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In [[special relativity]], '''four-momentum''' is the generalization of the [[classical three-dimensional momentum]] to [[four-dimensional spacetime]]. Momentum is a vector in [[three dimensions]]; similarly four-momentum is a [[four-vector]] in [[spacetime]].  The [[contravariant vector|contravariant]] four-momentum of a particle with relativistic energy {{math|''E''}} and three-momentum {{math|1='''p''' = (''p''<sub>x</sub>, ''p''<sub>y</sub>, ''p''<sub>z</sub>) = ''γm'''''v'''}}, where {{math|'''v'''}} is the particle's three-velocity and {{mvar|γ}} the [[Lorentz factor]], is

:<math>
p = (p^0 , p^1 , p^2 , p^3 ) = \left({E \over c} , p_x , p_y , p_z\right).
</math>

The quantity {{math|''m'''''v'''}} of above is ordinary [[Momentum#Single particle|non-relativistic momentum]] of the particle and {{mvar|m}} its [[rest mass]]. The four-momentum is useful in relativistic calculations because it is a [[Lorentz covariant]] vector.  This means that it is easy to keep track of how it transforms under [[Lorentz transformation]]s.

The above definition applies under the coordinate convention that {{math|1=''x''<sup>0</sup> = ''ct''}}. Some authors use the convention {{math|1=''x''<sup>0</sup> = ''t''}}, which yields a modified definition with {{math|1=''p''<sup>0</sup> = ''E''/''c''<sup>2</sup>}}. It is also possible to define [[covariant vector|covariant]] four-momentum {{math|''p''<sub>''μ''</sub>}} where the sign of the energy is reversed.

== Minkowski norm ==

Calculating the [[Minkowski space#Mathematical structure|Minkowski norm squared]] of the four-momentum gives a [[Lorentz invariant]] quantity equal (up to factors of the [[speed of light]] {{math|''c''}}) to the square of the particle's [[proper mass]]:

:<math>p \cdot p = \eta_{\mu\nu} p^\mu p^\nu = p_\nu p^\nu = -{E^2 \over c^2} + |\mathbf p|^2 = -m^2c^2 </math>

where

:<math> \eta_{\mu\nu} = \left(\begin{matrix}
  -1 & 0 & 0 & 0\\
   0 & 1 & 0 & 0\\
   0 & 0 & 1 & 0\\
   0 & 0 & 0 & 1
\end{matrix}\right) </math>

is the metric tensor of [[special relativity]] with [[metric signature]] for definiteness chosen to be {{math|(–1, 1, 1, 1)}}. The negativity of the norm reflects that the momentum is a [[timelike]] four-vector for massive particles. The other choice of signature would flip signs in certain formulas (like for the norm here). This choice is not important, but once made it must for consistency be kept throughout.

The Minkowski norm is Lorentz invariant, meaning its value is not changed by Lorentz transformations/boosting into different frames of reference. More generally, for any two four-momenta {{mvar|p}} and {{mvar|q}}, the quantity {{math|''p'' ⋅ ''q''}} is invariant.

== Relation to four-velocity ==

For a massive particle, the four-momentum is given by the particle's [[invariant mass]] ''m'' multiplied by the particle's [[four-velocity]],

:<math>p^\mu = m u^\mu,</math>

where the four-velocity {{math|''u''}} is

:<math>
  u = \left(u^0 , u^1 , u^2 , u^3\right) = \gamma_v \left(c , v_x , v_y , v_z\right),
</math>

and

:<math>\gamma_v = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}</math>

is the Lorentz factor (associated with the speed {{math|''v''}}), {{math|''c''}} is the [[speed of light]].

== Derivation ==
There are several ways to arrive at the correct expression for four-momentum. One way is to first define the four-velocity {{math|1=''u'' = ''dx''/''dτ''}} and simply define {{math|1=''p'' = ''mu''}}, being content that it is a four-vector with the correct units and correct behavior. Another, more satisfactory, approach is to begin with the [[principle of least action]] and use the [[Lagrangian mechanics|Lagrangian framework]] to derive the four-momentum, including the expression for the energy.<ref>{{harvnb|Landau|Lifshitz|2002|pp=25&ndash;29}}</ref> One may at once, using the observations detailed below, define four-momentum from the [[Action (physics)#Single relativistic particle|action]] {{mvar|S}}. Given that in general for a closed system with [[generalized coordinate]]s {{math|''q''<sub>''i''</sub>}} and [[canonical momentum|canonical momenta]] {{math|''p''<sub>''i''</sub>}},<ref>{{harvnb|Landau|Lifshitz|1975|pp=139}}</ref>

