Kerr–Sen Black Hole Solution

• Kerr–Sen solution is a rotating, electrically charged black hole in heterotic string theory that incorporates nontrivial dilaton, Maxwell, and axion fields.
• Its metric deforms Kerr’s radial functions via a charge-dependent parameter, enabling detailed studies of geodesics, shadows, and superradiance.
• Recent research extends the solution to AdS, ultraspinning limits, and higher-derivative corrections, clarifying its impact on multipole moments and observational probes.

Searching arXiv for recent and foundational Kerr-Sen papers to support the article. The Kerr–Sen solution is a stationary, axisymmetric, rotating, electrically charged black-hole solution of the low-energy effective theory of heterotic string theory, often described as the string-theoretic analogue of Kerr–Newman, but with nontrivial dilaton, Maxwell, and antisymmetric tensor or axion structure (Siahaan, 2015). In the four-dimensional low-energy description, the relevant bosonic sector includes the graviton, a $U(1)$ gauge field, a dilaton, and a Kalb–Ramond field or its axionic dual (Narang et al., 2020). Across the literature, the solution is presented in several equivalent coordinate and frame conventions, but a common feature is the deformation of the Kerr radial functions by a charge-dependent parameter such as $b=Q^2/(2M)$ or closely related notation, together with the appearance of accompanying non-gravitational fields (Siahaan, 2015). The Kerr–Sen geometry has served as a central model for investigations of geodesics, shadows, superradiance, test-particle thought experiments, merger estimates, AdS thermodynamics, and higher-derivative heterotic corrections (Uniyal et al., 2017, Cayuso et al., 2019, Xavier et al., 2020, Hu et al., 25 Jun 2025).

1. Heterotic-string origin and field content

The Kerr–Sen black hole was found by Sen in the low-energy heterotic string theory and is repeatedly characterized as the rotating, electrically charged solution of that theory (Siahaan, 2015). One form of the underlying action is

$$S = \int d^4x\, \sqrt{|\tilde g|}\, e^{-\tilde\Phi}\left(R-\frac18 F^2 + \tilde g^{\mu\nu}\partial_\mu \tilde\Phi\,\partial_\nu \tilde\Phi -\frac1{12}H^2\right),$$

with $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$, dilaton $\tilde\Phi$, and antisymmetric tensor sector entering through $H_{\kappa\mu\nu}$ (Siahaan, 2015). Closely related string-frame and Einstein-frame formulations appear in later analyses, all emphasizing the same matter content: metric, gauge field, dilaton, and antisymmetric tensor or axion (Cayuso et al., 2019, Narang et al., 2020).

In one four-dimensional Einstein-frame presentation, the effective action is written as

$$S=\int d^4 x \sqrt{g} \left( R -\frac{1}{12}e^{-2\tilde\phi}\tilde H^2 -\partial_\mu\tilde\phi\,\partial^\mu\tilde\phi -\frac{1}{8}e^{-\tilde\phi}\tilde F^2 \right),$$

making explicit the dilaton dressing of both the antisymmetric tensor and gauge-field sectors (Narang et al., 2020). Another formulation used in Proca-field analysis is

$$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\,e^{\Phi}\left(R + g^{ab}\partial_a\Phi\,\partial_b\Phi - F_{ab}F^{ab} - \frac{1}{12}H_{abc}H^{abc}\right),$$

with

$H = d\mathcal B - 2A\wedge F,\qquad F=dA,$

which is particularly suited to the hidden-symmetry structure of the solution (Cayuso et al., 2019).

A persistent theme in the literature is that Kerr–Sen is not an electrovacuum geometry. Its charge is tied to the heterotic gauge sector together with dilaton and axion or Kalb–Ramond structure, distinguishing it from Kerr–Newman even when some metric expressions appear formally similar (Siahaan, 2015, Xavier et al., 2020). This is why shadow, multipole, superradiant, and thermodynamic comparisons between Kerr–Sen and Kerr–Newman generally show close analogies but not exact coincidence (Xavier et al., 2020, Hu et al., 25 Jun 2025).

