In the weak-field, slow-motion limit (linearized GR), we keep only first-order wiggles of the time potential \(\Phi(t,\mathbf{x})\) and the shift (frame-drag) vector \(\boldsymbol{\omega}(t,\mathbf{x})\). In this regime, free-fall looks almost Newtonian, so it’s convenient to package gravity into fields that act like forces. The first is the gravito-electric field:
\[ \boxed{\,\mathbf{E}_g \;=\; -\,\nabla \Phi \;-\; \tfrac12\,\partial_t \boldsymbol{\omega}\, }. \]
Operationally: for slow test particles, \(\mathbf{E}_g\) acts like an acceleration field—you can think “drop a pebble and this is what it feels,” up to linear order. In Step 21 we’ll define the companion gravito-magnetic field \(\mathbf{B}_g=\nabla\times\boldsymbol{\omega}\), which couples to motion (via a $ _g$ term).
| Symbol | Name | Meaning | Value / Units | Metaphor | ||||
|---|---|---|---|---|---|---|---|---|
| \(\mathbf{E}_g\) | gravito-electric field | Newtonian-like “pull,” plus induction from changing frames | acceleration-like (geom. units: 1/length) | “Downhill tug of the time landscape” | ||||
| \(\Phi(t,\mathbf{x})\) | time potential | sets lapse \(N=e^{\Phi}\); sources the static part of \(\mathbf{E}_g\) | dimensionless | “Altitude of time” | ||||
| \(\boldsymbol{\omega}(t,\mathbf{x})\) | frame-drag potential (shift) | encodes how spatial frames flow/drag | velocity-like (geom. units) | “River-flow of space” | ||||
| \(\nabla\) | gradient | spatial slope operator on a slice | 1/length | “Compass that points downhill” | ||||
| \(\partial_t\) | time derivative | rate of change at fixed position | 1/time | “How fast the river’s current is changing” | ||||
| (linearized) | weak-field limit | keep only first-order perturbations | small ( | , | ) | “Small ripples on calm water” |