The lorentz package: special relativity in R

Overview

The lorentz package furnishes some R-centric functionality for special relativity. Lorentz transformations of four-vectors are handled and some functionality for the stress energy tensor is given. The package deals with four-momentum and has facilities for dealing with photons and mirrors in relativistic situations. A detailed vignette is provided in the package.

The original motivation for the package was the investigation of the (nonassociative) gyrogroup structure of relativistic three-velocities under Einsteinian velocity composition. Natural R idiom may be used to manipulate vectors of three-velocities, although one must be careful with brackets.

Installation

To install the most recent stable version on CRAN, use install.packages() at the R prompt:

R> install.packages("lorentz")

To install the current development version use devtools:

R> devtools::install_github("RobinHankin/lorentz")

And then to load the package use library():

 library("lorentz")

The `lorentz` package in use

The package furnishes natural R idiom for working with three-velocities, four-velocities, and Lorentz transformations as four-by-four matrices. Although natural units in which [埋込みオブジェクト:https://latex.codecogs.com/png.latex?c%3D1] are used by default, this can be changed.

 u <- as.3vel(c(0.6,0,0)) # define a three-velocity, 0.6c to the right
 u
 #> A vector of three-velocities (speed of light = 1)
 #> x y z
 #> [1,] 0.6 0 0
 
 as.4vel(u) # convert to a four-velocity:
 #> A vector of four-velocities (speed of light = 1)
 #> t x y z
 #> [1,] 1.25 0.75 0 0
 gam(u) # calculate the gamma term
 #> [1] 1.25
 
B <- boost(u) # give the Lorentz transformation
B
 #> t x y z
 #> t 1.25 -0.75 0 0
 #> x -0.75 1.25 0 0
 #> y 0.00 0.00 1 0
 #> z 0.00 0.00 0 1

The boost matrix can be used to transform arbitrary four-vectors:

B %*% (1:4) # Lorentz transform of an arbitrary four-vector
 #> [,1]
 #> t -0.25
 #> x 1.75
 #> y 3.00
 #> z 4.00

But it can also be used to transform four-velocities:

v <- as.4vel(c(0,0.7,-0.2))
B %*% t(v)
 #> [,1]
 #> t 1.823312
 #> x -1.093987
 #> y 1.021055
 #> z -0.291730

The classical parallelogram law for addition of velocities is incorrect when relativistic effects are included. To combine [埋込みオブジェクト:https://latex.codecogs.com/png.latex?u] and [埋込みオブジェクト:https://latex.codecogs.com/png.latex?v] in terms of successive boosts we would simply multiply the boost matrices:

 boost(u) %*% boost(v)
 #> t x y z
 #> t 1.823312 -0.75 -1.2763187 0.3646625
 #> x -1.093987 1.25 0.7657912 -0.2187975
 #> y -1.021055 0.00 1.4240348 -0.1211528
 #> z 0.291730 0.00 -0.1211528 1.0346151

and note that the result depends on the order:

 boost(v) %*% boost(u)
 #> t x y z
 #> t 1.8233124 -1.0939874 -1.0210549 0.2917300
 #> x -0.7500000 1.2500000 0.0000000 0.0000000
 #> y -1.2763187 0.7657912 1.4240348 -0.1211528
 #> z 0.3646625 -0.2187975 -0.1211528 1.0346151

Vectorization

The package is fully vectorized and can deal with vectors whose entries are three-velocities or four-velocities:

 set.seed(0)
 options(digits=3)
 # generate 5 random three-velocities:
 (u <- r3vel(5))
 #> A vector of three-velocities (speed of light = 1)
 #> x y z
 #> [1,] 0.230 0.0719 0.314
 #> [2,] -0.311 0.4189 -0.277
 #> [3,] -0.185 0.5099 -0.143
 #> [4,] -0.739 -0.4641 0.129
 #> [5,] -0.304 -0.2890 0.593
 # calculate the gamma correction term:
 gam(u)
 #> [1] 1.09 1.24 1.21 2.13 1.46
 
 # add a velocity of 0.9c in the x-direction:
 v <- as.3vel(c(0.9,0,0))
 v+u
 #> A vector of three-velocities (speed of light = 1)
 #> x y z
 #> [1,] 0.936 0.026 0.113
 #> [2,] 0.818 0.253 -0.168
 #> [3,] 0.858 0.267 -0.075
 #> [4,] 0.480 -0.605 0.168
 #> [5,] 0.820 -0.174 0.356
 
 
 # convert u to a four-velocity:
 as.4vel(u)
 #> A vector of four-velocities (speed of light = 1)
 #> t x y z
 #> [1,] 1.09 0.250 0.0783 0.341
 #> [2,] 1.24 -0.385 0.5190 -0.343
 #> [3,] 1.21 -0.223 0.6160 -0.173
 #> [4,] 2.13 -1.571 -0.9862 0.273
 #> [5,] 1.46 -0.443 -0.4209 0.864
 
