Playing with complex notation for relativistic applications in a plane
Posted by peeterjoot on April 19, 2011
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.
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
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 , finishing the game for the day.
 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.