# Lorentz transforms via Clifford algebra

#### Robin K. S. Hankin

To cite the `clifford` package in publications please use Hankin (2022b). In this short document I show how Clifford algebra may be used to effect Lorentz transforms, and showcase the `clifford` R package. Throughout, we use units in which \(c=1\). Notation follows Snygg (2010). Consider the following four-vector:

    (fourvec <- c(1,5,3,2))  # a four-vector

    ## [1] 1 5 3 2

    u <- c(0.2,0.3,0.4)  # a three-velocity

We wish to consider the effect of a Lorentz transformation of `s`. This is done by the `boost()` function of the `lorentz` package (Hankin 2022a):

    (Bmat <- boost(u))  # Bmat = "B-matrix"

    ##            t           x           y           z
    ## t  1.1867817 -0.23735633 -0.35603450 -0.47471266
    ## x -0.2373563  1.02576299  0.03864448  0.05152597
    ## y -0.3560345  0.03864448  1.05796672  0.07728896
    ## z -0.4747127  0.05152597  0.07728896  1.10305195

The transformation itself is simply matrix multiplication:

    Bmat %*% fourvec

    ##        [,1]
    ## t -2.017529
    ## x  5.110444
    ## y  3.165666
    ## z  2.220888

We will effect this operation using Clifford algebra. Conceptually I am following Snygg but using a somewhat modified notation for consistency with the `clifford` and `lorentz` packages.

## Lorentz transforms in terms of rapidity

The general form for a Lorentz transform of speed \(u\) in the \(x\)-direction is

\[ \begin{pmatrix} \overline{t}\\ \overline{x} \end{pmatrix} = \begin{pmatrix} \gamma&-\gamma v\\ -\gamma v&\gamma \end{pmatrix} \begin{pmatrix} t\\x\end{pmatrix} \]

where \(\gamma=(1-u^2)^{-1/2}\). Writing \(\cosh\phi=\gamma\) and noting that \(\phi\) is real (sometimes \(\phi\) is known as the rapidity) we get

\[ \begin{pmatrix} \cosh\phi&-\sinh\phi\\ -\sinh\phi &\cosh\phi \end{pmatrix} \] for the transformation, and we can see that the matrix has unit determinant.

## Lorentz transforms in Clifford algebra

Above we considered the four-vector \(s=(1,5,3,2)\). In Clifford formalism this appears as

    (scliff <- as.1vector(fourvec))

    ## Element of a Clifford algebra, equal to
    ## + 1e_1 + 5e_2 + 3e_3 + 2e_4

Algebraically this would be \(1\mathbf{e}_1+5\mathbf{e}_2+3\mathbf{e}_3+2\mathbf{e}_4\) (Snygg would write \(1\mathbf{e}_0+5\mathbf{e}_1+3\mathbf{e}_2+2\mathbf{e}_3\); we cannot use that notation here because basis vectors are numbered from 1 in the package, not zero). Also note that the vectors appear in implementation-specific order, as per `disordR` discipline (Hankin 2022c). The metric would be

\[ \begin{pmatrix} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{pmatrix} \]

[NB in relativity, the word “signature” refers to the eigenvalues of the metric, so the signature of the above matrix would be \((1,3)\) [or sometimes \({+}{-}{-}{-}\)], because it has one positive and three negative eigenvalues. In package idiom, “signature” means the number of basis vectors that square to \(+1\) and \(-1\), so we would implement this metric using a signature of \((1,3)\)].

The squared interval for our four-vector would be given by

    M <- diag(c(1,-1,-1,-1))
    t(fourvec) %*% M %*% fourvec

    ##      [,1]
    ## [1,]  -37

We might use the slightly slicker and more efficient idiom `quad.form()` from the `quadform` package:

    quad.form(M,fourvec)

    ##      [,1]
    ## [1,]  -37

The Clifford equivalent would be `scalprod()` [remembering to set the signature to 1]:

    signature(1,3)
    scalprod(scliff,scliff)

    ## [1] -37

We seek a boost \(B\in{\mathcal C}_{1,3}\) such that \(\overline{s}=B^{-1}sB\) (juxtaposition indicating geometric product). We will start with a boost in the \(x\)-direction with rapidity \(\phi\). This would be \(B=\cosh(\phi/2)+{\mathbf e}_{12}\sinh(\phi/2)\). We note that \(B^{-1}=\cosh(\phi/2)-{\mathbf e}_{12}\sinh(\phi/2)\). Numerically:

    phi <- 2.1234534   # just a made-up random value
    B <- cosh(phi/2) + sinh(phi/2)*e(1:2)
    Binv <- rev(B) # cosh(phi/2)- sinh(phi/2)*e(1:2)
    B*Binv

    ## Element of a Clifford algebra, equal to
    ## scalar ( 1 )

We may verify that rapidities add:

    B <- function(phi){cosh(phi/2) + sinh(phi/2)*e(1:2)}
    B(0.26) * B(1.33)

