The discussion centers on the properties of Killing Vector Fields (KVFs) in Minkowski spacetime, particularly focusing on the number of independent KVFs and the implications of the Lie derivative in this context. Participants explore theoretical aspects and mathematical formulations related to symmetries in spacetime.
Participants express differing views on the implications of the Lie derivative and the conditions under which vector fields can be considered KVFs. There is no consensus on the broader implications of these discussions, particularly regarding the nature of smooth vector fields in relation to KVFs.
Participants highlight limitations in understanding the linearity of the Lie derivative and its implications for KVFs, as well as the potential for confusion arising from index notation and mathematical derivations.
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 ?Orodruin said:$$
\mathcal L_\xi T = \lim_{s\to 0}[(\phi_{s\xi}^*T - T)/s].
$$ 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.
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##.vanhees71 said:Sure, this is a definition completely independent of any choice of coordinates. Since all the manipulations used to define the limit are manifestly covariant