Theory of relativity

Two-dimensional projection of a three-dimensional analogy of spacetime curvature described in general

relativity

The theory of relativity, or simply relativity in physics, usually encompasses two theories by Albert Einstein:
special relativity and general relativity. (The word relativity can also be used in the context of an older theory,
that of Galilean invariance.)

Concepts introduced by the theories of relativity include:

 Measurements of various quantities are relative to the velocities of observers. In particular, space and
time can dilate.
 Spacetime: space and time should be considered together and in relation to each other.
 The speed of light is nonetheless invariant, the same for all observers.

The term "theory of relativity" was based on the expression "relative theory" (German: Relativtheorie) used by
Max Planck in 1906, who emphasized how the theory uses the principle of relativity. In the discussion section
of the same paper Alfred Bucherer used for the first time the expression "theory of relativity" (German:
Relativitätstheorie).

Scope
The theory of relativity transformed theoretical physics and astronomy during the 20th century. When first
published, relativity superseded a 200-year-old theory of mechanics created primarily by Isaac Newton.[4][5][6]

In the field of physics, relativity catalyzed and added an essential depth of knowledge to the science of
elementary particles and their fundamental interactions, along with ushering in the nuclear age. With relativity,
cosmology and astrophysics predicted extraordinary astronomical phenomena such as neutron stars, black holes,
and gravitational waves.

Two-theory view

The theory of relativity was representative of more than a single new physical theory. There are some
explanations for this. First, special relativity was published in 1905, and the final form of general relativity was
published in 1916.[4]

Second, special relativity applies to elementary particles and their interactions, whereas general relativity
applies to the cosmological and astrophysical realm, including astronomy.

Third, special relativity was accepted in the physics community by 1920. This theory rapidly became a
significant and necessary tool for theorists and experimentalists in the new fields of atomic physics, nuclear
physics, and quantum mechanics. Conversely, general relativity did not appear to be as useful. There appeared
to be little applicability for experimentalists as most applications were for astronomical scales. It seemed limited
to only making minor corrections to predictions of Newtonian gravitation theory.

Finally, the mathematics of general relativity appeared to be very difficult. Consequently, it was thought that a
small number of people in the world, at that time, could fully understand the theory in detail, but this has been
discredited by Richard Feynman. Then, at around 1960 a critical resurgence in interest occurred which has
resulted in making general relativity central to physics and astronomy. New mathematical techniques applicable
to the study of general relativity substantially streamlined calculations. From this, physically discernible
concepts were isolated from the mathematical complexity. Also, the discovery of exotic astronomical
phenomena in which general relativity was crucially relevant, helped to catalyze this resurgence. The
astronomical phenomena included quasars (1963), the 3-kelvin microwave background radiation (1965), pulsars
(1967), and the discovery of the first black hole candidates (1981).[4]

On the theory of relativity

Einstein stated that the theory of relativity belongs to a class of "principle-theories". As such it employs an
analytic method. This means that the elements which comprise this theory are not based on hypothesis but on
empirical discovery. The empirical discovery leads to understanding the general characteristics of natural
processes. Mathematical models are then developed which separate the natural processes into theoretical-
mathematical descriptions. Therefore, by analytical means the necessary conditions that have to be satisfied are
deduced. Separate events must satisfy these conditions. Experience should then match the conclusions.

The special theory of relativity and the general theory of relativity are connected. As stated below, special
theory of relativity applies to all physical phenomena except gravity. The general theory provides the law of
gravitation, and its relation to other forces of nature.

Special relativity

USSR stamp dedicated to Albert Einstein

Special relativity is a theory of the structure of spacetime. It was introduced in Einstein's 1905 paper "On the
Electrodynamics of Moving Bodies" (for the contributions of many other physicists see History of special
relativity). Special relativity is based on two postulates which are contradictory in classical mechanics:

1. The laws of physics are the same for all observers in uniform motion relative to one another (principle of
relativity).
2. The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the
motion of the light source.
The resultant theory copes with experiment better than classical mechanics, e.g. in the Michelson–Morley
experiment that supports postulate 2, but also has many surprising consequences. Some of these are:

 Relativity of simultaneity: Two events, simultaneous for one observer, may not be simultaneous for
another observer if the observers are in relative motion.
 Time dilation: Moving clocks are measured to tick more slowly than an observer's "stationary" clock.
 Relativistic mass
 Length contraction: Objects are measured to be shortened in the direction that they are moving with
respect to the observer.
 Mass–energy equivalence: E = mc2, energy and mass are equivalent and transmutable.
 Maximum speed is finite: No physical object, message or field line can travel faster than the speed of
light in a vacuum.

The defining feature of special relativity is the replacement of the Galilean transformations of classical
mechanics by the Lorentz transformations. (See Maxwell's equations of electromagnetism).

General relativity
General relativity is a theory of gravitation developed by Einstein in the years 1907–1915. The development of
general relativity began with the equivalence principle, under which the states of accelerated motion and being
at rest in a gravitational field (for example when standing on the surface of the Earth) are physically identical.
The upshot of this is that free fall is inertial motion: an object in free fall is falling because that is how objects
move when there is no force being exerted on them, instead of this being due to the force of gravity as is the
case in classical mechanics. This is incompatible with classical mechanics and special relativity because in
those theories inertially moving objects cannot accelerate with respect to each other, but objects in free fall do
so. To resolve this difficulty Einstein first proposed that spacetime is curved. In 1915, he devised the Einstein
field equations which relate the curvature of spacetime with the mass, energy, and momentum within it.

Some of the consequences of general relativity are:

 Clocks run more slowly in deeper gravitational wells.[8] This is called gravitational time dilation.
 Orbits precess in a way unexpected in Newton's theory of gravity. (This has been observed in the orbit
of Mercury and in binary pulsars).
 Rays of light bend in the presence of a gravitational field.
 Rotating masses "drag along" the spacetime around them; a phenomenon termed "frame-dragging".
 The universe is expanding, and the far parts of it are moving away from us faster than the speed of light.

Technically, general relativity is a theory of gravitation whose defining feature is its use of the Einstein field
equations. The solutions of the field equations are metric tensors which define the topology of the spacetime
and how objects move inertially.

Experimental evidence
Tests of special relativity
A diagram of the Michelson–Morley experiment

Like all falsifiable scientific theories, relativity makes predictions that can be tested by experiment. In the case
of special relativity, these include the principle of relativity, the constancy of the speed of light, and time
dilation. The predictions of special relativity have been confirmed in numerous tests since Einstein published
his paper in 1905, but three experiments conducted between 1881 and 1938 were critical to its validation. These
are the Michelson–Morley experiment, the Kennedy–Thorndike experiment, and the Ives–Stilwell experiment.
Einstein derived the Lorentz transformations from first principles in 1905, but these three experiments allow the
transformations to be induced from experimental evidence.

