Re: Algebra of Einstein velocity addition

*To*: mathgroup at smc.vnet.net*Subject*: [mg53818] Re: Algebra of Einstein velocity addition*From*: "Ray Koopman" <koopman at sfu.ca>*Date*: Fri, 28 Jan 2005 02:44:10 -0500 (EST)*References*: <ctai4a$c0d$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

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) > > 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 Will you settle for einPlus[x_,y_] = Tanh[ArcTanh[x] + ArcTanh[y]] and einMul[x_,k_] = Tanh[k*ArcTanh[x]] ?