Black Hole Dragging: Frame & Environmental Effects

• Black hole dragging is a relativistic phenomenon where rotating black holes twist inertial frames, influencing particle trajectories and electromagnetic fields.
• The effect is quantified by off-diagonal metric components that modify gyroscopic precession, horizon kinematics, and the dynamics of accreting matter.
• Environmental drag mechanisms, including dynamical friction and shock-cone deflection, introduce nonlinear momentum transfer in complex astrophysical settings.

Black hole dragging denotes several distinct but related phenomena in relativistic astrophysics. In its classical general-relativistic sense, it is frame dragging: the rotation of a Kerr black hole twists local inertial frames through the off-diagonal metric component $g_{t\phi}$, enforcing co-rotation in the ergoregion and modifying the dynamics of particles, photons, electromagnetic fields, and accreting fluids. In other contexts, the same phrase is used for drag generated by external media or for matter carried with a black hole, as in environmental dynamical friction, common-envelope shock-cone deflection, or stars bound to a recoiling supermassive black hole. The literature therefore uses one term for mechanisms that are physically distinct but all involve momentum transfer, azimuthal advection, or co-motion induced by a black hole (Karas et al., 2012, Xin et al., 25 Jul 2025, Khonji et al., 29 Sep 2025).

1. Geometric basis: Kerr frame dragging and horizon rotation

In Kerr spacetime, frame dragging is encoded by the coupling of time and azimuth. In Boyer–Lindquist coordinates,

$$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$

with

$$\Delta = r^2 - 2Mr + a^2,\qquad \Sigma = r^2 + a^2\cos^2\theta,\qquad A = (r^2+a^2)^2 - a^2\Delta\sin^2\theta.$$

The local angular velocity of zero-angular-momentum observers is

$$\omega(r,\theta) = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2aMr}{A},$$

and the outer horizon and horizon angular velocity are

$$r_+ = M + \sqrt{M^2-a^2},\qquad \Omega_H = \frac{a}{2Mr_+}.$$

The ergosphere boundary is determined by

$r_{\rm erg}(\theta)=M+\sqrt{M^2-a^2\cos^2\theta}.$

Inside the ergosphere, all timelike observers must co-rotate; in the weak-field limit, the dragging reduces to the Lense–Thirring form

$\Omega_{\rm LT}=\frac{2GJ}{c^2 r^3}.$

These relations underlie essentially all strong-field dragging effects discussed below (Karas et al., 2012, Gutiérrez-Ruiz et al., 2018).

This geometric structure has two immediate consequences. First, any local notion of “non-rotation” becomes observer-dependent but is anchored by the ZAMO congruence. Second, the same $g_{t\phi}$ term that governs gyroscope precession also enters photon propagation, fluid angular velocity, and the measured electric and magnetic fields in local tetrads. A common misconception is to identify black hole dragging exclusively with spin-induced precession of test bodies. In the strong-field regime, the phenomenon is broader: it changes horizon kinematics, shifts null geodesics, alters effective potentials, and can reorganize entire magnetospheric or hydrodynamic structures.

2. Horizon dragging in composed black-hole systems

Frame dragging is not confined to vacuum Kerr geometry. In the composed black-hole–ring system, the angular velocity of the black-hole horizon is modified by the angular momentum of an external ring. The vacuum Kerr horizon relation used in that setting is

$\Omega_H^{\mathrm{Kerr}(J_H)}=\frac{J_H}{2M^2R_H},$

with

$$R_H\equiv r_+ = M+\sqrt{M^2-a^2},\qquad a\equiv \frac{J_H}{M}.$$

For a central black hole with $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$0, Will’s perturbative result gives

$$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$1

valid in the regime $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$2, with subleading corrections of order $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$3. In the near-horizon limit $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$4, this becomes

$$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$5

After adiabatic assimilation, one has $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$6, $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$7, and the final Kerr value is again

$$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$8

so the evolution of $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$9 is continuous (Hod, 2015).

