The Headlight Effect
This Demonstration shows how nine light rays (emitted at angles from to in steps in the rocket frame) are measured by the laboratory observer as the rocket flies past at speed .
A standard problem in special relativity asks you to consider light emitted uniformly in all directions from both the rocket frame (where the light is emitted from) and the laboratory frame (watching the rocket fly past). From the point of view of an observer in the laboratory frame, the light emitted from the rocket frame (in the forward hemisphere) is measured to be compressed into a cone in the forward direction. In the rocket frame, a particular light ray is emitted at an angle θ' such that
Such a ray will be measured to have a different angle, , in the laboratory frame, defined by
Then use the Lorentz transformations , to replace and , yielding
or, more simply,
To visualize this effect, define a ray as , where the slope, , is given by the tangent of the angle .