In
physics,
mass–energy
equivalence is the concept that the
mass of a body is a measure of its
energy content. The mass of a body as measured on a
scale is always equal to the total energy inside, divided by a
constant
c^{2} that changes the units
appropriately:
 :E = mc^2 \,\!
where
E is energy,
m is
mass,
and
c is the
speed of light in a vacuum, which is .
Mass–energy equivalence was proposed in
Albert Einstein's 1905 paper, "Does the
inertia of a body depend upon its energycontent?", one of his
Annus Mirabilis ("Miraculous
Year") Papers. Einstein was not the first to propose a mass–energy
relationship, and various similar formulas appeared before
Einstein's theory with incorrect numerical coefficients and an
incomplete interpretation. Einstein was the first to propose the
simple formula and the first to interpret it correctly: as a
general principle which follows from the
relativistic symmetries of
space and time.
In the formula,
c^{2} is the
conversion factor required to convert from
units of mass to
units of energy. The formula does
not depend on a specific
system
of units. Using the
International System of Units,
joules are used to measure energy,
kilograms for mass,
meters per second for speed. Note that
1 joule equals 1
kg·
m^{2}/
s^{2}. In
unitspecific terms,
E (in
joules) =
m (in
kilograms) multiplied by (
299,792,458 m/s)
^{2}. In
natural
units, the speed of light is set equal to 1, and the formula
becomes an identity.
Conservation of mass and energy
The concept of mass–energy equivalence unites the concepts of
conservation of mass and
conservation of energy,
allowing
rest mass to be converted to
other forms of energy, like
kinetic
energy, heat, or light. Kinetic energy or light can also be
converted to particles which have
mass. The
total amount of mass–energy in a closed system remains constant
because energy cannot be created or destroyed and, in all of its
forms, trapped energy has mass. According to the theory of
relativity, mass and energy as commonly understood are two names
for the same thing, and one is not changed to the other. Rather,
neither one appears without the other. When energy changes type and
leaves a system, it takes its mass with it.
Fastmoving objects and systems of objects
When an object is pushed in the direction of motion, it gains
momentum and energy, but when the object is already traveling near
the
speed of light, it cannot move
much faster, no matter how much energy it absorbs. Its momentum and
energy continue to increase without bounds, whereas its speed
approaches a constant value—the speed of light. This implies that
in relativity the momentum of an object cannot be not a constant
times the velocity, nor can the
kinetic energy
be a constant times the square of the velocity.
The
relativistic mass is defined
as the ratio of the momentum of an object to its velocity, and it
depends on the motion of the object. If the object is moving
slowly, the relativistic mass is nearly equal to the
rest mass and both are nearly equal to the usual
Newtonian mass. If the object is moving quickly, the relativistic
mass is greater than the rest mass by an amount equal to the mass
associated with the
kinetic energy of
the object. As the object approaches the speed of light, the
relativistic mass becomes infinite, because the
kinetic energy becomes infinite and this
energy is associated with mass.
The relativistic mass is always equal to the total energy (rest
energy plus kinetic energy) divided by
c^{2}.
Because the relativistic mass is exactly proportional to the
energy, relativistic mass and relativistic energy are nearly
synonyms; the only difference between them is the
units. If length and time are measured
in
natural units, the speed of light
is equal to 1, and even this difference disappears. Then mass and
energy have the same units and are always equal, so it is redundant
to speak about relativistic mass, because it is just another name
for the energy. This is why physicists usually reserve the useful
short word "mass" to mean restmass.
For things made up of many parts, like a
nucleus,
planet, or
star, the relativistic mass is the sum of the
relativistic masses (or energies) of the parts, because energies
are additive in closed systems. This is not true in systems which
are open, however, if energy is subtracted. For example, if a
system is bound by attractive forces and the work they do in
attraction is removed from the system, mass will be lost. Such work
is a form of energy which itself has mass, and thus mass is removed
from the system, as it is bound. For example, the mass of an atomic
nucleus is less than the total mass of the protons and neutrons
that make it up, but this is only true after the energy (work) of
binding has been removed in the form of a gamma ray (which in this
system, carries away the mass of binding). This mass decrease is
also equivalent to the energy required to break up the nucleus into
individual protons and neutrons (in this case, work and mass would
need to be supplied). Similarly, the mass of the solar system is
slightly less than the masses of sun and planets
individually.
The relativistic mass of a moving object is bigger than the
relativistic mass of an object that isn't moving, because a moving
object has extra kinetic energy. The
rest mass of an
object is defined as the mass of an object when it is at rest, so
that the rest mass is always the same, independent of the motion of
the observer: it is the same in all
inertial frames.
For a system of particles going off in different directions, the
invariant mass of the system is the
analog of the rest mass, and is the same for all observers. It is
defined as the total energy (divided by
c^{2}) in
the
center of mass frame (where
by definition, the system total momentum is zero). A simple example
of an object with moving parts but zero total momentum, is a
container of gas. In this case, the mass of the container is given
by its total energy (including the kinetic energy of the gas
molecules), since the system total energy and invariant mass are
the same in the reference frame where the momentum is zero, and
this reference frame is also the only frame in which the object can
be weighed.
Meanings of the mass–energy equivalence formula
Mass–energy equivalence states that any object has a certain
energy, even when it isn't moving. In
Newtonian mechanics, a motionless body
has no
kinetic energy, and it may or
may not have other amounts of internal stored energy, like
chemical energy or
thermal energy, in addition to any
potential energy it may have from its
position in a
field of force. In
Newtonian mechanics, all of these energies are much smaller than
the mass of the object times the speed of light squared.
In relativity, all of the energy that moves along with an object
adds up to the total mass of the body, which measures how much it
resists deflection. Each potential and kinetic energy makes a
proportional contribution to the mass. Even a single
photon traveling in empty space has a relativistic
mass, which is its energy divided by
c^{2}. If a
box of ideal mirrors contains light, the mass of the box is
increased by the energy of the light, since the total energy of the
box is its mass.
In relativity, removing energy is removing mass, and the formula
m =
E/
c^{2} tells you
how much mass is lost when energy is removed. In a chemical or
nuclear reaction, the mass of the atoms that come out is less than
the mass of the atoms that go in, and the difference in mass shows
up as heat and light which has the same relativistic mass as the
difference (and also the same
invariant
mass in the center of mass frame of the system). In this case,
the
E in the formula is the energy released and removed,
and the mass
m is how much the mass goes down. In the same
way, when any kind of energy is added, the increase in the mass is
equal to the added energy divided by
c^{2}. For
example, when water is heated in a
microwave oven, the oven adds about of mass
for every joule of heat added to the water.
