A conserved Bach current (Q2768101)

Gravitational theory allows a conserved current \(J^\alpha_{\text{ein}}\) to be formed from the Einstein tensor with a symmetry generated by a Killing vector. In analogy, a so-called Bach current is defined, \(J^\alpha_{ \text{bach}}: =B^\alpha_\beta \xi^\beta\), where \(B_{\alpha \beta}\) is the Bach tensor and \(\xi^\beta\) is a generator of conformal maps which include Killing symmetry and homothety as special cases. Since the Bach tensor is not only symmetric and divergence-free, but also trace-free, the Bach current is conserved in the case of such ``conformal Killing vectors'', too. The Bach current gives rise to a quasilocal 2-surface expression for power per luminosity distance in the Vaidya exterior collapsing fluid interiors. This is interpreted in terms of thermodynamics. In particular, it is shown that the luminosity gradient can be identified with the entropy production in the Vaidya exterior.