novel prediction in special
relativity. For in Newtonian physics, there is no relationship at all
between the inertial mass of a body and the amount of energy it
radiates or absorbs. This is why Einstein was led to conclude that
"If the theory agrees with the facts, then radiation transmits
inertia between emitting and absorbing bodies" (Einstein 1905b).
If special relativity is supported by empirical evidence, the inertial
mass of an object can change, not because we have chopped off a piece
of the object or attached more stuff to it, but merely because the
object has radiated or absorbed energy. To physicists and philosophers
trained exclusively in the Newtonian tradition, this result would have
seemed 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 special
relativity between a change in the rest energy of a body \(E\_o\) and a
change in its mass \(m\). However, Einstein also emphasized two subtly
different "readings" of his equation. First, as early as
1906, Einstein argued that when one considers physical systems in
which there are electromagnetic processes, such as a
"complex" of light being emitted from the inside wall of a
freely-floating box and absorbed by the opposite wall, one could avoid
a fundamental conflict with the laws of mechanics "if one
ascribes 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, for
a detailed discussion of this example). This insight, sometimes
expressed with talk about the "inertia of energy," was an
important step in the development of general relativity, because that
theory uses the principle of equivalence, which states very roughly
that inertial mass and gravitational mass are directly proportional,
as a foundational principle. 



At a time when physicists were moving toward regarding energy-carrying
fields, such as the electromagnetic field, as entities in their own
right, combining the principle of equivalence with the "inertia
of energy" led Einstein to the insight that fields themselves
could gravitate. So, for example, Einstein claims in 1907 to show that
"radiation enclosed in a cavity possesses not only inertia but
also weight" (Einstein 1907b, p. 288). A more contemporary and
fairly common way of making the same point is to say that a mirror box
with perfectly reflecting walls filled with light (conceived here for
convenience as an energy-carrying disturbance in the electromagnetic
field) is attracted by gravity by an amount that is greater than the
rest mass \(M\) of the box itself. Specifically, if the energy of the
light in the box is \(E\_L\), gravity acts on the box not as if the box
has a mass \(M\) but as if the box has a mass \(M + E\_L /c^2\).
Although the difference is numerically tiny, strictly speaking a
balance with an empty mirror box on one side and an identical mirror
box 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 his
review article from 1907, he says, "with respect to inertia, a
mass \(\mu\) is equivalent to an energy content of magnitude \(\mu
c^2\)" (Einstein 1907b, p. 287). Einstein uses the phrase
"energy content" (translated from the original German
"Energieinhalt") to convey a notion that is new in special
relativity: A physical object contains within it energy that, at least
in principle, could be transformed into other forms of energy such as
kinetic energy (as we now know only too well). Because this rest
energy is contained within the boundaries of a body when we treat the
object 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 to
ask: Where does all that "internal energy" reside (Rindler
1991, p. 75)? Rindler's answer to this question assumes that we
wish 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 the
> molecules constituting the particle, and can be given up as heat; a
> part resides in the intermolecular and interatomic cohesion forces,
> and some of that can be given up in chemical explosions; another part
> may reside in excited atoms and escape in the form of radiation; much
> more resides in nuclear bonds and can also sometimes be set free, as
> in the atomic bomb. But by far the largest part of the energy (about
> 99 per cent) resides simply in the mass of the ultimate particles and
> cannot be further explained. Nevertheless, it too can be liberated
> under suitable conditions, e.g., when matter and antimatter annihilate
> each other (Rindler 1991, p.75).
> 



However, when developing a theoretical description of a macroscopic
object, it is seldom practical to construct that description by
treating subatomic quantum objects as the fundamental constituents, as
Rindler suggests. 



Instead, when one analyzes a physical object theoretically by looking
at its component parts, one has to decide the level of granularity at
which one wishes to analyze the object. For example, if the object we
are analyzing is a chunk of iron, and we are studying magnetism, it
may be sufficient to analyze the iron by treating its magnetic domains
as fundamental constituents. However, a magnetic domain is a chunk of
iron that contains many, many atoms. For other purposes, we may elect
to analyze the sample of iron all the way down to the atomic level. Of
course, the atoms themselves have parts. So, for yet different
purposes, we may elect to analyze the sample of iron at the sub-atomic
level of quantum objects. 



As Rindler's description of where the rest energy resides makes
clear, at each level of analysis, there are two main contributors to
the rest energy of the "macroscopic" sample we are
analyzing: (1) the energy equivalent of the sum of the rest-masses of
the constituent elements considered as fundamental and (2) the sum of
the energy "stored" or "carried" by the
constituent elements. To perform this kind of analysis for a concrete
object is a rather subtle affair. So physicists and philosophers
writing about mass-energy equivalence tend to focus on the highly
idealized notion of an ideal gas. 



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



When viewed from a Newtonian perspective, and assuming that the vessel
containing the gas is itself massless, the mass of the vessel of gas
is simply equal to the sum of the masses of the molecules. From a
relativistic point of view, this last assertion states
*incorrectly* that the rest-mass of the vessel of gas is equal
to the sum of the rest-masses of the molecules. Yet, from a
relativistic point of view, the rest-mass of the vessel of gas is
equal to the sum of the rest-masses of the molecules *plus* the
kinetic energy of the molecules divided by \(c^2\). Since, according
to the Kinetic Theory of Gases, the temperature of the gas is
proportional to the average kinetic energy of its molecules, if the
gas temperature increases or decreases, the rest-mass of the vessel of
gas increases or decreases accordingly by a tiny amount. 



Assuming the principle of equivalence, which entails that we can
measure inertial mass by measuring gravitational mass with a balance,
we can illustrate the difference between the Newtonian and
relativistic understanding of the ideal gas as follows. Imagine that
we have two otherwise identical massless vessels filled with exactly
the same amount and type of gas. In one vessel, the gas is at a
temperature very near absolute zero, so its molecules have very little
kinetic energy. In the other vessel, the gas is at a temperature of
500deg C. Place these two vessels of gas on the ends of a balance.
According to Newtonian physics, the balance will be level, because
both gas samples have exactly the same mass. According to relativity,
the balance will not be level and will be tipped on the side of the
hot gas, because the high kinetic energy of the molecules contributes
to the rest energy of the gas, which contributes, through
Einstein'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's
equation concern collisions among sub-atomic objects. For our
purposes, it is safe to treat atomic and sub-atomic objects as
particles involved in collisions where the total number of particles
may or may not be conserved.



