Four velocities
Create and test for four-velocities.
as.4vel(u) is.consistent.4vel(U, give=FALSE, TOL=1e-10) inner4(U,V=U) to3(U)
u |
A vector of three-velocities |
U,V |
A vector of four-velocities |
give |
In function |
TOL |
Small positive value used for tolerance |
Function as.4vel() takes a three-velocity and returns a
four-velocity.
Given a four-vector V, function inner4() returns the
Lorentz invariant V^i.V_i. This
quantity is unchanged under Lorentz transforms. Note that function
inner4() works for any four-vector, not just four-velocities.
It will work for (eg) a four-displacement, a four-momentum vector or a
four-frequency. In electromagnetism, we could have a four-current or
a four-potential. If U is a four-velocity, then
U^iU_i=-c^2; if U is a 4-displacement, then U^iU_i is
the squared interval. If P is the four-momentum of a photon
then P^iP_i=0.
Function to3() is a low-level helper function used when
as.3vel() is given a four-velocity.
Function is.consistent.4vel() checks for four-velocities being
consistent in the sense that U.U=-c^2. Giving this
function a vector, for example, is.consistent.4vel(1:5), will
return an error.
Compare the functions documented here with boost(), which
returns a 4*4 transformation matrix (which also
includes rotation information).
Robin K. S. Hankin
a <- r3vel(10) as.4vel(a) # a four-velocity as.3vel(as.4vel(a))-a # zero to numerical precision inner4(as.4vel(a)) # -1 to numerical precision stopifnot(all(is.consistent.4vel(as.4vel(a)))) ## check Lorentz invariance of dot product: U <- as.4vel(r3vel(10)) V <- as.4vel(r3vel(10)) B <- boost(as.3vel(1:3/10)) frame1dotprod <- inner4(U, V) frame2dotprod <- inner4(U %*% B, V %*% B) max(abs(frame1dotprod-frame2dotprod)) # zero to numerical precision
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