Generalized velocity addition and aberration formulas

[PAllen]

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.

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:

√(u² + v² - 2uvcosθ - u²v²sin²θ) / (1 - uv cosθ)

Of interest is that the corresponding classical formula results if the relativistic correction terms are removed. The sin² 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):

√(u² + v² - u²v²)

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)
]

 

[Mentz114]

Interesting. I haven't seen either of the first two in that form, if at all.

In the final expression, I assume u,v are the speeds of two things from a third frame ? That could be useful.

 

[PAllen]

Interesting. I haven't seen either of the first two in that form, if at all.

In the final expression, I assume u,v are the speeds of two things from a third frame ? That could be useful.

Yes, that's right.

 

[Mentz114]

Yes, that's right.

If one solves for $V_r$ with $\cos(\theta)=1$ it is the velocity addition formula as it should be. The $\cos(\theta)$ adds some value though.

 

[PAllen]

An example of the use of these formulas is:

Consider observer A who sees B moving .9c to the right (for example). B reports that some body is moving .9c in a direction towards A but 45 degrees away from the direction of A per B. What is the speed and direction of this body per A?

1) Classically (note, in the formulas there are identified relativistic corrections that can simply be removed), the result is that the body is moving at speed .68883c per A, and at an angle of 112.5 decrees away from the the B to A line. That is, moving away from A 22.5 degrees beyond orthogonal. Note that this direction and the ratio .68883/.9 would hold for any situation where A to B relative speed and speed per B are equal, and the direction per B is 45 degrees. Relativistically, this becomes ever more true for low speeds, but high speeds differ.

2) Relativistically, for the case given, the relative speed of the body per A is about .895674c. The direction is 133.54 degrees away from B to A vector, so moving away from A almost 45 degrees beyond orthogonal.