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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
PC4274A: Mathematical Methods in Physics III
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
PC4274A: Mathematical Methods in Physics III
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
PC4274A: Mathematical Methods in Physics III
and, in addition to the left-hand end-point requirement u(x1) = u1, the right-hand end-point condition
PC4274A: Mathematical Methods in Physics III
(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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