Kerr

General Relativity · Part 4

184 KB10 sections4 key equationsLaTeX typeset
Download PDFRead Online (Free)

Table of Contents

  1. 1.The Boyer-Lindquist Metric
  2. 2.The Newman-Janis Algorithm
  3. 3.Event Horizons
  4. 4.The Ergosphere
  5. 5.Important Limiting Cases
  6. 6.Metric Determinant and Inverse
  7. 7.The Kerr Metric
  8. 8.Newman-Janis procedure:
  9. 9.Kerr Horizons
  10. 10.Outer horizon r₊

Key Equations

$$ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right)dt^2 - \frac{4Mar\sin^2\theta}{\Sigma}\,dt\,d\phi + \frac{\Sigma}{\Delta}\,dr^2 + \Sigma\,d\theta^2 + \frac{\mathcal{A}\sin^2\theta}{\Sigma}\,d\phi^2$$
$$\mathcal{A} = (r^2 + a^2)^2 - a^2\Delta\sin^2\theta$$
$$g_{tt} = -\left(1 - \frac{2Mr}{\Sigma}\right) > 0 \quad \Longrightarrow \quad r < r_{\text{ergo}}(\theta)$$
$$ds^2 = -dt^2 + \frac{r^2 + a^2\cos^2\theta}{r^2 + a^2}\,dr^2 + (r^2 + a^2\cos^2\theta)\,d\theta^2 + (r^2+a^2)\sin^2\theta\,d\phi^2$$

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...

← Gravitational Waves
Pp Waves →