Fourmomentum
In special relativity, fourmomentum (also called momentumenergy or momenergy[1] ) is the generalization of the classical threedimensional momentum to fourdimensional spacetime. Momentum is a vector in three dimensions; similarly fourmomentum is a fourvector in spacetime. The contravariant fourmomentum of a particle with relativistic energy E and threemomentum p = (p_{x}, p_{y}, p_{z}) = γmv, where v is the particle's threevelocity and γ the Lorentz factor, is
Special relativity 


The quantity mv of above is ordinary nonrelativistic momentum of the particle and m its rest mass. The fourmomentum 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.
The above definition applies under the coordinate convention that x^{0} = ct. Some authors use the convention x^{0} = t, which yields a modified definition with p^{0} = E/c^{2}. It is also possible to define covariant fourmomentum p_{μ} where the sign of the energy (or the sign of the threemomentum, depending on the chosen metric signature) is reversed.
Minkowski norm
Calculating the Minkowski norm squared of the fourmomentum gives a Lorentz invariant quantity equal (up to factors of the speed of light c) to the square of the particle's proper mass:
where
is the metric tensor of special relativity with metric signature for definiteness chosen to be (–1, 1, 1, 1). The negativity of the norm reflects that the momentum is a timelike fourvector 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 fourmomenta p and q, the quantity p ⋅ q is invariant.
Relation to fourvelocity
For a massive particle, the fourmomentum is given by the particle's invariant mass m multiplied by the particle's fourvelocity,
where the fourvelocity u is
and
is the Lorentz factor (associated with the speed v), c is the speed of light.
Derivation
There are several ways to arrive at the correct expression for fourmomentum. One way is to first define the fourvelocity u = dx/dτ and simply define p = mu, being content that it is a fourvector 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 fourmomentum, including the expression for the energy.[2] One may at once, using the observations detailed below, define fourmomentum from the action S. Given that in general for a closed system with generalized coordinates q_{i} and canonical momenta p_{i},[3]
it is immediate (recalling x^{0} = ct, x^{1} = x, x^{2} = y, x^{3} = z and x_{0} = −x^{0}, x_{1} = x^{1}, x_{2} = x^{2}, x_{3} = x^{3} in the present metric convention) that
is a covariant fourvector with the threevector part being the (negative of) canonical momentum.
Consider initially a system of one degree of freedom 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 variation of the action,
The assumption is then that the varied paths satisfy δq(t_{1}) = δq(t_{2}) = 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 δq(t_{2}) = 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 δq(t_{2}), but t_{2} is still fixed. The above equation becomes with S = S(q), and defining δq(t_{2}) = δq, and letting in more degrees of freedom,
Observing that
one concludes
In a similar fashion, keep endpoints fixed, but let t_{2} = 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
by the fundamental theorem of calculus. Compute using the above expression for canonical momenta,
Now using
where H is the Hamiltonian, leads to, since E = H in the present case,
Incidentally, using H = H(q, p, t) with p = ∂S/∂q in the above equation yields the Hamilton–Jacobi equations. In this context, S is called Hamilton's principal function.
The action S is given by
where L is the relativistic Lagrangian for a free particle. From this,
The variation of the action is
To calculate δds, observe first that δds^{2} = 2dsδds and that
So
or
and thus
which is just
where the second step employs the field equations du^{μ}/ds = 0, (δx^{μ})_{t1} = 0, and (δx^{μ})_{t2} ≡ δx^{μ} as in the observations above. Now compare the last three expressions to find
with norm −m^{2}c^{2}, and the famed result for the relativistic energy,
where m_{r} 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 threemomentum and relating them gives the energy–momentum relation,
Substituting
in the equation for the norm gives the relativistic Hamilton–Jacobi equation,[4]
It is also possible to derive the results from the Lagrangian directly. By definition,[5]
which constitute the standard formulae for canonical momentum and energy of a closed (timeindependent Lagrangian) system. With this approach it is less clear that the energy and momentum are parts of a fourvector.
The energy and the threemomentum are separately conserved quantities for isolated systems in the Lagrangian framework. Hence fourmomentum is conserved as well. More on this below.
More pedestrian approaches include expected behavior in electrodynamics.[6] 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 fourvector.
It is also possible to avoid electromagnetism and use well tuned experiments of thought involving welltrained physicists throwing billiard balls, utilizing knowledge of the velocity addition formula and assuming conservation of momentum.[7][8] This too gives only the threevector part.
Conservation of fourmomentum
As shown above, there are three conservation laws (not independent, the last two imply the first and vice versa):
 The fourmomentum p (either covariant or contravariant) is conserved.
 The total energy E = p^{0}c is conserved.
 The 3space momentum is conserved (not to be confused with the classic nonrelativistic momentum ).
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 centerofmass frame and potential energy 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 physics of the conservation of the invariant mass 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} ⋅ P_{C} = 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 colliders, 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 fourmomentum and corresponding fouracceleration A^{μ} is simply 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 a charged particle of charge q, moving in an electromagnetic field given by the electromagnetic fourpotential:
where φ is the scalar potential and A = (A_{x}, A_{y}, A_{z}) the vector potential, the components of the (not gaugeinvariant) canonical momentum fourvector P is
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
 Fourforce
 Fourgradient
 Pauli–Lubanski pseudovector
References
 Taylor, Edwin; Wheeler, John (1992). Spacetime physics introduction to special relativity. New York: W. H. Freeman and Company. p. 191. ISBN 9780716723271.
 Landau & Lifshitz 2002, pp. 25–29
 Landau & Lifshitz 1975, pp. 139
 Landau & Lifshitz 1975, p. 30
 Landau & Lifshitz 1975, pp. 15–16
 Sard 1970, Section 3.1
 Sard 1970, Section 3.2
 Lewis & Tolman 1909 Wikisource version
 Goldstein, Herbert (1980). Classical mechanics (2nd ed.). Reading, Mass.: Addison–Wesley Pub. Co. ISBN 9780201029185.
 Landau, L. D.; Lifshitz, E. M. (1975) [1939]. Mechanics. Translated from Russian by J. B. Sykes and J. S. Bell. (3rd ed.). Amsterdam: Elsevier. ISBN 9780750628969.
 Landau, L.D.; Lifshitz, E.M. (2000). The classical theory of fields. 4th rev. English edition, reprinted with corrections; translated from the Russian by Morton Hamermesh. Oxford: Butterworth Heinemann. ISBN 9780750627689.
 Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford: Oxford University Press. ISBN 9780198539520.
 Sard, R. D. (1970). Relativistic Mechanics  Special Relativity and Classical Particle Dynamics. New York: W. A. Benjamin. ISBN 9780805384918.
 Lewis, G. N.; Tolman, R. C. (1909). "The Principle of Relativity, and NonNewtonian Mechanics". Phil. Mag. 6. 18 (106): 510–523. doi:10.1080/14786441008636725. Wikisource version