Regular Black String: Geometric Regularization

• Regular black strings are extended black objects that eliminate singular cores through geometric regularization techniques such as bounce deformations, compact bubbles, and nonlocal source smearing.
• They are constructed via distinct mechanisms including black-bounce deformations, Kaluza–Klein compactifications, and string-theoretic α' corrections to ensure finite curvature invariants.
• These structures offer valuable insights into thermodynamics, phase transitions, and causal geometries while serving as laboratories for singularity resolution in higher-dimensional gravity.

Searching arXiv for recent and foundational papers on regular black strings to ground the article in the literature. A regular black string is a black-string geometry—typically cylindrically symmetric, or obtained by extending a lower-dimensional black hole along an extra direction—in which the would-be singular core is removed and replaced by a smooth cap, a de Sitter-like core, a Kaluza–Klein bubble, or a bounce throat, so that curvature invariants remain finite or the full higher-dimensional bulk is regular. In the recent literature, regular black strings appear in several technically distinct settings: five-dimensional toy models for four-dimensional regular black holes (Estrada, 2022), Kaluza–Klein reductions with compact extra dimensions (Hung et al., 2023), bilocal and noncommutative effective geometries (Muniz et al., 2022, Singh et al., 2017), unimodular gravity supported by Maxwell fields (Alencar et al., 31 Mar 2026), Simpson–Visser black-bounce deformations of Lemos-type strings (Lima et al., 2022, Lima et al., 2023, Lima et al., 2023), dynamical braneworld bulk completions (Bazeia et al., 2014), and string-theoretic loop or complete $\alpha'$ corrections (Ying, 2022, Ying, 2022).

1. Conceptual setting and relation to ordinary black strings

Black strings are extended black objects with one or more translational directions. In the homogeneous AdS solutions of Cisterna and Oliva, the line element is written in $D=d+p$ dimensions with transverse Einstein space $\Sigma_{d-2,\gamma}$ and flat brane directions $x^i$; the event horizon is the largest real root of $F(r)=0$, and the true curvature singularity lies at $r=0$, shielded by the horizon when $\mu>0$ (Cisterna et al., 2017). In that class the asymptotic geometry is $AdS_d\times\mathbb{R}^p$, and the regularity statement applies only for $r>r_+$, not at the center (Cisterna et al., 2017).

This singular behavior is also present in broad families of non-regular strings. The five-parameter black string of Compère, de Buyl, Stotyn, and Virmani in minimal five-dimensional ungauged supergravity carries magnetic one-brane charge, smeared electric zero-brane charge, boost, non-extremality, and transverse rotation; it admits BTZ$\times S^2$ decoupling and reproduces the macroscopic entropy through the Maldacena–Strominger–Witten CFT in appropriate parameter ranges, but it is not presented as a regularized core geometry (Compère et al., 2010).

Against this background, a regular black string is defined operationally by the removal of the $D=d+p$0 pathology. In different constructions this is implemented by smearing the source, replacing $D=d+p$1 by $D=d+p$2, introducing a compact bubble where the extra circle pinches off smoothly, modifying the gravitational sector, or incorporating string-loop or all-orders $D=d+p$3 corrections. This suggests that “regularity” in the black-string literature is not tied to a single mechanism, but to the elimination of curvature blow-ups and, in some works, to regularity of the full higher-dimensional bulk.

2. Principal construction mechanisms

The existing literature realizes regular black strings through several non-equivalent technical routes.

Mechanism	Representative papers	Defining feature
Black-bounce deformation	(Lima et al., 2022, Lima et al., 2023, Lima et al., 2023)	$D=d+p$4 replaces the singular core by a throat
Kaluza–Klein compactification and bubble	(Hung et al., 2023, Estrada, 2022)	A compact $D=d+p$5 direction ends smoothly or supports a regular 5D completion
Smeared or nonlocal source	(Muniz et al., 2022, Singh et al., 2017, Jusufi, 2022)	The point source is replaced by a bilocal, Gaussian, or zero-point-length distribution
Modified gravitational dynamics	(Alencar et al., 31 Mar 2026, Bazeia et al., 2014)	Regularity follows from unimodular $D=d+p$6 or variable brane tension
String-theoretic corrections	(Ying, 2022, Ying, 2022)	Non-local dilaton potentials or complete $D=d+p$7 corrections smooth the interior

In the Simpson–Visser branch, the starting point is the Lemos black string, whose metric function $D=d+p$8 is regularized by the replacement $D=d+p$9. The resulting static geometry has

$\Sigma_{d-2,\gamma}$0

and the same prescription extends to charged and rotating cases (Lima et al., 2022, Lima et al., 2023, Lima et al., 2023).

