Charged traversable wormholes: charge without charge

1. Problem Statement

The paper addresses the theoretical construction and physical viability of charged, traversable wormholes within the framework of General Relativity coupled with Maxwell's equations (Einstein-Maxwell theory).

• The "Charge without Charge" Concept: The authors aim to realize Misner and Wheeler's concept where electric field lines enter one side of a wormhole and exit the other, creating the appearance of charge without a net source (Gaussian surface enclosing both universes yields zero net flux).
• The Challenge of Rotation: While static wormhole solutions exist, constructing rotating wormhole solutions is mathematically difficult. Standard methods like the Newman-Janis (NJ) algorithm are well-defined for static black holes where the metric functions satisfy $-g_{tt} = g_{rr}$ (i.e., radial pressure equals negative energy density). However, traversable wormholes supported by anisotropic matter typically violate this condition ($-g_{tt} \neq g_{rr}$), rendering the standard NJ algorithm inapplicable.
• Physical Plausibility: The study seeks to determine if these geometries are physically traversable by analyzing tidal forces, the flare-out condition, and observational signatures (light deflection) compared to black holes.

2. Methodology

A. Static Charged Wormhole Construction

1. Action and Field Equations: The authors start with the Einstein-Hilbert action coupled to a source-free Maxwell field and an effective anisotropic matter field ($L_{am}$).
   • The stress-energy tensor includes contributions from the electromagnetic field and anisotropic matter with radial pressure $p_r = w_1 \epsilon$ and transverse pressure $p_t = w_2 \epsilon$.
2. Metric Ansatz: A static, spherically symmetric metric is assumed:
   $$ds^2 = -f(r)dt^2 + \frac{1}{g(r)}dr^2 + r^2(d\theta^2 + \sin^2\theta d\psi^2)$$
   where $f(r)$ is the redshift function and $g(r)$ incorporates the shape function $b(r)$.
3. Analytical Solution:
   • Solving the Maxwell equations yields the electric field $E \propto Q/r^2$.
   • Solving the Einstein equations with the anisotropic matter constraints yields explicit forms for the shape function $b(r)$, energy density $\epsilon$, and pressures.
   • The solution is parameterized by the throat radius $r_o$, charge $Q$, and equation of state parameter $w_1$.

B. Traversability Analysis

• Flare-out Condition: Verified by embedding the geometry and ensuring $d^2r/dz^2 > 0$ at the throat.
• Energy Conditions: Analysis of the "exoticity function" $\zeta$ to confirm the violation of the Null Energy Condition (NEC), which is necessary for wormhole stability.
• Tidal Forces: Calculation of the Riemann tensor components in the traveler's frame to ensure tidal accelerations ($\Delta a$) remain within humanly tolerable limits (comparable to Earth's gravity).
• Observational Signatures:
  • Photon Sphere: Determination of unstable circular photon orbits.
  • Capture Cross-section: Calculation of the impact parameter ($r_{ip}$) for light capture.
  • Deflection Angle: Integration of the geodesic equation to find the light deflection angle $\alpha$.

C. Rotating Generalization (Modified NJ Algorithm)

• Algorithm Modification: The authors propose a modification to the Newman-Janis algorithm to handle static geometries where $-g_{tt} \neq g_{rr}$.
• Procedure:
  1. Transform to Eddington-Finkelstein coordinates.
  2. Introduce complex coordinate transformations ($u' = u - ia\cos\theta$, $r' = r + ia\cos\theta$).
  3. Crucial Deviation: Unlike the black hole case where $\bar{\rho}^2 = r^2 + a^2\cos^2\theta$, the wormhole case utilizes a modified definition $\bar{\rho}^2 = r'^2 - a^2\cos^2\theta$ to maintain the throat structure.
  4. Derive the rotating metric in Boyer-Lindquist coordinates.
• Limitations: The authors acknowledge they could not analytically prove that the $G_{r\theta}$ component of the Einstein tensor vanishes for the rotating case, nor did they solve the Maxwell equations for the rotating geometry (leaving this for future work).

3. Key Results

Static Solutions

• Geometry: A concrete analytic solution for a charged wormhole supported by anisotropic matter was derived.
• "Charge without Charge": The solution demonstrates that electric flux enters one universe and exits the other. A Gaussian surface enclosing both asymptotic regions shows zero net charge, despite the presence of a non-zero charge parameter $Q$ in the metric.
• Throat Constraints: The charge $Q$ reduces the throat size ($r_o$). For the solution to be valid, $r_o > Q$ and specific constraints on $w_1$ ($w_1 > 0$ or $w_1 < -1$) are required to satisfy the flare-out condition.
• Tidal Forces: Numerical analysis (Figure 2) shows that for specific ranges of $w_1$ and moderate charges, tidal forces at the throat can be kept below $1g$, making the wormhole traversable for a human-sized traveler.
• Light Deflection:
  • Photon Sphere: Unlike black holes where the photon sphere is outside the horizon, the photon sphere for this wormhole is located at the throat ($r = r_o$).
  • Capture Cross-section: The capture cross-section depends on $Q$ and $b_o$. For fixed $r_o$, smaller $Q$ yields a larger capture cross-section.
  • Deflection Angle: The deflection angle diverges as the impact parameter approaches the throat, similar to black holes, but the divergence behavior differs slightly in the approach to the photon sphere.

Rotating Solutions

• Metric Construction: A rotating metric was successfully constructed using the modified NJ algorithm.
• Throat Independence: The location of the rotating wormhole throat is independent of the angular coordinate $\theta$ (a key feature for traversability).
• Parameter Effects:
  • Rotation ($a$): Tends to increase the size of the throat.
  • Charge ($Q$): Tends to decrease the size of the throat.
• Constraints: New constraints on the equation of state parameter $w_1$ were derived to satisfy the flare-out condition in the presence of rotation and charge.
• Singularities: The authors note potential physical singularities at the poles ($\theta = 0, \pi$) where the metric functions behave differently than in the Kerr black hole case, requiring further investigation.

4. Significance and Contributions

1. Realization of "Charge without Charge": The paper provides a rigorous mathematical realization of the Misner-Wheeler concept, showing how a traversable wormhole geometry can support an electric field without a net source charge.
2. Extension of the NJ Algorithm: The work represents a significant methodological advancement by modifying the Newman-Janis algorithm to apply to geometries where the radial pressure does not equal the negative energy density ($-g_{tt} \neq g_{rr}$). This opens the door to generating rotating solutions for a broader class of exotic matter configurations.
3. Observational Differentiation: The study highlights distinct observational signatures between wormholes and black holes:
   • The location of the photon sphere (at the throat vs. outside the horizon).
   • The behavior of the deflection angle near the throat.
   • The specific dependence of the capture cross-section on charge and rotation.
4. Traversability Criteria: By explicitly calculating tidal forces and providing numerical bounds on parameters ($w_1, Q, r_o, v$), the paper moves wormhole physics from purely theoretical existence proofs to models with potential physical plausibility for travel.

5. Conclusion

The authors successfully constructed a family of charged, traversable wormhole solutions supported by anisotropic matter. They demonstrated that these geometries satisfy the "charge without charge" paradigm and are traversable under specific conditions regarding tidal forces. Furthermore, they extended the Newman-Janis algorithm to generate rotating counterparts, identifying how rotation and charge compete to determine the throat size. While challenges remain regarding the rigorous verification of the rotating Einstein equations and the Maxwell field in the rotating case, the work provides a robust framework for future investigations into the observational and theoretical properties of charged wormholes.