Gravitational lensing and deflection angles of generalised Ellis-Bronnikov wormhole embedded in a warped braneworld background

This paper investigates gravitational lensing and deflection angles in generalized Ellis-Bronnikov wormholes within a five-dimensional warped braneworld, deriving analytic and numerical results that demonstrate how the throat steepness parameter and extra-dimensional effects produce distinct signatures in lensing observables like the Einstein ring radius and image positions.

The Gist

Imagine the universe as a giant, stretchy fabric. Usually, we think of gravity like a heavy bowling ball sitting on a trampoline, curving the fabric and making things roll toward it. But what if there were a "shortcut" through that fabric? A tunnel connecting two distant points in space? That's a wormhole.

This paper explores a specific type of theoretical wormhole called the Generalized Ellis-Bronnikov (GEB) wormhole and asks two big questions:

1. How does light bend when it passes near this tunnel?
2. What happens if our universe is actually part of a larger, 5-dimensional structure (like a slice of bread in a loaf)?

Here is a simple breakdown of their findings using everyday analogies.

1. The Shape of the Tunnel (The "Steepness" Parameter)

Think of a standard wormhole as a smooth, gentle funnel. But the authors are studying a "generalized" version that can be shaped differently. They introduced a dial called $m$ (the steepness parameter).

• $m = 2$: This is the classic, smooth funnel shape (the standard Ellis-Bronnikov wormhole).
• $m > 2$: This makes the funnel much sharper and flatter at the bottom, like a steep, narrow canyon that suddenly opens up.

The Finding: The shape of this tunnel changes how light behaves.

• If the tunnel is sharper (higher $m$), light passing far away from the center is barely affected. It's like driving on a highway far from a deep canyon; the canyon doesn't pull your car much.
• However, if light gets very close to the sharp throat, it gets "grabbed" much more intensely than in the smooth version. It spirals around the tunnel more times before escaping. The authors found that the sharper the tunnel, the more dramatic the "spinning" effect becomes.

2. The Hidden Dimension (The "Warped" Extra Dimension)

Now, imagine our universe isn't just a flat sheet, but a 3D slice floating inside a 5D "bulk" (like a 2D drawing existing inside a 3D room). This is the Warped Braneworld idea.

In this scenario, light (photons) doesn't just move through our 3D space; it can also have a tiny bit of "momentum" or movement along that hidden 5th dimension. The authors call this $\delta$.

The Finding: This hidden movement acts like a "blur" or a "widening" effect.

• The Analogy: Imagine throwing a ball at a target. In a normal world, if you aim at a specific spot, the ball goes there. In this 5D world, the ball has a secret side-to-side wobble (the extra dimension) that makes its path slightly different.
• The Result: Because of this wobble, the "effective" distance the light feels from the wormhole changes. It makes the region where light gets trapped (the "photon sphere") appear broader. Instead of a sharp, single ring of light, the image gets slightly smeared out or widened.

3. The Cosmic Lens (Gravitational Lensing)

When a massive object (like a wormhole) sits between us and a distant star, it acts like a lens, bending the star's light. This creates images, rings, or multiple copies of the star.

The authors calculated exactly how much the light bends (the deflection angle) and where the images appear.

• The "Steepness" Signature: By measuring how much the light bends, astronomers could theoretically tell if the wormhole is "smooth" ($m=2$) or "sharp" ($m>2$). A sharper wormhole bends light differently than a smooth one.
• The "Extra Dimension" Signature: The hidden dimension doesn't just shift the image; it changes the rules of the game. In a normal 4D world, light can only get so close to the wormhole before it gets stuck. But in this 5D warped world, light can get closer than that limit and still escape. This creates a unique signature: the "Einstein Ring" (a perfect circle of light) would be slightly smaller and the images slightly different than if the extra dimension didn't exist.

Summary of the "Detective Work"

The paper is essentially a guide for future astronomers on how to spot these wormholes if they exist.

• If you see a wormhole: You can measure the "steepness" of its throat by looking at how much light spirals around it.
• If you see a "blurry" or widened ring of light: It might be a sign that our universe has a hidden, warped 5th dimension that light is interacting with.

The Bottom Line:
The authors didn't find a real wormhole (we don't have one yet!). Instead, they built a mathematical map. They showed that if we ever find a wormhole, the way light bends around it will tell us two things: how "sharp" the tunnel is, and whether our universe is secretly part of a larger, multi-dimensional structure. The "steepness" leaves a clear fingerprint, and the "extra dimension" acts like a subtle widening of the lens.

Technical Summary

1. Problem Statement

The paper addresses the challenge of observationally distinguishing traversable wormholes from black holes and other compact objects. Specifically, it investigates the Generalized Ellis-Bronnikov (GEB) wormhole, a theoretical model that extends the standard Ellis-Bronnikov (EB) wormhole by introducing a steepness parameter $m \ge 2$. This parameter controls the throat geometry and partially mitigates the violation of classical energy conditions.

Furthermore, the authors explore the embedding of this 4D GEB geometry into a five-dimensional warped braneworld (WGEB) background. In this scenario, the extra dimension is characterized by a warp factor and a parameter $\delta$, representing the photon's momentum along the extra dimension. The core problem is to determine how the parameters $m$ (throat steepness), $b_0$ (throat radius), and $\delta$ (extra-dimensional momentum) manifest in observable gravitational lensing signatures, such as deflection angles, Einstein rings, and image positions.

2. Methodology

The authors employ a rigorous analytical and numerical approach to study null geodesics (photon trajectories) in both 4D and 5D spacetimes.

