Celerity and rapidity

The celerity and rapidity of an object (experimental functionality)

Usage

# S3 method for class '3vel'
celerity(u)
# S3 method for class '4vel'
celerity(u)
celerity_ur(d)
# S3 method for class '3vel'
rapidity(u)
# S3 method for class '4vel'
rapidity(u)
rapidity_ur(d)
as.3cel(x)
cel_to_vel(x)
vel_to_cel(x)

Arguments

u,x

Speed: either a vector of speeds or a vector of three-velocities or four-velocities

d

In functions celerity_ur() and rapidity_ur(), deficit of speed; speed of light minus speed of object

Details

The celerity corresponding to speed \(u\) is defined as \(u\gamma\) and the rapidity is \(c\cdot\mathrm{atanh}(u/c)\).

Functions celerity_ur() and rapidity_ur() are used for the ultrarelativistic case where speeds are very close to the speed of light. Its argument d is the deficit, that is, \(d=c-v\) where \(v\) is the speed of the transformation. Algebraically, celerity_ur(c-v) == celerity(v), but if \(d=1-v/c\) is small the result of celerity_ur() is more accurate than that of celerity().

Things get a bit sticky for celerity and rapidity if \(c\neq 1\). The guiding principle in the package is to give the celerity and rapidity the same units as \(c\), so if \(u\ll c\) we have that all three of celerity(u), rapidity(u) and u are approximately equal. Note carefully that, in contrast, \(\gamma\) is dimensionless. Also observe that d in functions celerity_ur() and rapidity_ur() has the same units as \(c\).

Author

Robin K. S. Hankin

See also

gam

Examples

u <- 0.1  # c=1
c(u,celerity(u),rapidity(u))
#> [1] 0.1000000 0.1005038 0.1003353

omgp <- 4.9e-24  # speed deficit of the Oh-My-God particle
c(celerity_ur(omgp),rapidity_ur(omgp))
#> [1] 3.194383e+11 2.718298e+01


sol(299792458)                 # use SI units
#> [1] 299792458
u <- 3e7  # ~0.1c
c(u,celerity(u),rapidity(u))
#> [1] 30000000 30151345 30100745


snail <- 0.00275
c(snail,celerity(snail),rapidity(snail))
#> [1] 0.00275 0.00275 0.00275


omgp <- omgp*sol() 
c(celerity_ur(omgp),rapidity_ur(omgp))
#> [1] 9.576519e+19 8.149252e+09


sol(1)
#> [1] 1