Re: Algebra of Einstein velocity addition
- To: mathgroup at smc.vnet.net
- Subject: [mg53817] Re: [mg53787] Algebra of Einstein velocity addition
- From: "Dr. Wolfgang Hintze" <weh at snafu.de>
- Date: Fri, 28 Jan 2005 02:44:08 -0500 (EST)
- References: <NDBBJGNHKLMPLILOIPPOMEJDEFAA.djmp@earthlink.net>
- Sender: owner-wri-mathgroup at wolfram.com
David, thanks for your messages and the interesting hints and links (which I'll follow soon). I didn't know the of Bondi factor but my explicit solution for n-fold Einstein addition of velocity v is einMul[v_, n_] := (1 - ( (1 - v)/(1 + v) )^n ) / (1 + ( (1 - v)/(1 + v))^n ) which contains the Bondi factor. By the way, I found the fornula by studying the polynomials in the examples below, identifying Binomial coefficients, and finally observing that odd powers of v appear in the nominator and even powers in the denominator. Another thread I'm studying utilizes complex velocities... Regards, Wolfgang David Park wrote: >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 > > > >