2. Metric forms, parameters, and horizons

A standard Einstein-frame Boyer–Lindquist form used in several works is

$$g_{tt}=-\frac{\Delta-a^2\sin^2\theta}{\rho^2},\qquad g_{t\phi}=-\frac{2Mar\sin^2\theta}{\rho^2},$$

$b=Q^2/(2M)$0

with

$b=Q^2/(2M)$1

(Siahaan, 2015). Equivalent notations include $b=Q^2/(2M)$2 in place of $b=Q^2/(2M)$3 and $b=Q^2/(2M)$4 in place of $b=Q^2/(2M)$5 (Zhang et al., 2020). Another widely used form is

$b=Q^2/(2M)$6

$b=Q^2/(2M)$7

with

$b=Q^2/(2M)$8

(Xavier et al., 2020).

The physical parameters are mass $b=Q^2/(2M)$9, angular momentum $$S = \int d^4x\, \sqrt{|\tilde g|}\, e^{-\tilde\Phi}\left(R-\frac18 F^2 + \tilde g^{\mu\nu}\partial_\mu \tilde\Phi\,\partial_\nu \tilde\Phi -\frac1{12}H^2\right),$$0, electric charge $$S = \int d^4x\, \sqrt{|\tilde g|}\, e^{-\tilde\Phi}\left(R-\frac18 F^2 + \tilde g^{\mu\nu}\partial_\mu \tilde\Phi\,\partial_\nu \tilde\Phi -\frac1{12}H^2\right),$$1, and spin parameter $$S = \int d^4x\, \sqrt{|\tilde g|}\, e^{-\tilde\Phi}\left(R-\frac18 F^2 + \tilde g^{\mu\nu}\partial_\mu \tilde\Phi\,\partial_\nu \tilde\Phi -\frac1{12}H^2\right),$$2, together with the charge deformation parameter $$S = \int d^4x\, \sqrt{|\tilde g|}\, e^{-\tilde\Phi}\left(R-\frac18 F^2 + \tilde g^{\mu\nu}\partial_\mu \tilde\Phi\,\partial_\nu \tilde\Phi -\frac1{12}H^2\right),$$3 (Siahaan, 2015). In some solution-generating parametrizations, one also encounters $$S = \int d^4x\, \sqrt{|\tilde g|}\, e^{-\tilde\Phi}\left(R-\frac18 F^2 + \tilde g^{\mu\nu}\partial_\mu \tilde\Phi\,\partial_\nu \tilde\Phi -\frac1{12}H^2\right),$$4 and $$S = \int d^4x\, \sqrt{|\tilde g|}\, e^{-\tilde\Phi}\left(R-\frac18 F^2 + \tilde g^{\mu\nu}\partial_\mu \tilde\Phi\,\partial_\nu \tilde\Phi -\frac1{12}H^2\right),$$5, with corresponding relations for $$S = \int d^4x\, \sqrt{|\tilde g|}\, e^{-\tilde\Phi}\left(R-\frac18 F^2 + \tilde g^{\mu\nu}\partial_\mu \tilde\Phi\,\partial_\nu \tilde\Phi -\frac1{12}H^2\right),$$6, $$S = \int d^4x\, \sqrt{|\tilde g|}\, e^{-\tilde\Phi}\left(R-\frac18 F^2 + \tilde g^{\mu\nu}\partial_\mu \tilde\Phi\,\partial_\nu \tilde\Phi -\frac1{12}H^2\right),$$7, and $$S = \int d^4x\, \sqrt{|\tilde g|}\, e^{-\tilde\Phi}\left(R-\frac18 F^2 + \tilde g^{\mu\nu}\partial_\mu \tilde\Phi\,\partial_\nu \tilde\Phi -\frac1{12}H^2\right),$$8 (Uniyal et al., 2017, Narang et al., 2020).

