Random relativistic velocities
r3vel.RdGenerates random three-velocities or four-velocities, optionally specifiying a magnitude
Details
Function r3vel() returns a random three-velocity. Function
r4vel() is a convenience wrapper for as.4vel(r3vel()).
Function rboost() returns a random \(4\times 4\) Lorentz
boost matrix, drawn from the connected component. If given
r=0, then a transform corresponding to a random rotation will
be returned.
Note
If the speed of light is infinite, these functions require a specified
argument for r.
It is not entirely trivial to sample uniformly from the unit ball or unit sphere, but it is not hard either.
Examples
r3vel()
#> A vector of three-velocities (speed of light = 1)
#> x y z
#> [1,] -0.45646828 -0.14605170 0.58970407
#> [2,] 0.24172040 0.52732322 0.40075655
#> [3,] -0.13891530 0.52066535 -0.56540025
#> [4,] 0.88848514 0.03080818 -0.43238351
#> [5,] -0.27141412 0.54346958 0.27619711
#> [6,] -0.01135609 -0.65786259 -0.07659117
#> [7,] -0.35032850 0.81683309 0.33793502
a <- r3vel(10000)
b <- r3vel(1000,0.8)
u <- as.3vel(c(0,0,0.9))
pairs(unclass(u+a),asp=1)
pairs(unclass(a+u),asp=1)
is.consistent.boost(rboost())
#> [1] TRUE
sol(299792458) # switch to SI units
#> [1] 299792458
sound <- 343 # speed of sound in metres per second
r3vel(100,343) # random 3-velocities with speed = 343 m/s
#> A vector of three-velocities (speed of light = 299792458)
#> x y z
#> [1,] 223.3637271 -254.074729 -56.6010367
#> [2,] 295.9875562 153.450280 80.5815004
#> [3,] 174.7920729 102.514653 276.7444259
#> [4,] 324.9495888 105.319219 -31.0584433
#> [5,] -0.1038609 244.593553 240.4640989
#> [6,] 226.7075156 253.355045 -45.4304266
#> [7,] -186.3741656 245.827043 149.9424395
#> [8,] 140.5583043 237.562485 -203.6085184
#> [9,] 340.9010218 26.507370 27.0712519
#> [10,] 55.8010975 -31.394440 336.9712550
#> [11,] 71.8624435 -308.323889 131.9892738
#> [12,] 63.0429839 -200.927521 -270.7447386
#> [13,] 296.5805652 171.836706 -12.6930956
#> [14,] -222.7324858 119.783992 231.7132605
#> [15,] 54.5412062 330.091707 75.5891625
#> [16,] -154.7347782 71.075970 -297.7488118
#> [17,] 256.5838815 62.027114 219.0122118
#> [18,] 72.4701149 241.222192 -232.8281262
#> [19,] -129.5365142 -281.960402 -146.1766850
#> [20,] 198.3639932 -56.170712 -274.1269365
#> [21,] -124.9898118 55.209131 314.6084850
#> [22,] -178.4840530 94.440229 -277.2606823
#> [23,] 111.3058995 198.478312 256.6444165
#> [24,] -166.1107486 -188.965840 233.1268549
#> [25,] 159.1668167 60.985050 -297.6503790
#> [26,] 19.7868900 250.899966 233.0379502
#> [27,] 161.9906476 281.049044 -111.4426543
#> [28,] -101.2627047 134.476626 -298.8492961
#> [29,] 38.3897542 67.689065 -334.0560091
#> [30,] -52.9485477 -192.224848 -279.0968630
#> [31,] 41.4137920 334.082646 65.7471174
#> [32,] -150.4809062 105.952929 -289.4451135
#> [33,] 128.3907123 4.057077 -318.0383076
#> [34,] -259.4231391 -189.206130 120.6220345
#> [35,] -323.1727689 -104.581318 -47.6561579