:<math>p_i = \frac{\partial S}{\partial q_i}, \quad E = -\frac{\partial S}{\partial t},</math>

it is immediate (recalling {{math|''x''<sup>0</sup> {{=}} ''ct'', ''x''<sup>1</sup> {{=}} ''x''}}, {{math|''x''<sup>2</sup> {{=}} ''y''}}, {{math|''x''<sup>3</sup> {{=}} ''z''}} and {{math|''x''<sub>0</sub> {{=}} −''x''<sup>0</sup>}}, {{math|''x''<sub>1</sub> {{=}} ''x''<sup>1</sup>}}, {{math|''x''<sub>2</sub> {{=}} ''x''<sup>2</sup>}}, {{math|''x''<sub>3</sub> {{=}} ''x''<sup>3</sup>}} in the present metric convention) that

:<math>p_\mu = -\frac{\partial S}{\partial x^\mu} = \left({E \over c}, -\mathbf p\right)</math>

is a covariant four-vector with the three-vector part being the (negative of) canonical momentum.
{{Hidden begin
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Consider initially a system of one degree of freedom {{math|''q''}}. In the derivation of the [[equations of motion]] from the action using [[Hamilton's principle]], one finds (generally) in an intermediate stage for the [[Calculus of variation|variation]] of the action,

:<math>\delta S = \left. \left[ \frac{\partial L}{\partial \dot q}\delta q\right]\right|_{t_1}^{t_2} + \int_{t_1}^{t_2}\left( \frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot q}\right)\delta q dt.</math>

The assumption is then that the varied paths satisfy {{math|1=''δq''(''t''<sub>1</sub>) = ''δq''(''t''<sub>2</sub>) = 0}}, from which [[Lagrange's equations]] follow at once. When the equations of motion are known (or simply assumed to be satisfied), one may let go of the requirement {{math|1=''δq''(''t''<sub>2</sub>) = 0}}. In this case the path is ''assumed'' to satisfy the equations of motion, and the action is a function of the upper integration limit {{math|''δq''(''t''<sub>2</sub>)}}, but {{math|''t''<sub>2</sub>}} is still fixed. The above equation becomes with {{math|1=''S'' = ''S''(''q'')}}, and defining {{math|1=''δq''(''t''<sub>2</sub>) = ''δq''}}, and letting in more degrees of freedom,

:<math>\delta S = \sum_i \frac{\partial L}{\partial \dot{q}_i}\delta q_i = \sum_i p_i \delta q_i.</math>

Observing that

:<math>\delta S = \sum_i \frac{\partial S}{\partial {q}_i}\delta q_i,</math>

one concludes

:<math>p_i = \frac{\partial S}{\partial q_i}.</math>

In a similar fashion, keep endpoints fixed, but let {{math|''t''<sub>2</sub> {{=}} ''t''}} vary. This time, the system is allowed to move through configuration space at "arbitrary speed" or with "more or less energy", the field equations still assumed to hold and variation can be carried out on the integral, but instead observe

:<math>\frac{dS}{dt} = L</math>

by the [[fundamental theorem of calculus]]. Compute using the above expression for canonical momenta,

:<math>
  \frac{dS}{dt} = \frac{\partial S}{\partial t} + \sum_i \frac{\partial S}{\partial q_i}\dot{q}_i =
  \frac{\partial S}{\partial t} + \sum_i p_i\dot{q}_i = L.
</math>

Now using

:<math>H = \sum_i p_i\dot{q}_i - L,</math>

where {{mvar|H}} is the [[Hamiltonian function|Hamiltonian]], leads to, since {{math|1=''E'' = ''H''}} in the present case,

:<math>E = H = -\frac{\partial S}{\partial t}.</math>

Incidentally, using {{math|1=''H'' = ''H''(''q'', ''p'', ''t'')}} with {{math|1=''p'' = {{sfrac|∂''S''|∂''q''}}}} in the above equation yields the [[Hamilton&ndash;Jacobi equations]]. In this context, {{mvar|S}} is called [[Hamilton's principal function]].
----
{{Hidden end}}

