gyr {lorentz}

Gyr function

Description

Relativistic addition of three velocities

Usage

gyr(u, v, x)
gyr.a(u, v, x)
gyrfun(u, v)

Arguments

u, v, x

Three-velocities, objects of class 3vel

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().

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))

Author(s)

Robin K. S. Hankin

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

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

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