section_path: Relativistic/Lorentz invariance
section_name: Lorentz invariance
paragraph_text: 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 c is invariant. As a result, position and time in two reference frames are related by the Lorentz transformation instead of the Galilean transformation.
  
  Consider, for example, one reference frame moving relative to another at velocity v in the x direction. The Galilean transformation gives the coordinates of the moving frame as
  
  t ′ = t x ′ = x − v t {\displaystyle {\begin{aligned}t'&=t\\x'&=x-vt\end{aligned}}}
  
  while the Lorentz transformation gives
  
  t ′ = γ (t − v x c 2) x ′ = γ (x − v t) {\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\,\end{aligned}}}
  
  where γ is the Lorentz factor:
  
  γ = 1 1 − v 2 / c 2. {\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}.}
  
  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 m of an object a function of velocity:
  
  m = γ m 0; {\displaystyle m=\gamma m_{0}\,;}
  
  m is the object's invariant mass.
  
  The modified momentum,
  
  p = γ m 0 v, {\displaystyle \mathbf {p} =\gamma m_{0}\mathbf {v} \,,}
  
  obeys Newton's second law:
  
  F = d p d t. {\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}\,.}
  
  Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum: at low velocity, γ m v is approximately equal to m v, the Newtonian expression for momentum.