[Paper Review] The role of the boundary term in $f(Q,B)$ symmetric teleparallel gravity

Salvatore Capozzıello, Vittorio De Falco+1 more|arXiv (Cornell University)|Jul 25, 2023

Cosmology and Gravitation TheoriesPhysics and Astronomy3 citations

TL;DR

This paper investigates $f(Q,B)$ symmetric teleparallel gravity, where $f$ is a function of the non-metricity scalar $Q$ and a boundary term $B$. It shows that $f(Q,B) = f(Q - B)$ models are dynamically equivalent to $f(R)$ gravity, unifying $f(R)$, $f(T,\tilde{B})$, and $f(Q,B)$ under the Extended Geometric Trinity of Gravity, with consistent field equations and conservation laws derived via a variational approach.

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 $B o0$. 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 $f( ilde{B}-T)$ gravity (where $B eq ilde{B}$). 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 $f(R)$, $f(T, ilde{B})$, 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.

Motivation & Objective

• To investigate the role of the boundary term $B$ in $f(Q,B)$ symmetric teleparallel gravity within metric-affine gravity.
• To derive consistent field equations for $f(Q,B)$ gravity using a variational principle.
• To establish dynamical equivalence between $f(Q,B) = f(Q - B)$ models and $f(R)$ gravity.
• To unify $f(R)$, $f(T,\tilde{B})$, and $f(Q,B)$ gravity under the Extended Geometric Trinity of Gravity framework.
• To compare the Gibbons-Hawking-York boundary term in GR with the $B$ term in $f(Q,B)$ gravity.

Proposed method

• Adopt a variational approach to derive field equations for $f(Q,B)$ gravity, treating $Q$ and $B$ as independent variables.
• Define the non-metricity scalar $Q = Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}$ and the boundary term $B = \partial_\alpha (Q^{\alpha\mu\nu}g_{\mu\nu})$.
• Use the Levi-Civita connection $\accentset{\circ}{\Gamma}$ and the non-metricity tensor $Q_{\alpha\mu\nu} = \nabla_\alpha g_{\mu\nu}$ to express geometric quantities.
• Derive the field equations via variation of the action with respect to the metric $g^{\mu\nu}$, yielding $\accentset{\circ}{G}_{\mu\nu}f_Q - \frac{1}{2}g_{\mu\nu}(f - f_Q Q - f_B B) + \partial_\lambda(f_Q + f_B)2P^\lambda_{\ \mu\nu} - g_{\mu\nu}\accentset{\circ}{\Box}f_B + \accentset{\circ}{\nabla}_\mu\accentset{\circ}{\nabla}_\nu f_B = \chi\Theta_{\mu\nu}$.
• Analyze the limit $B \to 0$ to recover $f(Q)$ gravity and compare with $f(R)$ gravity.
• Establish conservation laws and verify consistency of the field equations through tensorial identities and covariant derivatives.

Experimental results

Research questions

• RQ1 How does the inclusion of the boundary term $B$ affect the field equations in $f(Q,B)$ gravity?
• RQ2 What is the dynamical equivalence between $f(Q,B) = f(Q - B)$ and $f(R)$ gravity?
• RQ3 How does the boundary term $B$ in $f(Q,B)$ gravity compare to the Gibbons-Hawking-York boundary term in General Relativity?
• RQ4 Can $f(Q,B)$ gravity be unified with $f(R)$ and $f(T,\tilde{B})$ gravity under a single geometric framework?
• RQ5 What are the conservation laws and consistency conditions in $f(Q,B)$ gravity derived from the variational principle?

Key findings

• The field equations of $f(Q,B)$ gravity are derived via a consistent variational principle, with the boundary term $B$ playing a crucial role in the dynamics.
• The model $f(Q,B) = f(Q - B)$ is dynamically equivalent to $f(R)$ gravity, extending the Geometric Trinity of Gravity to include symmetric teleparallel formulations.
• In the limit $B \to 0$, $f(Q,B)$ gravity reduces to $f(Q)$ gravity, recovering known results from the literature.
• The boundary term $B$ in $f(Q,B)$ gravity is not equivalent to the boundary term $\tilde{B}$ in teleparallel gravity, despite both being related to boundary contributions.
• Conservation laws are derived, showing that the energy-momentum tensor $\Theta_{\mu\nu}$ is conserved in the theory.
• The field equations reduce to those of General Relativity in the $f(Q,B) = Q + B$ case, confirming consistency with the standard model in the appropriate limit.