Editing Momentum (section)

====Lorentz invariance====
Newtonian physics assumes that [[absolute time and space]] exist outside of any observer; this gives rise to [[Galilean invariance]]. It also results in a prediction that the [[speed of light]] can vary from one reference frame to another. This is contrary to what has been observed. In the [[special theory of relativity]], Einstein keeps the postulate that the equations of motion do not depend on the reference frame, but assumes that the speed of light {{mvar|c}} is invariant. As a result, position and time in two reference frames are related by the [[Lorentz transformation]] instead of the [[Galilean transformation]].<ref name=RindlerCh2>{{harvnb|Rindler|1986|loc=Chapter 2}}</ref>

Consider, for example, one reference frame moving relative to another at velocity {{mvar|v}} in the {{mvar|x}} direction. The Galilean transformation gives the coordinates of the moving frame as

<math display="block">\begin{align}
  t' &= t  \\
  x' &= x - v t
\end{align}</math>

while the Lorentz transformation gives<ref name=FeynmanCh15>[https://feynmanlectures.caltech.edu/I_15.html#Ch15-S2 ''The Feynman Lectures on Physics''] Vol. I Ch. 15-2: The Lorentz transformation</ref>

<math display="block">\begin{align}
  t' &= \gamma \left( t - \frac{v x}{c^2} \right)  \\
  x' &= \gamma \left( x - v t \right)\,
\end{align}</math>

where {{mvar|γ}} is the [[Lorentz factor]]:

<math qid=Q599404 display="block">\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}. </math>

Newton's second law, with mass fixed, is not invariant under a Lorentz transformation. However, it can be made invariant by making the ''inertial mass'' {{mvar|m}} of an object a function of velocity:

<math display="block">m = \gamma m_0\,;</math>

{{math|{{var|m}}{{sub|0}}}} is the object's [[invariant mass]].<ref name=Rindler>{{harvnb|Rindler|1986|pp=77–81}}</ref>

The modified momentum,

<math display="block"> \mathbf{p} = \gamma m_0 \mathbf{v}\,,</math>

obeys Newton's second law:

<math display="block"> \mathbf{F} = \frac{d \mathbf{p}}{dt}\,.</math>

Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum: at low velocity, {{math|{{var|γ}}{{var|m}}{{sub|0}}'''v'''}} is approximately equal to {{math|{{var|m}}{{sub|0}}'''v'''}}, the Newtonian expression for momentum.