• Metric Formulation:
  • 4D GEB: The metric is defined in tortoise coordinates $l$, with a shape function $r(l) = (b_0^m + l^m)^{1/m}$.
  • 5D WGEB: The metric includes a warp factor $e^{2f(y)}$ and an extra dimension $y$. The authors analyze both growing and decaying warp factors, focusing on the decaying case ($f(y) = -\log[\cosh(y/y_0)]$) which reduces energy condition violations.
• Geodesic Equations: The authors derive the geodesic equations for null trajectories using the Euler-Lagrange formalism. They identify constants of motion (energy $k$ and angular momentum $h$) and derive the radial equation of motion.
• Deflection Angle Calculation:
  • The total deflection angle $\alpha$ is calculated by integrating the trajectory equation from infinity to the point of closest approach ($l_0$) and back.
  • Analytic Solutions: Exact analytic solutions are derived for the standard case ($m=2$) using complete elliptic integrals of the first kind.
  • Approximations: For general $m > 2$, where exact analytic solutions are unavailable, the authors derive asymptotic approximations for both weak-field ($b \gg b_0$) and strong-field ($b \to b_0$) limits.
• Lensing Equations: Standard gravitational lensing equations are applied to relate the deflection angle to observable quantities like the Einstein ring radius ($\theta_E$) and image positions, considering both non-relativistic and relativistic (multiple winding) regimes.

3. Key Contributions

• Generalization of EB Wormholes: The paper provides the first systematic analysis of gravitational lensing for the generalized GEB family ($m \ge 2$) and its 5D warped embedding.
• Analytic Expressions for General $m$: The authors derive new analytic approximations for the deflection angle in both weak and strong lensing regimes for arbitrary $m$.
  • Weak Lensing: $\alpha \sim \frac{b_0^m}{b^m}$.
  • Strong Lensing: A transition from logarithmic divergence (for $m=2$) to power-law divergence (for $m > 2$).
• Identification of Extra-Dimensional Signatures: The study explicitly quantifies how the extra-dimensional parameter $\delta$ modifies the effective impact parameter ($b_w = b/\sqrt{1-\delta^2}$), leading to distinct observational signatures compared to pure 4D models.
• Photon Sphere Broadening: The work demonstrates that the presence of the extra dimension causes a "broadening" of the photon sphere and lensed images, a feature absent in standard 4D wormhole models.

4. Key Results

A. Deflection Angle Behavior

• Effect of Steepness ($m$):
  • As $m$ increases, the throat geometry becomes "sharper" (flatter away from the throat).
  • Weak Lensing: The deflection angle decreases as $m$ increases. The angle scales as $b_0^m / b^m$.
  • Strong Lensing: For $m > 2$, the deflection angle exhibits a power-law divergence ($\sim 1/x_0^{m/2-1}$) as the impact parameter approaches the throat, which is significantly stronger than the logarithmic divergence seen in the standard $m=2$ EB wormhole. This implies higher winding numbers and more relativistic images for steeper throats.
• Effect of Extra Dimension ($\delta$):
  • The effective impact parameter in 5D is $b_w = b/\sqrt{1-\delta^2}$. Since $b_w > b$, the deflection angle for a given physical impact parameter $b$ is smaller in the 5D case compared to the 4D case.
  • Crucially, the divergence of the deflection angle (photon sphere) can occur at impact parameters $b < b_0$ in the 5D scenario, whereas in 4D it is strictly bounded by $b_0$.

B. Gravitational Lensing and Einstein Rings

• Einstein Radius: The angular Einstein radius $\theta_E$ is derived for both weak and strong lensing. The steepness parameter $m$ appears explicitly in the relation between the non-relativistic and relativistic Einstein ring radii, offering a potential way to constrain $m$ observationally.
• Image Broadening: The parameter $\delta$ does not simply shift the Einstein ring radius but introduces an effective spread in image positions. Different combinations of $b$ and $\delta$ can yield the same deflection angle, leading to a broadening of the photon sphere and the resulting lensed images.
• Distinguishing from Black Holes: The relationship between the angular sizes of the relativistic Einstein ring ($\theta_\infty$) and the non-relativistic ring ($\theta_0$) differs quantitatively between GEB/WGEB wormholes and Schwarzschild black holes, providing a diagnostic tool for future high-resolution observations (e.g., Event Horizon Telescope).

C. Numerical Validation

Numerical simulations confirm the analytic approximations. Plots of deflection angle $\alpha$ vs. normalized impact parameter $b/b_0$ show:

1. Increasing $m$ leads to steeper divergence curves.
2. Increasing $\delta$ shifts the divergence point to lower values of $b/b_0$ (allowing $b < b_0$) and reduces the overall deflection magnitude for a fixed $b$.

5. Significance

This research is significant for several reasons:

1. Observational Viability: It provides concrete, testable predictions (deflection angles, ring radii, image broadening) that could allow astronomers to distinguish between standard black holes, standard wormholes, and generalized/warped wormholes using current and future gravitational lensing data.
2. Energy Conditions: By linking the geometric parameter $m$ and the extra dimension $\delta$ to reduced energy condition violations, the paper strengthens the theoretical plausibility of traversable wormholes within modified gravity frameworks.
3. Extra Dimensions: It offers a specific mechanism (momentum along the extra dimension) by which higher-dimensional physics could leave imprints on 4D astrophysical observations, bridging the gap between braneworld cosmology and compact object phenomenology.
4. Methodological Framework: The derivation of analytic approximations for general $m$ fills a gap in the literature, enabling future studies to model a wider class of wormhole geometries without relying solely on numerical integration.

In conclusion, the paper establishes that the steepness of the wormhole throat ($m$) and the presence of warped extra dimensions ($\delta$) leave distinct, measurable signatures in gravitational lensing, particularly in the behavior of the deflection angle near the throat and the structure of Einstein rings.