Applying the Lorentz Transformation Matrix on Polar Coordinates

The Lorentz transformation is a fundamental concept in the theory of special relativity. It is used to reconcile the different observations of time and space by observers moving at a constant velocity relative to each other. While the transformation is generally applied in Cartesian coordinates, understanding its application in polar coordinates can provide unique insights. This article will guide you through the process of using the Lorentz transformation on polar coordinates, illustrating this through a specific example.

Lorentz Transformation Basics

The Lorentz transformation for a boost along the x-axis in Cartesian coordinates is given by:

[begin{pmatrix} CT X Y Z end{pmatrix} begin{pmatrix} gamma -betagamma 0 0 -betagamma gamma 0 0 0 0 1 0 0 0 0 1 end{pmatrix} begin{pmatrix} CT X Y Z end{pmatrix}]

Where:

(gamma frac{1}{sqrt{1 - beta^2}}) is the Lorentz factor, with (beta frac{v}{c}) being the velocity ratio of the moving frame to the speed of light (c).

Understanding Polar Coordinates

In polar coordinates, the spatial coordinates are typically expressed as:

(X rcosphi) (Y rsinphi) (Z z)

Where:

(r) is the radial coordinate. (phi) is the angle in the XY-plane. (z) is the height in the Z-direction.

Applying the Lorentz Transformation

Given a polar coordinate representation as (CT, 0, sinphi, cosphi), this can be interpreted as:

(CT) is the temporal part. (X 0) indicating no radial movement in the X-direction. (Y rsinphi) (Z rcosphi)

To apply the Lorentz transformation to this representation, we need to express the spatial coordinates in Cartesian coordinates, substitute them into the transformation matrix, and then convert back to polar coordinates if necessary.

Convert to Cartesian Coordinates

From the polar coordinates, we have:

(X 0) (Y rsinphi) (Z rcosphi)

Apply the Lorentz Transformation

Using the Lorentz transformation matrix, we only apply it to (CT) and (X) since (Y) and (Z) do not change under a boost in the X-direction:

[begin{pmatrix} CT 0 rsinphi rcosphi end{pmatrix} begin{pmatrix} gamma -betagamma 0 0 -betagamma gamma 0 0 0 0 1 0 0 0 0 1 end{pmatrix} begin{pmatrix} CT 0 rsinphi rcosphi end{pmatrix}]

This yields:

[begin{pmatrix} gamma CT 0 rsinphi rcosphi end{pmatrix}]

The transformed coordinates are:

(CT gamma CT) (Y rsinphi) (Z rcosphi)

The (X) coordinate remains zero since it was initially zero.

Final Polar Representation

Finally, if we want to express the results back in polar coordinates, we can use:

The transformed radial distance can be calculated using the Pythagorean theorem, but since the transformation does not change (Y) and (Z), they can be directly used in their original form.

In summary, the Lorentz transformation affects the time and the x-component of the coordinates while leaving the radial and angular coordinates unchanged in this particular case.

Understanding the Lorentz transformation in polar coordinates provides a deeper insight into how special relativity affects the representation of space and time, especially in contexts where polar coordinates might more naturally describe the system under study.

Keywords: Lorentz Transformation, Polar Coordinates, Spacetime Transformation