Double Wick-Rotated BTZ Black Hole

• The double Wick-rotated BTZ black hole is a geometry obtained by exchanging Euclidean time and angular directions in rotating BTZ, yielding a smooth Riemannian manifold.
• It preserves key thermodynamic and holographic properties such as energy, entropy, and two-point functions, while maintaining a local AdS3 structure via quotienting hyperbolic space.
• Its Lorentzian interpretation is subtle, exhibiting closed timelike curves and emphasizing that it behaves as an alternative torus filling rather than a standard black hole.

The double Wick-rotated BTZ black hole is a geometry obtained from rotating BTZ by exchanging Euclidean time and the angular direction, together with an exchange of the BTZ parameters. In the 2025 Euclidean treatment, it is analyzed entirely in Euclidean signature as a smooth Riemannian manifold rather than as a Lorentzian black hole geometry, and its central property is that, once the relevant periodicities are matched to those of Euclidean rotating BTZ, the thermodynamics, total energy, holographic stress tensor, and geodesic two-point functions agree with those of rotating BTZ (Dai et al., 14 Apr 2025). Later work places the same construction in a broader quotient and modular framework, emphasizing that the geometry is locally $AdS_3$, that its observables can be mapped to ordinary rotating BTZ after exchange of cycles and parameters, and that its Lorentzian interpretation is subtler than the term “black hole” suggests (Dai et al., 17 Apr 2026).

1. Euclidean construction from rotating BTZ

The starting point is the Euclidean rotating BTZ black hole. In the conventions of the Euclidean analysis, the action is

$$I_E[g] = \frac{1}{16\pi G_3}\int d^3x\,\sqrt{g}\,(R-2\Lambda) -\frac{1}{8\pi G_3}\int d^2x\,\sqrt{h}\,(K-k), \qquad \Lambda=-\frac{1}{L^2},$$

with the standard counterterm $k=1/L$. Euclidean rotating BTZ is obtained from Lorentzian rotating BTZ by

$t\to -iT_E,\qquad J\to -iJ_E,$

equivalently $u_- \to -i\,u_-$, which is the continuation required to obtain a positive-definite metric (Dai et al., 14 Apr 2025).

In these conventions, the Euclidean BTZ metric is

$$ds^2 = L^2\left[ u^2\left(dx-\frac{u_+u_-}{u^2}\,dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_+^2)(u^2+u_-^2)} +\frac{(u^2+u_-^2)(u^2-u_+^2)}{u^2}\,dT_E^2 \right],$$

with

$$x\sim x+2\pi, \qquad 8G_3 M=u_+^2-u_-^2, \qquad J_E=-\,\frac{L u_+u_-}{4G_3}.$$

The double Wick-rotated geometry is then defined by

$$T_E \leftrightarrow x, \qquad u_+ \leftrightarrow u_-.$$

This yields

$$ds^2_{\rm DW} = L^2\left[ u^2\left(\frac{u_+u_-}{u^2}\,dx-dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_-^2)(u^2+u_+^2)} +\frac{(u^2-u_-^2)(u^2+u_+^2)}{u^2}\,dx^2 \right],$$

again with $x\sim x+2\pi$. The metric remains a solution of the same Einstein equations, but the Euclidean analysis stresses that it is positive definite and should be viewed as a Riemannian manifold. Although it inherits a degeneration locus at $$I_E[g] = \frac{1}{16\pi G_3}\int d^3x\,\sqrt{g}\,(R-2\Lambda) -\frac{1}{8\pi G_3}\int d^2x\,\sqrt{h}\,(K-k), \qquad \Lambda=-\frac{1}{L^2},$$0, the same work explicitly remarks that it “is not a black hole but rather a global spacetime with angular momentum in the Lorentzian frame” (Dai et al., 14 Apr 2025).

