In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, the energy–momentum relation, or relativistic dispersion relation, is the
relativistic equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...
relating total
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
(which is also called relativistic energy) to
invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
(which is also called rest mass) and
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
. It is the extension of
mass–energy equivalence
In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicis ...
for bodies or systems with non-zero momentum. It can be written as the following equation:
This equation holds for a
body or
system
A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and express ...
, such as one or more
particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...
s, with total energy , invariant mass , and momentum of
magnitude ; the constant is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
. It assumes the
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
case of
flat spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
.
Total energy is the sum of
rest energy
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
and
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
, while invariant mass is mass measured in a
center-of-momentum frame
In physics, the center-of-momentum frame (also zero-momentum frame or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The ''center of momentum'' of a system is ...
.
For bodies or systems with zero momentum, it simplifies to the mass–energy equation
, where total energy in this case is equal to
rest energy
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
(also written as ).
The
Dirac sea
The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by th ...
model, which was used to predict the existence of
antimatter
In modern physics, antimatter is defined as matter composed of the antiparticles (or "partners") of the corresponding particles in "ordinary" matter. Antimatter occurs in natural processes like cosmic ray collisions and some types of radioac ...
, is closely related to the energy–momentum relation.
Connection to
Einstein Triangle
The energy–momentum relation is consistent with the familiar
mass–energy relation in both its interpretations: relates total energy to the (total)
relativistic mass
The word " mass" has two meanings in special relativity: '' invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity ...
(alternatively denoted or ), while relates
rest energy
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
to (invariant) rest mass .
Unlike either of those equations, the energy–momentum equation () relates the ''total'' energy to the ''rest'' mass . All three equations hold true simultaneously.
Special cases
# If the body is a
massless particle
In particle physics, a massless particle is an elementary particle whose invariant mass is zero. There are two known gauge boson massless particles: the photon (carrier of electromagnetism) and the gluon (carrier of the strong force). However, g ...
(), then () reduces to . For
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
s, this is the relation, discovered in 19th century
classical electromagnetism
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
, between radiant momentum (causing
radiation pressure
Radiation pressure is the mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is a ...
) and
radiant energy
Radiant may refer to:
Computers, software, and video games
* Radiant (software), a content management system
* GtkRadiant, a level editor created by id Software for their games
* Radiant AI, a technology developed by Bethesda Softworks for ''Th ...
.
# If the body's speed is much less than , then () reduces to ; that is, the body's total energy is simply its classical
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
() plus its rest energy.
# If the body is at rest (), i.e. in its
center-of-momentum frame
In physics, the center-of-momentum frame (also zero-momentum frame or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The ''center of momentum'' of a system is ...
(), we have and ; thus the energy–momentum relation and both forms of the mass–energy relation (mentioned above) all become the same.
A more
general form of relation () holds for
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
.
The
invariant mass (or rest mass) is an invariant for all
frames of reference
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathe ...
(hence the name), not just in
inertial frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration ...
s in flat spacetime, but also
accelerated frames traveling through curved spacetime (see below). However the total energy of the particle and its relativistic momentum are frame-dependent; relative motion between two frames causes the observers in those frames to measure different values of the particle's energy and momentum; one frame measures and , while the other frame measures and , where and , unless there is no relative motion between observers, in which case each observer measures the same energy and momenta. Although we still have, in flat spacetime:
:
The quantities , , , are all related by a
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
. The relation allows one to sidestep Lorentz transformations when determining only the
magnitudes of the energy and momenta by equating the relations in the different frames. Again in flat spacetime, this translates to;
:
Since does not change from frame to frame, the energy–momentum relation is used in
relativistic mechanics
In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...
and
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
calculations, as energy and momentum are given in a particle's rest frame (that is, and as an observer moving with the particle would conclude to be) and measured in the
lab frame
In theoretical physics, a local reference frame (local frame) refers to a coordinate system or frame of reference that is only expected to function over a small region or a restricted region of space or spacetime.
The term is most often used in t ...
(i.e. and as determined by particle physicists in a lab, and not moving with the particles).
In
relativistic quantum mechanics
In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light  ...
, it is the basis for constructing
relativistic wave equations, since if the relativistic wave equation describing the particle is consistent with this equation – it is consistent with relativistic mechanics, and is
Lorentz invariant
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
. In
relativistic quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, it is applicable to all particles and fields.
Origins and derivation of the equation
The was first established by
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
in 1928 under the form
, where ''V'' is the amount of potential energy.
The equation can be derived in a number of ways, two of the simplest include:
# From the relativistic dynamics of a massive particle,
# By evaluating the norm of the
four-momentum
In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
of the system. This method applies to both massive and massless particles, and can be extended to multi-particle systems with relatively little effort (see below).
