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Re: Algebra of Einstein velocity addition

On 1/27/05 at 5:41 AM, weh at (Dr. Wolfgang Hintze) wrote:

>100 years ago Albert Einstein discovered his law of adding two
>velocities x and y which is different from the ordinary sum, namely

>einPlus[x_, y_] := (x + y)/(1 + x*y)

Since Einstein was more a theorectical physicist rather than experimental, I would not have used the word discover when describing Einstein's formula for composition of velocity

>It is easy to show that velocities x, y, z, ... with a module less
>than unity form a commutative group under this law.

>I became interested in the algebraic properties of this law, and
>possible extensions. Here's the beginning of it.

>What about multiple additions of the same velocity, i.e.

>einPlus[x,x], einPlus[x,einPlus[x,x]], ...?

<code snipped>

It is fairly simple to show closure. First, note that all valid velocities (more correctly speeds as it is the magnitude of velocity that is used in this formula) are strictly greater than -1 and strictly less than 1. (Here I am talking of anything other than a photon and the units are such that c = 1). Next note that the function (x+y)/(1+x y) is a stricly increasing function in x for any fixed y between -1 and 1. At the extremes, the value of this function becomes -1 or 1. Therefore, any valid speeds input into this function result in a valid speed output.

Another way to come to the same conclusion is to realize the Einstein composition formula for velocity is a straight forward derivation from the Lorentz transform. The Lorentz transform is a group which requires closure. Since speed is nothing more than the ratio of spatial separation and time, that is the the coordinates being transformed by the Lorentz transform, it follows immediately that composition of two speeds must result in another speed.

>Question 1: can you devise an explicit formula for einMul[x,k]?
>(I've found one) 

>Question 2: is it possible to extend the commutative group to a vector
>space using the Einstein Multiple k as the scalar factor?

Definitely. If it were not, this would have little relevance to physics and special relativity would have been relegated to the dust bin ages ago (at least from a physics point of view).

>Question 3: can the algebraic properties be extended further? 

I've no idea what you have in mind here.

>Question 4: is it possible to define an Einstein multiplication which
>is consistent with the multiple studied here?

Here too, I've no idea what you have in mind.

Note (in case it is not apparent) my comments are from a physics point of view. My formal training is physics. Multiplication of velocities makes no physical sense in general. 
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