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Bachian gravity in three dimensions

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2019
Tek, Mustafa
Modified theories in 3-dimensions such as the topologically massive gravity (TMG), new massive gravity (NMG) or Born-Infeld extension of NMG arise from the vari-ations of diffeomorphism invariant actions; hence the resulting field equations aredivergence free. Namely, the rank two tensor defining the field equations satisfy aBianchi identity for all smooth metrics. However there are some recently constructedtheories that do not identically satisfy Bianchi identities for all metrics, but only forthe solutions of the theory. These are called on-shell consistent theories of whichexamples are the minimal massive gravity (MMG) and the exotic massive gravity(EMG). We work out the generic on-shell consistent model in 3-dimensions as amodified Einstein gravity theory which is based on the analog of the Bach tensor,hence we name it as the Bachian gravity. Conserved charges are found by using thelinearization about maximally symmetric backgrounds for the Bañados-Teitelboim-Zanelli (BTZ)-black hole metric. It is complicated to solve the field equations of thegravity theory and hence very few solutions with only maximal symmetry are known. We use the projection formalism to obtain a reduction of the some relevant 2-tensorsdefining the field equations with the help of the Geroch’s reduction method.