Four-momentum

In special relativity, four-momentum is a four-vector that replaces classical momentum; the four-momentum of a particle is defined as the particle's mass times the particle's four-velocity.

<math> P^a := mU^a= m\left( \gamma c , \gamma u_x , \gamma u_y ,\gamma u_z \right) = \left( \gamma m c^2 /c , \gamma m u_x , \gamma m u_y ,\gamma m u_z \right) = \left( {E \over c} , \gamma p_x , \gamma p_y ,\gamma p_z \right)</math>

where

<math> \gamma m c^2 = E \,\!</math>

is the energy of the moving body, and c is the speed of light.

Calculating the Minkowski norm of the four-momentum gives:

<math> P^aP_a \ = {E^2 \over c^2} - {\gamma}^2 m^2 u^2 = m^2c^2 </math>

Since c is a constant, we may say that, choosing units of measurement in which c = 1, the Minkowski norm of the four-momentum is equal to the body's mass.

The conservation of the four-momentum yields three laws of "classical" conservation:

1. The energy (p⁰) is conserved.
2. The classical momentum is conserved.
3. The norm of the four-momentum is conserved.

In reactions between an isolated handful of particles, four-momentum is conserved. The mass of a system of particles may be more than the sum of the particle's masses, since kinetic energy counts as mass. As an example, two particles with the four-momentums {5, 4, 0, 0} and {5, -4, 0, 0} both have the mass 3, but their total mass is 10. Note that the length of the four-vector {t, x, y, z} is <math>\sqrt{t^2-x^2-y^2-z^2}</math>.

The Minkowski inner product of a four-momentum and the corresponding four-acceleration is always 0.

See also

• four-vector
• four-velocity
• four-acceleration
• four-force

References

• Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd), Oxford: Oxford University Press. ISBN 0198539525.es:Cuadrimomento

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