Maxwell's equations – the foundation of classical electromagnetism – describe light as a wave which moves
with a characteristic velocity. The modern view is that light needs no medium of transmission, but Maxwell and
his contemporaries were convinced that light waves were propagated in a medium, analogous to sound
propagating in air, and ripples propagating on the surface of a pond. This hypothetical medium was called the
luminiferous aether, at rest relative to the "fixed stars" and through which the Earth moves. Fresnel's partial
ether dragging hypothesis ruled out the measurement of first-order (v/c) effects, and although observations of
second-order effects (v2/c2) were possible in principle, Maxwell thought they were too small to be detected with
then-current technology.

The Michelson–Morley experiment was designed to detect second order effects of the "aether wind" – the
motion of the aether relative to the earth. Michelson designed an instrument called the Michelson interferometer
to accomplish this. The apparatus was more than accurate enough to detect the expected effects, but he obtained
a null result when the first experiment was conducted in 1881, and again in 1887. Although the failure to detect
an aether wind was a disappointment, the results were accepted by the scientific community. In an attempt to
salvage the aether paradigm, Fitzgerald and Lorentz independently created an ad hoc hypothesis in which the
length of material bodies changes according to their motion through the aether. This was the origin of
Fitzgerald-Lorentz contraction, and their hypothesis had no theoretical basis. The interpretation of the null result
of the Michelson–Morley experiment is that the round-trip travel time for light is isotropic (independent of
direction), but the result alone is not enough to discount the theory of the aether or validate the predictions of
special relativity.
The Kennedy–Thorndike experiment shown with interference fringes.

While the Michelson–Morley experiment showed that the velocity of light is isotropic, it said nothing about
how the magnitude of the velocity changed (if at all) in different inertial frames. The Kennedy–Thorndike
experiment was designed to do that, and was first performed in 1932 by Roy Kennedy and Edward Thorndike.
They obtained a null result, and concluded that "there is no effect ... unless the velocity of the solar system in
space is no more than about half that of the earth in its orbit". That possibility was thought to be too
coincidental to provide an acceptable explanation, so from the null result of their experiment it was concluded
that the round-trip time for light is the same in all inertial reference frames.

The Ives–Stilwell experiment was carried out by Herbert Ives and G.R. Stilwell first in 1938 and with better
accuracy in 1941. It was designed to test the transverse Doppler effect – the redshift of light from a moving
source in a direction perpendicular to its velocity – which had been predicted by Einstein in 1905. The strategy
was to compare observed Doppler shifts with what was predicted by classical theory, and look for a Lorentz
factor correction. Such a correction was observed, from which was concluded that the frequency of a moving
atomic clock is altered according to special relativity.[15][16]

Those classic experiments have been repeated many times with increased precision. Other experiments include,
for instance, relativistic energy and momentum increase at high velocities, time dilation of moving particles,
and modern searches for Lorentz violations.

Tests of general relativity

General relativity has also been confirmed many times, the classic experiments being the perihelion precession
of Mercury's orbit, the deflection of light by the Sun, and the gravitational redshift of light. Other tests
confirmed the equivalence principle and frame dragging.

History
Main articles: History of special relativity and History of general relativity

The history of special relativity consists of many theoretical results and empirical findings obtained by Albert
Michelson, Hendrik Lorentz, Henri Poincaré and others. It culminated in the theory of special relativity
proposed by Albert Einstein, and subsequent work of Max Planck, Hermann Minkowski and others.

General relativity (GR) is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915,
with contributions by many others after 1915.

Currently, it can be said that far from being simply of theoretical scientific interest or requiring experimental
verification, the analysis of relativistic effects on time measurement is an important practical engineering
concern in the operation of the global positioning systems such as GPS, GLONASS, and the forthcoming
Galileo, as well as in the high precision dissemination of time.[21] Instruments ranging from electron
microscopes to particle accelerators simply will not work if relativistic considerations are omitted.

Minority views
Einstein's contemporaries did not all accept his new theories at once. However, the theory of relativity is now
considered as a cornerstone of modern physics, see Criticism of relativity theory.
Although it is widely acknowledged that Einstein was the creator of relativity in its modern understanding,
some believe that others deserve credit for it, see Relativity priority dispute.

Einstein field equations

General relativity

Introduction
Mathematical formulation
Resources  · Tests

Fundamental concepts[show]

Phenomena[show]

Equations[hide]

Linearized gravity
Post-Newtonian formalism
Einstein field equations
Geodesic equation
Mathisson–Papapetrou–Dixon equations
Friedmann equations
ADM formalism
BSSN formalism
Hamilton–Jacobi–Einstein equation

Advanced theories[show]

Solutions[show]

Scientists[show]

Spacetime[show]
  v
 t
 e

The Einstein field equations (EFE) or Einstein's equations are a set of 10 equations in Albert Einstein's
general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime
being curved by matter and energy.[1] First published by Einstein in 1915 as a tensor equation, the EFE equate
local spacetime curvature (expressed by the Einstein tensor) with the local energy and momentum within that
spacetime (expressed by the stress–energy tensor).[3]

Similar to the way that electromagnetic fields are determined using charges and currents via Maxwell's
equations, the EFE are used to determine the spacetime geometry resulting from the presence of mass-energy
and linear momentum, that is, they determine the metric tensor of spacetime for a given arrangement of stress–
energy in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to
be written as a set of non-linear partial differential equations when used in this way. The solutions of the EFE
are the components of the metric tensor. The inertial trajectories of particles and radiation (geodesics) in the
resulting geometry are then calculated using the geodesic equation.

As well as obeying local energy-momentum conservation, the EFE reduce to Newton's law of gravitation where
the gravitational field is weak and velocities are much less than the speed of light.[4]

Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry. Special classes
of exact solutions are most often studied as they model many gravitational phenomena, such as rotating black
holes and the expanding universe. Further simplification is achieved in approximating the actual spacetime as
flat spacetime with a small deviation, leading to the linearised EFE. These equations are used to study
phenomena such as gravitational waves.

Mathematical form
The Einstein field equations (EFE) may be written in the form:

where is the Ricci curvature tensor, the scalar curvature, the metric tensor, is the cosmological
constant, is Newton's gravitational constant, the speed of light in vacuum, and the stress–energy tensor.

The EFE is a tensor equation relating a set of symmetric 4×4 tensors. Each tensor has 10 independent
components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the
metric with four gauge fixing degrees of freedom, which correspond to the freedom to choose a coordinate
system.

Although the Einstein field equations were initially formulated in the context of a four-dimensional theory,
some theorists have explored their consequences in n dimensions. The equations in contexts outside of general
relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when T is
identically zero) define Einstein manifolds.

Despite the simple appearance of the equations they are actually quite complicated. Given a specified
distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations
for the metric tensor , as both the Ricci tensor and scalar curvature depend on the metric in a complicated
nonlinear manner. In fact, when fully written out, the EFE are a system of 10 coupled, nonlinear, hyperbolic-
elliptic partial differential equations.