In the Kaluza–Klein construction of the regular black string in five-dimensional Einstein–Maxwell theory, the metric is written as

$\Sigma_{d-2,\gamma}$1

with $\Sigma_{d-2,\gamma}$2, $\Sigma_{d-2,\gamma}$3, and $\Sigma_{d-2,\gamma}$4. Here $\Sigma_{d-2,\gamma}$5 is a regular event horizon and $\Sigma_{d-2,\gamma}$6 is a smooth Kaluza–Klein bubble where the $\Sigma_{d-2,\gamma}$7-circle pinches off (Hung et al., 2023).

In bilocal gravity, the regularization is encoded in an entire form factor $\Sigma_{d-2,\gamma}$8 and in the smeared density

$\Sigma_{d-2,\gamma}$9

leading to a four-dimensional black string that is regular at the origin, asymptotically approaches the ordinary static uncharged black string, and can possess event and inner horizons depending on the mass density (Muniz et al., 2022). Closely related source-smearing appears in the noncommutative rotating black string, where $x^i$0 is replaced by an incomplete-gamma mass profile, and in the zero-point-length regularized charged black string, where both gravitational and electromagnetic potentials are modified by $x^i$1 (Singh et al., 2017, Jusufi, 2022).

In unimodular gravity, the black-string metric is parametrized by

$x^i$2

while the effective vacuum contribution becomes a radial function

$x^i$3

For Bardeen-type and Hayward-type mass functions, $x^i$4 near $x^i$5, which removes the singularity and keeps curvature scalars finite (Alencar et al., 31 Mar 2026).

Finally, in string theory the regularization may arise directly from the effective action. In the three-dimensional case, a non-local dilaton potential $x^i$6 yields a regular black-string interior with finite Ricci and Kretschmann scalars (Ying, 2022). In the two-dimensional string black hole and its three-dimensional uplift, the Hohm–Zwiebach all-orders $x^i$7 organization produces exact non-perturbative solutions for which $x^i$8 and $x^i$9 stay finite over the full interval $F(r)=0$0 (Ying, 2022).

3. Geometry, topology, and causal structure

Regular black strings do not share a single global topology. In the five-dimensional toy model for representing a four-dimensional regular black hole “at the black string style,” the origin has topology $F(r)=0$1 with compact $F(r)=0$2; the complete five-dimensional geometry has topology $F(r)=0$3; and the asymptotic region approaches Minkowski$F(r)=0$4 (Estrada, 2022). The same work emphasizes that this differs from the Kaluza–Klein black string, for which the origin corresponds to the product between the Schwarzschild singularity and $F(r)=0$5 or $F(r)=0$6, depending on whether $F(r)=0$7 is noncompact or compact (Estrada, 2022).

In the Einstein–Maxwell Kaluza–Klein solution, the pair of radii $F(r)=0$8 separates two geometrically distinct loci: a regular event horizon at $F(r)=0$9 and a smooth bubble at $r=0$0. Under four-dimensional reduction, the five-dimensional magnetic charge becomes a four-dimensional charge $r=0$1 together with a dilaton $r=0$2, and the horizon area has topology $r=0$3 (Hung et al., 2023).

The black-bounce family makes the black-string/wormhole interpolation explicit. For the static Lemos-type case, the horizons are located at

$r=0$4

If $r=0$5, the geometry has two real horizons and is a one-way regular black string; if $r=0$6, the throat at $r=0$7 is extremal; and if $r=0$8, no zeros of $r=0$9 remain and the spacetime is a two-way traversable wormhole of cylindrical symmetry (Lima et al., 2022). The charged and rotating generalizations preserve the same qualitative trichotomy: a two-horizon regular string for sufficiently small $\mu>0$0, an extremal throat at the transition value, and a traversable wormhole for larger $\mu>0$1 (Lima et al., 2023, Lima et al., 2023).

The rotating black-bounce metric in Boyer–Lindquist-type form,

$\mu>0$2

has horizon radii

$\mu>0$3

and interpolates between a two-sided black bounce, an extremal one-way bounce, and a traversable AdS wormhole as $\mu>0$4 crosses the critical value $\mu>0$5 (Lima et al., 2023).

