McVittie & Kottler Metrics in Cosmology

Updated 24 October 2025

McVittie and Kottler metrics are foundational general relativity solutions that model spherically symmetric masses embedded in either dynamic FLRW or static de Sitter cosmological settings.
The Kottler metric represents a static black hole in a universe with a cosmological constant, while the McVittie metric generalizes this to a time-dependent Hubble function, leading to evolving horizon structures.
These metrics provide crucial insights into causal structures, gravitational lensing, entropy bounds, and tidal effects, making them key tools for understanding black hole formation within cosmological contexts.

The McVittie and Kottler metrics are central solutions in the paper of inhomogeneous cosmological models in general relativity, describing spherically symmetric mass concentrations embedded in expanding (or contracting) universes. While the Kottler metric (also known as the Schwarzschild–de Sitter metric) represents the static embedding of a black hole in a universe with cosmological constant, the McVittie metric extends this construction to arbitrary FLRW backgrounds with time-dependent scale factor. These spacetimes provide essential theoretical laboratories for understanding horizon structure, causal properties, tidal dynamics, gravitational lensing, thermodynamics, and the embedding of local inhomogeneities in cosmological settings. Differences between their apparent horizon structures and implications for black hole formation, entropy bounds, and causal diagrams are especially rich when interpreted against different cosmic backgrounds and matter content.

1. Metric Definitions and Geometric Structure

Kottler (Schwarzschild–de Sitter) Metric

The Kottler metric describes a static, spherically symmetric spacetime incorporating both a central mass $m$ and a cosmological constant $\Lambda$ : $ds^2 = -\left[1 - \frac{2m}{r} - H^2 r^2\right] dt^2 + \left[1 - \frac{2m}{r} - H^2 r^2\right]^{-1} dr^2 + r^2 d\Omega^2,$ where $H = \sqrt{\Lambda/3}$ and $d\Omega^2$ is the metric on the unit 2-sphere.

Horizon positions are located by solving: $1 - \frac{2m}{r} - H^2 r^2 = 0,$ which generically yields two positive roots for $mH < 1/(3\sqrt{3})$ (black hole and cosmological horizons), a coincident root in the Nariai limit, and complex roots (no physical horizons) otherwise.

McVittie Metric

The McVittie metric generalizes Kottler by allowing a time-dependent Hubble function $H(t)$ , providing a model of a mass $m$ embedded in an FLRW background: $ds^2 = -[1 - (2m/r) - H^2(t) r^2] dt^2 - \frac{2H(t) r}{\sqrt{1 - 2m/r}} dt\, dr + r^2 d\Omega^2,$ where $H(t) = \dot{a}(t)/a(t)$ with $a(t)$ the cosmological scale factor.

Alternatively, in isotropic coordinates, the metric may be written as: $ds^2 = -\left(\frac{1 - \frac{m}{2 a(t) r}}{1 + \frac{m}{2 a(t) r}}\right)^2 dt^2 + a^2(t) \left(1 + \frac{m}{2 a(t) r}\right)^4 [dr^2 + r^2 d\Omega^2].$ The McVittie metric reduces to Kottler when $a(t)$ is exponential and to FLRW when $m = 0$ (Faraoni et al., 2012, Nolan, 2017).

2. Horizon Structure and Dynamics

Kottler Horizons

Horizons are static for constant $H$ ; apparent and event horizons coincide. The locations are fixed, determined by the polynomial above. The area and entropy of the horizons are determined directly by the solution (Faraoni et al., 2012).

McVittie Apparent Horizons

Apparent horizons in the McVittie spacetime are given (for any background) by the time-dependent roots of: $1 - \frac{2m}{r} - H^2(t) r^2 = 0.$ These evolve with cosmic time, with distinct behaviors in different backgrounds:

Dust-dominated: $H(t) \propto t^{-1}$ is initially large, precluding any real apparent horizons (naked singularity). At a critical $t_*$ , two horizons appear simultaneously and then split: a black hole horizon shrinks toward $r = 2m$ , while the cosmological horizon expands ( $r \sim 1/H(t)$ ) (Faraoni et al., 2012).
Phantom-dominated: $H(t)$ increases toward a future “Big Rip”. Horizons initially exist, merge at a critical $t_*$ , and then vanish entirely, exposing a naked singularity; the horizon area (entropy) can decrease discontinuously, reflecting unconventional thermodynamic properties in such spacetimes.

These horizon dynamics illustrate the contrast between the static Kottler scenario and the rich, time-dependent phenomenology of McVittie geometries. In both cases, the mathematical structure is determined by an effective quartic or cubic equation, but their physical interpretations diverge due to the dynamical background.

3. Causal Structure and Global Properties

Causal properties of McVittie spacetimes strongly depend on the asymptotic behavior of the expansion rate $H(t)$ (Silva et al., 2012, Nolan, 2017):

For $H(t)\to H_0>0$ (e.g., $\Lambda$ -dominated), the causal boundary mimics Kottler: black hole and cosmological horizons persist, separating regions accessible to different classes of null geodesics.
The detailed classification theorem of (Silva et al., 2012) establishes that the existence of a “black-hole-only” structure or a “black and white hole” structure depends on the integrability of certain comparison integrals involving $H(t)$ and the horizon’s limiting radius. This resolves previous literature discrepancies.
In non-flat (open or closed) FLRW backgrounds, the domain of McVittie spacetimes can be more intricate. For example, in closed universes the allowed region at fixed time is bounded above and below, with boundaries (at fixed $r$ ) determined by the sign change of $1 - (2M/r) - r^2/a^2(t)$ (Nolan, 2017). The global structure and completeness properties (e.g., singularities, horizon reachability by geodesics) are determined by both curvature and expansion.

