Re: Algebra of Einstein velocity addition

*To*: mathgroup at smc.vnet.net*Subject*: [mg53825] Re: [mg53787] Algebra of Einstein velocity addition*From*: Bob Hanlon <hanlonr at cox.net>*Date*: Fri, 28 Jan 2005 02:44:32 -0500 (EST)*Reply-to*: hanlonr at cox.net*Sender*: owner-wri-mathgroup at wolfram.com

Addressing only your first question: "Question 1: can you devise an explicit formula for einMul[x,k]? (I've found one)" Clear[einPlus, einMul]; einPlus[x_,y_]:=(x+y)/(1+x*y); einMul[x_,1]=x; einMul[x_,k_Integer?Positive]:= einPlus[x,einMul[x,k-1]]; Needs["DiscreteMath`RSolve`"]; RSolve[ {a[x,n]==einPlus[x,a[x,n-1]],a[x,1]==x}, a[x,n], n]//FullSimplify {{a[x, n] -> 2/((2/(x + 1) - 1)^n + 1) - 1}} And@@Table[einMul[x,k]==2/((2/(x+1)-1)^k+1)-1, {k,1,12}]//Simplify True Consequently, Clear[einMul]; einMul[x_,k_Integer?Positive]:= 2/((2/(x+1)-1)^k+1)-1; Bob Hanlon > > From: "Dr. Wolfgang Hintze" <weh at snafu.de> To: mathgroup at smc.vnet.net > Date: 2005/01/27 Thu AM 05:41:27 EST > To: mathgroup at smc.vnet.net > Subject: [mg53825] [mg53787] Algebra of Einstein velocity addition > > 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 > >