Solve Relativistic Lens Problem: Focal Length & Gamma

[AUK 1138]

How would you go about solving a problem involving a lens moving at relativistic speeds, or an object being relativistic being viewed through a lens? instead of coming up with a specific problem, does anyone know what happens to lens' focal length and such as its velocity becomes large? not so much as a limit, but in general. i would assume its focal length increases in both converging and diverging lenses, but by what factor? does it simply change by gamma? any help appreciated.

 

[danskal]

The easiest way to think about it is if you have a solid rod held by an observer alongside it, moving at the same speed.

The observer won't notice anything relativistic, because from their point of view nothing is moving at relativistic speeds (except the surroundings, which, for this thought experiment are just empty space).

So if the observer sees a focused spot of light from the lens near the end of the rod, so will anyone else who can see the rod (because we must keep causality).

Which means that the focal length of the lens just follows http://en.wikipedia.org/wiki/Length_contraction" like the rod does, assuming that you have an inertial (constant velocity) frame of reference. So you just divide the length by gamma.

regards, Michael

 

[AUK 1138]

Thank you for the response, but I'm adamant in thinking the focal length gets longer. it seems like for an observer who's viewing the lens on it's side, as it travels transversely with it's focal lengths that it would eventually appear to a be a single sheet of glass, which could be viewed as a lens with an infinite focal length, which therefore doesn't bend the light. meaning that the focal length was getting longer as its velocity increases. basically, i'd think the new focal length would be the old one times gamma, not divided by it, as the focal length is a property of the lens and not a physical distance in and of itself.

 

[utesfan100]

Thank you for the response, but I'm adamant in thinking the focal length gets longer. it seems like for an observer who's viewing the lens on it's side, as it travels transversely with it's focal lengths that it would eventually appear to a be a single sheet of glass, which could be viewed as a lens with an infinite focal length, which therefore doesn't bend the light. meaning that the focal length was getting longer as its velocity increases. basically, i'd think the new focal length would be the old one times gamma, not divided by it, as the focal length is a property of the lens and not a physical distance in and of itself.

This sounds to me like an argument that the index of refraction of the material is not invariant, as the point where the lightray's converge should be invariant.

In fact, the index of refraction of empty space being increased by a gravitational field is one explanation of the fact that relative to an outside observer the speed of light varies as one decends into a gravitational well (always locally c, but dime dilated relative to the outside observer.)