Symmetries of the Energy-Momentum Tensor: Some Basic Facts

It has been pointed by Hall et al. [1] that matter collinations can be defined by using three different methods. But there arises the question of whether one studies matter collineations by using the ${\mathcal{L}}_{\xi }{T}_{ab}=0$, or ${\mathcal{L}}_{\xi }{T}^{ab}=0$ or ${\mathcal{L}}_{\xi }{T}_a^b=0$. These alternative conditions are, of course, not generally equivalent. This problem has been explored by applying these three definitions to general static spherically symmetric spacetimes. We compare the results with each definition.