# The Equivalence of Mass and Energy

*First published Wed Sep 12, 2001; substantive revision Mon Sep 13, 2004*

Einstein correctly described the equivalence of mass and energy as
"the most important upshot of the special theory of relativity"
(Einstein, 1919), for this result lies at the core of modern physics.
According to Einstein's famous equation
*E* = *mc*^{2}, the energy
(*E*) of a physical system is numerically equal to the product
of its mass (*m*) and the speed of light (*c*)
squared. It is customary to refer to this result as "the equivalence
of mass and energy," or simply "mass-energy equivalence," because one
can choose units in which *c* = 1, and hence
*E* = *m*. An important consequence of
*E* = *mc*^{2} is that a change in the
rest-energy of a physical system is accompanied by a corresponding
change to its inertial mass. (This is discussed further in Section 1.)
This has led many philosophers to argue that mass-energy equivalence
has profound ontological consequences. ( Philosophical interpretations
of *E* = *mc*^{2} are discussed
further in Section 2.) Recently, the history of
*E* = *mc*^{2} has also attracted the
attention of some philosophers. This is primarily, though not
exclusively, because this history shows that
*E* = *mc*^{2} is a direct consequence
of changes to the structure of spacetime brought about by Special
Relativity. (This is discussed further in Section 3.)

- 1. Mass-Energy Equivalence: The Result
- 2. Philosophical Interpretations of Mass-Energy Equivalence
- 3. History of Derivations of Mass-Energy Equivalence
- 4. Experimental Verification of Mass-Energy Equivalence
- Bibliography
- Other Internet Resources
- Related Entries

## 1. Mass-Energy Equivalence: The Result

The equation *E* = *mc*^{2} has two
distinct physical consequences. To see this, one needs first to
distinguish between the rest-mass and the relativistic mass of a
physical system. Let us suppose the physical system we are considering
consists of an ordinary, mid-sized object. In mature classical
physics, the inertial mass of such an object is a measure of that
body's resistance to acceleration. This is the notion of mass one uses
in everyday life when one talks about, for example, 1 kg. of salt.
Furthermore, in classical physics, the inertial mass of a body is
independent of its relative state of motion. Because this is no longer
the case in relativistic physics, one can identify two notions of mass
in Special Relativity (SR). The *rest-mass* of a body is the
inertial mass of that body when it is at rest relative to an inertial
frame. The term *m* in the equation
*E*=*mc*^{2} does not represent rest-mass; it
represents *relativistic mass*, which is the inertial mass of a
body when it is in a state of motion relative to an inertial frame. If
we use *m*_{o} to
designate the rest-mass of a body, then we can re-write Einstein's
equation in the following way:

These equations entail that:

(I)In the frame of reference in which a body is at rest, its energy (in this case called therest-energy) is equal to the product of its rest-massm_{o}and the speed of light squared. This is because in this casev= 0, so the Lorentz factor is one.

(II)In a frame of reference in which a body moves with velocityv, the energy of the body is equal to the product of its rest-mass, the speed of light squared, and the Lorentz factor.

From **(I)** it follows that if there is a change in
the rest-energy of a body, there must be a corresponding change in its
rest-mass. For example, if a body is heated, and thereby absorbs a
small amount of energy Δ*E* (as measured in the frame of
reference in which the body is at rest), its rest-mass will increase by
a very small amount equal to Δ*E*/*c*^{2}.
This increase is tiny because of the high numerical value of the speed
of light. Indeed, for mid-sized objects, such an increase in mass would
be too small to measure with even the most accurate balance. For
example, if a 1 kg block of gold is heated so that its temperature
increases by 10 °C, then, *ceteris paribus*, its mass would
increase by as much as 1.4 × 10^{-14} kg; a cube of gold
of this additional tiny mass would have sides smaller than one
one-thousandths of a millimeter. Similarly, if a body emits an amount
of energy Δ*E*, say in the form of light or heat, its
rest-mass will decrease by a tiny amount
Δ*E*/*c*^{2}. In both cases, the important
and novel claim made by SR is that the inertial mass of a body can
change depending on whether it absorbs or emits energy. This is true,
of course, not just for isolated mid-sized objects but for any physical
system. For example, as Einstein (1907) first showed, if we consider a
physical system composed of point-particles, such as an ideal gas, the
entire system can be considered as a single point-particle whose
inertial mass increases as the kinetic energies of the component
particles increase.

