Four velocities

Create and test for four-velocities.

Usage

as.4vel(u)
is.consistent.4vel(U, give=FALSE, TOL=1e-10)
inner4(U,V=U)
to3(U)

Arguments

u

A vector of three-velocities

U,V

A vector of four-velocities

give

In function is.consistent.4vel(), Boolean with TRUE meaning to return \(U\cdot U+c^2\), which is zero for a four-velocity, and default FALSE meaning to return whether the four-velocity is consistent to numerical precision

TOL

Small positive value used for tolerance

Details

Function as.4vel() takes a three-velocity and returns a four-velocity.

Given a four-vector \(V\), function inner4() returns the Lorentz invariant \(V^iV_i=\eta_{ij}V^iV^j\). This quantity is unchanged under Lorentz transformations. 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 \(U^iU_i=-c^2\). 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 \(4\times 4\) transformation matrix (which also includes rotation information).

Author

Robin K. S. Hankin

See also

boost

Examples

a <- r3vel(10)
as.4vel(a)     # a four-velocity
#> A vector of four-velocities (speed of light = 1)
#>              t            x           y           z
#>  [1,] 1.075881 -0.389660255  0.01479755 -0.07393753
#>  [2,] 1.803134 -0.278264930 -1.00876432  1.07529303
#>  [3,] 1.252070  0.751268944  0.05064054  0.02664360
#>  [4,] 1.155579 -0.009211468 -0.16467223  0.55512340
#>  [5,] 4.898861  0.616188783 -1.07658729 -4.63250633
#>  [6,] 2.698658 -2.326281454 -0.33930196  0.86950880
#>  [7,] 2.435266 -1.633167767 -1.39156737 -0.57168464
#>  [8,] 1.707531  0.460130562 -0.60417358  1.15711616
#>  [9,] 1.737318 -0.334341873  1.28482113  0.50569274
#> [10,] 1.124769  0.463450013 -0.14627545  0.17006881

as.3vel(as.4vel(a))-a   # zero to numerical precision
#> A vector of three-velocities (speed of light = 1)
#>                   x             y             z
#>  [1,]  2.007979e-17 -2.509973e-19  5.019946e-18
#>  [2,]  0.000000e+00 -9.024148e-17  0.000000e+00
#>  [3,] -8.702369e-17 -2.719490e-18 -4.079235e-18
#>  [4,]  5.791217e-19  1.389892e-17 -1.853190e-17
#>  [5,]  0.000000e+00  0.000000e+00 -2.664407e-15
#>  [6,]  0.000000e+00  0.000000e+00  0.000000e+00
#>  [7,]  0.000000e+00  0.000000e+00  1.646050e-16
#>  [8,]  4.046296e-17 -8.092592e-17  1.618518e-16
#>  [9,] -4.188698e-17  0.000000e+00  0.000000e+00
#> [10,]  1.755687e-17  0.000000e+00 -2.194609e-17

inner4(as.4vel(a))   #  -1 to numerical precision
#>  [1] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

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
#> [1] 1.776357e-15