The same continuity argument extends to nonzero black-hole spin. At capture,

$$\Delta = r^2 - 2Mr + a^2,\qquad \Sigma = r^2 + a^2\cos^2\theta,\qquad A = (r^2+a^2)^2 - a^2\Delta\sin^2\theta.$$0

and the near-horizon angular velocity just before assimilation satisfies

$$\Delta = r^2 - 2Mr + a^2,\qquad \Sigma = r^2 + a^2\cos^2\theta,\qquad A = (r^2+a^2)^2 - a^2\Delta\sin^2\theta.$$1

The asymptotic deviation from the vacuum Kerr value is therefore universal,

$$\Delta = r^2 - 2Mr + a^2,\qquad \Sigma = r^2 + a^2\cos^2\theta,\qquad A = (r^2+a^2)^2 - a^2\Delta\sin^2\theta.$$2

independent of $$\Delta = r^2 - 2Mr + a^2,\qquad \Sigma = r^2 + a^2\cos^2\theta,\qquad A = (r^2+a^2)^2 - a^2\Delta\sin^2\theta.$$3. The paper also proposes a compact interpolation for generic radii,

$$\Delta = r^2 - 2Mr + a^2,\qquad \Sigma = r^2 + a^2\cos^2\theta,\qquad A = (r^2+a^2)^2 - a^2\Delta\sin^2\theta.$$4

and explicitly labels this as a conjectured formula rather than a derived one (Hod, 2015).

The significance of this result is conceptual as well as technical. The horizon angular velocity is not determined solely by the black hole’s own spin when external rotating matter is present; near the horizon, the leading correction is linear in $$\Delta = r^2 - 2Mr + a^2,\qquad \Sigma = r^2 + a^2\cos^2\theta,\qquad A = (r^2+a^2)^2 - a^2\Delta\sin^2\theta.$$5 and universal in $$\Delta = r^2 - 2Mr + a^2,\qquad \Sigma = r^2 + a^2\cos^2\theta,\qquad A = (r^2+a^2)^2 - a^2\Delta\sin^2\theta.$$6. This sharply separates composed-system dragging from the vacuum Kerr relation and gives a controlled perturbative example of horizon kinematics shaped by external angular momentum.

3. Electromagnetic restructuring, magnetic nulls, and particle acceleration

Near a rotating black hole, frame dragging strongly deforms external magnetic fields. For an aligned asymptotically uniform field, Wald’s vacuum solution has

$$\Delta = r^2 - 2Mr + a^2,\qquad \Sigma = r^2 + a^2\cos^2\theta,\qquad A = (r^2+a^2)^2 - a^2\Delta\sin^2\theta.$$7

and the black hole acquires the Wald charge

$$\Delta = r^2 - 2Mr + a^2,\qquad \Sigma = r^2 + a^2\cos^2\theta,\qquad A = (r^2+a^2)^2 - a^2\Delta\sin^2\theta.$$8

For oblique fields, the Bičák–Janiš solution introduces explicit $$\Delta = r^2 - 2Mr + a^2,\qquad \Sigma = r^2 + a^2\cos^2\theta,\qquad A = (r^2+a^2)^2 - a^2\Delta\sin^2\theta.$$9-dependent twisting through

$$\omega(r,\theta) = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2aMr}{A},$$0

with the logarithmic term encoding the azimuthal winding produced by frame dragging (Karas et al., 2012).

The local physical fields are defined by

$$\omega(r,\theta) = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2aMr}{A},$$1

and the invariants

$$\omega(r,\theta) = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2aMr}{A},$$2

diagnose electric dominance and parallel electric fields. In the oblique case, Kerr frame dragging twists and layers the field near the ergosphere and horizon, producing X-type topologies and magnetic null points where $$\omega(r,\theta) = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2aMr}{A},$$3 in the local frame. The induced electric field develops a non-vanishing component threading the null, and electric field lines rise out of the equatorial plane and pass through the null vertically, providing an accelerating direction for charged particles (Karas et al., 2011, Karas et al., 2014).