    ## Element of a Clifford algebra, equal to
    ## + 1.333011 + 0.8814299e_12

    B(0.26 + 1.33) # should match

    ## Element of a Clifford algebra, equal to
    ## + 1.333011 + 0.8814299e_12

We may formally write \(B=\exp({\mathbf e}_{12}\phi/2)\) on the grounds that

\[ \begin{eqnarray} \exp({\mathbf e}_{12}x) &=&1+\mathbf{e}_{12}x + \frac{(\mathbf{e}_{12}x)^2}{2!} + \frac{(\mathbf{e}_{12}x)^3}{3!}+\frac{(\mathbf{e}_{12}x)^4}{4!}+\cdots\\ &=& (1+x^2/2+x^4/4!+\cdots) + \mathbf{e}_{12}(x+\frac{x^3}{3!}+\cdots)\\ &=& \cosh x + \mathbf{e}_{12}\sinh x \end{eqnarray} \]

and note that this exponential obeys the usual rules for the regular exponential function \(e^x,x\in\mathbb{R}\). More generally, if we have a transform of rapidity \(\phi\) and direction cosines \(k_x,k_y,k_z\) then the transform would be

\[ B_{xyz}= \cosh(\phi/2) +k_x\mathbf{e}_{12}\sinh(\phi/2) +k_y\mathbf{e}_{13}\sinh(\phi/2) +k_z\mathbf{e}_{14}\sinh(\phi/2) \]

and we can use standard Clifford algebra (together with the fact that \(k_x^2+k_y^2+k_z^2=1\)) to demonstrate the transformations. Numerically:

    B3 <- function(phi,k){cosh(phi/2) + (
         +k[1]*sinh(phi/2)*e(c(1,2))
         +k[2]*sinh(phi/2)*e(c(1,3))
         +k[3]*sinh(phi/2)*e(c(1,4))
       )}
    k <- function(kx,ky){c(kx, ky, sqrt(1-kx^2-ky^2))}
    kx <- +0.23
    ky <- -0.38

    k1 <- k(kx=0.23, ky=-0.38)
    sum(k1^2) # verify; should be = 1

    ## [1] 1

    zap(B3(0.3,k1)*B3(1.9,k1))  # zap() kills terms with small coefficients

    ## Element of a Clifford algebra, equal to
    ## + 1.668519 + 0.3071989e_12 - 0.507546e_13 + 1.196654e_14

    zap(B3(0.3+1.9,k1)) # should match previous line (up to numerical accuracy)

    ## Element of a Clifford algebra, equal to
    ## + 1.668519 + 0.3071989e_12 - 0.507546e_13 + 1.196654e_14

But if the two boosts have different direction cosines, the result is more complicated:

    k2 <- k(-0.5,0.1)
    zap(B3(2.4,k1) * B3(1.9,k2))

    ## Element of a Clifford algebra, equal to
    ## + 3.716216 - 0.479412e_12 - 0.653413e_13 + 0.277158e_23 + 3.722481e_14 -
    ## 1.071824e_24 + 0.691205e_34

Above, we see new terms not present in the pure boosts which correspond to rotation.

Now we consider a general four-vector \(s=s^1\mathbf{e}_1+s^2\mathbf{e}_2+s^3\mathbf{e}_3+s^4\mathbf{e}_4\) and calculate \(B^{-1}sB\). This is made easier if we use the facts that \(\mathbf{e}_{12}\) commutes with \(\mathbf{e}_3\) and \(\mathbf{e}_4\) as well as scalars, and anticommutes with \(\mathbf{e}_1\) and \(\mathbf{e}_2\). Noting that \(\exp(\mathbf{e}_{12})\) is a linear combination of a scalar and \(\mathbf{e}_{12}\) we have

\[ \begin{eqnarray} B^{-1}sB &=& \exp(-\mathbf{e}_{12}\phi/2)(s^1\mathbf{e}_1+s^2\mathbf{e}_2+s^3\mathbf{e}_3+s^4\mathbf{e}_4)\exp(\mathbf{e}_{12}\phi/2)\\ &=& (\mathbf{e}_1s^1+\mathbf{e}_2s^2)\exp(\mathbf{e}_{12}\phi/2)\exp(\mathbf{e}_{12}\phi/2)+\mathbf{e}_3s^3 + \mathbf{e}_4s^4\\ &=& \mathbf{e}_1(s^1\cosh\phi-s^2\sinh\phi)+\mathbf{e}_2(s^2\cosh\phi-s^1\sinh\phi) +\mathbf{e}_3s^3+\mathbf{e}_4s^4 \end{eqnarray}\]

as required (it matches the matrix version). If we have two boosts \(B_1\) and \(B_2\) then the combined boost is either \(B_1B_2\) (for \(B_1\) followed by \(B_2\)) or \(B_2B_1\) (for \(B_2\) followed by \(B_1\)). Numerical methods are straightforward as I will demonstrate below.

