An alternative algebra for Lorentz boosts

doug  <sweetser@world.com>

Introduction
The tools of special relativity
Using quaternions to do special relativity
Using quaternions in practice
Implications

Introduction

Many problems in physics are expressed efficiently as differential equations whose solutions are dictated by calculus.  The foundations of calculus were shown in turn to rely on the properties of fields (the mathematical variety, not the ones in physics).  According to the theorem of Frobenius, there are only three finite dimensional fields: the real numbers (1D), the complex numbers (2D), and the quaternions (4D).  Special relativity stresses the importance of 4-dimensional Minkowski spaces: spacetime, energy-momentum, and the electromagnetic potential.  In this notebook, events in spacetime will be treated as the 4-dimensional field of quaternions.  It will be shown that problems involving boosts along an axis of a reference frame can be solved with this approach.

The tools of special relativity

Three mathematical tools are required to solve problems that arise in special relativity.  Events are represented as 4-vectors, which can be add or subtracted, or multiplied by a scalar.  To form an inner product between two vectors requires the Minkowski metric, which can be represented by the following matrix (where c = 1).

(Subscript[M, metric]) = (({{1, 0, 0, 0}, {0, -1, 0, 0}, {0, 0, -1, 0}, {0, 0, 0, -1}}));
{t, x, y, z}.(Subscript[M, metric]) . {t, x, y, z}(t^2) - (x^2) - (y^2) - (z^2)

((Subscript[g, µ])^ν)= (({{1, 0, 0, 0}, {0, -1, 0, 0}, {0, 0, -1, 0}, {0, 0, 0, -1}}))
{t, x, y, z}.((Subscript[g, µ])^ν) . {t, x, y, z}  = 
      (t^2) - (x^2) - (y^2) - (z^2)

The Lorentz group is defined as the set of matrices that preserves the inner product of two 4-vectors.  A member of this group is for boosts along the x axis, which can be easily defined.

γ[β_] := (1/(Sqrt[1  -  (β^2)]))(Subscript[Λ, x])[β_] :=(({{γ[β], -β γ[β], 0, 0}, {-β γ[β], γ[β], 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}))

γ[β_] := (1/(Sqrt[1  -  (β^2)]))(Subscript[Λ, x])[β_] :=(({{γ[β], -β γ[β], 0, 0}, {-β γ[β], γ[β], 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}))

Τhe boosted 4-vector is

(Subscript[Λ, x])[β] . {t, x, y, z}{(t/(Sqrt[1 - (β^2)])) - ((x β)/(Sqrt[1 - (β^2)])), 
      (x/(Sqrt[1 - (β^2)])) - ((t β)/(Sqrt[1 - (β^2)])), y, z}

\(Λ\_x[β]\ \((t, x, y, z)\)
 = 
      \((t\/\@\(1 - β\^2\) - \(x\ β\)\/\@\(1 - β\^2\), 
        x\/\@\(1 - β\^2\) - \(t\ β\)\/\@\(1 - β\^2\), y, z)
        \)\)

To demonstrate that the interval has been preserved, calculate the inner product.

\(Simplify[
	
      Λ\_x[β]\ .\ {t, x, y, z}.M\_metric.
        Λ\_x[β]\ .\ {t, x, y, z}]\)(t^2) - (x^2) - (y^2) - (z^2)

\(Λ\_x[β]\ \((t, x, y, z)\)\ g\_µ\^ν\ \ 
        Λ\_x[β]\ \ \((t, x, y, z)\)
 = 
      t\^2 - x\^2 - y\^2 - z\^2\)

Starting from a 4-vector, this is the only way to boost a reference frame along the x axis to another 4-vector and preserve the inner product.  However, it is not clear why one must necessarily start from a 4-vector.

Using quaternions in special relativity

Events are treated as quaternions, a skew field or division algebra that is 4 dimensional.  Any tool built to manipulate quaternions will also be a quaternion.  In this way, although events play a different role from operators, they are made of identical mathematical fabric.