A complementary formulation describes double Wick rotation as an exchange of Euclidean space and Euclidean time directions, $$I_E[g] = \frac{1}{16\pi G_3}\int d^3x\,\sqrt{g}\,(R-2\Lambda) -\frac{1}{8\pi G_3}\int d^2x\,\sqrt{h}\,(K-k), \qquad \Lambda=-\frac{1}{L^2},$$1, or on the boundary $$I_E[g] = \frac{1}{16\pi G_3}\int d^3x\,\sqrt{g}\,(R-2\Lambda) -\frac{1}{8\pi G_3}\int d^2x\,\sqrt{h}\,(K-k), \qquad \Lambda=-\frac{1}{L^2},$$2, with the result that the natural boundary object becomes a transition matrix rather than an ordinary thermal density matrix (Dai et al., 17 Apr 2026).

2. Periodicities, regularity, and thermodynamic parameters

For Euclidean rotating BTZ, regularity at the Euclidean horizon $$I_E[g] = \frac{1}{16\pi G_3}\int d^3x\,\sqrt{g}\,(R-2\Lambda) -\frac{1}{8\pi G_3}\int d^2x\,\sqrt{h}\,(K-k), \qquad \Lambda=-\frac{1}{L^2},$$3 imposes the thermal and rotational identification

$$I_E[g] = \frac{1}{16\pi G_3}\int d^3x\,\sqrt{g}\,(R-2\Lambda) -\frac{1}{8\pi G_3}\int d^2x\,\sqrt{h}\,(K-k), \qquad \Lambda=-\frac{1}{L^2},$$4

so that

$$I_E[g] = \frac{1}{16\pi G_3}\int d^3x\,\sqrt{g}\,(R-2\Lambda) -\frac{1}{8\pi G_3}\int d^2x\,\sqrt{h}\,(K-k), \qquad \Lambda=-\frac{1}{L^2},$$5

The Euclidean analysis of the double Wick-rotated geometry finds that smoothness at the degeneration locus $$I_E[g] = \frac{1}{16\pi G_3}\int d^3x\,\sqrt{g}\,(R-2\Lambda) -\frac{1}{8\pi G_3}\int d^2x\,\sqrt{h}\,(K-k), \qquad \Lambda=-\frac{1}{L^2},$$6 imposes exactly the same combined identification,

$$I_E[g] = \frac{1}{16\pi G_3}\int d^3x\,\sqrt{g}\,(R-2\Lambda) -\frac{1}{8\pi G_3}\int d^2x\,\sqrt{h}\,(K-k), \qquad \Lambda=-\frac{1}{L^2},$$7

with the same $$I_E[g] = \frac{1}{16\pi G_3}\int d^3x\,\sqrt{g}\,(R-2\Lambda) -\frac{1}{8\pi G_3}\int d^2x\,\sqrt{h}\,(K-k), \qquad \Lambda=-\frac{1}{L^2},$$8 and $$I_E[g] = \frac{1}{16\pi G_3}\int d^3x\,\sqrt{g}\,(R-2\Lambda) -\frac{1}{8\pi G_3}\int d^2x\,\sqrt{h}\,(K-k), \qquad \Lambda=-\frac{1}{L^2},$$9 (Dai et al., 14 Apr 2025).

This matching of periodicities is the central structural observation. Both Euclidean rotating BTZ and the double Wick-rotated geometry fill the same boundary torus data, and the temperature and angular potential of the double Wick-rotated geometry are therefore

$k=1/L$0

the same as those of Euclidean rotating BTZ once $k=1/L$1 are identified. The 2025 analysis is explicit that these quantities are not derived from a Lorentzian causal horizon in the usual sense, but from Euclidean regularity and thermal identifications (Dai et al., 14 Apr 2025).

A closely related Euclidean formulation writes the original BTZ regularity data as

$k=1/L$2

while the double Wick-rotated Euclidean geometry instead obeys

$k=1/L$3

so that the roles of Euclidean time period and spatial period are exchanged (Fujita et al., 2022).

