In RobinHankin/schwarzschild: General Relativity in R

set.seed(1)
knitr::opts_chunk$set(echo = TRUE)
library("schwarzschild")

We present a new form of coordinates for a black hole.

f <- function(r){
  A <- sqrt(r)
  B <- sqrt(r-1)
  A*B + log((A+B)/(A-B))/2
}

f_inv_single <- function(s){uniroot(function(x){f(x)-s},c(1,61))$root}
f_inv <- function(s){sapply(s,f_inv_single)}
fdash <- function(r,dr=1e-6){ # df/dr
(f(r+dr/2)-f(r-dr/2))/dr
}

r <- seq(from=0,to=10,by=0.1)
par(pty="s")
plot(r,f(r),asp=1,xlab="Schwarzschild r",ylab="string length",
xlim=c(0,10),ylim=c(0,10))

s <- seq(from=0,to=10,by=0.1)
par(pty="s")
plot(s,f_inv(s),asp=1,ylab="Schwarzschild r",xlab="string length")

df <- 0.01
string <- seq(from=0.04, to=6,by=df)

r <- f_inv(string)
tee <- df*cumsum(fdash(r) * (1-1/r))
par(pty="s")
plot(string,tee,asp=1,xlim=c(0,4),ylim=c(0,4),type="n")
for(i in -6:6){
points(string,i+tee,type="l",col="red")  # ingoing
points(string,i-tee,type="l",col="blue") # outgoing
}