Introduction
Velocities do not add simply in relativity. The relativistic velocity addition formula ensures no combined velocity exceeds c. This is required by the second postulate and is verified in particle physics experiments. Classical velocity addition fails at high speeds.
Velocity Transformation
If object has velocity u' in frame S' moving at v relative to S, its velocity in S is: u = (u' + v)/(1 + u'v/c2). For motion along same direction. Denominator ensures u < c even when u' and v approach c. Inverse transformation: u' = (u - v)/(1 - uv/c2).
Properties
If u' = c (light), then u = c regardless of v - consistent with postulate. If u' << c and v << c: denominator ≈ 1, recovers Galilean addition u ≈ u' + v. As u' or v approaches c, result approaches c but never exceeds it. For opposite direction, signs change appropriately.
Perpendicular Components
If u' has components: u'_x and u'_y in S'. Then in S: u_x = (u'_x + v)/(1 + vu'_x/c2), u_y = u'_y/(γ(1 + vu'_x/c2)). Perpendicular components are affected by motion of frame. Velocity vector rotates between frames (aberration).
\nIntroduction
Velocities do not add simply in relativity. The relativistic velocity addition formula ensures no combined velocity exceeds c. This is required by the second postulate and is verified in particle physics experiments. Classical velocity addition fails at high speeds.
Velocity Transformation
If object has velocity u' in frame S' moving at v relative to S, its velocity in S is: u = (u' + v)/(1 + u'v/c2). For motion along same direction. Denominator ensures u < c even when u' and v approach c. Inverse transformation: u' = (u - v)/(1 - uv/c2).
Properties
If u' = c (light), then u = c regardless of v - consistent with postulate. If u' << c and v << c: denominator ≈ 1, recovers Galilean addition u ≈ u' + v. As u' or v approaches c, result approaches c but never exceeds it. For opposite direction, signs change appropriately.
Perpendicular Components
If u' has components: u'_x and u'_y in S'. Then in S: u_x = (u'_x + v)/(1 + vu'_x/c2), u_y = u'_y/(γ(1 + vu'_x/c2)). Perpendicular components are affected by motion of frame. Velocity vector rotates between frames (aberration).
\nAberration of Light
Relativistic velocity addition changes direction of light between frames. Starlight appears to come from different direction due to Earth's motion (stellar aberration). Requires relativistic formula at high precision; was known classically but correctly explained only by SR. Aberration angle depends on relative velocity.
Applications
Particle collisions: velocity addition in center-of-mass and lab frames. Doppler effect derivation. Light propagation in moving media (Fizeau experiment - partial drag of light by moving water). High-speed spacecraft navigation. Velocity measurements in particle accelerators.
\nSolved Example: Particle Accelerator
A proton moves at 0.9c in lab frame. An electron is fired toward it at 0.8c relative to lab (opposite direction). (a) What is electron's velocity relative to proton? (b) What if both moving same direction at these speeds? Solution: (a) Opposite directions: let proton frame be S (v = 0.9c), electron in lab is u = -0.8c (opposite). In proton frame: u' = (u - v)/(1 - uv/c2) = (-0.8c - 0.9c)/(1 - (-0.8)(0.9)) = (-1.7c)/(1 + 0.72) = -1.7c/1.72 = -0.988c. Electron approaches proton at 0.988c, not 1.7c. (b) Same direction: electron at 0.8c, proton at 0.9c in lab. In proton frame: u' = (0.8c - 0.9c)/(1 - (0.8)(0.9)) = (-0.1c)/(1 - 0.72) = -0.1c/0.28 = -0.357c. Electron appears to move backward at 0.357c in proton frame because proton is faster. Classically: 0.8c - 0.9c = -0.1c. Relativistic effect makes apparent speed larger (0.357c > 0.1c). If both at 0.99c same direction: classical difference 0, relativistic: u' = (0.99-0.99)/(1-0.9801) = 0 (but second order shows tiny difference). For relativistic beams colliding head-on, effective collision energy is much higher than simple sum because velocities don't simply add.
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