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 $$B\rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . It is possible to show that $$f(Q,B)=f(Q-B)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>-</mml:mo> <mml:mi>B</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> models are dynamically equivalent to f ( R ) gravity as in the case of teleparallel $$f(\tilde{B}-T)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>-</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> gravity (where $$B\ne \tilde{B}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>≠</mml:mo> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> </mml:math> ). 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 ), $$f(T,\tilde{B})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , 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.

Topics & Concepts