These constructions show that regularity need not mean the preservation of the causal structure of a classical black string. In some cases the singularity is replaced by a smooth cap behind a horizon; in others the interior opens into a wormhole throat or terminates on a bubble where the compact circle shrinks to zero size.

4. Regularity criteria, matter content, and energy conditions

The minimal regularity criterion used across the literature is finiteness of curvature invariants. In the static bounce geometry, the Ricci scalar, Ricci-tensor squared, and Kretschmann scalar remain finite at $\mu>0$6 for every $\mu>0$7 and approach constant values at large $\mu>0$8 (Lima et al., 2022). The charged bounce similarly has $\mu>0$9 with finite $AdS_d\times\mathbb{R}^p$0 whenever $AdS_d\times\mathbb{R}^p$1, and its Ricci scalar and curvature-tensor components remain bounded (Lima et al., 2023). For the rotating bounce, explicit expressions for $AdS_d\times\mathbb{R}^p$2 and $AdS_d\times\mathbb{R}^p$3 show that $AdS_d\times\mathbb{R}^p$4 and $AdS_d\times\mathbb{R}^p$5 are finite provided $AdS_d\times\mathbb{R}^p$6 (Lima et al., 2023).

A second notion of regularity concerns the full bulk rather than only scalar invariants of an induced metric. In braneworld models with variable brane tension obeying the Eötvös law $AdS_d\times\mathbb{R}^p$7, the extra $AdS_d\times\mathbb{R}^p$8-derivative terms in the five-dimensional curvature can cancel the divergences of the four-dimensional soft invariant $AdS_d\times\mathbb{R}^p$9. For radiation- and matter-dominated eras, the five-dimensional geometry becomes completely regular after a critical time $r>r_+$0, and the black string acquires finite bulk extent $r>r_+$1 (Bazeia et al., 2014).

The required sources vary sharply by model. In the charged black-string bounce, the field-theory realization combines a phantom scalar, signaled by $r>r_+$2, with nonlinear electrodynamics and an electric field $r>r_+$3 (Lima et al., 2023). In the rotating black-bounce solution, the orthonormal-frame stresses $r>r_+$4, $r>r_+$5, and $r>r_+$6 satisfy $r>r_+$7 for all $r>r_+$8, so the null, weak, and strong energy conditions fail everywhere (Lima et al., 2023). The static bounce likewise has $r>r_+$9 for all $\times S^2$0, implying global NEC violation (Lima et al., 2022).

Other mechanisms avoid this specific pattern. In bilocal gravity, the string is sourced by a nonlocal energy–momentum tensor and stabilized by a perfect cosmological fluid with $\times S^2$1 throughout space, though the analysis does not exclude an exotic substance near the string (Muniz et al., 2022). In unimodular gravity, by contrast, the ordinary Maxwell Lagrangian $\times S^2$2 is sufficient because the nonconservation equation $\times S^2$3 transfers part of the burden to the radial vacuum function $\times S^2$4; the positivity of the geometric function $\times S^2$5 guarantees a real electric field $\times S^2$6 in the Bardeen-type and Hayward-type regular strings (Alencar et al., 31 Mar 2026).

A related but distinct phenomenon appears in supersymmetric “coiffured” black strings. There the geometry can remain regular at the horizon even when the matter fields carry nontrivial oscillating density profiles along the string. The metric is generically not infinitely differentiable; after imposing the coiffuring conditions $\times S^2$7 and $\times S^2$8, the Riemann tensor remains finite at the horizon and both neutral and charged probes can cross without divergent forces (Bena et al., 2014). This suggests that regularity at the level of curvature need not coincide with analyticity or even with very high differentiability of the metric.

5. Thermodynamics, phase structure, and microscopic interpretation

Thermodynamics is a central diagnostic of regular black strings, and in several models it remains tractable in closed form. In the five-dimensional toy model that represents a four-dimensional regular black hole in black-string style, the $\times S^2$9 components of the five-dimensional equations of motion take the form of the four-dimensional equations, the five-dimensional conservation equation adopts the four-dimensional form, and the first law is recovered on the induced four-dimensional geometry with the correct values of temperature and entropy (Estrada, 2022).

