Lorentz transformation of time

Let us consider two inertial reference frames \(S\) (lab frame) and \(S'\) (proper frame). The proper frame moves with speed \(v\) relative to the lab frame. Both frames contain identical fixed (relative to their own frame) clocks that are synchronized according to the Einstein rule. Let \(x, y, z, t\) be the coordinates and time of some event in frame \(S\), and \(x', y', z', t'\) be the coordinates and time of the same event in frame \(S'\). Assuming that the space is uniform and isotropic and that the time is uniform, there exists a linear dependence between \(x, y, z, t\) and \(x', y', z', t'\), which is called the Lorentz transformation of space and time.

Notation:

  1. \(c\) (c) is speed_of_light.

Notes

  1. Lab frame \(S\) is usually thought as stationary, and proper frame \(S'\) is the one that is considered to be moving relative to lab frame and the moving object in question is at rest in the proper frame.

  2. In this law, the Lorentz transformation from the lab frame \(S\) into the proper frame \(S'\) is described. In order to get an opposite transformation (from the proper frame \(S'\) into the lab frame \(S\)), replace all primed variables with unprimed ones and vice verce, and replace \(v\) with \(-v\). This is consistent with the fact that frame \(S\) can be viewed as moving with speed \(-v\) relative to frame \(S'\), and hence the same Lorentz transformation can be applied.

  3. In the limit \(v/c \ll 1\) the formula reduces to the classical Galilean transformation \(t' = t\).

Conditions:

  1. Space is uniform and isotropic.

  2. Time is uniform.

  3. The relative frame velocity is parallel to the \(x\)-axis.

Links:

  1. Wikipedia, first formula in box.

time_in_proper_frame

time in proper frame \(S'\).

Symbol:

t'

Latex:

\(t'\)

Dimension:

time

time_in_lab_frame

time in lab frame \(S\).

Symbol:

t

Latex:

\(t\)

Dimension:

time

position_in_lab_frame

position in lab frame \(S\).

Symbol:

x

Latex:

\(x\)

Dimension:

length

proper_frame_speed_in_lab_frame

speed of proper frame \(S'\) relative to lab frame \(S\).

Symbol:

v

Latex:

\(v\)

Dimension:

velocity

law

t' = (t - v * x / c^2) / sqrt(1 - (v / c)^2)

Latex:
\[t' = \frac{t - \frac{v x}{c^{2}}}{\sqrt{1 - \left(\frac{v}{c}\right)^{2}}}\]