BTZ Wormhole: Geometries & Applications

• BTZ wormhole is a 2+1D geometric construction derived from BTZ black hole backgrounds, featuring both non-traversable Einstein–Rosen bridges and traversable models.
• Key methods include holographic double-trace deformations, thin-shell surgery, and modified gravity frameworks that illuminate black hole thermodynamics and stability.
• Quantum probe analyses reveal distinct correlator structures, entanglement patterns, and information-transfer properties that differentiate BTZ wormholes from standard BTZ black holes.

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 (Hirano et al., 2019, Harvey et al., 2023, Tsukamoto et al., 2018, Furtado et al., 2022, Bazeia et al., 2016).

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 $R$0 (Hirano et al., 2019).

In the rotating case, the BTZ line element can be written

$R$1

with

$R$2

and horizon radii

$R$3

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 (Tsukamoto et al., 2018).

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 (Hirano et al., 2019).

A closely related construction rewrites the static BTZ black hole using

$R$4

which yields the “null-like wormhole spacetime”

$R$5

with $R$6, $R$7, and two congruent universes connected at $R$8. In this geometry the radial null geodesic speed at the throat vanishes,

$R$9

so classical traversability is obstructed, although quantum-field correlations can still be shared across the bridge (Liu et al., 5 Oct 2025).

A different Einstein–Rosen BTZ wormhole construction introduces

$_3$0

and obtains

$_3$1

Here the throat sits at $_3$2, where $_3$3 and $_3$4, so the throat coincides with a horizon. The same work describes this as a one-way traversable wormhole, derives the Hawking temperature

$_3$5

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 (Anand et al., 2024).

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 $_3$6, the deformation produces a null shock on $_3$7,

$_3$8

and choosing $_3$9 gives negative null energy and opens a traversable window

$\ell$0

for a message sent as a positive-energy shock at $\ell$1. Exact null-shell patching yields

$\ell$2

$\ell$3

so the message backreaction partially closes the wormhole. Combining this geometry with an uncertainty-principle estimate gives

$\ell$4

namely an $\ell$5 information-transfer capacity, and the same analysis finds delayed scrambling,

$\ell$6

which supports the claim that traversable wormholes act as fast decoders (Hirano et al., 2019).

A distinct eternal construction in three dimensions couples a large number of light matter fields dual to relevant operators through an eternal, negative double-trace interaction. The resulting stationary bulk geometry is

$\ell$7

with throat radius $\ell$8, no horizons, and a global time coordinate. In the associated boundary-graviton effective theory the fluctuation spectrum contains masses $\ell$9, and perturbative stability requires

$$ds^2 = -\frac{r^2 - R^2}{\ell^2}\,d\tau^2 + \frac{\ell^2}{r^2 - R^2}\,dr^2 + r^2\,d\phi^2, \qquad \phi\sim\phi+2\pi,$$0

This construction is presented as the AdS$$ds^2 = -\frac{r^2 - R^2}{\ell^2}\,d\tau^2 + \frac{\ell^2}{r^2 - R^2}\,dr^2 + r^2\,d\phi^2, \qquad \phi\sim\phi+2\pi,$$1 analogue of the Maldacena–Qi eternal traversable wormhole in AdS$$ds^2 = -\frac{r^2 - R^2}{\ell^2}\,d\tau^2 + \frac{\ell^2}{r^2 - R^2}\,dr^2 + r^2\,d\phi^2, \qquad \phi\sim\phi+2\pi,$$2 (Harvey et al., 2023).

