3vel                    Three velocities
4mom                    Four momentum
4vel                    Four velocities
Extract.3vel            Extract or replace parts of three-velocity
Ops.3vel                Arithmetic Ops Group Methods for 3vel objects
as.matrix.3vel          Coerce 3-vectors and 4-vectors to a matrix
boost                   Lorentz transformations
c.3vel                  Combine vectors of three-velocities and
                        four-velocities into a single vector
comm_fail               Failure of commutativity and associativty using
                        visual plots
coordnames              Coordinate names for relativity
cosines                 Direction cosines
galileo                 Classical mechanics; Newtonian approximation;
                        infinite speed of light
gam                     Gamma correction
gyr                     Gyr function
lorentz-package         The Lorentz Transform in Relativistic Physics
photon                  Photons
print.3vel              Print methods for three-velocities and
                        four-velocities
r3vel                   Random relativstic velocities
reflect                 Mirrors
seq.3vel                seq method for three velocities
sol                     Speed of light and Minkowski metric
transform               The energy-momentum tensor

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