An object moves with different speed in different frames, depending
on the motion of the observer, so the kinetic energy in both
Newtonian mechanics and relativity is
frame dependent.
This means that the amount of energy, and therefore the amount of
relativistic mass, that an object is measured to have depends on
the observer. The
rest mass is defined as the mass that an
object has when it isn't moving. It also applies to the invariant
mass of systems when the system as a whole isn't "moving" (has no
net momentum). The rest and invariant masses are the smallest
possible value of the mass of the object or system. They are
conserved quantities, so long as the system is closed.
The rest mass is almost never additive: the rest mass of an object
is not the sum of the rest masses of its parts. The rest mass of an
object is the total energy of all the parts, including kinetic
energy, as measured by an observer that sees the center of the mass
of the object to be standing still. The rest mass adds up only if
the parts are standing still and don't attract or repel, so that
they don't have any extra kinetic or potential energy. The other
possibility is that they have a positive kinetic energy and a
negative potential energy that exactly cancels.
The difference between the rest mass of a bound system and of the
unbound parts is exactly proportional to the
binding energy of the system. A water
molecule weighs a little less than two free hydrogen atoms and an
oxygen atom; the minuscule mass difference is the energy that is
needed to split the molecule into three individual atoms (divided
by c
^{2}), and which was given off as heat when the
molecule formed (this heat had mass). Likewise, a stick of dynamite
weighs a little bit more than the fragments after the explosion, so
long as the fragments are cooled and the heat removed; the mass
difference is the energy/heat that is released when the dynamite
explodes (when it escapes, the mass associated with it escapes, but
total mass is conserved). The change in mass only happens when the
system is open, and the energy escapes. If a stick of dynamite is
blown up in a hermetically sealed chamber, the mass of the chamber
and fragments, the heat, sound, and light would still be equal to
the original mass of the chamber and dynamite. This would in theory
also happen, even with a nuclear bomb, if it could be kept in a
chamber which did not rupture.
Massless particles
In relativity, all energy moving along with a body adds up to the
total energy, which is exactly proportional to the relativistic
mass. Even a single
photon,
graviton, or
neutrino
traveling in empty space has a relativistic mass, which is its
energy divided by
c^{2}. But the rest mass of a
photon is slightly subtler to define in terms of physical
measurements, because a photon is always moving at the speed of
light—it is never at rest.
If you run away from a photon, having it chase you, by moving fast
enough in the same direction, when the photon catches up to you the
photon would be seen as having less energy, and even less the
faster you were traveling when it caught you. As you approach the
speed of light, the photon looks redder and redder, by
doppler shift (although for a photon the
Doppler shift is relativistic), and the energy of a very
longwavelength photon approaches zero. This is why a photon is
massless; this means that the rest mass of a photon is
zero. A massless particle in relativity is the limit of a particle
with very small mass, but which is moving so close to the speed of
light, so that it has a nonnegligible total energy.
Two photons moving in different directions can't both be made to
have arbitrarily small total energy by changing frames, by chasing
them. The reason is that in a twophoton system, the energy of one
photon is decreased by chasing it, but the energy of the other will
increase. Two photons not moving in the same direction have an
inertial frame where the combined
energy is smallest, but not zero. This is called the
center of mass frame or the
center of momentum frame; these terms are
almost synonyms (the center of mass frame is the special case of a
center of momentum frame where the center of mass is put at the
origin). If you move at the same direction and speed as the center
of mass of the two photons, the total momentum of the photons is
zero. Their combined energy E in this frame gives them, as a
system, a mass equal to the energy divided by
c^{2}. This mass is called the
invariant mass of the pair of photons
together. It is the smallest mass and energy the system may be seen
to have by any observer.
If the photons formed by the collision of a particle and an
antiparticle, the invariant mass is the same as the total energy of
the particle and antiparticle (their rest energy plus the kinetic
energy), in the center of mass frame, where they will automatically
be moving in equal and opposite directions (since they have equal
momentum in this frame). If the photons are formed by the
disintegration of a
single particle with a welldefined
rest mass, like the neutral
pion, the invariant
mass of the photons is equal to rest mass of the pion. In this
case, the center of mass frame for the pion is just the frame where
the pion is at rest, and the center of mass doesn't change. After
the two photons are formed, their center of mass is still moving
the same way the pion did, and their total energy in this frame
adds up to the mass energy of the pion. So the invariant mass of
the photons is equal to the pion's rest energy. So by calculating
the
invariant mass of pairs of
photons in a particle detector, pairs can be identified that were
probably produced by pion disintegration.
Are photons massless?
The photon is currently believed to be strictly massless, but this
is an experimental question. If the photon is not a strictly
massless particle, it would not move at the exact speed of light.
Its speed would be lower and depend on its frequency. Relativity
would be unaffected by this; the "speed of light",
c,
would then not be the actual speed at which light moves, but a
constant of nature which is the maximum speed that any object could
theoretically attain. It would still be the speed of
gravitons, but it would not be the speed of
photons.
A massive photon would have other effects as well.
Coulomb's law would be modified and the
electromagnetic field would have an extra physical degree of
freedom. These effects yield more sensitive experimental probes of
the photon mass than the frequency dependence of the speed of
light. If Coulomb's law is not exactly valid, then that would cause
the presence of an
electric field
inside a hollow conductor when it is subjected to an external
electric field. This thus allows one to test Coulomb's law to very
high precision. . A null result of such an experiment has set a
limit of m\lesssim 10^{14} eV..
Sharper upper limits have been obtained in experiments designed to
detect effects caused by the Galactic
vector potential. Although the galactic
vector potential is very large because the galactic
magnetic field exists on very long length
scales, only the magnetic field is observable if the photon is
massless. In case of a massive photon, the mass term \frac{1}{2}
m^2 A_{\mu}A^{\mu} would affect the galactic plasma. The fact that
no such effects are seen implies an upper bound on the photon mass
of m 3\times 10^{27} eV. The galactic vector potential can also be
probed directly by measuring the torque exerted on a magnetized
ring.. Such methods were used to obtain the sharper upper limit of
10^{18}eV. given by the Particle Data Group
These sharp limits from the nonobservation of the effects caused
by the galactic vector potential have been shown to be model
dependent. If the photon mass is generated via the Higgs mechanism
then the upper limit of m\lesssim 10^{14} eV from the test of
Coulomb's law is valid.