One can write the EFE in a more compact form by defining the Einstein tensor

which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as

Using geometrized units where G = c = 1, this can be rewritten as

The expression on the left represents the curvature of spacetime as determined by the metric; the expression on
the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of
equations dictating how matter/energy determines the curvature of spacetime.

These equations, together with the geodesic equation,[5] which dictates how freely-falling matter moves through
space-time, form the core of the mathematical formulation of general relativity.

Sign convention

The above form of the EFE is the standard established by Misner, Thorne, and Wheeler. The authors analyzed
all conventions that exist and classified according to the following three signs (S1, S2, S3):

The third sign above is related to the choice of convention for the Ricci tensor:

With these definitions Misner, Thorne, and Wheeler classify themselves as , whereas Weinberg
(1972) is , Peebles (1980) and Efstathiou (1990) are while Peacock (1994), Rindler (1977),
Atwater (1974), Collins Martin & Squires (1989) are .
Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the
sign of the constant on the right side being negative

The sign of the (very small) cosmological term would change in both these versions, if the +−−− metric sign
convention is used rather than the MTW −+++ metric sign convention adopted here.

Equivalent formulations

Taking the trace of both sides of the EFE one gets

which simplifies to

If one adds times this to the EFE, one gets the following equivalent "trace-reversed" form

Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in
some cases (for example, when one is interested in weak-field limit and can replace in the expression on the
right with the Minkowski metric without significant loss of accuracy).

The cosmological constant

Einstein modified his original field equations to include a cosmological term proportional to the metric

The constant is the cosmological constant. Since is constant, the energy conservation law is unaffected.

The cosmological constant term was originally introduced by Einstein to allow for a static universe (i.e., one
that is not expanding or contracting). This effort was unsuccessful for two reasons: the static universe described
by this theory was unstable, and observations of distant galaxies by Hubble a decade later confirmed that our
universe is, in fact, not static but expanding. So was abandoned, with Einstein calling it the "biggest blunder
[he] ever made".[6] For many years the cosmological constant was almost universally considered to be 0.

Despite Einstein's misguided motivation for introducing the cosmological constant term, there is nothing
inconsistent with the presence of such a term in the equations. Indeed, recent improved astronomical techniques
have found that a positive value of is needed to explain the accelerating universe.[7][8]
Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can
also be moved algebraically to the other side, written as part of the stress–energy tensor:

The resulting vacuum energy is constant and given by

The existence of a cosmological constant is thus equivalent to the existence of a non-zero vacuum energy. The
terms are now used interchangeably in general relativity.

Features
Conservation of energy and momentum

General relativity is consistent with the local conservation of energy and momentum expressed as

.
Derivation of local energy-momentum conservation

which expresses the local conservation of stress–energy. This conservation law is a physical requirement. With
his field equations Einstein ensured that general relativity is consistent with this conservation condition.

Nonlinearity

The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For
example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields, and charge and
current distributions (i.e. the sum of two solutions is also a solution); another example is Schrödinger's equation
of quantum mechanics which is linear in the wavefunction.

The correspondence principle

The EFE reduce to Newton's law of gravity by using both the weak-field approximation and the slow-motion
approximation. In fact, the constant G appearing in the EFE is determined by making these two approximations.

Derivation of Newton's law of gravity

Vacuum field equations

A Swiss commemorative coin showing the vacuum field equations with zero cosmological constant (top).

If the energy-momentum tensor is zero in the region under consideration, then the field equations are also
referred to as the vacuum field equations. By setting in the trace-reversed field equations, the vacuum
equations can be written as

In the case of nonzero cosmological constant, the equations are

The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest
example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution.

Manifolds with a vanishing Ricci tensor, , are referred to as Ricci-flat manifolds and manifolds with a
Ricci tensor proportional to the metric as Einstein manifolds.

Einstein–Maxwell equations
If the energy-momentum tensor is that of an electromagnetic field in free space, i.e. if the electromagnetic
stress–energy tensor

is used, then the Einstein field equations are called the Einstein–Maxwell equations (with cosmological constant
Λ, taken to be zero in conventional relativity theory):

Additionally, the covariant Maxwell Equations are also applicable in free space:
where the semicolon represents a covariant derivative, and the brackets denote anti-symmetrization. The first
equation asserts that the 4-divergence of the two-form F is zero, and the second that its exterior derivative is
zero. From the latter, it follows by the Poincaré lemma that in a coordinate chart it is possible to introduce an
electromagnetic field potential Aα such that

in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell
equation from which it is derived.[9] However, there are global solutions of the equation which may lack a
globally defined potential.[10]

Solutions
The solutions of the Einstein field equations are metrics of spacetime. These metrics describe the structure of
the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear,
they cannot always be completely solved (i.e. without making approximations). For example, there is no known
complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star
system, for example). However, approximations are usually made in these cases. These are commonly referred
to as post-Newtonian approximations. Even so, there are numerous cases where the field equations have been
solved completely, and those are called exact solutions.[11]

The study of exact solutions of Einstein's field equations is one of the activities of cosmology. It leads to the
prediction of black holes and to different models of evolution of the universe.

The linearised EFE

The nonlinearity of the EFE makes finding exact solutions difficult. One way of solving the field equations is to
make an approximation, namely, that far from the source(s) of gravitating matter, the gravitational field is very
weak and the spacetime approximates that of Minkowski space. The metric is then written as the sum of the
Minkowski metric and a term representing the deviation of the true metric from the Minkowski metric, with
terms that are quadratic in or higher powers of the deviation being ignored. This linearisation procedure can be
used to investigate the phenomena of gravitational radiation.

Polynomial form
One might think that EFE are non-polynomial since they contain the inverse of the metric tensor. However, the
equations can be arranged so that they contain only the metric tensor and not its inverse. First, the determinant
of the metric in 4 dimensions can be written:

using the Levi-Civita symbol; and the inverse of the metric in 4 dimensions can be written as:
Substituting this definition of the inverse of the metric into the equations then multiplying both sides by det(g)
until there are none left in the denominator results in polynomial equations in the metric tensor and its first and
second derivatives. The action from which the equations are derived can also be written in polynomial form by
suitable redefinitions of the fields.[12]

Mass–energy equivalence

4-meter-tall sculpture of Einstein's 1905 E = mc2 formula at the 2006 Walk of Ideas, Berlin, Germany.
Part of a series on
Special relativity

Explication

In physics, mass–energy equivalence is the concept that the mass of an object or system is a measure of its
energy content. For instance, adding 25 kilowatt-hours (90 megajoules) of any form(s) of energy to any object
increases its mass by 1 microgram. If you had a sensitive enough mass balance or scale, this mass increase of
the object could be verified.