**(I)** also entails that there are physical
interactions in which masses no longer combine by simple addition, as
they do in pre-relativistic physics. For example, suppose two bodies
*A* and *B* collide to produce a single, more massive
body *AB*. Suppose further that a net amount of energy
*E* is emitted in this inelastic collision, say in the form of
heat. **(I)** entails that the rest-mass of *AB*
will be *less* than the rest-mass of *A* plus the
rest-mass of *B* by an amount equal to
*E*/*c*^{2}. This stands in sharp contrast to the
pre-relativistic prediction that the rest-mass of *AB* will be
*equal* to the rest-mass of *A* plus the rest-mass of
*B*. So, for example, suppose a meteor (*A*) struck the
earth (*B*). After the crash, the earth (*AB*) would have
a mass that is a tiny bit less than the mass of the meteor plus the
mass of the earth prior to the crash. This is because during the
collision the meteor loses part of its kinetic energy as heat
radiation. This energy loss corresponds to a loss of mass. It is worth
emphasizing that, according to SR, it is the *inertial* mass of
bodies that is no longer simply added in collisions such as these. In
other words, SR predicts that the resulting body *AB* will
resist acceleration a tiny bit less than one would have predicted
according to pre-relativistic physics. There is an analogous result for
cases where a single body disintegrates into two or more bodies.

These consequences of **(I)** also illustrate how the
classical conservation principles are modified by SR. According to
Newtonian physics, all physical interactions are separately governed by
the principles of conservation of mass and conservation of energy. So,
for example, according to pre-relativistic physics the mass of the
block of gold discussed above must remain the same as it is heated.
However, as we have seen, this is not the case in relativistic physics,
because the energy absorbed by the block of gold contributes to an
increase in its rest-mass. Similarly, Newtonian physics predicts that
mass is conserved when the meteor crashes into the earth in the above
example. However, according to relativistic physics, some of the mass
is radiated away as energy in the form of heat. In both of these
examples, it is the total mass *and* energy of the entire system
that is conserved in these interactions. In general, in SR physical
interactions no longer satisfy the two classical conservation
principles separately. Instead, these two principles are fused into a
single principle: the principle of conservation of mass-energy. It is
these consequences of **(I)**, and indirectly the fusing
of the two classical conservation principles, that have motivated
different philosophical interpretations of
*E* = *mc*^{2} (see Section 2,
Philosophical Interpretations of Mass-Energy Equivalence).

From **(II)** it follows that no bounded amount of
energy is sufficient to accelerate a body to the speed of light. This
is because as the speed of a body approaches the speed of light its
relativistic mass increases without bound. But this means that the
body's resistance to acceleration, as measured in the inertial frame
relative to which it is moving, also increases without bound. In
practice, this means that it takes more and more energy to achieve
proportionally smaller increases in the speed of a body. For example,
suppose an electron requires an amount of energy *E* to reach
50% of the speed of light. The electron requires twice that amount of
energy to reach 90% of the speed of light, roughly six times *E*
to reach 99% of the speed of light, and nearly two hundred times
*E* to reach 99.999% of the speed of light! This consequence of
*E* = *mc*^{2} is thus crucial in the
design and operation of particle accelerators, and it is often
emphasized in the popular media (e.g., in popular science books and
films). However, its philosophical import is relatively minor because
the increase in relativistic mass does not result in a change to the
body. In the frame of reference in which the body is at rest, its
inertial mass continues to be *m*_{o}.