The square of a quaternion is

\(Simplify[

      \((\(q[t, \ x, \ y, \ z]\ .\ q[t, \ x, \ y, \ z]\  + \ 

              q[t, \ x, \ y, \ z]\ .\ q[t, \ x, \ y, \ z]\)\/2)\).{1, 0, 0, 
          0}]\){(t^2) - (x^2) - (y^2) - (z^2), 2 t x, 2 t y, 2 t z}

(((t,  (Overscript[X,  ⇀]) ))^2) = 
      ((t^2) - (Overscript[X,  ⇀]) .(Overscript[X,  ⇀]) , 
        2 t (Overscript[X,  ⇀]) )

The first term of squaring a quaternion is the invariant interval squared.  There is implicitly, a form of the Minkowski metric that is part of the rules of quaternion multiplication.  The vector portion is frame-dependent.  If a set of quaternions can be found that do not alter the interval, then that set would serve the same role as the Lorentz group, acting on quaternions, not on 4-vectors.  If two 4-vectors x and x' are known to have the property that their intervals are identical, then the first term of squaring q[x] and q[x'] will be identical.  Because quaternions are a division ring, there must exist a quaternion L such that L q[x] = q[x'] since L = q[x'] q[x]^-1.   The inverse of a quaternion is its transpose divided by the square of the norm (which is the first term of transpose of a quaternion times itself).  Apply this approach to determine L for 4-vectors boosted along the x axis.

\(Simplify[

      \(\(q[γ[β]\ t\  - \ β\ γ[β]\ x, \ 
                \(-\ β\)\ γ[β]\ t\  + \ 
                  γ[β]\ x, \ y, z]\ .\ 

              Transpose[q[t, \ x, \ y, \ z]]\ 
            \)\/\((Transpose[q[t, \ x, \ y, \ z]]\ .q[t, \ x, \ y, \ z])\)[[
              1, 1]]\).\ 
	{1, 0, 0, 0}]\)\({\(t\^2 + x\^2 - 2\ t\ x\ β + 
          \((y\^2 + z\^2)\)\ \@\(1 - β\^2\)\)\/\(\((
            t\^2 + x\^2 + y\^2 + z\^2)\)\ \@\(1 - β\^2\)\), 
      \(\((\(-t\^2\) + x\^2)\)\ β\)\/\(\((t\^2 + x\^2 + y\^2 + z\^2)\)\ 
          \@\(1 - β\^2\)\), 
	
      \(\(-t\)\ \((y + z\ β - y\ \@\(1 - β\^2\))\) + 
          x\ \((z + y\ β - z\ \@\(1 - β\^2\))\)\)\/\(\((
            t\^2 + x\^2 + y\^2 + z\^2)\)\ \@\(1 - β\^2\)\), 
	
      \(x\ \((z\ β + y\ \((\(-1\) + \@\(1 - β\^2\))\))\) + 
          t\ \((y\ β + z\ \((\(-1\) + \@\(1 - β\^2\))\))
            \)\)\/\(\((t\^2 + x\^2 + y\^2 + z\^2)\)\ \@\(1 - β\^2\)\)}
      \)

\(\((γ[β]\ t - β\ γ[β]\ x, 
          \(-\ β\)\ γ[β]\ t + γ[β]\ x, \ y, z)
          \) \((t, \ x, \ y, \ z)\)\^\(-1\)

 = 
      \((t\^2 + x\^2 - 2\ t\ x\ β + 
                \((y\^2 + z\^2)\)\ \@\(1 - β\^2\), 
              \((\(-t\^2\) + x\^2)\)\ β, 
	
              \(-t\)\ \((y + z\ β - y\ \@\(1 - β\^2\))\) + 
                x\ \((z + y\ β - z\ \@\(1 - β\^2\))\), 
	
              x\ \((z\ β + y\ \((\(-1\) + \@\(1 - β\^2\))\))\) + 
                t\ \((y\ β + z\ \((\(-1\) + \@\(1 - β\^2\))\))\))
              \)
/\((t\^2 + x\^2 + y\^2 + z\^2)\)\ \@\(1 - β\^2\) 
        ≡ L\)

Define the Lorentz boost quaternion L along x using this equations.  L depends on the relative velocity and position, making it "local" in a sense. See if  L q[x] = q[x'].

\(Simplify[
      Expand[\((
L[t, \ x, \ y, \ z, \ β]\ .\ q[t, \ x, \ y, \ z])\)\ .
          \ {1, 0, 0, 0}]]\){(t - x β) γ[β], 
      (x - t β) γ[β], y, z}

\(L[t, \ x, \ y, \ z, \ β] \((t, \ x, \ y, \ z)\)
 = \ 
      \((γ\ t\  - \ γ\ β\ x\ , 
        \(-\ γ\)\ β\ t\  + \ γ\ x, \ y, \ z)\)\)

This is a quaternion composed of the boosted 4-vector.  At this point, it can be said that _any_ problem that can be solved using 4-vectors, the Minkowski metric and a Lorentz boost along the x axis can also be solved using the above quaternion for boosting the event quaternion.  This is because both techniques transform the same set of 4 numbers to the same new set of 4 numbers using the same variable beta.  To see this work in practice, please examine the problem sets.

