Solving higher curvature gravity theories

TL;DR

This paper demonstrates that higher curvature gravity theories, like $f(R)$ models, can be effectively solved by translating their field equations into equivalent Einstein-Hilbert plus scalar field equations, simplifying the solution process.

Contribution

It establishes the equivalence of higher curvature gravity field equations to Einstein-Hilbert plus scalar field equations, even on lower dimensional hypersurfaces, providing a new method for solving such theories.

Findings

Equivalence of higher curvature and scalar-tensor field equations for regular spacetimes.

Explicit examples showing solutions correspond between the two formulations.

Extension of equivalence to lower dimensional hypersurfaces.

Abstract

Solving field equations in the context of higher curvature gravity theories is a formidable task. However in many situations, e.g., in the context of $f(R)$ theories the higher curvature gravity action can be written as Einstein-Hilbert action plus a scalar field action. We show that not only the action but the field equations derived from the action are also equivalent provided the spacetime is regular. We also demonstrate that such equivalence continues to hold even when gravitational field equations are projected on a lower dimensional hypersurface. We have further depicted explicit examples in which the solutions for Einstein-Hilbert and a scalar field system lead to solutions of the equivalent higher curvature theory. The same, but on the lower dimensional hypersurface, has been illustrated in the reverse order as well. We conclude with a brief discussion on this technique of solving higher curvature field equations.