A common misconception surrounding *E* =
*mc*^{2} is that it entails that the entire rest-mass of
a body can become energy. Strictly speaking, mass-energy equivalence
only entails that a *change* in the rest-energy of a body is
invariably accompanied by a corresponding *change* in the
rest-mass of the body. For example, a body may lose a bit of its mass
because it radiates a bit of energy. The stronger claim that a body may
lose *all* of its rest-mass as it radiates energy is
*not* a consequence of SR. However, this stronger claim is very
well confirmed by experiments in atomic physics. Many
particle-antiparticle collisions have been observed, such as collisions
between electrons and positrons, where the entire mass of the particles
is radiated away as energy in the form of light. Nevertheless, SR
leaves open the possibility that a form of matter exists whose mass
cannot become energy. This is significant because it emphasizes that
mass-energy equivalence is not a consequence of a theory of matter; it
is instead a direct consequence of changes to the structure of
spacetime imposed by SR (see Section 3,
Derivations of Mass-Energy Equivalence: History).

## 2. Philosophical Interpretations of Mass-Energy Equivalence

There are two main philosophical interpretations of mass-energy
equivalence. According to one common interpretation,
*E* = *mc*^{2} implies that mass and
energy, which are treated as distinct properties in Newtonian physics,
are actually the same. I will refer to this view, which is the weaker
of the two, as the *same-property* interpretation hereafter. The
second interpretation of mass-energy equivalence is that it entails
that there is only one fundamental stuff in the world. I will call this
view the *one-stuff* interpretation hereafter.

### 2.1 The *Same-Property* Interpretation of *E* = *mc*^{2}

Most physicists and philosophers regard the terms "mass" and
"energy" as designating properties of physical systems. Thinkers such
as Eddington (1929), and more recently Torretti (1983), argue that
since mass and energy are numerically equivalent according to
Einstein's famous equation, the properties mass and energy are the
same. 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.

Torretti (1983) argues along similar lines when he responds to the
opposing view, which is held by a minority (e.g., Bunge, 1967; Sachs,
1981). This minority holds that the numerical equivalence of mass and
energy is not sufficient to conclude that the two properties are the
same. However, according to Torretti, "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" (1983, p. 307, fn. 13). Like Eddington, Torretti
points out that mass and energy seem to be different properties because
they are measured in different units. But the units of mass and energy
are different only if one uses different units for space and time,
which one need not do. For Torretti, the apparent difference between
mass and energy is thus an illusion that arises from "the convenient
but deceitful act of the mind by which we abstract time and space from
nature" (1983, p. 307, fn. 13). Unfortunately, Torretti does not
elaborate on the nature of this "deceitful act of the mind." However,
he seems to be suggesting that in our perception of the world spatial
and temporal dimensions merely *appear* to be distinct. We
perceive spatial intervals as different in kind from temporal
intervals. Consequently, we use different types of units to measure
spatial and temporal intervals, which has the consequence that mass and
energy have different types of units. Since it is customary to regard
quantities measured in different types of units as measuring different
properties, we conclude that mass and energy are different properties
of physical systems. Thus, for Torretti, our mistaken judgment that
mass and energy are distinct properties arises from the peculiar way in
which we, as humans, perceive space and time.

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.

### 2.2 The *One-Stuff* Interpretation of *E* = *mc*^{2}

Interpretations in the second group establish a connection between the terms "mass" and "energy," which are again treated as terms designating properties, and the two basic constituents in the ontology of physics: matter and fields. The equivalence of mass and energy is then taken to show that we can no longer distinguish between matter and fields. Einstein and Infeld (1938) offer a clear articulation of this interpretation. According to Einstein and Infeld, in pre-relativistic physics one can distinguish matter from fields by their properties. Specifically, matter has energy and mass, whereas fields only have energy. Since mass and energy are distinct in pre-relativistic physics, there are physical criteria that allow us to distinguish matter from fields qualitatively. So it is reasonable to adopt an ontology that contains both matter and fields. However, in relativistic physics, the qualitative distinction between matter and fields is lost because of the equivalence of mass and energy. Consequently, Einstein and Infeld argue, the distinction between matter and fields is no longer a qualitative one in relativistic physics. Instead, it is merely a quantitative difference, since "matter is where the concentration of energy is great, field where the concentration of energy is small"(1938, p. 242). Thus, Einstein and Infeld conclude, mass-energy equivalence entails that we should adopt an ontology consisting only of fields.

