Coordinate transformations involving vectors and covectors utilize Jacobian Matrices, which are inverses of each other. In the case of simple translations, the Jacobian Matrix becomes trivial, providing no information about the extent of the translation. The equivalence principle asserts that all gauges yield the same results regardless of position or relative velocity, making it impossible to determine relative distances from identical experiments conducted at different locations. This discussion clarifies the distinction between coordinate translations and transformations of geometric objects at a single point.
PREREQUISITESPhysicists, mathematicians, and students studying differential geometry or general relativity, particularly those interested in the behavior of vectors and covectors under coordinate transformations.
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.Shadumu said: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?
Shadumu said:Vectors and covectors transform differently with Jacobian Matrices inverse of each other.
Shadumu said:what is the general coordinate transformation is a simple translation of coordinates