The horizon structure is determined by the roots of $$S = \int d^4x\, \sqrt{|\tilde g|}\, e^{-\tilde\Phi}\left(R-\frac18 F^2 + \tilde g^{\mu\nu}\partial_\mu \tilde\Phi\,\partial_\nu \tilde\Phi -\frac1{12}H^2\right),$$9: $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$0 (Siahaan, 2015). The event horizon exists iff

$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$1

with equality giving the extremal Kerr–Sen black hole (Düztaş, 2018). Equivalent extremality conditions include

$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$2

in the conventions of the corresponding papers (Siahaan, 2015). In the notation of equatorial escape-probability analysis, extremality is written as

$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$3

with $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$4 (Zhang et al., 2020).

Several limiting cases recur throughout the literature. Setting $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$5 gives Kerr (Siahaan, 2015). Setting $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$6, followed in some conventions by a radial shift, yields the Gibbons–Maeda–Garfinkle–Horowitz–Strominger black hole (Siahaan, 2015). These reductions are structurally important because many Kerr–Sen properties interpolate between Kerr-like rotational behavior and static dilatonic behavior.

3. Associated fields, symmetries, and hidden structure

The Kerr–Sen background includes non-gravitational fields in addition to the metric. A standard set is

$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$7

$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$8

$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$9

(Siahaan, 2015). In another notation,

$\tilde\Phi$0

together with the dilaton

$\tilde\Phi$1

(Huang et al., 2017).

Because the spacetime is stationary and axisymmetric, it admits the Killing vectors

$\tilde\Phi$2

which generate conserved energy and azimuthal angular momentum for test-particle motion (Zhang et al., 2020). This is the starting point for separability in geodesic and field equations.

An important geometric distinction from Kerr and Kerr–Newman is that Kerr–Sen does not possess the usual principal tensor, but instead a principal tensor with torsion, identified with the background three-form $\tilde\Phi$3 (Cayuso et al., 2019). In that formulation,

$\tilde\Phi$4

and the generalized principal tensor $\tilde\Phi$5 obeys

$\tilde\Phi$6

(Cayuso et al., 2019). The explicit torsionful hidden-symmetry tensor is given by

$\tilde\Phi$7

(Cayuso et al., 2019).

This hidden symmetry underlies the separability of the Proca equations on Kerr–Sen (Cayuso et al., 2019). A plausible implication is that the solution’s integrability properties are more subtle than those of type-D electrovacuum backgrounds, and this helps explain why Kerr–Sen often preserves separability in contexts where its algebraic classification differs from Kerr–Newman (Xavier et al., 2020).

4. Geodesics, photon regions, and escape phenomena

The Hamilton–Jacobi or equivalent first-integral treatment yields separated radial and polar potentials for geodesics. One frequently used form is

$\tilde\Phi$8

with

$\tilde\Phi$9

$H_{\kappa\mu\nu}$0

(Uniyal et al., 2017). A central observation of that analysis is that the charge does not enter the $H_{\kappa\mu\nu}$1-motion, so latitudinal motion is the same as in Kerr for the same $H_{\kappa\mu\nu}$2, while charge modifies radial motion (Uniyal et al., 2017).

For null motion, the radial function becomes

$H_{\kappa\mu\nu}$3

(Uniyal et al., 2017). Spherical photon orbits are obtained from

$H_{\kappa\mu\nu}$4

(Uniyal et al., 2017). The physically acceptable family is identified in that work, and the allowed spherical-photon region is found to shrink as the charge increases (Uniyal et al., 2017).

The impact-parameter formulation used in shadow calculations writes the Kerr–Sen null potentials as

$H_{\kappa\mu\nu}$5

$H_{\kappa\mu\nu}$6

(Xavier et al., 2020). This separability is noteworthy because the spacetime is stated there to be Petrov type I rather than type D (Xavier et al., 2020).

The same geodesic structure supports analyses of observables. For an observer at infinity,

$H_{\kappa\mu\nu}$7

describe the shadow edge in image-plane coordinates (Xavier et al., 2020). Related expressions for celestial coordinates are

$H_{\kappa\mu\nu}$8

(Narang et al., 2020).