The bombardment of a Lithium nucleus by protons is a historically
significant and useful example for discussing mass-energy equivalence
in collisions where the number of particles is conserved. Cockcroft
and 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:



 
\[
p + {^7}\Li \rightarrow \alpha + \alpha
\]


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



In the bombardment of Lithium reaction above, the sum of the
rest-masses of the reactants (the proton and the \({^7}\Li\) nucleus) is
*greater than* the sum of the rest-masses of the products (the
two \(\alpha\)-particles). However, the total kinetic energy of the
reactants is *less than* the total kinetic energy of the
products. Cockcroft and Walton's experiment is routinely
interpreted as demonstrating that the difference in the rest-masses of
the products and reactants (times \(c^2)\) is equal to the difference
in the kinetic energies of the products and reactants (but see
 Section 4
 for further discussion of this experiment as a confirmation of
mass-energy equivalence).



Descriptions of collisions among sub-atomic particles such as the
bombardment of Lithium make it seem as though one must admit that mass
is *converted* into energy. However, influenced perhaps by the
widely-known discussion of mass-energy equivalence by Bondi and
Spurgin (1987) (see
 Section 2.3.1),
 physicists now explain such reactions not as cases of mass being
converted into energy, but merely as cases where energy has changed
forms. Typically, in these types of reactions, the potential energy
that "contributes" to the rest-mass of one (or possibly)
more of the reactants is *transformed* in a non-controversial
way to the kinetic energy of the products. As Baierlein (2007, p. 322)
explains, in the case of the bombardment of \({^7}\Li\) with protons and
its subsequent decomposition into two \(\alpha\)-particles, the
apparently "excess" kinetic energy of the
\(\alpha\)-particles did not simply "appear" out of
nowhere. Instead, that energy was there all along as the potential and
kinetic energy of the nucleons. In other words, one can explain the
change in mass and energy in the bombardment reaction by saying (i)
that the potential and kinetic energies of the nucleons that make up
the \({^7}\Li\) nucleus contribute to its rest-mass and (ii) that the
vast amount of energy of the \(\alpha\)-particles was not
"created" in the reaction, or "converted" from
mass, but was simply transformed from the various forms of energy the
nucleons possess.



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



 
\[
e^- + e^+ \rightarrow \gamma + \gamma
\]


According to the currently accepted Standard Model of particle
physics, electrons and photons are both "fundamental
particles," by which physicists mean that such particles have no
structure, i.e., such particles are not composed of other, smaller
particles. Furthermore, the photons that are the products in the
annihilation reaction have zero rest-mass. Thus, the rest-masses of
the incoming electron and positron seems to "disappear"
and an equivalent amount of energy "appears" as the energy
of the outgoing photons. Of course, Einstein's famous equation
makes all of the correct predictions concerning the relevant masses
and energies involved in this reaction. So, for example, the total
energy of the two photons is equal to the sum of the kinetic energies
of the electron and positron *plus* the sum of the rest-masses
of the electron and positron multiplied by \(c^2\).



Finally, although mass and energy seem to "disappear" and
"appear" respectively when we focus on the individual
constituents of the physical system containing the incoming
electron-positron pair *and* the outgoing photons, the mass and
energy *of the entire system* remains the same throughout the
interaction. Before the collision, the rest-mass of the system is
simply the sum of the rest-masses of the electron and positron plus
the mass-equivalent of the total kinetic energy of the particles.
Consequently, the entire system (if we draw the boundary of the system
around the reactants and products--which is, of course, a spatial
and temporal boundary), has a non-zero rest-mass prior to the
collision. However, after the collision, the system, which now
consists of two photons moving in non-parallel directions, also has a
non-zero rest-mass (for a detailed discussion concerning the rest-mass
of systems of photons, see Taylor and Wheeler, 1992, p. 232).



### 1.5 Why *Does* \(E\_o\) Equal \(mc^2\)?



A common way in which the question "Why *does* \(E\_o\)
equal \(mc^2\)?" is interpreted by philosophers and physicists
is that it is a request for a derivation that shows that given certain
physical principles, such as the principle of relativity and the
principle of conservation of energy, Einstein's equation is a
logical consequence of those assumptions. We will discuss briefly the
history of derivation of Einstein's equation in
 Section 3.
 



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



As we will see, if Baierlein's assessment of his own derivation
is correct, the price to pay for this particular simplification is
that the derivation ceases to be relativistic, in the sense that none
of the core principles at the heart of special relativity seem to be
required to derive Einstein's equation. Nevertheless, it may
help readers get a "feel" for why \(E\_o = mc^2\) and it
does familiarize those not already conversant in the methods of
relativistic physics with a style of reasoning that is common in that
field. Those interested in a presentation of Einstein's own 1946
derivation, which explicitly shows all of the relativistic assumptions
and key approximation steps Einstein takes, may be interested in
consulting the exposition by Fernflores (Fernflores 2018, vol. II,
SS3.3). 



Like Einstein, many physicists and philosophers who wish to derive
Einstein's equation do so by considering an idealized physical
configuration. Typically, one considers a physical object \(B\)
involved in a symmetric physical interaction. For example, in
Einstein's original derivation from 1905, \(B\) emits two
equally energetic pulses of light in opposite directions. Mermin and
Feigenbaum have shown how one can also derive Einstein's
equation by considering the case where \(B\) emits two physical
bodies, 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 goal
in 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 the
physical interaction first in the inertial reference frame in which
\(B\) is at rest, which is sometimes called the "rest
frame." One then compares the mathematical description of this
interaction to the corresponding description from the perspective of a
different inertial reference frame that moves with a constant velocity
relative to the rest frame. Finally, using dynamical principles, such
as the conservation of energy or conservation of momentum, and the two
principles at the core of special relativity (i.e., the relativity
principle and the light principle), one shows that the dynamical
principles require that after \(B\) suffers the interaction in which
it absorbs or emits energy, its mass (i.e., rest-mass) must change by
an amount given by Einstein's equation. 



