Kerr–Sen AdS Black Hole

• The Kerr–Sen AdS black hole is a rotating and electrically charged solution from heterotic string theory, incorporating dilaton and axion fields alongside a U(1) gauge field.
• It introduces a string-induced deformation parameter that alters the radial structure compared to Kerr–Newman–AdS and supports dyonic as well as ultraspinning generalizations.
• Thermodynamic studies reveal van der Waals–like phase behavior, holographic dualities, and insights into chaotic dynamics and efficient energy extraction mechanisms.

A Kerr–Sen AdS black hole is the asymptotically anti-de Sitter extension of the Kerr–Sen solution, namely a rotating, electrically charged black hole of the low-energy heterotic string effective theory, or equivalently of a gauged Einstein–Maxwell–Dilaton–Axion (EMDA) system. Unlike Kerr–Newman–AdS, it is supported not only by the metric and a $U(1)$ gauge field, but also by a dilaton and an axion or Kalb–Ramond sector, with a string-induced parameter that deforms the radial structure of the geometry. The literature also treats dyonic and ultraspinning generalizations, and uses the solution as a laboratory for black-hole chemistry, restricted phase space thermodynamics, near-horizon holography, thermodynamic topology, chaos diagnostics, and energy-extraction mechanisms (Ali et al., 2023, Wu et al., 2020).

1. Theoretical origin and field content

The Kerr–Sen family originates in the low-energy effective action of heterotic string theory. In the Einstein frame, the bosonic sector contains the metric $g_{\mu\nu}$, a dilaton scalar $\phi$, an axion $\chi$, and an Abelian gauge field $A_\mu$ with field strength $F=dA$. In the AdS generalization, the scalar potential fixes a negative cosmological constant, and one convenient schematic form of the action is

$$S=\int d^4 x\,\sqrt{-g}\Big[\,R-\tfrac12 (\nabla\phi)^2-\tfrac12 e^{2\phi}(\nabla\chi)^2-e^{-\phi}F^2\Big] +\tfrac{\chi}{2}\,\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda} +\sqrt{-g}\,\frac{4+e^{-\phi}+e^{\phi}(1+\chi^2)}{\ell^2}\,,$$

with $\Lambda=-3/\ell^2$ in four dimensions (Ali et al., 2023). Closely related gauged EMDA presentations write

$$I=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\Big(R-\tfrac{1}{2}(\partial\phi)^2-\tfrac{1}{2}e^{2\phi}(\partial\chi)^2-e^{-\phi}F^2+\tfrac{\chi}{2}F\tilde{F}\Big)+I_\Lambda,$$

$$I_\Lambda=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\frac{4+e^{-\phi}+e^\phi(1+\chi^2)}{l^2},$$

making explicit the axion topological coupling and the scalar potential induced by gauging (Prihadi et al., 2023).

The axion is related to a two-form $g_{\mu\nu}$0 through a three-form field strength. One formulation uses

$g_{\mu\nu}$1

while the dyonic EMDA literature also introduces a magnetic dual potential $g_{\mu\nu}$2 through

$g_{\mu\nu}$3

These structures distinguish Kerr–Sen–AdS from minimally coupled Einstein–Maxwell black holes and are responsible for the appearance of dilatonic and axionic charge parameters in the metric functions and thermodynamics (Ali et al., 2023, Prihadi et al., 2023).

A standard statement in this literature is that Kerr–Sen–AdS generalizes Kerr–Newman–AdS by replacing minimal electromagnetic coupling with dilaton/axion couplings and by introducing a string-theoretic deformation parameter. In the purely electric presentation this parameter is $g_{\mu\nu}$4, with

$g_{\mu\nu}$5

whereas dyonic EMDA presentations employ

$g_{\mu\nu}$6

so that the scalar sector is encoded directly by the electric and magnetic charge parameters $g_{\mu\nu}$7 and $g_{\mu\nu}$8 (Ali et al., 2023, Wu et al., 2020).

2. Geometry and parameterizations

The purely electric Kerr–Sen–AdS metric is commonly written in Boyer–Lindquist-type coordinates $g_{\mu\nu}$9 as

$\phi$0

with

$\phi$1

$\phi$2

Here $\phi$3 is a mass parameter, $\phi$4 is the rotation parameter, and $\phi$5 is the dilatonic charge parameter inherited from string theory (Ali et al., 2023).

