Four-momentum

 ( Four-vectors ) 

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 a four-vector in spacetime. The contravariant four-momentum of a particle with relativistic energy and three-momentum , where is the particle's three-velocity and the Lorentz factor, is $$p\; =\; \backslash left(p^0\; ,\; p^1\; ,\; p^2\; ,\; p^3\backslash right)\; =\; \backslash left(\backslash frac\; E\; c\; ,\; p\_x\; ,\; p\_y\; ,\; p\_z\backslash right).$$

The quantity of above is the ordinary non-relativistic momentum of the particle and 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 transformations.

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Minkowski norm Calculating the Minkowski norm squared of the four-momentum gives a Lorentz invariant quantity equal (up to factors of the speed of light ) to the square of the particle's proper mass:

$$p\; \backslash cdot\; p\; =\; \backslash eta\_\{\backslash mu\backslash nu\}\; p^\backslash mu\; p^\backslash nu\; =\; p\_\backslash nu\; p^\backslash nu\; =\; -\{E^2\; \backslash over\; c^2\}\; +\; |\backslash mathbf\; p|^2\; =\; -m^2\; c^2$$ where the following denote:

$p$, the four-momentum vector of a particle,

$p\; \backslash cdot\; p$, the Minkowski inner product of the four-momentum with itself,

$p^\backslash mu$and $p^\backslash nu$, the contravariant components of the four-momentum vector,

$p\_\backslash nu$, the covariant form,

$E$, the energy of the particle,

$c$, the speed of light,

$|\backslash mathbf\; p|$, the magnitude of the four-momentum vector,

$m$, the invariant mass (rest) of the particle,

and $$\backslash eta\_\{\backslash mu\backslash nu\}\; =\; \backslash begin\{pmatrix\}$$

 -1 & 0 & 0 & 0\\
  0 & 1 & 0 & 0\\
  0 & 0 & 1 & 0\\
  0 & 0 & 0 & 1
     

\end{pmatrix} is the metric tensor of special relativity with metric signature for definiteness chosen to be . 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 and , the quantity is invariant.

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Relation to four-velocity For a massive particle, the four-momentum is given by the particle's invariant mass multiplied by the particle's four-velocity, $$p^\backslash mu\; =\; m\; u^\backslash mu,$$ where the four-velocity is $$u\; =\; \backslash left(u^0\; ,\; u^1\; ,\; u^2\; ,\; u^3\backslash right)\; =\; \backslash gamma\_v\; \backslash left(c\; ,\; v\_x\; ,\; v\_y\; ,\; v\_z\backslash right),$$ and $$\backslash gamma\_v\; :=\; \backslash frac\{1\}\{\backslash sqrt\{1\; -\; \backslash frac\{v^2\}\{c^2\}\}\}$$ is the Lorentz factor (associated with the speed $v$), is the speed of light.

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Derivation There are several ways to arrive at the correct expression for four-momentum. One way is to first define the four-velocity and simply define , 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 framework to derive the four-momentum, including the expression for the energy. One may at once, using the observations detailed below, define four-momentum from the action . Given that in general for a closed system with generalized coordinates and canonical momenta , $$p\_i\; =\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; q\_i\}\; =\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; x\_i\},\; \backslash quad\; E\; =\; -\backslash frac\{\backslash partial\; S\}\{\backslash partial\; t\}\; =\; -\; c\; \backslash cdot\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; x^\{0\}\},$$ it is immediate (recalling , , , and , , , in the present metric convention) that $$p\_\backslash mu\; =\backslash frac\{\backslash partial\; S\}\{\backslash partial\; x^\backslash mu\}\; =\; \backslash left(-\{E\; \backslash over\; c\},\; \backslash mathbf\; p\backslash right)$$ is a covariant four-vector with the three-vector part being the canonical momentum.