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



> 
> 
> For a particular idealized physical interaction, if:
> 
> 
> * certain conservation principles are true, such as the principle of
> conservation of energy and the principle of conservation of
> momentum,
> * 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 amount
> of 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 these
approaches. For example, since the analysis is based on a particular
physical interaction \(\boldsymbol{I}\), one cannot immediately
conclude that a body's mass will change according to
Einstein's equation in a different type of physical interaction
***I'***, for instance one in which
electromagnetism plays no role. 



Reasoning along these lines, Ohanian (2009, 2012) has argued that
Einstein should not be credited with proving his famous equation. On
the other hand, the physicist N. David Mermin (2011, 2012) claims that
Ohanian's demands for what counts as a "proof" in
physics are too stringent. Even though we are considering a very
specific interaction \(\boldsymbol{I}\), Mermin might say, we can
understand that it is such a generic situation that we can confidently
expect the result in all circumstances. Einstein himself believed that
if \(\boldsymbol{I}\) involved a physical process in which there was
an interaction between an electromagnetic field (say in the form of
light) and an ordinary physical object, then \(\boldsymbol{I}\) was
too specific to support the general conclusion that in *all*
cases, a change in the rest energy of a body is accompanied by a
change to its mass (i.e., rest-mass). 



With all of these caveats in place, we are almost ready to understand
Baierlein's derivation of Einstein's equation. Because
Baierlein's derivation involves analyzing a physical interaction
in which an object emits light, i.e., electromagnetic radiation, we
need first to state two dynamical properties of electromagnetic
radiation. First, considered as an electromagnetic wave, like all
waves, light carries energy. This is quite familiar to us today at an
everyday level, especially because of all the "solar"
devices we use. Second, light also carries momentum. As Baierlein
reports, in the late nineteenth century, James Clerk Maxwell had
already determined that a burst of light of energy \(E\) had momentum
\(E/c\). So, for example, if a laser beam strikes a freely-floating
mirror in outer space, the collision of light against mirror will
impart a finite, non-zero momentum to the mirror. One can calculate
exactly how the mirror will have its state of motion changed as a
result of this interaction by using the principle of conservation of
momentum. 



Baierlein asks us to consider an atom that emits two photons of equal
energy "back-to-back" in opposite directions. Although
Baierlein uses the Planck-Einstein expression \(E = hf\) for the
energy of a photon, where \(f\) is its frequency, and \(h\) is
Planck's constant, as he himself points out the right-hand side
of that expression does not figure significantly in his derivation.
Baierlein's derivation, he tells us, works in exactly the same
way if we just think of an object that emits two "bursts"
of light construed as classical electromagnetic radiation. We will
adopt this latter approach and consider a body \(B\) that emits two
bursts 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 two
different inertial frames. The first inertial frame we consider is
simply the inertial frame in which \(B\) is at rest. Since we will not
be performing many calculations using quantities defined relative to
this inertial frame, we will label it \(K'\). The direction of
motion of the second inertial frame, which we will label \(K\), is
carefully chosen to simplify the calculations. 



First, we choose \(K'\) so that \(B\) emits the two bursts of
light along \(z'\)-axis in opposite directions. We then choose
\(K\) so that \(K\) moves with velocity \(v\) in the negative
direction along the \(x'\)-axis. Using standard conventions,
this means that an observer at rest in \(K'\) (Alice) judges
that the light emitted by \(B\) travels up and down the page. For an
observer at rest in \(K\) (Bob), the atom \(B\) moves to the right
with velocity \(v\) and the light is emitted toward the right making
an 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 the
emission of light. However, as we will see shortly, the momentum of
\(B\) does change in both \(K\) and \(K'\) because the light it
emits carries away momentum. If we assume the classical definition for
the momentum of \(B\) as the product of its mass \(m\) and its
velocity \(v\), and \(v\) does not change, it follows that in order
for 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 the
inertial frame \(K\). Relative to \(K, B\) moves (to the right) with
velocity \(v\). Because relative to \(K'\) the bursts of light
are collinear, the \(x\)-component of the velocity of the light as
measured in \(K\) must be \(v\). However, the velocity of the light
also has a vertical component, i.e., a \(z\)-component. This is
significant, because we wish to calculate the change in the momentum
of \(B\). 



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 \(x\)-direction
is: 



\[
\frac{E}{2c} \cdot \cos \theta = \frac{E}{2c} \cdot \frac{v}{c}.
\]


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



\[
\frac{E}{c} \cdot \frac{v}{c},
\]


which, by the principle of conservation of momentum, must be equal to
the 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, 



\[
m \cdot v = \frac{E}{c^2} \cdot v,
\]


or simply, 



\[
E = mc^2.
\]


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



\[
E\_o = mc^2
\]


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 a
state 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 that
Einstein's own derivations do not constitute a
"proof" of mass-energy equivalence because they consider
physical configurations that are too specific, a new concern arises
when one considers "simplified" versions of derivations of
\(E\_o = mc^2\) such as the one we have just reviewed. Baierlein
concludes his derivation by praising, as one of its merits, that it
"requires only the ratio of momentum to energy for
electromagnetic radiation" (Baierlein 1991, p. 172). We have
already used this aspect of the derivation in our presentation above.




