Symmetries at stationary Killing horizons

It has often been suggested (especially by S. Carlip) that spacetime symmetries in the neighborhood of a black hole horizon may be relevant to a statistical understanding of the Bekenstein–Hawking entropy. A prime candidate for this type of symmetry is that one which is exhibited by the Einstein tensor. More precisely, it is now known that this tensor takes on a strongly constrained (block-diagonal) form as it approaches any stationary, non-extremal Killing horizon. Presently, exploiting the geometrical properties of such horizons, we provide a particularly elegant argument that substantiates this highly symmetric form for the Einstein tensor. It is, however, duly noted that, on account of a “loophole,” the argument does fall just short of attaining the status of a rigorous proof.