Consider initially a system of one degree of freedom . In the derivation of the equations of motion from the action using Hamilton's principle, one finds (generally) in an intermediate stage for the variation of the action, $$\backslash delta\; S\; =\; \backslash left.\; \backslash left\backslash right|\_\{t\_1\}^\{t\_2\}\; +\; \backslash int\_\{t\_1\}^\{t\_2\}\; \backslash left(\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; q\}\; -\; \backslash frac\{d\}\{dt\}\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\; q\}\backslash right)\backslash delta\; q\; dt.$$

The assumption is then that the varied paths satisfy , 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 . In this case the path is assumed to satisfy the equations of motion, and the action is a function of the upper integration limit , but is still fixed. The above equation becomes with , and defining , and letting in more degrees of freedom, $$\backslash delta\; S\; =\; \backslash sum\_i\; \backslash frac\{\backslash partial\; L\}\{\backslash partial\; \backslash dot\{q\}\_i\}\backslash delta\; q\_i\; =\; \backslash sum\_i\; p\_i\; \backslash delta\; q\_i.$$

Observing that $$\backslash delta\; S\; =\; \backslash sum\_i\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; \{q\}\_i\}\backslash delta\; q\_i,$$ one concludes $$p\_i\; =\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; q\_i\}.$$

In a similar fashion, keep endpoints fixed, but let 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 $$\backslash frac\{dS\}\{dt\}\; =\; L$$ by the fundamental theorem of calculus. Compute using the above expression for canonical momenta, $$$$

 \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.
     

Now using $$H\; =\; \backslash sum\_i\; p\_i\; \backslash dot\{q\}\_i\; -\; L,$$ where is the Hamiltonian, leads to, since in the present case, $$E\; =\; H\; =\; -\backslash frac\{\backslash partial\; S\}\{\backslash partial\; t\}.$$

Incidentally, using with in the above equation yields the Hamilton–Jacobi equations. In this context, is called Hamilton's principal function.

The action is given by $$S\; =\; -mc\backslash int\; ds\; =\; \backslash int\; L\; dt,\; \backslash quad\; L\; =\; -mc^2\backslash sqrt\{1\; -\; \backslash frac\{v^2\}\{c^2\}\},$$ where is the relativistic Lagrangian for a free particle. From this,

The variation of the action is $$\backslash delta\; S\; =\; -mc\backslash int\; \backslash delta\; ds.$$

To calculate , observe first that and that $$\backslash delta\; ds^2$$

 = \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.
     

So $$\backslash delta\; ds\; =\; \backslash eta\_\{\backslash mu\backslash nu\}\; \backslash delta\; dx^\backslash mu\; \backslash frac\{dx^\backslash nu\}\{ds\}\; =\; \backslash eta\_\{\backslash mu\backslash nu\}\; d\backslash delta\; x^\backslash mu\; \backslash frac\{dx^\backslash nu\}\{ds\},$$ or $$\backslash delta\; ds\; =\; \backslash eta\_\{\backslash mu\backslash nu\}\; \backslash frac\{d\backslash delta\; x^\backslash mu\}\{d\backslash tau\}\; \backslash frac\{dx^\backslash nu\}\{cd\backslash tau\}d\backslash tau,$$ and thus $$\backslash delta\; S\; =$$

 -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[\frac{d}{d\tau} \left(\delta x^\mu u^\nu\right) - \delta x^\mu\frac{d}{d\tau}u^\nu\right] d\tau
     

which is just $$\backslash delta\; S\; =\; \backslash left-mu\_\backslash mu\backslash delta\_\{t\_1\}^\{t\_2\}\; +\; m\; \backslash int\_\{t\_1\}^\{t\_2\}\; \backslash delta\; x^\backslash mu\backslash frac\{du\_\backslash mu\}\{ds\}ds$$

$$\backslash delta\; S\; =\; \backslash left\_\{t\_1\}^\{t\_2\}\; +\; m\backslash int\_\{t\_1\}^\{t\_2\}\backslash delta\; x^\backslash mu\backslash frac\{du\_\backslash mu\}\{ds\}ds\; =\; -mu\_\backslash mu\backslash delta\; x^\backslash mu\; =\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; x^\backslash mu\}\backslash delta\; x^\backslash mu\; =\; -p\_\backslash mu\backslash delta\; x^\backslash mu,$$

