--- title: 'BTZ Wormhole: Geometries & Applications' url: https://www.emergentmind.com/topics/btz-wormhole type: topic --- # BTZ Wormhole: Geometries & Applications Searching arXiv for recent and foundational papers on BTZ wormholes to ground the article in published work. Searching arXiv for "BTZ wormhole traversable BTZ wormhole" A BTZ wormhole is a wormhole geometry built from the Bañados–Teitelboim–Zanelli background or from a deformation, gluing, or generalization of that background in \(2+1\)-dimensional gravity with negative cosmological constant. In the literature, the term covers several distinct objects: the non-traversable Einstein–Rosen bridge of the eternal BTZ black hole, traversable wormholes generated by inter-boundary couplings and negative null energy, thin-shell wormholes constructed by cut-and-paste methods, black-bounce and Einstein–Rosen-type geometries with a throat at or outside the BTZ horizon, and horizonless microstate-like or metric-affine/nonlinear-electrodynamic generalizations [1906.10715], [2302.14049], [1807.01528], [2210.06608], [1609.05827]. ## 1. BTZ geometry as the underlying background In the non-rotating case, the BTZ black hole of horizon radius \(R\) in AdS\(_3\) with radius \(\ell\) has Schwarzschild-form metric \[ ds^2 = -\frac{r^2 - R^2}{\ell^2}\,d\tau^2 + \frac{\ell^2}{r^2 - R^2}\,dr^2 + r^2\,d\phi^2, \qquad \phi\sim\phi+2\pi, \] with \[ M = \frac{R^2}{8 G_N \ell^2}, \qquad S = \frac{\pi R}{2 G_N}, \qquad T = \frac{R}{2\pi \ell^2}, \qquad \beta=\frac{2\pi\ell^2}{R}. \] In Kruskal coordinates \((U,V,\phi)\), \[ ds^2 = \frac{-4\ell^2\,dU\,dV + R^2(1-UV)^2\,d\phi^2}{(1+UV)^2}, \] the two asymptotic boundaries sit at \(UV=-1\), while the future horizons are at \(U=0\) and \(V=0\) [1906.10715]. In the rotating case, the BTZ line element can be written \[ ds^2 = -N^2(r)\, dt^2 + N^{-2}(r)\, dr^2 + r^2\big(d\phi + N^{\phi}(r)\, dt\big)^2, \] with \[ N^2(r) = -M + \frac{r^2}{l^2} + \frac{J^2}{4 r^2}, \qquad N^{\phi}(r) = - \frac{J}{2 r^2}, \] and horizon radii \[ r_{\pm}^2 = \frac{l^2}{2}\left(M \pm \sqrt{M^2 - \frac{J^2}{l^2}}\right). \] This lower-dimensional AdS setting is repeatedly used as an ideal laboratory for wormhole constructions because the geometry retains black-hole thermodynamics, rotation, and AdS asymptotics while simplifying junction conditions and holographic control [1807.01528]. ## 2. Einstein–Rosen bridges and horizon-based BTZ wormholes The eternal BTZ black hole is dual to the thermofield double state of two decoupled CFTs on the left and right boundaries. Its Einstein–Rosen bridge is non-traversable when the CFTs remain decoupled: no signal can be sent from one boundary to the other without violating bulk causality or the averaged null energy condition [1906.10715]. A closely related construction rewrites the static BTZ black hole using \[ u^2=r-r_h,\qquad r_h=l\sqrt{M}, \] which yields the “null-like wormhole spacetime” \[ \mathrm{d}s^2=-\frac{u^2(u^2+2u_0)}{l^2}\,\mathrm{d}t^2+\frac{4l^2}{u^2+2u_0}\,\mathrm{d}u^2+(u^2+u_0)^2\,\mathrm{d}\phi^2, \] with \(u_0=r_h\), \(u\in(-\infty,+\infty)\), and two congruent universes connected at \(u=0\). In this geometry the radial null geodesic speed at the throat vanishes, \[ \left.\frac{\mathrm{d}u}{\mathrm{d}t}\right|_{u=0}=0, \] so classical traversability is obstructed, although quantum-field correlations can still be shared across the bridge [2510.04005]. A different Einstein–Rosen BTZ wormhole construction introduces \[ r^2=|u|+\ell^2 M,\qquad u\in(-\infty,\infty), \] and obtains \[ ds^2=-\frac{|u|}{\ell^2}\,dt^2+\frac{\ell^2\,du^2}{4|u|(|u|+\ell^2 M)}+(|u|+\ell^2 M)\,d\phi^2. \] Here the throat sits at \(u=0\), where \(g_{tt}\to0\) and \(g_{uu}\to\infty\), so the throat coincides with a horizon. The same work describes this as a one-way traversable wormhole, derives the Hawking temperature \[ T_H=\frac{\sqrt{M}}{2\pi\,\ell}, \] in both BTZ-like and Kruskal-like coordinates, and identifies a distributional exotic string source with negative tension at the throat together with ANEC violation there [2410.11907]. Across these constructions, “BTZ wormhole” therefore includes both the standard non-traversable Einstein–Rosen bridge and horizon-based geometries whose throat coincides with the BTZ horizon. This suggests