Casimir traversable wormholes in Gauss-Bonnet gravity

Technical Summary: Casimir Traversable Wormholes in Gauss-Bonnet Gravity

Problem Statement
The primary challenge in constructing traversable wormholes within General Relativity (GR) is the requirement for "exotic matter" that violates the Null Energy Condition (NEC) to keep the wormhole throat open and prevent gravitational collapse. While quantum effects like the Casimir effect can provide negative energy densities, their ability to support macroscopic wormhole geometries within standard GR is limited. This paper investigates whether higher-order curvature corrections, specifically within the Einstein-Gauss-Bonnet (EGB) gravity framework, can facilitate traversable wormhole solutions supported by Casimir energy while satisfying energy conditions (Weak and Strong) that are typically violated in GR.

Methodology
The authors analyze static, spherically symmetric wormhole spacetimes in $n$-dimensional EGB gravity. The study proceeds through the following steps:

1. Field Equations: The authors utilize the EGB action, which includes the standard Einstein-Hilbert term and the Gauss-Bonnet (GB) quadratic curvature term. They derive the field equations for a general metric defined by a redshift function $\phi(r)$ and a shape function $b(r)$.
2. Casimir Source: Instead of assuming a specific equation of state, the energy density $\rho(r)$ is prescribed as a general power-law form characteristic of Casimir effects in various geometries: $\rho(r) = \lambda/r^m$, where $\lambda$ and $m$ are constants dependent on boundary conditions and field types.
3. Solution Construction: The authors solve the differential equation for the shape function $b(r)$ derived from the energy density constraint. They classify solutions based on a parameter $\eta = (n-2)(m-n+1)$, distinguishing between cases where $\eta \neq 0$ and $\eta = 0$.
4. Energy Conditions: The study evaluates the Weak Energy Condition (WEC) and Strong Energy Condition (SEC) for two scenarios:
   • Zero Tidal Force: Assuming a constant redshift function ($\phi' = 0$).
   • Non-Zero Tidal Force: Assuming a specific asymptotically flat redshift function $\phi(r) = \phi_0 (r_0/r)^k$.
5. Stability Analysis: The stability of the solutions is examined using the Tolman-Oppenheimer-Volkoff (TOV) equation, analyzing the balance between gravitational, hydrostatic, and anisotropic forces.
6. Geodesic Motion: The trajectories of null (massless) and timelike (massive) particles are studied using the Lagrangian formalism to determine effective potentials and orbital behaviors.

Key Contributions and Results

• Generalized Solutions: The paper derives exact analytical solutions for the shape function $b(r)$ in $n$-dimensions supported by a general Casimir energy density. These solutions include asymptotically flat, asymptotically non-flat, and de-Sitter (small wormhole) configurations.
• Role of Redshift Function:
  • For zero tidal force solutions (constant $\phi$), the energy conditions (WEC and SEC) are generally violated throughout the spacetime, consistent with standard GR limitations.
  • For non-zero tidal force solutions (variable $\phi$), the authors demonstrate that by choosing suitable model parameters (specifically the GB coupling constant $\alpha$ and the redshift function parameters), the WEC and SEC can be satisfied in the vicinity of the throat and at spatial infinity. Notably, satisfying these conditions often requires a negative GB coupling constant ($\alpha < 0$) and specific constraints on the throat radius relative to the coupling strength.
• Singularity Avoidance: The authors verify that the Kretschmann and Weyl curvature scalars remain finite at the wormhole throat, provided $r_0^2 \neq -2\alpha$, ensuring the absence of spacetime singularities at the throat.
• Stability: The TOV analysis confirms that the gravitational, hydrostatic, and anisotropic forces balance each other ($F_g + F_h + F_a = 0$), indicating that the derived wormhole configurations are in a state of equilibrium and stable against radial perturbations.
• Particle Trajectories:
  • Null Geodesics: The effective potential for photons exhibits a local maximum at the throat for certain parameter ranges, suggesting the existence of photon spheres at or outside the throat.
  • Timelike Geodesics: Depending on the angular momentum and model parameters, the effective potential can exhibit local minima at the throat (allowing for stable bound orbits) or maxima. The analysis identifies three types of trajectories: particles traversing the throat to another universe, particles reflected back by a potential barrier, and particles in bound oscillatory orbits.

Significance and Claims
The paper claims to extend previous studies on Casimir wormholes by generalizing the energy density to $n$-dimensions and relaxing the assumption of a zero redshift function. The central finding is that the higher-order curvature terms inherent in EGB gravity, when combined with a specific radial-dependent redshift function, can effectively mimic the properties of exotic matter. This allows for the construction of traversable wormhole geometries where the energy conditions are satisfied in significant portions of the spacetime, including the throat region, thereby reducing or eliminating the need for exotic matter sources that are unphysical in classical contexts. The work provides a robust framework for exploring wormhole stability and observational signatures (such as photon spheres) in modified gravity theories without relying solely on quantum field theory exotic sources.