Several quantitative trends are established. Nulls typically appear for spins $$\omega(r,\theta) = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2aMr}{A},$$4. As $$\omega(r,\theta) = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2aMr}{A},$$5 increases, the null recedes outward. For nearly extreme spin $$\omega(r,\theta) = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2aMr}{A},$$6, the null can cross outside the ergosphere in the equatorial plane. With combined fast rotation and translatory motion, the null can be found as far as

$$\omega(r,\theta) = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2aMr}{A},$$7

for $$\omega(r,\theta) = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2aMr}{A},$$8 and $$\omega(r,\theta) = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2aMr}{A},$$9, while in the extreme-spin case nulls appear already for modest transverse motion $$r_+ = M + \sqrt{M^2-a^2},\qquad \Omega_H = \frac{a}{2Mr_+}.$$0 (Karas et al., 2011). Boosting the black hole through the external field further deforms the field lines and the $$r_+ = M + \sqrt{M^2-a^2},\qquad \Omega_H = \frac{a}{2Mr_+}.$$1 surfaces, adding a motional contribution to the purely gravitomagnetic twist (Karas et al., 2012).

This mechanism differs from force-free Blandford–Znajek electrodynamics. In the latter, large-scale Poynting flux is central; in the oblique vacuum or low-density configurations considered here, the salient effect is local magnetic reconnection geometry and direct acceleration near nulls with nonzero $$r_+ = M + \sqrt{M^2-a^2},\qquad \Omega_H = \frac{a}{2Mr_+}.$$2. The literature therefore treats black hole dragging not merely as a source of global field-line rotation, but as a generator of local accelerator sites in the strong-gravity region.

4. Dynamical stabilization, chaos suppression, and dragged tori

Frame dragging also reshapes orbital dynamics beyond the integrable Kerr case. In the PRS family of exact stationary axisymmetric Einstein–Maxwell solutions, the local dragging strength can be measured by the spacetime vorticity scalar

$$r_+ = M + \sqrt{M^2-a^2},\qquad \Omega_H = \frac{a}{2Mr_+}.$$3

which in Ernst variables becomes

$$r_+ = M + \sqrt{M^2-a^2},\qquad \Omega_H = \frac{a}{2Mr_+}.$$4

At fixed gravitational quadrupole $$r_+ = M + \sqrt{M^2-a^2},\qquad \Omega_H = \frac{a}{2Mr_+}.$$5, increasing the dimensionless spin from $$r_+ = M + \sqrt{M^2-a^2},\qquad \Omega_H = \frac{a}{2Mr_+}.$$6 to $$r_+ = M + \sqrt{M^2-a^2},\qquad \Omega_H = \frac{a}{2Mr_+}.$$7 reconstructs KAM tori from initially chaotic configurations: nonmonotonic rotation curves and resonant plateaus disappear, and the inferred correlation dimension collapses toward $$r_+ = M + \sqrt{M^2-a^2},\qquad \Omega_H = \frac{a}{2Mr_+}.$$8 (Gutiérrez-Ruiz et al., 2018).

In magnetized or charged configurations the same tendency survives but competes with chaos-inducing deformations. For $$r_+ = M + \sqrt{M^2-a^2},\qquad \Omega_H = \frac{a}{2Mr_+}.$$9 and $r_{\rm erg}(\theta)=M+\sqrt{M^2-a^2\cos^2\theta}.$0, increasing $r_{\rm erg}(\theta)=M+\sqrt{M^2-a^2\cos^2\theta}.$1 to $r_{\rm erg}(\theta)=M+\sqrt{M^2-a^2\cos^2\theta}.$2 shortens resonant plateaus and regular motion predominates; for $r_{\rm erg}(\theta)=M+\sqrt{M^2-a^2\cos^2\theta}.$3, chaotic zones remain because the charge’s prolate contribution partly offsets the stabilizing vorticity (Gutiérrez-Ruiz et al., 2018). This identifies frame dragging as a mechanism that can suppress rather than generate chaos in non-Kerr backgrounds.

