Gyr function
Relativistic addition of three velocities
gyr(u, v, x) gyr.a(u, v, x) gyrfun(u, v)
u,v,x |
Three-velocities, objects of class |
Function gyr(u,v,x) returns the three-vector
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().
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))
Robin K. S. Hankin
Ungar 2006. “Thomas precession: a kinematic effect of the algebra of Einstein's velocity addition law. Comments on ‘Deriving relativistic momentum and energy: II. Three-dimensional case’”. European Journal of Physics, 27:L17-L20.
Sbitneva 2001. “Nonassociative geomery of special relativity”. International Journal of Theoretical Physics, volume 40, number 1, pages 359–362
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 f <- gyrfun(u,v) g <- gyrfun(v,u) f(x) f(r3vel(10)) f(g(x)) - x # zero, by eqn 9 g(f(x)) - x # zero, by eqn 9 (x+y) - f(y+x) # zero by eqn 10 (u+(v+w)) - ((u+v)+f(w)) # zero by eqn 11 # Following taken from Sbitneva 2001: rbind(x+(y+(x+z)) , (x+(y+x))+z) # left Bol property rbind((x+y)+(x+y) , x+(y+(y+x))) # left Bruck property sol(299792458) # speed of light in SI as.3vel(c(1000,3000,1000)) + as.3vel(c(1000,3000,1000)) ## should be close to Galilean result sol(1) # revert to default c=1
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.