A new look inside the BTZ black hole

Exploring the BTZ black hole in (2+1)-dimensional gravity took me down a fascinating rabbit hole, connecting ideas I never expected—like black holes and topological phases in quantum matter! When I swapped the roles of space and time in the equations (it felt like turning my map upside down when I was lost in a new city), I discovered an interior version of the solution existing alongside the familiar exterior, each with its own thermofield double state.

What surprised me was how these states seem to communicate, even bridging regions where orientation flips—like walking through a door and suddenly left is right—reminding me of getting turned around on a mountain hike until I saw the landscape from a new perspective.

Digging deeper, I found that the weirdness of black holes with swapped space and time is connected to non-orientable spacetimes and topological invariants, revealing deep ties between gravity and the strange properties of quantum materials that emerge when you flip orientation.

In my recent research published in Physics Letters B, I explored the geometry of the BTZ black hole from a new angle by interchanging the spatial and temporal coordinates. The BTZ (Bañados-Teitelboim-Zanelli) black hole is a fundamental model in lower-dimensional gravity that helps us probe black hole physics, holographic dualities, and aspects of quantum gravity with relative mathematical simplicity.

My key insight was to examine what happens when the usual roles of space and time coordinates in the BTZ metric are swapped, giving rise to a richer geometric and quantum description of this black hole, and offering fresh perspectives on its interior structure, holographic states, and the topology of spacetime itself.

I began by deriving a new BTZ metric where space and time effectively exchange their characteristics. Usually, the BTZ metric clearly delineates time from space, particularly across the event horizon: outside the horizon, time flows as we expect, while spatial dimensions behave conventionally; inside the horizon, however, the temporal and spatial roles switch their roles, with time acting like a spatial coordinate. By explicitly interchanging these coordinates, I constructed a maximal extension of the black hole’s interior.

Remarkably, the line element describing this interior metric closely resembles that of the exterior solution, but with space and time swapped. This surprising symmetry suggested to me a sort of duality between the inside and outside regions of the black hole, providing an innovative way to analyze black hole interiors and their holographic duals. This new interior metric thus opens up novel avenues to probe black hole physics beyond the traditional horizon.

Building on this geometric insight, I proposed to associate two distinct thermofield double (TFD) states with the BTZ black hole. Typically, the TFD state is an entangled quantum state describing an eternal black hole holographically by coupling two copies of a conformal field theory (CFT). This state encapsulates the black hole’s exterior region, which connects two boundaries.

However, by considering the space-time interchange framework, I found that a complete quantum description requires two independent TFD states: one corresponding to the conventional exterior region, and the second encoding the interior region characterized by reversed spatial and temporal roles.

These two TFD states complement one another, collectively encoding the full bulk geometry. This richer dual-TFD structure expands the traditional holographic dictionary and may provide new insights into the quantum microstructure of black holes and the long-standing information puzzle.

Next, I analyzed the partition function that corresponds to the entire BTZ black hole geometry, now viewed as the combination of interior and exterior regions under the coordinate interchange. The partition function is fundamental in quantum statistical mechanics and quantum field theory because it encodes the full thermodynamic and spectral information of the system.

What I discovered is that the resulting partition function describes a non-orientable spacetime—a topology where one cannot consistently assign a global orientation across the manifold. This observation challenges the conventional assumption in gravitational physics that spacetimes are orientable, revealing a profound topological novelty.

Such non-orientable geometries might play an essential role in uncovering new quantum gravitational effects, especially within the enigmatic regime of black hole interiors.

I went further and constructed a thermofield double–like state that mediates between spacetime sectors with reversed space and time orientations. This state functions as a bridge between the two TFD states assigned to the exterior and interior regions, embodying a sort of temporal-spatial duality in the gravitational dual theory.

This construction points to deeper algebraic and geometric structures underlying holographic dualities—namely that black holes cannot be fully described by just one boundary state but instead by interconnected sectors distinguished by orientation reversals in time and space.

This insight underscores the important role of temporal-spatial dualities in gravitational physics and suggests new approaches for describing the quantum relationships among different regions of spacetime historically treated as separate.

Perhaps the most fascinating part of this work is the connection I found between the black hole’s partition function and topological invariants known from condensed matter physics—specifically those that classify many-body topological phases protected by orientation-reversing symmetries. In condensed matter, symmetry-protected topological (SPT) phases represent exotic quantum states robust against local perturbations, distinguished by global topological properties rather than local order parameters.

My findings reveal that the non-orientable spacetime geometry and its partition function link naturally to these topological invariants, suggesting that quantum states of black holes might be understood through the mathematical frameworks developed for topological quantum matter.

This interdisciplinary bridge opens a promising path for integrating ideas from quantum gravity, holography, and condensed matter physics, hinting that black hole interiors share striking similarities with symmetry-protected topological phases.

These insights enrich our conceptual tools for approaching quantum gravity, holography, black hole interiors, and the interplay of topology with quantum information. They also inspire future directions across quantum field theory, gravitational physics, and topological quantum matter—bringing us closer to a more cohesive understanding of the quantum nature of spacetime.

This story is part of Science X Dialog, where researchers can report findings from their published research articles. Visit this page for information about Science X Dialog and how to participate.

Ovidiu Racorean is a physicist specializing in quantum physics and holography, with a focus on black hole geometry and quantum information. The research explores the intersections of quantum mechanics, gravitational physics, and condensed matter concepts. Racorean has worked extensively on topics such as the quantum mechanics of financial markets and the relationship between thermodynamic arrows of time and quantum entanglement in holographic frameworks. Based in Bucharest, Romania, Racorean actively contributes to academic research through publications, investigating foundational questions in quantum gravity, black hole dualities, and temporal-spatial dualities in gravity.