For the Kaluza–Klein regular black string reduced to four dimensions, the mass, temperature, and entropy are

$D=d+p$00

and the first law $D=d+p$01 holds (Hung et al., 2023). The same system exhibits a fully developed topological thermodynamic classification. The heat capacity at fixed charge diverges at $D=d+p$02, signaling a second-order phase transition; the Ruppeiner scalar curvature is strictly negative on both branches and diverges at the same critical point; and the thermodynamic phases are classified by winding numbers $D=d+p$03 and $D=d+p$04, corresponding respectively to locally stable and locally unstable branches (Hung et al., 2023). The on-shell free energy shows a swallowtail degeneracy that disappears at the critical temperature $D=d+p$05, while the cusp in the $D=d+p$06 plane acts as a “generation point” where the two phases coalesce (Hung et al., 2023).

The static bounce regularization modifies Lemos thermodynamics by a simple factor. The Hawking temperature and heat capacity become

$D=d+p$07

while the entropy and Helmholtz free energy acquire logarithmic corrections. As $D=d+p$08, $D=d+p$09, $D=d+p$10, and $D=d+p$11 vanish (Lima et al., 2022). In bilocal gravity, the temperature rises linearly for large horizon radius, reaches a maximum, and then falls to zero at a finite remnant radius; this gives a mass remnant associated with vanishing Hawking temperature (Muniz et al., 2022). The noncommutative rotating black string is likewise curvature-regular and thermodynamically stable, with a finite minimal-mass remnant and strictly positive heat capacity (Singh et al., 2017).

String-theoretic regularizations retain the horizon while smoothing the interior. In the three-dimensional loop-corrected solution, the event horizon remains at $D=d+p$12, the near-horizon geometry is Rindler-like, and the Hawking temperature is $D=d+p$13 with $D=d+p$14 (Ying, 2022). In the complete $D=d+p$15-corrected two-dimensional black hole and its three-dimensional uplift, the horizon is again at $D=d+p$16, the Penrose diagram becomes that of an eternal non-singular black hole with two exterior regions and an interior bridge of finite proper extent, and the Hawking temperature reduces to the classical value in the $D=d+p$17 limit (Ying, 2022).

A plausible implication is that regular black strings preserve much of the black-string thermodynamic apparatus—first law, Euclidean temperature, free energy, state-space geometry, and in some cases microscopic entropy matching—while modifying the endpoint structure of evaporation and the topology of the interior.

6. Scope, limitations, and open directions

The current literature does not support a single universal definition of “regular black string.” In some works regularity means that all scalar invariants are finite at $D=d+p$18 after a bounce deformation (Lima et al., 2022, Lima et al., 2023, Lima et al., 2023). In others it means that the full five-dimensional bulk is free of singularities over cosmological time evolution (Bazeia et al., 2014), or that a four-dimensional regular black hole can be represented by a five-dimensional regular black-string-style geometry with compact extra direction and controlled topology (Estrada, 2022). This suggests that the term is family-dependent and should always be read together with the underlying regularity criterion.

The same caution applies to matter content. Several explicit bounce geometries require NEC violation everywhere, and the charged realization uses a phantom scalar (Lima et al., 2022, Lima et al., 2023, Lima et al., 2023). By contrast, unimodular gravity shifts part of the regularizing role to a radial vacuum function $D=d+p$19 while retaining linear Maxwell electrodynamics (Alencar et al., 31 Mar 2026), and bilocal gravity regularizes the source through nonlocal smearing (Muniz et al., 2022). Consequently, it is inaccurate to identify regular black strings with a single “exotic matter” paradigm.

A further limitation concerns differentiability and dynamical stability. The supersymmetric coiffured solutions show that a horizon can be regular in the sense of finite Riemann tensor and finite probe forces while the metric is not infinitely differentiable (Bena et al., 2014). Standard homogeneous AdS black strings are expected to suffer Gregory–Laflamme–type long-wavelength instability for sufficiently small $D=d+p$20, but no analytic threshold is known there (Cisterna et al., 2017). For regularized charged strings obtained from zero-point-length potentials, it is suggested that classical Gregory–Laflamme instabilities could be softened by the removal of the singular core, although a full perturbative analysis is left for future work (Jusufi, 2022).

Recent work therefore points in two complementary directions. One direction treats regular black strings as laboratories for singularity resolution, using compact extra dimensions, bilocality, unimodular vacuum functions, or stringy corrections. The other treats them as thermodynamic systems with nontrivial phase topology, remnants, and, in some cases, wormhole transitions. Taken together, these results establish the regular black string as a broad research category rather than a single solution: an extended black object whose classical core singularity is replaced by a regular geometric structure, while its horizon physics, asymptotics, and matter sector remain strongly model-dependent.