3. Local $k=1/L$4 structure and quotient interpretation

A key result of the Euclidean treatment is that the double Wick-rotated metric is locally hyperbolic space $k=1/L$5, just like Euclidean BTZ. With

$k=1/L$6

the coordinate transformation

$k=1/L$7

$k=1/L$8

$k=1/L$9

brings the metric to

$t\to -iT_E,\qquad J\to -iJ_E,$0

The geometry is therefore a quotient or filling of Euclidean $t\to -iT_E,\qquad J\to -iJ_E,$1, with $t\to -iT_E,\qquad J\to -iJ_E,$2 inducing a torus-like boundary structure (Dai et al., 14 Apr 2025).

The same work introduces

$t\to -iT_E,\qquad J\to -iJ_E,$3

and states that smoothness at the $t\to -iT_E,\qquad J\to -iJ_E,$4-axis, corresponding to $t\to -iT_E,\qquad J\to -iJ_E,$5, removes a conical singularity and fixes the combined identification of $t\to -iT_E,\qquad J\to -iJ_E,$6. The geometry is thus characterized not by new local curvature data but by a different organization of the quotient cycles.

Later work places this in an explicitly modular framework. The double Wick-rotated BTZ geometry is described as belonging to an $t\to -iT_E,\qquad J\to -iJ_E,$7 family of quotients of $t\to -iT_E,\qquad J\to -iJ_E,$8, related by

$t\to -iT_E,\qquad J\to -iJ_E,$9

with quotient identification

$u_- \to -i\,u_-$0

After analytic continuation and parameter exchange

$u_- \to -i\,u_-$1

followed by

$u_- \to -i\,u_-$2

the metric is rewritten into the ordinary rotating BTZ form. The observable equivalence is summarized as

$u_- \to -i\,u_-$3

which makes the exchange of torus cycles the essential operation (Dai et al., 17 Apr 2026).

4. Thermodynamics, conserved quantities, and central charge

The Euclidean thermodynamic analysis parallels that of rotating BTZ. Using the Brown–York type quasilocal expression

$u_- \to -i\,u_-$4

with reference background $u_- \to -i\,u_-$5, the total energy of Euclidean rotating BTZ is

$u_- \to -i\,u_-$6

Performing the same computation for the double Wick-rotated geometry gives

$u_- \to -i\,u_-$7

so the total energy agrees exactly with rotating BTZ (Dai et al., 14 Apr 2025).

The same paper computes conserved quantities from the holographic stress tensor after analytically continuing back $u_- \to -i\,u_-$8. For Euclidean rotating BTZ one finds

$u_- \to -i\,u_-$9

and for the double Wick-rotated geometry the same method yields

$$ds^2 = L^2\left[ u^2\left(dx-\frac{u_+u_-}{u^2}\,dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_+^2)(u^2+u_-^2)} +\frac{(u^2+u_-^2)(u^2-u_+^2)}{u^2}\,dT_E^2 \right],$$0

The interpretation given there is that the dual CFT sees the same stress tensor data because the two Euclidean geometries have the same boundary periodicity (Dai et al., 14 Apr 2025).

The entropy is taken from the Bekenstein–Hawking formula. For rotating BTZ,

$$ds^2 = L^2\left[ u^2\left(dx-\frac{u_+u_-}{u^2}\,dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_+^2)(u^2+u_-^2)} +\frac{(u^2+u_-^2)(u^2-u_+^2)}{u^2}\,dT_E^2 \right],$$1

and for the double Wick-rotated geometry the final result is again

$$ds^2 = L^2\left[ u^2\left(dx-\frac{u_+u_-}{u^2}\,dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_+^2)(u^2+u_-^2)} +\frac{(u^2+u_-^2)(u^2-u_+^2)}{u^2}\,dT_E^2 \right],$$2

