Born-Infeld new massive gravity completes new massive gravity to all orders in curvature through the determinant of the metric shifted by the Einstein tensor. We recast it with an independent auxiliary metric $q_{\mu \nu }$, whose algebraic equation of motion $q_{\mu \nu }=g_{\mu \nu }+\frac{\sigma }{m^2}{G}_{\mu \nu }(g)$ recovers the determinant action exactly on the regular branch and resums the infinite curvature series into a single relation. In the desensitized variable $P^{\mu \nu }=\sqrt{-q}{q}^{\mu \nu }$, the three-dimensional action is polynomial, with all derivative dependence carried by the coupling $P^{\mu \nu }{G}_{\mu \nu }(g)$. The formulation makes known properties follow with substantially less algebra: the unique vacuum follows in one line, and the quadratic action yields a single Pauli-Fierz massive spin-2 field with the Fierz-Pauli tuning generated rather than imposed. On locally anti–de Sitter backgrounds, the conserved charges, Bañados–Teitelboim–Zanelli mass and angular momentum, central charge, and entropy reduce to the Einstein results times a common factor. The formulation also isolates the nonlinear degree-of-freedom problem in the right variables, leaving the full Dirac count to separate work.

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• Alternative gravity theories
• Gravity in dimensions other than four