Strictly speaking, Einstein and Infeld's conclusion concerning the
ontology of modern physics does not follow from
*E* = *mc*^{2} alone. As we have noted
toward the end of Section 1, mass-energy equivalence by itself does not
entail that a chunk of what we ordinarily regard as material can be
completely converted into energy. Thus, even if
*E* = *mc*^{2} is true, it is still
logically possible that a theory whose basic ontology consists of both
matter and fields might be required. What speaks against this option is
a generalized hypothesis concerning the nature of matter based on the
empirical observation that some sub-atomic particles can radiate all of
their mass. Finally, the development of quantum field theories
subsequent to Einstein and Infeld's interpretation lend further support
to their view, since these empirically successful theories treat the
basic constituents of matter (such as electrons) as quantizations of a
field.

Among philosophers, Russell interprets mass-energy equivalence in a
way that *prima facie* seems similar to Einstein and Infeld.
According to Russell, "a unit of matter tends more and more to be
something like an electromagnetic field filling all space, though
having its intensity in a small region" (1915, p. 121). In his later
work, Russell continues to hold this view. For example, in *Human
Knowledge, Its Scope and Limits*, he points out that "atoms" are
merely small regions in which there is a great deal of energy.
Furthermore, these regions are precisely the regions where one would
have said, in pre- relativistic physics, that there was matter. For
Russell, these considerations suggest that "mass is only a form of
energy, and there is no reason why matter should not be dissolved into
other forms of energy. It is energy, not matter, that is fundamental in
physics" (1948, p. 291). Russell is proposing that mass is reducible to
energy in the sense that the world consists only of energy. Thus, for
Russell, "mass" and "matter" are otiose in modern physics. Several
physicists have held a similar position, though this view is less
common now. For example, after a discussion particle-antiparticle
annihilation experiments in 1951, Wolfgang Pauli states: "Taking the
existence of all these transmutations into account, what remains of the
old idea of matter and of substance? The answer is energy. This is the
true substance, that which is conserved; only the form in which it
appears is changing" (1951, p. 31).

Russell and Pauli's interpretations are, despite the superficial
similarity, importantly different from Einstein and Infeld's. Russell
(in some places) and Pauli both treat the term "energy" as though it
designates a substance, whereas Einstein and Infeld clearly regard
energy as a property. This is an important difference. Treating energy
as a term designating a substance is now widely regarded as a remnant
of an untenable nineteenth century view. Nevertheless, some
philosophers have continued to promulgate, albeit inadvertantly at
times, the view that energy is a substance. A fairly recent example of
this, which is part of an interpretation of mass-energy equivalence, is
contained in Zahar's (1989) *Einstein's Revolution: A Study in
Heuristic*.

According to Zahar, energy in pre-relativistic physics occupies a distinct "ontological level" from matter primarily because the former is regarded as dependent on the latter, but not vice versa. In relativistic physics, however, Einstein's famous equation shows that these two ontological levels are in fact identical. According to Zahar, Einstein showed "that ‘energy’ and ‘mass' could be treated as two names for the same basic entity. The stuff which appears to the senses as hard extended substance and the quantity of energy which characterises a process are in fact one and the same thing" (1989, p. 262). For Zahar, the apparent difference between mass and energy arises from the contingent fact that our senses perceive mass and energy differently. On this reading, mass-energy equivalence has the metaphysical implication that what is real, "is no longer the familiar hard substance but a new entity which can be interchangeably called matter or energy" (1989, p. 263). Thus, Zahar holds that the fundamental stuff of physics is a sort of "I- know-not-what" that we can call either "mass" or "energy."