Likewise, the on-shell Euclidean action gives the same free energy in both cases,

$$ds^2 = L^2\left[ u^2\left(dx-\frac{u_+u_-}{u^2}\,dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_+^2)(u^2+u_-^2)} +\frac{(u^2+u_-^2)(u^2-u_+^2)}{u^2}\,dT_E^2 \right],$$3

with

$$ds^2 = L^2\left[ u^2\left(dx-\frac{u_+u_-}{u^2}\,dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_+^2)(u^2+u_-^2)} +\frac{(u^2+u_-^2)(u^2-u_+^2)}{u^2}\,dT_E^2 \right],$$4

The agreement is therefore explicit for $$ds^2 = L^2\left[ u^2\left(dx-\frac{u_+u_-}{u^2}\,dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_+^2)(u^2+u_-^2)} +\frac{(u^2+u_-^2)(u^2-u_+^2)}{u^2}\,dT_E^2 \right],$$5, $$ds^2 = L^2\left[ u^2\left(dx-\frac{u_+u_-}{u^2}\,dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_+^2)(u^2+u_-^2)} +\frac{(u^2+u_-^2)(u^2-u_+^2)}{u^2}\,dT_E^2 \right],$$6, $$ds^2 = L^2\left[ u^2\left(dx-\frac{u_+u_-}{u^2}\,dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_+^2)(u^2+u_-^2)} +\frac{(u^2+u_-^2)(u^2-u_+^2)}{u^2}\,dT_E^2 \right],$$7, $$ds^2 = L^2\left[ u^2\left(dx-\frac{u_+u_-}{u^2}\,dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_+^2)(u^2+u_-^2)} +\frac{(u^2+u_-^2)(u^2-u_+^2)}{u^2}\,dT_E^2 \right],$$8, $$ds^2 = L^2\left[ u^2\left(dx-\frac{u_+u_-}{u^2}\,dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_+^2)(u^2+u_-^2)} +\frac{(u^2+u_-^2)(u^2-u_+^2)}{u^2}\,dT_E^2 \right],$$9, and $$x\sim x+2\pi, \qquad 8G_3 M=u_+^2-u_-^2, \qquad J_E=-\,\frac{L u_+u_-}{4G_3}.$$0, once $$x\sim x+2\pi, \qquad 8G_3 M=u_+^2-u_-^2, \qquad J_E=-\,\frac{L u_+u_-}{4G_3}.$$1 are identified (Dai et al., 14 Apr 2025).

The boundary central charge is unchanged. Using the holographic Weyl anomaly,

$$x\sim x+2\pi, \qquad 8G_3 M=u_+^2-u_-^2, \qquad J_E=-\,\frac{L u_+u_-}{4G_3}.$$2

one finds

$$x\sim x+2\pi, \qquad 8G_3 M=u_+^2-u_-^2, \qquad J_E=-\,\frac{L u_+u_-}{4G_3}.$$3

for both geometries. The Euclidean discussion stresses that $$x\sim x+2\pi, \qquad 8G_3 M=u_+^2-u_-^2, \qquad J_E=-\,\frac{L u_+u_-}{4G_3}.$$4 depends only on $$x\sim x+2\pi, \qquad 8G_3 M=u_+^2-u_-^2, \qquad J_E=-\,\frac{L u_+u_-}{4G_3}.$$5 and $$x\sim x+2\pi, \qquad 8G_3 M=u_+^2-u_-^2, \qquad J_E=-\,\frac{L u_+u_-}{4G_3}.$$6, not on $$x\sim x+2\pi, \qquad 8G_3 M=u_+^2-u_-^2, \qquad J_E=-\,\frac{L u_+u_-}{4G_3}.$$7 or $$x\sim x+2\pi, \qquad 8G_3 M=u_+^2-u_-^2, \qquad J_E=-\,\frac{L u_+u_-}{4G_3}.$$8, so neither Euclidean analytic continuation nor double Wick rotation changes the central charge (Dai et al., 14 Apr 2025).