A physical system has a property called energy and a corresponding property called mass; the two properties are
equivalent in that they are always both present in the same (i.e. constant) proportion to one another. Mass–
energy equivalence arose originally from special relativity, as developed by Albert Einstein, who proposed this
equivalence in 1905 in one of his Annus Mirabilis papers entitled "Does the inertia of an object depend upon its
energy content?"[1] The equivalence is described by the famous equation:

where E is energy, m is mass, and c is the speed of light. Thus, this mass–energy relation states that the
universal proportionality factor between equivalent amounts of energy and mass is equal to the speed of light
squared. This also serves to convert units of mass to units of energy, no matter what system of measurement
units is used.

If a body is stationary, it still has some internal or intrinsic energy, called its rest energy. Rest mass and rest
energy are equivalent and remain proportional to one another. When the body is in motion (relative to an
observer), its total energy is greater than its rest energy. The rest mass (or rest energy) remains an important
quantity in this case because it remains the same regardless of this motion, even for the extreme speeds or
gravity considered in special and general relativity; thus it's also called the invariant mass.

On the one hand, the equation E = mc2 can be applied to rest mass (m or m0) and rest energy (E0) to show their
proportionality as E0 = m0c2.[2]

On the other hand, it can also be applied to the total energy (Etot or simply E) and total mass of a moving body.
The total mass is also called the relativistic mass mrel, because it isn't noticeably greater than the rest mass until
the speed approaches that of light, where we have to use special relativity in order to describe the motion. So,
the total energy and total mass are related by E = mrelc2.[3]

Thus, the mass–energy relation E = mc2 can be used to relate the rest energy to the rest mass, or to relate the
total energy to the total mass. To instead relate the total energy or mass to the rest energy or mass, a
generalization of the mass–energy relation is required: the energy–momentum relation.

E = mc2 has sometimes been used as an explanation for the origin of energy in nuclear processes, but such
processes can be understood as simply converting nuclear potential energy, without the need to invoke mass–
energy equivalence. Instead, mass–energy equivalence merely indicates that the large amounts of energy
released in such reactions may exhibit enough mass that the mass loss may be measured, when the released
energy (and its mass) have been removed from the system. For example, the loss of mass to an atom and a
neutron, as a result of the capture of the neutron and the production of a gamma ray, has been used to test mass–
energy equivalence to high precision, as the energy of the gamma ray may be compared with the mass defect
after capture. In 2005, these were found to agree to 0.0004%, the most precise test of the equivalence of mass
and energy to date. This test was performed in the World Year of Physics 2005, a centennial celebration of
Einstein's achievements in 1905.[4]

Einstein was not the first to propose a mass–energy relationship (see the History section). However, Einstein
was the first scientist to propose the {{{1}}} formula and the first to interpret mass–energy equivalence as a
fundamental principle that follows from the relativistic symmetries of space and time.

Nomenclature
The formula was initially written in many different notations, and its interpretation and justification was further
developed in several steps.[5][6]

 In ″Does the inertia of a body depend upon its energy content?″ (1905), Einstein used V to mean the
speed of light in a vacuum and L to mean the energy lost by a body in the form of radiation.[1]
Consequently, the equation E = mc2 was not originally written as a formula but as a sentence in German
 saying that if a body gives off the energy L in the form of radiation, its mass diminishes by L/V2. A
remark placed above it informed that the equation was approximated by neglecting "magnitudes of
fourth and higher orders" of a series expansion.[7]
 In May 1907, Einstein explained that the expression for energy ε of a moving mass point assumes the
simplest form, when its expression for the state of rest is chosen to be ε0 = μV2 (where μ is the mass),
which is in agreement with the "principle of the equivalence of mass and energy". In addition, Einstein
used the formula μ = E0/V2, with E0 being the energy of a system of mass points, in order to describe the
energy and mass increase of that system when the velocity of the differently moving mass points is
increased.[8]
 In June 1907, Max Planck rewrote Einstein's mass-energy relationship as M = (E0 + pV0)/c2, where p is
the pressure and V the volume, in order to express the relation between mass, its "latent energy", and
thermodynamic energy within the body.[9] Subsequently in October 1907, this was rewritten as M0 =
E0/c2 and given a quantum interpretation by Johannes Stark, who assumed its validity and correctness
(Gültigkeit).[10]
 In December 1907, Einstein expressed the equivalence in the form M = μ + E0/c2 and concluded: A mass
μ is equivalent, as regards inertia, to a quantity of energy μc2. [..] It appears far more natural to
consider every inertial mass as a store of energy.[11][12]
 In 1909, Gilbert N. Lewis and Richard C. Tolman used two variations of the formula: m = E/c2 and m0 =
E0/c2, with E being the energy of a moving body, E0 its rest energy, m the relativistic mass, and m0 the
invariant mass.[13] The same relations in different notation were used by Hendrik Lorentz in 1913
(published 1914), though he placed the energy on the left-hand side: ε = Mc2 and ε0 = mc2, with ε being
the energy of a moving material point, ε0 its rest energy, M the relativistic mass, and m the invariant
mass.[14]
 In 1911, Max von Laue gave a more comprehensive proof of M0 = E0/c2 from the Stress–energy tensor,
[15]
which was later (1918) generalized by Felix Klein.[16]
 Einstein returned to the topic once again after World War II and this time he wrote E = mc2 in the title of
his article[17] intended as an explanation for a general reader by analogy.[18]

Conservation of mass and energy

Main article: Conservation of energy
Main article: Conservation of mass

Mass and energy can be seen as two names (and two measurement units) for the same underlying, conserved
physical quantity.[19] Thus, the laws of conservation of energy and conservation of (total) mass are equivalent
and both hold true.[20][21]

On the other hand, if the conservation of mass law is interpreted as conservation of rest mass, this does not hold
true in general. The rest energy (equivalently, rest mass) of a particle can be converted, not "to energy" (it
already is energy (mass)), but rather to other forms of energy (mass) which require motion, such as kinetic
energy, thermal energy, or radiant energy; similarly, kinetic or radiant energy can be converted to other kinds of
particles which have rest energy (rest mass). In the transformation process, neither the total amount of mass nor
the total amount of energy changes, since both are properties which are connected to each other via a simple
constant.[22] This view requires that if either energy or (total) mass disappears from a system, it will always be
found that both have simply moved off to another place, where they may both be measured as an increase of
both energy and mass corresponding to the loss in the first system.

Fast-moving objects and systems of objects

When an object is pulled 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 a constant times the
velocity, nor can the kinetic energy be a constant times the square of the velocity.

A property called the relativistic mass is defined as the ratio of the momentum of an object to its velocity.[23]
Relativistic mass depends on the motion of the object, so that different observers in relative motion see different
values for it. 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 grows infinitely, because the kinetic energy grows infinitely
and this energy is associated with mass.

The relativistic mass is always equal to the total energy (rest energy plus kinetic energy) divided by c2.[3]
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 rest mass, or invariant
mass, and not relativistic mass.