Consequences for nuclear physics
Max Planck pointed out that the
mass–energy equivalence formula implied that bound systems would
have a mass less than the sum of their constituents, once the
binding energy had been allowed to escape. However, Planck was
thinking about chemical reactions, where the binding energy is too
small to measure. Einstein suggested that radioactive materials
such as radium would provide a test of the theory, but even though
a large amount of energy is released per atom, only a small
fraction of the atoms decay.
Once the nucleus was discovered, experimenters realized that the
very high binding energies of the atomic nuclei should allow
calculation of their binding energies from mass differences. But it
was not until the discovery of the
neutron
in 1932, and the measurement of its mass, that this calculation
could actually be performed (see
nuclear binding energy for example
calculation). A little while later, the first
transmutation reactions (such as
^{7}Li +
p → 2
^{4}He) verified Einstein's
formula to an accuracy of ±0.5%.
The mass–energy equivalence formula was used in the development of
the
atomic bomb. By measuring the mass
of different
atomic nuclei and
subtracting from that number the total mass of the
protons and
neutrons as they
would weigh separately, one gets the exact
binding energy available in an
atomic nucleus. This is used to calculate the
energy released in any
nuclear
reaction, as the difference in the total mass of the nuclei
that enter and exit the reaction.
In
quantum chromodynamics,
the modern theory of the nuclear force, most of the mass of the
proton and the
neutron
is explained by special relativity. The mass of the proton is about
eighty times greater than the sum of the rest masses of the
quarks that make it up, while the
gluons have zero rest mass. The extra energy of the
quarks and
gluons in a
region within a proton, as compared to the energy of the quarks and
gluons in the
QCD vacuum, accounts for
over 98% of the mass.
The internal dynamics of the proton are complicated, because they
are determined by the quarks exchanging gluons, and interacting
with various vacuum condensates.
Lattice
QCD provides a way of calculating the mass of the proton
directly from the theory to any accuracy, in principle. The most
recent calculations claim that the mass is determined to better
than 4% accuracy, arguably accurate to 1% (see Figure S5 in Dürr
et al.). These claims are still controversial, because the
calculations cannot yet be done with quarks as light as they are in
the real world. This means that the predictions are found by a
process of
extrapolation, which can
introduce systematic errors. It is hard to tell whether these
errors are controlled properly, because the quantities that are
compared to experiment are the masses of the
hadrons, which are known in advance.
These recent calculations are performed by massive supercomputers,
and, as noted by Boffi and Pasquini: “a detailed description of the
nucleon structure is still missing because ... longdistance
behavior requires a nonperturbative and/or numerical
treatment..."More conceptual approaches to the structure of the
proton are: the
topological soliton
approach originally due to
Tony Skyrme
and the more accurate
AdS/QCD approach which
extends it to include a
string theory
of gluons, various QCD inspired models like the
bag model and the
constituent quark model, which were
popular in the 1980s, and the SVZ sum rules which allow for rough
approximate mass calculations. These methods don't have the same
accuracy as the more brute force lattice QCD methods, at least not
yet.
But all these methods are consistent with
special relativity, and so calculate the
mass of the proton from its total energy.
Practical examples
Einstein used the
CGS system of units
(centimeters, grams, seconds, dynes, and ergs), but the formula is
independent of the system of units. In
natural units, the speed of light is defined
to equal 1, and the formula expresses an identity:
E =
m. In the
SI system (expressing the
ratio
E /
m in
joules per kilogram using the value of
c in
meters per second):
 E / m =
c^{2} = (299,792,458 m/s)^{2} =
89,875,517,873,681,764 J/kg
(≈9.0 × 10^{16} joules per kilogram)
So one
gram of mass is equivalent to the
following amounts of energy:
 89.9 terajoules
 24.9 million kilowatthours
(≈25 GW·h)
 21.5 billion kilocalories
(≈21 Tcal)^{ }Conversions
used: 1956 International (Steam) Table (IT) values where one
calorie ≡ 4.1868 J and one BTU
≡ 1055.05585262 J. Weapons designers’ conversion value of
one gram TNT ≡ 1000 calories used.^{ }
 21.5 kilotons of TNTequivalent energy
(≈21 kt)^{ }
 85.2 billion BTUs
Any time energy is generated, the process can be evaluated from an
E =
mc^{2} perspective.
For
instance, the "Gadget"style bomb
used in the Trinity
test and the bombing of
Nagasaki had an explosive yield equivalent to 21 kt of
TNT. About 1 kg of the approximately 6.15 kg of
plutonium in each of these bombs fissioned into lighter elements
totaling almost exactly one gram less, after cooling [The heat,
light, and electromagnetic radiation released in this explosion
carried the missing one gram of mass.] This occurs because nuclear
binding energy is released whenever
elements with more than 62 nucleons fission.
Another example is
hydroelectric
generation.
The electrical energy produced by Grand Coulee
Dam’s turbines every
3.7 hours represents one gram of mass. This mass passes
to the electrical devices which are powered by the generators (such
as lights in cities), where it appears as a gram of heat and light.
Turbine designers look at their equations in terms of pressure,
torque, and RPM. However, Einstein’s equations show that all energy
has mass, and thus the electrical energy produced by a dam's
generators, and the heat and light which result from it, all retain
their mass, which is equivalent to the energy. The potential
energy—and equivalent mass—represented by the waters of the
Columbia River as it descends to the
Pacific Ocean would be converted to heat due to
viscous friction and the
turbulence of white water rapids and waterfalls
were it not for the dam and its generators. This heat would remain
as mass on site at the water, were it not for the equipment which
converted some of this potential and kinetic energy into electrical
energy, which can be moved from place to place (taking mass with
it).
Whenever energy is added to a system, the system gains mass.
 A spring's mass increases whenever it is put into compression
or tension. Its added mass arises from the added potential energy
stored within it, which is bound in the stretched chemical
(electron) bonds linking the atoms within the spring.
