Re: Algebra of Einstein velocity addition

*To*: mathgroup at smc.vnet.net*Subject*: [mg53806] Re: [mg53787] Algebra of Einstein velocity addition*From*: "David Park" <djmp at earthlink.net>*Date*: Fri, 28 Jan 2005 02:43:51 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

Wolfgang, You may be interested in looking at the SpacetimeGeometry Mathematica notebook at my web site below. It gives an introduction to special relativity using lots of graphics and animation and is based in part on two recent papers Dieter Brill & Ted Jacobson, Spacetime and Euclidean Geometry, arXiv:gr-qc/0407022 v1 6 Jul 2004 http://arxiv.org/abs/gr-qc/0407022 N. David Mermin, From Einstein's 1905 Postulates to the Geometry of Flat Space-Time, arXiv:gr-qc/0411069 v1 15 Nov 2004 http://arxiv.org/abs/gr-qc/0411069 In any case, have you heard of the Bondi k factor? The Bondi k factor relates two intersecting inertial lines just as velocity v does. The relation between the two is... k -> Sqrt[(1 + v)/(1 - v)] and v -> (k^2 - 1)/(k^2 + 1) The nice thing is that when k's are composed, they are just multiplied. So for n velocity boosts of v we have the equations and solutions... eqns = {kn == k^n /. k -> Sqrt[(1 + v)/(1 - v)], v[n] == (kn^2 - 1)/(kn^2 + 1)}; Solve[eqns, v[n], kn][[1,1]]; FullSimplify[%, -1 < v < 1] Table[Simplify[%], {n, 2, 4}] v[n] -> 1 - 2/(1 + ((1 + v)/(1 - v))^n) {v[2] -> (2*v)/(1 + v^2), v[3] -> (v*(3 + v^2))/(1 + 3*v^2), v[4] -> (4*(v + v^3))/(1 + 6*v^2 + v^4)} So using k instead of v we just a commutative group of real positive numbers under multiplication. David Park djmp at earthlink.net http://home.earthlink.net/~djmp/ From: Dr. Wolfgang Hintze [mailto:weh at snafu.de] To: mathgroup at smc.vnet.net 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