Question

What is the difference between a local and global coordinate transformation?

Answer 1

In the case of fields, it is clear what a local transformation in the internal space of the field is:

ϕ(x)→ϕ′(x′)=G(x)ϕ(x),ϕ(x)→ϕ′(x′)=G(x)ϕ(x),

as opposed to a global transformation, where GGwould not depend on xx.

But I don't understand the difference between local and global coordinate transformations.

It is said that general coordinate transformations,

xμ→x′μ=xμ+ϵμ(x)xμ→x′μ=xμ+ϵμ(x)

are local, while rotations

xμ→x′μ=xμ+σμνxνxμ→x′μ=xμ+σνμxν

are global.

Answer 2

A transformation is said to be global if it acts the same at every point; it is said to be local otherwise.

hope it helps:-)