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README.md
In lorentz: 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
#> 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")
Try the lorentz package in your browser
Any scripts or data that you put into this service are public.
lorentz documentation built on Oct. 23, 2020, 5:50 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
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
#> 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")
Try the lorentz package in your browser
Any scripts or data that you put into this service are public.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
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