#> [36,] -270.4329806 -188.002157 95.7611204
#> [37,] -200.2002195 -60.379922 -271.8880967
#> [38,] -269.5107055 -206.496273 -48.7059446
#> [39,] -200.9015116 145.189925 -237.0811428
#> [40,] -222.9571964 -118.826718 -231.9898695
#> [41,] 271.0710787 -166.645334 -128.0578107
#> [42,] -326.7156279 -32.804370 -99.1452054
#> [43,] 0.6571765 88.038690 -331.5083063
#> [44,] 167.1621621 150.170658 -259.1420169
#> [45,] -120.8995477 -83.630189 309.9004530
#> [46,] 187.7939173 -282.198817 -52.4048878
#> [47,] -109.1324069 -91.350793 312.0803588
#> [48,] -283.9026443 192.026051 13.2016720
#> [49,] 131.6985705 -157.153262 -274.9678867
#> [50,] 120.2380853 319.790239 -30.4303420
#> [51,] 231.4134483 253.171996 0.8697855
#> [52,] -53.8973644 -239.581740 -239.4674593
#> [53,] 14.4945628 31.201831 341.2702058
#> [54,] 289.2158328 180.910046 -35.7037461
#> [55,] -82.9041710 -225.386346 -244.9018039
#> [56,] 68.8252150 177.694782 -285.1958173
#> [57,] 245.9550061 -237.970629 22.9153779
#> [58,] 10.9365679 -184.285427 -289.0817753
#> [59,] -61.1548670 -222.938879 -253.3916704
#> [60,] 224.1149912 -219.491841 138.7256371
#> [61,] -245.6392277 -149.027637 187.3529647
#> [62,] -278.1466626 198.858752 -27.1777667
#> [63,] -316.3065087 -63.138369 116.6736427
#> [64,] -293.6583269 157.517795 81.2522694
#> [65,] 190.3211425 -107.283145 -264.4185877
#> [66,] -2.2221735 271.560622 209.5206212
#> [67,] 146.0051262 -181.017715 -252.1191983
#> [68,] -103.1493296 284.047501 162.2536070
#> [69,] 250.0363911 -192.341521 134.6682674
#> [70,] 50.4533834 339.180380 7.7540916
#> [71,] 95.5431531 46.518756 -326.1234601
#> [72,] 62.4903065 -234.062273 242.8143611
#> [73,] -315.7474918 10.784228 -133.5523188
#> [74,] 209.6173551 -216.127125 -164.3125987
#> [75,] 85.2519089 90.578311 -319.6508745
#> [76,] 2.4643883 -238.454130 -246.5411823
#> [77,] -147.7155147 -278.074961 -136.0273597
#> [78,] 279.1687341 183.367200 78.0402975
#> [79,] 276.1017311 176.974072 100.4838886
#> [80,] 39.5506973 318.735366 -120.3848353
#> [81,] -217.5315978 -181.690723 -193.1773410
#> [82,] -189.5233145 -251.636508 135.6797011
#> [83,] 227.3832621 -71.231441 -246.7223824
#> [84,] -247.7408332 -15.813754 236.6926377
#> [85,] 281.5670855 192.132821 38.1307658
#> [86,] -294.9147680 64.072197 163.0001017
#> [87,] 306.6753502 85.924681 127.3427610
#> [88,] 206.2407041 -243.971577 124.8664943
#> [89,] 269.9749933 125.112785 170.6144604
#> [90,] -144.0253575 99.421114 294.9934551
#> [91,] 49.3198382 135.889372 311.0476363
#> [92,] 209.5728550 270.403356 24.7031070
#> [93,] 154.4194922 -224.922568 -207.8784718
#> [94,] 219.5527314 165.794695 -204.8358302
#> [95,] -127.8892224 -184.104180 -259.6131693
#> [96,] 112.7666997 309.326201 -96.1767779
#> [97,] 55.2166206 90.036071 326.3336189
#> [98,] 205.3759527 196.140746 192.3500082
#> [99,] -171.5250023 29.095191 -295.6038624
#> [100,] 42.8275552 32.163329 338.7924450
sol(1) # return to default c=1
#> [1] 1