General Relativity: An Introduction for Physicists · Problem 17.25

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · lens-thirring-precession-gravitomagnetic

Problem

In electromagnetism, the magnetic dipole moment of a current density distribution j⃗(y⃗)\vec{j}(\vec{y}) is defined by m⃗=12∫(y⃗×j⃗) d3y⃗\vec{m} = \frac{1}{2}\int(\vec{y}\times\vec{j})\, d^3\vec{y}, and the force and torque on the dipole in a magnetic field B⃗\vec{B} are given by F⃗=(m⃗⋅∇⃗)B⃗\vec{F} = (\vec{m}\cdot\vec{\nabla})\vec{B} and T⃗=m⃗×B⃗\vec{T} = \vec{m}\times\vec{B} respectively. Hence deduce that, in linearised gravity, the force and torque exerted by the gravitomagnetic field B⃗g\vec{B}_\mathrm{g} on a spinning body with spin angular momentum s⃗\vec{s} are given respectively by F⃗g=12(s⃗⋅∇⃗)B⃗gandT⃗g=12(s⃗×B⃗g).\vec{F}_\mathrm{g} = \tfrac{1}{2}(\vec{s}\cdot\vec{\nabla})\vec{B}_\mathrm{g} \qquad \text{and} \qquad \vec{T}_\mathrm{g} = \tfrac{1}{2}(\vec{s}\times\vec{B}_\mathrm{g}). Thus show that the spin angular momentum of the body will evolve as ds⃗dt=12(s⃗×B⃗g)\frac{d\vec{s}}{dt} = \tfrac{1}{2}(\vec{s}\times\vec{B}_\mathrm{g}) and therefore that s⃗\vec{s} precesses about B⃗g\vec{B}_\mathrm{g} with angular velocity Ω=−12∣B⃗g∣\Omega = -\frac{1}{2}|\vec{B}_\mathrm{g}| (i.e. in the negative sense). This is called the Lens--Thirring precession.
Intuitive explanation

The key insight is recognizing the deep analogy between electromagnetism and linearized gravity: a current distribution creates a magnetic field just as a spinning mass creates a gravitomagnetic field, and spin angular momentum s⃗\vec{s} plays the role that magnetic dipole moment m⃗\vec{m} plays electromagnetically. By mapping the electromagnetic force and torque formulas onto their gravitational counterparts with the substitution m⃗→12s⃗\vec{m} \to \frac{1}{2}\vec{s}, you inherit the same dynamical equations—the torque equation then directly gives the precession equation, showing that spin precesses around the gravitomagnetic field much like a magnetic dipole precesses in a magnetic field.

Intuitive explanation

The key insight is recognizing the deep analogy between electromagnetism and linearized gravity: a current distribution creates a magnetic field just as a spinning mass creates a gravitomagnetic field, and spin angular momentum s⃗\vec{s} plays the role that magnetic dipole moment m⃗\vec{m} plays electromagnetically. By mapping the electromagnetic force and torque formulas onto their gravitational counterparts with the substitution m⃗→12s⃗\vec{m} \to \frac{1}{2}\vec{s}, you inherit the same dynamical equations—the torque equation then directly gives the precession equation, showing that spin precesses around the gravitomagnetic field much like a magnetic dipole precesses in a magnetic field.

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