General relativity predicts a deflection angle of 4GM/(bc²) for light passing a point mass, while a Newtonian calculation treating photons as massive particles gives 2GM/(bc²). Why is the GR result exactly twice the Newtonian value?
AThe factor of 2 comes from the photon's relativistic mass being twice its Newtonian effective mass
BIn GR, both the temporal curvature (g_{tt}) and the spatial curvature (g_{rr}) contribute equally to the deflection, whereas the Newtonian calculation only accounts for the temporal part
CThe Newtonian calculation is incorrect because it uses the wrong value of G
DThe factor of 2 arises from frame-dragging effects near the mass
In the weak-field limit, the Schwarzschild metric has perturbations in both g_{tt} and g_{rr}. The Newtonian calculation (equivalently, a purely temporal metric perturbation) captures only half the effect. The spatial curvature (the deviation of g_{rr} from unity) contributes an equal amount to the light deflection. For slowly moving massive particles, the spatial curvature contribution is suppressed by v²/c², but for photons traveling at c, both contributions are equal. This factor of 2 was the key prediction distinguishing GR from competing theories and was confirmed by Eddington's 1919 eclipse expedition.
Question 2 True / False
An Einstein ring is observed when the source, lens, and observer are perfectly aligned.
TTrue
FFalse
Answer: True
When a background source, a gravitational lens, and the observer are collinear, the lensing geometry has perfect axial symmetry. Light from the source is deflected around all sides of the lens equally, forming a complete ring — the Einstein ring. The angular radius of the ring is θ_E = √(4GM D_{LS}/(c² D_L D_S)), where D_L, D_S, and D_{LS} are the angular diameter distances to the lens, to the source, and from the lens to the source. Perfect alignment is rare, so partial arcs are much more commonly observed than complete rings.
Question 3 Short Answer
Explain how weak gravitational lensing is used to map the distribution of dark matter in galaxy clusters.
Think about your answer, then reveal below.
Model answer: Weak lensing measures the small, coherent distortions (shear) in the shapes of thousands of background galaxies caused by the gravitational field of a foreground mass distribution. Individual galaxies have intrinsic random shapes, but the lensing-induced shear is correlated — galaxies behind a massive cluster are preferentially stretched tangentially to the cluster center. By statistically averaging over many background galaxies, the shear field is reconstructed, and from it the projected mass distribution of the lens is inferred through inversion. This technique is sensitive to all mass, not just luminous matter, making it a direct probe of dark matter distribution.
Weak lensing has become one of the most powerful tools in observational cosmology. It has confirmed that galaxy clusters contain far more mass than their visible components, mapped the filamentary structure of dark matter on cosmological scales, and provides constraints on dark energy through its effect on the growth of structure.
Question 4 Short Answer
Calculate the deflection angle for starlight grazing the Sun's limb, given M_☉ ≈ 2 × 10³⁰ kg and R_☉ ≈ 7 × 10⁸ m.
Think about your answer, then reveal below.
Model answer: α = 4GM/(Rc²) = 4(6.67 × 10⁻¹¹)(2 × 10³⁰)/((7 × 10⁸)(3 × 10⁸)²) = 8.49 × 10⁻⁶ radians ≈ 1.75 arcseconds. This is the value predicted by GR and confirmed by Eddington's 1919 eclipse observations (within the measurement uncertainty of about 20%). The Newtonian prediction would be half this: 0.87 arcseconds. Modern VLBI measurements of radio source deflection by the Sun confirm the GR prediction to better than 0.01% accuracy.
1.75 arcseconds is tiny — roughly the angular size of a quarter seen from 3 km away — which is why a total solar eclipse (blocking the Sun's overwhelming brightness) was necessary for the optical measurement. Radio interferometry now achieves far greater precision without needing an eclipse.