Re: Algebra of Einstein velocity addition

• To: mathgroup at smc.vnet.net
• Subject: [mg53829] Re: Algebra of Einstein velocity addition
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Sat, 29 Jan 2005 06:02:37 -0500 (EST)
• Organization: The University of Western Australia
• References: <ctai4a\$c0d\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <ctai4a\$c0d\$1 at smc.vnet.net>,
"Dr. Wolfgang Hintze" <weh at snafu.de> 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)
>
> 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.
>
>
> einPlus[x,x], einPlus[x,einPlus[x,x]], ...?

In special relativity, the (vector) rapidity parameter, Xi, is defined by

Xi = v/Norm[v] ArcTanh[Norm[v]]

where the vector v is the relative velocity (in units where c = 1)
between moving reference frames (see The Mathematica Journal 8(3) page
402). The benefit of this representation is the trivial formula for the
addition of colinear rapidities. In particular, note that

Simplify[(Tanh[a] + Tanh[b])/(1 + Tanh[a] Tanh[b])]

Tanh[a + b]

> Defining the k-fold Einstein Multiple recursively as
>
> einMul[x_, k_] := einPlus[x, einMul[x, k - 1]]
> einMul[x_, 1] = x;

These can be obtained as follows:

Table[{n, Factor[TrigExpand[Tanh[n y]] /.
{Tanh[y] -> x, Sinh[y] -> x Cosh[y]}]}, {n, 6}]

> we find the first few terms using
>
> In[232]:=
> Table[{k, Simplify[einMul[x, k]]}, {k, 2, 10}]

NestList[Factor[einPlus[x, #]] & , x, 9]

> as
>
> Out[232]=
> {{2, (2*x)/(1 + x^2)}, {3, (x*(3 + x^2))/(1 + 3*x^2)}, {4, (4*(x +
> x^3))/(1 + 6*x^2 + x^4)},
>    {5, (x*(5 + 10*x^2 + x^4))/(1 + 10*x^2 + 5*x^4)},
>    {6, (6*x + 20*x^3 + 6*x^5)/(1 + 15*x^2 + 15*x^4 + x^6)},
>    {7, (x*(7 + 35*x^2 + 21*x^4 + x^6))/(1 + 21*x^2 + 35*x^4 + 7*x^6)},
>    {8, (8*(x + 7*x^3 + 7*x^5 + x^7))/(1 + 28*x^2 + 70*x^4 + 28*x^6 + x^8)},
>    {9, (x*(9 + 84*x^2 + 126*x^4 + 36*x^6 + x^8))/(1 + 36*x^2 + 126*x^4 +
> 84*x^6 + 9*x^8)},
>    {10, (2*x*(5 + 60*x^2 + 126*x^4 + 60*x^6 + 5*x^8))/(1 + 45*x^2 +
> 210*x^4 + 210*x^6 + 45*x^8 +
>       x^10)}}
>
> 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?
> Question 3: can the algebraic properties be extended further?
> Question 4: is it possible to define an Einstein multiplication which is
> consistent with the multiple studied here?

I think you can answer all these questions using the rapidity
representation.

Cheers,
Paul

--
Paul Abbott                                   Phone: +61 8 6488 2734
School of Physics, M013                         Fax: +61 8 6488 1014
The University of Western Australia      (CRICOS Provider No 00126G)
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Crawley WA 6009                      mailto:paul at physics.uwa.edu.au
AUSTRALIA                            http://physics.uwa.edu.au/~paul

```

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