General Relativity

5.

Exact Solutions to the Einstein Field Equations

5.1.

Spherically Symmetric Solutions

5.1.1.

The Schwarzschild Solution

5.1.1.1.

Spherically Symmetric, Static, Uncharged Mass

5.1.1.2.

Derivation of the Metric

5.1.1.3.

The Schwarzschild Metric

5.1.1.3.1.

5.1.1.3.2.

Isotropic Coordinates

5.1.1.4.

The Schwarzschild Radius

5.1.1.4.1.

5.1.1.4.2.

Gravitational Radius

5.1.1.5.

Gravitational Singularity

5.1.1.5.1.

Curvature Singularity

5.1.1.5.2.

Coordinate Singularities

5.1.1.6.

Orbits in Schwarzschild Spacetime

5.1.1.6.1.

Effective Potential

5.1.1.6.2.

Circular Orbits

5.1.1.6.3.

Innermost Stable Circular Orbit

5.1.1.6.4.

Precession of Orbits

5.1.2.

The Reissner-Nordström Solution

5.1.2.1.

Spherically Symmetric, Static, Charged Mass

5.1.2.2.

Derivation and Metric

5.1.2.3.

Inner and Outer Horizons

5.1.2.4.

Extremal and Super-extremal Cases

5.1.2.5.

5.2.

Rotating Solutions

5.2.1.

The Kerr Solution

5.2.1.1.

Axially Symmetric, Stationary, Rotating Mass

5.2.1.2.

Boyer-Lindquist Coordinates

5.2.1.3.

The Kerr Metric

5.2.1.3.1.

Metric Components

5.2.1.3.2.

Ring Singularity

5.2.1.4.

The Ergosphere and Frame-Dragging

5.2.1.4.1.

Lense-Thirring Effect

5.2.1.4.2.

Dragging of Inertial Frames

5.2.1.5.

Penrose Process

5.2.1.5.1.

Energy Extraction

5.2.1.5.2.

5.2.2.

The Kerr-Newman Solution

5.2.2.1.

Rotating, Charged Mass

5.2.2.2.

Properties and Horizons

5.2.2.3.

Electromagnetic Field

5.3.

Cosmological Solutions

5.3.1.

Friedmann-Lemaître-Robertson-Walker Metrics

5.3.1.1.

Homogeneous and Isotropic Spacetimes

5.3.1.2.

Scale Factor Evolution

5.3.1.3.

Spatial Curvature Types

5.3.2.

de Sitter and Anti-de Sitter Spacetimes

5.3.2.1.

Maximally Symmetric Spacetimes

5.3.2.2.

Cosmological Constant Dominated

5.4.

Other Important Solutions

5.4.1.

Plane Wave Solutions

5.4.1.1.

Gravitational Wave Spacetimes

5.4.1.2.

pp-Wave Metrics

5.4.2.

Gödel Solution

5.4.2.1.

Rotating Universe

5.4.2.2.

Closed Timelike Curves

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4. The Einstein Field Equations

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6. Black Holes