Overspinning BTZ Geometries in AdS3 Gravity

• Overspinning BTZ geometries are configurations where angular momentum exceeds mass, leading to naked singularity states distinct from regular black holes.
• They are constructed via mixed boost-rotation identifications in AdS3 and exhibit integrable geodesic equations without closed timelike curves in the physical domain.
• Perturbative analyses show that modified absorption bounds, including induced horizon angular velocity, can restore cosmic censorship by blocking overspinning transitions.

Overspinning BTZ geometries are members of the Bañados–Teitelboim–Zanelli family in $2+1$-dimensional anti–de Sitter gravity for which angular momentum dominates over mass strongly enough to violate the black-hole extremality bound. In the standard BTZ parameterization this is the regime

$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$

which lies outside the ordinary black-hole sector and yields a geometry with no event horizon. The topic occupies a junction between classical global geometry, weak cosmic censorship thought experiments, semiclassical backreaction, and Euclidean path-integral questions. Across these settings, overspinning BTZ configurations have been described as naked singularities, smooth quotients, inadmissible complex saddles, and candidate states in AdS$_3$ quantum gravity, depending on which structure is being probed (Düztaş, 2017, Briceño et al., 2021).

1. BTZ family, extremality, and the overspinning regime

The rotating BTZ metric can be written as

$$ds^2=-N^2dt^2+N^{-2}dr^2+r^2\left(N^\phi dt+d\phi\right)^2,$$

with

$$N^2=-M+\frac{r^2}{\ell^2}+\frac{J^2}{4r^2}, \qquad N^\phi=-\frac{J}{2r^2},$$

where $M$ and $J$ are the mass and angular momentum parameters and $\Lambda=-1/\ell^2$ (Düztaş, 2017). The horizons are located by $N^2=0$, equivalently

$$r^2=\frac{M\ell^2}{2}\left(1\pm\sqrt{1-\frac{J^2}{M^2\ell^2}}\right),$$

so a BTZ black hole exists only if

$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$0

with extremality at

$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$1

Extending the same local solution beyond that bound produces naked-singularity sectors. A standard partition of parameter space is the following (Briceño et al., 2024, Briceño et al., 2021):

Sector	Condition	Characterization
Black hole	$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$2	Horizons present
Extremal boundary	$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$3	Degenerate horizon
Overspinning	$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$4	No horizon, naked singularity
Conical singularity	$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$5	Spinning conical singularity

A closely related notation introduces the dimensionless spin parameter

$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$6

so that subextremal, extremal, and overextremal configurations correspond to $$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$7, $$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$8, and $$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$9, respectively (Düztaş, 2024). In overspinning BTZ geometries the roots of $_3$0 are complex rather than real, so there is no ordinary horizon structure (Briceño et al., 2021). The geometry remains locally AdS$_3$1 away from the center, but its global identification data place it outside the black-hole regime.

2. Lorentzian geometry, quotient structure, and geodesics

The overspinning sector can be constructed as a quotient of the universal cover of AdS$_3$2 by a mixed boost-rotation identification. In the embedding-space description used for the BTZ family, the overspinning case is generated by a mixed boost-rotation Killing vector and corresponds to the type $_3$3 sector in the BTZ classification (Briceño et al., 2021). One patch covers the whole overspinning spacetime, and within the physical coordinate range

$_3$4

the geometry is treated as a classical spacetime with a central boundary at $_3$5.

A central result of the geodesic analysis is that, for the physically relevant domain $_3$6, the overspinning geometry has no closed timelike curves. In ADM variables,

$_3$7

and in the overspinning regime $_3$8 everywhere on that domain. The causal argument given in the literature is that a causal curve returning to the same $_3$9 with $$ds^2=-N^2dt^2+N^{-2}dr^2+r^2\left(N^\phi dt+d\phi\right)^2,$$0 returning to its initial value would require $$ds^2=-N^2dt^2+N^{-2}dr^2+r^2\left(N^\phi dt+d\phi\right)^2,$$1 somewhere; the causal inequality then forces $$ds^2=-N^2dt^2+N^{-2}dr^2+r^2\left(N^\phi dt+d\phi\right)^2,$$2, yielding a contradiction. On that basis, claims that overspinning BTZ necessarily contains closed timelike curves are explicitly rejected for the physical manifold $$ds^2=-N^2dt^2+N^{-2}dr^2+r^2\left(N^\phi dt+d\phi\right)^2,$$3 (Briceño et al., 2021).