A different manifestation appears in perfect-fluid accretion tori near the Kerr ergoregion. In the Polish-doughnut construction, equilibrium is encoded in the effective potential $r_{\rm erg}(\theta)=M+\sqrt{M^2-a^2\cos^2\theta}.$4, with torus centers at minima and cusps at maxima. “Dragged” configurations are defined as corotating tori entirely contained in the outer ergoregion, while “partially contained” tori straddle the outer ergosurface. The outer ergoregion is

$r_{\rm erg}(\theta)=M+\sqrt{M^2-a^2\cos^2\theta}.$5

and on the equatorial plane $r_{\rm erg}(\theta)=M+\sqrt{M^2-a^2\cos^2\theta}.$6. The paper argues that dragged tori can be observed only for extremely rapid rotation,

$r_{\rm erg}(\theta)=M+\sqrt{M^2-a^2\cos^2\theta}.$7

with maximum equatorial elongation

$r_{\rm erg}(\theta)=M+\sqrt{M^2-a^2\cos^2\theta}.$8

for tori entirely within $r_{\rm erg}(\theta)=M+\sqrt{M^2-a^2\cos^2\theta}.$9 (Pugliese et al., 2022).

The same analysis connects dragged tori to QPO production and to a proposed instability. Smaller dragged tori can be subjected to a characteristic instability, effect of frame dragging, which can lead to “disk exfoliation”: destruction of the torus combined with accretion and processes near the horizon. This suggests that ergoregion-confined fluid structures are not merely geometric curiosities but dynamically fragile configurations whose variability may be directly tied to strong dragging (Pugliese et al., 2022).

5. Environmental drag: scalar fields, gas, and common envelopes

A separate usage of black hole dragging refers to environmental drag rather than frame dragging. In relativistic scalar dark matter, the relevant forces arise from dynamical friction and accretion-induced drag: a black hole moving through a scalar field builds a gravitational wake, experiences a force opposite to its motion, and accretes scalar momentum through the horizon. This environmental drag is explicitly distinguished from Lense–Thirring frame dragging in vacuum (Xin et al., 25 Jul 2025).

For binary black holes in a scalar medium, the drag is not a simple superposition of two isolated black holes. Wake–wake interactions, tidal distortions, and interference make the total force and torque nonlinear, with distinct regimes set by the ratio of de Broglie wavelength to binary separation: $\Omega_{\rm LT}=\frac{2GJ}{c^2 r^3}.$0 gives strong interference and drag suppression, $\Omega_{\rm LT}=\frac{2GJ}{c^2 r^3}.$1 gives the strongest nonlinearities, and $\Omega_{\rm LT}=\frac{2GJ}{c^2 r^3}.$2 yields weaker wakes. In the simulations, the drag force opposing the wind grows approximately linearly with $\Omega_{\rm LT}=\frac{2GJ}{c^2 r^3}.$3 around $\Omega_{\rm LT}=\frac{2GJ}{c^2 r^3}.$4 and tends to saturate above $\Omega_{\rm LT}=\frac{2GJ}{c^2 r^3}.$5; the net torque is consistently negative; and a transverse “Magnus-like” force is observed, with $\Omega_{\rm LT}=\frac{2GJ}{c^2 r^3}.$6 for $\Omega_{\rm LT}=\frac{2GJ}{c^2 r^3}.$7 in representative cases (Xin et al., 25 Jul 2025).

In gaseous media, hypersonic dynamical friction admits a ballistic analytic treatment. For a point mass moving with speed $\Omega_{\rm LT}=\frac{2GJ}{c^2 r^3}.$8 through gas of density $\Omega_{\rm LT}=\frac{2GJ}{c^2 r^3}.$9, the accretion rate is

$g_{t\phi}$0

and the total drag takes the Coulomb-log form

$g_{t\phi}$1

with the lower cutoff fixed by the nonlinear inner wake,

$g_{t\phi}$2

The analytic model reproduces both the Bondi–Hoyle–Lyttleton accretion rate and the total drag in axisymmetric simulations to within the quoted accuracy (Canto et al., 2011).

Common-envelope simulations extend this environmental-drag picture to relativistic hydrodynamics. In a nonrotating background, increasing the density-gradient parameter $g_{t\phi}$3 drags the shock cone progressively toward the direction of motion; the deflection angle is fit by

$g_{t\phi}$4

so the cone can become orthogonal or even point upstream for sufficiently large gradients (Cruz-Osorio et al., 2020). In the Kerr common-envelope problem, the dominant dragging of the shock cone is still produced by the envelope’s transverse pressure and density gradients, but highly rotating black holes add “additional dragging near the black hole,” most evident for $g_{t\phi}$5 and supersonic flow (Cruz-Osorio et al., 8 Jul 2026).

The observational significance of these environmental effects is heterogeneous. For scalar-field drag, the estimated gravitational-wave dephasing is well below detectability for ground-based detectors at typical galactic dark-matter densities, but dense environments such as dark-matter spikes or superradiant clouds may produce appreciable effects in the LISA band (Xin et al., 25 Jul 2025). In common envelopes, enhanced drag and momentum accretion directly affect orbital decay and secular binary evolution (Cruz-Osorio et al., 2020, Cruz-Osorio et al., 8 Jul 2026).

6. Electromagnetic and timing observables of frame dragging

The dragging of light in Kerr spacetime is intrinsically asymmetric. For equatorial null geodesics with signed impact parameter $g_{t\phi}$6, the exact bending angle can be written in terms of elliptic integrals, and its weak-deflection expansion is

$g_{t\phi}$7

The terms odd in $g_{t\phi}$8 produce the prograde–retrograde asymmetry, and the signed critical impact parameter is

$g_{t\phi}$9

This asymmetry is the lensing imprint of frame dragging and underlies the skew of Kerr shadows and photon-ring structure (Iyer, 2018).

The same dragging enters spectroscopic inference of black-hole parameters. For circular equatorial emitters, defining $\Omega_H^{\mathrm{Kerr}(J_H)}=\frac{J_H}{2M^2R_H},$0 and $\Omega_H^{\mathrm{Kerr}(J_H)}=\frac{J_H}{2M^2R_H},$1 at the two extrema, the mass follows from

$\Omega_H^{\mathrm{Kerr}(J_H)}=\frac{J_H}{2M^2R_H},$2

while the central bending parameter for a line-of-sight photon is

$\Omega_H^{\mathrm{Kerr}(J_H)}=\frac{J_H}{2M^2R_H},$3

The total shifts depend on azimuth $\Omega_H^{\mathrm{Kerr}(J_H)}=\frac{J_H}{2M^2R_H},$4 through a dragging-modified bending parameter $\Omega_H^{\mathrm{Kerr}(J_H)}=\frac{J_H}{2M^2R_H},$5, allowing the mass and spin of a Kerr black hole to be expressed analytically in terms of redshift, blueshift, orbital radius, and, if necessary, the black hole’s peculiar motion (Banerjee et al., 2022).

In pulsar timing around Sgr A*, frame dragging appears as a propagation delay of photons on Kerr null geodesics. The exact time-of-flight is expressed in terms of elliptic integrals, and the frame-dragging delay is defined as the difference between exact Kerr and exact Schwarzschild propagation delays at fixed orbital geometry. For edge-on orbits, the delay is asymmetric around superior conjunction because the primary ray switches from counter-rotating to co-rotating before conjunction. The exact treatment remains finite where post-Newtonian formulas diverge or overestimate the signal, and for $\Omega_H^{\mathrm{Kerr}(J_H)}=\frac{J_H}{2M^2R_H},$6 the frame-dragging contribution is in the $\Omega_H^{\mathrm{Kerr}(J_H)}=\frac{J_H}{2M^2R_H},$7–$\Omega_H^{\mathrm{Kerr}(J_H)}=\frac{J_H}{2M^2R_H},$8 range for $\Omega_H^{\mathrm{Kerr}(J_H)}=\frac{J_H}{2M^2R_H},$9 and spins $$R_H\equiv r_+ = M+\sqrt{M^2-a^2},\qquad a\equiv \frac{J_H}{M}.$$0–$$R_H\equiv r_+ = M+\sqrt{M^2-a^2},\qquad a\equiv \frac{J_H}{M}.$$1 (Ben-Salem et al., 2022).

Horizon-scale imaging introduces a complementary set of diagnostics. In GRMHD plus polarized radiative-transfer simulations, retrograde accretion flows are forced by strong frame dragging to flip the sign of their tangential velocity as they plunge, producing “S”-shaped total-intensity streams and a radial flip in the handedness of the EVPA pattern. The flip radius moves outward with increasing $$R_H\equiv r_+ = M+\sqrt{M^2-a^2},\qquad a\equiv \frac{J_H}{M}.$$2, and the linear-polarization diagnostic based on the coefficient $$R_H\equiv r_+ = M+\sqrt{M^2-a^2},\qquad a\equiv \frac{J_H}{M}.$$3 is particularly clean in retrograde MAD models (Ricarte et al., 2022). Semi-analytical imaging of thin retrograde equatorial flows gives a related three-radius hierarchy,

$$R_H\equiv r_+ = M+\sqrt{M^2-a^2},\qquad a\equiv \frac{J_H}{M}.$$4

where $$R_H\equiv r_+ = M+\sqrt{M^2-a^2},\qquad a\equiv \frac{J_H}{M}.$$5 is the flow turning radius in spacetime, $$R_H\equiv r_+ = M+\sqrt{M^2-a^2},\qquad a\equiv \frac{J_H}{M}.$$6 the polarization-flip radius, and $$R_H\equiv r_+ = M+\sqrt{M^2-a^2},\qquad a\equiv \frac{J_H}{M}.$$7 the source radius whose primary image turns on the screen (Wang et al., 21 Aug 2025).

7. Horizon-wave signatures and post-merger stellar dragging

Gravitational waves offer a direct probe of near-horizon frame dragging. For a Kerr remnant, the horizon quantities are

$$R_H\equiv r_+ = M+\sqrt{M^2-a^2},\qquad a\equiv \frac{J_H}{M}.$$8

The source-driven horizon mode is

$$R_H\equiv r_+ = M+\sqrt{M^2-a^2},\qquad a\equiv \frac{J_H}{M}.$$9

so the direct wave oscillates near $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$00 because of frame dragging and decays at a rate set by $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$01 because of gravitational redshift. In GW250114, the remnant parameters were measured as

$$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$02

and the direct-wave component was detected with matched-filter signal-to-noise ratio

$$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$03

in Hanford and

$$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$04

in Livingston, with frequency and damping consistent with Kerr predictions (Lu et al., 1 Oct 2025).

In yet another usage, “black hole dragging” describes the co-motion of stars with a recoiling supermassive black hole after a merger. A gravitational-wave kick of speed $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$05 leaves some stars bound within the characteristic scale

$$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$06

while the black hole’s fallback through the stellar core also ejects and heats background stars through dynamical friction. At intermediate recoil velocities of $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$07–$$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$08, both effects become important, and the density of bound stars can exceed that of the background stellar core, producing a visible nuclear star cluster (Khonji et al., 29 Sep 2025).

The resulting clusters have realistic sizes, masses, and velocity dispersions when measured analogously to observed systems, and the predicted stellar populations are distinctive: colors, ages, and chemistry should be indistinguishable from non-NSC central stars, while ellipticities remain low (Khonji et al., 29 Sep 2025). This usage is physically separate from Kerr frame dragging, but it preserves the core idea of matter being carried by black-hole motion in a way that imprints itself on galactic structure.

Taken together, these lines of work show that black hole dragging is not a single mechanism but a family of strong-gravity and black-hole–environment interactions. In vacuum Kerr geometry it is the dragging of inertial frames, encoded by $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$09 and measured through $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$10 and $$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right) dt^2 - \frac{4aMr\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}dr^2 + \Sigma\,d\theta^2 + \frac{A\sin^2\theta}{\Sigma}\,d\phi^2,$$11. In composed systems it shifts horizon rotation away from the vacuum Kerr relation. In electromagnetic settings it twists field lines, creates nulls, and opens acceleration channels. In dynamical systems it can regularize phase space or destabilize ergoregion tori. In external media it becomes environmental drag through wakes, accretion, and shock-cone deflection. In observations it appears in lensing asymmetries, spectroscopic redshift patterns, pulsar timing delays, horizon-scale imaging, merger-waveform structure, and even the post-merger transport of stars in galactic nuclei.