that the term is geometric rather than uniquely causal: traversability depends on the matter sector, boundary couplings, or gluing prescription, not on the BTZ ancestry alone. ## 3. Traversable BTZ wormholes from holographic couplings In the traversable setup of “Information Transfer and Black Hole Evaporation via Traversable BTZ Wormholes,” the two boundary CFTs are coupled by an infinite-boost limit of a specific double-trace deformation. For conformal dimension \(\Delta=1/2\), the deformation produces a null shock on \(U=0\), \[ \langle T_{UU}\rangle = \frac{\alpha}{4\pi G_N}\,\delta(U), \] and choosing \(\alpha<0\) gives negative null energy and opens a traversable window \[ 01\) there is a specific radius \[ a_0 = \frac{|J|}{\sqrt{2\,M}} \] at which the shell is linearly stable irrespective of the barotropic equation of state. The same analysis states that the equilibrium shell violates the weak energy condition because \(\sigma_0<0\) [1807.01528]. A non-rotating thin-shell BTZ wormhole can be constructed from the metric \[ ds^2 = -\left(K^2 r^2 - M_0\right)\,dt^2 + \left(K^2 r^2 - M_0\right)^{-1}\,dr^2 + r^2\,d\theta^2, \] with throat outside \[ r_h=\frac{\sqrt{M_0}}{K}. \] For a static shell at \(a_0\), \[ \sigma_0 = -\frac{1}{4\pi a_0}\sqrt{K^2 a_0^2 - M_0}, \qquad p_0 = \frac{1}{4\pi}\frac{K^2 a_0}{\sqrt{K^2 a_0^2 - M_0}}, \] and the total amount of exotic matter is \[ \Omega_\sigma = 2\pi a\,\sigma(a) = -\frac{1}{2}\sqrt{K^2 a^2 - M_0}. \] With a generalized Chaplygin gas on the shell, \[ p=-\frac{A}{\sigma^\alpha}, \] linear stability is obtained for \[ \alpha>1, \] while a more general perturbative treatment with \(\beta^2=\partial p/\partial \sigma\) gives the bound \[ \beta_0^2 < \frac{K^2 a_0^2}{K^2 a_0^2 - M_0}. \] This work presents the construction as the first example of a stable thin-shell wormhole in \(2+1\) dimensions [1109.0976]. The same cut-and-paste strategy extends to noncommutative BTZ geometry with \[ f(r)=M\left[2e^{-\frac{r^{2}}{4\theta}}-1\right]-\Lambda r^{2}. \] Here the shell matter has \(\sigma<0\), while the paper reports \(\sigma+p\ge 0\) and \(\sigma+2p\ge 0\). For a dark-energy equation of state \(p=\omega\sigma\), stable wormholes exist for phantom energy at the shell, \(\omega<-1\), and the total exotic matter \[ \Omega = 2\pi a\,\sigma(a) \] decreases when \(M\) increases or \(\theta\) decreases [1501.07161]. Thin-shell BTZ wormholes therefore form a separate branch of the subject: they are not holographic traversable wormholes opened by negative null energy on a pre-existing Einstein–Rosen bridge, but geodesically complete manifolds created by identifying exterior BTZ regions across a timelike shell. ## 5. Regularized, NLED, and modified-gravity BTZ wormholes The black-bounce construction replaces the BTZ areal radius by \(\sqrt{r^2+a^2}\) and uses \[ ds^2 = f(r)\,dt^2 - \frac{dr^2}{f(r)} - \big(r^2 + a^2\big)\,d\phi^2. \] For the uncharged case, \[ f(r) = -\,M + \frac{r^2 + a^2}{\ell^2}, \] so the throat sits at \(r=0\) with areal radius \(R_{\rm th}=a\). The parameter \(a\) controls the global structure: \[ a^2<\ell^2M \ \text{black hole},\qquad a^2=\ell^2M \ \text{one-way wormhole},\qquad a^2>\ell^2M \ \text{traversable wormhole}. \] In the wormhole regime the NEC and WEC are violated at the throat, and in the uncharged case there are no stable circular orbits, whereas in the charged case stable massive and massless orbits can appear [2210.06608]. In Palatini Born–Infeld gravity coupled to a static, nonrotating electric field, exact BTZ-type solutions were found for which the wormhole throat is \[ r_{\mathrm{th}}=r_{\min}=\frac{r_c}{\sqrt{\lambda}}. \] Two branches are geodesically complete and nonsingular in the sense of geodesic completeness, even though curvature invariants can diverge at the throat. In the GR limit \(\epsilon\to0\), the throat pinches off and the charged BTZ geometry is recovered. The paper emphasizes that these wormholes arise without exotic matter because the Maxwell field is standard and the effective violations needed for the wormhole come from the metric-affine structure of the gravitational sector [1609.05827]. Einstein gravity coupled to nonlinear electrodynamics yields further rotating BTZ generalizations with black-hole or wormhole interpretation