Komar superpotentials in extended theories of gravity

Abstract

This paper investigates Lagrangians with arbitrary dependence on the Riemann tensor and any number of derivatives. Noether's second theorem is applied separately to ${\cal L}_{G}$ and ${\cal L}_{M}$, where the matter Lagrangian depends on a scalar field. For ${\cal L}_{G}$ diffeomorphism invariance requires that the generalized Einstein tensor satisfy $\nabla_αE^{αβ}=0$ and that there is a current satisfying $0=\partial_α(\sqrt{g}\, J^α_{G2})$. The main result is an explicit formula for the superpotential $Φ^{[αμ]}$ that solves $\partial_μΦ^{[αμ]}=\sqrt{g}\,J^α_{G2}$. Some applications to $f(R)$ gravity and various quadratic gravity theories are discussed. There are analogous results, including superpotentials, for the matter sector whenever the couplings to the matter fields contain derivatives of the metric.