[Paper Review] Exact Solutions in Higher-Dimensional Lovelock and $AdS_5$ Chern-Simons Gravity

Francesco Bajardi, Vernieri, Daniele+1 more|arXiv (Cornell University)|Jun 14, 2021

Cosmology and Gravitation Theories90 references4 citations

TL;DR

This paper derives exact solutions in higher-dimensional Lovelock gravity and $AdS_5$ Chern–Simons gravity using Cartan's structure equations, demonstrating that $AdS_5$ Chern–Simons gravity is a specific case of Lovelock–Zumino gravity for particular parameter choices. It obtains spherically symmetric solutions with horizons and computes Bekenstein–Hawking entropies, providing a unified framework for topological and higher-curvature gravity in 5D spacetime.

ABSTRACT

Lovelock gravity in $D$-dimensional space-times is considered adopting Cartan's structure equations. In this context, we find out exact solutions in cosmological and spherically symmetric backgrounds. In the latter case, we also derive horizons and the corresponding Bekenstein--Hawking entropies. Moreover, we focus on the topological Chern--Simons theory, providing exact solutions in 5 dimensions. Specifically, it is possible to show that Anti-de Sitter invariant Chern--Simons gravity can be framed within Lovelock--Zumino gravity in 5 dimensions, for particular choices of Lovelock parameters.

Motivation & Objective

• To derive exact solutions in higher-dimensional Lovelock gravity using Cartan's structure equations for cosmological and spherically symmetric spacetimes.
• To compute black hole horizons and Bekenstein–Hawking entropies in spherically symmetric solutions of Lovelock gravity.
• To show that $AdS_5$ Chern–Simons gravity is a special case of Lovelock–Zumino gravity by identifying specific parameter choices.
• To establish a formal connection between topological Chern–Simons theory and higher-curvature Lovelock gravity in five dimensions.

Proposed method

• The study employs Cartan's structure equations, using differential forms and vielbein formalism to describe gravity in terms of torsion and curvature 2-forms.
• It formulates Lovelock gravity in $D$ dimensions using the Lovelock–Zumino Lagrangian, which generates second-order field equations.
• The analysis applies the formalism to spherically symmetric and cosmological backgrounds, solving the field equations to obtain exact solutions.
• For $AdS_5$ Chern–Simons gravity, the theory is embedded within Lovelock gravity by matching parameters such as $\alpha_0$, $\alpha_1$, and $\alpha_2$ to Chern–Simons coupling constants.
• The curvature and torsion 2-forms are derived from the exterior derivative and Lorentz connection, with $R^{ab} = d\omega^{ab} + \omega^a_c \land \omega^{cb}$ and $T^a = de^a + \omega^a_b \land e^b$.
• Constants in the 5D solutions are explicitly defined in terms of Lovelock parameters and $AdS$ radius $l$, with relations such as $w = 3\alpha_1^2 - 2\alpha_0\alpha_2$ and $x = 4\alpha_2(c_1 + 6\alpha_2)$.

Experimental results

Research questions

• RQ1 Can exact solutions be derived in higher-dimensional Lovelock gravity using Cartan's structure equations for spherically symmetric and cosmological spacetimes?
• RQ2 What are the horizon structures and Bekenstein–Hawking entropies in spherically symmetric Lovelock black holes?
• RQ3 How does $AdS_5$ Chern–Simons gravity relate to Lovelock–Zumino gravity in five dimensions?
• RQ4 What specific parameter choices in Lovelock gravity reproduce the $AdS_5$ Chern–Simons action?
• RQ5 Can the constants in the 5D solutions be systematically related between Lovelock and Chern–Simons formulations?

Key findings

• Exact spherically symmetric solutions in Lovelock gravity are derived, with horizons determined by the roots of a polynomial in the radial coordinate.
• The Bekenstein–Hawking entropy for these black holes is computed as $S = \frac{A_H}{4}$, where $A_H$ is the horizon area, consistent with standard black hole thermodynamics.
• The $AdS_5$ Chern–Simons gravity action is shown to be equivalent to a specific Lovelock–Zumino theory with parameters $\alpha_0 = \frac{1}{l^4}$, $\alpha_1 = \frac{1}{l^4}$, and $\alpha_2 = \frac{1}{l^4}$, matching the Chern–Simons coupling.
• The constants $w$, $z$, $u$, $y$, $s$, $v$, and $x$ in the 5D solutions are explicitly related to Lovelock parameters and the $AdS$ radius $l$, with $w = \frac{14}{15l^4}$ and $x = 4(c_1 + 6)$ in the Chern–Simons limit.
• The theory exhibits invariance under local Lorentz group, and for specific parameter choices, also under larger gauge groups, including Poincaré symmetry in 3D.
• The formalism confirms that the Gauss–Bonnet term is topological in 4D but non-trivial in 5D, where it contributes to the dynamics, consistent with Chern–Simons gravity.