The Kerr spacetime: A brief introduction (0706.0622v3)

Abstract: This chapter provides a brief introduction to the Kerr spacetime and rotating black holes, touching on the most common coordinate representations of the spacetime metric and the key features of the geometry -- the presence of horizons and ergospheres. The coverage is by no means complete, and serves chiefly to orient oneself when reading subsequent chapters.

• The paper demonstrates how incorporating angular momentum extends the Schwarzschild solution to model rotating black holes.
• The paper details multiple coordinate systems, including Boyer-Lindquist and Kerr-Schild, to simplify complex geometrical analyses.
• The paper highlights astrophysical implications such as horizon structures and ergospheres, influencing gravitational wave research and black hole thermodynamics.

Overview of "The Kerr Spacetime: A Brief Introduction" by Matt Visser

The paper by Matt Visser provides a concise introduction to the Kerr spacetime, a fundamental solution to the Einstein field equations representing rotating black holes. Discovered in 1963, the Kerr solution addresses the limitations of the Schwarzschild solution by incorporating angular momentum. This solution not only has deep theoretical implications in general relativity but also holds significant astrophysical relevance, given that most celestial bodies, such as stars and black holes, rotate.

Key Features of Kerr Spacetime

The Kerr spacetime is characterized by several coordinate systems, each offering unique insights into its complex geometry. This paper explores various coordinate representations:

1. Kerr's Original Coordinates: These are advanced Eddington-Finkelstein coordinates, which demonstrate the rotating nature of the spacetime but involve considerable computational complexity due to off-diagonal metric components.
2. Kerr-Schild "Cartesian" Coordinates: Here, the Kerr metric is expressed in a form that makes calculations involving the Kerr-Schild decomposition more straightforward, although the implicit nature of the radius coordinate complicates exact computations.
3. Boyer-Lindquist Coordinates: This is perhaps the most widely utilized form for the Kerr metric due to its relatively simple structure, reducing the number of off-diagonal metric components. It is particularly effective for asymptotic analyses and identifying event horizons and ergospheres.
4. Rational Polynomial Coordinates: By redefining coordinate systems to eliminate trigonometric components, computations become more efficient, particularly when working with symbolic manipulation software.
5. Doran Coordinates: These aim to simplify the ingoing/painlevé form of the Kerr metric by aligning closely with the Painlevé–Gullstrand form of the Schwarzschild metric. They reveal certain properties of Kerr spacetime while preserving coordinate time as a proper time experienced by freely falling observers.

Horizons and Ergospheres

The Kerr solution fundamentally differs from the Schwarzschild solution by introducing new features like ergospheres, regions outside the event horizon where no static observer can remain stationary due to the rotation. The presence of outer and inner horizons, as well as ergospheres between the static limit surface and the event horizon, reveals the rich structure of rotating black holes. The absence of a Birkhoff theorem means that outside a rotating body, the Kerr solution is only an asymptotic descriptor.

Implications and Further Developments

The implications of studying Kerr spacetime are profound:

• Astrophysics: Understanding the dynamics around rotating black holes supports studies of gravitational wave emissions and accretion phenomena, crucial for interpreting data from detections such as those by LIGO and Virgo.
• Black Hole Thermodynamics: Kerr black holes influence the paper of black hole thermodynamics, including laws pertaining to the conservation of energy and entropy.
• Singularity theorems & Cosmic Censorship: The paper of Kerr spacetime is intimately connected with theoretical considerations about singularities and the cosmic censorship conjecture.

Future research will likely explore the anti-de Sitter and de Sitter extensions of Kerr spacetimes, the role of Kerr spacetimes in string theory, and the potential existence of azimuthal instabilities or other phenomena that might emerge in the strong-field regime. The complexity and non-intuitive nature of Kerr spacetimes ensure that they remain a rich area for ongoing research in both theoretical and observational cosmology.