A different but complementary question concerns local escape from the black hole neighborhood. Assuming a source at rest in a locally non-rotating frame on the equatorial plane, the escape probability of emitted photons and massive particles can be defined from the width of the allowed escape cone (Zhang et al., 2020). In the extreme case, the near-horizon photon escape probability becomes

$H_{\kappa\mu\nu}$9

which decreases with increasing $$S=\int d^4 x \sqrt{g} \left( R -\frac{1}{12}e^{-2\tilde\phi}\tilde H^2 -\partial_\mu\tilde\phi\,\partial^\mu\tilde\phi -\frac{1}{8}e^{-\tilde\phi}\tilde F^2 \right),$$0 and hence, within the extremal family, indicates that larger rotation enhances escape (Zhang et al., 2020). In the non-extreme case, the qualitative trend reverses: increasing charge increases the escape probability, equivalently increasing angular momentum suppresses escape (Zhang et al., 2020). The same work also reports that the escape probability as a function of emission radius can be non-monotonic, so the horizon is not always the hardest place from which a particle can escape (Zhang et al., 2020).

5. Superradiance, clouds, and stability questions

Superradiance on Kerr–Sen backgrounds has been studied for scalar, vector, and cloud configurations. For neutral scalar test fields, the relevant superradiance threshold is

$$S=\int d^4 x \sqrt{g} \left( R -\frac{1}{12}e^{-2\tilde\phi}\tilde H^2 -\partial_\mu\tilde\phi\,\partial^\mu\tilde\phi -\frac{1}{8}e^{-\tilde\phi}\tilde F^2 \right),$$1

equivalently $$S=\int d^4 x \sqrt{g} \left( R -\frac{1}{12}e^{-2\tilde\phi}\tilde H^2 -\partial_\mu\tilde\phi\,\partial^\mu\tilde\phi -\frac{1}{8}e^{-\tilde\phi}\tilde F^2 \right),$$2 for absorption (Düztaş, 2018). In nearly extremal Kerr–Sen, a scalar field can overspin the black hole if its mode frequency lies in the window

$$S=\int d^4 x \sqrt{g} \left( R -\frac{1}{12}e^{-2\tilde\phi}\tilde H^2 -\partial_\mu\tilde\phi\,\partial^\mu\tilde\phi -\frac{1}{8}e^{-\tilde\phi}\tilde F^2 \right),$$3

so nearly extremal Kerr–Sen can be overspun into a naked singularity at the test-field level (Düztaş, 2018). By contrast, for extremal Kerr–Sen the dangerous scalar modes lie below the superradiance threshold and are not absorbed, so superradiance protects the horizon in that setup (Düztaş, 2018).

For massive vector bosons, the Proca equations can be fully separated on Kerr–Sen by exploiting the torsionful hidden symmetry (Cayuso et al., 2019). The separated radial and angular equations arise from the Lunin–Frolov–Krtouš–Kubizňák ansatz

$$S=\int d^4 x \sqrt{g} \left( R -\frac{1}{12}e^{-2\tilde\phi}\tilde H^2 -\partial_\mu\tilde\phi\,\partial^\mu\tilde\phi -\frac{1}{8}e^{-\tilde\phi}\tilde F^2 \right),$$4

with

$$S=\int d^4 x \sqrt{g} \left( R -\frac{1}{12}e^{-2\tilde\phi}\tilde H^2 -\partial_\mu\tilde\phi\,\partial^\mu\tilde\phi -\frac{1}{8}e^{-\tilde\phi}\tilde F^2 \right),$$5

(Cayuso et al., 2019). The instability condition takes the standard form

$$S=\int d^4 x \sqrt{g} \left( R -\frac{1}{12}e^{-2\tilde\phi}\tilde H^2 -\partial_\mu\tilde\phi\,\partial^\mu\tilde\phi -\frac{1}{8}e^{-\tilde\phi}\tilde F^2 \right),$$6