 Raising the temperature of an object (increasing its heat
energy) increases its mass. If the temperature of the
platinum/iridium "international prototype" of the kilogram—the world’s primary mass standard—is
allowed to change by 1°C, its mass will change by 1.5 picograms
(1 pg = 1 × 10^{12} g).Assuming a
90/10 alloy of Pt/Ir by weight, a C_{p} of 25.9
for Pt and 25.1 for Ir, a Ptdominated average
C_{p} of 25.8, 5.134 moles of metal, and
132 J.K^{1} for the prototype. A variation of
±1.5 picograms is of course, much smaller than the actual
uncertainty in the mass of the international prototype, which is
±2 micrograms.
 A spinning ball will weigh more than a ball that is not
spinning.
Note that no net mass or energy is really created or lost in any of
these scenarios. Mass/energy simply moves from one place to
another. These are some examples of the
transfer of energy
and mass in accordance with the
principle of mass–energy
conservation.
Note further that in accordance with Einstein’s Strong Equivalence
Principle (SEP), all forms of mass
and energy produce a
gravitational field in the same way.Earth’s gravitational
selfenergy is 4.6 × 10
^{10} that of Earth’s
total mass, or 2.7 trillion metric tons. Citation:
The
Apache Point Observatory Lunar LaserRanging Operation
(APOLLO), T. W. Murphy, Jr.
et al. University of
Washington, Dept. of Physics (
132 kB PDF, here.). So all radiated and
transmitted energy
retains its mass. Not only does the
matter comprising Earth create gravity, but the gravitational field
itself has mass, and that mass contributes to the field too. This
effect is accounted for in ultraprecise laser ranging to the Moon
as the Earth orbits the Sun when testing Einstein’s
general theory of relativity.
According to
E=
mc^{2}, no
closed
system (any system treated and observed as a whole) ever loses
mass, even when rest mass is converted to energy. This statement is
more than an abstraction based on the principle of equivalence—it
is a realworld effect.
All types of energy contribute to mass, including potential
energies. In relativity, interaction potentials are always due to
local fields, not to direct nonlocal
interactions, because signals can't travel faster than light. The
field energy is stored in field gradients or, in some cases (for
massive fields), where the field has a nonzero value. The mass
associated with the potential energy is the mass–energy of the
field energy. The mass associated with field energy can be
detected, in principle, by gravitational experiments, by checking
how the field attracts other objects gravitationally.
The energy in the gravitational field itself is different. There
are several consistent ways to define the location of the energy in
a gravitational field, all of which agree on the total energy when
space is mostly flat and empty. But because the gravitational field
can be made to vanish locally by choosing a freefalling frame, it
is hard to avoid making the location dependent on the observer's
frame of reference. The gravitational field energy is the familiar
Newtonian gravitational potential energy in the Newtonian
limit.
Efficiency
In nuclear reactions, typically only a small fraction of the total
mass–energy is converted into heat, light, radiation and motion,
into a form which can be used. When an atom fissions, it loses only
about 0.1% of its mass, and in a bomb or reactor not all the atoms
can fission. In a fission based atomic bomb, the
efficiency is only 40%, so
only 40% of the fissionable atoms actually fission, and only 0.04%
of the total mass appears as energy in the end. In nuclear fusion,
more of the mass is released as usable energy, roughly 0.3%. But in
a fusion bomb (see
nuclear weapon
yield), the bomb mass is partly casing and nonreacting
components, so that again only about 0.03% of the total mass is
released as usable energy.
In theory, it should be possible to convert all the mass in matter
into heat and light, but none of the theoretically known methods
are practical. One way to convert all restmass into usable energy
is to annihilate matter with
antimatter.
But
antimatter is rare in our
universe, and must be made first. Making the antimatter
requires more energy than would be released.
Since most of the mass of ordinary objects is in protons and
neutrons, in order to convert all the mass in ordinary matter to
useful energy, the protons and neutrons must be converted to
lighter particles. In the
standard model
of particle physics, the
number of
protons plus neutrons is nearly exactly conserved. Still,
Gerardus 't Hooft showed that
there is a process which will convert protons and neutrons to
antielectrons and neutrinos. This is the weak SU(2)
instanton proposed by Belavin Polyakov Schwarz and
Tyupkin. This process, can in principle convert all the mass of
matter into neutrinos and usable energy, but it is normally
extraordinarily slow. Later it became clear that this process will
happen at a fast rate at very high temperatures, since then
instantonlike configurations will be copiously produced from
thermal fluctuations. The
temperature required is so high that it would only have been
reached shortly after the
big bang.
Many extensions of the standard model contain
magnetic monopoles, and in some models of
grand unification, these
monopoles catalyze proton decay, a process known as the
Callan–Rubakov effect. This process would be an efficient
mass–energy conversion at ordinary temperatures, but it requires
making
monopoles and antimonopoles first.
The energy required to produce monopoles is believed to be
enormous, but magnetic charge is conserved, so that the lightest
monopole is stable. All these properties are deduced in theoretical
models—magnetic monopoles have never been observed, nor have they
been produced in any experiment so far.
The third known method of total mass–energy conversion is using
gravity, specifically black holes.
Stephen Hawking theorized that black holes
radiate thermally with no regard to how they are formed. So it is
theoretically possible to throw matter into a black hole and use
the emitted heat to generate power. According to the theory of
Hawking radiation, however, the
black hole used will radiate at a higher rate the smaller it is,
producing usable powers at only small black hole masses, where
usable may for example be something greater than the local
background radiation. It is also worth noting that the ambient
irradiated power would change with the mass of the black hole,
increasing as the mass of the black hole decreases, or decreasing
as the mass increases, at a rate where power is proportional to the
inverse square of the mass. In a "practical" scenario, mass and
energy could be dumped into the black hole to regulate this growth,
or keep its size, and thus power output, near constant.
Background
E=
mc^{2} where
m stands for
rest mass (
invariant mass) m_0, applies most simply to
single particles viewed in an inertial frame where they have no
momentum. But it also applies to ordinary
objects composed of many particles so long as the particles are
moving in different directions so the "net" or total momentum is
zero. The rest mass of the object includes contributions from heat
and sound, chemical binding energies, and trapped radiation.
Familiar examples are a tank of gas, or a hot poker. The kinetic
energy of their particles, the heat motion and radiation,
contribute to their weight on a scale according to
E=
mc^{2}.
The formula is the special case of the
relativistic energy–momentum relationship:
 :E^2  (pc)^2 = (m_0 c^2)^2.\,
This equation gives the rest mass of an object which has an
arbitrary amount of momentum and energy. The interpretation of this
equation is that the rest mass is the relativistic length of the
energy–momentum
fourvector.
