Schwarzschild Solution for a Black Hole

1. Schwarzschild Metric: The Schwarzschild solution is the simplest solution to Einstein’s field equations in general relativity. It describes the spacetime geometry surrounding a spherical, non-rotating, and uncharged mass such as a static black hole. The Schwarzschild metric in Schwarzschild coordinates (t,r,θ,ϕ)(t, r, \theta, \phi) is given by:

ds2=−(1−2GMc2r)c2dt2+(1−2GMc2r)−1dr2+r2(dθ2+sin⁡2θ dϕ2),ds^2 = -\left(1 – \frac{2GM}{c^2r}\right)c^2 dt^2 + \left(1 – \frac{2GM}{c^2r}\right)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta \, d\phi^2),

where:

  • GG is the gravitational constant,
  • MM is the mass of the black hole,
  • cc is the speed of light,
  • tt is the time coordinate,
  • rr is the radial coordinate,
  • θ\theta and ϕ\phi are the angular coordinates.

2. Event Horizon: The Schwarzschild radius rs=2GMc2r_s = \frac{2GM}{c^2} is the radius of the event horizon of the black hole. At r=rsr = r_s, the metric component gttg_{tt} goes to zero, and grrg_{rr} becomes infinite, indicating the event horizon beyond which no information can escape.

3. Singularities:

  • Coordinate Singularities: At r=rsr = r_s, the metric has a coordinate singularity, which can be removed by changing coordinates (e.g., using Kruskal-Szekeres coordinates).
  • Physical Singularities: At r=0r = 0, there is a true physical singularity where spacetime curvature becomes infinite.

Scalar Wave Equation in a Schwarzschild Background

1. Scalar Wave Equation: The scalar wave equation for a massless scalar field Φ\Phi in a curved spacetime is given by the Klein-Gordon equation:

□Φ=1−g∂μ(−g gμν∂νΦ)=0,\Box \Phi = \frac{1}{\sqrt{-g}} \partial_\mu (\sqrt{-g} \, g^{\mu\nu} \partial_\nu \Phi) = 0,

where □\Box is the d’Alembertian operator in curved spacetime, gg is the determinant of the metric tensor gμνg_{\mu\nu}, and gμνg^{\mu\nu} are the components of the inverse metric tensor.