and the numerical results show that the instability is strongest for rapidly rotating, weakly charged black holes, while increasing charge suppresses the growth rate (Cayuso et al., 2019). The same comparison finds Kerr–Sen slightly more unstable than Kerr–Newman for the same asymptotic $$S=\int d^4 x \sqrt{g} \left( R -\frac{1}{12}e^{-2\tilde\phi}\tilde H^2 -\partial_\mu\tilde\phi\,\partial^\mu\tilde\phi -\frac{1}{8}e^{-\tilde\phi}\tilde F^2 \right),$$7, $$S=\int d^4 x \sqrt{g} \left( R -\frac{1}{12}e^{-2\tilde\phi}\tilde H^2 -\partial_\mu\tilde\phi\,\partial^\mu\tilde\phi -\frac{1}{8}e^{-\tilde\phi}\tilde F^2 \right),$$8, and $$S=\int d^4 x \sqrt{g} \left( R -\frac{1}{12}e^{-2\tilde\phi}\tilde H^2 -\partial_\mu\tilde\phi\,\partial^\mu\tilde\phi -\frac{1}{8}e^{-\tilde\phi}\tilde F^2 \right),$$9 (Cayuso et al., 2019).

Charged massive scalar clouds furnish stationary bound states at the superradiant threshold

$$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\,e^{\Phi}\left(R + g^{ab}\partial_a\Phi\,\partial_b\Phi - F_{ab}F^{ab} - \frac{1}{12}H_{abc}H^{abc}\right),$$0

(Huang et al., 2017). The cloud condition is

$$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\,e^{\Phi}\left(R + g^{ab}\partial_a\Phi\,\partial_b\Phi - F_{ab}F^{ab} - \frac{1}{12}H_{abc}H^{abc}\right),$$1

with

$$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\,e^{\Phi}\left(R + g^{ab}\partial_a\Phi\,\partial_b\Phi - F_{ab}F^{ab} - \frac{1}{12}H_{abc}H^{abc}\right),$$2

(Huang et al., 2017). The allowed cloud region in $$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\,e^{\Phi}\left(R + g^{ab}\partial_a\Phi\,\partial_b\Phi - F_{ab}F^{ab} - \frac{1}{12}H_{abc}H^{abc}\right),$$3 space is bounded, so the scalar mass and charge are confined to a finite range (Huang et al., 2017). A distinctive extremal Kerr–Sen result is

$$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\,e^{\Phi}\left(R + g^{ab}\partial_a\Phi\,\partial_b\Phi - F_{ab}F^{ab} - \frac{1}{12}H_{abc}H^{abc}\right),$$4

independent of the black-hole angular velocity $$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\,e^{\Phi}\left(R + g^{ab}\partial_a\Phi\,\partial_b\Phi - F_{ab}F^{ab} - \frac{1}{12}H_{abc}H^{abc}\right),$$5, unlike in extremal Kerr–Newman (Huang et al., 2017). The static GMGHS limit cannot support these stationary scalar clouds (Huang et al., 2017).

A separate line of inquiry concerns chronology protection in the dyonic Kerr–Sen interior. In the region inside the inner horizon where closed timelike curves exist, exact scalar-mode solutions yield quasinormal frequencies with positive imaginary parts for the relevant outgoing modes, implying exponential growth and instability of the causality-violating region (Bunyaratavej et al., 2024). This is presented as support for chronology protection in the dyonic Kerr–Sen spacetime (Bunyaratavej et al., 2024).

6. Comparisons, observational probes, and dynamical applications

Kerr–Sen has been compared extensively with Kerr and Kerr–Newman through shadows and lensing. For the same asymptotic $$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\,e^{\Phi}\left(R + g^{ab}\partial_a\Phi\,\partial_b\Phi - F_{ab}F^{ab} - \frac{1}{12}H_{abc}H^{abc}\right),$$6 and viewing conditions, the Kerr–Sen shadow is reported to be always slightly larger than the Kerr–Newman shadow, with the difference typically at the percent level or below (Xavier et al., 2020). The explanation offered there is that the Kerr–Newman horizon dimensionless spin $$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\,e^{\Phi}\left(R + g^{ab}\partial_a\Phi\,\partial_b\Phi - F_{ab}F^{ab} - \frac{1}{12}H_{abc}H^{abc}\right),$$7 is always larger than the Kerr–Sen horizon spin $$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\,e^{\Phi}\left(R + g^{ab}\partial_a\Phi\,\partial_b\Phi - F_{ab}F^{ab} - \frac{1}{12}H_{abc}H^{abc}\right),$$8 for the same asymptotic parameters, and increasing spin tends to shrink and deform the shadow (Xavier et al., 2020).