## Numerical methods: Lorentz transforms using the Clifford package

Above we considered boost `Bmat`, and here I will show the effect of this boost in terms of Clifford objects, using a specialist function `f()`:

    f <- function(u){
        phi <- acosh(gam(u))               # rapidity
        k <- cosines(u)                    # direction cosines
        return(
               cosh(phi/2)                 # t
        + k[1]*sinh(phi/2)*basis(c(1,2))   # x
        + k[2]*sinh(phi/2)*basis(c(1,3))   # y
        + k[3]*sinh(phi/2)*basis(c(1,4))   # z
        )
    }

Thus we can express the Lorentz transform as a Clifford object:

    u <- as.3vel(-c(0.2,0.3,0.4))  # negative (passive transform)
    options(digits=5)
    (B <- f(u))

    ## Element of a Clifford algebra, equal to
    ## + 1.0457 - 0.1135e_12 - 0.17025e_13 - 0.22699e_14

The first thing to do is to verify that the inverse of `B` behaves as expected:

    B*rev(B)

    ## Element of a Clifford algebra, equal to
    ## scalar ( 1 )

Then we can apply the transformation \(\overline{s}=B^{-1}sB\):

    zap(rev(B)*scliff*B)

    ## Element of a Clifford algebra, equal to
    ## - 2.0175e_1 + 5.1104e_2 + 3.1657e_3 + 2.2209e_4

Comparing with the result from the `lorentz` package

    Bmat %*% fourvec

    ##      [,1]
    ## t -2.0175
    ## x  5.1104
    ## y  3.1657
    ## z  2.2209

we see agreement to within numerical precision. We can further verify that the squared interval is unchanged:

    jj <- rev(B)*scliff*B
    scalprod(jj,jj)

    ## [1] -37

matching the untransformed square interval.

## Multiple boosts

Successive Lorentz boosts can induce a rotation as well as a translation.

    u <- as.3vel(c(0.2, 0.3,  0.4))
    v <- as.3vel(c(0.5, 0.0, -0.4))
    w <- as.3vel(c(0.0, 0.7,  0.1))
    Buvw <- f(u)*f(v)*f(w)
    zap(Buvw)

    ## Element of a Clifford algebra, equal to
    ## + 1.2914 + 0.45284e_12 + 0.69409e_13 - 0.14103e_23 + 0.07821e_14 + 0.07767e_24
    ## + 0.02902e_34 - 0.04011e_1234

In the above, note that Clifford object `Buvw` has a nonzero scalar component, and also a nonzero `e_1234` component. However, it represents a consistent Lorentz transformation:

    zap(Buvw*rev(Buvw))

    ## [1] 1

We can now apply this transform to a four-velocity:

    n <- as.1vector(c(1,0,0,0))
    zap(rev(Buvw) * n * Buvw)

    ## Element of a Clifford algebra, equal to
    ## + 2.3891e_1 + 0.98367e_2 + 1.9312e_3 + 0.10269e_4

## Algebra of Clifford representations

We can shed some light on this representation of Lorentz transforms as follows:

    signature(1,3)
    L <- list(
        C     = basis(numeric()),
        e12   = basis(c(1,2)), e13 = basis(c(1,3)),
        e14   = basis(c(1,4)), e23 = basis(c(2,3)),
        e24   = basis(c(2,4)), e34 = basis(c(3,4)),
        e1234 = basis(1:4)
    )

    out <- noquote(matrix("",8,8))
    rownames(out) <- names(L)
    colnames(out) <- names(L)
    for(i in 1:8){
      for(j in 1:8){
        out[i,j] <- gsub('[_ ]','',as.character(L[[i]]*L[[j]]))
      }
    }
    options("width" = 110)
    out

    ##       C       e12     e13     e14     e23     e24     e34     e1234
    ## C     1       +1e12   +1e13   +1e14   +1e23   +1e24   +1e34   +1e1234
    ## e12   +1e12   1       -1e23   -1e24   -1e13   -1e14   +1e1234 +1e34
    ## e13   +1e13   +1e23   1       -1e34   +1e12   -1e1234 -1e14   -1e24
    ## e14   +1e14   +1e24   +1e34   1       +1e1234 +1e12   +1e13   +1e23
    ## e23   +1e23   +1e13   -1e12   +1e1234 -1      +1e34   -1e24   -1e14
    ## e24   +1e24   +1e14   -1e1234 -1e12   -1e34   -1      +1e23   +1e13
    ## e34   +1e34   +1e1234 +1e14   -1e13   +1e24   -1e23   -1      -1e12
    ## e1234 +1e1234 +1e34   -1e24   +1e23   -1e14   +1e13   -1e12   -1

Thus we can see, for example, that `e12*e13 = -e23` and `e13*e12 = +e23`.

## References

Hankin, R. K. S. 2022a. “A Systematic Search for a Three-Velocity Gyrodistributive Law in Special Relativity with the Lorentz R Package.” arXiv. https://doi.org/10.48550/ARXIV.2212.07005.
———. 2022b. “Clifford Algebra in R.” arXiv. https://doi.org/10.48550/ARXIV.2209.13659.
———. 2022c. “Disordered vectors in R: introducing the `disordR` package.” arXiv, https://arxiv.org/abs/2210.03856; arXiv. https://doi.org/10.48550/ARXIV.2210.03856.
Snygg, J. 2010. A New Approach to Differential Geometry Using Clifford’s Geometric Algebra. Birkhäuser.