Wormholes exact solutions in high dimensions General Relativity

TL;DR

The paper develops exact five-dimensional vacuum wormhole solutions in General Relativity, parameterized by $p$ and $q$, using a flat-subspace construction that yields an $\hat{g}_p$ metric with a throat whose properties depend on the derived quantity $l = 1 + \frac{3 q^2 - p^2}{4}$. For even $p$, the solutions are asymptotically flat wormholes featuring ergoregions and a throat located at $R_G$, with regularity and causal structure governed by $l$ and $q$; notably, $l \le 0$ or certain $q$-bounds enforce Wormhole Cosmic Censorship (WCCC) by cloaking singularities and CTC regions. The work analyzes singularities via the Kreschman invariant, demonstrates the absence of event horizons, and provides detailed geodesic dynamics and embedded diagrams, including a focused study of the $p=2$ case and specific $q$ values (e.g., $q=0$ and $q=6\sqrt{3}$). The results yield a coherent picture in which higher-dimensional wormholes can be regular and causality-respecting under precise parameter regimes, with rich phenomenology for geodesics traversing the throat. This advances the understanding of wormhole solutions in higher-dimensional vacuum GR and highlights conditions under which WCCC can hold in such spacetimes.

Abstract

In the present work, we develop and examine a series of exact solutions to Einstein's 5-dimensional field equations in the vacuum, which depend on two constant parameters, $p$ and $q$, which generalize the solutions of Lü and Mei [8] belonging to our class $p=2$. This category of solutions can be split into two sections: when $p$ is odd, it represents a compact object that may have naked singularities. However, the intriguing outcome occurs when $p$ is even, as asymptotically Ricci flat wormholes emerge in this scenario. The Kreschman invariant of these solutions depends on the constant parameter $l = 1 + \frac{3 q^2 - p^2}{4}$. When $l = 0$ and $l \leq -\frac{1}{2}$, the solutions are regular. For the specific cases where $l \leq 0$, or $l > 0$ such that $q < 0$ for $p \geq 2$, $q \leq \frac{1}{3}$ for $p = 2$, and $q \leq \frac{1 + \sqrt{2}}{3}$ for $ p \geq 4$, this class of wormholes adheres to Wormhole Cosmic Censorship, implying that the throat effectively obscures all causal anomalies and singularities. In our analysis, we investigated the embedded geometry, geodesics, singularities, potential event horizons, ergoregions, and the wormhole throat.

Figures (12)

• Figure 1: The volume function, along with its first and second derivatives, has been determined through numerical methods for the parameters $p=2$, $q=6\sqrt{3}$, and $r_0=1$. By applying these numerical methods, it is found that in this scenario, the throat is $R_G=3.47197$ ($V_{,r}(R_G)=0$ and $V_{,rr}(R_G)>0$) that corresponds to dashed black line.
• Figure 2: The embedding associated with $h=\phi,u=r,v_0=\theta_0$ pertains to the profile diagram of the compact object. Reference $z>0$ corresponds to one universe, while reference $z<0$ may refer either to another universe or potentially the same one. Each colour line represents a distinct constant angle.
• Figure 3: The embedding associated with $h=\phi,u=\theta,v_0=r_0$ pertains to the shape diagram of the compact object. Each colour line represents a distinct constant radius.
• Figure 5: The behavior of $\dot{t}$, fixing the constants $J=0$, $E=10$, and $r_0=2$, reveals $\dot{t}>0$ for all values of $r>r_0$. Consequently, causality violations occur when $\dot{\psi}=-\dot{t}>0$. In other words, the red line represents a violation of causality and spacelike geodesics, whereas the blue region indicates causal geodesics.
• Figure 6: The solutions concerning the momenta and variables associated with \ref{['Solucion p2 q0']}, using $J=0$, $E=10$, $r_0=2$, $p_\phi =1$, and \ref{['Condiciones iniciales p=2 pi12']}. The black dotted line signifies the moment when the null geodesic touch the throat of the wormhole located in $R_G=r_0$, and this line separates universe 1 from universe 2.