In Kottler, the static nature allows the event and apparent horizons to coincide precisely, simplifying causal analysis.

4. Physical Interpretation and Astrophysical Context

Kottler spacetimes serve as static prototypes for black holes in universes with nonzero cosmological constant and have been widely employed to estimate gravitational lensing in dark energy-dominated universes and corrections to thermodynamical relations (e.g., entropy–area).

McVittie spacetimes model local inhomogeneities embedded in evolving cosmologies, offering:

Theoretical settings for investigating black hole evolution in expanding backgrounds.
A means to probe how cosmic expansion affects accretion disk formation, gravitational lensing, and shadow size (Nolan, 2014, Tsupko et al., 2019, Piattella, 2015).
A framework for understanding the interplay between local and global features, such as whether the embedded mass generates a persistent event horizon (as in the static Kottler case) or only transient apparent horizons subject to cosmological evolution (Faraoni et al., 2012, Esfandiar et al., 21 Oct 2025).

However, an important caveat is that, according to the trapping horizon analysis (Esfandiar et al., 21 Oct 2025), the standard McVittie metric in a matter-dominated universe generally lacks a suitable future outer trapping horizon (FOTH) and hence fails to fully represent a “cosmological black hole” by the standards of Hayward’s dynamical formalism, in contrast to certain Culetu and Sultana–Dyer-type spacetimes.

5. Extensions, Modifications, and Mathematical Techniques

Charged and Modified Gravity Generalizations

Charged McVittie: Incorporates an electric charge parameter, yielding a two-parameter family that reduces to Reissner–Nordström or FLRW as limits. Horizon structure is determined by a quartic in areal radius, and extremal and naked singularity regimes are accessible depending on $|Q|/m$ (Faraoni et al., 2014).
Horndeski and $f(T)$ Gravity: The McVittie metric can be naturally embedded as an exact solution in generalized scalar–tensor frameworks (Horndeski) (Miranda et al., 2022) and in $f(T)$ gravity, where teleparallel torsion replaces curvature in the Lagrangian. The survival of the solution often depends on a judicious (null) tetrad selection (Bejarano et al., 2017).

Analytical and Numerical Tools

Dynamical Systems: Analysis of geodesic equations using stable/center manifold and Lyapunov methods shows the existence of future-complete bound orbits, supporting the formation of accretion disks in McVittie backgrounds, with late-time asymptotics matching those of Schwarzschild–(de Sitter) (Nolan, 2014).
Junction Conditions and Bubble Evolution: The mathematical formalism for matching two McVittie or Schwarzschild–FLRW regions across a timelike hypersurface utilizes a unified boost function to efficiently represent extrinsic curvature and surface stress-energy (Tang et al., 26 Dec 2024). This allows for a systematic exploration of gravitational collapse and the evolution of shells (“cosmic bubbles”) under cosmic expansion.
Matched Asymptotic Expansions: Provides the analytical bridge between the inner (black hole-dominated) and outer (cosmology-dominated) regions for observables such as shadow size, yielding composite formulas valid from near to far field (Tsupko et al., 2019).

6. Thermodynamics, Entropy Bounds, and Lensing

Entropy and Holography: The correspondence between the D-bound and Bekenstein bound is satisfied at all epochs in de Sitter (Kottler) backgrounds, but fails generically for quintessence or phantom dark energy models except in special epochs. The presence of a central inhomogeneity discretizes the allowable cosmological horizon radius and imposes $R < 1/H$ (Hadi et al., 2019).
Cavity Temperature: For McVittie black holes in cavities (finite regions bounded by isothermal boxes), the equilibrium temperature experienced at the cavity wall is determined by applying the Tolman relation to the local horizon temperature, and, in the Kottler/de Sitter limit, matches results from the Euclidean action method (Kong, 30 Aug 2025).
Lensing: At leading (post-Newtonian) order, cosmological expansion modeled using either McVittie or Kottler backgrounds does not affect the bending angle of light by an isolated mass, once cosmological distance conversions are properly incorporated (Piattella, 2015). In isotropic static coordinates, distinctions in Fermat metrics (effective refractive indices) can produce differences in null geodesic trajectories for Kottler versus Schwarzschild (Solanki, 2021).

7. Tidal Dynamics and Local Peculiar Motions

Tidal Forces: Kottler spacetimes exhibit radius-dependent sign changes in radial and angular tidal components—phenomena absent in the Schwarzschild limit. Tidal sign reversals are characteristic of Kottler/Schwarzschild–de Sitter with negative or sufficiently large positive cosmological constant, and are captured analytically via elliptic integrals (Vandeev et al., 2022).
Local Deviations: In McVittie, the tidal field experienced by comoving (or Fermi-normal) observers combines contributions from the FLRW background, the local mass $m$ , and cross terms, producing nontrivial dynamics for peculiar velocities relative to the Hubble flow. This results in qualitative differences relative to dust LTB models, where observers follow geodesics and the local pressure gradient is absent (Molaei et al., 2 Oct 2024).

In summary, the McVittie and Kottler metrics provide foundational models for studying central masses and black holes in cosmological settings. While Kottler serves as the static embedding in de Sitter space, McVittie captures time-dependent effects and allows for the investigation of horizon formation, causal structure, thermodynamics, embedding in modified gravity, and the nuanced influence of cosmic expansion on local dynamics. Differences in horizon structure, physical interpretation (particularly regarding “cosmological black holes”), and applications to observational signatures such as lensing and shadow size are central themes in the literature, continuously motivating further generalizations and critical analysis.