The relativistic mass of a moving object is larger than the relativistic mass of an object that is not 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 things and systems made up of many parts, like an atomic nucleus, planet, or star, the relativistic mass is the
sum of the relativistic masses (or energies) of the parts, because energies are additive in isolated 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 energy gained due to the forces of attraction in excess of the work done is removed
from the system, then mass will be lost with this removed energy. 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 this energy from
binding has been removed in the form of a gamma ray (which in this system, carries away the mass of the
energy 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.

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, even those in relative motion. It is defined as the total energy
(divided by c2) 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 any reference frame where the momentum is zero, and such a
reference frame is also the only frame in which the object can be weighed. In a similar way, the theory of
special relativity posits that the thermal energy in all objects (including solids) contributes to their total masses
and weights, even though this energy is present as the kinetic and potential energies of the atoms in the object,
and it (in a similar way to the gas) is not seen in the rest masses of the atoms that make up the object.

In a similar manner, even photons (light quanta), if trapped in a container space (as a photon gas or thermal
radiation), would contribute a mass associated with their energy to the container. Such an extra mass, in theory,
could be weighed in the same way as any other type of rest mass. This is true in special relativity theory, even
though individually photons have no rest mass. The property that trapped energy in any form adds weighable
mass to systems that have no net momentum is one of the characteristic and notable consequences of relativity.
It has no counterpart in classical Newtonian physics, in which radiation, light, heat, and kinetic energy never
exhibit weighable mass under any circumstances.

Just as the relativistic mass of isolated system is conserved through time, so also is its invariant mass. It is this
property which allows the conservation of all types of mass in systems, and also conservation of all types of
mass in reactions where matter is destroyed (annihilated), leaving behind the energy that was associated with it
(which is now in non-material form, rather than material form). Matter may appear and disappear in various
reactions, but mass and energy are both unchanged in this process.

Applicability of the strict mass–energy equivalence formula, E = mc2

As is noted above, two different definitions of mass have been used in special relativity, and also two different
definitions of energy. The simple equation E = mc2 is not generally applicable to all these types of mass and
energy, except in the special case that the total additive momentum is zero for the system under consideration.
In such a case, which is always guaranteed when observing the system from either its center of mass frame or
its center of momentum frame, E = mc2 is always true for any type of mass and energy that are chosen. Thus, for
example, in the center of mass frame, the total energy of an object or system is equal to its rest mass times c2, a
useful equality. This is the relationship used for the container of gas in the previous example. It is not true in
other reference frames where the center of mass is in motion. In these systems or for such an object, its total
energy will depend on both its rest (or invariant) mass, and also its (total) momentum.[24]

In inertial reference frames other than the rest frame or center of mass frame, the equation E = mc2 remains true
if the energy is the relativistic energy and the mass the relativistic mass. It is also correct if the energy is the rest
or invariant energy (also the minimum energy), and the mass is the rest mass, or the invariant mass. However,
connection of the total or relativistic energy (Er) with the rest or invariant mass (m0) requires consideration
of the system total momentum, in systems and reference frames where the total momentum has a non-zero
value. The formula then required to connect the two different kinds of mass and energy, is the extended version
of Einstein's equation, called the relativistic energy–momentum relation:[25]

or

Here the (pc)2 term represents the square of the Euclidean norm (total vector length) of the various momentum
vectors in the system, which reduces to the square of the simple momentum magnitude, if only a single particle
is considered. This equation reduces to E = mc2 when the momentum term is zero. For photons where m0 = 0,
the equation reduces to Er = pc.

View a video explanation of the full equation at Minute Physics on Youtube

Meanings of the strict mass–energy equivalence formula, E = mc2

The mass–energy equivalence formula was displayed on Taipei 101 during the event of the World Year of
Physics 2005.

Mass–energy equivalence states that any object has a certain energy, even when it is stationary. 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 (that is, all the energy which is present in the
object's rest frame) contributes to the total mass of the body, which measures how much it resists acceleration.
Each potential and kinetic energy makes a proportional contribution to the mass. As noted above, even if a box
of ideal mirrors "contains" light, then the individually massless photons still contribute to the total mass of the
box, by the amount of their energy divided by c2.[26]

In relativity, removing energy is removing mass, and for an observer in the center of mass frame, the formula m
= E/c2 indicates how much mass is lost when energy is removed. In a 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 decreases. In the same way, when any sort of energy is added to an isolated system, the
increase in the mass is equal to the added energy divided by c2. For example, when water is heated it gains
about 1.11×10−17 kg 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
relativistic 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 is not moving (or when an inertial
frame is chosen such that it is not moving). The term also applies to the invariant mass of systems when the
system as a whole is not "moving" (has no net momentum). The rest and invariant masses are the smallest
possible value of the mass of the object or system. They also are conserved quantities, so long as the system is
isolated. Because of the way they are calculated, the effects of moving observers are subtracted, so these
quantities do not change with the motion of the observer.
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 do not attract or repel, so that they do not 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.

Binding energy and the "mass defect"

Whenever any type of energy is removed from a system, the mass associated with the energy is also removed,
and the system therefore loses mass. This mass defect in the system may be simply calculated as Δm = ΔE/c2,
and this was the form of the equation historically first presented by Einstein in 1905. However, use of this
formula in such circumstances has led to the false idea that mass has been "converted" to energy. This may be
particularly the case when the energy (and mass) removed from the system is associated with the binding
energy of the system. In such cases, the binding energy is observed as a "mass defect" or deficit in the new
system.

The fact that the released energy is not easily weighed in many such cases, may cause its mass to be neglected
as though it no longer existed. This circumstance has encouraged the false idea of conversion of mass to energy,
rather than the correct idea that the binding energy of such systems is relatively large, and exhibits a measurable
mass, which is removed when the binding energy is removed. This energy is often released in the form of light
and heat, which is too quickly and widely dispersed to be easily weighed, though it does carry mass.

The difference between the rest mass of a bound system and of the unbound parts is the binding energy of the
system, if this energy has been removed after binding. For example, 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 c2), and which was given off as heat when the
molecule formed (this heat had mass). Likewise, a stick of dynamite in theory weighs a little bit more than the
fragments after the explosion, but this is true only so long as the fragments are cooled and the heat removed. In
this case the mass difference is the energy/heat that is released when the dynamite explodes, and when this heat
escapes, the mass associated with it escapes, only to be deposited in the surroundings which absorb the heat (so
that total mass is conserved).