The dyonic EMDA literature presents the AdS solution in an alternative parameterization: $\phi$6 where

$\phi$7

$\phi$8

$\phi$9

In this form the gauge potentials, dilaton, and axion are all nontrivial, and the scalar charges $\chi$0 and $\chi$1 encode the dyonic EMDA couplings (Prihadi et al., 2023).

Several limiting cases are standard. In the $\chi$2 or $\chi$3 limit one recovers the asymptotically flat Kerr–Sen solution; for $\chi$4 one recovers the charged GMGHS sector; and in the $\chi$5 limit the metric functions reduce to those of Kerr–AdS (Ali et al., 2023). In the dyonic literature, turning off the dyonic sector reduces the gauged solution to Kerr–Sen–AdS, while equal electric and magnetic parameters imply vanishing dilaton charge $\chi$6 (Sakti et al., 2022).

Because Boyer–Lindquist AdS coordinates rotate at infinity, frame issues are intrinsic to the geometry. In one convention,

$\chi$7

while another writes

$\chi$8

This is not a contradiction but a difference of notation using $\chi$9. The physically relevant angular velocity is therefore defined relative to a nonrotating frame at infinity (Prihadi et al., 2023, Lee et al., 1 Jun 2025).

3. Conserved charges and horizon quantities

For the purely electric AdS normalization, the conserved charges are

$A_\mu$0

If $A_\mu$1 is the largest root of $A_\mu$2, then

$A_\mu$3

The corresponding horizon thermodynamic quantities are

$A_\mu$4

$A_\mu$5

$A_\mu$6

The $A_\mu$7 formula contains the usual AdS correction $A_\mu$8 associated with the nonrotating frame at infinity (Ali et al., 2023).

For the dyonic Kerr–Sen–AdS$A_\mu$9 solution, the charges and horizon data become

$F=dA$0

$F=dA$1

$F=dA$2

$F=dA$3

Here $F=dA$4 and $F=dA$5 are the electric and magnetic horizon potentials, and the dyonic scalar sector modifies all horizon quantities through $F=dA$6 and $F=dA$7 (Prihadi et al., 2023).

A recurrent technical point is that thermodynamics is simplest in a rest frame at infinity. In the dyonic AdS literature, the coordinate shift $F=dA$8 yields

$F=dA$9

after which the standard first law and Smarr relation take their conventional form (Wu et al., 2020). The extended-phase-space treatment of the purely electric solution uses the same logic: the AdS boundary rotation must be subtracted to identify the physical thermodynamic angular velocity (Ali et al., 5 Jan 2026).

4. Thermodynamic formulations

Two thermodynamic frameworks dominate the Kerr–Sen–AdS literature. In extended phase space, $$S=\int d^4 x\,\sqrt{-g}\Big[\,R-\tfrac12 (\nabla\phi)^2-\tfrac12 e^{2\phi}(\nabla\chi)^2-e^{-\phi}F^2\Big] +\tfrac{\chi}{2}\,\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda} +\sqrt{-g}\,\frac{4+e^{-\phi}+e^{\phi}(1+\chi^2)}{\ell^2}\,,$$0 varies through the pressure

$$S=\int d^4 x\,\sqrt{-g}\Big[\,R-\tfrac12 (\nabla\phi)^2-\tfrac12 e^{2\phi}(\nabla\chi)^2-e^{-\phi}F^2\Big] +\tfrac{\chi}{2}\,\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda} +\sqrt{-g}\,\frac{4+e^{-\phi}+e^{\phi}(1+\chi^2)}{\ell^2}\,,$$1

and the mass is interpreted as enthalpy. In the rest frame at infinity the first law and Smarr relation are

$$S=\int d^4 x\,\sqrt{-g}\Big[\,R-\tfrac12 (\nabla\phi)^2-\tfrac12 e^{2\phi}(\nabla\chi)^2-e^{-\phi}F^2\Big] +\tfrac{\chi}{2}\,\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda} +\sqrt{-g}\,\frac{4+e^{-\phi}+e^{\phi}(1+\chi^2)}{\ell^2}\,,$$2

