The role of the boundary term in f(Q, B) symmetric teleparallel gravity

Abstract

Abstract In the framework of metric-affine gravity, we consider the role of the boundary term in Symmetric Teleparallel Gravity assuming f(Q, B) models where f is a smooth function of the non-metricity scalar Q and the related boundary term B. Starting from a variational approach, we derive the field equations and compare them with respect to those of f(Q) gravity in the limit of $$B\rightarrow 0$$ B → 0 . It is possible to show that $$f(Q,B)=f(Q-B)$$ f ( Q , B ) = f ( Q - B ) models are dynamically equivalent to f(R) gravity as in the case of teleparallel $$f(\tilde{B}-T)$$ f ( B ~ - T ) gravity (where $$B\ne \tilde{B}$$ B ≠ B ~ ). Furthermore, conservation laws are derived. In this perspective, considering boundary terms in f(Q) gravity represents the last ingredient towards the Extended Geometric Trinity of Gravity, where f(R), $$f(T,\tilde{B})$$ f ( T , B ~ ) , and f(Q, B) can be dealt under the same standard. In this perspective, we discuss also the Gibbons–Hawking–York boundary term of General Relativity comparing it with B in f(Q, B) gravity.