gyr: Gyr function

Description Usage Arguments Details Note Author(s) References Examples

Description

Relativistic addition of three velocities

Usage

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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 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

Examples

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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

RobinHankin/gyrogroup documentation built on Feb. 19, 2019, 3:20 a.m.