Mirrors
reflect.RdPlane mirrors in special relativity
Value
Takes a four-momentum and returns the four-momentum after reflection. Will handle objects or photons.
Note
All four-momenta are measured in the rest frame of the mirror, but it is easy to reflect from moving mirrors; see examples.
However, note that the ref argument is designed to work with
photons only, where it is conceptually the percentage of photons
reflected and not absorbed by the mirror. If ref is less than
unity, odd results are given for four momenta of nonzero restmass
objects.
Examples
## We will reflect some photons from an oblique mirror moving at half
## the speed of light.
## First create 'A', a bunch of photons all moving roughly along the x-axis:
A <- as.photon(as.3vel(cbind(0.9,runif(10)/1000,runif(10)/1000)))
## Now create 'm', a mirror oriented perpendicular to c(1,1,1):
m <- c(1,1,1)
## Reflect the photons in the mirror:
reflect(A,m)
#> E p_x p_y p_z
#> [1,] 1 0.3321928 -0.6668874 -0.6670151
#> [2,] 1 0.3322844 -0.6669715 -0.6668854
#> [3,] 1 0.3327337 -0.6669269 -0.6667060
#> [4,] 1 0.3326268 -0.6673622 -0.6663235
#> [5,] 1 0.3323483 -0.6670075 -0.6668175
#> [6,] 1 0.3329772 -0.6667594 -0.6667519
#> [7,] 1 0.3332999 -0.6666554 -0.6666947
#> [8,] 1 0.3325008 -0.6666548 -0.6670941
#> [9,] 1 0.3326675 -0.6671699 -0.6664958
#> [10,] 1 0.3327153 -0.6671589 -0.6664829
## Reflect the photons in a series of mirrors:
A |> reflect(m) |> reflect(1:3) |> reflect(3:1)
#> E p_x p_y p_z
#> [1,] 1 -0.6468273 -0.7476385 0.1505029
#> [2,] 1 -0.6466903 -0.7477518 0.1505287
#> [3,] 1 -0.6464284 -0.7480372 0.1502354
#> [4,] 1 -0.6461089 -0.7482207 0.1506953
#> [5,] 1 -0.6466150 -0.7478175 0.1505259
#> [6,] 1 -0.6464150 -0.7481070 0.1499449
#> [7,] 1 -0.6462912 -0.7482694 0.1496683
#> [8,] 1 -0.6468292 -0.7477146 0.1501163
#> [9,] 1 -0.6462545 -0.7481351 0.1504958
#> [10,] 1 -0.6462322 -0.7481618 0.1504589
## To reflect from a moving mirror we need to transform to a frame in
## which the mirror is at rest, then transform back to the original
## frame. First create B, a boost representing the mirror's movement
## along the x-axis at speed c/2:
B <- boost(as.3vel(c(0.5,0,0)))
## Transform to the mirror's rest frame:
A %*% t(B)
#> t x y z
#> [1,] 0.5773507 0.5773494 9.190760e-04 7.913814e-04
#> [2,] 0.5773506 0.5773496 7.434578e-04 8.295925e-04
#> [3,] 0.5773504 0.5773500 3.392551e-04 5.601556e-04
#> [4,] 0.5773506 0.5773496 1.041656e-05 1.049097e-03
#> [5,] 0.5773506 0.5773496 6.436446e-04 8.336927e-04
#> [6,] 0.5773503 0.5773502 2.633771e-04 2.708369e-04
#> [7,] 0.5773503 0.5773503 4.476737e-05 5.434626e-06
#> [8,] 0.5773505 0.5773498 8.439073e-04 4.046268e-04
#> [9,] 0.5773505 0.5773499 1.622607e-04 8.363282e-04
#> [10,] 0.5773505 0.5773499 1.254527e-04 8.014256e-04
## NB: in the above, take a transpose because the *rows* of A are 4-vectors.
## Then reflect the photons in the mirror:
reflect(A %*% t(B),m)
#> E p_x p_y p_z
#> [1,] 0.5773507 0.1913095 -0.3851208 -0.3852485
#> [2,] 0.5773506 0.1914012 -0.3852049 -0.3851188
#> [3,] 0.5773504 0.1918504 -0.3851604 -0.3849395
#> [4,] 0.5773506 0.1917435 -0.3855957 -0.3845570
#> [5,] 0.5773506 0.1914650 -0.3852410 -0.3850510
#> [6,] 0.5773503 0.1920939 -0.3849929 -0.3849854
#> [7,] 0.5773503 0.1924166 -0.3848889 -0.3849282
#> [8,] 0.5773505 0.1916176 -0.3848883 -0.3853276
#> [9,] 0.5773505 0.1917842 -0.3854034 -0.3847293
#> [10,] 0.5773505 0.1918320 -0.3853924 -0.3847164
## Now transform back to the original rest frame (NB: active transform):
A |> tcrossprod(B) |> reflect(m) |> tcrossprod(solve(B))
#> t x y z
#> [1,] 0.7771197 0.5542388 -0.3851208 -0.3852485
#> [2,] 0.7771726 0.5543446 -0.3852049 -0.3851188
#> [3,] 0.7774317 0.5548632 -0.3851604 -0.3849395
#> [4,] 0.7773702 0.5547399 -0.3855957 -0.3845570
#> [5,] 0.7772094 0.5544182 -0.3852410 -0.3850510
#> [6,] 0.7775722 0.5551443 -0.3849929 -0.3849854
#> [7,] 0.7777585 0.5555169 -0.3848889 -0.3849282
#> [8,] 0.7772974 0.5545944 -0.3848883 -0.3853276
#> [9,] 0.7773936 0.5547868 -0.3854034 -0.3847293
#> [10,] 0.7774212 0.5548420 -0.3853924 -0.3847164