# NOT RUN {
u <- as.3vel(c(0.3,0.6,-0.1)) # u is a three-velocity
gam(u) # relativistic gamma term for u
U <- as.4vel(u) # U is a four-velocity
B1 <- boost(u) # B1 is the Lorentz transform matrix for u
B1 %*% c(1,0,0,0) # Lorentz transform of zero 4-velocity (=-u)
B2 <- boost(as.3vel(c(-0.1,0.8,0.3)))
B3 <- boost(as.3vel(c(-0.1,0.1,0.9))) # more boosts
Bi <- B1 %*% B2 # Bi is the boost for successive Lorentz transforms
pureboost(Bi) # Decompose Bi into a pure boost...
orthog(Bi) # and an orthogonal matrix
Bj <- B2 %*% B1 # B1 and B2 do not commute...
(B1 %*% B2) %*% B3
B1 %*% (B2 %*% B3) # ...but composition *is* associative
## Three velocities and the gyrogroup
## Create some random three-velocities:
u <- r3vel(10)
v <- r3vel(10)
w <- r3vel(10)
u+v
v+u # Three-velocity addition is not commutative...
u+(v+w) # ... nor associative
(u+v)+w
# }
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