Coordinate conversion at constant velocity

Let the frame of reference move relative to the other frame of reference at a constant speed along the X axis, and the origin of the spatial coordinates coincide at the initial moment of time in both systems. Then there are simple transformations that can be used to get the x coordinate in one frame of reference, knowing the x coordinate in another frame of reference.

Notation:

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

Links:

1. Wikipedia, second formula in box.

position_in_proper_frame

position in the second frame of reference.

Symbol:

x_2

Latex:

\(x_{2}\)

Dimension:

length

position_in_lab_frame

position in the first frame of reference.

Symbol:

x_1

Latex:

\(x_{1}\)

Dimension:

length

proper_frame_speed_in_lab_frame

speed of the second reference frame relative to the first one.

Symbol:

v

Latex:

\(v\)

Dimension:

velocity

time_in_lab_frame

time in the first frame of reference.

Symbol:

t_1

Latex:

\(t_{1}\)

Dimension:

time

law

x_2 = (x_1 - v * t_1) / sqrt(1 - (v / c)^2)

Latex:

\[x_{2} = \frac{x_{1} - v t_{1}}{\sqrt{1 - \left(\frac{v}{c}\right)^{2}}}\]