Final answer:
The Lorentz transformation equations relate positions and times of events between inertial frames, leading to the relativistic velocity transformation rules.
These rules, together with Einstein's second postulate about the speed of light being constant in vacuum for all observers regardless of their relative motion, ensure the consistency of the speed of light across inertial frames.
Step-by-step explanation:
The Lorentz transformation provides the mathematical rules for transforming positions and times of events from one inertial frame to another.
For two frames, O and O', where O' moves with a velocity v relative to O in the x-direction, the Lorentz transformation equations are:
- x' = γ(x - vt)
- y' = y
- z' = z
- t' = γ(t - vx/c^2)
Here, γ is the Lorentz factor, γ = 1 / √(1 - v^2/c^2), where c is the speed of light.
To derive the relativistic velocity transformation rule, one considers the displacement dx' and the time interval dt' in the moving frame O'.
Applying the Lorentz transformation, one finds the corresponding changes in the stationary frame O, allowing the derivation of how velocity components transform between the inertial frames.
The result shows the transformed velocity components as:
- ux = (ux' + v) / (1 + ux'v/c^2)
- uy = uy' / γ(1 + ux'v/c^2)
- uz = uz' / γ(1 + ux'v/c^2)
It is a cornerstone of Einstein's special theory of relativity, supported by the Michelson-Morley experiment, that the speed of light is constant and does not depend on the relative motion of the source and observer.
This experimental result is encapsulated in the second postulate of special relativity.
It ensures that, despite the transformations of velocities and other relativistic effects, all observers will measure the speed of light in vacuum to be c.