Relativistic Aberration Formula & Lorentz Transformation

[Tahmeed]

Let's assume that a light source is moving parralel to x-axis and is in point x,y,z in lab frame. Suppose it emits a light ray. In the rest frame that coincides with the lab frame, the light source is in point x',y and z.
However, because of relativistic aberration the two light rays will make different angle with origin. Obviously, in the rest frame the light ray will create a straight line from origin to the source x',y,z that has a certain slope theta. This theta will be transformed by relativistic aberration equation. But will this transformed ray pass through point x,y,z in the lab frame? If that's the case, can't we use analytical geometry of straight lines to derive the aberration formula instead of using typical velocity addition process??

 

[pervect]

I'm not sure I understand the question. Conceptually, if you have a single light ray, it follows a specific path through space-time which can be regarded as being independent of the observer.

One observer might use (t,x,y,z) coordinates, the other observer might use (t', x', y', z') coordinates, but they are just different descriptions (labels) by different observers of the same "reality". The last point is a bit philsophical, of course, but that's a convenient description of something that can be expressed in observer independent terms.

One can regard the light ray as consisting of a set of points (events) in space-time. One can transform the individual points (t,x,y,z) which comprise this set of points which represent the light ray from the unprimed coordinates to primed coordinates (t', x', y', z') via the Lorentz transform. So knowing the path in one coordinate system allows one to compute the path in any other coordinate system as long as the new coordinate system is adequately specified. In this example, sepcifying the relative velocity of the two coordinate systems and one shared point is sufficient to define the relationship between the primed and unprimed coordinates.

The angle is an observer-dependent quantity which then can be calculated from the set of points that make up the light ray. Contrast the observer dependence of the angle, with the non-observer dependent quantities previously discussed.

 

[Tahmeed]

I'm not sure I understand the question. Conceptually, if you have a single light ray, it follows a specific path through space-time which can be regarded as being independent of the observer.

One observer might use (t,x,y,z) coordinates, the other observer might use (t', x', y', z') coordinates, but they are just different descriptions (labels) by different observers of the same "reality". The last point is a bit philsophical, of course, but that's a convenient description of something that can be expressed in observer independent terms.

One can regard the light ray as consisting of a set of points (events) in space-time. One can transform the individual points (t,x,y,z) which comprise this set of points which represent the light ray from the unprimed coordinates to primed coordinates (t', x', y', z') via the Lorentz transform. So knowing the path in one coordinate system allows one to compute the path in any other coordinate system as long as the new coordinate system is adequately specified. In this example, sepcifying the relative velocity of the two coordinate systems and one shared point is sufficient to define the relationship between the primed and unprimed coordinates.

The angle is an observer-dependent quantity which then can be calculated from the set of points that make up the light ray. Contrast the observer dependence of the angle, with the non-observer dependent quantities previously discussed.

Yes, you got my question right. So, the relativistic aberration formula can be derived the way I suggested then?

 

[pervect]

Yes, you got my question right. So, the relativistic aberration formula can be derived the way I suggested then?

I'd say that it should work - the real proof is in actually carrying out the calculations and comparing the answers, though.