The action {{mvar|S}} is given by

:<math>S = -mc\int ds = \int L dt, \quad L = -mc^2\sqrt{1 - \frac{v^2}{c^2}},</math>

where {{mvar|L}} is the relativistic [[Lagrangian mechanics|Lagrangian]] for a free particle. From this,
{{Hidden begin
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The variation of the action is

:<math>\delta S = -mc\int \delta ds.</math>

To calculate {{math|''δds''}}, observe first that {{math|1= ''δds''<sup>2</sup> = 2''dsδds''}} and that

:<math>\delta ds^2 = \delta \eta_{\mu\nu}dx^\mu dx^\nu = \eta_{\mu\nu} \left(\delta \left(dx^\mu\right) dx^\nu + dx^\mu \delta \left(dx^\nu\right)\right)
= 2\eta_{\mu\nu} \delta \left(dx^\mu\right) dx^\nu.</math>

So
:<math>\delta ds = \eta_{\mu\nu} \delta dx^\mu \frac{dx^\nu}{ds} = \eta_{\mu\nu} d\delta x^\mu \frac{dx^\nu}{ds},</math>

or
:<math>\delta ds = \eta_{\mu\nu} \frac{d\delta x^\mu}{d\tau} \frac{dx^\nu}{cd\tau}d\tau,</math>

and thus
:<math>\delta S =
  -m\int \eta_{\mu\nu}  \frac{d\delta x^\mu}{d\tau} \frac{dx^\nu}{d\tau}d\tau =
  -m\int \eta_{\mu\nu} \frac{d\delta x^\mu}{d\tau} u^\nu d\tau =
  -m\int \eta_{\mu\nu} \left\lbrack\frac{d}{d\tau}\left(\delta x^\mu u^\nu\right) - \delta x^\mu\frac{d}{d\tau}u^\nu\right\rbrack d\tau
</math>

which is just
:<math>\delta S = \left[ -mu_\mu\delta x^\mu\right]_{t_1}^{t_2} + m\int_{t_1}^{t_2}\delta x^\mu\frac{du_\mu}{ds}ds</math>
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{{Hidden end}}
:<math>\delta S = \left[ -mu_\mu\delta x^\mu\right]_{t_1}^{t_2} + m\int_{t_1}^{t_2}\delta x^\mu\frac{du_\mu}{ds}ds = -mu_\mu\delta x^\mu = \frac{\partial S}{\partial x^\mu}\delta x^\mu = -p_\mu\delta x^\mu,</math>

where the second step employs the field equations {{math|''du''<sup>''μ''</sup>/''ds'' {{=}} 0}}, {{math|(''δx''<sup>''μ''</sup>)<sub>''t''<sub>1</sub></sub> {{=}} 0}}, and {{math|(''δx''<sup>''μ''</sup>)<sub>''t''<sub>2</sub></sub> ≡ ''δx''<sup>''μ''</sup>}} as in the observations above. Now compare the last three expressions to find

:<math>p^\mu = -\partial^\mu[S] = -\frac{\partial S}{\partial x_\mu} = mu^\mu = m\left(\frac{c}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_x}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_y}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_z}{\sqrt{1 - \frac{v^2}{c^2}}}\right),</math>

with norm {{math|−''m''<sup>2</sup>''c''<sup>2</sup>}}, and the famed result for the relativistic energy,

{{Equation box 1
 |indent =:
 |title=
 |equation = <math>E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} = m_{r}c^2,</math>
 |cellpadding = 6
 |border
 |border colour = #0073CF
 |bgcolor = #F9FFF7
}}

where {{math|''m''<sub>''r''</sub>}} is the now unfashionable [[Mass in special relativity#Relativistc mass|relativistic mass]], follows. By comparing the expressions for momentum and energy directly, one has

{{Equation box 1
 |indent =:
 |title=
 |equation = <math>\mathbf p = E\frac{\mathbf v}{c^2},</math>
 |cellpadding= 6
 |border
 |border colour = #0073CF
 |bgcolor = #F9FFF7
}}

that holds for massless particles as well. Squaring the expressions for energy and three-momentum and relating them gives the [[energy–momentum relation]],

{{Equation box 1
|indent =:
|title=
|equation = <math>\frac{E^2}{c^2} = \mathbf p \cdot \mathbf p + m^2c^2.</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}