depending on parameter ranges. One exact family is characterized by \(M\), \(J\), \(\Lambda\), and an electromagnetic parameter, reduces to the stationary BTZ black hole when the field is turned off, and admits a traversable asymptotically AdS wormhole interpretation in a regime with \(\hat{J}>1\) and \(\hat{Q}<1\), where the flare-out condition holds and the NEC is violated at or near the throat [2002.00890]. A related Einstein–AdS–NLED family is characterized by five parameters \(\{M,J,\Lambda,q_\alpha,q_\beta\}\), becomes a traversable wormhole when \[ M>0,\qquad \Lambda<0,\qquad M^2+\Lambda J^2>0,\qquad q_\beta^2-J^2 q_\alpha^2<0, \] and reduces to the rotating BTZ black hole when \(q_\alpha=0\). Fine-tuning the cosmological constant to \[ \Lambda_0=-\frac{M^2}{J^2} \] produces an extremal BTZ black hole, which the paper interprets as the boundary between black-hole and wormhole regimes [1810.12111]. These regularized and modified-gravity constructions broaden the meaning of “BTZ wormhole.” Some are horizonless and geodesically complete, some are one-way geometries with a horizon at the throat, and some are rotating NLED solutions that interpolate continuously between BTZ black holes and traversable wormholes. ## 6. Correlators, quantum probes, and entanglement structure Probe observables sharply distinguish BTZ wormholes from BTZ black holes. In the Solodukhin wormhole corresponding to the non-rotating BTZ black hole, the metric is modified to \[ X(r)=\frac{r^2-r_\lambda^2}{R^2},\qquad Y(r)=\frac{r^2-r_+^2}{R^2},\qquad r_\lambda^2=r_+^2(1-\lambda^2), \] so the geometry is horizonless and traversable, with effective length \[ L_\lambda=\frac{R^2}{r_+}\log\!\left(\frac{16}{\lambda^2}\right). \] Its retarded correlator has poles on the real axis instead of BTZ quasinormal poles in the lower half-plane, the normal-mode frequencies satisfy \[ \omega_n\approx\frac{n\pi}{L_\lambda} \quad\text{or}\quad \omega_n\approx\frac{(n\pm\frac12)\pi}{L_\lambda}, \] and late-time behavior is governed by echo recurrences rather than exponential thermal decay. A hybrid WKB treatment reproduces the exact real-time holographic correlator accurately at moderate and high frequencies [2112.08197]. Quantum-harvesting observables probe another aspect of BTZ wormholes. For a null-like BTZ wormhole with Hartle–Hawking vacuum, two static Unruh–DeWitt detectors on opposite sides of the throat have reduced density matrix entries \(\mathcal{L}_{DD}\), \(\mathcal{L}_{AB}\), and \(\mathcal{M}\) determined by image-sum Wightman functions. Under symmetric placement, \(\mathcal{L}_{AA}=\mathcal{L}_{BB}=\mathcal{L}_{DD}\) and \(\mathcal{L}_{AB}=\mathcal{L}_{DD}\), so \[ I_{AB}=2\ln 2\,\mathcal{L}_{DD}. \] The mutual information and concurrence peak at image-symmetric points across the throat, and both vanish as \(u_0\to0\). The same study stresses that correlation harvesting does not imply signal transmission or traversability: the protocol probes vacuum correlations across classically disconnected regions [2510.04005]. From the entanglement-thread perspective, the two-sided planar BTZ black hole is described by thread bundles that either remain within one exterior or cross the wormhole horizon. The horizon-crossing flux between elementary regions is encoded by \[ F_{i\bar{j}}=\frac{c}{6}\ln \left[\frac{\cosh \left(\frac{\pi(a_i + \tilde{a} + a_j)}{\beta}\right)\cosh \left(\frac{\pi \tilde{a}}{\beta}\right)}{\cosh \left(\frac{\pi(a_i + \tilde{a})}{\beta}\right)\cosh \left(\frac{\pi(a_j + \tilde{a})}{\beta}\right)}\right], \] and the corresponding horizon entropy density is \[ \rho=\frac{\pi c}{3\beta_{CFT}}. \] This formulation resolves the von Neumann entropy of a boundary subregion into internal and wormhole-crossing contributions and is argued to imply a “perfect-type” entanglement jointly formed by crossing and internal threads [2508.16977]. Taken together, these probe analyses show that BTZ wormholes are not characterized solely by a throat condition or by a specific metric ansatz. They are equally distinguished by their correlator structure, their information-transfer channel, their scrambling properties, and the pattern by which entanglement is distributed across the two asymptotic regions. Source: https://www.emergentmind.com/topics/btz-wormhole