The geodesic equations are completely integrable in elementary functions. Writing

$$ds^2=-N^2dt^2+N^{-2}dr^2+r^2\left(N^\phi dt+d\phi\right)^2,$$4

the first integrals are

$$ds^2=-N^2dt^2+N^{-2}dr^2+r^2\left(N^\phi dt+d\phi\right)^2,$$5

and

$$ds^2=-N^2dt^2+N^{-2}dr^2+r^2\left(N^\phi dt+d\phi\right)^2,$$6

Because $$ds^2=-N^2dt^2+N^{-2}dr^2+r^2\left(N^\phi dt+d\phi\right)^2,$$7 is positive definite for $$ds^2=-N^2dt^2+N^{-2}dr^2+r^2\left(N^\phi dt+d\phi\right)^2,$$8 in the overspinning regime, its radial behavior differs sharply from both black holes and conical singularities. Null and spacelike geodesics may reach infinity after turning around a closest approach to the singularity, may extend from the singularity to infinity, or may terminate at the singularity after a finite radial excursion. Timelike geodesics never reach infinity; they either remain bounded between turning points or fall into the singularity. Null circular geodesics do not exist in the overspinning regime, while timelike circular geodesics can occur, including special closed or self-intersecting families determined by rationality conditions on $$ds^2=-N^2dt^2+N^{-2}dr^2+r^2\left(N^\phi dt+d\phi\right)^2,$$9 (Briceño et al., 2021).

3. Overspinning as a weak-cosmic-censorship thought experiment

A major line of work asks whether an initially subextremal BTZ black hole can be driven into the overextremal regime by throwing in particles or fields. The analysis parallel to the Kerr literature starts from a nearly extremal black hole parameterized by

$$N^2=-M+\frac{r^2}{\ell^2}+\frac{J^2}{4r^2}, \qquad N^\phi=-\frac{J}{2r^2},$$0

For test particles, the future-directedness condition outside the horizon yields

$$N^2=-M+\frac{r^2}{\ell^2}+\frac{J^2}{4r^2}, \qquad N^\phi=-\frac{J}{2r^2},$$1

which at the horizon becomes

$$N^2=-M+\frac{r^2}{\ell^2}+\frac{J^2}{4r^2}, \qquad N^\phi=-\frac{J}{2r^2},$$2

Rocha and Cardoso used the BTZ horizon relation to obtain, for an extremal hole,

$$N^2=-M+\frac{r^2}{\ell^2}+\frac{J^2}{4r^2}, \qquad N^\phi=-\frac{J}{2r^2},$$3

If an initially extremal black hole absorbs a test particle, the final spin parameter is

$$N^2=-M+\frac{r^2}{\ell^2}+\frac{J^2}{4r^2}, \qquad N^\phi=-\frac{J}{2r^2},$$4

and the test-particle expansion gives

$$N^2=-M+\frac{r^2}{\ell^2}+\frac{J^2}{4r^2}, \qquad N^\phi=-\frac{J}{2r^2},$$5

With $$N^2=-M+\frac{r^2}{\ell^2}+\frac{J^2}{4r^2}, \qquad N^\phi=-\frac{J}{2r^2},$$6 and $$N^2=-M+\frac{r^2}{\ell^2}+\frac{J^2}{4r^2}, \qquad N^\phi=-\frac{J}{2r^2},$$7, one obtains $$N^2=-M+\frac{r^2}{\ell^2}+\frac{J^2}{4r^2}, \qquad N^\phi=-\frac{J}{2r^2},$$8, so an extremal BTZ black hole cannot be overspun by a test particle (Düztaş, 2017).

For nearly extremal initial data, the situation changes. The horizon-capture condition becomes, to second order,

$$N^2=-M+\frac{r^2}{\ell^2}+\frac{J^2}{4r^2}, \qquad N^\phi=-\frac{J}{2r^2},$$9

and the post-absorption spin expands as

$M$0

If one chooses

$M$1

while also taking

$M$2

then $M$3, so the BTZ hole is overspun. In the test-particle approximation, nearly extremal BTZ black holes therefore admit an overspinning window that is absent in the strictly extremal case (Düztaş, 2017).

