FOUR-MOMENTUM

(Redirected from 4-momentum)
In special relativity, 'four-momentum' is the generalization of the classical three-dimensional momentum to four-dimensional space-time. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in space-time. The covariant four-momentum of a particle with three-momentum and energy $E$ is
:$$
egin{pmatrix}
p_0 \ p_1 \ p_2 \ p_3
end{pmatrix} =
egin{pmatrix}
-E \ p_x \ p_y \ p_z
end{pmatrix}

The four-momentum 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 transformations.

Contents

Minkowski norm: p2

Relation to four-velocity

Conservation of four-momentum

Canonical momentum in the presence of an electromagnetic potential

See also

References

Minkowski norm: p²

Calculating the Minkowski norm of the four-momentum gives a Lorentz invariant quantity equal (up to factors of the speed of light ''c'') to the square of the particle's proper mass:
:$-\; |p|^2\; =\; -\; eta^\{mu$
u} p_mu p_
u = {E^2 over c^2} - |ec p|^2 = m^2c^2
where we use the SI units convention that
:
-1/c^2 & 0 & 0 & 0\
0 & 1 & 0 & 0\
0 & 0 & 1 & 0\
0 & 0 & 0 & 1
end{pmatrix}
is the reciprocal of the metric tensor of special relativity. Because $|p|^2!$ is Lorentz invariant, its value is not changed by Lorentz transformations, i.e. boosts into different frames of reference.

Relation to four-velocity

For a massive particle, the four-momentum is given by the particle's invariant mass times the particle's four-velocity:
:$p\_mu\; =\; m\; ,\; eta\_\{mu$
u} U^
u!
where the four-velocity is
:$$
egin{pmatrix}
U^0 \ U^1 \ U^2 \ U^3
end{pmatrix} =
egin{pmatrix}
gamma \ gamma v^x \ gamma v^y \ gamma v^z
end{pmatrix}

and is the Lorentz factor and ''c'' is the speed of light.

Conservation of four-momentum

The conservation of the four-momentum yields two conservation laws for "classical" quantities:
# The total energy E = - p₀ is conserved.
# The classical three-momentum is conserved.
Note that the 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 counts as system mass. As an example, two particles with the four-momentums {-5 Gev, 4 Gev/''c'', 0, 0} and {-5 Gev, -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 four-momenta p^A and p^B of two daughter particles produced in the decay of a heavier particle with four-momentum q to find the mass of the heavier particle. Conservation of four-momentum gives q_μ = p^A_μ + p^B_μ, while the mass M of the heavier particle is given by -|q|² = M²c². 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 M. This technique is used, e.g., in experimental searches for Z' bosons at high-energy 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 an object's mass does not change, the Minkowski inner product of its four-momentum and corresponding four-acceleration A^μ is zero. The acceleration is proportional to the time derivative of the momentum divided by the particle's mass, so
:
u} p_{
u}}{m} = rac{1}{2m} rac{d}{dt} |p|^2 = rac{1}{2m} rac{d}{dt} (-m^2c^2) = 0 .

Canonical momentum in the presence of an electromagnetic potential

For applications in relativistic quantum mechanics, it is useful to define a "canonical" momentum four-vector, $P\_mu$, which is the sum of the four-momentum and the product of the electric charge with the four-vector potential:
:$P\_\{mu\}\; =\; p\_\{mu\}\; +\; q\; A\_\{mu\}\; !$
where the four-vector potential is a result of combining the scalar potential and the vector potential:
:$$
egin{pmatrix}
A_0 \ A_1 \ A_2 \ A_3
end{pmatrix} =
egin{pmatrix}
-phi \ A_x \ A_y \ A_z
end{pmatrix}

This 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 into the Schroedinger equation.

See also

★ Momentum

★ Four-vector

★ Special relativity

References

★ Introduction to Special Relativity (2nd), Rindler, Wolfgang, , , Oxford University Press, 1991, ISBN 0-19-853952-5