However, Baierlein then goes on to state: 



> 
> Moreover, this derivation--unlike Einstein's 1905
> derivation--makes no use of Lorentz transformations or other
> results 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 that
> would have led him to search for a connection between energy and
> inertia (Baierlein 1991, p. 172).
> 



This is a remarkable conclusion, for, if correct, it suggests that
when we use this kind of simplified version of Einstein's 1946
derivation, we are not displaying how Einstein's equation is a
consequence of special relativity. By contrast, Einstein's own
1946 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 that
whenever there is a change in the rest energy of an object, there is a
corresponding change in its mass (i.e., rest-mass). Although we have
not exactly demonstrated it, Einstein's equation is also
interpreted 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 motions
of idealized point particles that move in accordance with the theory
of special relativity, an object's total energy \(E\), which is
defined as the sum of its kinetic energy and its rest energy, is given
by the equation: 



\[
E = m \gamma(v) c^2,
\]


where \(\gamma(v)\) is to so-called "Lorentz factor." This
total energy \(E\) differs from the rest energy \(E\_o\) for any object
that moves with some velocity \(v\) relative to a given inertial
frame. In such an inertial frame, an object will have a non-zero
relativistic kinetic energy and its *total* energy \(E\) is
given by the equation above. In the inertial frame in which such an
object is at rest, however, the value of \(\gamma(v)\) becomes 1 and
the total energy is equal to the rest energy \(E\_o\), one might say,
precisely because in that inertial frame the object's
relativistic kinetic energy is zero. 



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



\[
m = m\_o \gamma(v).
\]


Using *these* notational conventions, the iconic equation \(E =
mc^2\) is the equation for the total energy \(E\) of a body as a
function of its *relativitic* mass. *This* equation is
not really of interest to us in discussing mass-energy equivalence.
Furthermore, physicists today have generally given up this notational
convention and deprecate the notion of "relativistic
mass," which, as Griffiths quips, "has gone the way of the
two dollar bill" (1999, p. 510 fn. 8). 


## 2. Philosophical interpretations of \(E\_o = mc^2\)



There are three main philosophical questions concerning the
interpretation of \(E\_o = mc^2\) that have occupied philosophers and
physicists:


1. Are mass and energy the *same* property of physical systems
and is that what is meant by asserting that they are
"equivalent"?
2. Is mass "converted" into energy in some physical
interactions, 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 according
to how they answer questions (1) and (2) above (Flores 2005). As we
will see (in
 Section 2.5),
 interpretations that answer question (3) affirmatively assume that
the answer to question (1) is yes.



The only combination of answers to questions (1) and (2) that is
inconsistent is to say that mass and energy are the same property of
physical systems but that the conversion of mass into energy (or vice
versa) is a genuine physical process. All the other three combinations
of answers to questions (1) and (2) are viable options and have been
held, at one time or another, by physicists or philosophers as
indicated by the examples given in
 Table 2.





|  |  |  |
| --- | --- | --- |
|  | *Conversion* | *No Conversion* |
| *Same Property* | X | Torretti (1996), Eddington (1929) |
| *Different Properties* | Rindler (1977)

(conversion is *possible*) | Bondi & Spurgin (1987) |


Table 2. Interpretations of mass-energy equivalence




In this section, we will describe the merits and demerits of each of
the interpretations in
 Table 2.
 Beyond these interpretations, we will also discuss two other types of
interpretations of mass-energy equivalence that do not fit neatly in
 Table 2.
 First, we will discuss Lange's (2001, 2002) interpretation,
which holds that only mass is a *real* property of physical
systems and that *we* convert mass into energy when we shift
the level at which we analyze physical systems. Second, we will
discuss two interpretations (one by Einstein and Infeld, 1938 and the
other by Zahar, 1989), which we will call *ontological*
interpretations, that attempt to answer question (3) above
affirmatively. However, we begin this section by addressing what has
formerly been a fairly common misconception concerning mass-energy
equivalence.


### 2.1 Misconceptions about \(E\_o = mc^2\)



Although it is far less common today, one still sometimes hears of
Einstein's equation entailing that matter can be converted into
energy. Strictly speaking, this constitutes an elementary category
mistake. In relativistic physics, as in classical physics, mass and
energy are both regarded as *properties* of physical systems or
properties of the constituents of physical systems. If one wishes to
talk about the physical *stuff* that is the bearer of such
properties, then one typically talks about either "matter"
or "fields." The distinction between "matter"
and "fields" in modern physics is itself rather subtle in
no 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, it
will have to be a "conversion" between *mass* and
energy, and not between *matter* and energy. Finally, our
observation 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 reactions
does not really help our philosophical understanding of mass-energy
equivalence and is perhaps partly to blame for some of the
misconceptions 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 energy
*are* the same property of physical systems. Consequently,
there is no sense in which one of the properties is ever physically
converted 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 and
energy are two ways of measuring what is essentially the same thing,
in the same sense that the parallax and distance of a star are two
ways of expressing the same property of location" (1929, p.
146). According to Eddington, the distinction between mass and energy
is artificial. We treat mass and energy as different properties of
physical systems because we routinely measure them using different
units. However, one can measure mass and energy using the same units
by choosing units in which \(c = 1\), i.e., units in which distances
are measured in units of time (e.g., light-years). Once we do this,
Eddington claims, the distinction between mass and energy
disappears.



Like Eddington, Torretti points out that mass and energy *seem*
to be different properties because they are measured in different
units. Speaking against Bunge's (1967) view that their numerical
equivalence does *not* entail that mass and energy "are
the same thing," Torretti explains:



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



For Torretti, the very existence of physical processes in which the
emission of energy by an object is correlated with the decrease in the
object's mass in accordance with Einstein's equation
speaks strongly against the view that mass and energy are somehow
distinct properties of physical systems. Torretti continues:



> 
> Of course, if lengths and times are measured with different, unrelated
> units, the 'mass'... differs conceptually from the
> 'energy.' But this difference can be understood as a
> consquence of the convenient but deceitful act of the mind by which we
> abstract time and space from nature (1996, p. 307, fn. 13).
> 



Thus, this footnote in his masterly *Relativity and Geometry*
suggests that, for Torretti, we are misled into using different units
for 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 by
adopting the convention Torretti himself uses of selecting units in
which \(c = 1\) (pp. 88-89). However, it may be useful to
remember that merely using the same units for spatial and temporal
intervals does not entail that space and time are treated "on a
par" in special relativity; they are not, as is evident from the
signature of the Minkwoski metric.



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



Interpretations such as Torretti's and Eddington's draw no
further ontological conclusions from mass-energy equivalence. For
example, neither Eddington nor Torretti make any explicit claim
concerning whether properties are best understood as universals, or
whether one ought to be a realist about such properties. Finally, by
saying that mass and energy are the same, these thinkers are
suggesting that the denotation of the terms "mass" and
"energy" is the same, though they recognize that the
connotation of these terms is clearly different.


