Schwarzschild and Kerr Solutions of Einstein's Field Equation -- an introduction

Abstract

Starting from Newton's gravitational theory, we give a general introduction into the spherically symmetric solution of Einstein's vacuum field equation, the Schwarzschild(-Droste) solution, and into one specific stationary axially symmetric solution, the Kerr solution. The Schwarzschild solution is unique and its metric can be interpreted as the exterior gravitational field of a spherically symmetric mass. The Kerr solution is only unique if the multipole moments of its mass and its angular momentum take on prescribed values. Its metric can be interpreted as the exterior gravitational field of a suitably rotating mass distribution. Both solutions describe objects exhibiting an event horizon, a frontier of no return. The corresponding notion of a black hole is explained to some extent. Eventually, we present some generalizations of the Kerr solution.

arXiv comment96 pages, 17 figures, pdflatex. Invited review article. To appear in Wei-Tou Ni (editor) "One Hundred Years of General Relativity: Cosmology and Gravity," World Scientific, Singapore (2015)

References in corpus (53)

• The Confrontation between General Relativity and Experiment
• Gravity Probe B: Final Results of a Space Experiment to Test General Relativity
• Stationary Black Holes: Uniqueness and Beyond
• Black Hole Thermodynamics
• General relativistic gravitational field of a rigidly rotating disk of dust: Solution in terms of ultraelliptic functions
• Phenomenology of the Lense-Thirring effect in the Solar System
• A new form of the Kerr solution
• The Cotton tensor in Riemannian spacetimes
• Kretschmann Scalar for a Kerr-Newman Black Hole
• The Kerr spacetime: A brief introduction
• Second Order Scalar Invariants of the Riemann Tensor: Applications to Black Hole Spacetimes
• Uniqueness properties of the Kerr metric
• Searches for the role of spin and polarization in gravity
• Myers-Perry black holes with scalar hair and a mass gap
• On the uniqueness of smooth, stationary black holes in vacuum
• Motion of spinning test bodies in Kerr spacetime
• Spacetime geometry of static fluid spheres
• Spin-torsion coupling and gravitational moments of Dirac fermions: theory and experimental bounds
• Progress in relativistic gravitational theory using the inverse scattering method
• Gravitomagnetism and the Clock Effect
• A spacetime characterization of the Kerr metric
• Quasi-Spherical Light Cones of the Kerr Geometry
• On the Gravitomagnetic Time Delay
• An Assessment of the Systematic Uncertainty in Present and Future Tests of the Lense-Thirring Effect with Satellite Laser Ranging
• An axially symmetric solution of metric-affine gravity
• Constructive proof of the Kerr-Newman black hole uniqueness including the extreme case
• Separability of Black Holes in String Theory
• Heuristic approach to the Schwarzschild geometry
• Rotating Black Holes in Metric-Affine Gravity
• The Einstein 3-form G_a and its equivalent 1-form L_a in Riemann-Cartan space
• Clocks around Sgr A*
• Observational Evidence for Black Holes
• Computer algebra in gravity: Programs for (non-)Riemannian spacetimes. I
• Rigidly rotating ZAMO surfaces in the Kerr spacetime
• Discovering the Kerr and Kerr-Schild metrics
• Nonstationary Kerr Congruences
• Emergence of the Dirac Equation in the Solitonic Source of the Kerr Spinning Particle
• A space-time characterization of the Kerr-Newman metric
• Generalized gravitomagnetic clock effect
• Rotation, Equivalence Principle, and GP-B Experiment
• A novel derivation of the rotating black hole metric
• Constructing "non-Kerrness" on compact domains
• General Wahlquist Metrics in All Dimensions
• A novel derivation for Kerr metric in Papapetrou gauge
• The Post-Newtonian Approximation of the Rigidly Rotating Disc of Dust to Arbitrary Order
• The Use of Computer Algebra in Maxwell's Theory
• Complex Structure of the Four-Dimensional Kerr Geometry: Stringy System, Kerr Theorem, and Calabi-Yau Twofold
• On the discovery of Birkhoff's theorem
• Charged Rotating Black Holes in Higher Dimensions
• On non-disk geometry of r = 0 in Kerr-de Sitter and Kerr-Newman-de Sitter spacetimes
• Rigid Rotation and the Kerr Metric
• Race for the Kerr field
• On the Relation Between Quadratic and Linear Curvature Lagrangians in Poincaré Gauge Gravity

Cited by in corpus (17)

• Editorial for the Special Issue 100 Years of Chronogeometrodynamics: The Status of the Einstein's Theory of Gravitation in Its Centennial Year
• Zero mass limit of Kerr spacetime is a wormhole
• Exact Solutions in Poincaré Gauge Gravity Theory
• Source of the Kerr-Newman Solution as a Supersymmetric Domain-Wall Bubble: 50 years of the problem
• Conservation of energy-momentum of matter as the basis for the gauge theory of gravitation
• Hamiltonian approach to black hole entropy: Kerr-like spacetimes
• Physically motivated ansatz for the Kerr spacetime
• Proposed method of combining continuum mechanics with Einstein Field Equations
• Painleve-Gullstrand coordinates versus Kerr spacetime geometry
• Gravitational lensing and shadow of charged black holes in the low-energy limit of string theory
• Consistent derivation of the Hawking effect for both non-extremal and extremal Kerr black holes
• Hawking effect in an extremal Kerr black hole spacetime
• On the Kerr metric in a synchronous reference frame
• Rapid Spectral Parameter Prediction for Black Hole X-Ray Binaries using Physicalised Autoencoders
• Thermal radiation in curved spacetime using influence functional formalism
• Zero Mass limit of Kerr-MOG Black Hole Equals Wormhole
• A New Black Hole Solution In Dilaton Gravity Inspired By Power-Law Electrodynamics