On Physical Equivalence between Nonlinear Gravity Theories

TL;DR

This paper demonstrates that in nonlinear gravity theories, the physically meaningful variables are those in the Einstein frame, and establishes conditions under which the energy is positive, linking physical viability to the existence of this frame.

Contribution

It clarifies the physical variables in nonlinear gravity theories and proves positive energy conditions using the Einstein frame, especially in the presence of matter.

Findings

The Einstein frame variables are the true physical variables in nonlinear gravity.

The ADM energy is positive for solutions close to flat space under certain Lagrangians.

The existence of the Einstein frame is crucial for the positive energy theorem.

Abstract

We argue that in a nonlinear gravity theory, which according to well-known results is dynamically equivalent to a self-gravitating scalar field in General Relativity, the true physical variables are exactly those which describe the equivalent general-relativistic model (these variables are known as Einstein frame). Whenever such variables cannot be defined, there are strong indications that the original theory is unphysical. We explicitly show how to map, in the presence of matter, the Jordan frame to the Einstein one and backwards. We study energetics for asymptotically flat solutions. This is based on the second-order dynamics obtained, without changing the metric, by the use of a Helmholtz Lagrangian. We prove for a large class of these Lagrangians that the ADM energy is positive for solutions close to flat space. The proof of this Positive Energy Theorem relies on the existence of the Einstein frame, since in the (Helmholtz--)Jordan frame the Dominant Energy Condition does not hold and the field variables are unrelated to the total energy of the system.