The field-theoretic version uses a mode

$M$4

with

$M$5

Overspinning requires

$M$6

which, for $M$7, becomes

$M$8

The horizon angular velocity is

$M$9

so the superradiant threshold is

$J$0

For a nearly extremal BTZ black hole one has $J$1. If the field is bosonic and superradiance is present, overspinning is possible only in the narrow range

$J$2

If there is no superradiance, the lower cutoff is absent and the range extends to

$J$3

Because $J$4, the ratio of angular momentum to energy diverges as $J$5, so overspinning becomes generic in that case and extends even to extremal BTZ black holes. The paper explicitly presents this as the BTZ analogue of the Kerr story, including the contrast between bosonic superradiant protection and non-superradiant, neutrino-like behavior (Düztaş, 2017).

4. Modified absorption bounds and the induced horizon angular velocity

A later development revisits these thought experiments by modifying the absorption criterion itself. For perturbations obeying the null energy condition, Needham’s absorption condition in the neutral case is

$J$6

or, for a test body,

$J$7

Using only this uncorrected criterion, nearly extremal BTZ black holes can be overspun. In the notation

$J$8

the dangerous range for test bodies is

$J$9

while for scalar fields

$\Lambda=-1/\ell^2$0

the overspinning window is

$\Lambda=-1/\ell^2$1

(Düztaş, 2024).

The central claim of the 2024 analysis is that the perturbation induces an increase in the horizon angular velocity before absorption, following the idea of Will’s classic analysis. One must therefore replace $\Lambda=-1/\ell^2$2 by

$\Lambda=-1/\ell^2$3

and strengthen the absorption condition to

$\Lambda=-1/\ell^2$4

For test bodies, $\Lambda=-1/\ell^2$5 is found to be large enough that the modified absorption condition is saturated at the lower edge of the previous overspinning window; perturbations with larger $\Lambda=-1/\ell^2$6 then fail the inequality and are reflected rather than absorbed. For scalar fields, the threshold becomes

$\Lambda=-1/\ell^2$7

and the upper end of the previously dangerous frequency band lies below this modified threshold, so the entire overspinning window is excluded (Düztaş, 2024).

This mechanism is presented as distinct from self-energy and distinct from gravitational radiation. Its significance lies in the claim that cosmic censorship can be restored in BTZ spacetime without invoking mechanisms that are problematic or unavailable in $\Lambda=-1/\ell^2$8 dimensions. In this formulation, nearly extremal BTZ black holes can be overspun only if one neglects the induced increase in horizon angular velocity; once that effect is included, both test bodies and scalar fields are blocked from crossing the horizon.

5. Complex saddles and semiclassical backreaction

Overspinning BTZ geometries also arise in Euclidean and semiclassical analyses that do not begin from a censorship thought experiment. Under the quasi-Euclidean continuation

$\Lambda=-1/\ell^2$9

with $N^2=0$0 held fixed, the metric becomes

$N^2=0$1

In the overspinning regime $N^2=0$2, one of $N^2=0$3 is real and the other imaginary, so $N^2=0$4 are complex conjugates. The smoothness analysis shows that the overspinning quotient is always smooth on the real slice; it is not ruled out by ordinary geometric regularity. However, Witten’s admissibility criterion imposes additional positivity conditions. For the BTZ quasi-Euclidean metric these reduce to

$N^2=0$5

and

$N^2=0$6

In region 3, the overspinning region, condition $N^2=0$7 always fails somewhere along the real $N^2=0$8-interval, so smooth overspinning quasi-Euclidean BTZ geometries are never admissible as complex saddles. The regularized on-shell action then contributes only black-hole and thermal AdS-type saddles, not overspinning geometries (Basile et al., 2023).