Heuristic approach for massive particles
For a massive object moving at three-velocity with magnitude in the
lab frame
In theoretical physics, a local reference frame (local frame) refers to a coordinate system or frame of reference that is only expected to function over a small region or a restricted region of space or spacetime.
The term is most often used in t ...
:
:
is the total energy of the moving object in the lab frame,
:
is the three dimensional
relativistic momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...
of the object in the lab frame with magnitude . The relativistic energy and momentum include the
Lorentz factor
The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
defined by:
:
Some authors use
relativistic mass
The word " mass" has two meanings in special relativity: '' invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity ...
defined by:
:
although rest mass has a more fundamental significance, and will be used primarily over relativistic mass in this article.
Squaring the 3-momentum gives:
:
then solving for and substituting into the Lorentz factor one obtains its alternative form in terms of 3-momentum and mass, rather than 3-velocity:
:
Inserting this form of the Lorentz factor into the energy equation gives:
:
followed by more rearrangement it yields (). The elimination of the Lorentz factor also eliminates implicit velocity dependence of the particle in (), as well as any inferences to the "relativistic mass" of a massive particle. This approach is not general as massless particles are not considered. Naively setting would mean that and and no energy–momentum relation could be derived, which is not correct.
Norm of the four-momentum
Special relativity
In
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
, energy (divided by ) and momentum are two components of a Minkowski
four-vector
In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
, namely the
four-momentum
In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
;
:
(these are the
contravariant components).
The
Minkowski inner product
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
of this vector with itself gives the square of the
norm of this vector, it is
proportional to the square of the rest mass of the body:
:
a
Lorentz invariant quantity, and therefore independent of the
frame of reference
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
. Using the
Minkowski metric
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
with
metric signature
In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
, the inner product is
:
and
:
so
:
:
or, in natural units where = 1,
:
.
:
General relativity
In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the 4-momentum is a four-vector defined in a local coordinate frame, although by definition the inner product is similar to that of special relativity,
:
in which the Minkowski metric is replaced by the
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathe ...
tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
:
:
solved from the
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
. Then:
:
Performing the summations over indices followed by collecting "time-like", "spacetime-like", and "space-like" terms gives:
:
where the factor of 2 arises because the metric is a
symmetric tensor
In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments:
:T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_)
for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of orde ...
, and the convention of Latin indices taking space-like values 1, 2, 3 is used. As each component of the metric has space and time dependence in general; this is significantly more complicated than the formula quoted at the beginning, see
metric tensor (general relativity)
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. The m ...
for more information.
Units of energy, mass and momentum
In
natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
where , the energy–momentum equation reduces to
:
In
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, energy is typically given in units of
electron volt
In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacuum ...
s (eV), momentum in units of eV·
−1, and mass in units of eV·
−2. In
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, and because of relativistic invariance, it is useful to have the
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
and the
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
in the same unit (
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
), using the
cgs (Gaussian) system of units, where energy is given in units of
erg
The erg is a unit of energy equal to 10−7joules (100 nJ). It originated in the Centimetre–gram–second system of units (CGS). It has the symbol ''erg''. The erg is not an SI unit. Its name is derived from (), a Greek word meaning 'work' o ...
, mass in
gram
The gram (originally gramme; SI unit symbol g) is a unit of mass in the International System of Units (SI) equal to one one thousandth of a kilogram.
Originally defined as of 1795 as "the absolute weight of a volume of pure water equal to th ...
s (g), and momentum in g·cm·s
−1.
Energy may also in theory be expressed in units of grams, though in practice it requires a large amount of energy to be equivalent to masses in this range. For example, the first
atomic bomb
A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either fission (fission bomb) or a combination of fission and fusion reactions ( thermonuclear bomb), producing a nuclear explosion. Both bomb ...
liberated about 1 gram of
heat
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
, and the largest
thermonuclear bombs have generated a
kilogram
The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially. ...
or more of heat. Energies of thermonuclear bombs are usually given in tens of
kiloton
TNT equivalent is a convention for expressing energy, typically used to describe the energy released in an explosion. The is a unit of energy defined by that convention to be , which is the approximate energy released in the detonation of a ...
s and megatons referring to the energy liberated by exploding that amount of
trinitrotoluene
Trinitrotoluene (), more commonly known as TNT, more specifically 2,4,6-trinitrotoluene, and by its preferred IUPAC name 2-methyl-1,3,5-trinitrobenzene, is a chemical compound with the formula C6H2(NO2)3CH3. TNT is occasionally used as a reage ...
