The idea was elegant: take a BTZ black hole (the three-dimensional analog of a regular black hole) and smooth its horizon with a special function. The hope was that inside, the spacetime signature might change — time becoming space, and space becoming time. But the calculations showed otherwise.
Instead of a signature change, the geometry turned into a 'black trampoline' (black hole with a traversable throat). If you switch to the coordinates of an infalling observer, spacetime continues beyond the horizon: there lies a second region, an exact copy of the exterior. The minimal throat radius (the narrowest point) sits at the horizon, and the singularity disappears. This is not curvature leading nowhere, but a bridge between two worlds.
The authors computed the effective matter source that supports such a geometry and proved that all energy conditions are satisfied. Near the throat, spacetime becomes AdS₂ × S¹ — the product of two-dimensional anti-de Sitter space and a circle. Scalar perturbations are stable, but there is a peculiar Aretakis-type instability: the leading transverse derivative is conserved, while the next one grows linearly with time.
The work did not confirm the signature-change hypothesis, but it provided an exact recipe for constructing a traversable wormhole from a BTZ black hole. This is important for understanding quantum gravity and the thermodynamics of horizons.
as written by the authors arXiv:2608.04461v1
We study a static, circularly symmetric deformation of the non-rotating BTZ black hole obtained by inserting a smooth transition function into the \emph{inverse} radial metric component, $$g^{rr}=S_\delta(r)F(r)$$ with $$S_\delta=\tanh[(r-r_h)/\delta]$$, leaving $$g_{tt}=-F$$ untouched. This was motivated by the proposal that such a construction realizes a Lorentzian-to-Riemannian signature change at the horizon; we show that it does not. In coordinates $$r-r_h=q^2$$ with an advanced time, the metric extends real-analytically across $$r=r_h$$, and the extension is Lorentzian: $$q=0$$ is a regular null hypersurface, a degenerate Killing horizon with vanishing surface gravity, beyond which lies a second, isometric copy of the exterior. The areal radius has a minimum there, so the geometry is a black bounce; the would-be Riemannian branch is a separate geometry the Lorentzian sector never reaches. We give the effective source in closed form, an invariant account of the energy conditions, and identify the near-throat geometry as AdS$$_2\times S^1$$. The scalar effective potential is proven strictly positive for every mode, and the throat circle is a minimal surface whose length gives an entropy $$\pi r_h/2G$$, reproduced independently by the Wald--Noether charge and by a Cardy estimate from the computed Brown--York mass -- concordant results for which no first law is available since $$\kappa=0$$. The throat carries an Aretakis-type instability, with a conserved leading transverse derivative and a linearly growing subleading one. We also record a negative result: smoothing $$g_{tt}$$ instead, as in the Lorentzian-Euclidean Schwarzschild proposal, is singular at the horizon for any finite smoothing width. We state explicitly what the construction does not establish.
The entropy of the throat of a BTZ black hole is proportional to the circumference of the throat, not the area, as in the four-dimensional case. In three dimensions, the horizon is a circle, not a sphere.
The idea of wormholes connecting different regions of space is familiar from the movie 'Interstellar', where heroes travel through a wormhole in search of a new home for humanity.