$$S=\int d^4 x\,\sqrt{-g}\Big[\,R-\tfrac12 (\nabla\phi)^2-\tfrac12 e^{2\phi}(\nabla\chi)^2-e^{-\phi}F^2\Big] +\tfrac{\chi}{2}\,\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda} +\sqrt{-g}\,\frac{4+e^{-\phi}+e^{\phi}(1+\chi^2)}{\ell^2}\,,$$3

for the dyonic case (Wu et al., 2020). A compact Christodoulou–Ruffini-like formula in that framework is

$$S=\int d^4 x\,\sqrt{-g}\Big[\,R-\tfrac12 (\nabla\phi)^2-\tfrac12 e^{2\phi}(\nabla\chi)^2-e^{-\phi}F^2\Big] +\tfrac{\chi}{2}\,\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda} +\sqrt{-g}\,\frac{4+e^{-\phi}+e^{\phi}(1+\chi^2)}{\ell^2}\,,$$4

which reproduces the thermodynamic conjugates upon differentiation (Wu et al., 2020).

Restricted phase space thermodynamics (RPST) instead keeps the AdS radius fixed and varies Newton’s constant through the central charge

$$S=\int d^4 x\,\sqrt{-g}\Big[\,R-\tfrac12 (\nabla\phi)^2-\tfrac12 e^{2\phi}(\nabla\chi)^2-e^{-\phi}F^2\Big] +\tfrac{\chi}{2}\,\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda} +\sqrt{-g}\,\frac{4+e^{-\phi}+e^{\phi}(1+\chi^2)}{\ell^2}\,,$$5

removing the $$S=\int d^4 x\,\sqrt{-g}\Big[\,R-\tfrac12 (\nabla\phi)^2-\tfrac12 e^{2\phi}(\nabla\chi)^2-e^{-\phi}F^2\Big] +\tfrac{\chi}{2}\,\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda} +\sqrt{-g}\,\frac{4+e^{-\phi}+e^{\phi}(1+\chi^2)}{\ell^2}\,,$$6 term and replacing it with a chemical pair $$S=\int d^4 x\,\sqrt{-g}\Big[\,R-\tfrac12 (\nabla\phi)^2-\tfrac12 e^{2\phi}(\nabla\chi)^2-e^{-\phi}F^2\Big] +\tfrac{\chi}{2}\,\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda} +\sqrt{-g}\,\frac{4+e^{-\phi}+e^{\phi}(1+\chi^2)}{\ell^2}\,,$$7. In this formulation,

$$S=\int d^4 x\,\sqrt{-g}\Big[\,R-\tfrac12 (\nabla\phi)^2-\tfrac12 e^{2\phi}(\nabla\chi)^2-e^{-\phi}F^2\Big] +\tfrac{\chi}{2}\,\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda} +\sqrt{-g}\,\frac{4+e^{-\phi}+e^{\phi}(1+\chi^2)}{\ell^2}\,,$$8

and $$S=\int d^4 x\,\sqrt{-g}\Big[\,R-\tfrac12 (\nabla\phi)^2-\tfrac12 e^{2\phi}(\nabla\chi)^2-e^{-\phi}F^2\Big] +\tfrac{\chi}{2}\,\epsilon^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda} +\sqrt{-g}\,\frac{4+e^{-\phi}+e^{\phi}(1+\chi^2)}{\ell^2}\,,$$9 is a first-order homogeneous function of the extensive variables while the intensive variables are homogeneous of degree zero (Ali et al., 2023). In the notation used for RPST topology, the exact mass function is

$\Lambda=-3/\ell^2$0

with

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

This formulation is explicitly motivated as a holographically natural alternative to varying $\Lambda=-3/\ell^2$2 itself (Hazarika et al., 2024).

The extended and restricted formulations lead to different but related thermodynamic applications. In extended phase space, the Joule–Thomson expansion of the AdS Kerr–Sen black hole exhibits cooling and heating regions separated by a single-branched positively sloped inversion curve, and for

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

the paper reports

$\Lambda=-3/\ell^2$4

The same study emphasizes that the ratio depends on $\Lambda=-3/\ell^2$5 and $\Lambda=-3/\ell^2$6, even though near-$\Lambda=-3/\ell^2$7 behavior is recovered for particular small values of those parameters (Alipour et al., 2024).