Such a change in mass may only happen when the system is open, and the energy and mass escapes. Thus, 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. If sitting on a
scale, the weight and mass would not change. This would in theory also happen even with a nuclear bomb, if it
could be kept in an ideal box of infinite strength, which did not rupture or pass radiation.[27] Thus, a 21.5 kiloton
(9 x 1013 joule) nuclear bomb produces about one gram of heat and electromagnetic radiation, but the mass of
this energy would not be detectable in an exploded bomb in an ideal box sitting on a scale; instead, the contents
of the box would be heated to millions of degrees without changing total mass and weight. If then, however, a
transparent window (passing only electromagnetic radiation) were opened in such an ideal box after the
explosion, and a beam of X-rays and other lower-energy light allowed to escape the box, it would eventually be
found to weigh one gram less than it had before the explosion. This weight loss and mass loss would happen as
the box was cooled by this process, to room temperature. However, any surrounding mass which had absorbed
the X-rays (and other "heat") would gain this gram of mass from the resulting heating, so the mass "loss" would
represent merely its relocation. Thus, no mass (or, in the case of a nuclear bomb, no matter) would be
"converted" to energy in such a process. Mass and energy, as always, would both be separately conserved.

Massless particles
Massless particles have zero rest mass. Their relativistic mass is simply their relativistic energy, divided by c2,
or m(relativistic) = E/c2.[28][29] The energy for photons is E = hf where h is Planck's constant and f is the photon
frequency. This frequency and thus the relativistic energy are frame-dependent.

If an observer runs away from a photon in the direction it travels from a source, having it catch up with the
observer, then when the photon catches up it will be seen as having less energy than it had at the source. The
faster the observer is traveling with regard to the source when the photon catches up, the less energy the photon
will have. As an observer approaches the speed of light with regard to the source, the photon looks redder and
redder, by relativistic Doppler effect (the Doppler shift is the relativistic formula), and the energy of a very
long-wavelength photon approaches zero. This is why a photon is massless; this means that the rest mass of a
photon is zero.

Massless particles contribute rest mass and invariant mass to systems

Two photons moving in different directions cannot both be made to have arbitrarily small total energy by
changing frames, or by moving toward or away from them. The reason is that in a two-photon system, the
energy of one photon is decreased by chasing after it, but the energy of the other will increase with the same
shift in observer motion. Two photons not moving in the same direction will exhibit 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). The most that chasing a pair of photons can accomplish to
decrease their energy is to put the observer in frame where the photons have equal energy and are moving
directly away from each other. In this frame, the observer is now moving in the same direction and speed as the
center of mass of the two photons. The total momentum of the photons is now zero, since their momentums are
equal and opposite. In this frame the two photons, as a system, have a mass equal to their total energy divided
by c2. 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. It is only the invariant mass of a two-photon system that can
be used to make a single particle with the same rest mass.

If the photons are 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 well-
defined 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
does not change after it disintegrates into two photons. 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. Thus, by calculating the invariant mass of pairs of photons in a particle detector, pairs can be identified
that were probably produced by pion disintegration.

A similar calculation illustrates that the invariant mass of systems is conserved, even when massive particles
(particles with rest mass) within the system are converted to massless particles (such as photons). In such cases,
the photons contribute invariant mass to the system, even though they individually have no invariant mass or
rest mass. Thus, an electron and positron (each of which has rest mass) may undergo annihilation with each
other to produce two photons, each of which is massless (has no rest mass). However, in such circumstances, no
system mass is lost. Instead, the system of both photons moving away from each other has an invariant mass,
which acts like a rest mass for any system in which the photons are trapped, or that can be weighed. Thus, not
only the quantity of relativistic mass, but also the quantity of invariant mass does not change in transformations
between "matter" (electrons and positrons) and energy (photons).

Relation to gravity
In physics, there are two distinct concepts of mass: the gravitational mass and the inertial mass. The
gravitational mass is the quantity that determines the strength of the gravitational field generated by an object,
as well as the gravitational force acting on the object when it is immersed in a gravitational field produced by
other bodies. The inertial mass, on the other hand, quantifies how much an object accelerates if a given force is
applied to it. The mass–energy equivalence in special relativity refers to the inertial mass. However, already in
the context of Newton gravity, the Weak Equivalence Principle is postulated: the gravitational and the inertial
mass of every object are the same. Thus, the mass–energy equivalence, combined with the Weak Equivalence
Principle, results in the prediction that all forms of energy contribute to the gravitational field generated by an
object. This observation is one of the pillars of the general theory of relativity.

The above prediction, that all forms of energy interact gravitationally, has been subject to experimental tests.
The first observation testing this prediction was made in 1919.[30] During a solar eclipse, Arthur Eddington
observed that the light from stars passing close to the Sun was bent. The effect is due to the gravitational
attraction of light by the sun. The observation confirmed that the energy carried by light indeed is equivalent to
a gravitational mass. Another seminal experiment, the Pound–Rebka experiment, was performed in 1960.[31] In
this test a beam of light was emitted from the top of a tower and detected at the bottom. The frequency of the
light detected was higher than the light emitted. This result confirms that the energy of photons increases when
they fall in the gravitational field of the earth. The energy, and therefore the gravitational mass, of photons is
proportional to their frequency as stated by the Planck's relation.

Consequences for nuclear physics

Task Force One, the world's first nuclear-powered task force. Enterprise, Long Beach and Bainbridge in
formation in the Mediterranean, 18 June 1964. Enterprise crew members are spelling out Einstein's Mass–
Energy Equivalence formula E=mc2 on the flight deck.

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 in radium, due to the half-life of the substance (1602 years), only a small
fraction of radium atoms decay over an experimentally measurable period of time.

Once the nucleus was discovered, experimenters realized that the very high binding energies of the atomic
nuclei should allow calculation of their binding energies, simply from mass differences. But it was not until the
discovery of the neutron in 1932, and the measurement of the neutron 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[32] the Cockcroft–Walton experiment: 7Li + p → 2 4He) verified Einstein's formula to an
accuracy of ±0.5%. In 2005, Rainville et al. published a direct test of the energy-equivalence of mass lost in the
binding energy of a neutron to atoms of particular isotopes of silicon and sulfur, by comparing the mass lost to
the energy of the emitted gamma ray associated with the neutron capture. The binding mass-loss agreed with the
gamma ray energy to a precision of ±0.00004 %, the most accurate test of E = mc2 to date.[4]

The mass–energy equivalence formula was used in the understanding of nuclear fission reactions, and implies
the great amount of energy that can be released by a nuclear fission chain reaction, used in both nuclear
weapons and nuclear power. 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.

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):[citation needed]

E / m = c2 = (299,792,458 m/s or 983,571,056 ft/s)2 = 89,875,517,873,681,764 J/kg (≈9.0 × 1016 joules

per kilogram).

