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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
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