Schwarzschild Solution

Black Holes · Part 1

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Table of Contents

  1. 1.Introduction
  2. 2.Step 1: Symmetry Assumptions
  3. 3.Step 2: Einstein Field Equations in Vacuum
  4. 4.Step 3: Christoffel Symbols
  5. 5.Step 4: Ricci Tensor Components
  6. 6.Step 6: The Schwarzschild Metric
  7. 7.Birkhoff's Theorem
  8. 8.Physical Interpretation
  9. 9.Key Assumptions
  10. 10.For the Sun:

Key Equations

$$ds^2 = -e^{2\alpha(r)} c^2 dt^2 + e^{2\beta(r)} dr^2 + r^2 d\Omega^2$$
$$\Gamma^\theta_{r\theta} = \Gamma^\phi_{r\phi} = \frac{1}{r}, \quad \Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta$$
$$R_{\theta\theta} = e^{-2\beta}\left[r(\beta' - \alpha') - 1\right] + 1$$
$$\Rightarrow e^{2\beta} = \frac{1}{1 - r_s/r}$$

Equations are rendered with MathJax in the PDF with professional LaTeX typesetting.

Course Context

This PDF is part of the Black Holes course on CoursesHub.World. Study the physics of black holes from Schwarzschild geometry to Hawking radiation. Covers rotating and charged black holes, thermodynamics, the information paradox, holographic principle, and astrophy...

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