Substituting
:<math>p_\mu \leftrightarrow -\frac{\partial S}{\partial x^\mu}</math>

in the equation for the norm gives the relativistic [[Hamilton&ndash;Jacobi equation]],<ref>{{harvnb|Landau|Lifshitz|1975|p=30}}</ref>

{{Equation box 1
|indent =:
|title=
|equation = <math>\eta^{\mu\nu}\frac{\partial S}{\partial x^\mu}\frac{\partial S}{\partial x^\nu} = -m^2c^2.</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}

It is also possible to derive the results from the Lagrangian directly. By definition,<ref>{{harvnb|Landau|Lifshitz|1975|pp=15&ndash;16}}</ref>

:<math>\begin{align}
  \mathbf p &= \frac{\partial L}{\partial \mathbf v} 
             = \left({\partial L\over \partial \dot x},{\partial L\over\partial \dot y},{\partial L\over\partial \dot z}\right)
             = m(\gamma v_x, \gamma v_y, \gamma v_z) = m\gamma \mathbf v
             = m \mathbf u ,\\
          E &= \mathbf p \cdot \mathbf v - L = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}},
\end{align}</math>

which constitute the standard formulae for canonical momentum and energy of a closed (time-independent Lagrangian) system. With this approach it is less clear that the energy and momentum are parts of a four-vector.

The energy and the three-momentum are ''separately conserved'' quantities for isolated systems in the Lagrangian framework. Hence four-momentum is conserved as well. More on this below.

More pedestrian approaches include expected behavior in electrodynamics.<ref>{{harvnb|Sard|1970|loc=Section 3.1}}</ref> In this approach, the starting point is application of [[Lorentz force law]] and [[Newton's second law]] in the rest frame of the particle. The transformation properties of the electromagnetic field tensor, including invariance of [[electric charge]], are then used to transform to the lab frame, and the resulting expression (again Lorentz force law) is interpreted in the spirit of Newton's second law, leading to the correct expression for the relativistic three- momentum. The disadvantage, of  course, is that it isn't immediately clear that the result applies to all particles, whether charged or not, and that it doesn't yield the complete four-vector.

It is also possible to avoid electromagnetism and use well tuned experiments of thought involving well-trained physicists throwing billiard balls, utilizing knowledge of the [[velocity addition formula]] and assuming conservation of momentum.<ref>{{harvnb|Sard|1970|loc=Section 3.2}}</ref><ref>{{harvnb|Lewis|Tolman|1909}} [[s:The Principle of Relativity, and Non-Newtonian Mechanics|Wikisource version]]</ref> This too gives only the three-vector part.

== Conservation of four-momentum ==
As shown above, there are three (not independent, the last two imply the first) conservation laws:
* The four-[[momentum]] {{math|''p''}} (either covariant or contravariant) is conserved.
* The total [[energy]] {{math|1=''E'' = ''p''<sup>0</sup>''c''}} is conserved.
* The [[3-space]] [[momentum]] {{math|'''p'''}} 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 energy]] in the system center-of-mass frame and [[potential energy]] from forces between the particles contribute to the invariant mass. As an example, two particles with four-momenta {{nowrap|(5 GeV/''c'', 4 GeV/''c'', 0, 0)}} and {{nowrap|(5 GeV/''c'', −4 GeV/''c'', 0, 0)}} each have (rest) mass 3 GeV/''c''<sup>2</sup> separately, but their total mass (the system mass) is 10 GeV/''c''<sup>2</sup>. If these particles were to collide and stick, the mass of the composite object would be 10 GeV/''c''<sup>2</sup>.

One practical application from [[particle physics]] of the conservation of the [[invariant mass]] involves combining the four-momenta {{math|''p''<sub>A</sub>}} and {{math|''p''<sub>B</sub>}} of two daughter particles produced in the decay of a heavier particle with four-momentum {{math|''p''<sub>C</sub>}} to find the mass of the heavier particle. Conservation of four-momentum gives {{math|1=''p''<sub>C</sub><sup>''μ''</sup> = ''p''<sub>A</sub><sup>''μ''</sup> + ''p''<sub>B</sub><sup>''μ''</sup>}}, while the mass {{math|''M''}} of the heavier particle is given by {{math|1=−''P''<sub>C</sub> ⋅ ''P''<sub>C</sub> = ''M''<sup>2</sup>''c''<sup>2</sup>}}. By measuring the energies and three-momenta of the daughter particles, one can reconstruct the invariant mass of the two-particle system, which must be equal to {{math|''M''}}. This technique is used, e.g., in experimental searches for [[Z′ boson]]s at high-energy particle [[collider]]s, where the Z′ boson would show up as a bump in the invariant mass spectrum of [[electron]]–[[positron]] or [[muon]]–antimuon pairs.

