Question

Use special relativity and Newton’s law of gravitation to show that a photon of mass m = E/c² just grazing the Sun will be deflected by an angle ∆θ given by ∆θ = 2GM/c²R, where G is the gravitational constant, R and M are the radius and mass of the Sun, and c is the speed of light. Put in values and show ∆θ = 0.87". (General Relativity predicts an angle twice as large, 1.74".)

Accepted Answer

Step 1: Start by understanding the problem. The goal is to derive the deflection angle (∆θ) of a photon grazing the Sun using Newtonian gravity and special relativity. The photon is treated as having an equivalent mass m = E/c², where E is its energy and c is the speed of light. Step 2: Use Newton's law of gravitation to calculate the gravitational force acting on the photon. The gravitational force is given by F = GMm/R², where G is the gravitational constant, M is the mass of the Sun, R is the radius of the Sun, and m is the photon's equivalent mass (m = E/c²). Substituting m = E/c² into the equation gives F = GME/(c²R²). Step 3: Relate the force to the photon's deflection. The deflection angle ∆θ can be approximated by considering the transverse acceleration of the photon due to the gravitational force as it passes near the Sun. The transverse acceleration is a = F/m = GM/R². Since the photon travels at the speed of light c, the time it spends near the Sun is approximately t = 2R/c (assuming a straight-line path). The transverse displacement is then d = (1/2)at² = (1/2)(GM/R²)(2R/c)². Step 4: Use the small-angle approximation to find the deflection angle. The deflection angle ∆θ is approximately the transverse displacement divided by the distance of closest approach (R). Substituting d and simplifying gives ∆θ = 2GM/(c²R). This is the desired result for the deflection angle using Newtonian gravity and special relativity. Step 5: To verify the numerical value of ∆θ, substitute the known values: G = 6.674 × 10⁻¹¹ m³/kg·s², M = 1.989 × 10³⁰ kg (mass of the Sun), R = 6.96 × 10⁸ m (radius of the Sun), and c = 3.00 × 10⁸ m/s. Calculate ∆θ in radians and convert to arcseconds (1 radian = 206,265 arcseconds). The result should be approximately 0.87 arcseconds, which matches the prediction from Newtonian gravity. Note that General Relativity predicts a deflection angle twice as large, 1.74 arcseconds.

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Key Concepts

Special Relativity

Newton's Law of Gravitation

Deflection of Light by Gravity