Rotation shows up as a swirl of the time-slicing frames. Package that swirl into the gravito-magnetic field
\[ \boxed{\;\mathbf{B}_g \;\equiv\; \nabla \times \boldsymbol{\omega}\;} \]
where \(\boldsymbol{\omega}(t,\mathbf{x})\) is the (linearized) shift / frame-drag potential. If \(\boldsymbol{\omega}\) is the “river” of space, then \(\mathbf{B}_g\) is the river’s vorticity—how much the flow tries to twist you.
What it means physically:
So \(\mathbf{E}_g\) (Step 20) captures “downhill in time,” while \(\mathbf{B}_g\) captures “swirl of frames.” Together they give the tidy gravito-EM picture for weak, time-varying situations.
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(\mathbf{B}_g\) | gravito-magnetic field | curl of frame-drag potential; measures rotational frame twist | \(\nabla\times\boldsymbol{\omega}\) (geom units: 1/length) | “Vorticity of the space-river” |
| \(\boldsymbol{\omega}\) | frame-drag (shift) potential | linearized 3-vector encoding how slices flow | velocity-like | “Current of the river of space” |
| \(\nabla\times\) | curl operator | measures local rotation of a vector field | 1/length | “Twist detector” |
| \(\boldsymbol{\Omega}_{\mathrm{LT}}\) | Lense–Thirring precession | gyro precession from frame drag (linear) | \(\approx \tfrac12\,\mathbf{B}_g\) (1/time) | “Slow spin a compass feels in a whirlpool” |
| \(\mathbf{v}\times\mathbf{B}_g\) | magnetic-like force term (preview) | velocity-coupled piece of test-body acceleration | acceleration-like | “Sideways shove from the swirl as you move” |