Abstract
Starting from a divergence-free rank-4 tensor of which the trace is the cosmological Einstein tensor, we give a construction of conserved charges in Einstein's gravity and its higher derivative extensions for asymptotically anti-de Sitter spacetimes. The current yielding the charge is explicitly gauge-invariant, and the charge expression involves the linearized Riemann tensor at the boundary. Hence, to compute the mass and angular momenta in these spacetimes, one just needs to compute the linearized Riemann tensor. We give two examples.
Highlights
Let us start with a seemingly innocent question which will have far-reaching consequences for the conserved charges of gravity theories
Λgμσ, with the condition that this four-index tensor has the symmetries of the Riemann tensor and it is divergence free just like the Einstein tensor?
Conserved charges of generic gravity theory in asymptotically AdS spacetimes were constructed in Ref. [4] as an extension of the Abbott-Deser charges [5] of the cosmological Einstein theory
Summary
Let us start with a seemingly innocent question which will have far-reaching consequences for the conserved charges of gravity theories. Note that this still leaves an ambiguity in the P-tensor, since one can add an arbitrary constant times gμσgβν, but that part can be fixed by demanding that the P-tensor has the symmetries of the Riemann tensor and vanishes for constant curvature backgrounds, which we assumed. This tensor turns out to be extremely useful in finding conserved charges of Einstein’s gravity for asymptotically AdS spacetimes for n > 3 dimensions. In Ref. [3], we studied the Kerr-AdS solution, and we shall not repeat it here
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