In
special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
,
fourmomentum is the generalization of the classical threedimensional
momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
to fourdimensional
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being threedimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. Momentum is a vector in three dimensions; similarly fourmomentum is a
fourvectorIn the theory of relativity, a fourvector is a vector in a fourdimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the fourvector name tacitly assumes that its components refer to a standard basis...
in
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being threedimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. The contravariant fourmomentum of a particle with threemomentum
and energy
is
The fourmomentum is useful in relativistic calculations because it is a Lorentz vector. This means that it is easy to keep track of how it transforms under
Lorentz transformationIn physics, the Lorentz transformation or LorentzFitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
s.
(The above definition applies under the coordinate convention that
. Some authors use the convention
which yields a modified definition with
. It is also possible to define covariant fourmomentum
where the sign of
is reversed.)
Minkowski norm
Calculating the Minkowski norm of the fourmomentum gives a Lorentz invariant quantity equal (up to factors of the
speed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
c) to the square of the particle's proper mass:
where we use the convention that
is the reciprocal of the metric tensor of
special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
.
is Lorentz invariant, meaning its value is not changed by Lorentz transformations/boosting into different frames of reference.
Relation to fourvelocity
For a massive particle, the fourmomentum is given by the particle's
invariant massThe invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
m multiplied by the particle's
fourvelocityIn physics, in particular in special relativity and general relativity, the fourvelocity of an object is a fourvector that replaces classicalvelocity...
:
where the fourvelocity is
and
is the
Lorentz factorThe Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...
and
c is the
speed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
.
Conservation of fourmomentum
The conservation of the fourmomentum yields two conservation laws for "classical" quantities:
 The total energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
E = P^{0}c is conserved.
 The classical threemomentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
is conserved.
Note that the invariant mass of a system of particles may be more than the sum of the particles' rest masses, since
kinetic energyThe kinetic energy of an object is the energy which 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 acceleration, the body maintains this kinetic energy unless its speed changes...
in the system centerofmass frame and
potential energyIn physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
from forces between the particles contribute to the invariant mass. As an example, two particles with fourmomenta (−5 GeV/
c, 4 GeV/
c, 0, 0) and (−5 GeV/
c, −4 GeV/
c, 0, 0) each have (rest) mass 3 GeV/
c^{2} separately, but their total mass (the system mass) is 10 GeV/
c^{2}. If these particles were to collide and stick, the mass of the composite object would be 10 GeV/
c^{2}.
One practical application from
particle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
of the conservation of the
invariant massThe invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
involves combining the fourmomenta
P(
A) and
P(
B) of two daughter particles produced in the decay of a heavier particle with fourmomentum
P(
C) to find the mass of the heavier particle. Conservation of fourmomentum gives
P(
C)
^{μ} =
P(
A)
^{μ} +
P(
B)
^{μ}, while the mass
M of the heavier particle is given by 
P(
C)
^{2} =
M^{2}c^{2}. By measuring the energies and threemomenta of the daughter particles, one can reconstruct the invariant mass of the twoparticle system, which must be equal to M. This technique is used, e.g., in experimental searches for Z' bosons at highenergy particle
colliderA collider is a type of a particle accelerator involving directed beams of particles.Colliders may either be ring accelerators or linear accelerators.Explanation:...
s, where the Z' boson would show up as a bump in the invariant mass spectrum of
electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

positronThe positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...
or
muonThe muon mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...
antimuon pairs.
If an object's mass does not change, the Minkowski inner product of its fourmomentum and corresponding
fouraccelerationIn special relativity, fouracceleration is a fourvector and is defined as the change in fourvelocity over the particle's proper time:whereandand \gamma_u is the Lorentz factor for the speed u...
A^{μ} is zero. The fouracceleration is proportional to the proper time derivative of the fourmomentum divided by the particle's mass, so
Canonical momentum in the presence of an electromagnetic potential
For applications in relativistic quantum mechanics, it is useful to define a "canonical" momentum fourvector,
, which is the sum of the fourmomentum and the product of the
electric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
with the
electromagnetic fourpotentialThe electromagnetic fourpotential is a potential from which the electromagnetic field can be derived. It combines both the electric scalar potential and the magnetic vector potential into a single spacetime fourvector. In a given reference frame, the first component is the scalar potential and...
:
where the fourvector potential is a result of combining the
scalar potentialA scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...
and the
vector potentialIn vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
:
This allows the potential energy from the charged particle in an electrostatic potential and the
Lorentz forceIn physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...
on the charged particle moving in a magnetic field to be incorporated in a compact way into the
Schrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
.