If the equation
E=
mc^{2} is used with the
rest mass or
invariant mass of the object, the E given by
the equation will be the
rest energy of
the object, and will change according to the object's internal
energy, heat and sound and chemical binding energies (all of which
must be added or subtracted from the object), but will not change
with the object's overall motion (in the case of systems, the
motion of its center of mass). However, if a system is closed, its
invariant mass does not vary between different inertial observers
(different
inertial frames), and is
also constant, and conserved.
If the equation
E=
mc^{2} is used with the
relativistic mass of the object,
the energy will be the total energy of the object, which is
also conserved so long as no energy is added to or
subtracted from the object, However, like the kinetic energy, this
total energy will depend on the velocity of the object, and is
different in different inertial frames. Thus, this quantity is not
invariant between different inertial observers, even though it is
constant over time for
any single observer. As in the case
of
rest energy, these relationships for
total energy are also true for
systems of objects, so long
as the system is closed.
 Mass–Velocity Relationship
In developing
special relativity,
Einstein found that the kinetic energy of a moving body is
 :K.E. = \frac{m_0 c^2}\sqrt{1\frac{v^2}{c^2}}  m_0 c^2,
with v the
velocity, and m_0 the rest
mass.
He included the second term on the right to make sure that for
small velocities, the energy would be the same as in classical
mechanics:
 :K.E. = \frac{1}{2}m_0 v^2 + ...
Without this second term, there would be an additional contribution
in the energy when the particle is not moving.
Einstein found that the total momentum of a moving particle
is:
 :P = \frac{m_0 v}\sqrt{1\frac{v^2}{c^2}}.
and it is this quantity which is conserved in collisions. The ratio
of the momentum to the velocity is the
relativistic mass, m.
 :m = \frac{m_0}{\sqrt{1\frac{v^2}{c^2}}}
And the relativistic mass and the relativistic kinetic energy are
related by the formula:
 :K.E. = m c^2  m_0 c^2. \,
Einstein wanted to omit the unnatural second term on the righthand
side, whose only purpose is to make the energy at rest zero, and to
declare that the particle has a total energy which obeys:
 : E = m c^2 \,
which is a sum of the rest energy m_0 c^2 and the kinetic energy.
This total energy is mathematically more elegant, and fits better
with the momentum in relativity. But to come to this conclusion,
Einstein needed to think carefully about collisions. This
expression for the energy implied that matter at rest has a huge
amount of energy, and it is not clear whether this energy is
physically real, or just a mathematical artifact with no physical
meaning.
In a collision process where all the restmasses are the same at
the beginning as at the end, either expression for the energy is
conserved. The two expressions only differ by a constant which is
the same at the beginning and at the end of the collision. Still,
by analyzing the situation where particles are thrown off a heavy
central particle, it is easy to see that the inertia of the central
particle is reduced by the total energy emitted. This allowed
Einstein to conclude that the inertia of a heavy particle is
increased or diminished according to the energy it absorbs or
emits.
Relativistic mass
After Einstein first made his proposal, it became clear that the
word mass can have two different meanings. The rest mass is what
Einstein called
m, but others defined the
relativistic
mass with an explicit index:
 :m_{\mathrm{rel}} = \frac{m_0}{\sqrt{1\frac{v^2}{c^2}}}\,\,
.
This mass is the ratio of momentum to velocity, and it is also the
relativistic energy divided by (it is not Lorentzinvariant, in
contrast to m_0). The equation holds for moving objects. When the
velocity is small, the relativistic mass and the rest mass are
almost exactly the same.
 E=mc^{2} either means
E=m_{0}c^{2} for an
object at rest, or
E=m_{rel}c^{2} when the
object is moving.
Also Einstein (following
Hendrik
Lorentz and
Max Abraham) used
velocity—and directiondependent mass concepts (
longitudinal and
transverse mass) in his 1905 electrodynamics paper and in
another paper in 1906.
However, in his first paper on
E=
mc^{2}
(1905) he treated
m as what would now be called the
rest mass. Some claim that (in later years) he did not
like the idea of "relativistic mass." When modern physicists
say "mass", they are usually talking about rest mass, since if they
meant "relativistic mass", they would just say "energy".
Considerable debate has ensued over the use of the concept
"relativistic mass" and the connection of "mass" in relativity to
"mass" in Newtonian dynamics. For example, one view is that only
rest mass is a viable concept and is a property of the particle;
while relativistic mass is a conglomeration of particle properties
and properties of spacetime. A perspective that avoids this debate,
due to Kjell Vøyenli, is that the Newtonian concept of mass as a
particle property and the relativistic concept of mass have to be
viewed as embedded in their own theories and as having no precise
connection.
Lowspeed expansion
We can rewrite the expression
E =
γm_{0}c^{2} as a
Taylor series:
 E = m_0 c^2 \left[1 + \frac{1}{2} \left(\frac{v}{c}\right)^2 +
\frac{3}{8} \left(\frac{v}{c}\right)^4 + \frac{5}{16}
\left(\frac{v}{c}\right)^6 + \ldots \right].
For speeds much smaller than the speed of light, higherorder terms
in this expression get smaller and smaller because
v/
c is small. For low speeds we can ignore all
but the first two terms:
 E \approx m_0 c^2 + \frac{1}{2} m_0 v^2 .
The total energy is a sum of the rest energy and the
Newtonian kinetic energy.
The classical energy equation ignores both the
m_{0}c^{2} part, and the
highspeed corrections. This is appropriate, because all the
highorder corrections are small. Since only
changes in
energy affect the behavior of objects, whether we include the
m_{0}c^{2} part makes no
difference, since it is constant. For the same reason, it is
possible to subtract the rest energy from the total energy in
relativity. By considering the emission of energy in different
frames, Einstein could show that the rest energy has a real
physical meaning.
The higherorder terms are extra correction to Newtonian mechanics
which become important at higher speeds. The Newtonian equation is
only a lowspeed approximation, but an extraordinarily good one.
All of the calculations used in putting astronauts on the moon, for
example, could have been done using Newton's equations without any
of the higherorder corrections.
History
While Einstein was the first to have correctly deduced the
mass–energy equivalence formula, he was not the first to have
related energy with mass. But nearly all previous authors thought
that the energy which contributes to mass comes only from
electromagnetic fields.