Unfortunately, Zahar's interpretation suffers from a rather
imprecise use of the terms "mass," "matter," and "energy." For example,
Zahar uses both "mass" and "matter" to designate a substance, when he
clearly seems to intend only for the latter to designate a substance
and for the former to designate a property. This equivocation can be
easily corrected. His use of the term "energy," however, is more
difficult to repair unless we introduce the notion of a field. So, for
example, when Zahar talks about energy occupying a different
"ontological level" from matter, what he should be saying is that
fields occupied such a different level. According to mature classical
physics (without the aether), it is fields that are "produced" by
matter. Consequently, Zahar would have to say that it is fields and
matter that are on the same ontological level, and hence that as a
result of *E* = *mc*^{2}, we can no
longer really distinguish between the two. Thus, a charitable
interpretation of Zahar, which uses the terms "matter," "mass,"
"field," and "energy" a bit more carefully, reduces to Einstein and
Infeld's position.

Despite the difference in the ontological claims made by the two leading interpretations, there is one significant similarity. Both interpretations implicitly claim that mass-energy equivalence changes our knowledge concerning the extensions of the terms "mass" and "energy." Whereas the terms "mass" and "energy" had different extensions and intensions in pre-relativistic physics, SR teaches us that the extension of the two terms is actually the same. This is analogous to the discovery that the referents of "the morning star" and "the evening star" are the same. We can push the analogy a bit further. Just as it is possible to verify empirically that the planet Venus is the referent of both "the morning star" and "the evening star," it is possible to verify empirically that the extensions of "mass" and "energy" are the same. (See Section 4, Experimental Verification of Mass- Energy Equivalence.) Thus, the one-stuff interpretation merely goes farther than the same-property interpretation by drawing a conclusion concerning not just the properties of physical systems, but also about their very constituents.

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

Einstein first derived mass-energy equivalence from the principles
of SR 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 Subsection 3.1,
Derivations of *E* = *mc*^{2} that Use Maxwell's Theory.)
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 principles that govern them. (See Subsection
3.2,
Purely Dynamical Derivations of *E* = *mc*^{2}.)

### 3.1 Derivations of *E* = *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 ×
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 *E* = *mc*^{2}
in Fox (1965)).

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 Δ*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. Recently, 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 Δ*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 (1983) 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* = *mc*^{2} in SR, 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, 1983, p. 112).

### 3.2 Purely Dynamical Derivations of *E* = *mc*^{2}

Purely dynamical derivations of *E* =
*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).

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 SR. 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* = *mc*^{2}
by analyzing a collision, one must first define relativistic momentum
(**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
**p**_{rel} 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,
**p**_{rel} 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 SR
will inherit the empirical success of Newtonian physics. Once the
definitions of **p**_{rel} and
*T*_{rel} are obtained, the derivation of mass-energy
equivalence is straight-forward. (For a more detailed discussion of
Einstein's (1935), see Flores, (1998).)

## 4. Experimental Verification of Mass-Energy Equivalence

Cockcroft and Walton (1932) are routinely credited with the first experimental verification of mass-energy equivalence. Einstein (1905b) had conjectured that the equivalence of mass and energy could be tested by "weighing" an atom before and after it undergoes radioactive decay. But there was no way of performing this experiment or another experiment that would directly confirm mass-energy equivalence at the time. Technological developments allowed Cockcroft and Walton to take a different approach. They studied the bombardment of a lithium atom (Li) by a proton (p), which produces two alpha particles (α). This reaction is symbolized by the following equation:

p + Li = 2α

In this reaction, there is a *decrease* in the total
rest-mass as the reaction proceeds from left to right: the total
rest-mass of proton and the Lithium atom is greater than the total
rest-mass of the two alpha particles. Furthermore, there is also an
*increase* in the total kinetic energy: the kinetic energy of
the proton is less than the total kinetic energy of the two alpha
particles. (One only considers the kinetic energy of the proton because
the Lithium atom is considered at rest, and hence has zero kinetic
energy.) Cockcroft and Walton were able to measure the kinetic energies
of the incident proton and the out-going alpha particles very
precisely. They found that the decrease in rest-mass corresponds to the
increase in kinetic energy according to Einstein's famous equation
*E* = *mc*^{2} (to an accuracy of better
than 1%). Hence, the total mass *and* energy of the entire
system is conserved.

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