Special relativity · Kinematics §5
Speeds do not
simply add.
Why doesn't adding speeds preserve light speed, and what happens when the motions are not along one line?
Ideal model, host calculation
Why speeds do not simply add
Ordinary speeds do not add. Two motions slower than light still compose to a motion slower than light, and a ray of light stays a ray of light.
The dashed circle is light speed. The Galilean sum is labeled and is not the model. Composed speed 0.882353 c; Galilean 1.200000 c.
| U_x / c | 0.882353 |
|---|---|
| U_y / c | 0.000000 |
| U / c | 0.882353 |
| Printed §5 U / c | 0.882353 |
| Galilean |v + w| / c | 1.200000 |
| Shortfall 1 − U/c | 0.11764706 |
| Wigner rotation (degrees) | A single boost along one line has no Wigner rotation. |
| Composition increment (m/s) | 3.08676358 |
|---|---|
| Fresnel first order (m/s) | 3.08676363 |
Same scientific action without the plot
Type v/c, w/c, and the angle. Read U_x, U_y, U, the shortfall from light speed, and, in two-boosts mode, the rotation angle from the table. Ask whether the result is still below light speed.
Not modeled: accelerated motion; spin dynamics and Thomas precession of real bodies (only the kinematic rotation is shown); dispersion and the medium's own physics in Fizeau-type setups; gravity.
Presets
Section 5 of the 1905 kinematics paper gives the composition from the Lorentz transformation's differentials. Along a line, U = (v + w)/(1 + vw/c²). At an angle the printed formula is used. Two successive boosts that are not parallel also rotate the spatial axes; that rotation is shown as a matrix fact, not as a force.
Type the two speeds as fractions of c. The Galilean sum is drawn for contrast and is labeled as such. No inertial observer is admitted at or beyond c; that is a refusal, not a silent clamp. Rapidity, if shown, is Minkowski's 1908 aid: collinear boosts add rapidities. It is not the paper's presentation.
The Wigner rotation of two non-collinear boosts is a later geometric name for a 1905 kinematic fact. Fizeau's first-order drag is compared here as a later interpretation (Laue 1907); the 1851 dataset is not this instrument.
Worked case (readable without JavaScript)
Two collinear motions of 0.6c compose to exactly 15/17 of light speed, about 0.882353c, not 1.2c. The Galilean sum exceeds light speed; the relativistic composition does not.
At a right angle the printed formula gives about 0.768375c. Two successive perpendicular boosts of 0.6c give the same speed and a spatial rotation of about 12.6804 degrees. That rotation is a kinematic fact of the composition, not a force on a gyroscope.