### 2.3 Different-properties interpretations of \(E\_o = mc^2\)



As we have displayed in
 Table 2,
 interpretations of mass-energy equivalence that hold that mass and
energy are different properties disagree concerning whether there is
some physical process by which mass is converted into energy (or vice
versa). Although superficially Lange's (2001, 2002)
interpretation seems to fall in this category, as he certainly treats
mass and energy as different properties, he differs from others in
this category because Lange explicitly argues that only mass is a real
property of physical systems. Consequently, we will discuss
Lange's interpretation separately below (in
 Section 2.3.3).



We will begin with a discussion of Bondi and Spurgin's
interpretation (in
 Section 2.3.1).
 They hold that mass and energy are distinct properties and that there
is no such thing as the conversion of mass and energy. We will then
discuss Rindler's interpretation (in
 Section 2.3.2).
 He maintains that mass and energy are different properties but that
genuine conversions of mass and energy are at least permitted by
mass-energy equivalence.


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



Bondi and Spurgin's (1987) interpretation of mass-energy
equivalence has been influential especially among physicists concerned
with physics education. In an article where they complained about how
students often misunderstand Einstein's famous equation, Bondi
and Spurgin argued that Einstein's equation does not entail that
mass and energy are the same property any more than the equation \(m =
\varrho V\) (where \(m\) is mass, \(V\) is volume, and \(\varrho\) is
density) entails that mass and volume are the same. Just as in the
case of mass and volume, Bondi and Spurgin argue, mass and energy have
different *dimensions*. Ultimately, this reduces to a
disagreement with philosophers such as Torretti who would argue that
time, as a dimension, is no different than any one of the spatial
dimensions. Note well that this is not an issue about the
*units* we use for measuring mass (or energy).



Everyone agrees that according to special relativity one can measure
spatial intervals in units of time. We can do this because of the
postulate of special relativity that states that the speed of light
has the same value in all inertial frames. If we perform what amounts
to 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 hold
that there is nevertheless a fundamental distinction between space and
time as dimensions. On such a view, which is the view that Bondi and
Spurgin seem implicitly to be defending, while time is distinct from
any given spatial dimension, the contingent fact that \(c\) has the
same value in all inertial frames allows us to perform the relevant
substitution of variables. However, it does not follow from this that
we ought to treat time on a par with any spatial dimension, or that we
ought to treat the saptio-temporal interval as more fundamental (in
the way Torretti does).



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



Explanations of purported "conversions" along the lines
suggested by Bondi and Spurgin are now commonplace in the physics
literature. These explanations have the merit of emphasizing that in
many cases the mysteries of mass-energy equivalence do *not*
concern one physical property magically being transfigured into
another. However, the Bondi-Spurgin interpretation of mass-energy
equivalence has the demerit that it fails to address reactions such as
the electron-positron annihilation reaction. In such reactions, not
only is the number of particles not conserved, but *all* of the
particles involved are, by hypothesis, indivisible wholes. Thus, the
energy liberated in such reactions cannot be explained as resulting
from a transformation of the energy that was originally possessed by
the constituents of the reacting particles. Of course, Bondi and
Spurgin may simply be hoping that physics will reveal that particles
such as electrons and positrons are not indivisible wholes after all.
Indeed, they may even use annihilation reactions combined with their
interpretation of mass-energy equivalence to argue that it cannot be
the case that such particles are indivisible. Thus, we witness here
explicitly just how closely related interpretations concerning
mass-energy equivalence can be to views concerning the nature of
matter.



The second demerit of the Bondi-Spurgin interpretation, which it
shares with all other interpretations of mass-energy equivalence that
hold that mass and energy are different properties, is that it remains
silent about a central feature of physical systems it uses in
explaining apparent conversions of mass and energy. In order to
explain purported conversions along the lines suggested by
Bondi-Spurgin, one must make the familiar assumption that the energy
of the constituents of a system, be it potential energy or kinetic
energy, "contributes" to the rest-mass of the system.
Thus, for example, in the bombardment of \({^7}\Li\) reaction Bondi and
Spurgin must explain the rest-mass of the \({^7}\Li\) in the familiar
way, as arising from *both* the sum of the rest-masses of the
nucleons, and the mass-equivalents of their energies. However, the
Bondi-Spurgin interpretation offers no explanation concerning why the
energy of the constituents of a physical system, be it potential
energy or kinetic energy, manifest itself as part of the
*inertial* mass of the system as a whole. Of course, one can
always reply that even to ask for this type of explanation is to
refuse to accept relativistic thinking fully: the potential and
kinetic energies of the constituents contributes to the rest energy of
the whole, and because of Einstein's equation, contributes to
the rest-mass of the whole.



As we shall see, Rindler's interpretation of mass-energy
equivalence attempts to address the first demerit of the Bondi-Spurgin
interpretation, while Lange's interpretation brings to the
foreground 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 a
slightly, though importantly, modified version of the Bondi-Spurgin
interpretation. Rindler (for example, in 1977), agrees that there are
many purported conversions that are best understood as mere
transformations of one kind of energy into a different kind of energy.




However, for Rindler, there is nothing within special relativity
itself that rules out the possibility that there exists fundamental,
structureless particles (i.e., particles that are "atomic"
in the *philosophical* sense of the term). If such particles
exist, it is possible according to Einstein's equation that some
or all of the mass of such particles "disappears" and an
equivalent amount of energy "appears" within the relevant
physical system. Thus, Rindler seems to be suggesting that we should
confine our interpretation of mass-energy equivalence to what we can
deduce from special relativity. Thus, we should hold that
Einstein's equation at least allows for genuine conversions of
mass into energy, in the sense that there *may* be cases where
a certain amount of inertial mass "disappears" from
*within* a physical system and a corresponding amount of energy
"appears." Furthermore, in such cases we cannot explain
the reaction as merely involving a transformation of one kind of
energy into another.