5. Phase structure and thermodynamic topology

Within RPST, critical points are located from the inflection conditions on the fixed-$\Lambda=-3/\ell^2$8 temperature–entropy curve,

$\Lambda=-3/\ell^2$9

Because the algebra is cumbersome, the critical values are obtained numerically. Below criticality, $$I=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\Big(R-\tfrac{1}{2}(\partial\phi)^2-\tfrac{1}{2}e^{2\phi}(\partial\chi)^2-e^{-\phi}F^2+\tfrac{\chi}{2}F\tilde{F}\Big)+I_\Lambda,$$0 shows a van der Waals-like oscillatory segment and the Helmholtz free energy $$I=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\Big(R-\tfrac{1}{2}(\partial\phi)^2-\tfrac{1}{2}e^{2\phi}(\partial\chi)^2-e^{-\phi}F^2+\tfrac{\chi}{2}F\tilde{F}\Big)+I_\Lambda,$$1 exhibits a swallowtail; at the transition temperature small and large black holes are stable while the intermediate branch is metastable. At criticality the oscillation disappears and the swallowtail terminates in a cusp, signaling a second-order critical point. The resulting phenomenology closely parallels the RPST behavior of RN–AdS and Kerr–AdS black holes, which the authors interpret as evidence for an underlying universality (Ali et al., 2023).

In extended phase space, the same qualitative structure reappears in a more conventional black-hole chemistry setting. The 2026 analysis introduces the dimensionless parameter

$$I=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\Big(R-\tfrac{1}{2}(\partial\phi)^2-\tfrac{1}{2}e^{2\phi}(\partial\chi)^2-e^{-\phi}F^2+\tfrac{\chi}{2}F\tilde{F}\Big)+I_\Lambda,$$2

and fits the critical data through functions $$I=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\Big(R-\tfrac{1}{2}(\partial\phi)^2-\tfrac{1}{2}e^{2\phi}(\partial\chi)^2-e^{-\phi}F^2+\tfrac{\chi}{2}F\tilde{F}\Big)+I_\Lambda,$$3 such that

$$I=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\Big(R-\tfrac{1}{2}(\partial\phi)^2-\tfrac{1}{2}e^{2\phi}(\partial\chi)^2-e^{-\phi}F^2+\tfrac{\chi}{2}F\tilde{F}\Big)+I_\Lambda,$$4

The same work reports oscillatory $$I=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\Big(R-\tfrac{1}{2}(\partial\phi)^2-\tfrac{1}{2}e^{2\phi}(\partial\chi)^2-e^{-\phi}F^2+\tfrac{\chi}{2}F\tilde{F}\Big)+I_\Lambda,$$5, swallowtail $$I=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\Big(R-\tfrac{1}{2}(\partial\phi)^2-\tfrac{1}{2}e^{2\phi}(\partial\chi)^2-e^{-\phi}F^2+\tfrac{\chi}{2}F\tilde{F}\Big)+I_\Lambda,$$6, no reentrant or triple-point behavior, and an Ehrenfest analysis in which $$I=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\Big(R-\tfrac{1}{2}(\partial\phi)^2-\tfrac{1}{2}e^{2\phi}(\partial\chi)^2-e^{-\phi}F^2+\tfrac{\chi}{2}F\tilde{F}\Big)+I_\Lambda,$$7, $$I=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\Big(R-\tfrac{1}{2}(\partial\phi)^2-\tfrac{1}{2}e^{2\phi}(\partial\chi)^2-e^{-\phi}F^2+\tfrac{\chi}{2}F\tilde{F}\Big)+I_\Lambda,$$8, and $$I=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\Big(R-\tfrac{1}{2}(\partial\phi)^2-\tfrac{1}{2}e^{2\phi}(\partial\chi)^2-e^{-\phi}F^2+\tfrac{\chi}{2}F\tilde{F}\Big)+I_\Lambda,$$9 all diverge at the critical point while the Prigogine–Defay ratio satisfies

$$I_\Lambda=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\frac{4+e^{-\phi}+e^\phi(1+\chi^2)}{l^2},$$0

That establishes the endpoint transition as second order (Ali et al., 5 Jan 2026).

