Gravitational Lensing & Geodesic Precession
Gravitational Lensing & Geodesic Precession
General relativity predicts that light bends in gravitational fields — confirmed by Eddington in 1919. GR also modifies orbital mechanics: Mercury's perihelion precesses 43"/century beyond Newtonian predictions, and gyroscopes undergo geodetic and Lense-Thirring precession due to spacetime curvature and frame-dragging.
Deflection Angle & Einstein Radius
Light deflection by mass \(M\) at impact parameter \(b\): \(\hat\alpha = \frac{4GM}{c^2 b} = \frac{2r_s}{b}\). For a point lens the Einstein radius: \(\theta_E = \sqrt{\frac{4GM}{c^2}\frac{D_{ls}}{D_l D_s}}\), where \(D_l, D_s, D_{ls}\) are angular diameter distances. Sources within \(\theta_E\) form an Einstein ring.
Perihelion Precession
For a nearly circular orbit in Schwarzschild, the perihelion advances per orbit: \(\Delta\phi = \frac{6\pi G M}{a(1-e^2)c^2}\) radians. For Mercury: \(a=5.79\times10^{10}\,\text{m}\), \(e=0.206\), giving \(\approx 43''\)/century — a landmark confirmation of GR.
Example 1: Solar Light Deflection
Find the deflection of starlight grazing the Sun (\(b = R_\odot = 6.96\times10^8\,\text{m}\)).
Solution: \(\hat\alpha = 4GM_\odot/(c^2 R_\odot) \approx 8.48\times10^{-6}\,\text{rad} \approx 1.75''\). Eddington's 1919 measurement: \(1.75'' \pm 0.09''\). Newtonian gravity predicts only half this value.
Example 2: Shapiro Time Delay
A radar signal passes the Sun at \(b \approx R_\odot\). Estimate the extra travel time.
Solution: \(\Delta t = \frac{2r_s}{c}\ln\!\left(\frac{4r_e r_r}{b^2}\right)\approx 240\,\mu\text{s}\) for an Earth-Sun-planet geometry. Measured by Cassini to \(0.002\%\) precision.
Practice
- Derive \(\hat\alpha = 4GM/(c^2 b)\) from the null geodesic equation in Schwarzschild.
- Compute \(\theta_E\) for a galaxy cluster \(M=10^{14}M_\odot\) at \(D_l=1\,\text{Gpc}\), \(D_s=2\,\text{Gpc}\).
- Show that Newtonian gravity predicts only half the GR deflection angle.
- Explain the geodetic (de Sitter) precession and its measurement by Gravity Probe B.
Show Answer Key
1. Null geodesic in Schwarzschild with impact parameter $b$: the deflection angle $\hat{\alpha} = \int (\text{bending integrand})\,d\phi - \pi$. In the weak-field limit ($b \gg r_s$): $\hat{\alpha} = \frac{4GM}{c^2 b} = \frac{2r_s}{b}$. This is twice the Newtonian prediction, famously confirmed by Eddington in 1919.
2. Einstein radius: $\theta_E = \sqrt{\frac{4GM}{c^2}\frac{D_{ls}}{D_l D_s}}$. $M = 10^{14}M_\odot = 2\times10^{44}$ kg, $D_l = 1$ Gpc $= 3.09\times10^{25}$ m, $D_s = 2$ Gpc, $D_{ls} = 1$ Gpc. $\theta_E = \sqrt{\frac{4\times6.67\times10^{-11}\times2\times10^{44}}{9\times10^{16}}\cdot\frac{1}{3.09\times10^{25}}} \approx 30''$ (arcseconds). Galaxy cluster lensing produces arcs of this angular scale.
3. Newtonian deflection (treating photon as a particle with $v=c$): $\hat{\alpha}_N = \frac{2GM}{c^2b}$. This accounts only for the gravitational acceleration component. GR gives $\hat{\alpha}_{GR} = \frac{4GM}{c^2b} = 2\hat{\alpha}_N$. The extra factor of 2 comes from spatial curvature (the spatial metric contributes equally to the time-time component). Eddington's 1919 eclipse measurement confirmed the GR prediction.
4. Geodetic (de Sitter) precession: a gyroscope in orbit around a massive body has its spin axis precess due to spacetime curvature at rate $\Omega_{dS} = \frac{3GM}{2c^2r}\omega_{\text{orb}}$ (for circular orbit). Gravity Probe B (2004–2011) measured this for Earth-orbiting gyroscopes: predicted 6606 mas/yr, measured $6601.8 \pm 18.3$ mas/yr — confirming GR to 0.3%.