Relativistic sum of velocities

In relativistic mechanics, the speed of the body in the lab reference frame is no longer the sum of the speed of the body’s proper reference frame and its speed in the proper frame.

Notation:

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

Conditions:

  1. The velocity of the body relative to the proper frame and the velocity of the proper frame relative to the lab frame must be parallel to each other.

Links:

  1. Wikipedia.

body_speed_in_lab_frame

speed of the body relative to the lab frame.

Symbol:

v_OL

Latex:

\(v_{OL}\)

Dimension:

velocity

body_speed_in_proper_frame

speed of the body relative to the proper reference frame.

Symbol:

v_OP

Latex:

\(v_{OP}\)

Dimension:

velocity

proper_frame_speed_in_lab_frame

speed of the proper frame relative to the lab frame.

Symbol:

v_PL

Latex:

\(v_{PL}\)

Dimension:

velocity

law

v_OL = (v_OP + v_PL) / (1 + v_OP * v_PL / c^2)

Latex:
\[v_{OL} = \frac{v_{OP} + v_{PL}}{1 + \frac{v_{OP} v_{PL}}{c^{2}}}\]