Lorentz transform as a complex function

In summary, the quaternion system seems simpler and more compatible with special relativity than the vector system.
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quaternion
4
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I tried to represent the Lorentz transform which converts a pair of space-time co-ordinates (ct,x) to (ct',x') as a function of a complex variable i.e

ct' + ix' = f(ct + ix)

Unfortunately the rules of complex algebra do not permit this because the complex product is defined as

(a + ib)(c +id) = (ac - bd) + i(ad + bc)

Consequently the required function is prohibited by the Cauchy-Riemann conditions.

I have been investigating what happens if the rules of complex algebra are changed so that the product rule looks the same in form as the Lorentz transform i.e.

(a + ib)(c + id) = (ac - bd) + i(ad - bc)

This results in a non-commutative algebra in which the Lorentz transform can be represented as multiplication by a complex constant which depends on the relative velocity.
This seems to be a convenient language for doing these problems as Lorentz invariant quantitites can be represented as complex functions of the space-time co-ordinate without reference to any particular inertial frame.
The algebra has some unusual features e.g factorisation of zero and an infinite number of square roots.
Following W.R. Hamilton I extended the algebra to four dimensions [historically the origin of the modern vector calculus] making the same modification to the quaternion algebra as above. I defined a differential calculus for this algebra and formed the first derivitave of a scalar + vector potential [this looks like a Minkowski four-vector but the time component is real and the space components are imaginary].

The result is a mathematical object with 16 partial derivative terms six of these look like the expression for the electric field, six look like the magnetic field and the remaining four look like the Lorentz gauge condition. This suggests that the algebra can be used to make field equations again using only Lorentz invariant quantities.

Has anyone seen this approach before?
 
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  • #3
hyperbolic numbers

Thanks for that. The algebra I have got in mind not only permits division but it is possible to define a four-dimensional differential calculus.

If I make a four-dimensional potential function and take the first derivative I get a mathematical object consisting of sixteen partial derivatives, six of these look like the expression for the magnetic field six look like the electric field and the remaining four look like the Lorentz gauge condition. Does this mean any thing to you?

It occurred to me that it might be possible to write the laws of electromagnetism as four-dimensional differential equations.
The dimensionality is embedded in the algebra so the process of solving them looks like a one-dimensional equations.
As Hamilton said we could solve these problems 'without the aid of a co-ordinate system' [On Quaternions]. Do you think this makes any sense?
 
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  • #4
I'm not sure if this is anywhere near what you have in mind, but the complex quaternions can used to do special relativity. In this formalism: a 4-vector is a scalar plus a (3-)vector; Maxwell's equations can be written as one equation.

Regards,
George
 
  • #5
Maxwell's equations

Thanks George.
Can you tell me what the equations look like in the quaternion algebra.

Do you know why all the textbooks use partial derivatives and vector calculus? The quaternion system seems so much simpler.
 
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Home page

Hi,
If anyone is interested I posted my thoughts on this at

http://www.members.aol.com/csborrell/Index.html [Broken]
 
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1. What is the Lorentz transform as a complex function?

The Lorentz transform as a complex function is a mathematical tool used in special relativity to describe the relationship between measurements of time and space in different reference frames. It is a complex-valued function that accounts for the effects of time dilation and length contraction.

2. How is the Lorentz transform derived?

The Lorentz transform is derived from the principles of special relativity, which state that the laws of physics should appear the same to all observers moving at a constant velocity. By applying the laws of physics to a system of two reference frames in relative motion, the Lorentz transform can be derived mathematically.

3. What is the significance of the complex nature of the Lorentz transform?

The complex nature of the Lorentz transform allows for the inclusion of imaginary numbers, which represent the effects of time dilation and length contraction. This allows for a more accurate description of the relationship between time and space in different reference frames, particularly at high speeds.

4. Can the Lorentz transform be applied to objects with mass?

Yes, the Lorentz transform can be applied to objects with mass. In fact, it is a crucial component of special relativity, which applies to all objects regardless of their mass. The Lorentz transform is used to calculate the effects of time dilation and length contraction on objects with mass as they move at high speeds.

5. How does the Lorentz transform relate to the theory of general relativity?

The Lorentz transform is a fundamental concept in special relativity, which is a subset of the broader theory of general relativity. While special relativity deals with the effects of constant velocities on time and space, general relativity extends this to include the effects of acceleration and gravity. The Lorentz transform is an important tool in both theories, but it is particularly useful in special relativity.

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