5. Holographic correlators, entanglement entropy, and transition matrices

The Euclidean analysis computes the holographic two-point function by the geodesic approximation. For Euclidean rotating BTZ, after summing over images from the quotient $$x\sim x+2\pi, \qquad 8G_3 M=u_+^2-u_-^2, \qquad J_E=-\,\frac{L u_+u_-}{4G_3}.$$9, the boundary correlator is

$$T_E \leftrightarrow x, \qquad u_+ \leftrightarrow u_-.$$0

Repeating the same analysis for the double Wick-rotated geometry gives exactly the same large-$$T_E \leftrightarrow x, \qquad u_+ \leftrightarrow u_-.$$1 chordal distance and the same image-summed two-point function,

$$T_E \leftrightarrow x, \qquad u_+ \leftrightarrow u_-.$$2

The equality is exact and is attributed to the common quotient periodicity (Dai et al., 14 Apr 2025).

A Lorentzian entanglement analysis for the double Wick-rotated rotating BTZ geometry uses the HRT prescription and the fact that the bulk is locally $$T_E \leftrightarrow x, \qquad u_+ \leftrightarrow u_-.$$3. For an interval of length $$T_E \leftrightarrow x, \qquad u_+ \leftrightarrow u_-.$$4 at constant Lorentzian time, the entropy is

$$T_E \leftrightarrow x, \qquad u_+ \leftrightarrow u_-.$$5

which factorizes as

$$T_E \leftrightarrow x, \qquad u_+ \leftrightarrow u_-.$$6

with

$$T_E \leftrightarrow x, \qquad u_+ \leftrightarrow u_-.$$7

where

$$T_E \leftrightarrow x, \qquad u_+ \leftrightarrow u_-.$$8

The same work shows agreement with the dual CFT computation obtained from a conformal transformation of twist-operator correlators (Fujita et al., 2022).

The 2026 extension reframes the boundary object after double Wick rotation as a transition matrix,

$$T_E \leftrightarrow x, \qquad u_+ \leftrightarrow u_-.$$9

which can be rewritten in the “usual” form

$$ds^2_{\rm DW} = L^2\left[ u^2\left(\frac{u_+u_-}{u^2}\,dx-dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_-^2)(u^2+u_+^2)} +\frac{(u^2-u_-^2)(u^2+u_+^2)}{u^2}\,dx^2 \right],$$0

after exchanging torus cycles, with an imaginary chemical potential. In this framework, ordinary rotating BTZ entanglement entropy

$$ds^2_{\rm DW} = L^2\left[ u^2\left(\frac{u_+u_-}{u^2}\,dx-dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_-^2)(u^2+u_+^2)} +\frac{(u^2-u_-^2)(u^2+u_+^2)}{u^2}\,dx^2 \right],$$1

maps to the geometric entropy

$$ds^2_{\rm DW} = L^2\left[ u^2\left(\frac{u_+u_-}{u^2}\,dx-dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_-^2)(u^2+u_+^2)} +\frac{(u^2-u_-^2)(u^2+u_+^2)}{u^2}\,dx^2 \right],$$2

The same paper also defines a time-like entanglement entropy and its late-time growth coefficient

$$ds^2_{\rm DW} = L^2\left[ u^2\left(\frac{u_+u_-}{u^2}\,dx-dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_-^2)(u^2+u_+^2)} +\frac{(u^2-u_-^2)(u^2+u_+^2)}{u^2}\,dx^2 \right],$$3

derived from the linear late-time behavior of the analytically continued entropy (Dai et al., 17 Apr 2026).