where the second step employs the field equations , , and as in the observations above. Now compare the last three expressions to find $$p^\backslash mu\; =\; \backslash partial^\backslash muS\; =\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; x\_\backslash mu\}\; =\; mu^\backslash mu\; =\; m\backslash left(\backslash frac\{c\}\{\backslash sqrt\{1\; -\; \backslash frac\{v^2\}\{c^2\}\}\},\; \backslash frac\{v\_x\}\{\backslash sqrt\{1\; -\; \backslash frac\{v^2\}\{c^2\}\}\},\; \backslash frac\{v\_y\}\{\backslash sqrt\{1\; -\; \backslash frac\{v^2\}\{c^2\}\}\},\; \backslash frac\{v\_z\}\{\backslash sqrt\{1\; -\; \backslash frac\{v^2\}\{c^2\}\}\}\backslash right),$$ with norm , and the famed result for the relativistic energy,

} = m_{r}c^2,

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where is the now unfashionable relativistic mass, follows. By comparing the expressions for momentum and energy directly, one has

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

Substituting $$p\_\backslash mu\; \backslash leftrightarrow\; -\backslash frac\{\backslash partial\; S\}\{\backslash partial\; x^\backslash mu\}$$ in the equation for the norm gives the relativistic Hamilton–Jacobi equation,

It is also possible to derive the results from the Lagrangian directly. By definition, $$\backslash begin\{align\}$$

 \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 , \\[3pt]
         E &= \mathbf p \cdot \mathbf v - L = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}},
     

\end{align} 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. 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. This too gives only the three-vector part.

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Conservation of four-momentum As shown above, there are three conservation laws (not independent, the last two imply the first and vice versa):

• The four-momentum (either covariant or contravariant) is conserved.
• The total energy is conserved.
• The 3-space momentum $\backslash mathbf\{p\}\; =\; \backslash left(p^1,\; p^2,\; p^3\backslash right)$ is conserved (not to be confused with the classic non-relativistic momentum $m\backslash mathbf\{v\}$).

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 and each have (rest) mass 3GeV/ c² separately, but their total mass (the system mass) is 10GeV/ c². If these particles were to collide and stick, the mass of the composite object would be 10GeV/ c².

One practical application from particle physics of the conservation of the invariant mass involves combining the four-momenta and of two daughter particles produced in the decay of a heavier particle with four-momentum to find the mass of the heavier particle. Conservation of four-momentum gives , while the mass of the heavier particle is given by . 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 . This technique is used, e.g., in experimental searches for Z′ bosons at high-energy particle , 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 is simply zero. The four-acceleration is proportional to the proper time derivative of the four-momentum divided by the particle's mass, so $$p^\backslash mu\; A\_\backslash mu\; =\; \backslash eta\_\{\backslash mu\backslash nu\}\; p^\backslash mu\; A^\backslash nu\; =\; \backslash eta\_\{\backslash mu\backslash nu\}\; p^\backslash mu\; \backslash frac\{d\}\{d\backslash tau\}\; \backslash frac\{p^\{\backslash nu\}\}\{m\}\; =\; \backslash frac\{1\}\{2m\}\; \backslash frac\{d\}\{d\backslash tau\}\; p\; \backslash cdot\; p\; =\; \backslash frac\{1\}\{2m\}\; \backslash frac\{d\}\{d\backslash tau\}\; \backslash left(-m^2c^2\backslash right)\; =\; 0\; .$$

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Canonical momentum in the presence of an electromagnetic potential For a charged particle of electric charge , moving in an electromagnetic field given by the electromagnetic four-potential: $$A\; =\; \backslash left(A^0\; ,\; A^1\; ,\; A^2\; ,\; A^3\backslash right)\; =\; \backslash left(\{\backslash phi\; \backslash over\; c\},\; A\_x\; ,\; A\_y\; ,\; A\_z\backslash right)$$ where ϕ is the scalar potential and the vector potential, the components of the (not gauge invariance) canonical momentum four-vector is $$P^\backslash mu\; =\; p^\backslash mu\; +\; q\; A^\backslash mu.$$

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.