A distinct but increasingly important line of work studies the thermodynamic topology of Kerr–Sen–AdS black holes. In extended phase space, the generalized off-shell free energy

$$I_\Lambda=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\frac{4+e^{-\phi}+e^\phi(1+\chi^2)}{l^2},$$1

defines a vector field whose zeros correspond to on-shell black-hole states. For representative parameter choices, the Kerr–Sen–AdS solution exhibits three branches—small, intermediate, and large black holes—with winding numbers

$$I_\Lambda=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\frac{4+e^{-\phi}+e^\phi(1+\chi^2)}{l^2},$$2

so that the total topological charge is

$$I_\Lambda=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\frac{4+e^{-\phi}+e^\phi(1+\chi^2)}{l^2},$$3

The same analysis finds that varying the dilaton parameter $$I_\Lambda=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\frac{4+e^{-\phi}+e^\phi(1+\chi^2)}{l^2},$$4 moves the zeroes but does not change the total topological class, whereas rotation is crucial for the emergence of the three-branch structure (Rehan et al., 25 Mar 2026).

In RPST, the topological classification becomes ensemble dependent. The off-shell construction yields $$I_\Lambda=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\frac{4+e^{-\phi}+e^\phi(1+\chi^2)}{l^2},$$5 in the fixed $$I_\Lambda=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\frac{4+e^{-\phi}+e^\phi(1+\chi^2)}{l^2},$$6, fixed $$I_\Lambda=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\frac{4+e^{-\phi}+e^\phi(1+\chi^2)}{l^2},$$7, and fixed $$I_\Lambda=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\frac{4+e^{-\phi}+e^\phi(1+\chi^2)}{l^2},$$8 ensembles, while the fixed $$I_\Lambda=\frac{1}{16\pi G_N}\int d^4x\,\sqrt{|g|}\,\frac{4+e^{-\phi}+e^\phi(1+\chi^2)}{l^2},$$9 and fixed $g_{\mu\nu}$00 ensembles can produce total charges $g_{\mu\nu}$01, $g_{\mu\nu}$02, or $g_{\mu\nu}$03, depending on the thermodynamic parameters. In the cases with $g_{\mu\nu}$04, the same vector-field method identifies both a Hawking–Page point and a Davies point, with topological charges $g_{\mu\nu}$05 and $g_{\mu\nu}$06, respectively (Hazarika et al., 2024).

6. Extremality, holography, and the ultraspinning sector

Near extremality, the Kerr–Sen family supports several holographic descriptions. For the asymptotically flat Kerr–Sen black hole, the near-horizon near-extremal geometry contains an AdS$g_{\mu\nu}$07 throat and an AdS$g_{\mu\nu}$08/CFT$g_{\mu\nu}$09 analysis gives

$g_{\mu\nu}$10

with the Cardy formula reproducing

$g_{\mu\nu}$11

The same framework yields the Hawking temperature from the CFT holomorphic flux (Button et al., 2013). This construction is not itself AdS$g_{\mu\nu}$12 black-hole chemistry, but it provides the near-horizon benchmark from which later Kerr–Sen–AdS holography is developed.

For the dyonic Kerr–Sen and Kerr–Sen–AdS families, the near-horizon extremal geometry has $g_{\mu\nu}$13 isometry, enabling a Kerr/CFT analysis. In the ungauged dyonic case the extremality condition produces two mass branches $g_{\mu\nu}$14, and the left-moving central charges are

$g_{\mu\nu}$15

The paper then shows exact agreement between the Bekenstein–Hawking entropy and the CFT entropy in both branches. The same duality is stated to remain robust for nonzero AdS length, and the extremal dyonic Kerr–Sen–AdS black hole as well as its ultraspinning counterpart both reproduce the expected entropy through the Cardy formula (Sakti et al., 2022).

