Bachian gravity in three dimensions

Alkaç, Gökhan
Tek, Mustafa
Tekin, Bayram
In three dimensions, there exist modifications of Einstein's gravity akin to the topologically massive gravity that describe massive gravitons about maximally symmetric backgrounds. These theories are built on the three-dimensional version of the Bach tensor (a curl of the Cotton-York tensor) and its higher derivative generalizations; and they are on-shell consistent without a Lagrangian description based on the metric tensor alone. We give a generic construction of these models, find the spectra and compute the conserved quantities for the Banados-Teitelboim-Zanelli black hole.
Physical Review D


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Modifications to Einstein's vacuum equations for gravitation in 2+1 dimensions are studied. The addition of the Schouten-Eisenhart 2-forms to the field equations admits gravitational wave solutions although no non-trivial static rotationally symmetric metrics exist. Higher-order derivative models for the metric are discussed together with a 2+1 Brans-Dicke theory. The latter is solved for a static metric exhibiting singularities.
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Citation Formats
G. Alkaç, M. Tek, and B. Tekin, “Bachian gravity in three dimensions,” Physical Review D, 2018, Accessed: 00, 2020. [Online]. Available: