Speed of light and Minkowski metric
sol.RdGetting and setting the speed of light
Arguments
- c
Scalar, speed of light. If missing, return the speed of light
- downstairs
Boolean, with default
TRUEmeaning to return the covariant metric tensor \(g_{ij}\) with two downstairs indices, andFALSEmeaning to return the contravariant version \(g^{ij}\) with two upstairs indices- to_natural,change_time
Boolean, specifying the nature of the passive transform matrix
Details
In the context of an R package, the symbol “c” presents
particular problems. In the lorentz package, the speed of light
is denoted “sol”, for ‘speed of light’. You can set the
speed of light with sol(x), and query it with sol(); see
the examples. An infinite speed of light is sometimes useful for
Galilean transforms.
The speed of light is a global variable, governed by
options("c"). If NULL, define c=1. Setting
showSOL to TRUE makes sol() change the prompt to
display the speed of light which might be useful.
Function eta() returns the Minkowski flat-space metric
$$\mathrm{diag}\left(-c^2,1,1,1\right).$$
Note that the top-left element of eta() is \(-c^2\), not
\(-1\).
Function ptm() returns a passive transformation matrix that
converts displacement vectors to natural units
(to_natural=TRUE) or from natural units
(to_natural=FALSE). Argument change_time specifies
whether to change the unit of time (if TRUE) or the unit of
length (if FALSE).
Note
Typing “sol(299792458)” is a lot easier than typing
“options("c"=299792458)”, which is why the package uses
the idiom that it does.
In a R-devel discussion about options for printing, Martin Maechler makes the following observation: “Good programming style for functions according to my book is to have them depend only on their arguments, and if a global option really (really? think twice!) should influence behavior, there should be arguments of the function which have a default determined by the global option”
I think he is right in general, but offer the observation that the
speed of light depends on the units chosen, and typically one fixes
one's units once and for all, and does not subsequently change them.
This would indicate (to me at least) that a global option would be
appropriate. Further, there is a default, \(c=1\), which is
returned by sol() if the option is unset. This is not just a
“default”, though: it is used in the overwhelming majority of
cases. Indeed, pedagogically speaking, one learning objective from
the package is that units in which \(c\neq 1\) are difficult,
awkward, and unnatural. In the package R code, the
only place the speed of light option is accessed is via
sol(). Similar arguments are presented in the clifford
package at signature.Rd.
Examples
sol() # returns current speed of light
#> [1] 1
sol(299792458) # use SI units
#> [1] 299792458
sol() # speed of light now SI value
#> [1] 299792458
eta() # note [t,t] term
#> [,1] [,2] [,3] [,4]
#> [1,] -8.987552e+16 0 0 0
#> [2,] 0.000000e+00 1 0 0
#> [3,] 0.000000e+00 0 1 0
#> [4,] 0.000000e+00 0 0 1
u <- as.3vel(c(100,200,300)) # fast terrestrial speed, but not relativistic
boost(u) # boost matrix practically Galilean
#> t x y z
#> t 1 -1.112650e-15 -2.22530e-15 -3.337950e-15
#> x -100 1.000000e+00 1.11265e-13 1.668975e-13
#> y -200 1.112650e-13 1.00000e+00 3.337950e-13
#> z -300 1.668975e-13 3.33795e-13 1.000000e+00
is.consistent.boost(boost(u)) # should be TRUE
#> [1] TRUE
sol(1) # revert to relativistic units
#> [1] 1