The ultraspinning Kerr–Sen–AdS$g_{\mu\nu}$16 black hole is obtained by redefining the azimuth and taking the $g_{\mu\nu}$17 limit. One standard prescription is $g_{\mu\nu}$18 followed by $g_{\mu\nu}$19, after which the azimuthal direction is compactified with a dimensionless period $g_{\mu\nu}$20 (Prihadi et al., 2023). The resulting horizon is noncompact but of finite area, and the near-pole geometry is that of a quotient of $g_{\mu\nu}$21, so the horizon is a “black spindle” rather than a compact $g_{\mu\nu}$22 (Wu et al., 2020).

Thermodynamically, the ultraspinning solution satisfies

$g_{\mu\nu}$23

together with the chirality condition

$g_{\mu\nu}$24

and an ultraspinning Christodoulou–Ruffini-like mass formula

$g_{\mu\nu}$25

(Wu et al., 2020).

A common misconception is that ultraspinning Kerr–Sen–AdS$g_{\mu\nu}$26 black holes are always super-entropic. The literature answers this negatively. The isoperimetric ratio can be smaller than, equal to, or greater than unity depending on the solution parameters. In the dyonic formulation,

$g_{\mu\nu}$27

whereas

$g_{\mu\nu}$28

Equivalent criteria are written as $g_{\mu\nu}$29 and $g_{\mu\nu}$30. This behavior is explicitly contrasted with ultraspinning Kerr–Newman–AdS$g_{\mu\nu}$31, which always violates the reverse isoperimetric inequality (Wu et al., 2020, Wu et al., 2020).

7. Chaotic dynamics and energy extraction

Kerr–Sen–AdS black holes have also become a testbed for dynamical probes of chaos. In the holographic shock-wave analysis of the dyonic Kerr–Sen–AdS$g_{\mu\nu}$32 background, the scrambling time for large entropy behaves as

$g_{\mu\nu}$33

supporting fast scrambling. The corresponding instantaneous minimal Lyapunov index is bounded by

$g_{\mu\nu}$34

where $g_{\mu\nu}$35 is the angular chemical potential in the stationary frame and $g_{\mu\nu}$36 is the shock angular momentum per unit energy. The same work reports that this bound becomes tight near extremality, but for small AdS radius $g_{\mu\nu}$37 and sufficiently large $g_{\mu\nu}$38 the Lyapunov exponent can exceed $g_{\mu\nu}$39; the electric and magnetic charges of the shock delay scrambling by

$g_{\mu\nu}$40

Analogous results are obtained for the ultraspinning geometry, where the bound is numerically obeyed (Prihadi et al., 2023).

A distinct notion of instability arises from charged-particle motion near unstable circular orbits. In the Kerr–Sen–AdS background, the local Lyapunov exponent is defined by

$g_{\mu\nu}$41

and is tested against

$g_{\mu\nu}$42

The 2025 analysis finds that the bound is often violated, especially near extremality, and that violations are enhanced for aligned charges $g_{\mu\nu}$43 and anti-aligned particle angular momentum and black-hole spin $g_{\mu\nu}$44. It also reports that a more negative cosmological constant shrinks the region where unstable orbits exist, but can strengthen the magnitude of the violations where those orbits persist (Lee et al., 1 Jun 2025).

Energy extraction has been analyzed both thermodynamically and through plasma processes. In the extended-phase-space Penrose discussion, the efficiency

$g_{\mu\nu}$45

approaches $g_{\mu\nu}$46 as $g_{\mu\nu}$47 and $g_{\mu\nu}$48 as $g_{\mu\nu}$49, while in the asymptotically flat uncharged limit it reproduces the familiar Kerr value $g_{\mu\nu}$50 (Ali et al., 5 Jan 2026). In the magnetized Kerr–Sen–AdS$g_{\mu\nu}$51 background, magnetic reconnection is found to extract energy even at relatively modest spins. In the circular-orbit region, increasing the dilatonic scalar charge $g_{\mu\nu}$52 and decreasing the AdS radius $g_{\mu\nu}$53 lower the spin threshold; the paper quotes an allowed spin window $g_{\mu\nu}$54 for one parameter set. In the plunging region the threshold can fall to $g_{\mu\nu}$55, and the extraction power and efficiency are reported to exceed those of the circular region. The same study also compares the reconnection power to the Blandford–Znajek process and finds parameter ranges in which reconnection is larger (Zeng et al., 14 Jul 2025).