The Lorentz Transform in Relativistic Physics.
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 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 and
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,] -1.45e-15 1.73e-15 1.08e-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,] 0.0e+00 8.16e-17 -6.53e-16
#> [2,] -3.8e-15 2.85e-15 9.49e-16
#> [3,] 0.0e+00 3.21e-15 1.60e-15
#> [4,] 0.0e+00 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")