New conserved tensors and Brans–Dicke-type field equation, using integrability condition

Abstract

In this paper, we explore some new off-shell and on-shell conserved quantities for a scalar field in Minkowski space, using integrability condition. The off-shell conserved tensors are related to the kinematics of the field, while a linear combination of the off-shell and the on-shell conserved tensors ends up with the energy–momentum tensor for the scalar field. In the curved background, using Ricci and Bianchi identities, scalar–tensor theory of gravity, that resembles somewhat with Brans–Dicke-type field equations, emerges without requiring the principle of equivalence. Further, starting from the curvature scalar and using these identities, the field equations for modified gravity (Einstein–Hilbert action in the presence of higher-order terms) follow.