 # use four-velocities to effect the same transformation:
 w <- as.4vel(u) %*% boost(-v)
 as.3vel(w)
 #> A vector of three-velocities (speed of light = 1)
 #> x y z
 #> [1,] 0.936 0.026 0.113
 #> [2,] 0.818 0.253 -0.168
 #> [3,] 0.858 0.267 -0.075
 #> [4,] 0.480 -0.605 0.168
 #> [5,] 0.820 -0.174 0.356

Three-velocities

Three-velocities behave in interesting and counter-intuitive ways.

 u <- as.3vel(c(0.2,0.4,0.1)) # single three-velocity
 v <- r3vel(4,0.9) # 4 random three-velocities with speed 0.9
 w <- as.3vel(c(-0.5,0.1,0.3)) # single three-velocity

The three-velocity addition law is given by Ungar.

Then we can see that velocity addition is not commutative:

 u+v
 #> A vector of three-velocities (speed of light = 1)
 #> x y z
 #> [1,] 0.702 -0.113 0.567
 #> [2,] -0.679 0.580 0.102
 #> [3,] -0.046 0.879 0.364
 #> [4,] 0.312 0.407 0.788
 v+u
 #> A vector of three-velocities (speed of light = 1)
 #> x y z
 #> [1,] 0.624 -0.378 0.543
 #> [2,] -0.823 0.358 0.045
 #> [3,] -0.234 0.832 0.401
 #> [4,] 0.228 0.190 0.892
 (u+v)-(v+u)
 #> A vector of three-velocities (speed of light = 1)
 #> x y z
 #> [1,] 0.243 0.506 0.1190
 #> [2,] 0.201 0.490 0.1206
 #> [3,] 0.503 0.245 -0.0519
 #> [4,] 0.242 0.564 -0.1105

Observe that the difference between u+v and v+u is not "small" in any sense. Commutativity is replaced with gyrocommutatitivity:

 # Compare two different ways of calculating the same thing:
 (u+v) - gyr(u,v,v+u) 
 #> A vector of three-velocities (speed of light = 1)
 #> x y z
 #> [1,] 3.53e-15 -1.20e-15 2.89e-15
 #> [2,] 2.89e-16 -3.18e-15 -1.08e-16
 #> [3,] -4.26e-15 1.09e-13 4.67e-14
 #> [4,] 1.67e-15 4.76e-16 1.91e-15
 
 # The other way round:
 (v+u) - gyr(v,u,u+v)
 #> A vector of three-velocities (speed of light = 1)
 #> x y z
 #> [1,] 3.21e-15 -6.42e-16 2.89e-15
 #> [2,] 3.76e-15 -1.73e-15 -2.53e-16
 #> [3,] 1.47e-14 -4.07e-14 -2.03e-14
 #> [4,] 9.05e-15 6.43e-15 3.24e-14

(that is, zero to numerical accuracy)

Nonassociativity of three-velocities

It would be reasonable to expect that u+(v+w)==(u+v)+w. However, this is not the case:

 ((u+v)+w) - (u+(v+w))
 #> A vector of three-velocities (speed of light = 1)
 #> x y z
 #> [1,] 0.00613 0.0794 -0.001467
 #> [2,] -0.11096 -0.1508 -0.031226
 #> [3,] -0.10748 -0.1022 0.000795
 #> [4,] -0.05772 -0.0631 -0.007364

(that is, significant departure from associativity). Associativity is replaced with gyroassociativity:

 (u+(v+w)) - ((u+v)+gyr(u,v,w))
 #> A vector of three-velocities (speed of light = 1)
 #> x y z
 #> [1,] 0 8.16e-17 -6.53e-16
 #> [2,] 0 -9.49e-16 0.00e+00
 #> [3,] 0 3.21e-15 1.60e-15
 #> [4,] 0 0.00e+00 0.00e+00
 ((u+v)+w) - (u+(v+gyr(v,u,w)))
 #> A vector of three-velocities (speed of light = 1)
 #> x y z
 #> [1,] 0.00e+00 4.03e-17 -1.29e-15
 #> [2,] -1.81e-15 9.07e-16 0.00e+00
 #> [3,] 0.00e+00 1.37e-14 5.48e-15
 #> [4,] 0.00e+00 -1.84e-15 -1.84e-15

(zero to numerical accuracy).

References

The most concise reference is

A. A. Ungar 2006. Thomas precession: a kinematic effect of the algebra of Einstein’s velocity addition law. Comments on "Deriving relativistic momentum and energy: II, Three-dimensional case. European Journal of Physics, 27:L17-L20

Further information

For more detail, see the package vignette

vignette("lorentz")