The merit of Rindler's interpretation is that it confines the
interpretation of Einstein's equation to what we can validly
infer from the postulates of special relativity. Unlike the
interpretation proposed by Bondi and Spurgin, Rindler's
interpretation makes no assumptions about the constitution of matter
but 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 of
mass-energy equivalence. Lange begins his interpretation by arguing
that rest-mass is the only *real* property of physical systems.
This claim by itself suggests that there can be no such thing as a
physical process by which mass is converted into energy, for as Lange
asks "in what sense can mass be *converted* into energy
when mass and energy are not on a par in terms of their
reality?" (2002, p. 227, emphasis in original). Lange then
goes on to argue that a careful analysis of purported conversions of
mass-energy equivalence reveals that there is no physical process by
which mass is ever converted into energy. Instead, Lange argues, the
apparent conversion of mass into energy (or vice versa) is an illusion
that arises when we shift our level of analysis in examining a
physical system.



Lange seems to use a familiar argument from the Lorentz invariance of
certain physical quantities to their "reality." For Lange,
if a physical quantity is not Lorentz invariant, then it is not real
in the sense that it does not represent "the objective facts, on
which all inertial frames agree" (2002, p. 209). Thus Lange
uses Lorentz invariance as a *necessary* condition for the
reality of a physical quantity. However, in several other places, for
example 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 a
*sufficient* condition for the reality of a physical quantity.
However, if Lange adopts Lorentz-invariance as both a necessary and
sufficient condition for the reality of a physical quantity, then he
is committed to the view that rest energy is real for the very same
reasons he is committed to the view that rest-mass is real. Thus,
Lange's original suggestion that there can be no physical
process of conversion between mass and energy because they have
different ontological status seems challenged.



As it happens, Lange's overall position is not seriously
challenged by the ontological status of rest energy. Lange could
easily grant that rest energy is a real property of physical systems
and still argue (i) that there is no such thing as a physical process
of conversion between mass and energy and (ii) that purported
conversions result from shifting levels of analysis when we examine a
physical system. It is his observations concerning (ii) that force us
to face again the question of why the energy of the constituents of a
physical system manifests itself as the mass of the system, though
admittedly the question itself may simply reveal a failure fully to
appreciate a relativistic description of composite systems. 



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



> 
> ... we have just seen that this "conversion" of energy
> into mass is not a real physical process at all. *We*
> "converted" energy into mass simply by *changing our
> perspective* on the gas: shifting from initially treating it as
> many bodies to treating it as a single body [emphasis in original]
> (p. 236, 2002).
> 



Unfortunately, Lange's characterization threatens to leave
readers with the impression that if "we" had not shifted
our perspective in the analysis of the gas, no change to the inertial
mass of the gas sample would have ensued. Of course, it is unlikely
that Lange means this. Lange would likely agree that even if no human
beings are around to analyze a gas sample, the gas sample will respond
in any physical interaction differently as a whole *after* it
has absorbed some energy precisely because its inertial mass will have
increased.



The merits of Lange's view concerning the
"conversion" of mass-energy equivalence are essentially
the same as the merits of both the Bondi-Spurgin interpretation and
Rindler's interpretation. All these interpretations agree that
there are important cases where we have now learned enough to assert
confidently that purported "conversions" of mass and
energy are merely cases where energy of one kind is transformed into
energy of another kind. Aside from the comparatively minor issue
concerning the "reality" of rest energy, the main demerit
of Lange's view is that it might potentially mislead
unsuspecting readers.


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



The relationship between mass-energy equivalence and hypotheses
concerning the nature of matter is twofold. First, as we have
suggested, some of the interpretations of mass-energy equivalence seem
to assume implicitly certain features of matter. Second, some
philosophers and physicists, notably Einstein and Infeld (1938) and
Zahar (1989), have argued that mass-energy equivalence has
*consequences* concerning the nature of matter. In this
section, we will discuss the first of these two relationships between
\(E\_o = mc^2\) and hypotheses concerning the nature of matter. We
discuss the second relationship in the next section
 (Section 2.5).



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




\[\tag{7}
E\_o = (m - q)c^2 + K,
\]


where \(K\) is merely an additive factor that fixes the zero-point of
energy and is conventionally set to zero and \(q\) is also routinely
set to zero. However, unlike the convention to set \(K\) to zero,
setting \(q = 0\) involves a hypothesis concerning the nature of
matter, because it rules out the possibility that there exists matter
that 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 rests
squarely on the assumption that \(q = 0\). Mass and energy cannot be
the same property if there exists matter that has mass some of which
cannot ever, under any conditions, be "converted" into
energy. However, one could argue that although the same-property
interpretation makes this assumption, it is not an
*unjustified* assumption. Currently, physicists do not have any
evidence that there exists matter for which \(q\) is not equal to
zero. Nevertheless, it seems important, from a philosophical point of
view, to recognize that the same-property interpretation depends not
only 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 are
distinct properties of physical systems need not, of course, assume
that \(q\) is different from zero. Such interpretations can simply
leave the value of \(q\) to be determined empirically, for as we have
seen such interpretations argue for treating mass and energy as
distinct properties on different grounds. Nevertheless, the
Bondi-Spurgin interpretation does seem to adopt implicitly a
hypothesis concerning the nature of matter.



According to Bondi and Spurgin, all purported conversions of mass and
energy are cases where one type of energy is transformed into another
kind of energy. This in turn assumes that we can, in all cases,
understand a reaction by examining the constituents of physical
systems. If we focus on reactions involving sub-atomic particles, for
example, Bondi and Spurgin seem to assume that we can always explain
such reactions by examining the internal structure of sub-atomic
particles. However, if we ever find good evidence to support the view
that some particles have *no* internal structure, as it now
seems to be the case with electrons for example, then we either have
to give up the Bondi-Spurgin interpretation or use the interpretation
itself to argue that such seemingly structureless particles actually
do contain an internal structure. Thus, it seems that the
Bondi-Spurgin interpretation assumes something like the infinite
divisibility 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 the
Einstein-Infeld and Zahar interpretations begin by adopting the
*same-property* interpretation of \(E\_o = mc^2\). Thus,
according to both interpretations, mass and energy are the same
properties of physical systems. Furthermore, both the Einstein-Infeld
and Zahar interpretations use a rudimentary distinction between
"matter" and "fields." According to this
somewhat dated distinction, classical physics includes two fundamental
substances: matter, by which one means ponderable material stuff, and
fields, by which one means physical fields such as the electromagnetic
field. For both Einstein and Infeld and Zahar, matter and fields in
classical 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 mass
and energy are really the same physical property after all, say
Einstein and Infeld and Zahar, one can no longer distinguish between
matter and fields, as both now have both mass *and* energy.



