Information About

Four-momentum
  CATEGORIES ABOUT FOUR-MOMENTUM
   
minkowski spacetime
   
relativity
   
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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 .

: P^a := mU^a= m\left( \gamma c , \gamma u_x , \gamma u_y ,\gamma u_z ight) = \left( \gamma m c^2 /c , \gamma m u_x , \gamma m u_y ,\gamma m u_z ight) = \left( {E \over c} , \gamma p_x , \gamma p_y ,\gamma p_z ight)
where
: \gamma m c^2 = E \,\!
is the energy of the moving body, and ''c'' is the Speed Of Light .

Calculating the Minkowski Norm of the four-momentum gives:
: P^aP_a \ = {E^2 \over c^2} - {\gamma}^2 m^2 u^2 = m^2c^2

Since ''c'' is a constant, we may say that, choosing Units Of Measurement in which ''c = 1'', the Minkowski norm (essentially a length) of the four-momentum, is equal to the body's Mass .

The conservation of the four-momentum yields three laws of "classical" conservation:
# The Energy (''p0'') is conserved.
# The classical Momentum is conserved.
# The Norm of the four-momentum (mass) is conserved.

In reactions between an isolated handful of particles, the total four-momentum is Conserved , and therefore, so is the system mass (this is Mass in the ordinary sense of Rest Mass , as used for m\,\! above). 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, 4, 0, 0} and {5, -4, 0, 0} both have (rest) mass 3, but their total mass (the system mass) is 10. If these particles were to collide and stick, the mass of the composite object would be 10.

Note that the length (norm) of the four-vector {t, x, y, z} is \sqrt{t^2-x^2-y^2-z^2}. Similar rules apply to the 4-momentum vector. The length of these 4-vectors is unchanged in Lorentz transformations.

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


SEE ALSO



REFERENCES