On the uplift of 4D wormholes in Braneworld models and their 5D structure

TL;DR

This work develops and applies the General Embedding Algorithm (GEA) and its extended form (eGEA) to uplift a broad class of 4D wormholes into 5D single-brane braneworlds with warped extra dimensions. It derives general 5D conditions for wormhole structure, clarifies how the circumferential radius $r(u)$ governs the uplift, and shows that warping fundamentally reshapes the 5D throat geometry while localizing curvature near the brane. The authors provide explicit 5D constructions for four seed spacetimes (CFM, BK, SV, EB) in RS-II and thick brane models, analyzing curvature, flare-out, and energy conditions, and illustrating how embedding diagrams encode bulk warping effects. The results reveal that warped extra dimensions induce nontrivial 5D wormhole features and stress the importance of warp profiles in determining bulk topology and energy-condition behavior, with implications for stability and extensions to multi-brane or multi-warp scenarios.

Abstract

Recent developments for the consistent embedding of general 4D static and spherically-symmetric spacetimes in arbitrary single-brane braneworld models in the form of the General Embedding Algorithm (GEA) [Phys.Rev.D 109 (2024) 4, L041501], initiated the program of studying the bulk structure of braneworld wormholes. In this article, adopting a completely generic approach, we derive the general conditions that the metric functions of any braneworld spacetime must satisfy to describe a wormhole structure in the bulk. Particular emphasis is placed on clarifying the proper uplift of 4D spacetimes, expressed in terms of arbitrary radial coordinates on the brane, and we demonstrate the important role of the circumferential radius metric function $r(u)$ for the embedding. To ensure applicability even when $r(u)$ is non-invertible, we develop an extended formulation of the GEA. Additionally, the flare-out conditions for braneworld wormholes are presented for the first time and are found to differ from the case of flat extra dimensions. To illustrate the method, we first perform the uplift into both thin (Randall-Sundrum II) and thick braneworld models for four well-known 4D wormhole spacetimes: the effective braneworld wormhole solutions of Casadio-Fabbri-Mazzacurati and Bronnikov-Kim, the Simpson-Visser spacetime, and the Ellis-Bronnikov or "anti-Fisher" solution. Subsequently, we study their bulk features by means of curvature invariants, flare-out conditions, energy conditions and embedding diagrams. Our analysis reveals that the assumption of a warped extra dimension has non-trivial implications for the structure of 5D wormholes.

Figures (11)

• Figure 1: The ${\rm 5D}$ bulk extension of the Schwarzschild BH in the RS-II model using the CHR method (left panel) and the GEA (right panel). The light-blue planes correspond to identical copies of the seed brane that continuously extend the geometry into the bulk prior to warping. The dashed blue curves correspond to the locations of the event horizon in each of the replica branes, while the red dots correspond to the locations of the curvature singularities. The green surfaces correspond to the ${\rm 5D}$ event horizon of the final warped bulk geometry. The CHR method yields a black string with the curvature singularity and the event horizon extending indefinitely into the bulk, while the GEA generates a BH with a finite-area event horizon and a curvature singularity that is localized strictly on the brane.
• Figure 2: Coordinates in the bulk. The light-purple region is not part of spacetime, since the radial coordinate $\rho\geqslant R_\Join$. However this does not restrict the domains for the coordinates $r \in [0,+\infty)$ and $z\in (-\infty,+\infty)$.
• Figure 3: Schematic representations of both the General Embedding Algorithm (GEA) for the Morris-Thorne (MT) frame and the extended General Embedding Algorithm (eGEA) for the non-Morris-Thorne (nMT) frame.
• Figure 4: The Ricci curvature of the Casadio-Fabbri-Mazzacurati wormhole embedded in the RS-II model for a fixed value of the dimensionless parameter $\alpha/(2M)=1.5$ and $k\alpha=\{0.015$, $0.15$, $0.75\}$. In all cases, the purple lines indicate the location of the wormhole throat as it extends into the bulk ($\rho=\alpha$), the continuous red line is for $r/\alpha=1$, while the dashed red line is for $y/\alpha=0$ (on the 3-brane). All depicted quantities are dimensionless.
• Figure 5: The graphs of the quantities defining the energy conditions on the hyperthroat ($\rho=\alpha$) of the CFM wormhole embedded in the RS-II braneworld, in terms of the $\chi$-coordinate, for $\alpha/(2M)=1.5$ and $k\alpha=0.15$ (left panel), $0.75$ (right panel). All depicted quantities are dimensionless.