Lorentz transformations — boost • lorentz

Lorentz transformations: boosts and rotations

Usage

boost(u=0)
rot(u,v,space=TRUE)
is.consistent.boost(L, give=FALSE, TOL=1e-10)
is.consistent.boost.galilean(L, give=FALSE, TOL=1e-10)
pureboost(L,include_sol=TRUE)
orthog(L)
pureboost.galilean(L, tidy=TRUE)
orthog.galilean(L)

Arguments

u,v: Three-velocities, coerced to class 3vel. In function boost(), if u takes the special default value 0, this is interpreted as zero three velocity
L: Lorentz transform expressed as a \(4\times 4\) matrix
TOL: Numerical tolerance
give: Boolean with TRUE meaning to return the transformed metric tensor (which should be the flat-space eta(); qv) and default FALSE meaning to return whether the matrix is a consistent boost or not
space: Boolean, with default TRUE meaning to return just the spatial component of the rotation matrix and FALSE meaning to return the full \(4\times 4\) matrix transformation
tidy: In pureboost.galilean(), Boolean with default TRUE meaning to return a “tidy” boost matrix with spatial components forced to be a \(3\times 3\) identity matrix
include_sol: In function pureboost(), Boolean with default TRUE meaning to correctly account for the speed of light, and FALSE meaning to assume \(c=1\). See details

Value

Function boost() returns a \(4\times 4\) matrix; function rot() returns an orthogonal matrix.

Details

Arguments u,v are coerced to three-velocities.

A rotation-free Lorentz transformation is known as a boost (sometimes a pure boost), here expressed in matrix form. Pure boost matrices are symmetric if \(c=1\). Function boost(u) returns a \(4\times 4\) matrix giving the Lorentz transform of an arbitrary three-velocity u.

Boosts can be successively applied with regular matrix multiplication. However, composing two successive pure boosts does not in general return a pure boost matrix: the product is not symmetric in general. Also note that boost matrices do not commute. The resulting matrix product represents a Lorentz transform.

It is possible to decompose a Lorentz transform \(L\) into a pure boost and a spatial rotation. Thus \(L=OP\) where \(O\) is an orthogonal matrix and \(P\) a pure boost matrix; these are returned by functions orthog() and pureboost() respectively. If the speed of light is not equal to 1, the functions still work but can be confusing.

Functions pureboost.galilean() and orthog.galilean() are the Newtonian equivalents of pureboost() and orthog(), intended to be used when the speed of light is infinite (which causes problems for the relativistic functions).

As noted above, the composition of two pure Lorentz boosts is not necessarily pure. If we have two successive boosts corresponding to \(u\) and \(v\), then the composed boost may be decomposed into a pure boost of boost(u+v) and a rotation of rot(u,v).

The reason argument include_sol exists is that function orthog() needs to call pureboost() in an environment where we pretend that \(c=1\).

References

Ungar 2006. “Thomas precession: a kinematic effect...”. European Journal of Physics, 27:L17-L20
Sbitneva 2001. “Nonassociative geometry of special relativity”. International Journal of Theoretical Physics, volume 40, number 1, pages 359–362
Wikipedia contributors 2018. “Wigner rotation”, Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/w/index.php?title=Wigner_rotation&oldid=838661305. Online; accessed 23 August 2018

Author

Robin K. S. Hankin

Note

Function rot() uses crossprod() for efficiency reasons but is algebraically equivalent to

boost(-u-v) %*% boost(u) %*% boost(v).

Examples

boost(as.3vel(c(0.4,-0.2,0.1)))
#>            t           x           y           z
#> t  1.1250879 -0.45003516  0.22501758 -0.11250879
#> x -0.4500352  1.09530507 -0.04765253  0.02382627
#> y  0.2250176 -0.04765253  1.02382627 -0.01191313
#> z -0.1125088  0.02382627 -0.01191313  1.00595657

u <- r3vel(1)
v <- r3vel(1)
w <- r3vel(1)

boost(u) - solve(boost(-u))  # should be zero
#>               t             x             y            z
#> t -2.220446e-16 -1.110223e-16  1.110223e-16 0.000000e+00
#> x -5.551115e-17 -4.440892e-16 -2.081668e-17 3.469447e-17
#> y  0.000000e+00 -2.775558e-17  2.220446e-16 2.775558e-17
#> z  0.000000e+00  3.469447e-17  0.000000e+00 0.000000e+00

boost(u) %*% boost(v)   # not a pure boost (not symmetrical)
#>            t          x           y          z
#> t  1.9785365  0.6997728  0.09466761 -1.5543367
#> x  0.6577317  1.1507970 -0.06296776 -0.3229743
#> y  0.4431298  0.1902924  0.99271578 -0.4179333
#> z -1.5118305 -0.3593565  0.13968679  1.7711527
boost(u+v)  # not the same!
#>            t           x           y          z
#> t  1.9785365  0.65773173  0.44312981 -1.5118305
#> x  0.6577317  1.14524282  0.09785361 -0.3338482
#> y  0.4431298  0.09785361  1.06592635 -0.2249216
#> z -1.5118305 -0.33384815 -0.22492159  1.7673673
boost(v+u)  # also not the same!
#>             t           x           y           z
#> t  1.97853648  0.69977283  0.09466761 -1.55433672
#> x  0.69977283  1.16440357  0.02224106 -0.36517351
#> y  0.09466761  0.02224106  1.00300885 -0.04940189
#> z -1.55433672 -0.36517351 -0.04940189  1.81112407

