In RobinHankin/gyrogroup: The Lorentz Transform in Relativistic Physics

![](`r system.file("help/figures/lorentz.png", package = "lorentz")`){width=10%}

set.seed(0)
knitr::opts_chunk$set(echo = TRUE)
library("lorentz")
library("magrittr")
options("digits" = 5)

![](`r system.file("help/figures/lorentz.png", package = "lorentz")`){width=10%}

In special relativity, the headlight effect (also known as relativistic beaming and light aberration) occurs when a source of light moves relative to an observer. Rays of light from the source which reach the observer are tilted towards the direction of the source's motion (relative to the observer). It is as if light emitted by a moving object is concentrated towards its direction of motion. Also, light received by a moving object (e.g. the view from a very fast spacecraft) also appears concentrated towards its direction of motion; see the wikipedia entry.

First we will create 10 photons emitted from a stationary object, in random directions:

p1 <- as.photon(r3vel(n=10))
p1

See how they all have unit energy (column 1). We can calculate the photons' angle with respect the the $x$-axis:

angle <- function(P){acos(P[,2]/sqrt(rowSums(P[,-1]^2)))}
angle(p1)
summary(angle(p1))

and we see that the angles are reasonably uniformly distributed on the interval $[0,\pi)$. But now suppose these photons are emitted from an object moving at say $0.7c$ along the $x$-axis. This is most easily effected by a passive Lorentz transform

B <- boost(c(-0.7,0,0))
p2 <- p1 %>% tcrossprod(B)
p2

above we see that the photons' energy has changed---as one might expect from Doppler shifting. Although most photons have increased their energy, some have decreased: this is because they were emitted backwards compared with the path of the object. Photon number 4, for example, was emitted almost exactly along the (negative) $x$-axis and this photon has about half its energy in the transformed frame. What angle do the photons make with the x-axis?

angle(p2)
summary(angle(p2))

we see that the mean angle changes from about 1.4 to about 0.8: relativistic aberration of light. The effect may be illustrated more dramatically using a larger number of photons.

p1 <- as.photon(r3vel(n=10000))
hist(angle(p1))

```r
p2 <- p1 %>% tcrossprod(B)
hist(angle(p2))

Figures \@ref(fig:unshiftedphotons) and \@ref(fig:shiftedphotons) show that the photons have a smaller angle in the shifted frame than the unshifted frame. This may be illustrated by plotting one angle against the other:

par(pty='s')
plot(angle(p1),angle(p2),asp=1,xlim=c(0,pi),ylim=c(0,pi))
abline(0,1,col='gray')

par(pty='s')
plot(NA,xlim=c(-1.5,3),ylim=c(-1,1),asp=1,xlab="x",ylab="y")
p1[,4] <- 0
p2[,4] <- 0
for(i in seq_len(100)){
   segments(x0=0,y0=0,x1=p1[i,1]*cos(angle(p1)[i]),y1=p1[i,1]*sin(angle(p1)[i])*sign(p1[i,3]),col='black')
   segments(x0=0,y0=0,x1=p2[i,1]*cos(angle(p2)[i]),y1=p2[i,1]*sin(angle(p2)[i])*sign(p2[i,3]),col='red'  )
}