## Playing with complex notation for relativistic applications in a plane

Posted by peeterjoot on April 19, 2011

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# Motivation.

In the electrodynamics midterm we had a question on circular motion. This screamed for use of complex numbers to describe the spatial parts of the spacetime trajectories.

Let’s play with this a bit.

# Our invariant.

Suppose we describe our spacetime point as a paired time and complex number

Our spacetime invariant interval in this form is thus

Not much different than the usual coordinate representation of the spatial coordinates, except that we have a replacing the usual .

Taking the spacetime distance between and another point, say motivates the inner product between two points in this representation

It’s clear that it makes sense to define

consistent with our original starting point

Let’s also introduce a complex inner product

Our dot product can now be written

# Change of basis.

Our standard basis for our spatial components is , but we are free to pick any other basis should we choose. In particular, if we rotate our basis counterclockwise by , our new basis, still orthonormal, is .

In any orthonormal basis the coordinates of a point with respect to that basis are real, so just as we can write

we can extract the coordinates in the rotated frame, also simply by taking inner products

The values , and are the (real) coordinates of the point in this rotated basis.

This is enough that we can write the Lorentz boost immediately for a velocity at an arbitrary angle in the plane

Let’s translate this to coordinates as a check. For the spatial component parallel to the boost direction we have

and the perpendicular components are

Grouping the two gives

The boost equation in terms of the cartesian coordinates is thus

Writing

the boost matrix is found to be (after a bit of work)

A final bit of regrouping gives

This is consistent with the result stated in [1], finishing the game for the day.

# References

[1] Wikipedia. Lorentz transformation — wikipedia, the free encyclopedia [online]. 2011. [Online; accessed 20-April-2011]. http://en.wikipedia.org/w/index.php?title=Lorentz_transformation&oldid=424507400.

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