Step 23. Mass currents source frame-drag

\(\nabla^{2}\boldsymbol{\omega} = -\,16\pi\,\mathbf{J}\)

In the same weak, slowly varying limit as Step 22, the source of frame-drag (the shift \(\boldsymbol{\omega}\)) is the mass-current density \(\mathbf{J}\) (matter flowing through space). This is the gravitational analogue of Ampère’s law in electromagnetism: electric currents source the vector potential \(\mathbf{A}\); here, mass currents source the frame-flow potential \(\boldsymbol{\omega}\). In a convenient “gravito-Coulomb” gauge (\(\nabla\!\cdot\!\boldsymbol{\omega}=0\)), Einstein’s equations reduce to the vector Poisson equation

\[ \boxed{ \ \nabla^{2}\boldsymbol{\omega}(\mathbf{x}) \;=\; -\,16\pi\,\mathbf{J}(\mathbf{x}) \ } . \]

• Static matter (\(\mathbf{J}=0\)) ⇒ no frame-drag: \(\boldsymbol{\omega}=0\) (to leading order).
• Moving/rotating matter (\(\mathbf{J}\neq 0\)) ⇒ nonzero \(\boldsymbol{\omega}\), and its curl \(\mathbf{B}_g=\nabla\times\boldsymbol{\omega}\) gives the local vorticity of inertial frames (Lense–Thirring effect).

Green’s-function view (boundary \(\boldsymbol{\omega}\!\to\!0\) at infinity): because the source is \(-16\pi\,\mathbf{J}\), the solution is four times the EM analogue,

\[ \boldsymbol{\omega}(\mathbf{x}) \;=\; 4 \int \frac{\mathbf{J}(\mathbf{x}')}{|\mathbf{x}-\mathbf{x}'|}\,d^{3}x' \quad (\text{quasi-static}). \]

For a slowly rotating body with total angular momentum \(\mathbf{S}\), this yields the familiar \(1/r^{3}\) falloff outside: \(|\boldsymbol{\omega}|\sim 2\,|\mathbf{S}|/r^{3}\) (geometric units), and \(\mathbf{B}_g=\nabla\times\boldsymbol{\omega}\) sets the Lense–Thirring precession rate \(\boldsymbol{\Omega}_{\rm LT}\approx \tfrac12\,\mathbf{B}_g\).

Mini-Glossary