Minkowski Spacetime KVF Symmetries

[cianfa72]

$$
\mathcal L_\xi T = \lim_{s\to 0}[(\phi_{s\xi}^*T - T)/s].
$$ As the pullback defines a map between different points in the manifold, we are not in need of a connection in order to compute the Lie derivative.

A doubt about the limit involved in the Lie derivative definition. At first sight that limit makes sense when we pick a chart in the differentiable atals and do the calculation there. Does it imply a notion of limit from an invariant coordinate-free point of view ?

 

[vanhees71]

Sure, this is a definition completely independent of any choice of coordinates. Since all the manipulations used to define the limit are manifestly covariant, it maps the arbitrary tensor field $T$ to another tensor field of the same rank.

 

[cianfa72]

Sure, this is a definition completely independent of any choice of coordinates. Since all the manipulations used to define the limit are manifestly covariant

Ah ok, indeed the operations involved in the definition are part of tensor space structure that exist at each point of the manifold. Namely a difference between (0,2) tensors defined at the same point (i.e. elements of the same tensor space) and a scalar multiplication for $1/s$.