Newton: Matter and light
In 1717
Isaac Newton speculated that
light particles and matter particles were interconvertible in
"Query 30" of the
Opticks, where he
asks:
Since Newton did not understand light as the motion of a field, he
was not speculating about the conversion of motion into matter.
Since he did not know about energy, he could not have understood
that converting light to matter is turning work into mass.
Electromagnetic rest mass
There were many attempts in the 19th and the beginning of the 20th
century—like those of
J. J. Thomson
(1881),
Oliver Heaviside (1888),
and
George Frederick
Charles Searle (1897)—to understand how the mass of a charged
object depends on the electrostatic field. Because the
electromagnetic field carries part of the momentum of a moving
charge, it was also suspected that the mass of an electron would
vary with velocity near the speed of light. Searle calculated that
it is impossible for a charged object to supersede the velocity of
light because this would require an infinite amount of
energy.
Following Thomson and Searle (1896),
Wilhelm Wien (1900),
Max
Abraham (1902), and
Hendrik
Lorentz (1904) argued that this relation applies to the
complete mass of bodies, because all inertial mass is
electromagnetic in origin. The formula of the mass–energyrelation
given by them was m=(4/3)E/c^2. Wien went on by stating, that if it
is assumed that gravitation is an electromagnetic effect too, then
there has to be a strict proportionality between (electromagnetic)
inertial mass and (electromagnetic) gravitational mass. This
interpretation is in the now discredited
electromagnetic worldview, and the
formulas that they discovered always included a factor of 4/3 in
the proportionality. For example, the formulas given by Lorentz in
1904 for the prerelativistic longitudinal and transverse masses
were (in modern notation):

m_{L}=\frac{m_{0}}{\left(\sqrt{1\frac{v^{2}}{c^{2}}}\right)^{3}},\quad
m_{T}=\frac{m_{0}}{\sqrt{1\frac{v^{2}}{c^{2}}}} , where
m_{0}=\frac{4}{3}\frac{E_{em}}{c^{2}}
In July 1905 (published 1906), nearly at the same time when
Einstein found the simple relation from relativity, Poincaré was
able to explain the reason that the electromagnetic mass
calculations always had a factor of 4/3. In order for a particle
consisting of positive or negative charge to be stable, there must
be some sort of attractive force of nonelectrical nature which
keeps it together. If the mass–energy of this force field is
included in a way which is consistent with relativity theory, the
attractive contribution adds an amount (1/3)E/c^2 to the energy of
the bodies, and this explains the discrepancy between the pure
electromagnetic theory and relativity.
Inertia of energy and radiation
James Clerk Maxwell (1874) and
Adolfo Bartoli (1876) found out that
the existence of tensions in the ether like the
radiation pressure follows from the
electromagnetic theory. However, Lorentz (1895) recognized that
this led to a conflict between the
action/reaction principle
and
Lorentz's ether
theory.
 Poincaré
In 1900
Henri Poincaré studied
this conflict and tried to determine whether the
center of gravity still moves with a
uniform velocity when electromagnetic fields are included. He
noticed that the action/reaction principle does not hold for matter
alone, but that the electromagnetic field has its own momentum. The
electromagnetic field energy behaves like a fictitious
fluid ("fluide fictif") with a mass density of E/c^2
(in other words
m =
E/
c^{2}). If the
center of mass frame is defined
by both the mass of matter
and the mass of the fictitious
fluid, and if the fictitious fluid is indestructible—it is neither
created or destroyed—then the motion of the center of mass frame
remains uniform. But electromagnetic energy can be converted into
other forms of energy. So Poincaré assumed that there exists a
nonelectric energy fluid at each point of space, into which
electromagnetic energy can be transformed and which also carries a
mass proportional to the energy. In this way, the motion of the
center of mass remains uniform. Poincaré said that one should not
be too surprised by these assumptions, since they are only
mathematical fictions.
But Poincaré's resolution led to a paradox when changing frames: if
a Hertzian oscillator radiates in a certain direction, it will
suffer a
recoil from the inertia of the
fictitious fluid. In the framework of
Lorentz ether theory Poincaré performed
a
Lorentz boost to the frame of the
moving source. He noted that energy conservation holds in both
frames, but that the law of conservation of momentum is violated.
This would allow a
perpetuum
mobile, a notion which he abhorred. The laws of nature would
have to be different in the frames of reference, and the relativity
principle would not hold. Poincaré's paradox was resolved by
Einstein's insight that a body losing energy as radiation or heat
was losing a mass of the amount m=E/c^2. The Hertzian oscillator
loses mass in the emission process, and momentum is conserved in
any frame. Einstein noted in 1906 that Poincaré's solution to the
center of mass problem and his own were mathematically equivalent
(see below).
Poincaré came back to this topic in "Science and Hypothesis" (1902)
and "
The Value of Science"
(1905). This time he rejected the possibility that energy carries
mass: "... [the recoil] is contrary to the principle of Newton
since our projectile here has no mass, it is not matter, it is
energy". He also discussed two other unexplained effects: (1)
nonconservation of mass implied by Lorentz's variable mass \gamma
m, Abraham's theory of variable mass and
Kaufmann's experiments on the
mass of fast moving electrons and (2) the nonconservation of
energy in the radium experiments of
Madame
Curie.
 Abraham and Hasenöhrl
Following Poincaré,
Max Abraham in 1902
introduced the term "electromagnetic momentum" to maintain the
action/reaction principle. Poincaré's result was verified by him,
whereby the field density of momentum per cm
^{3} is E/c^2
and E/c per cm
^{2}.
In 1904,
Friedrich
Hasenöhrl specifically associated inertia with
radiation in a paper, which was according to his own words
very similar to some papers of Abraham. Hasenöhrl suggested that
part of the mass of a body (which he called
apparent mass)
can be thought of as radiation bouncing around a cavity. The
apparent mass of radiation depends on the temperature (because
every heated body emits radiation) and is proportional to its
energy, and he first concluded that m=(8/3)E/c^2. However, in 1905
Hasenöhrl published a summary of a letter, which was written by
Abraham to him. Abraham concluded that Hasenöhrl's formula of the
apparent mass of radiation is not correct, and on the basis of his
definition of electromagnetic momentum and longitudinal
electromagnetic mass Abraham changed it to m=(4/3)E/c^2, the same
value for the electromagnetic mass for a body at rest. Hasenöhrl
recalculated his own derivation and verified Abraham's result. He
also noticed the similarity between the apparent mass and the
electromagnetic mass. However, Hasenöhrl stated that this
energy–apparentmass relation
only holds as long a body
radiates, i.e. if the temperature of a body is greater than 0
K.
