Schwarzschild

General Relativity · Part 4

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Table of Contents

  1. 1.The Spherically Symmetric Ansatz
  2. 2.Solving the Vacuum Field Equations
  3. 3.The Metric Components
  4. 4.Birkhoff's Theorem
  5. 5.Coordinate vs. True Singularities
  6. 6.Physical Schwarzschild Radii
  7. 7.The Schwarzschild Metric
  8. 8.Coordinate Singularity at r = 2M
  9. 9.True Singularity at r = 0
  10. 10.Kretschner Scalar Divergence

Key Equations

$$ds^2 = -e^{2\alpha(r)} c^2 \, dt^2 + e^{2\beta(r)} dr^2 + r^2 \left( d\theta^2 + \sin^2\theta \, d\phi^2 \right)$$
$$R_{tt} = e^{2(\alpha-\beta)} \left[ \alpha'' + (\alpha')^2 - \alpha'\beta' + \frac{2\alpha'}{r} \right] = 0$$
$$\Longrightarrow \quad e^{-2\beta}\left(1 + 2r\alpha'\right) = 1$$
$$g_{tt} = -\left(1 - \frac{2M}{r}\right)$$

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

Course Context

This PDF is part of the General Relativity course on CoursesHub.World. Explore Einstein's theory of gravity as curved spacetime. Covers differential geometry, curvature tensors, Einstein field equations, classic solutions including Schwarzschild and Kerr black holes, gra...

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