# 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
#>        x y z
#> [1,] 0.6 0 0

as.4vel(u)    # convert to a four-velocity:
#>         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))
#>           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
#>          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)
#>         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)
#>          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-velocites 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
#>           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
#>           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)
#>          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)
#>              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)
#>             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))
#>             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))
#>      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)))
#>              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")``