The Lorentz Transform in modified complex algebra
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Author |
Carl Borrell |
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carlborrell@btinternet.com |
The Lorentz transform in one spatial dimension:
w' = w.cosh(a) - x.sinh(a)
x' = x.cosh(a) - w.sinh(a)
e = w + ix
e' = w' + ix'
e' = L(e) = [w.cosh(a) - x.sinh(a)] + i[x.cosh(a) - w.sinh(a)]
[Where tanh(a) = V/c; sinh(a) = 1/(1 - V2/c2)1/2 ; w = ct; c = speed of light ; V = relative velocity]
cannot be expressed as a function of a complex variable because the Cauchy-Reimann conditions are not satisfied. However if we change the product rule for complex multiplication:
a = a0 + ia1
b = b0 + ib1
a.b = (a0.b0 - a1b1) + i(a0b1 - b0a1)
We can still regard i as (-1)1/2 because,
(0 + i).(0 + i) = (0.0 - i.i) + i(0.i - i.0) = -1
The algebra is non-commutative:
a.b = (b.a)*
The square of any complex variable is always a purely real number in this algebra:
a2 = (a0a0 - a1a1) + i(a0a1 - a0a1) = a02 - a12
So the square is defined unambiguously even though the algebra is non-commutative.
This leads to a definition of the reciprocal function as:
1/a = a/a2 = (a0 + ia1) / (a02 - a12)
The Lorentz transform can be expressed as multiplication by a complex constant:
L = cosh(a) + isinh(a)
e' = L.e = [w.cosh(a) - x.sinh(a)] + i[x.cosh(a) - w.sinh(a)]
This suggests the following proof of Lorentz invariance:
e'2 = (L.e)2
= L2.e2
w'2 - x'2 = (cosh2(a) - sinh2(a)).(w2 - x2)
= w2 - x2 = e2
[Because cosh2(a) - sinh2(a) = 1 for all a].
This shows the invariance of the interval w2 - x2 under this transform. In fact all purely real numbers have an infinite number of square roots corresponding to the space-time co-ordinates of the same event viewed from each of the infinite number of inertial frames in standard configuration.
In terms of the velocity vector v = (c + iV) the Lorentz factor can be expressed as follows:
L = 1/(1 - V2/c2)1/2 + i(V/c(1 - V2/c2)1/2)
= c/(c2 - V2)1/2 + i(V/(c2 - V2)1/2)
= (c + iV)/(c2 - V2)1/2
= v/(v2)1/2
Now v2 is purely real therefore (v2)1/2 has an infinite number of solutions. In each case L is analogous to unity and L2 = 1.
Consequences for the algebra
Some other consequences for the algebra are as follows:
Theorem: Zero factorisation:
0 = (a + ia).(b + ib)
In terms of special relativity where ct = x the interval corresponds to two events joined by a light pulse.
Theorem: All even powers are purely real:
a = w + ix
a2 = (w2 - x2)
a2n = (w2 - x2)n
Theorem: Two definitions for odd powers:
a = w + ix
a(2n + 1) = (w2 - x2)n . a or
a(2n + 1) = a.(w2 - x2)n
These two possibilities are complex conjugates of each other, the time component is the same but the space component is reversed. Perhaps this is related to chirality?
Binomial
(a + b)2 = a2 + a.b + b.a + b2
We cannot collect the cross products because the algebra is non-commutative, therefore
(a + b)2 = a2 + a.b + (a.b)* + b2
= a2 + 2Re(a.b) + b2
So (a + b)2 is purely real as required.
Derivatives and partial derivatives
Let x = x0 + ix1 ; f(x) = f0 (x0,x1) + if1(x0,x1)
What is the relationship between the derivative df(x)/dx
and the partial derivatives d
[i.e Cauchy-Reimann conditions do not seem to be satisfied there appears to be a relationship with the curl and divergence operators]
Path Integrals
What are the consequences of the zero factorisation?
Taylor Series
Let s = x2
Then for infinitely differentiable function p(x),
p(x) = a0 + a1.x + a2.x2/2! + a3.x3 /3! ...
= f(s) + x.g(s)
Quaternions
A similar change can be made to the four-dimensional algebra defined by Sir W.R. Hamilton in the early 1840's.
[Hamilton's original papers are available at the history of mathematics web site, Trinity College, Dublin ref.
www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quatern1/Quatern1.html
www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quatern2/Quatern2.html ]
In Hamilton's quaternion algebra the product is defined as
i2 = j2 = k2 = -1
i.j = k
j.k = i
k.i = j
j.i = -k
k.j = -i
i.k = -j
a = a0 + a1i + a2j + a3k
b = b0 + b1i + b2j + b3k
a.b = (a0b0 + a1b1 + a2b2 + a3b3) +
(a0b1 + a1b0 + a2b3 - a3b2)i +
(a0b2 + a2b0 + a3b1 - a1b3)j +
(a0b3 + a3b0 + a1b2 - a2b1)k
Which is the origin of the more familiar vector algebra due to Gibbs.
The quaternion product can be expressed in vector algebra, if we regard a quaternion as the sum of a vector and a scalar quantity
a = a0 + v ; where v = a1i + a2j + a3k
b = b0 + u ; where u = b1i + b2j + b3k
Then a.b = a0b0 + v.u + a0u + b0v + v X u
The quaternion algebra can be modified in the same way as the complex algebra as follows:
a.b = (a0b0 + a1b1 + a2b2 + a3b3) +
(a0b1 - a1b0 + a2b3 - a3b2)i +
(a0b2 - a2b0 + a3b1 - a1b3)j +
(a0b3 - a3b0 + a1b2 - a2b1)k
[In vector algebra, if
a = a0 + v ; where v = a1i + a2j + a3k
b = b0 + u ; where u = b1i + b2j + b3k
Then a.b = a0b0 + v.u + a0u - b0v + v X u]
which gives
a2 = (a02 - a12 - a22 - a32)
i.e. the Lorentz invariant interval in four dimensions.
Field equations
Is it possible to express the laws of electromagnetism in terms of field equations in this 4-d algebra?
What is the relationship between the derivative and the curl and divergence operators?
Can the equations be transformed as follows?
f(s) = [p(x)]2 ; s = x2
This give a scalar function of a scalar variable in terms of Lorentz invariant quantities.