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Four-momentum in curved spacetime In the case when there is a moving physical system with a continuous distribution of matter in curved spacetime, the primary expression for four-momentum is a four-vector with covariant index:

$P\_\backslash mu\; =\; \backslash left(\backslash frac\; \{E\}\; \{c\}\; ,\; -\backslash mathbf\; P\; \backslash right).$

Four-momentum $P\_\backslash mu$ is expressed through the energy $E$ of physical system and relativistic momentum $\backslash mathbf\; P$. At the same time, the four-momentum $P\_\backslash mu$ can be represented as the sum of two non-local four-vectors of integral type:

$P\_\backslash mu\; =\; p\_\backslash mu\; +\; K\_\backslash mu.$

Four-vector $p\_\backslash mu$ is the generalized four-momentum associated with the action of fields on particles; four-vector $K\_\backslash mu$ is the four-momentum of the fields arising from the action of particles on the fields.

Energy $E$ and momentum $\backslash mathbf\; P$, as well as components of four-vectors $p\_\backslash mu$ and $K\_\backslash mu$ can be calculated if the Lagrangian density $\backslash mathcal\{L\}\; =\backslash mathcal\{L\}\_p\; +\; \backslash mathcal\{L\}\_f$ of the system is given. The following formulas are obtained for the energy and momentum of the system:

$E\; =\; \backslash int\_\{V\}\; \backslash frac\; \{\backslash partial\}\{\backslash partial\; \backslash mathbf\; v\}\; \backslash left(\; \backslash frac\; \{\; \backslash mathcal\{L\}\_p\; \}\{u^0\}\; \backslash right)\; \backslash cdot\; \backslash mathbf\; v\; u^0\; \backslash sqrt\; \{-g\}\; dx^1\; dx^2\; dx^3\; -\backslash int\_\{V\}\; \backslash left\; (\backslash mathcal\{L\}\_p\; +\; \backslash mathcal\{L\}\_f\; \backslash right\; )\; \backslash sqrt\; \{-g\}\; dx^1\; dx^2\; dx^3\; +\backslash sum\_\{n=1\}^N\; \backslash left(\; \backslash mathbf\; v\_n\; \backslash cdot\; \backslash frac\; \{\backslash partial\; L\_f\}\{\backslash partial\; \backslash mathbf\; v\_n\}\backslash right\; )\; .$

$\backslash mathbf\; P\; =\; \backslash int\_\{V\}\; \backslash frac\; \{\backslash partial\}\{\backslash partial\; \backslash mathbf\; v\}\; \backslash left(\; \backslash frac\; \{\; \backslash mathcal\{L\}\_p\; \}\{u^0\}\; \backslash right)\; u^0\; \backslash sqrt\; \{-g\}\; dx^1\; dx^2\; dx^3\; +\backslash sum\_\{n=1\}^N\; \backslash frac\; \{\backslash partial\; L\_f\}\{\backslash partial\; \backslash mathbf\; v\_n\}\; .$

Here $\backslash mathcal\{L\}\_p$ is that part of the Lagrangian density that contains terms with four-currents; $\backslash mathbf\; v$ is the velocity of matter particles; $u^0$ is the time component of four-velocity of particles; $g$ is determinant of metric tensor; $L\_f\; =\; \backslash int\_\{V\}\; \backslash mathcal\{L\}\_f\; \backslash sqrt\; \{-g\}\; dx^1\; dx^2\; dx^3$ is the part of the Lagrangian associated with the Lagrangian density $\backslash mathcal\{\; L\}\_f$; $\backslash mathbf\; v\_n$ is velocity of a particle of matter with number $n$.

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See also

• Four-force
• Four-gradient
• Pauli–Lubanski pseudovector

• Herbert Goldstein (1980). 9780201029185, Addison–Wesley Pub. Co.. ISBN 9780201029185
• (1975). 9780750628969, Elsevier. ISBN 9780750628969
• (2026). 9780750627689, Butterworth Heinemann. ISBN 9780750627689
• Rindler, Wolfgang (1991). 9780198539520, Oxford University Press. . ISBN 9780198539520
• R. D. Sard (1970). 9780805384918, W. A. Benjamin. . ISBN 9780805384918

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