However, Hasenöhrl did not include the pressure of the radiation on
the cavity shell. If he had included the shell pressure and inertia
as it would be included in the theory of relativity, the factor
would have been equal to 1 or m=E/c^2. This calculation assumes
that the shell properties are consistent with relativity, otherwise
the mechanical properties of the shell including the mass and
tension would not have the same transformation laws as those for
the radiation.
Nobel Prizewinner and
Hitler advisor
Philipp Lenard claimed that the mass–energy
equivalence formula needed to be credited to Hasenöhrl to make it
an
Aryan creation.
Einstein: Mass–energy equivalence
Albert Einstein did not formulate
exactly the formula in his 1905
Annus Mirabilis paper "Does the
Inertia of a Body Depend Upon Its Energy Content?"; rather, the
paper states that if a body gives off the energy
L in the
form of radiation, its mass diminishes by
L/
c^{2}. (Here, "radiation" means
electromagnetic radiation,
or light, and mass means the ordinary Newtonian mass of a
slowmoving object.) This formulation relates only a change
Δ
m in mass to a change
L in energy without
requiring the absolute relationship.
Objects with zero mass presumably have zero energy, so the
extension that all mass is proportional to energy is obvious from
this result. In 1905, even the hypothesis that changes in energy
are accompanied by changes in mass was untested. Not until the
discovery of the first type of antimatter (the
positron in 1932) was it found that all of the mass
of pairs of resting particles could be converted to
radiation.
 First correct derivation (1905)
Einstein considered a body at rest with mass
M. If the
body is examined in a frame moving with nonrelativistic velocity
v, it is no longer at rest and in the moving frame it has
momentum
P =
Mv.
Einstein supposed the body emits two pulses of light to the left
and to the right, each carrying an equal amount of energy
E/2. In its rest frame, the object remains at rest after
the emission since the two beams are equal in strength and carry
opposite momentum.
But if the same process is considered in a frame moving with
velocity
v to the left, the pulse moving to the left will
be
redshifted while the pulse moving to the
right will be
blue shifted. The blue
light carries more momentum than the red light, so that the
momentum of the light in the moving frame is not balanced: the
light is carrying some net momentum to the right.
The object hasn't changed its velocity before or after the
emission. Yet in this frame it has lost some rightmomentum to the
light. The only way it could have lost momentum is by losing mass.
This also solves Poincaré's radiation paradox, discussed
above.
The velocity is small, so the right moving light is blueshifted by
an amount equal to the nonrelativistic
Doppler shift factor . The momentum of the
light is its energy divided by
c, and it is increased by a
factor of
v/
c. So the right moving light is
carrying an extra momentum \Delta P given by:
\Delta P = {v \over c}{E \over 2c}.\,
The leftmoving light carries a little less momentum, by the same
amount \Delta P. So the total rightmomentum in the light is twice
\Delta P. This is the rightmomentum that the object lost.
2\Delta P = v {E\over c^2}.\,
The momentum of the object in the moving frame after the emission
is reduced by this amount:
P' = Mv  2\Delta P = \left(M  {E\over c^2}\right)v.\,
So the change in the object's mass is equal to the total energy
lost divided by c^2. Since any emission of energy can be carried
out by a two step process, where first the energy is emitted as
light and then the light is converted to some other form of energy,
any emission of energy is accompanied by a loss of mass. Similarly,
by considering absorption, a gain in energy is accompanied by a
gain in mass. Einstein concludes that all the mass of a body is a
measure of its energy content.
 1906—Relativistic centerofmass theorem
Like Poincaré, Einstein concluded in 1906 that the inertia of
electromagnetic energy is a necessary condition for the
centerofmass theorem to hold. On this occasion, Einstein referred
to Poincaré's 1900 paper and wrote:
In Einstein's more physical, as opposed to formal or mathematical,
point of view, there was no need for fictitious masses. He could
avoid the
perpetuum mobile
problem, because on the basis of the mass–energy equivalence he
could show that the transport of inertia which accompanies the
emission and absorption of radiation solves the problem. Poincaré's
rejection of the principle of action–reaction can be avoided
through Einstein's E=mc^2, because mass conservation appears as a
special case of the
energy
conservation law.
Others
During the nineteenth century there were several speculative
attempts to show that mass and energy were proportional in various
discredited ether theories. In particular, the writings of
Samuel Tolver Preston,Bjerknes:
S. Tolver Preston's Explosive Idea E =
mc^{2}. and a 1903 paper by
Olinto De Pretto, presented a mass–energy
relation. De Pretto's paper received recent press coverage when
Umberto Bartocci discovered that
there were only
three degrees
of separation linking De Pretto to Einstein, leading Bartocci
to conclude that Einstein was probably aware of De Pretto's
work.
Preston and De Pretto, following
Le Sage, imagined that the
universe was filled with an ether of tiny particles which are
always moving at speed
c. Each of these particles have a
kinetic energy of
mc^{2} up to a small numerical
factor. The nonrelativistic kinetic energy formula did not always
include the traditional factor of 1/2, since
Leibniz introduced kinetic energy without
it, and the 1/2 is largely conventional in prerelativistic physics.
By assuming that every particle has a mass which is the sum of the
masses of the ether particles, the authors would conclude that all
matter contains an amount of kinetic energy either given by
E =
mc^{2} or
2
E =
mc^{2} depending on the
convention. A particle ether was usually considered unacceptably
speculative science at the time, and since these authors didn't
formulate relativity, their reasoning is completely different from
that of Einstein, who used relativity to change frames.
Independently,
Gustave Le Bon in 1905
speculated that atoms could release large amounts of latent energy,
reasoning from an allencompassing qualitative philosophy of
physics.Bizouard:
Poincaré E = mc^{2} l’équation de Poincaré,
Einstein et Planck.
Radioactivity and nuclear energy
It was quickly noted after the discovery of
radioactivity in 1897, that the total energy
due to radioactive processes is about one
million times
greater than that involved in any known molecular change. However,
it raised the question where this energy is coming from. After
eliminating the idea of absorption and emission of some sort of
Lesagian ether
particles, the existence of a huge amount of latent energy,
stored within matter, was proposed by
Ernest Rutherford and
Frederick Soddy in 1903. Rutherford also
suggested that this internal energy is stored within normal matter
as well. He went on to speculate in 1904:
Einstein mentions in his 1905 paper that mass–energy equivalence
might perhaps be tested with radioactive decay, which releases
enough energy (the quantitative amount known roughly even by 1905)
to possibly be "weighed," when missing. But the idea that great
amounts of usable energy could be liberated from matter, however,
proved initially difficult to substantiate in a practical fashion.
