The Headlight Effect

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 .