Lense-Thirring precession ( $\dot{L} = Ω_{LT} \times L$ )

Lense-Thirring precession is the rotation of orbital angular momentum caused by the gravitational field of a spinning mass. In the secular weak-field picture, the precession vector is parallel to the central spin. Thus an annulus has torque proportional to

J_{h} \times δ J_{d}

, perpendicular to

J_{h}

. Summing such torques preserves this orthogonality even if their radial coefficients differ. Dissipation and the internal response of a warped accretion disk can supplement precession with an alignment torque.

Table of contents
- Frame dragging inside a slowly rotating spherical shell Lense-Thirring precession
  - Inertial coordinates inside a slowly rotating spherical shell Frame dragging inside a slowly rotating spherical shell
- Black-hole and disk alignment under internal torque Lense-Thirring precession
  - Counteralignment of a black hole and accretion disk Black-hole and disk alignment under internal torque
  - Persistent-torque criterion for asymptotic spin alignment Black-hole and disk alignment under internal torque

Frame dragging inside a slowly rotating spherical shell ( $ω = \frac{4 GM}{3 R} Ω$ )

 0  0

For a weakly gravitating thin spherical shell of mass

M

, radius

R

and uniform angular velocity

Ω

, its interior linearized gravity metric has constant scalar potentials

A = - GM / R

C = GM / R

and

g_{0 i} = ω (z \times x)_{i}

. The scalar mass shell is a monopole. Its mass current is a dipole: regular interior and decaying exterior radial solutions are proportional to

r

and

r^{- 2}

, and their derivative jump fixes the factor

4/3

. Local inertial axes co-rotate with the shell at

ω

relative to asymptotic nonrotating axes. This is the leading interior Lense-Thirring precession, neglecting quadratic rotation and higher gravitational orders.

Inertial coordinates inside a slowly rotating spherical shell ( $T = (1 - ϵ) t, X = (1 + ϵ) R_{z} (- ω t) x$ )

 0  0

At linear order in

ϵ = GM / R

and the leading rotation term, this coordinate change takes the interior shell metric to Minkowski spacetime. Consequently the retained interior Riemann curvature tensor vanishes although inertial axes rotate relative to distant reference axes. A freely moving particle has

X = X_{0} + V T

, while its original coordinates obey

d^{2} x / d t^{2} = 2 ω \times d x / d t

to first rotation order. Terms of order

ω^{2}

would include centrifugal acceleration and are outside this approximation. A gyroscope at rest to the retained order has its parallel-transported spin precessing at

ω

in the original coordinates.

Black-hole and disk alignment under internal torque ( $\dot{J}_{h} = - K_{1} J_{h} \times J_{d} - K_{2} J_{h} \times (J_{h} \times J_{d})$ )

 0  0

For the opposite internal torques on a black hole and its accretion disk, the total angular momentum

J_{t}

and the black-hole spin magnitude

J_{h}

are constant. If

A = K_{2} ∣ J_{h} \times J_{d} ∣^{2}

, then

d (J_{h} \cdot J_{t}) / d t = A

and

d J_{d}^{2} / d t = - 2 A

. The

K_{1}

term produces precession; the positive

K_{2}

term increases the alignment of the hole with the total vector while decreasing the disk angular-momentum magnitude. The exact direction equation is

\overset{c}{˙} = K_{2} J_{h} J_{t} (1 - c^{2})

for

c = cos ∠ (J_{h}, J_{t})

Counteralignment of a black hole and accretion disk ( $J_{t} < J_{h} ⟺ cos θ_{0} < - J_{d, 0} / (2 J_{h})$ )

 0  0

After persistent dissipative alignment of the hole with the conserved total angular momentum, the final vectors are

J_{h, f} = J_{h} J_{t}

and

J_{d, f} = (J_{t} - J_{h}) J_{t}

. The disk is therefore antiparallel to the hole if

J_{t} < J_{h}

. Squaring

J_{t}^{2} = J_{h}^{2} + J_{d, 0}^{2} + 2 J_{h} J_{d, 0} cos θ_{0}

gives the displayed initial-angle criterion. The disk and hole need not finish aligned with each other even though the hole aligns with the total. Equality

J_{t} = J_{h}

leaves zero final disk angular momentum rather than a defined disk spin direction.

Persistent-torque criterion for asymptotic spin alignment ( $\int_{0}^{\infty} K_{2} (t) d t = \infty$ )

 0  0

For nonzero

J_{h}, J_{t}

and a noncollinear initial configuration, the alignment equation integrates to

artanh c (t) = artanh c (0) + J_{h} J_{t} \int_{0}^{t} K_{2} (s) d s

. Hence the positive dissipative torque aligns the hole with the total vector if its integrated strength diverges. A fixed positive coefficient is sufficient. Pointwise positivity alone need not suffice for time-dependent coefficients: with