(a) 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. (b) Put in values and show ∆θ = 0.87" . (General Relativity predicts an angle twice as large, 1.74" .)

Verified step by step guidance

1. First, recognize that the mass of the photon can be expressed as m = E/c^2, where E is the energy of the photon and c is the speed of light. This relation comes from the equivalence of mass and energy in Einstein's theory of special relativity.
2. Next, apply Newton's law of universal gravitation, which states that the gravitational force F between two masses m1 and m2 separated by a distance r is given by F = G * m1 * m2 / r^2. For a photon passing near the Sun, m1 = m and m2 = M (mass of the Sun), and r = R (radius of the Sun).
3. Calculate the acceleration a of the photon due to gravity using Newton's second law, F = ma. Substituting the expression for F from step 2, we get a = G * M / R^2.
4. Using the small angle approximation for deflection, the angle ∆θ can be approximated by ∆θ = a * d / c^2, where d is the distance the photon travels near the Sun's influence. Since the photon just grazes the Sun, d is approximately equal to R, the radius of the Sun. Substituting the values, we get ∆θ = G * M / (c^2 * R).
5. Finally, simplify the expression for ∆θ to match the given formula: ∆θ = 2 * G * M / (c^2 * R). This shows that the deflection angle of the photon due to the Sun's gravity, using Newtonian mechanics, is ∆θ = 2GM/c²R.