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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
>
>
>
>
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