35.11 - Relativistic linear momentum
In discussing relativistic linear momentum, and the equations that describe it, we start
with three premises.
First, the same laws of physics hold true in any inertial reference frame. This is
Einstein’s first postulate. If a property like momentum is conserved in one inertial
reference frame, it must be conserved in any other inertial reference frame.
Second, Einstein’s second postulate, that the speed of light is constant. These two
postulates were used to deduce equations for time dilation, length contraction and so
on.
Third, we expect that at much lower speeds, the equations in this section will essentially
reduce to the “classical” ones that work perfectly well at low speeds. For instance, the
momentum of an object moving at say 30 m/s should very, very closely equal mv. This
has held true with prior equations in this chapter. At speeds much less than the speed
of light, the equations predict results that accord with “classic” mechanics equations.
These three premises are satisfied by the relativistic equation for momentum at the
right. This extended equation for momentum is required because it can be shown that if
Newtonian, or classical, momentum is conserved in one reference frame, then when the
velocities are converted to the values that would be observed in another frame, the total
momentum is never conserved in that second reference frame.
Specifically, if a collision occurs in a moving reference frame S ́, and the velocities are
converted to the values that would be observed in S, the quantity mv before and after
the collision is changed by the collision. Using the equation shown to the right ensures
this will not occur, and that momentum is conserved when the collision is observed from
any reference frame.
The larger the relative speed of the frames, the greater the discrepancy between the
classical momentum and the relativistic momentum will be. It turns out that the Lorentz
factorȖ appearing in the definition of momentum shown in Equation 1 provides exactly
the right corrective factor for momentum to be conserved in both S and S ́, no matter
how fast the frames move with respect to each other. (The variable u in the formula
refers to the speed of an object in its frame, not the speed of one frame with respect to
another.)
The extended momentum equation obeys the first two premises stated above. It also
obeys the third. At velocities much less than the speed of light, momentum approaches
mv, since dividing the square of a velocity like 300 m/s by the square of speed of light
means the relativistic effect is near nil.
As the example on the right shows, at half the speed of light, relativistic effects increase
momentum by 15% over its classical value. The effect increases significantly as an
object moves at speeds closer to the speed of light: At 99% of the speed of light,
momentum is about seven times its classical value.
This phenomenon is of great importance in the modern physics research done with
large particle accelerators, such as the Fermilab accelerator in Illinois, or the CERN
accelerator in France and Switzerland. Objects move near the speed of light in these
accelerators, and calculating their momentum requires the use of the equation shown in
this section.
Relativistic momentum
New equation for momentum required
for high velocities