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The Lorentz Transform in Relativistic Physics

lorentz-package.Rd

The Lorentz transform in special relativity; also the gyrogroup structure of three-velocities. Performs active and passive transforms and has the ability to use units in which the speed of light is not unity. Includes some experimental functionality for celerity and rapidity. For general relativity, see the 'schwarzschild' package.

Details

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Author

Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>)

Maintainer: Robin K. S. Hankin <hankin.robin@gmail.com>

References

Examples


u <- as.3vel(c(0.3,0.6,-0.1))  # u is a three-velocity
gam(u)                         # relativistic gamma term for u
#> [1] 1.360828
U <- as.4vel(u)                # U is a four-velocity
B1 <- boost(u)                 # B1 is the Lorentz transform matrix for u
B1 %*% c(1,0,0,0)              # Lorentz transform of zero 4-velocity (=-u)
#>         [,1]
#> t  1.3608276
#> x -0.4082483
#> y -0.8164966
#> z  0.1360828

B2 <- boost(as.3vel(c(-0.1,0.8,0.3)))  
B3 <- boost(as.3vel(c(-0.1,0.1,0.9)))  # more boosts

Bi <- B1 %*% B2  # Bi is the boost for successive Lorentz transforms


pureboost(Bi)      # Decompose Bi into a pure boost...
#>             t           x           y           z
#> t  3.78969964 -0.06713162 -3.54542995 -0.88726725
#> x -0.06713162  1.00094091  0.04969215  0.01243579
#> y -3.54542995  0.04969215  3.62439704  0.65677268
#> z -0.88726725  0.01243579  0.65677268  1.16436170
orthog(Bi)         # and an orthogonal matrix
#>               t             x             y             z
#> t  1.000000e+00 -1.046611e-14 -5.410925e-13 -1.350410e-13
#> x -1.236185e-13  9.794943e-01  1.895365e-01  6.831470e-02
#> y -5.484482e-13 -1.983854e-01  9.664761e-01  1.629943e-01
#> z -3.897417e-14 -3.513115e-02 -1.732047e-01  9.842591e-01

Bj <- B2 %*% B1    # B1 and B2 do not commute...

(B1 %*% B2) %*% B3 
#>            t          x          y          z
#> t  11.971728  1.0488714 -4.6614329 -10.931294
#> x  -2.357261  0.7672474  1.0039324   2.227135
#> y -11.332721 -1.2028617  4.6544280  10.311158
#> z  -2.887046 -0.2541382  0.2395334   3.035305
B1 %*% (B2 %*% B3)    # ...but composition *is* associative
#>            t          x          y          z
#> t  11.971728  1.0488714 -4.6614329 -10.931294
#> x  -2.357261  0.7672474  1.0039324   2.227135
#> y -11.332721 -1.2028617  4.6544280  10.311158
#> z  -2.887046 -0.2541382  0.2395334   3.035305



## Three velocities and the gyrogroup

## Create some random three-velocities:

u <- r3vel(10)
v <- r3vel(10)
w <- r3vel(10)

u+v
#> A vector of three-velocities (speed of light = 1)
#>                 x           y           z
#>  [1,] -0.56478934  0.25823932  0.49149635
#>  [2,]  0.34222078  0.49328155  0.26853820
#>  [3,]  0.81928617 -0.48504410  0.02809624
#>  [4,] -0.06886203  0.85086150 -0.50186720
#>  [5,] -0.89662990  0.21725228  0.12656127
#>  [6,] -0.66549350  0.17426286 -0.46591289
#>  [7,]  0.76717773  0.06379495 -0.61273061
#>  [8,]  0.68245485 -0.11176882  0.40877188
#>  [9,]  0.77223399  0.32665621 -0.48320928
#> [10,] -0.65375636  0.36263203  0.54726968
v+u        # Three-velocity addition is not commutative...
#> A vector of three-velocities (speed of light = 1)
#>                x          y           z
#>  [1,] -0.5629663  0.3130682  0.46076074
#>  [2,]  0.3231107  0.5216995  0.23660851
#>  [3,]  0.7071820 -0.5889199 -0.24567041
#>  [4,]  0.6605714  0.5145136 -0.52867678
#>  [5,] -0.6645414  0.6434610  0.10725765
#>  [6,] -0.5676908  0.3426581 -0.50063612
#>  [7,]  0.9468298  0.2513063 -0.09180836
#>  [8,]  0.4733079  0.1079067  0.64005243
#>  [9,]  0.3085770  0.7686311 -0.50052696
#> [10,] -0.5189814  0.4236121  0.64001137

u+(v+w)   # ... nor associative
#> A vector of three-velocities (speed of light = 1)
#>                x            y           z
#>  [1,] -0.1475623  0.796112386  0.41888092
#>  [2,]  0.0851946  0.935680929  0.31913792
#>  [3,]  0.8124212 -0.455730757 -0.05253253
#>  [4,] -0.0561954  0.851463111 -0.51949620
#>  [5,] -0.8049151  0.433754119 -0.35309972
#>  [6,] -0.7418702 -0.007149504 -0.49923309
#>  [7,]  0.8348374  0.058773808 -0.52618982
#>  [8,]  0.8569664 -0.222827177 -0.11727036
#>  [9,]  0.8262881  0.365942605 -0.40246178
#> [10,] -0.2715871  0.503695029  0.48807698
(u+v)+w 
#> A vector of three-velocities (speed of light = 1)
#>                  x           y           z
#>  [1,] -0.106783838  0.79515489  0.41054712
#>  [2,]  0.061286080  0.94386042  0.29878218
#>  [3,]  0.814280148 -0.43935546 -0.03085530
#>  [4,] -0.001697546  0.84255815 -0.53548132
#>  [5,] -0.797888806  0.26806231 -0.47923239
#>  [6,] -0.746013306  0.02691658 -0.51465350
#>  [7,]  0.835385914  0.04417690 -0.50303913
#>  [8,]  0.914693664 -0.21915463  0.06462596
#>  [9,]  0.805271704  0.40325896 -0.40763310
#> [10,] -0.316504550  0.45710983  0.38516756