New Cases of Universality Theorem for Gravitational Theories

TL;DR

This paper extends the Universality Theorem for gravity by introducing a new family of invariants R' in four dimensions, demonstrating their equivalence to existing f(R) theories and linking to the Holst Lagrangian in General Relativity.

Contribution

It introduces a one-parameter family of invariants R' inspired by the Barbero-Immirzi formulation, extending the Universality Theorem to include these invariants and establishing their equivalence to metric-affine f(R) theories.

Findings

f(R') theories are dynamically equivalent to metric-affine f(R) theories.

The standard equivalence between GR and the Holst Lagrangian is recovered for f(R)=R.

The extension applies to a generic class of analytic functions f(R).

Abstract

The "Universality Theorem" for gravity shows that f(R) theories (in their metric-affine formulation) in vacuum are dynamically equivalent to vacuum Einstein equations with suitable cosmological constants. This holds true for a generic (i.e. except sporadic degenerate cases) analytic function f(R) and standard gravity without cosmological constant is reproduced if f is the identity function (i.e. f(R)=R). The theorem is here extended introducing in dimension 4 a 1-parameter family of invariants R' inspired by the Barbero-Immirzi formulation of GR (which in the Euclidean sector includes also selfdual formulation). It will be proven that f(R') theories so defined are dynamically equivalent to the corresponding metric-affine f(R) theory. In particular for the function f(R)=R the standard equivalence between GR and Holst Lagrangian is obtained.