So the energy equivalent of one gram (1/1000 of a kilogram) of mass is equivalent to:

89.9 terajoules
25.0 million kilowatt-hours (≈25 GW·h)
21.5 billion kilocalories (≈21 Tcal) [33]
85.2 billion BTUs[33]

or to the energy released by combustion of the following:

21.5 kilotons of TNT-equivalent energy (≈21 kt) [33]

568,000 US gallons of automotive gasoline

Any time energy is generated, the process can be evaluated from an E = mc2 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 electromagnetic radiation and kinetic
energy (thermal and blast energy) released in this explosion carried the missing one gram of mass.[34] This
occurs because nuclear binding energy is released whenever elements with more than 62 nucleons fission.[citation
needed]

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 (such as lights in cities)
which are powered by the generators, where it appears as a gram of heat and light.[35] 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).[citation needed]
Whenever energy is added to a system, the system gains mass:[citation needed]

 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. For example,
consider the world's primary mass standard for the kilogram, made of platinum/iridium. If its
temperature is allowed to change by 1°C, its mass will change by 1.5 picograms (1 pg = 1 × 10−12 g).[36]
 A spinning ball will weigh more than a ball that is not spinning. Its increase of mass is exactly the
equivalent of the mass of energy of rotation, which is itself the sum of the kinetic energies of all the
moving parts of the ball. For example, the Earth itself is more massive due to its daily rotation, than it
would be with no rotation. This rotational energy (2.14 x 1029 J) represents 2.38 billion metric tons of
added mass.[37]

Note that no net mass or energy is really created or lost in any of these examples and 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.[citation needed]

Efficiency
Although mass cannot be converted to energy, in some reactions matter particles (which contain a form of rest
energy) can be destroyed and converted to other types of energy which are more usable and obvious as forms of
energy, such as light and energy of motion (heat, etc.). However, the total amount of energy and mass does not
change in such a transformation. Even when particles are not destroyed, a certain fraction of the ill-defined
"matter" in ordinary objects can be destroyed, and its associated energy liberated and made available as the
more dramatic energies of light and heat, even though no identifiable real particles are destroyed, and even
though (again) the total energy is unchanged (as also the total mass). Such conversions between types of energy
(resting to active energy) happen in nuclear weapons, in which the protons and neutrons in atomic nuclei lose a
small fraction of their average mass, but this mass loss is not due to the destruction of any protons or neutrons
(or even, in general, lighter particles like electrons). Also the mass is not destroyed, but simply removed from
the system. in the form of heat and light from the reaction.

In nuclear reactions, typically only a small fraction of the total mass–energy of the bomb is converted into the
mass–energy of heat, light, radiation and motion, which are "active" forms which can be used. When an atom
fissions, it loses only about 0.1% of its mass (which escapes from the system and does not disappear), 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 non-reacting components, so that in practicality, no
more than about 0.03% of the total mass of the entire weapon is released as usable energy (which, again, retains
the "missing" mass).

In theory, it should be possible to destroy matter and convert all of the rest-energy associated with matter into
heat and light (which would of course have the same mass), but none of the theoretically known methods are
practical. One way to convert all the energy within matter into usable energy is to annihilate matter with
antimatter. But antimatter is rare in our universe, and must be made first. Due to inefficient mechanisms of
production, making antimatter always requires far more usable energy than would be released when it was
annihilated.

Since most of the mass of ordinary objects resides in protons and neutrons, in order to convert all of the energy
of ordinary matter into a more useful type of energy, the protons and neutrons must be converted to lighter
particles, or else particles with no rest-mass at all. In the standard model of particle physics, the number of
protons plus neutrons is nearly exactly conserved. Still, Gerard 't Hooft showed that there is a process which
will convert protons and neutrons to antielectrons and neutrinos.[38] This is the weak SU(2) instanton proposed
by Belavin Polyakov Schwarz and Tyupkin.[39] This process, can in principle destroy matter and convert all the
energy 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,[40] since then instanton-like 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.[41] This process would
be an efficient mass–energy conversion at ordinary temperatures, but it requires making monopoles and anti-
monopoles 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.

A third known method of total matter–energy "conversion" (which again in practice only means converstion of
one type of energy into a different type of energy), is using gravity, specifically black holes. Stephen Hawking
theorized[42] 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. This could result from the fact that mass and energy are lost from the hole with its thermal radiation.

Background
Mass–velocity relationship

In developing special relativity, Einstein found that the kinetic energy of a moving body is

with the velocity, the rest mass, and γ the Lorentz factor.

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:

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:

and it is this quantity which is conserved in collisions. The ratio of the momentum to the velocity is the
relativistic mass, m.

And the relativistic mass and the relativistic kinetic energy are related by the formula:

Einstein wanted to omit the unnatural second term on the right-hand side, whose only purpose is to make the
energy at rest zero, and to declare that the particle has a total energy which obeys:

which is a sum of the rest energy m0c2 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 rest-masses 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:

This mass is the ratio of momentum to velocity, and it is also the relativistic energy divided by c2 (it is not
Lorentz-invariant, in contrast to ). The equation E = mrelc2 holds for moving objects. When the velocity is
small, the relativistic mass and the rest mass are almost exactly the same.

 E=mc2 either means E=m0c2 for an object at rest, or E=mrelc2 when the object is moving.

Also Einstein (following Hendrik Lorentz and Max Abraham) used velocity—and direction-dependent mass
concepts (longitudinal and transverse mass) in his 1905 electrodynamics paper and in another paper in 1906.[43]
[44]
However, in his first paper on E=mc2 (1905), he treated m as what would now be called the rest mass.[1]
Some claim that (in later years) he did not like the idea of "relativistic mass."[2]  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.[45][46]

Low-speed expansion

We can rewrite the expression E = γm0c2 as a Taylor series:

For speeds much smaller than the speed of light, higher-order 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:

The total energy is a sum of the rest energy and the Newtonian kinetic energy.

The classical energy equation ignores both the m0c2 part, and the high-speed corrections. This is appropriate,
because all the high-order corrections are small. Since only changes in energy affect the behavior of objects,
whether we include the m0c2 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 higher-order terms are extra correction to Newtonian mechanics which become important at higher speeds.
The Newtonian equation is only a low-speed 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 higher-order 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.[47][48][49][50]

Newton: matter and light

In 1717 Isaac Newton speculated that light particles and matter particles were inter-convertible in "Query 30" of
the Opticks, where he asks:

Are not the gross bodies and light convertible into one another, and may not bodies receive much of their
activity from the particles of light which enter their composition?
Swedenborg: matter composed of "pure and total motion"

In 1734 Emanuel Swedenborg in his Principia theorized that all matter is ultimately composed of dimensionless
points of "pure and total motion." He described this motion as being without force, direction or speed, but
having the potential for force, direction and speed everywhere within it.[51][52]

Electromagnetic 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), Wilhelm Wien (1900), Max
Abraham (1902), Hendrik Antoon Lorentz (1904) — to understand as to how the mass of a charged object
depends on the electrostatic field.[47][48] This concept was called electromagnetic mass, and was considered as
being dependent on velocity and direction as well. Lorentz (1904) gave the following expressions for
longitudinal and transverse electromagnetic mass:

where

Radiation pressure and inertia

Another way of deriving some sort of electromagnetic mass was based on the concept of radiation pressure. In
1900, Henri Poincaré associated electromagnetic radiation energy with a "fictitious fluid" having momentum
and mass

By that, Poincaré tried to save the center of mass theorem in Lorentz's theory, though his treatment led to
radiation paradoxes.[50]

Friedrich Hasenöhrl showed in 1904, that electromagnetic cavity radiation contributes the "apparent mass"

to the cavity's mass. He argued that this implies mass dependence on temperature as well.[53]

Einstein: mass–energy equivalence

Albert Einstein did not formulate exactly the formula E = mc2 in his 1905 Annus Mirabilis paper "Does the
Inertia of an object Depend Upon Its Energy Content?";[1] rather, the paper states that if a body gives off the
energy L in the form of radiation, its mass diminishes by L/c2. (Here, "radiation" means electromagnetic
radiation, or light, and mass means the ordinary Newtonian mass of a slow-moving object.) This formulation
relates only a change Δm 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.

