Not sure if this will be of interest to others, but, as an exercise, I decided to derive formulas for SR velocity addition for any angular relationship, and similar aberration of angle for any object speed and direction and observer relative velocity - using pure algebra/geometry. That is, no Lorentz transform or coordinate representations used. I also wanted formulations that could easily be compared to 3-vector pre-SR calculations.(adsbygoogle = window.adsbygoogle || []).push({});

Given observers A and B, with A having speed v relative to B (convention v>0 for A moving towards B, < 0 for A moving away), then if some object is moving at speed u relative to B at angle θ to the relative motion of A and B, per B, the relative speed of the object to A is:

√(u^{2}+ v^{2}- 2uvcosθ - u^{2}v^{2}sin^{2}θ) / (1 - uv cosθ)

Of interest is that the corresponding classical formula results if the relativistic correction terms are removed. The sin^{2}term is a correction for orthogonal motion component of object, the (1-uvcosθ) term for parallel component of motion of object. For angle of 0 and pi, the common velocity formulas immediately result (for both classical and SR case). For the object moving, per B, orthogonal to relative motion between A and B, an equally simple formula results, that seems much less quoted (at least in books I own):

√(u^{2}+ v^{2}- u^{2}v^{2})

The last term under the radical is the relativistic correction.

For aberration, we ask, given the problem definitions above, what angle does A observe between the object and the relative motion of B per A? We define v_{r}as the relative speed per A as given by the general formula above. Then the angle observed per A ( call it Φ ) is given by:

cos Φ = (1/v_{r}) (u cosθ - v) / ( 1 - uv cosθ)

The explicit SR correction is simply the the last term: ( 1 - uv cosθ). Implicitly, there is SR correction embedded in the relative speed. If you compute relative speed classically, and drop the explicit correction term, you have a formulation of general classical aberration. Note that, for light (u = 1, of course c=1), this leads to the less common cosine formulation of SR aberration (equivalent to the more common tangent formula):

cosΦ = (cosθ - v) / (1- vcosθ)

which is a cute formulation in that no gamma is involvled.

Also note that relatively simple algebra establishes that the relative speed formula leads to c=1 whenever u=v=1 and θ > 0. The θ=0 case is undefined because that is the case of A and the object moving in the same direction approaching c per B, and what is the speed of of the object per A? That is, it is asking about the speed of light in a frame comoving with that light.

[edit: I guess it is worth adding that if you just want γ corresponding to the relative speed of the object per A, that is very simple:

γ(v_{r}) = γ(u)γ(v)(1 - uv cosθ)

]

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# Generalized velocity addition and aberration formulas

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