lorentz (version 1.1-1)

gyr: 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

Author

Robin K. S. Hankin

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

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