Characteristic Formulation for Metric $f(R)$ Gravity

TL;DR

This paper develops a characteristic initial value formulation for vacuum $f(R)$ gravity, providing a foundation for numerical relativity simulations and analytic solutions in modified gravity theories.

Contribution

It formulates the characteristic initial value problem for $f(R)$ gravity without scalar-tensor equivalence and derives equations suitable for numerical implementation and analytical testing.

Findings

Derived the full hierarchy of hypersurface and evolution equations for $f(R)$ gravity.

Obtained analytic solutions for the dominant $ ext{l}=2$ mode.

Showed these solutions satisfy the constraints, serving as testbeds for numerical codes.

Abstract

In recent years, the Characteristic formulation of numerical relativity has found increasing use in the extraction of gravitational radiation from numerically generated spacetimes. In this paper, we formulate the Characteristic initial value problem for $f(R)$ gravity. We consider, in particular, the vacuum field equations of Metric $f(R)$ gravity in the Jordan frame, without utilising the dynamical equivalence with scalar-tensor theories. We present the full hierarchy of non-linear hypersurface and evolution equations necessary for numerical implementation in both tensorial and eth forms. Furthermore, we specialise the resulting equations to situations where the spacetime is almost Minkowski and almost Schwarszchild using standard linearization techniques. We obtain analytic solutions for the dominant $\ell=2$ mode and show that they satisfy the concomitant constraints. These results are ideally suited as testbed solutions for numerical codes. Finally, we point out that the Characteristic formulation can be used as a complementary analytic tool to the $1+1+2$ semi-tetrad formulation.