A more extreme holographic mechanism is provided by a two-sided $$ds^2 = -\frac{r^2 - R^2}{\ell^2}\,d\tau^2 + \frac{\ell^2}{r^2 - R^2}\,dr^2 + r^2\,d\phi^2, \qquad \phi\sim\phi+2\pi,$$3-type deformation that couples the left and right boundary stress tensors. In this case the deformed geometry is locally BTZ in each wedge but globally glues the right and left BTZ wedges along the horizons. For $$ds^2 = -\frac{r^2 - R^2}{\ell^2}\,d\tau^2 + \frac{\ell^2}{r^2 - R^2}\,dr^2 + r^2\,d\phi^2, \qquad \phi\sim\phi+2\pi,$$4, the Kruskal map collapses $$ds^2 = -\frac{r^2 - R^2}{\ell^2}\,d\tau^2 + \frac{\ell^2}{r^2 - R^2}\,dr^2 + r^2\,d\phi^2, \qquad \phi\sim\phi+2\pi,$$5 to a single point, the geometry is realized by uniform shock waves emanating from both boundaries, and the interior has effectively zero length, so every null geodesic launched from one boundary reaches the other (Bzowski, 2020). A related AdS$$ds^2 = -\frac{r^2 - R^2}{\ell^2}\,d\tau^2 + \frac{\ell^2}{r^2 - R^2}\,dr^2 + r^2\,d\phi^2, \qquad \phi\sim\phi+2\pi,$$6 analysis compares two complementary CFT constructions—Janus deformation with imaginary parameter and non-local double-trace, including non-local $$ds^2 = -\frac{r^2 - R^2}{\ell^2}\,d\tau^2 + \frac{\ell^2}{r^2 - R^2}\,dr^2 + r^2\,d\phi^2, \qquad \phi\sim\phi+2\pi,$$7—and emphasizes a sharp distinction: a global time slice of the wormhole is pure in the Janus case but mixed in the double-trace case, while in BTZ the cross-boundary geodesic can become null at finite time, signaling traversability (Kawamoto et al., 5 Feb 2025).

4. Thin-shell and cut-and-paste BTZ wormholes

One major use of the BTZ background is thin-shell surgery. In the rotating construction based on the Darmois–Israel method, two BTZ manifolds are joined at a timelike throat $$ds^2 = -\frac{r^2 - R^2}{\ell^2}\,d\tau^2 + \frac{\ell^2}{r^2 - R^2}\,dr^2 + r^2\,d\phi^2, \qquad \phi\sim\phi+2\pi,$$8. In a corotating frame the second junction condition forces

$$ds^2 = -\frac{r^2 - R^2}{\ell^2}\,d\tau^2 + \frac{\ell^2}{r^2 - R^2}\,dr^2 + r^2\,d\phi^2, \qquad \phi\sim\phi+2\pi,$$9

and the shell stress tensor becomes

$$M = \frac{R^2}{8 G_N \ell^2}, \qquad S = \frac{\pi R}{2 G_N}, \qquad T = \frac{R}{2\pi \ell^2}, \qquad \beta=\frac{2\pi\ell^2}{R}.$$0

with

$$M = \frac{R^2}{8 G_N \ell^2}, \qquad S = \frac{\pi R}{2 G_N}, \qquad T = \frac{R}{2\pi \ell^2}, \qquad \beta=\frac{2\pi\ell^2}{R}.$$1

The dimensionless stability criterion exhibits a critical angular momentum

$$M = \frac{R^2}{8 G_N \ell^2}, \qquad S = \frac{\pi R}{2 G_N}, \qquad T = \frac{R}{2\pi \ell^2}, \qquad \beta=\frac{2\pi\ell^2}{R}.$$2

and for $$M = \frac{R^2}{8 G_N \ell^2}, \qquad S = \frac{\pi R}{2 G_N}, \qquad T = \frac{R}{2\pi \ell^2}, \qquad \beta=\frac{2\pi\ell^2}{R}.$$3 there is a specific radius

$$M = \frac{R^2}{8 G_N \ell^2}, \qquad S = \frac{\pi R}{2 G_N}, \qquad T = \frac{R}{2\pi \ell^2}, \qquad \beta=\frac{2\pi\ell^2}{R}.$$4

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 $$M = \frac{R^2}{8 G_N \ell^2}, \qquad S = \frac{\pi R}{2 G_N}, \qquad T = \frac{R}{2\pi \ell^2}, \qquad \beta=\frac{2\pi\ell^2}{R}.$$5 (Tsukamoto et al., 2018).

A non-rotating thin-shell BTZ wormhole can be constructed from the metric

$$M = \frac{R^2}{8 G_N \ell^2}, \qquad S = \frac{\pi R}{2 G_N}, \qquad T = \frac{R}{2\pi \ell^2}, \qquad \beta=\frac{2\pi\ell^2}{R}.$$6

with throat outside

$$M = \frac{R^2}{8 G_N \ell^2}, \qquad S = \frac{\pi R}{2 G_N}, \qquad T = \frac{R}{2\pi \ell^2}, \qquad \beta=\frac{2\pi\ell^2}{R}.$$7

For a static shell at $$M = \frac{R^2}{8 G_N \ell^2}, \qquad S = \frac{\pi R}{2 G_N}, \qquad T = \frac{R}{2\pi \ell^2}, \qquad \beta=\frac{2\pi\ell^2}{R}.$$8,

$$M = \frac{R^2}{8 G_N \ell^2}, \qquad S = \frac{\pi R}{2 G_N}, \qquad T = \frac{R}{2\pi \ell^2}, \qquad \beta=\frac{2\pi\ell^2}{R}.$$9

and the total amount of exotic matter is

$(U,V,\phi)$0

With a generalized Chaplygin gas on the shell,

$(U,V,\phi)$1

linear stability is obtained for

$(U,V,\phi)$2

while a more general perturbative treatment with $(U,V,\phi)$3 gives the bound

$(U,V,\phi)$4

This work presents the construction as the first example of a stable thin-shell wormhole in $(U,V,\phi)$5 dimensions (Rahaman et al., 2011).

