Black Hole Atlas

The Black Hole Atlas is an object-first guide to a subject whose name can mean very different things: an exact spacetime, an astrophysical population, a particular source in the sky, a phenomenological metric, a laboratory horizon, or a proposed quantum construction. Each page begins by fixing that meaning. It then gives the minimum defining data, controlled limits, scientific status, and links needed to continue into a full derivation or a dated assessment.

Helpful background. Black-Hole Geometry and Mechanics supplies the horizon and charge language used throughout. Each page also introduces the local definitions needed to enter with a particular object or observation in mind.

Choose the object before the calculation

The six facets are not successive levels of approximation. They answer different questions and carry different kinds of evidence.

Facet	What counts as an entry	Question answered
Exact solutions and families	A metric and accompanying fields solving stated equations on a stated domain	Which geometry realizes the required dimension, asymptotics, charges, topology, or limit?
Astrophysical classes	A population defined through formation, mass scale, environment, or cosmological origin	How is the population formed and inferred, and how sharp is its boundary?
Named observed systems	A maintained benchmark source with complementary measurements	What was directly observed, what was inferred, and which assumptions dominate?
Effective models	A comparison metric or effective construction with an explicit regime	Which classical feature is modified, and what consistency test limits the model?
Analogue systems	Waves in a medium with a horizon or effective causal geometry	Which kinematic horizon phenomenon is reproduced, and where does the analogy stop?
Quantum-facing constructions	Restricted-theory solutions, microscopic proposals, endpoints, or horizonless alternatives	What is constructed, what is extrapolated, and what remains unproved?

A question such as “What is the temperature?” therefore needs an object class before it needs a formula. For a stationary exact black hole, temperature is related to a normalized horizon generator and its surface gravity. In an analogue medium it may describe a mode-conversion spectrum in laboratory time. For a proposed remnant it may be part of an endpoint hypothesis whose dynamics are not known. A shared symbol does not erase those differences.

Scientific status is part of the answer

Every page separates three questions that are often compressed into one.

1. Does the object exist mathematically? An exact solution must satisfy a declared set of field equations, boundary conditions, and regularity requirements. A phenomenological line element can be useful without being a solution of any supplied covariant action.
2. Is the relevant regime physically realized? Kerr is an exact vacuum family and an excellent macroscopic model, but an inference of its parameters still uses an astrophysical forward model. A laboratory horizon is physically realized as a medium configuration without solving Einstein’s equations.
3. How strong is the evidence for the interpretation? Direct observables, calibrated data products, fitted parameters, theory priors, and population claims occupy different stages of an inference chain.

This separation is crucial in the quantum-facing facet. A supersymmetric black hole may be exact in a controlled supergravity truncation while its relevance to a generic evaporating astrophysical hole remains conditional. A protected microstate count, an explicit smooth geometry, typical-state coverage, formation by collapse, and an outgoing-radiation calculation are five distinct achievements.

Read a page in layers

The opening table identifies the class, theory or medium, dimension, asymptotics or environment, defining parameters, and evidence status. The first equations are identifiers rather than a compressed textbook chapter: typically a metric function, horizon polynomial, dispersion relation, or measured quantity. The domain statement tells you where those equations support the prose.

Next come controlled limits and discriminating checks. For exact families, these can include uncharged, nonrotating, flat-boundary, extremal, or lower-dimensional limits. For effective models they include curvature, geodesic, signature, horizon, and field-equation checks. For observed systems they include an independent measurement channel or a change in the dominant calibration or prior. For analogue and quantum-facing objects they identify the approximation under which the shared mathematics survives.

Follow the links for detailed derivations. For Kerr, compare the coordinate forms in Kerr metric forms and parameters, the horizon analysis in Kerr horizon and ergoregion anatomy, and the response problem described by the Teukolsky equation. Together, these treatments connect the metric to its horizons and perturbations.

Useful routes through the Atlas

For a geometry-first route, compare Schwarzschild, Kerr, and Reissner–Nordström, then change the boundary condition with Schwarzschild–AdS or the dimension and topology with BTZ and black rings.

For an observation-first route, contrast stellar-orbit and image inference for Sagittarius A*, image and jet-environment inference for M87*, binary dynamics for Cygnus X-1, and maser dynamics for NGC 4258. These systems are useful precisely because their systematics differ.

For model discrimination, compare the Bardeen and Hayward metrics with a deformed Kerr metric or a black bounce. For horizon kinematics, compare acoustic and water-wave horizons. For microscopic structure, compare supersymmetric string black holes with fuzzballs, explicit microstate geometries, and endpoint proposals such as remnants.

Distinctions the Atlas preserves

The Atlas does not identify a coordinate singularity with a curvature singularity, a regular curvature scalar with geodesic completeness, an apparent horizon with an event horizon, a mass posterior with a direct observable, a ring-like image with a unique Kerr proof, a shared wave equation with shared dynamics, or a construction in a protected sector with a completed account of a generic evaporating black hole. Those distinctions are not cautions added at the end; they are part of defining each object.