Cosmological Black hole Candidates: A Detailed Analysis of McVittie, Culetu, Sultana-Dyer, and Glass-Mashhoon Spacetimes

TL;DR

This work assesses cosmological black hole viability via Hayward's trapping-horizon framework across several dynamical spacetimes embedded in FLRW backgrounds. It demonstrates that while spatially flat McVittie spacetimes fail to realize a cosmological BH (no simultaneous FOTH and PITH), the Culetu and Sultana–Dyer spacetimes can describe cosmological black holes in the matter-dominated early universe provided key energy conditions hold. The Glass–Mashhoon class generally does not yield a future outer trapping horizon, limiting its cosmological BH interpretation. Across isotropic and Painlevé–Gullstrand coordinates, the results are consistent: cosmological BHs arise only in specific spacetimes and parameter regimes, clarifying the viability of these models under Hayward's local-horizon criteria and highlighting the role of energy conditions in their physical interpretation.

Abstract

This paper investigates the existence of cosmological black holes by analyzing the properties of trapping horizons in detail, based on Hayward's formalism of future outer and past inner trapping horizons, in several dynamical spacetimes embedded in an expanding universe. Through a detailed examination of the McVittie, Culetu, and Sultana--Dyer metrics, as well as the generalized Glass--Mashhoon solution, we evaluate the existence and characteristics of trapping horizons and energy conditions. The Glass--Mashhoon solution provides an analytical model for spherical stellar collapse. However, it is shown that, as long as certain conditions are satisfied, it lacks suitable future outer trapping horizons, meaning it does not represent a cosmological black hole. As a result, the McVittie class of solutions also fails to describe a cosmological black hole. Conversely, the Culetu and Sultana--Dyer spacetimes can describe a cosmological black hole in the matter-dominated early universe, provided that the relevant energy conditions are satisfied.

Figures (16)

• Figure I: The time evolution of the THs of the matter-dominated spatially flat McVittie spacetime in isotropic coordinates $\tilde{\bar{r}}$. The upper branch corresponds to $\tilde{\bar{r}}_{-}^u$, while the lower branch corresponds to $\tilde{\bar{r}}_{-}^l$. The critical time $\tilde{t}_{*} = t_*/m_0$ represents the formation of the degenerate root of this spacetime, at which the two horizons coincide. At times before the critical time, there are no THs and after this time, the THs begin to separate from each other.
• Figure II: The time evolution of $\mathcal{L}_{+} \tilde{\theta}_{-}|_{\tilde{\theta}_{-}=0}$ for the matter dominated spatially flat McVittie spacetime on the THs, $\tilde{R}_1 \equiv R_{1}/m_0$ and $\tilde{R}_2 \equiv R_2/m_0$. $\tilde{t}_{*}=t_*/m_0=2\sqrt 3$, is the critical time when these horizons coincide. As it can be seen from Eqs. \ref{['new10']} and \ref{['new11']}, for $t_1 \sim 2t_*$, the corresponding graph for $R_2$ becomes zero and for $t=t_*$ both graphs approach to ${-1}/{(6\sqrt{3})}$.
• Figure III: The time evolution of the normalized squared norm of the normal vector of the THs $\tilde{R}_1 \equiv R_{1}/m_0$ and $\tilde{R}_2 \equiv R_2/m_0$ for the matter dominated spatially flat McVittie spacetime.
• Figure IV: The time evolution of the horizons in the matter-dominated Culetu spacetime with $a(t) = \tilde{t}^{2/3}$. The time and radius are normalized as $\tilde{t} = t/m$ and $\tilde{R} = R/m$, with $t_0 = m$, or equivalently $\tilde{t}_0 = 1$. In the interval $0 < \tilde{t} < (2/3)^3$, $0 < R_{+} < R_{-} < R_{\mathrm{EH}}$ holds. For $\tilde{t} > (2/3)^3$, the ordering becomes $0 < R_{+} < R_{\mathrm{EH}} < R_{-}$. The time $\tilde{t} = 2^3$ is the upper limit of the interval over which the NEC holds both on and outside the event horizon. Since $R_-$ lies outside the event horizon after a certain period of time, it follows from nielsen2009 that the NEC must be violated in some region of the Culetu spacetime.
• Figure V: The time evolution of the horizons in the matter-dominated Sultana-Dyer spacetime, $a(t)=\tilde{t}^{{2}/{3}}$. The time and radius of the horizons are normalized as $\tilde{t}={t}/{m}$, $\tilde{R}={R}/{m}$ and $t_0 = m$, with $\tilde{t}_0$ set to unity. In the interval $0< \tilde{t}<({4}/{3})^3$, we have $0<R_{-}<R_{+}<R_{EH}$ and $0<R_{-}<R_{EH}<R_{+}$ in the interval $\tilde{t}>({4}/{3})^3$. $\tilde{t}=({28}/{9})^3$ is the upper limit of $\eta$. Since $R_-$ lies outside the event horizon after a certain period of time, it follows from nielsen2009 that the NEC must be violated in some region of the Sultana-Dyer spacetime.