Coordinate Transformations

May 10, 2019August 12, 2019
by admin

With a change of coordinates, contravariant vectors transform the same way that the coordinate differential vector $d{x^\mu }$ does. In terms of the vector representing the same coordinate differential in a different set of coordinates, $d{x^{\mu '}}$ , the vector $d{x^\mu }$ is given using the transformation law between the two sets of coordinates, $x_{,\mu '}^\mu$ . So,

$d{x^\mu } = x_{,\mu '}^\mu d{x^{\mu '}}$

In the prior expression for the KDF (see the post entitled “Emergence of Points”), the entities, ${C^\mu }$ , ${f^\lambda }$ , and ${\partial _\nu }$ , are all referred to the same, unprimed, coordinate system;

$d{{x}^{\mu }}=\delta _{\nu }^{\mu }d{{x}^{\nu }}$

with the KDF given by

$\delta _{\nu }^{\mu }=\sum\nolimits_{f}{\left( \partial {{C}^{\mu }}/\partial {{f}^{\lambda }}\ \right)}f_{,\nu }^{\lambda }$

For the coordinate transformation law, the change of coordinate system can be incorporated into the quantities

$\left( \partial {{C}^{\mu }}/\partial {{f}^{{{\lambda }'}}}\ \right)$

This yields the transformation of the vector as follows:

$d{{x}^{\mu }}=x_{,{\mu }'}^{\mu }d{{x}^{{{\mu }'}}}$

with the transformation law given by

$x_{,\mu '}^{\mu }=\sum\nolimits_{f}{\left( \partial {{C}^{\mu }}/\partial {{f}^{{{\lambda }'}}}\ \right)}f_{,{\mu }'}^{{{\lambda }'}}$

This transformation law, then, follows from the requirement that the equation governing the emergence of a coordinate in a given system of coordinates can be a functional of the matter fields expressed in any other set of coordinates,

${x^\mu } = {C^\mu }\left[ {{F^{\lambda '}}\left( {{x^{\nu '}}} \right)} \right]$