(TNT).
Special cases
Centre-of-momentum frame (one particle)
For a body in its rest frame, the momentum is zero, so the equation simplifies to
:
where is the rest mass of the body.
Massless particles
If the object is massless, as is the case for a
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
, then the equation reduces to
:
This is a useful simplification. It can be rewritten in other ways using the
de Broglie relations:
:
if the
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
or
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
are given.
Correspondence principle
Rewriting the relation for massive particles as:
:
and expanding into
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
by the
binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
(or a
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
):
:
in the limit that , we have so the momentum has the classical form , then to first order in (i.e. retain the term for and neglect all terms for ) we have
:
or
:
where the second term is the classical
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
, and the first is the
rest energy
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
of the particle. This approximation is not valid for massless particles, since the expansion required the division of momentum by mass. Incidentally, there are no massless particles in classical mechanics.
Many-particle systems
Addition of four momenta
In the case of many particles with relativistic momenta and energy , where (up to the total number of particles) simply labels the particles, as measured in a particular frame, the four-momenta in this frame can be added;
:
and then take the norm; to obtain the relation for a many particle system:
:
where is the invariant mass of the whole system, and is not equal to the sum of the rest masses of the particles unless all particles are at rest (see
mass in special relativity
The word " mass" has two meanings in special relativity: '' invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity ...
for more detail). Substituting and rearranging gives the generalization of ();
The energies and momenta in the equation are all frame-dependent, while is frame-independent.
Center-of-momentum frame
In the
center-of-momentum frame
In physics, the center-of-momentum frame (also zero-momentum frame or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The ''center of momentum'' of a system is ...
(COM frame), by definition we have:
:
with the implication from () that the invariant mass is also the centre of momentum (COM) mass–energy, aside from the factor:
:
and this is true for ''all'' frames since is frame-independent. The energies are those in the COM frame, ''not'' the lab frame. However, many familiar bound systems have the lab frame as COM frame, since the system itself is not in motion and so the momenta all cancel to zero. An example would be a simple object (where vibrational momenta of atoms cancel) or a container of gas where the container is at rest. In such systems, all the energies of the system are measured as mass. For example the heat in an object on a scale, or the total of kinetic energies in a container of gas on the scale, all are measured by the scale as the mass of the system.
Rest masses and the invariant mass
Either the energies or momenta of the particles, as measured in some frame, can be eliminated using the energy momentum relation for each particle:
:
allowing to be expressed in terms of the energies and rest masses, or momenta and rest masses. In a particular frame, the squares of sums can be rewritten as sums of squares (and products):
:
:
so substituting the sums, we can introduce their rest masses in ():
:
The energies can be eliminated by:
:
similarly the momenta can be eliminated by:
:
where is the angle between the momentum vectors and .
Rearranging:
:
Since the invariant mass of the system and the rest masses of each particle are frame-independent, the right hand side is also an invariant (even though the energies and momenta are all measured in a particular frame).
Matter waves
Using the
de Broglie relations
Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wav ...
for energy and momentum for
matter wave
Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wav ...
s,
:
where is the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
and is the
wavevector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
with magnitude , equal to the
wave number
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
, the energy–momentum relation can be expressed in terms of wave quantities:
:
and tidying up by dividing by throughout:
This can also be derived from the magnitude of the
four-wavevector
In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
:
in a similar way to the four-momentum above.
Since the
reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
and the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
both appear and clutter this equation, this is where
natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
are especially helpful. Normalizing them so that , we have:
:
Tachyon and exotic matter
The velocity of a
bradyon
The physics technical term massive particle refers to a massful particle which has real non-zero rest mass (such as baryonic matter), the counter-part to the term massless particle. According to special relativity, the velocity of a massive particl ...
with the relativistic energy–momentum relation
:
can never exceed . On the contrary, it is always greater than for a
tachyon
A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such partic ...
whose energy–momentum equation is
:
By contrast, the hypothetical
exotic matter
There are several proposed types of exotic matter:
* Hypothetical particles and states of matter that have "exotic" physical properties that would violate known laws of physics, such as a particle having a negative mass.
* Hypothetical partic ...
has a
negative mass
In theoretical physics, negative mass is a type of exotic matter whose mass is of opposite sign to the mass of normal matter, e.g. −1 kg. Such matter would violate one or more energy conditions and show some strange properties such as t ...
and the energy–momentum equation is
:
See also
*
Mass–energy equivalence
In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicis ...
*
Four-momentum
In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
*
Mass in special relativity
The word " mass" has two meanings in special relativity: '' invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity ...
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{{DEFAULTSORT:Energy-Momentum Relation
Momentum
Special relativity