The shadow of Kerr–Sen has also been proposed as a direct observational probe of the heterotic-string charge sector. The deviation from circularity can constrain the charge, while the axion hair induces a frequency-independent polarization rotation,

$$S = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\,e^{\Phi}\left(R + g^{ab}\partial_a\Phi\,\partial_b\Phi - F_{ab}F^{ab} - \frac{1}{12}H_{abc}H^{abc}\right),$$9

which differs from plasma Faraday rotation because it does not scale with wavelength (Narang et al., 2020). That work emphasizes a correlation unique to Kerr–Sen: the same charge parameter controls both the shadow deformation and the axion-induced polarization rotation (Narang et al., 2020). Current EHT-level bounds do not strongly constrain the model, but improved circularity precision at the $H = d\mathcal B - 2A\wedge F,\qquad F=dA,$0 level is argued there to be potentially decisive (Narang et al., 2020).

Light deflection has also been analyzed using the material-medium approach. In the far-field approximation, the effective refractive index in Kerr–Sen spacetime is

$H = d\mathcal B - 2A\wedge F,\qquad F=dA,$1

with the second factor encoding frame dragging (Roy et al., 16 Apr 2025). The reported qualitative behavior is that increasing spin increases prograde bending and decreases retrograde bending, while increasing charge generally decreases the deflection angle (Roy et al., 16 Apr 2025).

Kerr–Sen has further been used as a background for approximate binary-merger estimates. In the generalized BKL prescription, charge affects the final spin indirectly through its effect on the ISCO and test-body angular momentum, rather than by appearing directly in the final-spin balance equation (Siahaan, 2019). The overall numerical trend is that larger charge lowers the final spin, while the neutral Kerr case gives the largest final spin (Siahaan, 2019). Light-ring-based quasinormal estimates show Kerr–Sen behavior broadly similar to Kerr–Newman, but with quantitative differences in the Lyapunov exponent and orbital frequency (Siahaan, 2019).

These observational and dynamical studies collectively suggest that Kerr–Sen effects are subtle rather than gross. The differences from Kerr or Kerr–Newman often appear in percent-level shadow shifts, modified multipoles, altered superradiant growth rates, or charge-dependent merger and escape properties. This suggests that the solution functions less as a replacement for Kerr in current phenomenology than as a controlled benchmark for string-inspired deviations.

7. AdS extensions, ultraspinning limits, and higher-derivative generalizations

The Kerr–Sen geometry admits AdS and ultraspinning generalizations in gauged Einstein–Maxwell–dilaton–axion theory. For the four-dimensional Kerr–Sen–AdS$H = d\mathcal B - 2A\wedge F,\qquad F=dA,$2 solution, one convenient metric form is

$H = d\mathcal B - 2A\wedge F,\qquad F=dA,$3

with

$H = d\mathcal B - 2A\wedge F,\qquad F=dA,$4

$H = d\mathcal B - 2A\wedge F,\qquad F=dA,$5

(Wu et al., 2020). In extended thermodynamics, the quantities

$H = d\mathcal B - 2A\wedge F,\qquad F=dA,$6

and

$H = d\mathcal B - 2A\wedge F,\qquad F=dA,$7

satisfy the first law and a Bekenstein–Smarr relation in the appropriate frame (Wu et al., 2020).