If the mass of an object does not change, the Minkowski inner product of its four-momentum and corresponding [[four-acceleration]] {{math|''A''<sup>''μ''</sup>}} is simply zero. The four-acceleration is proportional to the proper time derivative of the four-momentum divided by the particle's mass, so
:<math>p^\mu A_\mu = \eta_{\mu\nu} p^\mu A^\nu = \eta_{\mu\nu} p^\mu \frac{d}{d\tau} \frac{p^{\nu}}{m} = \frac{1}{2m} \frac{d}{d\tau} p \cdot p = \frac{1}{2m} \frac{d}{d\tau} \left(-m^2c^2\right) = 0 .</math>

== Canonical momentum in the presence of an electromagnetic potential ==
For a [[charged particle]] of [[electric charge|charge]] {{math|''q''}}, moving in an electromagnetic field given by the [[electromagnetic four-potential]]:

:<math>
  A = \left(A^0 , A^1 , A^2 , A^3\right) = \left({\phi \over  c}, A_x , A_y , A_z\right)
</math>

where {{mvar|Φ}} is the [[scalar potential]] and {{math|1='''A''' = (''A''<sub>x</sub>, ''A''<sub>y</sub>, ''A''<sub>z</sub>)}} the [[vector potential]], the components of the (not [[gauge invariance|gauge-invariant]]) canonical momentum four-vector {{mvar|P}} is

:<math> P^\mu = p^\mu + q A^\mu. \!</math>

This, in turn, allows the potential energy from the charged particle in an electrostatic potential and the [[Lorentz force]] on the charged particle moving in a magnetic field to be incorporated in a compact way, in [[relativistic quantum mechanics]].

== See also ==
{{portal|Physics}}
* [[Four-force]]
* [[Four-gradient]]
* [[Pauli–Lubanski pseudovector]]

== References ==
{{Reflist}}
*{{cite book|last=Goldstein|first=Herbert|title=Classical mechanics|year=1980|publisher=Addison–Wesley Pub. Co.|location=Reading, Mass.|isbn=978-0201029185|edition=2nd |ref=harv}}
*{{Cite book|ref=harv|last=Landau|first=L. D.|authorlink1=Lev Landau|last2=Lifshitz|first2=E. M.|authorlink2=E. M. Lifshitz|title=Mechanics|year=1975|edition=3rd|orig-year=1939|isbn=978-0-7506-28969|publisher=Elsevier|location=Amsterdam|others=Translated from Russian by J. B. Sykes and [[John Stewart Bell|J. S. Bell]].}} 
*{{cite book|last=Landau|first=L.D.|first2=E.M.|last2=Lifshitz|title=The classical theory of fields|year=2000|publisher=Butterworth Heinemann|location=Oxford|isbn=9780750627689 |others=4th rev. English edition, reprinted with corrections; translated from the Russian by Morton Hamermesh|ref=harv}}
*{{cite book | author = Rindler, Wolfgang | title=Introduction to Special Relativity |edition=2nd| location= Oxford | publisher=Oxford University Press | year=1991 | isbn=978-0-19-853952-0 |ref=harv}}
*{{cite book|ref=harv|last=Sard|first=R. D.|title=Relativistic Mechanics - Special Relativity and Classical Particle Dynamics|year=1970|publisher=W. A. Benjamin|location=New York|isbn=978-0805384918}}
*{{cite journal|ref=harv|first1=G. N.|last1=Lewis|authorlink1=Gilbert N. Lewis|first2=R. C.|last2=Tolman|authorlink2=Richard C. Tolman|title=The Principle of Relativity, and Non-Newtonian Mechanics|journal=Phil. Mag.|series=6|volume=18|issue=106|doi=10.1080/14786441008636725|pages=510&ndash;523|year=1909|url=https://zenodo.org/record/1430872}} [[s:The Principle of Relativity, and Non-Newtonian Mechanics|Wikisource version]]

[[Category:Four-vectors]]