Weak-Field Gravity and Classic Predictions

Weak-field gravity is where Einstein’s geometric theory first returns quantitative contact with familiar mechanics, clocks, rays, and gyroscopes. The chapter begins by separating physical curvature from coordinate gauge, recovers Newton’s equations under a controlled hierarchy, and then derives five classic observables. Its formulas apply to weak, slowly varying or stationary single-source fields; they are not substitutes for strong-field dynamics, modern binary expansions, or dated experimental inference.

Helpful background. Events, Intervals, and Minkowski Geometry supplies the background expansion, Tensor Fields and Natural Operations separates tensors from components, The Einstein Equation supplies the dynamics, and The Newtonian Gravitational Field and Potential supplies the comparison theory. Affine and Variational Geodesics and The Wave Equation and Characteristics are useful for the orbit and geometric-optics routes.

The approximation hierarchy

The common starting point is $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $\lvert h\rvert\ll1$. That amplitude condition is not enough by itself. Each result also selects the derivatives, source motion, probe motion, and boundary data that remain relevant.

• The Newtonian limit uses $v^2\sim\lvert\Phi\rvert\ll1$, negligible pressure, and time derivatives smaller than spatial derivatives.
• Stationary clock comparison requires a timelike Killing field and observers following its integral curves.
• Perihelion advance requires a bound test-body orbit with $GM/[a(1-e^2)]\ll1$ outside a spherical nonrotating source.
• Bending and delay require geometric optics and $GM/b\ll1$, with the impact parameter outside the source.
• Gyroscope formulas require weak stationary gravity, slow probe motion, a slowly rotating source, and a negligible test spin.

These restrictions are part of the answers. Removing one usually changes either the formula or the observable.

Six pages in derivational order

1. Weak Metric Perturbations and Gauge Freedom derives the first-order connection and Riemann tensor, then proves that $h_{\mu\nu}\mapsto h_{\mu\nu}-2\partial_{(\mu}\xi_{\nu)}$ leaves the linearized curvature invariant. Harmonic gauge simplifies the field equation without eliminating residual gauge freedom.

2. The Newtonian Limit of the Einstein Equation combines slow geodesic motion with the trace-reversed field equation to obtain

   $$g_{00}=-(1+2\Phi), \qquad \nabla^2\Phi=4\pi G\rho.$$

   Attraction for $\Phi=-GM/r$ checks the sign, while the field-equation trace checks the factor of two.

3. Gravitational Redshift and Clock Comparison first derives the exact stationary relation

   $$\frac{\omega_{\rm r}}{\omega_{\rm e}} =\sqrt{\frac{-g_{tt}(\mathrm e)}{-g_{tt}(\mathrm r)}},$$

   then expands it to $1+\Phi_{\rm e}-\Phi_{\rm r}$. The photon has a conserved Killing quantity, while each endpoint observer measures a different local energy.

4. Perihelion Precession isolates the Schwarzschild $r^{-3}$ effective-potential correction and solves the resonant orbit equation. The resulting secular advance per radial period is

   $$\Delta\varpi=\frac{6\pi GM}{a(1-e^2)}.$$

5. Light Deflection and Shapiro Delay keeps both temporal and spatial weak-metric perturbations. In GR they contribute equal halves to $\alpha=4GM/b$, while the same optical metric produces a positive logarithmic time delay.

6. Geodetic and Frame-Dragging Precession derives $d\mathbf S/dt=\boldsymbol\Omega\times\mathbf S$ and separates the mass-monopole term from the angular-momentum dipole term. Explicit cross-product order fixes the orientation, and $\mathbf J\mapsto-\mathbf J$ reverses only frame dragging.

Reall develops linearized gravity, the Newtonian hierarchy, and Schwarzschild geodesics in Reall 2013, §§ 8.1–8.2 and 12.2–12.4, printed pp. 77–83 and 134–141, PDF. The perturbative perihelion calculation is explicit in Guth–Turner 2017, §§ 7.2–8.2, printed pp. 33–36, PDF. Will’s post-Newtonian organization makes the temporal/spatial light split and the two gyroscope effects especially transparent in Will 2014, §§ 4.1–4.4, printed pp. 40–51, PDF.

Reader routes and recurring checks

For a first derivation route, read all six pages in order: gauge freedom prevents metric components from being mistaken for observables, and the Newtonian limit fixes notation used later. For a classic-tests route, review the approximation hierarchy and then read the redshift, perihelion, light, and spin pages; return to the first two whenever a sign or factor depends on the field equations.

Three checks recur throughout. First, every effect must vanish in its flat limit: $G\to0$, $M\to0$, or a pure-gauge perturbation. Second, signs are oriented physically—attraction points inward, an upward signal redshifts, perihelion advances, and spin precession follows a declared right-hand rule. Third, dimensions and normalization matter: logarithms have dimensionless arguments, precession vectors have inverse-time units, and $G$ remains explicit.

The chapter stops at durable leading predictions. Current bounds, source ephemerides, plasma corrections, instrument systematics, modern parameter estimation, gravitational radiation, and complete post-Newtonian binary dynamics require separate dated or specialized treatments.

Diagnose the approximation

A calculation uses only $g_{tt}=-(1-2GM/r)$ and predicts a bending angle $2GM/b$. A second calculation sends light from $r_{\rm e}$ to a larger $r_{\rm r}$ but predicts a blueshift. A third reverses the source spin while holding the orbit fixed and reverses both gyroscope-precession terms. Identify the error in each case.

Solution

The first calculation retained the temporal perturbation but omitted the spatial metric perturbation. It correctly finds the temporal half; in GR spatial curvature supplies another $2GM/b$, giving $4GM/b$.

For the second, $\Phi=-GM/r$ is more negative at the lower emitter. The weak ratio is $\omega_{\rm r}/\omega_{\rm e}=1+\Phi_{\rm e}-\Phi_{\rm r}<1$, so the signal redshifts. The predicted blueshift has the endpoint order or potential sign reversed.

For the third, the Lense–Thirring term is linear in $\mathbf J$ and must reverse. The geodetic term is proportional to $\mathbf r\times\mathbf v$ and the mass $M$; with the same orbit it does not reverse. Reversing both conflates source rotation with orbital orientation.

References

• Guth, Alan H., lectures; Andrew P. Turner, notes. 8.962 General Relativity, Spring 2017. Massachusetts Institute of Technology, 2017, §§ 7.2–8.2, printed pp. 33–36. PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013), §§ 8.1–8.2 and 12.2–12.4, printed pp. 77–83 and 134–141. PDF.
• Will, Clifford M. “The Confrontation between General Relativity and Experiment.” Living Reviews in Relativity 17, article 4 (2014), §§ 4.1–4.4, printed pp. 40–51. DOI; arXiv.