greatitself.
Continue
advanced30 min studyLesson 813 in Physics

Weak-field Newtonian limit

Recover acceleration and Poisson’s equation from a controlled weak metric, keeping the pressure and cosmological assumptions visible.

Before you begin
On this page

The question for today

What you’ll understand

  • Identify the weak, slow, and nearly static approximations separately.
  • Recover Poisson’s equation with the correct source coefficient.
  • Distinguish small potential, small speed, and small acceleration.

Newtonian gravity emerges from general relativity in a regime, not from deleting every component except the temporal one. The metric controls clocks, spatial measurements, and free trajectories together. We work in a nearly static weak field with slow test bodies, slowly moving source matter, and pressure small compared with its rest-energy density. A convenient scalar perturbation chart contains temporal potential Phi and spatial potential Psi. Coordinate choices matter, while measurable clock comparisons and tidal relative accelerations can be formulated independently of those choices.

Start with a metric and an ordered approximation

Take both potentials to have velocity squared units and magnitude much smaller than c squared. Neglect time derivatives relative to spatial derivatives, and keep only terms linear in the dimensionless potentials. The geodesic equation for a test body moving much slower than light is then dominated by the temporal connection component. Using coordinate time as the trajectory parameter gives acceleration minus the spatial gradient of Phi at this order. Velocity corrections and time dependence are not included in this reduced equation.

A static clock has proper-time rate approximately one plus Phi divided by c squared. Clock differences therefore probe potential differences once the time normalization is specified. Small potential does not require small local acceleration: a potential can vary over a short length. Likewise a slowly moving source does not make every possible test particle slow. Light propagation needs the spatial metric as well, which is why extending the slow-body acceleration formula to photons misses a relativistic contribution to bending.

ds2=−(1+2Φc2)c2dt2+(1−2Ψc2)δijdxidxj,d2xdt2≃−∇Φds^2=-\left(1+\frac{2\Phi}{c^2}\right)c^2dt^2+\left(1-\frac{2\Psi}{c^2}\right)\delta_{ij}dx^idx^j,\qquad \frac{d^2\boldsymbol x}{dt^2}\simeq-\boldsymbol\nabla\Phi
Phi and Psi have m²/s² units. The acceleration relation assumes a weak, nearly static metric and a slow test body; the displayed line element keeps only linear scalar perturbations.↔ Scroll wide equations horizontally.

The trace-reversed equation fixes the source factor

The temporal Ricci component is approximately the Laplacian of Phi divided by c squared. For nonrelativistic matter, the leading temporal stress is mass density times c squared, while the trace is approximately its negative. In the trace-reversed field equation these two contributions add to half that temporal energy density. The resulting coefficient is four pi G, not eight pi G. This factor is a useful diagnostic of a derivation that has overlooked the trace term.

With Lambda retained, its temporal trace-reversed contribution gives minus Lambda times c squared in Poisson’s equation. The ordinary local Newtonian limit drops it only when it is negligible compared with the relevant matter or tidal scale. When Lambda is negligible, negligible anisotropic stress and isolated decaying boundary conditions give equal scalar potentials at this order. A uniform additive potential can be absorbed into a local time normalization at leading order; gradients determine slow-body acceleration and second derivatives determine Newtonian tidal effects.

Rμν=8πGc4(Tμν−12gμνT)+Λgμν,∇2Φ≃4πGρ−Λc2R_{\mu\nu}=\frac{8\pi G}{c^4}\left(T_{\mu\nu}-\frac12g_{\mu\nu}T\right)+\Lambda g_{\mu\nu},\qquad \nabla^2\Phi\simeq4\pi G\rho-\Lambda c^2
rho is rest-mass density in kg/m³ and T is the metric trace. The Poisson approximation assumes pressure and source velocities are subleading; Lambda can be dropped only with an explicit scale comparison.↔ Scroll wide equations horizontally.

Boundary conditions and material domains still matter

Outside a spherical isolated source, neglecting Lambda and setting the potential to zero at infinity gives minus GM divided by radius. Inside a uniform-density sphere, spherical integration instead gives an acceleration proportional to radius, regular at the center. One cannot continue the exterior inverse square law through extended matter. Matching the potential and its radial derivative at the surface fixes integration constants when no singular surface layer has been inserted.

The weak-field criterion concerns the metric perturbation over the region being modeled, and the slow-motion criterion concerns the actual orbit. Neither condition alone proves that every correction is negligible for a high-precision experiment. A small correction can accumulate over many orbital periods, and pressure or anisotropic stress can spoil the simple source reduction. The planar field instrument below demonstrates a linear source response only. Its fixed softening and dimensionality prevent interpreting its arrows as a solved gravitational field of a spherical source.