6. Lorentzian interpretation, causal subtleties, and common misconceptions

The most persistent misconception is to treat the double Wick-rotated BTZ geometry as simply another Lorentzian black hole. The Euclidean 2025 analysis explicitly rejects that interpretation: its thermodynamic quantities are obtained from Euclidean regularity and the filling of a boundary torus, not from an ordinary Lorentzian horizon, and the cleanest interpretation is a smooth Euclidean saddle or alternative filling of the same boundary torus (Dai et al., 14 Apr 2025).

Lorentzian analyses sharpen the caveat. One formulation gives the double Wick-rotated Lorentzian metric

$$ds^2_{\rm DW} = L^2\left[ u^2\left(\frac{u_+u_-}{u^2}\,dx-dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_-^2)(u^2+u_+^2)} +\frac{(u^2-u_-^2)(u^2+u_+^2)}{u^2}\,dx^2 \right],$$4

and states that for real angular momentum the geometry is not an ordinary black hole and contains closed timelike curves; the usual BTZ ergosphere boundary

$$ds^2_{\rm DW} = L^2\left[ u^2\left(\frac{u_+u_-}{u^2}\,dx-dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_-^2)(u^2+u_+^2)} +\frac{(u^2-u_-^2)(u^2+u_+^2)}{u^2}\,dx^2 \right],$$5

is mapped to the boundary of the region with closed timelike curves (Dai et al., 17 Apr 2026).

A separate Lorentzian study states that, after analytic continuation back from Euclidean signature, the geometry has no Hawking temperature, the Euclidean time periodicity becomes effectively infinite, $$ds^2_{\rm DW} = L^2\left[ u^2\left(\frac{u_+u_-}{u^2}\,dx-dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_-^2)(u^2+u_+^2)} +\frac{(u^2-u_-^2)(u^2+u_+^2)}{u^2}\,dx^2 \right],$$6, $$ds^2_{\rm DW} = L^2\left[ u^2\left(\frac{u_+u_-}{u^2}\,dx-dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_-^2)(u^2+u_+^2)} +\frac{(u^2-u_-^2)(u^2+u_+^2)}{u^2}\,dx^2 \right],$$7 surfaces become timelike, Bekenstein entropy is not naturally defined, and the geometry is not a black hole in Lorentzian signature (Fujita et al., 2022). In that analysis the boundary stress tensor is

$$ds^2_{\rm DW} = L^2\left[ u^2\left(\frac{u_+u_-}{u^2}\,dx-dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_-^2)(u^2+u_+^2)} +\frac{(u^2-u_-^2)(u^2+u_+^2)}{u^2}\,dx^2 \right],$$8

so the Lorentzian dual state has negative energy, interpreted there as Casimir-like. For $$ds^2_{\rm DW} = L^2\left[ u^2\left(\frac{u_+u_-}{u^2}\,dx-dT_E\right)^2 +\frac{u^2\,du^2}{(u^2-u_-^2)(u^2+u_+^2)} +\frac{(u^2-u_-^2)(u^2+u_+^2)}{u^2}\,dx^2 \right],$$9, the same work states that the geometry becomes the $x\sim x+2\pi$0 soliton (Fujita et al., 2022).

The causal issue can also be seen directly from the norm of the spatial Killing vector: $x\sim x+2\pi$1 It is spacelike only for $x\sim x+2\pi$2 and timelike for $x\sim x+2\pi$3, so the geometry contains a closed timelike curve region. At $x\sim x+2\pi$4,

$x\sim x+2\pi$5

which is another manifestation of the failure of the standard Lorentzian black-hole interpretation (Fujita et al., 2022).

Accordingly, the term “double Wick-rotated BTZ black hole” is best understood as conventional nomenclature for a geometry derived from BTZ by exchanging temporal and angular roles. In Euclidean signature it is a smooth quotient of $x\sim x+2\pi$6 with the same boundary torus data as rotating BTZ; in Lorentzian signature it is globally delicate, can contain closed timelike curves, and is more naturally regarded as an alternative torus filling or smooth Euclidean saddle than as an independent thermal black-hole state (Dai et al., 14 Apr 2025).