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PC4274A: Mathematical Methods in Physics III Assignment, NUS, Singapore: The Schwarzschild metric for the static field of a non-rotating spherically symmetric
 University National University of Singapore (NUS) Subject PC4274A: Mathematical Methods in Physics III Assignment
Posted on: 17th Apr 2024

# PC4274A: Mathematical Methods in Physics III Assignment, NUS, Singapore: The Schwarzschild metric for the static field of a non-rotating spherically symmetric

1. [20 pts] The Schwarzschild metric for the static field of a non-rotating spherically symmetric black hole of mass M is given by

where t is the time coordinate, (r, θ, φ) are spherical-like coordinates, G is Newton’s gravitational constant and c is the speed of light. The path of a small test particle in this field can be described in term of proper time τ , i.e. (t(τ ), r(τ ), θ(τ ), φ(τ )).

(a) Considering only the motion confined to the so called equatorial plane θ(τ ) = π/2 and as- suming that the path of the small test particle is such as to make  ds stationary, find two first integrals of the equations of motion.

(b) From their Newtonian limits, in which GM/r, ̇r2 and r2 ̇φ2 are all ≪ c2 where ̇r ≡ dr/dτ and ̇φ ≡ dφ/dτ , identify the constants of integration.

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2. [30 pts]
(a) We seek to extremize the functional

with respect to functions which attain the value u1 for x = x1 and which satisfy the given relation g(x, u) = 0 at the upper limit of integration, as yet undetermined. Show that the stationary function u(x) satisfies the Euler equation

and, in addition to the left-hand end-point requirement u(x1) = u1, the right-hand end-point condition

(b) Find the shortest distance between the line f (x) = x 3 and the curve g(x) = ex. Also, identify the point on respective curves giving the shortest distance.
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