Coordinate Translation Vectors & Covectors

[Shadumu]

Vectors and covectors transform differently with Jacobian Matrices inverse of each other. However, what is the general coordinate transformation is a simple translation of coordinates, the Jacobian Matrix will be trivially a delta and contains no information of how much the translation is. How to describe such a translation of coordinates properly?

 

[Vrbic]

Vectors and covectors transform differently with Jacobian Matrices inverse of each other. However, what is the general coordinate transformation is a simple translation of coordinates, the Jacobian Matrix will be trivially a delta and contains no information of how much the translation is. How to describe such a translation of coordinates properly?

What do you mean by "properly". You have a definition of transformation (for example x'=x+a etc.). The fact that jacobian of the transformation is same for all translations is coming from equivalence principle (all gauges have same result independently of position or relative velocity). If I understand in good way, would you like to find out exact translation of coordinates of transformation from Jacobian? It is impossible and equivalence principle says it is generally imposible from same experiment on two different places find out their relative distance.

 

[PeterDonis]

Vectors and covectors transform differently with Jacobian Matrices inverse of each other.

More precisely: vectors and covectors at a given point transform with Jacobians the inverse of each other.

what is the general coordinate transformation is a simple translation of coordinates

A translation of coordinates is a different kind of transformation than the kind referred to above: it involves moving from one point to another, not transforming geometric objects at a single point.