Φout=−GMr,gin=−4πGρ3r(r<R)\Phi_{\mathrm{out}}=-\frac{GM}{r},\qquad \boldsymbol g_{\mathrm{in}}=-\frac{4\pi G\rho}{3}\boldsymbol r\quad(r<R)
The exterior formula assumes r exceeds the source radius R; the interior formula assumes uniform density. Both neglect Lambda and use the Newtonian regime, with g measured in m/s².↔ Scroll wide equations horizontally.

Work it through

Weak potential can coexist with strong acceleration

An ideal spherical source has geometric mass length m=GM/c2=1000 mm=GM/c^2=1000\,\mathrm m. At an exterior radius r=108 mr=10^8\,\mathrm m, take c=3×108 m/sc=3\times10^8\,\mathrm{m/s}. Estimate the potential ratio, static clock rate relative to infinity, and inward slow-body acceleration.

  1. The dimensionless potential is Φ/c2=−m/r=−10−5\Phi/c^2=-m/r=-10^{-5}. This is small, so the leading weak-field clock expansion is controlled at this radius.

  2. The clock rate is dτ/dt≃1+Φ/c2=0.99999d\tau/dt\simeq1+\Phi/c^2=0.99999. Here t is normalized by an asymptotically distant static clock.

  3. The acceleration magnitude is GM/r2=c2m/r2=9000 m/s2GM/r^2=c^2m/r^2=9000\,\mathrm{m/s^2}. Its large numerical value does not contradict the small potential; slow-motion validity must still be checked for the chosen trajectory.

The result

The potential ratio is −10⁻⁵, the leading clock rate is 0.99999, and the inward acceleration is 9000 m/s² under the stated exterior assumptions.

The living notebook

Explore the model

Interactive lab

Interactive lab · fields

Check linearity without claiming a gravity solution

Reset and set source strength q=1q=1, rotation Ω=0\Omega=0. At the lab’s readout point (1,0), predict Fx=5/6F_x=5/6, Fy=0F_y=0. Repeat from Reset with both controls q=2q=2, Ω=0\Omega=0; predict Fx=5/3F_x=5/3. Use numerical readouts: arrow lengths are compressed and need not double. This softened planar source is not the spherical inverse square field or an Einstein-equation solver.

Check linearity without claiming a gravity solutionThe softened illustration is q·r/(r² + 0.2) + Ω(−y, x). Moving markers indicate local direction, not integrated particle paths or a Maxwell solution. Current readings: Fx at (1, 0): 0.833; Fy at (1, 0): 0.5; Field magnitude there: 0.972. All parameters are adjustable with the labeled controls below.−3.2−1.601.63.2−1.5−0.7500.751.5xyq = 1
Local field directionSource strength
Fx at (1, 0)
0.833
Fy at (1, 0)
0.5
Field magnitude there
0.972
t = 0 s

The softened illustration is q·r/(r² + 0.2) + Ω(−y, x). Moving markers indicate local direction, not integrated particle paths or a Maxwell solution. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forLinearity in source strength can be tested in a toy field even when its spatial kernel differs from gravitational physics.

Pause & reason

Check your understanding

Which combination correctly identifies the elementary Newtonian limit used here?

Your turn

Try it yourself

A uniform sphere has density 6000 kg/m36000\,\mathrm{kg/m^3} and radius 106 m10^6\,\mathrm m. Using G=6.67×10−11 m3/(kg s2)G=6.67\times10^{-11}\,\mathrm{m^3/(kg\,s^2)}, find the Newtonian acceleration halfway from its center to its surface.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Only the enclosed mass contributes in spherical symmetry; its cubic radius dependence cancels two inverse powers.

Compare with the solution

At r=5×105 mr=5\times10^5\,\mathrm m, enclosed mass is 4πρr3/34\pi\rho r^3/3. Hence g=4πGρr/3≃0.8382 m/s2g=4\pi G\rho r/3\simeq0.8382\,\mathrm{m/s^2}, directed inward. The acceleration tends to zero at the center and matches the exterior value at the surface. The sphere’s surface compactness GM/(Rc2)GM/(Rc^2) is about 1.86×10−111.86\times10^{-11} for the stated c, consistent with a weak gravitational field; the material stresses and motion must also remain in the Newtonian regime.

Keep the idea

Take it with you

  • The temporal potential controls slow-body acceleration while spatial curvature remains relevant for light.
  • Trace reversal supplies the Newtonian source factor and exposes the Lambda correction.
  • Weak potential, slow motion, and source assumptions must each be checked on their own scales.

Read further

Original sources and deeper explanations to follow at your own pace.

Optional progress, saved in this browser when available. Revisit any lesson at any time.

Lesson not marked complete.