A distinct semiclassical question concerns Lorentzian quantum backreaction from a conformally coupled scalar field. In the overspinning sector $N^2=0$9, the renormalized stress-energy tensor is constructed by the method of images on the quotient spacetime, but the perturbative semiclassical program fails. Because the rotational parameter must be rational, there are infinitely many image numbers for which the relevant image separation becomes independent of $$r^2=\frac{M\ell^2}{2}\left(1\pm\sqrt{1-\frac{J^2}{M^2\ell^2}}\right),$$0, generating infinitely many additional divergent image contributions. The result is that the renormalized stress tensor is non-renormalizable in this approach, and the first-order corrections to the stationary axisymmetric ansatz become unbounded. The perturbative semiclassical solution therefore does not exist. This contrasts with earlier conical-naked-singularity analyses, where a finite RSET and a Planck-scale dressing horizon could emerge (Baake et al., 2023).

Taken together, these results sharply separate two questions. Smoothness of a complex or quotient geometry does not ensure admissibility as a gravitational saddle, and the existence of a classical overspinning background does not guarantee a controlled semiclassical expansion around it.

6. Singularity, holonomy, and holographic reinterpretations

The status of the overspinning geometry depends strongly on which diagnostic is used. In the holonomy-based analysis of the full BTZ family, the geometry is locally AdS$$r^2=\frac{M\ell^2}{2}\left(1\pm\sqrt{1-\frac{J^2}{M^2\ell^2}}\right),$$1 away from the origin: $$r^2=\frac{M\ell^2}{2}\left(1\pm\sqrt{1-\frac{J^2}{M^2\ell^2}}\right),$$2 yet generic loop holonomies around $$r^2=\frac{M\ell^2}{2}\left(1\pm\sqrt{1-\frac{J^2}{M^2\ell^2}}\right),$$3 are nontrivial. In the zero-radius limit this signals a delta-like source at the origin in curvature and/or torsion, schematically

$$r^2=\frac{M\ell^2}{2}\left(1\pm\sqrt{1-\frac{J^2}{M^2\ell^2}}\right),$$4

For overspinning geometries in particular, the paper states that the origin is still singular, that the local geometry near $$r^2=\frac{M\ell^2}{2}\left(1\pm\sqrt{1-\frac{J^2}{M^2\ell^2}}\right),$$5 is not regular in the sense of having a smooth tangent space there, and that only pure AdS$$r^2=\frac{M\ell^2}{2}\left(1\pm\sqrt{1-\frac{J^2}{M^2\ell^2}}\right),$$6,

$$r^2=\frac{M\ell^2}{2}\left(1\pm\sqrt{1-\frac{J^2}{M^2\ell^2}}\right),$$7

is genuinely nonsingular at the center. Even the special BPS-like cases

$$r^2=\frac{M\ell^2}{2}\left(1\pm\sqrt{1-\frac{J^2}{M^2\ell^2}}\right),$$8

for which the AdS$$r^2=\frac{M\ell^2}{2}\left(1\pm\sqrt{1-\frac{J^2}{M^2\ell^2}}\right),$$9 Wilson loop can reduce to the identity, do not generically eliminate the central singularity (Briceño et al., 2024).

A different perspective appears in recent work on the AdS$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$00 gravitational path integral and the Maloney–Witten–Keller negative-density problem. There, certain overspinning states above the black-hole threshold are identified not with classical spinning strings but with overspinning BTZ geometries understood as smooth pure-gravity quotients of AdS$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$01 with no fixed points. These overspinning geometries arise from mixed elliptic-hyperbolic identifications,

$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$02

and are said to contain no delta-function source or curvature singularity on the real section. At the same time, their Lorentzian continuations exhibit causal pathologies, and the backgrounds possess a right-moving temperature and right-moving quasinormal modes. One representative spectral choice is

$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$03

which gives

$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$04

These states lie above the BTZ threshold while still satisfying $$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$05, so they are overspinning and preserve the spectral gap (Li, 16 Apr 2026).

This suggests that the conceptual status of overspinning BTZ geometries is framework-dependent. In Lorentzian BTZ-family analyses they appear as naked, quasi-regular central defects; in the complex-metric path integral they are smooth but inadmissible saddles; in semiclassical backreaction they obstruct the perturbative program; and in recent spectral constructions they re-emerge as smooth pure-gravity quotients with problematic Lorentzian continuations. The unifying feature across these approaches is that overspinning lies precisely at the boundary where BTZ geometry stops behaving like an ordinary black hole while remaining locally AdS$$-|J|<M\ell<|J| \qquad\Longleftrightarrow\qquad J^2>M^2\ell^2,$$06.