The role of the boundary term in $f(Q,B)$ symmetric teleparallel gravity (2307.13280v2)

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 $B\to0$. It is possible to show that $f(Q,B)=f(Q-B)$ models are dynamically equivalent to $f(R)$ gravity as in the case of teleparallel $f(\tilde{B}-T)$ gravity (where $B\neq \tilde{B}$). Furtherrmore, 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 $f(R)$, $f(T,\tilde{B})$, and $f(Q,B)$ can be dealt with under the same standard. We also compare and discuss about the Gibbons-Hawking-York boundary term of General Relativity and the boundary term $B$ in $f(Q,B)$ gravity.

Analysis of the Boundary Term in $f(Q,B)$ Symmetric Teleparallel Gravity

The paper under discussion focuses on an advanced exploration within the framework of modified theories of gravity, specifically considering $f(Q,B)$ symmetric teleparallel gravity. As the interest in alternative theories to General Relativity (GR) grows due to its limitations at both macroscopic and quantum scales, such modifications are valuable for addressing current cosmological and theoretical challenges.

Summary of the Paper's Content

The authors, Capozziello, De Falco, and Ferrara, investigate the incorporation of the boundary term $B$ in $f(Q,B)$ gravity within a metric-affine framework. Here, they look into models where the function $f$ is dependent on both the non-metricity scalar $Q$ and the boundary term $B$. This work builds upon existing geometric reformulations of gravity, such as the Geometric Trinity of Gravity composed of curvature, torsion, and non-metricity representations; specifically, $f(R)$, $f(T)$, and $f(Q)$ where $R$, $T$, and $Q$ represent the Ricci, torsion, and non-metricity scalars, respectively.

Key Findings and Numerical Analysis

1. Dynamic Equivalence: The authors prove that $f(Q,B)=f(Q-B)$ models are dynamically equivalent to $f(R)$ gravity. This establishes freedom in choosing the gravitational action and the ability to switch between different geometric representations without altering physical predictions. This provides an analytical grounding to consider these modified gravity theories on equal footing.
2. Conservation Laws: Derived conservation laws within this framework illustrate how boundary terms affect the dynamics, ensuring consistency of $f(Q)$ gravity when $B\to 0$. These laws contribute to the conservation of energy-momentum and provide reliable criteria to confront such theories with observations.
3. Fourth-Order Dynamics: By extending the theory with the boundary term $B$, the authors effectively demonstrate a transition from second-order differential equations in $f(Q)$ gravity to fourth-order equations in $f(Q,B)$ gravity, paralleling the differential order of $f(R)$ gravity. This extension allows for richer phenomenology and potential resolutions to the issues present in GR's separability at large and quantum scales.
4. Extended Geometric Trinity of Gravity: The paper introduces the concept of the Extended Geometric Trinity of Gravity, composed of $f(R), f(T-B)$, and $f(Q-B)$, completing the cycle of symmetry between curvature, torsion, and non-metricity frameworks in modified theories.

Implications and Future Prospects

This paper's implications suggest a compelling argument for considering boundary terms as a significant factor in formulating modified theories of gravity. The inclusion of $B$ may offer new avenues in addressing problems related to dark matter and dark energy by providing extra degrees of freedom that can mimic these phenomena effectively. Additionally, these findings propose a restructuring of how gravitational interactions are understood, allowing further theoretical research and observational validation.

Future work could explore the cosmological applications of such an approach, examining the potential of $f(Q,B)$ gravity to describe the early universe's inflationary period or provide alternatives to dark energy models that align more closely with empirical data. Testing these theories against gravitational wave data could also yield significant insights, further cementing the role of teleparallel and symmetric teleparallel gravities in contemporary physics.

In conclusion, the development and analysis of $f(Q,B)$ gravity models are not only an academic exercise in mathematical physics but also offer promising routes towards new physical evidence and deeper understanding of the universe's fundamental workings. As such, this paper represents a meaningful contribution to the expanding field of modified gravity theories.