Although both Einstein and Infeld and Zahar use the same basic
argument, they reach slightly different conclusions. Zahar argues that
mass-energy equivalence entails that the fundamental stuff of physics
is a sort of "I-know-not-what" that can manifest itself as
either matter or field. Einstein and Infeld, on the other hand, in
places seem to argue that we can infer that the fundamental stuff of
physics is fields. In other places, however, Einstein and Infeld seem
a bit more cautious and suggest only that one can construct a physics
with only fields in its ontology.



The demerits of either ontological interpretation of mass-energy
equivalence are that it rests upon the *same-property*
interpretation of \(E\_o = mc^2\). As we have discussed above (see
 Section 2.4),
 while one *can* adopt the *same-property*
interpretation, to do so one must make additional assumptions
concerning the nature of matter. Furthermore, the ontological
interpretation rests on what nowadays seems like a rather crude
distinction between "matter" and "fields." To
be sure, mass-energy equivalence has figured prominently in
physicists' conception of matter in no small part because it
does open up the door to a description of what we ordinarily regard as
ponderable matter in terms of fields, since the energy of the field at
one level can manifest itself as mass one level up. However, the
inference from mass-energy equivalence to the fundamental ontology of
modern physics seems far more subtle than either Enstein and Infeld or
Zahar suggest.


## 3. History of Derivations of Mass-Energy Equivalence



Einstein first derived mass-energy equivalence from the principles of
special relativity in a small article titled "Does the Inertia
of a Body Depend Upon Its Energy Content?" (1905b). This
derivation, along with others that followed soon after (e.g., Planck
(1906), Von Laue (1911)), uses Maxwell's theory of
electromagnetism. (See
 Section 3.1.)
 However, as Einstein later observed (1935), mass-energy equivalence
is a result that should be independent of any theory that describes a
specific physical interaction. This is the main reason that led
physicists to search for "purely dynamical" derivations,
i.e., derivations that invoke only mechanical concepts such as energy
and momentum, and the conservation principles that govern them. (See
 Section 3.2)


### 3.1 Derivations of \(E\_o = mc^2\) that Use Maxwell's Theory



Einstein's original derivation of mass-energy equivalence is the
best known in this group. Einstein begins with the following
thought-experiment: a body at rest (in some inertial frame) emits two
pulses of light of equal energy in opposite directions. Einstein then
analyzes this "act of emission" from another inertial
frame, which is in a state of uniform motion relative to the first. In
this analysis, Einstein uses Maxwell's theory of
electromagnetism to calculate the physical properties of the light
pulses (such as their intensity) in the second inertial frame. By
comparing the two descriptions of the "act of emission",
Einstein arrives at his celebrated result: "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 \(L/9 \times 10^{20}\), the
energy being measured in ergs, and the mass in grammes" (1905b,
p. 71). A similar derivation using the same thought experiment
but appealing to the Doppler effect was given by Langevin (1913) (see
the discussion of the inertia of energy in Fox (1965, p. 8)).




Some philosophers and historians of science claim that
Einstein's first derivation is fallacious. For example, in
*The Concept of Mass*, Jammer says: "It is a curious
incident in the history of scientific thought that Einstein's
own derivation of the formula \(E = mc^2\), as published in his
article in *Annalen der Physik*, was basically fallacious. . .
  the result of a *petitio principii*, the conclusion
begging the question" (Jammer, 1961, p. 177). According to
Jammer, Einstein implicitly assumes what he is trying to prove, viz.,
that if a body emits an amount of energy \(L\), its inertial mass will
decrease by an amount \(\Delta m = L/c^2\). Jammer also accuses
Einstein of assuming the expression for the relativistic kinetic
energy of a body. If Einstein made these assumptions, he would be
guilty of begging the question. However, Stachel and Torretti (1982)
have shown convincingly that Einstein's (1905b) argument is
sound. They note that Einstein indeed derives the expression for the
kinetic energy of an "electron" (i.e., a structureless
particle with a net charge) in his earlier (1905a) paper. However,
Einstein nowhere uses this expression in the (1905b) derivation of
mass-energy equivalence. Stachel and Torretti also show that
Einstein's critics overlook two key moves that are sufficient to
make 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 is
a measure of its energy content" (1905b, p. 71) does not,
strictly speaking, follow from his argument. As Torretti (1996) and
other philosophers and physicists have observed, Einstein's
(1905b) argument allows for the possibility that once a body's
energy store has been entirely used up (and subtracted from the mass
using 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 special
relativity, that no "exotic matter" exists that is
*not* convertible into energy (see Ehlers, Rindler, Penrose,
(1965) for a discussion of this point). However, particle-antiparticle
anihilation experiments in atomic physics, which were first observed
decades after 1905, strongly support "Einstein's dauntless
extrapolation" (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 or
emits electromagnetic radiation (typically light) of total energy
\(E\_o\) in equal and opposite directions. Because light carries both
energy and momentum, one then uses the conservation principles for
those quantities and the standard heuristic in relativity of
considering the same physical process from two different inertial
frames that are in a state of relative motion to show that in order
for the conservation principles to be satisfied, the mass (i.e.,
rest-mass) of the emitting or absorbing object must increase or
decrease by an amount \(E\_o /c^2\). For a more detailed description of
a simplified derivation in this group, see
 Section 1.5




