• To: mathgroup at smc.vnet.net
• Subject: [mg53787] Algebra of Einstein velocity addition
• From: "Dr. Wolfgang Hintze" <weh at snafu.de>
• Date: Thu, 27 Jan 2005 05:41:27 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```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]], ...?

Defining the k-fold Einstein Multiple recursively as

einMul[x_, k_] := einPlus[x, einMul[x, k - 1]]
einMul[x_, 1] = x;

we find the first few terms using

In[232]:=
Table[{k, Simplify[einMul[x, k]]}, {k, 2, 10}]

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?

Regards,
Wolfgang

```

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