Algebra of Einstein velocity addition

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