The discussion centers on the identification of Killing Vector Fields (KVFs) in Minkowski spacetime, confirming that there are exactly 10 independent KVFs derived from the symmetry properties of the spacetime. The participants reference Carrol's chapter on geometry, noting that the KVFs arise from the linear combinations of four translation isometries and six rotation isometries. The Killing equation is discussed in detail, illustrating how the Lie derivative of the metric tensor vanishes along these vector fields, thereby establishing their role as generators of symmetry transformations in the context of the Poincare group.
PREREQUISITESThe discussion is beneficial for theoretical physicists, mathematicians specializing in differential geometry, and students studying general relativity, particularly those interested in the symmetries of spacetime and their implications in physics.
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