Confirm the interval is unchanged.

\(Simplify[
      \((
		L[t, \ x, \ y, \ z, \ β]\ .\ q[t, \ x, \ y, \ z]\ .
		
            L[t, \ x, \ y, \ z, \ β]\ .\ q[t, \ x, \ y, \ z])\)\ .
        \ {1, 0, 0, 0}]\)\({t\^2 - x\^2 - y\^2 - z\^2, 
      \(2\ \((t\^2\ β + x\^2\ β - t\ x\ \((1 + β\^2)\))
            \)\)\/\(\(-1\) + β\^2\), 
	
      \(2\ y\ \((t - x\ β)\)\)\/\@\(1 - β\^2\), 
      \(2\ z\ \((t - x\ β)\)\)\/\@\(1 - β\^2\)}\)

\(\((L[t, \ x, \ y, \ z, \ β]\ \((t, \ x, \ y, \ z)\))\)\^2

 = 
      \((t\^2 - x\^2 - y\^2 - z\^2, 
        \(2\ \((t\^2\ β + x\^2\ β - t\ x\ \((1 + β\^2)\))
              \)\)\/\(\(-1\) + β\^2\), 
	
        \(2\ y\ \((t - x\ β)\)\)\/\@\(1 - β\^2\), 
        \(2\ z\ \((t - x\ β)\)\)\/\@\(1 - β\^2\))\)\)

The first term is conserved as expected.  The vector portion of the square is frame dependent.

Using quaternions in practice

The boost quaternion L is too complex for simple calculations.  Mathematica does the grunge work.  A great many problems in special relativity do not involve angular momentum, which in effect sets y = z = 0.  Further, it is often the case that t = 0, or x = 0, or for Doppler shift problems, x = t.  In these cases, the boost quaternion L becomes a very simple.

If t = 0, then

L  ==  L[0, x, 0, 0, β] . {1, 0, 0, 0}L = 
      ((1/(Sqrt[1 - (β^2)])), (β/(Sqrt[1 - (β^2)])), 0, 0)

\(L = γ \((1, β, 0, 0)\)	\)q  ->  (q^′)  =  Lq(0, x, 0, 0)  ->  ((t^′), (x^′), 0, 0)  =  
      (-γ β x, γ x, 0, 0)

If x = 0, then

L  ==  L[t, 0, 0, 0, β] . {1, 0, 0, 0}L = 
      ((1/(Sqrt[1 - (β^2)])), -(β/(Sqrt[1 - (β^2)])), 0, 
        0)

L =  γ (1, -β, 0, 0)\(q\  -> \ q\^′\  = \ Lq	\)(t, (Overscript[0,  ⇀]) )  ->  
        ((t^′), (x^′), 0, 0) =  
      (γ t, -γ β t, 0, 0)

If t = x, then

L  ==  Simplify[L[t,  t, 0, 0, β] . {1, 0, 0, 0}]L == {((1 - β)/(Sqrt[1 - (β^2)])), 0, 0, 0}

L = γ (1 - β, 0, 0, 0)\(q\  -> \ q\^′\  = \ Lq	\)(t, x, 0, 0) -> ((t^′), (x^′), 0, 0) = 
      γ (1 - β) ( t, x, 0, 0)

Note: this is for blueshifts.  Redshifts have a plus instead of the minus.

Over 50 problems in a sophomore-level relativistic mechanics class have been solved using quaternions. 90% required this very simple form for the boost quaternion.

Implications

Problems in special relativity can be solved either using 4-vectors, the Minkowski metric and the Lorentz group, or using quaternions.  No experimental difference between the two methods has been presented.  At this point the difference is in the mathematical foundations.

An immense amount of work has gone into the study of metrics, particular in the field of general relativity.  A large effort has gone into group theory and its applications to particle physics.  Yet attempts to unite these two areas of study have failed.

There is no division between events, metrics and operators when solving problems using quaternions.  One must be judicious in choosing quaternions that will be relevant to a particular problem in physics and therein lies the skill.  Yet this creates hope that by using quaternions, the long division between between metrics (the Grassman inner product) and groups of transformations (sets of quaternions that preserve the Grassman inner product) may be bridged.