The same cut-and-paste strategy extends to noncommutative BTZ geometry with

$(U,V,\phi)$6

Here the shell matter has $(U,V,\phi)$7, while the paper reports $(U,V,\phi)$8 and $(U,V,\phi)$9. For a dark-energy equation of state $$ds^2 = \frac{-4\ell^2\,dU\,dV + R^2(1-UV)^2\,d\phi^2}{(1+UV)^2},$$0, stable wormholes exist for phantom energy at the shell, $$ds^2 = \frac{-4\ell^2\,dU\,dV + R^2(1-UV)^2\,d\phi^2}{(1+UV)^2},$$1, and the total exotic matter

$$ds^2 = \frac{-4\ell^2\,dU\,dV + R^2(1-UV)^2\,d\phi^2}{(1+UV)^2},$$2

decreases when $$ds^2 = \frac{-4\ell^2\,dU\,dV + R^2(1-UV)^2\,d\phi^2}{(1+UV)^2},$$3 increases or $$ds^2 = \frac{-4\ell^2\,dU\,dV + R^2(1-UV)^2\,d\phi^2}{(1+UV)^2},$$4 decreases (Bhar et al., 2015).

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 $$ds^2 = \frac{-4\ell^2\,dU\,dV + R^2(1-UV)^2\,d\phi^2}{(1+UV)^2},$$5 and uses

$$ds^2 = \frac{-4\ell^2\,dU\,dV + R^2(1-UV)^2\,d\phi^2}{(1+UV)^2},$$6

For the uncharged case,

$$ds^2 = \frac{-4\ell^2\,dU\,dV + R^2(1-UV)^2\,d\phi^2}{(1+UV)^2},$$7

so the throat sits at $$ds^2 = \frac{-4\ell^2\,dU\,dV + R^2(1-UV)^2\,d\phi^2}{(1+UV)^2},$$8 with areal radius $$ds^2 = \frac{-4\ell^2\,dU\,dV + R^2(1-UV)^2\,d\phi^2}{(1+UV)^2},$$9. The parameter $UV=-1$0 controls the global structure: $UV=-1$1 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 (Furtado et al., 2022).

In Palatini Born–Infeld gravity coupled to a static, nonrotating electric field, exact BTZ-type solutions were found for which the wormhole throat is

$UV=-1$2

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 $UV=-1$3, 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 (Bazeia et al., 2016).

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 $UV=-1$4, $UV=-1$5, $UV=-1$6, 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 $UV=-1$7 and $UV=-1$8, where the flare-out condition holds and the NEC is violated at or near the throat (Cañate et al., 2020). A related Einstein–AdS–NLED family is characterized by five parameters $UV=-1$9, becomes a traversable wormhole when

$U=0$0

and reduces to the rotating BTZ black hole when $U=0$1. Fine-tuning the cosmological constant to

$U=0$2

produces an extremal BTZ black hole, which the paper interprets as the boundary between black-hole and wormhole regimes (Cañate et al., 2018).

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

$U=0$3

so the geometry is horizonless and traversable, with effective length

$U=0$4

Its retarded correlator has poles on the real axis instead of BTZ quasinormal poles in the lower half-plane, the normal-mode frequencies satisfy

$U=0$5

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 (Dimitrov et al., 2021).

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 $U=0$6, $U=0$7, and $U=0$8 determined by image-sum Wightman functions. Under symmetric placement, $U=0$9 and $R$00, so

$R$01

The mutual information and concurrence peak at image-symmetric points across the throat, and both vanish as $R$02. The same study stresses that correlation harvesting does not imply signal transmission or traversability: the protocol probes vacuum correlations across classically disconnected regions (Liu et al., 5 Oct 2025).

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

$R$03

and the corresponding horizon entropy density is

$R$04

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 (Lin et al., 23 Aug 2025).

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.