u+v  # returns a three-velocity
#> A vector of three-velocities (speed of light = 1)
#>               x          y         z
#> [1,] -0.3324335 -0.2239685 0.7641156


boost(u) %*% boost(v) %*% boost(w)  # associative, no brackets needed
#>            t         x           y          z
#> t  2.4816250  1.843221 -0.04708488 -1.3261908
#> x  1.3585858  1.667719 -0.12705019 -0.2198357
#> y  0.5002695  0.432151  0.96273274 -0.3696766
#> z -1.7499822 -1.195585  0.24335345  1.6043047
boost(u+(v+w))  # not the same!
#>            t          x          y          z
#> t  2.4816250  1.3585858  0.5002695 -1.7499822
#> x  1.3585858  1.5301419  0.1952132 -0.6828711
#> y  0.5002695  0.1952132  1.0718830 -0.2514523
#> z -1.7499822 -0.6828711 -0.2514523  1.8796001
boost((u+v)+w)  # also not the same!
#>            t          x          y          z
#> t  2.4393350  1.3433462  0.4655575 -1.7114416
#> x  1.3433462  1.5246884  0.1818389 -0.6684602
#> y  0.4655575  0.1818389  1.0630191 -0.2316653
#> z -1.7114416 -0.6684602 -0.2316653  1.8516275


rot(u,v)
#>              x           y           z
#> x  0.996270498 -0.08387262  0.02026027
#> y  0.086184168  0.97863167 -0.18668783
#> z -0.004169345  0.18773769  0.98221035
rot(v,u)    # transpose (=inverse) of rot(u,v)
#>             x           y            z
#> x  0.99627050  0.08618417 -0.004169345
#> y -0.08387262  0.97863167  0.187737693
#> z  0.02026027 -0.18668783  0.982210352


rot(u,v,FALSE) %*% boost(v) %*% boost(u)
#>            t           x           y          z
#> t  1.9785365  0.65773173  0.44312981 -1.5118305
#> x  0.6577317  1.14524282  0.09785361 -0.3338482
#> y  0.4431298  0.09785361  1.06592635 -0.2249216
#> z -1.5118305 -0.33384815 -0.22492159  1.7673673
boost(u+v)     # should be the same.
#>            t           x           y          z
#> t  1.9785365  0.65773173  0.44312981 -1.5118305
#> x  0.6577317  1.14524282  0.09785361 -0.3338482
#> y  0.4431298  0.09785361  1.06592635 -0.2249216
#> z -1.5118305 -0.33384815 -0.22492159  1.7673673


orthog(boost(u) %*% boost(v)) - rot(u,v,FALSE)  # zero to numerical precision
#>               t             x             y             z
#> t -5.107026e-15  1.990922e-15  4.351751e-16 -4.531597e-15
#> x  5.309112e-15 -1.776357e-15 -2.636780e-16  4.562323e-15
#> y  2.411154e-15 -1.110223e-15 -2.220446e-16  1.942890e-15
#> z -7.803692e-15  3.932618e-15  3.885781e-16 -6.661338e-15
pureboost(boost(v) %*% boost(u)) - boost(u+v)   # ditto
#>               t             x             y             z
#> t -1.110223e-15  2.220446e-16 -5.551115e-17 -2.220446e-16
#> x  1.110223e-16  6.661338e-16 -5.551115e-17  0.000000e+00
#> y -1.110223e-16 -1.387779e-17  2.220446e-16  1.387779e-16
#> z  0.000000e+00  0.000000e+00  1.387779e-16  2.220446e-16


## Define a random-ish Lorentz transform
L <- boost(r3vel(1)) %*% boost(r3vel(1)) %*% boost(r3vel(1))

## check it:


if (FALSE)    # needs emulator package
quad.form(eta(),L)  # should be eta()
 # \dontrun{}

## More concisely:
is.consistent.boost(L)     # should be TRUE
#> [1] TRUE

## Decompose L into a rotation and a pure boost:
U <- orthog(L)
P <- pureboost(L)

L - U %*% P              # should be zero (L = UP)
#>               t             x             y             z
#> t  8.881784e-16  2.220446e-16 -4.440892e-16 -6.661338e-16
#> x  0.000000e+00  0.000000e+00 -2.220446e-16 -2.220446e-16
#> y -4.440892e-16 -2.220446e-16  0.000000e+00  4.440892e-16
#> z  4.996004e-16  4.302114e-16 -3.330669e-16 -2.220446e-16
crossprod(U)               # should be identity (U is orthogonal)
#>               t            x             y             z
#> t  1.000000e+00 5.566392e-15 -2.993074e-15 -3.072080e-15
#> x  5.566392e-15 1.000000e+00  2.668507e-15  2.528677e-15
#> y -2.993074e-15 2.668507e-15  1.000000e+00 -1.464153e-15
#> z -3.072080e-15 2.528677e-15 -1.464153e-15  1.000000e+00
P - t(P)                   # should be zero (P is symmetric)
#>              t             x             y             z
#> t 0.000000e+00 -2.220446e-16 -2.220446e-16  0.000000e+00
#> x 2.220446e-16  0.000000e+00 -1.110223e-16  1.665335e-16
#> y 2.220446e-16  1.110223e-16  0.000000e+00 -1.110223e-16
#> z 0.000000e+00 -1.665335e-16  1.110223e-16  0.000000e+00

## First row of P should be a consistent 4-velocity:
is.consistent.4vel(P[1,,drop=FALSE],give=TRUE)
#>             t 
#> -7.549517e-15