One of the few exceptions to this approach among derivations that use
Maxwell's theory is Einstein's 1906 derivation (Einstein
1906). In this derivation, Einstein considers a freely-floating box. A
burst of electromagnetic radiation of energy \(E\_o\) is emitted inside
the box from one wall toward a parallel wall. Einstein shows that the
principle of mechanics that says that the motion of the center of mass
of a body cannot change merely because of changes inside the body
would be violated if one did not attribute an inertial mass \(E\_o
/c^2\) to the burst of electromagnetic radiation (see Taylor and
Wheeler 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 by
analyzing an inelastic collision from the point of view of two
inertial frames in a state of relative motion (the centre-of-mass
frame, and an inertial frame moving with a relative velocity \(v)\).
One of the first papers to appear following this approach is
Perrin's (1932). According to Rindler and Penrose (1965),
Perrin's derivation was based largely on Langevin's
"elegant" lectures, which were delivered at the
College de France in Zurich around 1922. Einstein himself gave
a purely dynamical derivation (Einstein, 1935), though he nowhere
mentions either Langevin or Perrin. The most comprehensive derivation
of this sort was given by Ehlers, Rindler and Penrose (1965). More
recently, a purely dynamical version of Einstein's original
(1905b) thought experiment, where the particles that are emitted are
not photons, has been given by Mermin and Feigenbaum (1990) and Mermin
(2005).



Derivations in this group are distinctive because they demonstrate
that mass-energy equivalence is a consequence of the changes to the
structure of spacetime brought about by special relativity. The
relationship between mass and energy is independent of Maxwell's
theory or any other theory that describes a specific physical
interaction. We can get a glimpse of this by noting that to derive
\(E\_o = mc^2\) by analyzing a collision, one must first define the
four-momentum \(\mathbf{p}\), the "space-part" of which is
relativistic momentum \(\mathbf{p}\_{\rel}\), and relativistic kinetic
energy \(T\_{\rel}\), since one cannot use the old Newtonian notions of
momentum and kinetic energy. 



In Einstein's own purely dynamical derivation (1935), more than
half of the paper is devoted to finding the mathematical expressions
that define \(\mathbf{p}\) and \(T\_{\rel}\). This much work is required
to arrive at these expressions for two reasons. First, the changes to
the structure of spacetime must be incorporated into the definitions
of the relativistic quantities. Second, \(\mathbf{p}\) and \(T\_{\rel}\)
must be defined so that they reduce to their Newtonian counterparts in
the appropriate limit. This last requirement ensures, in effect, that
special relativity will inherit the empirical success of Newtonian
physics. Once the definitions of \(\mathbf{p}\) and \(T\_{\rel}\) are
obtained, one derives mass-energy equivalence in a straight-forward
way by analyzing a collision. (For a more detailed discussion of
Einstein's (1935), see Fernflores, 2018.)



At a very general level, purely dynamical derivations of
Einstein's equation and derivations that appeal to
Maxwell's theory really follow the same approach. In both styles
of derivations, although it may not seem like it at first glance, we
are dealing with one of the most basic dynamical interactions: a
collision. So, for example, we can construe the physical configuration
of Einstein's original 1905 derivation (Einstein 1905b) as a
collision in which the total number of objects is not conserved. This
is even easier to do if one adopts a "particle"
description of light. In both the purely dynamical derivations and the
derivations that appeal to an interaction with electromagnetic
radiation, one then examines the collision in question and shows that
in order for dynamical principles to be satisfied, the relationship
among the masses and energies of the objects involved in the
collisions must satsify Einstein's equation. 



The main difference between the two approaches to deriving
Einstein's equation is that in derivations that consider a
collision 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 (see
 Section 1.5),
 we must appeal to the expression for the momentum of a burst of
light. 


## 4. Experimental Verification of Mass-Energy Equivalence



Cockcroft and Walton (1932) are routinely credited with the first
experimental verification of mass-energy equivalence. Cockcroft and
Walton examined a variety of reactions where different atomic nulcei
are bombarded by protons. They focussed their attention primarily on
the bombardment of \({^7}\Li\) by protons (see
 Section 1.4).



In their famous paper, Cockcroft and Walton noted that the sum of the
rest-masses of the proton and the Lithium nucleus (i.e., the
reactants) was \(1.0072 + 7.0104 = 8.0176\) amu. However, the sum of
the 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 amu
has "disappeared" from the reactants. Cockcroft and Walton
also observed that the total energy (in the reference frame in which
the \({^7}\Li\) nucleus is at rest) for the reactants was 125 KeV.
However, the total kinetic energy of the \(\alpha\)-particles was
observed to be 17.2 MeV. Thus, it seems as if an amount of energy of
approximately 17 MeV has "appeared" in the reaction.



Implicitly referring to the equivalence of mass and energy, without
explicitly mentioning either the result or Einstein by name, Cockcroft
and Walton then simply assert that a mass 0.0154 amu "is
equivalent to an energy liberation of \((14.3 \pm 2.7) \times 10^6\)
Volts" (p. 236). They then implicitly suggest that this
inferred value for the kinetic energy of the two resulting
\(\alpha\)-particles is consistent with the *observed* value
for the kinetic energy of the \(\alpha\)-particles. Cockcroft and
Walton 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 Walton *use*
mass-energy equivalence to confirm their hypothesis about what happens
when \({^7}\Li\) is bombarded by protons. Hence, it does not seem we
ought to regard this experiment as a confirmation of \(E\_o = mc^2\).
However, if we take some of the other evidence that Cockcroft and
Walton provide concerning the identification of the products in the
bombardment reaction as sufficient to establish that the products are
indeed \(\alpha\)-particles, then we can interpret this experiment as
a confirmation of mass-energy equivalence, which is how this
experiment is often reported in the physics literature.



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


 
\[\begin{align}
n + {^{32}}\S &\rightarrow {^{33}}\S + \gamma \\
n + {^{28}}\Si &\rightarrow {^{29}}\Si + \gamma
\end{align}\]


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


## 5. Conclusion



In this entry, we have presented the physics of mass-energy
equivalence as widely understood by both physicists and philosophers.
We have also canvassed a variety of philosophical interpretations of
mass-energy equivalence. Along the way, we have presented the merits
and demerits of each interpretation. We have also presented a brief
history of derivations of mass-energy equivalence to emphasize that
the equivalence of mass and energy is a direct result to changes to
the structure of spacetime imposed by special relativity. Finally, we
have briefly and rather selectively discussed the empirical
confirmation of mass-energy equivalence.