Schwarzschild Solution for a Black Hole

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 $(t,r,\theta ,\varphi )$ 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),ds2=−(1−c2r2GM​)c2dt2+(1−c2r2GM​)−1dr2+r2(dθ2+sin2θdϕ2),

where:

• $G$ is the gravitational constant,
• $M$ is the mass of the black hole,
• $c$ is the speed of light,
• $t$ is the time coordinate,
• $r$ is the radial coordinate,
• $\theta$ and $\varphi$ are the angular coordinates.

2. Event Horizon: The Schwarzschild radius rs=2GMc2r_s = \frac{2GM}{c^2}rs​=c22GM​ is the radius of the event horizon of the black hole. At r=rsr = r_sr=rs​, the metric component gttg_{tt}gtt​ goes to zero, and grrg_{rr}grr​ becomes infinite, indicating the event horizon beyond which no information can escape.

3. Singularities:

• Coordinate Singularities: At r=rsr = r_sr=rs​, the metric has a coordinate singularity, which can be removed by changing coordinates (e.g., using Kruskal-Szekeres coordinates).
• Physical Singularities: At $r=0$, 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 $\Phi$ 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,□Φ=−g​1​∂μ​(−g​gμν∂ν​Φ)=0,

where $\square$ is the d’Alembertian operator in curved spacetime, $g$ is the determinant of the metric tensor gμνg_{\mu\nu}gμν​, and gμνg^{\mu\nu}gμν are the components of the inverse metric tensor.