Gyr function
gyr.RdRelativistic addition of three velocities
Usage
gyr(u, v, x)
gyr.a(u, v, x)
gyrfun(u, v)Details
Function gyr(u,v,x) returns the three-vector
\(\mathrm{gyr}[u,v]x\).
Function gyrfun(u,v) returns a function that returns a
three-vector; see examples.
The speed of light (1 by default) is not used directly by these
functions; set it with sol().
References
Ungar 2006. “Thomas precession: a kinematic effect of the algebra of Einstein's velocity addition law. Comments on ‘Deriving relativistic momentum and energy: I. Three-dimensional case’”. European Journal of Physics, 27:L17-L20.
Sbitneva 2001. “Nonassociative geometry of special relativity”. International Journal of Theoretical Physics, volume 40, number 1, pages 359–362
Note
Function gyr() is slightly faster than gyr.a(), which is
included for pedagogical reasons.
Function gyr() is simply
add3(neg3(add3(u,v)),add3(u,add3(v,x)))
while function gyr.a() uses the slower but more transparent
idiom
-(u+v) + (u+(v+x))
Examples
u <- r3vel(10)
v <- r3vel(10)
w <- r3vel(10)
x <- as.3vel(c(0.4,0.1,-0.5))
y <- as.3vel(c(0.1,0.2,-0.7))
z <- as.3vel(c(0.2,0.3,-0.1))
gyr(u,v,x) # gyr[u,v]x
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] 0.29688244 -0.284585217 -0.5008713
#> [2,] 0.36875817 0.065856906 -0.5288481
#> [3,] 0.53155065 0.074581061 -0.3631688
#> [4,] -0.04404694 -0.175241426 -0.6223747
#> [5,] 0.38537036 0.006363217 -0.5210079
#> [6,] 0.47435109 0.170964952 -0.4071388
#> [7,] 0.20106734 0.338903650 -0.5145058
#> [8,] -0.10634811 -0.387125922 -0.5087471
#> [9,] 0.20797919 0.015319935 -0.6136041
#> [10,] 0.45819629 -0.136356740 -0.4375649
f <- gyrfun(u,v)
g <- gyrfun(v,u)
f(x)
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] 0.29688244 -0.284585217 -0.5008713
#> [2,] 0.36875817 0.065856906 -0.5288481
#> [3,] 0.53155065 0.074581061 -0.3631688
#> [4,] -0.04404694 -0.175241426 -0.6223747
#> [5,] 0.38537036 0.006363217 -0.5210079
#> [6,] 0.47435109 0.170964952 -0.4071388
#> [7,] 0.20106734 0.338903650 -0.5145058
#> [8,] -0.10634811 -0.387125922 -0.5087471
#> [9,] 0.20797919 0.015319935 -0.6136041
#> [10,] 0.45819629 -0.136356740 -0.4375649
f(r3vel(10))
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] 0.07255247 -0.009937689 -0.97804879
#> [2,] -0.05467681 0.171111420 -0.87218568
#> [3,] 0.08386887 0.785488929 0.08851038
#> [4,] 0.41152071 0.404793224 -0.56714449
#> [5,] 0.32568173 0.407708661 0.36527154
#> [6,] 0.55753999 -0.309549737 -0.62490156
#> [7,] 0.19213693 -0.754186347 -0.61085947
#> [8,] -0.69261374 -0.233064131 0.47417197
#> [9,] -0.14808868 -0.914166267 0.03806476
#> [10,] 0.16215162 0.032779754 -0.38423240
f(g(x)) - x # zero, by eqn 9
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] 2.320940e-15 7.393511e-15 -5.742533e-16
#> [2,] 2.919121e-15 -6.705604e-15 -1.043227e-14
#> [3,] -2.078079e-13 -9.932787e-14 -2.452540e-14
#> [4,] -1.177219e-14 -1.433240e-14 1.086296e-14
#> [5,] -5.981805e-16 -6.041623e-16 -2.392722e-17
#> [6,] -1.028870e-15 -1.130561e-15 1.603124e-15
#> [7,] -3.349811e-16 -1.196361e-17 2.153450e-16
#> [8,] -3.100489e-13 -6.743349e-13 -8.060363e-13
#> [9,] 2.464504e-15 3.050721e-16 -1.890250e-15
#> [10,] -3.947991e-15 2.841357e-15 7.178166e-17
g(f(x)) - x # zero, by eqn 9
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] -2.105595e-15 -1.205932e-14 -1.272928e-14
#> [2,] 5.263988e-15 -9.618743e-15 -1.004943e-14
#> [3,] -1.520814e-13 -5.740738e-14 -2.316155e-14
#> [4,] -2.608067e-14 -3.953973e-15 9.499106e-15
#> [5,] -9.570888e-17 -9.570888e-17 -2.871266e-16
#> [6,] 4.067627e-16 4.665808e-16 2.153450e-16
#> [7,] -5.981805e-16 -5.981805e-18 4.067627e-16
#> [8,] 1.237994e-13 4.074626e-13 -5.915527e-13
#> [9,] 7.895983e-16 2.691812e-16 -5.024716e-16
#> [10,] 3.589083e-16 3.002866e-15 -2.249159e-15
(x+y) - f(y+x) # zero by eqn 10
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] 0.441410083 0.70243973 -0.47007056
#> [2,] 0.438160812 -0.02483539 0.08199839
#> [3,] -0.280162799 -0.29800937 -0.50249057
#> [4,] 0.786121767 0.36245653 -0.42410272
#> [5,] 0.386826527 0.31030591 0.09773939
#> [6,] 0.004981965 -0.37368903 -0.20381558
#> [7,] 0.715022430 -0.35065032 -0.39790221
#> [8,] 0.607567474 0.47310663 -0.61800986
#> [9,] 0.786071516 0.10098613 -0.07005284
#> [10,] -0.131355795 0.43315800 -0.10110650
(u+(v+w)) - ((u+v)+f(w)) # zero by eqn 11
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] 0.000000e+00 0.000000e+00 4.109056e-13
#> [2,] 1.804276e-14 1.443421e-13 1.443421e-13
#> [3,] 4.665010e-14 2.332505e-14 4.665010e-14
#> [4,] 1.514007e-14 1.514007e-14 -5.677528e-15
#> [5,] -1.058779e-15 -2.117558e-15 0.000000e+00
#> [6,] 1.216292e-16 4.865166e-16 0.000000e+00
#> [7,] 0.000000e+00 4.918352e-16 -6.147940e-17
#> [8,] 0.000000e+00 0.000000e+00 0.000000e+00
#> [9,] 0.000000e+00 0.000000e+00 -4.753304e-16
#> [10,] 0.000000e+00 0.000000e+00 -1.493291e-16
# Following taken from Sbitneva 2001:
rbind(x+(y+(x+z)) , (x+(y+x))+z) # left Bol property
#> [,1] [,2] [,3]
#> [1,] 0.4894115 0.2314344 -0.8300767
#> [2,] 0.4894115 0.2314344 -0.8300767
rbind((x+y)+(x+y) , x+(y+(y+x))) # left Bruck property
#> [,1] [,2] [,3]
#> [1,] 0.432766 0.209512 -0.8738679
#> [2,] 0.432766 0.209512 -0.8738679
sol(299792458) # speed of light in SI
#> [1] 299792458
as.3vel(c(1000,3000,1000)) + as.3vel(c(1000,3000,1000))
#> A vector of three-velocities (speed of light = 299792458)
#> x y z
#> [1,] 2000 6000 2000
## should be close to Galilean result
sol(1) # revert to default c=1
#> [1] 1