celerity {lorentz}

Celerity and rapidity

Description

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(s)

Robin K. S. Hankin

See Also

gam

Examples

u <- 0.1  # c=1
c(u,celerity(u),rapidity(u))

omgp <- 4.9e-24  # speed deficit of the Oh-My-God particle
c(celerity_ur(omgp),rapidity_ur(omgp))


sol(299792458)                 # use SI units
u <- 3e7  # ~0.1c
c(u,celerity(u),rapidity(u))


snail <- 0.00275
c(snail,celerity(snail),rapidity(snail))


omgp <- omgp*sol() 
c(celerity_ur(omgp),rapidity_ur(omgp))


sol(1)