Because it had been used as the basis of much speculation,
Rutherford himself, rejecting his ideas of 1904, was once reported
in the 1930s to have said that: "Anyone who expects a source of
power from the transformation of the atom is talking
moonshine."
This changed dramatically after the demonstration of energy
released from
nuclear fission after
the
atomic
bombings of Hiroshima and Nagasaki in 1945. The equation
E =
mc^{2} became directly
linked in the public eye with the power and peril of
nuclear weapons. The equation was featured as
early as page 2 of the
Smyth Report,
the official 1945 release by the US government on the development
of the atomic bomb, and by 1946 the equation was linked closely
enough with Einstein's work that the cover of
Time magazine prominently featured a
picture of Einstein next to an image of a
mushroom cloud emblazoned with the equation.
Einstein himself had only a minor role in the
Manhattan Project: he had
cosigned a letter to
the U.S. President in 1939 urging funding for research into atomic
energy, warning that an atomic bomb was theoretically possible. The
letter persuaded Roosevelt to devote a significant portion of the
wartime budget to atomic research. Without a security clearance,
Einstein's only scientific contribution was an analysis of an
isotope separation method based
on the rate of molecular diffusion through pores, a nowobsolete
process that was then competitive and contributed a fraction of the
enriched uranium used in the
project.
While
E =
mc^{2} is useful for
understanding the amount of energy released in a fission reaction,
it was not strictly necessary to develop the weapon. As the
physicist and Manhattan Project participant
Robert Serber put it: "Somehow the popular
notion took hold long ago that Einstein's theory of relativity, in
particular his famous equation
E =
mc^{2}, plays some
essential role in the theory of fission. Albert Einstein had a part
in alerting the United States government to the possibility of
building an atomic bomb, but his theory of relativity is not
required in discussing fission. The theory of fission is what
physicists call a nonrelativistic theory, meaning that
relativistic effects are too small to affect the dynamics of the
fission process significantly." However the association between
E =
mc^{2} and nuclear energy
has since stuck, and because of this association, and its simple
expression of the ideas of Albert Einstein himself, it has become
"the world's most famous equation".David Bodanis,
E = mc^{2}: A Biography of the World's Most
Famous Equation (New York: Walker, 2000).
While Serber's view of the strict lack of need to use mass–energy
equivalence in designing the atomic bomb is correct, it does not
take into account the pivotal role which this relationship played
in making the fundamental leap to the initial hypothesis that large
atoms could split into approximately equal halves. In late 1938,
while on the winter walk on which they solved the meaning of Hahn's
experimental results and introduced the idea that would be called
atomic fission,
Lise Meitner and
Otto Robert Frisch made direct
use of Einstein's equation to help them understand the quantitative
energetics of the reaction which overcame the "surface
tensionlike" forces holding the nucleus together, and allowed the
fission fragments to separate to a configuration from which their
charges could force them into an energetic "fission." To do this,
they made use of "packing fraction," or nuclear
binding energy values for elements, which
Meitner had memorized. These, together with use of
E =
mc^{2} allowed them to realize on the spot that
the basic fission process was energetically possible:
...We walked up and down in the snow, I on skis and she
on foot.
...and gradually the idea took shape... explained by
Bohr's idea that the nucleus is like a liquid drop; such a drop
might elongate and divide itself...
We knew there were strong forces that would resist,
..just as surface tension.
But nuclei differed from ordinary drops.
At this point we both sat down on a tree trunk and
started to calculate on scraps of paper.
...the Uranium nucleus might indeed be an unstable
drop, ready to divide itself...
But, ...when the two drops separated they would be
driven apart by electrical repulsion, about 200 MeV in
all.
Fortunately Lise Meitner remembered how to compute the
masses of nuclei... and worked out that the two nuclei formed...
would be lighter by about onefifth the mass of a
proton.
Now whenever mass disappears energy is created,
according to Einstein's formula E = mc2, and... the mass was just
equivalent to 200 MeV; it all fitted!
See also
References
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 See this news report and links
 The 6.2 kg core comprised 0.8% gallium by weight. Also,
about 20% of the Gadget’s yield was due to fast fissioning in its
natural uranium tamper. This resulted in 4.1 moles of Pu
fissioning with 180 MeV per atom actually contributing prompt
kinetic energy to the explosion. Note too that the term
"Gadget"style is used here instead of "Fat Man" because
this general design of bomb was very rapidly upgraded to a more
efficient one requiring only 5 kg of the Pu/gallium
alloy.
 Assuming the dam is generating at its peak capacity of
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 There is usually more than one possible way to define a field
energy, because any field can be made to couple to gravity in many
different ways. By general scaling arguments, the correct
answer at everyday distances, which are long compared to the
quantum gravity scale, should be minimal coupling, which
means that no powers of the curvature tensor appear. Any
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 G. 't Hooft, "Computation of the Effects Due to a Four
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 A. Belavin, A. M. Polyakov, A. Schwarz, Yu. Tyupkin,
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 S.W. Hawking "Black Holes Explosions?" Nature 248:30
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 . English translation.
 .
 See e.g. Lev B.Okun, The concept of Mass, Physics
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http://www.physicstoday.org/vol42/iss6/vol42no6p31_36.pdf
 .
 .
 .
 See also the partial English
translation.
 .
 .
 .
 . See also the English translation.
 .
 MathPages: Who Invented Relativity?
 Christian Schlatter: Philipp
Lenard et la physique aryenne.
 .
 Helge Kragh, "FindeSiècle Physics: A World Picture in Flux"
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 Preston, S. T., Physics of the Ether, E. & F. N. Spon,
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 De Pretto, O. Reale Instituto Veneto Di Scienze, Lettere Ed
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 Umberto Bartocci, Albert Einstein e Olinto De Pretto—La
vera storia della formula più famosa del mondo, editore
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 mathsyear2000.
 .
 John Worrall, review of the book Conceptions of Ether.
Studies in the History of Ether Theories by Cantor and Hodges,
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 http://homepage.mac.com/dtrapp/people/Meitnerium.html A quote
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External links