Gamma correction — gam • lorentz

Lorentz gamma correction term in special relativity

Usage

# S3 method for class '3vel'
speed(u)
# S3 method for class '4vel'
speed(u)
speedsquared(u)
gam(u)
gamm1(u)
# S3 method for class '3vel'
gam(u)
# S3 method for class '3cel'
gam(u)
# S3 method for class '4vel'
gam(u)
# S3 method for class '3vel'
gamm1(u)
# S3 method for class '4vel'
gamm1(u)
gam_ur(d)

Arguments

u: Speed: either a vector of speeds or a vector of three-velocities or four-velocities
d: In function gam_ur(), deficit of speed; speed of light minus speed of object

Details

Function speed(u) returns the speed of a 3vel object or 4vel object.

Function gam(u) returns the Lorentz factor $$\frac{1}{\sqrt{1-\mathbf{u}\cdot\mathbf{u}/c^2}}$$

Function gamm1(u) returns the Lorentz factor minus 1, useful for slow speeds when larger accuracy is needed (much like expm1()); to see the R idiom, type “gamm1.3vel” at the commandline. Function gamm1() is intended to work with 3vel objects or speeds. The function will take a 4-velocity, but this is not recommended as accuracy is lost (all it does is return the time component of the 4-velocity minus 1).

Function gam_ur() is used for the ultrarelativistic case where speeds are very close to the speed of light (the function is named for “gamma, ultrarelativistic”). Its argument d is the deficit, that is, $c-v$ where $v$ is the speed of the transformation. Algebraically, gam_ur(c-v) == gam(v), but if d is small compared to c the result is more accurate.

Function speedsquared(u) returns the square of the speed of a 3vel object. Use this to avoid taking a needless square root.

Author

Robin K. S. Hankin

Examples


gam(seq(from=0,by=0.1,len=10))
#>  [1] 1.000000 1.005038 1.020621 1.048285 1.091089 1.154701 1.250000 1.400280
#>  [9] 1.666667 2.294157
gam(r3vel(6,0.7))
#> [1] 1.40028 1.40028 1.40028 1.40028 1.40028 1.40028


x <- as.3vel(c(0.1,0.4,0.5))
speed(x)
#> [1] 0.6480741

gam(speed(x))  # works, but slow and inaccurate
#> [1] 1.313064
gam(x)         # recommended: avoids needless coercion
#> [1] 1.313064



## Use SI units and deal with terrestrial speeds.  Use gamm1() for this.
sol(299792458)
#> [1] 299792458
sound <- 343 # speed of sound in SI
gam(sound)
#> [1] 1
gam(sound)-1  
#> [1] 6.545875e-13
gamm1(sound)   # gamm1() gives much higher precision
#> [1] 6.545108e-13

snail <- as.3vel(c(0.00275,0,0)) # even the world's fastest snail...
gamm1(snail)                     # ...has only a small relativistic correction
#> [1] 4.207208e-23


## For the ultrarelativistic case of speeds very close to the speed of
## light, use gam_ur():

sol(1)           # revert to relativistic units
#> [1] 1

gam(0.99) - gam_ur(0.01) # zero to numerical accuracy
#> [1] -4.440892e-15

omgp <- 4.9e-24  # speed deficit of the Oh-My-God particle
gam(1-omgp)      # numeric overflow
#> [1] Inf
gam_ur(omgp)     # large but finite
#> [1] 319438282500