The first derivation by Einstein (1905)

Already in his relativity paper "On the electrodynamics of moving bodies", Einstein derived the correct
expression for the kinetic energy of particles:

Now the question remained open as to which formulation applies to bodies at rest. This was tackled by Einstein
in his paper "Does the inertia of a body depend upon its energy content?". Einstein used a body emitting two
light pulses in opposite directions, having energies of E0 before and E1 after the emission as seen in its rest
frame. As seen from a moving frame, this becomes H0 and H1. Einstein obtained:

then he argued that H − E can only differ from the kinetic energy K by an additive constant, which gives

Neglecting effects higher than third order in v/c this gives:

Thus Einstein concluded that the emission reduces the body's mass by E/c2, and that the mass of a body is a
measure of its energy content.

The correctness of Einstein's 1905 derivation of E = mc2 was criticized by Max Planck (1907), who argued that
it is only valid to first approximation. Another criticism was formulated by Herbert Ives (1952) and Max
Jammer (1961), asserting that Einstein's derivation is based on begging the question.[5][54] On the other hand,
John Stachel and Roberto Torretti (1982) argued that Ives' criticism was wrong, and that Einstein's derivation
was correct.[55] Hans Ohanian (2008) agreed with Stachel/Torretti's criticism of Ives, though he argued that
Einstein's derivation was wrong for other reasons.[56] For a recent review, see Hecht (2011).[6]

Alternative version

An alternative version of Einstein's thought experiment was proposed by Fritz Rohrlich (1990), who based his
reasoning on the Doppler effect.[57] Like Einstein, he 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. Then he 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 has not changed its velocity before or after the emission. Yet in this frame it has lost some right-
momentum 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 1 − v/c. 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 ΔP given by:

The left-moving light carries a little less momentum, by the same amount ΔP. So the total right-momentum in
the light is twice ΔP. This is the right-momentum that the object lost.

The momentum of the object in the moving frame after the emission is reduced by this amount:

So the change in the object's mass is equal to the total energy lost divided by c2. 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.

Relativistic center-of-mass theorem (1906)

Like Poincaré, Einstein concluded in 1906 that the inertia of electromagnetic energy is a necessary condition for
the center-of-mass theorem to hold. On this occasion, Einstein referred to Poincaré's 1900 paper and wrote:[58]

Although the merely formal considerations, which we will need for the proof, are already mostly contained in a
work by H. Poincaré2, for the sake of clarity I will not rely on that work.[59]

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 = mc2, 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 ether theories.[60] In 1873 Nikolay Umov pointed out a relation between mass and energy
for ether in the form of Е = kmc2, where 0.5 ≤ k ≤ 1.[61] The writings of Samuel Tolver Preston,[62][63] and a 1903
paper by Olinto De Pretto,[64][65] 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.[66]

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 mc2 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.
[67]
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 = mc2 or 2E = mc2
depending on the convention. A particle ether was usually considered unacceptably speculative science at the
time,[68] and since these authors did not 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 all-encompassing qualitative philosophy of physics.[69][70]

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:[71][72]

If it were ever found possible to control at will the rate of disintegration of the radio-elements, an enormous
amount of energy could be obtained from a small quantity of matter.

Einstein's equation is in no way an explanation of the large energies released in radioactive decay (this comes
from the powerful nuclear forces involved; forces that were still unknown in 1905). In any case, the enormous
energy released from radioactive decay (which had been measured by Rutherford) was much more easily
measured than the (still small) change in the gross mass of materials, as a result. Einstein's equation, by theory,
can give these energies by measuring mass differences before and after reactions, but in practice, these mass
differences in 1905 were still too small to be measured in bulk. Prior to this, the ease of measuring radioactive
decay energies with a calorimeter was thought possibly likely to allow measurement of changes in mass
difference, as a check on Einstein's equation itself. 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 by 1905) to possibly be "weighed," when missing from the system (having been given
off as heat). However, radioactivity seemed to proceed at its own unalterable (and quite slow, for radioactives
known then) pace, and even when simple nuclear reactions became possible using proton bombardment, the
idea that these great amounts of usable energy could be liberated at will with any practicality, proved difficult to
substantiate. It had been used as the basis of much speculation, causing Rutherford himself to later reject his
ideas of 1904; he was reported in 1933 to have said that: "Anyone who expects a source of power from the
transformation of the atom is talking moonshine."[73]
The popular connection between Einstein, E = mc2, and the atomic bomb was prominently indicated on the
cover of Time magazine in July 1946 by the writing of the equation on the mushroom cloud itself.

This situation changed dramatically in 1932 with the discovery of the neutron and its mass, allowing mass
differences for single nuclides and their reactions to be calculated directly, and compared with the sum of
masses for the particles that made up their composition. In 1933, the energy released from the reaction of
lithium-7 plus protons giving rise to 2 alpha particles (as noted above by Rutherford), allowed Einstein's
equation to be tested to an error of ± 0.5%. However, scientists still did not see such reactions as a source of
power.

After the very public demonstration of huge energies released from nuclear fission after the atomic bombings of
Hiroshima and Nagasaki in 1945, the equation E = mc2 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.[74] 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 in theoretical
terms. It was inconsequential, on account of Einstein not being given sufficient information (for security
reasons) to fully work on the problem.[75]

While E = mc2 is useful for understanding the amount of energy potentially released in a fission reaction, it was
not strictly necessary to develop the weapon, once the fission process was known, and its energy measured at
200 MeV (which was directly possible, using a quantitative Geiger counter, at that time). 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 = mc2, 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 non-relativistic theory, meaning that relativistic effects are too small to affect the dynamics of
the fission process significantly."[76] However the association between E = mc2 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".[77]
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 were energetically allowed to split into approximately equal parts
(before this energy was in fact measured). 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 tension-like" 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 = mc2 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 a very wobbly, 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 one-
fifth 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!