New Lorentzian Taub-NUT and Euclidean Eguchi-Hanson Solutions in $f(R)$ gravity

Technical Summary: New Lorentzian Taub-NUT and Euclidean Eguchi-Hanson Solutions in f(R) Gravity

Problem Statement
The paper addresses the construction of exact analytical solutions for modified theories of gravity, specifically $f(R)$ gravity, which generalize General Relativity (GR) by replacing the Ricci scalar $R$ in the Einstein-Hilbert action with a function $f(R)$. While gravitational instantons such as Taub-NUT and Eguchi-Hanson spaces are well-understood in the context of GR and supergravity, their existence and properties within $f(R)$ gravity remain largely unexplored. The authors aim to derive exact Taub-NUT and Eguchi-Hanson type solutions in $f(R)$ gravity, determine the specific functional form of $f(R)$ that admits these solutions, and analyze their physical properties, including thermodynamics and test particle orbits.

Methodology
The authors employ the metric formulation of $f(R)$ gravity, assuming the connection is the Levi-Civita connection. The action is defined as:
$$S_{f(R)} = \frac{1}{2\kappa} \int_M d^4x \sqrt{-g} f(R) + S_M$$
leading to the field equations:
$$f_R R_{\mu\nu} - \frac{1}{2}f(R)g_{\mu\nu} - [\nabla_\mu \nabla_\nu - g_{\mu\nu}\Box]f_R = \kappa T_{\mu\nu}$$

To solve these fourth-order differential equations, the authors utilize a specific metric ansatz for the Taub-NUT spacetime:
$$ds^2 = -B(r)e^{2\alpha(r)}(dt + 2n \cos\theta d\phi)^2 + \frac{dr^2}{B(r)} + (r^2 + n^2)d\Omega^2$$
where $n$ is the NUT charge. Instead of varying the action with respect to the metric components $g_{\mu\nu}$, the authors vary the action with respect to the metric functions $\alpha(r)$ and $B(r)$ directly. This approach reduces the problem to a system of two second-order differential equations rather than fourth-order expressions.

The authors utilize a "guess and check" approach for the function $\alpha(r)$ to decouple the system and solve for $B(r)$ and the unknown function $f(R)$. They also employ coordinate transformations to relate different solution sets and to map the Taub-NUT solution to an Eguchi-Hanson type geometry.

Key Contributions and Results

1. Taub-NUT Solutions in $f(R)$ Gravity:
   The authors derive two classes of solutions for the Taub-NUT metric:

   • Solution Set I: By assuming $\alpha(r) = 0$, they find a solution where the Ricci scalar $R$ allows for an inversion to determine the radial coordinate. This leads to the specific $f(R)$ form:
     $$f(R) = \eta \sqrt{R - 2\Lambda}$$
     where $\eta$ is a constant and $\Lambda$ is the cosmological constant.
   • Solution Set II: By solving a differential equation derived from the field equations for $\alpha(r)$, they find a more general solution involving a parameter $\epsilon$. Despite the apparent generalization, they demonstrate via coordinate transformations ($r \to \sqrt{r^2 - \epsilon m^2}$ and $t \to t/\sqrt{\epsilon+1}$) that Solution Set II is physically equivalent to Solution Set I.
   • Clifton-Barrow Connection: In the limit where the NUT charge $n \to 0$, the metric reduces to the $\delta = -1/2$ case of the Clifton-Barrow metric, extended by a cosmological constant. This establishes the new solution as a specific extension of known power-law $f(R)$ models.

2. Physical Properties of Taub-NUT Solutions:

   • Singularities: The Kretschmann invariant reveals singularities at $r=0$ (standard to Taub-NUT) and potentially at $r = \pm n \sqrt{-\epsilon/(1+\epsilon)}$ depending on the parameter $\epsilon$. To preserve the Lorentzian structure, the constraint $\epsilon > -1$ is mandated.
   • Thermodynamics: The authors derive expressions for temperature ($T$), entropy ($S$), and energy ($E$) using the first law of black hole thermodynamics adapted for $f(R)$ gravity. The horizon radius $r_+$ is determined implicitly via the Lambert W function.
   • Test Particle Orbits: The radial and polar trajectory equations are derived. The authors analyze the conditions for equatorial orbits and circular orbits, noting that analytical closed-form solutions for the trajectories are not possible due to the complexity of the potential, but constraints on the Carter constant and angular momentum are established.

3. Eguchi-Hanson Type-II Extension:
   Through a unique coordinate transformation relating the radial coordinate and time to the Eguchi-Hanson angle $\psi$, the authors construct an Eguchi-Hanson type-II solution.

   • The resulting metric describes a 4D spatial solution in the range $0 < r < 1/\xi$ and transitions to the 3+1D Taub-NUT solution for $r > 1/\xi$.
   • By allowing the parameter $\xi$ to be imaginary ($\chi = i\xi$), they obtain a proper full 4D Euclidean Eguchi-Hanson solution with domain $0 < r < \infty$.
   • In the limit $\xi \to 0$ (or $\chi \to 0$) and $\Lambda = 0$, the solution reduces to the standard Eguchi-Hanson metric.

Significance and Claims
The paper claims to present the first construction of Taub-NUT solutions in $f(R)$ gravity with a coordinate-dependent Ricci scalar. The work is motivated by the existence of gravitational instantons in four-dimensional GR and their utility in higher-dimensional gravity, M-theory, and black hole thermodynamics.

The authors position this work as a foundational step toward constructing rotating solutions with NUT twists in $f(R)$ gravity. They suggest that establishing these solutions is a prerequisite for investigating the existence (or non-existence) of conformal symmetry and the AdS/CFT duality for rotating NUT solutions in modified gravity. Specifically, they note that future work could explore whether the macroscopic Bekenstein-Hawking entropy of these rotating black holes matches the microscopic entropy calculated via the Cardy formula, a correspondence previously verified in GR. The paper concludes by identifying the construction of higher-dimensional Taub-NUT and Eguchi-Hanson solutions in $f(R)$ gravity as a direction for future research.