The ultraspinning limit $H = d\mathcal B - 2A\wedge F,\qquad F=dA,$8 yields a black spindle geometry with noncompact horizon but finite area (Wu et al., 2020). Unlike the Kerr–Newman–AdS$H = d\mathcal B - 2A\wedge F,\qquad F=dA,$9 super-entropic case, the ultraspinning Kerr–Sen–AdS$$g_{tt}=-\frac{\Delta-a^2\sin^2\theta}{\rho^2},\qquad g_{t\phi}=-\frac{2Mar\sin^2\theta}{\rho^2},$$0 black hole does not always violate the reverse isoperimetric inequality. Its isoperimetric ratio is

$$g_{tt}=-\frac{\Delta-a^2\sin^2\theta}{\rho^2},\qquad g_{t\phi}=-\frac{2Mar\sin^2\theta}{\rho^2},$$1

so $$g_{tt}=-\frac{\Delta-a^2\sin^2\theta}{\rho^2},\qquad g_{t\phi}=-\frac{2Mar\sin^2\theta}{\rho^2},$$2 may be smaller than, equal to, or larger than unity depending on whether $$g_{tt}=-\frac{\Delta-a^2\sin^2\theta}{\rho^2},\qquad g_{t\phi}=-\frac{2Mar\sin^2\theta}{\rho^2},$$3 is smaller than, equal to, or larger than $$g_{tt}=-\frac{\Delta-a^2\sin^2\theta}{\rho^2},\qquad g_{t\phi}=-\frac{2Mar\sin^2\theta}{\rho^2},$$4 (Wu et al., 2020). The dyonic extension shares the same non-universal super-entropic behavior (Wu et al., 2020).

A recent topological thermodynamics study of Kerr–Sen AdS reports three thermodynamic branches—small, intermediate, and large black holes—with winding numbers $$g_{tt}=-\frac{\Delta-a^2\sin^2\theta}{\rho^2},\qquad g_{t\phi}=-\frac{2Mar\sin^2\theta}{\rho^2},$$5, $$g_{tt}=-\frac{\Delta-a^2\sin^2\theta}{\rho^2},\qquad g_{t\phi}=-\frac{2Mar\sin^2\theta}{\rho^2},$$6, and $$g_{tt}=-\frac{\Delta-a^2\sin^2\theta}{\rho^2},\qquad g_{t\phi}=-\frac{2Mar\sin^2\theta}{\rho^2},$$7, giving total topological charge $$g_{tt}=-\frac{\Delta-a^2\sin^2\theta}{\rho^2},\qquad g_{t\phi}=-\frac{2Mar\sin^2\theta}{\rho^2},$$8 (Rehan et al., 25 Mar 2026). That analysis states that the total topological class is unchanged by variations of the dilaton charge parameter, while rotation is crucial for the multi-branch structure (Rehan et al., 25 Mar 2026).

At the level of effective theory, the most significant recent development is the construction of four-derivative corrections to Kerr–Sen in heterotic supergravity (Hu et al., 25 Jun 2025). In that work, Kerr is first embedded into heterotic supergravity, then corrected, and finally boosted by an $$g_{tt}=-\frac{\Delta-a^2\sin^2\theta}{\rho^2},\qquad g_{t\phi}=-\frac{2Mar\sin^2\theta}{\rho^2},$$9 transformation to obtain corrected Kerr–Sen (Hu et al., 25 Jun 2025). The resulting corrected solution reduces to the corrected GMGHS black hole in the static limit (Hu et al., 25 Jun 2025). Most notably, while two-derivative Kerr–Sen shares the Kerr gravitational multipole pattern when expressed in terms of physical $b=Q^2/(2M)$00 and $b=Q^2/(2M)$01, four-derivative corrections change the mass, current, electric, and magnetic multipoles in a way distinct from both Kerr and Kerr–Newman (Hu et al., 25 Jun 2025). This gives a concrete route, at least in principle, for distinguishing heterotic string corrections in gravitational-wave data (Hu et al., 25 Jun 2025).

A closely related later study shows that different consistent four-derivative heterotic truncations yield different corrected Kerr–Sen solutions, with distinct thermodynamics and multipole structures (Ma et al., 8 Sep 2025). This suggests that beyond the two-derivative level, “the Kerr–Sen solution” becomes a family of inequivalent string-corrected geometries rather than a unique object. A plausible implication is that future tests of heterotic black-hole physics may probe not just the existence of Kerr–Sen-like corrections, but also the precise effective-theory truncation that governs them.