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Can you fall into a McVittie black hole? Will you survive?

Brien C. Nolan

TL;DR

This paper shows that in expanding McVittie spacetimes, freely falling timelike geodesics can reach the black hole horizon in finite proper time, contrasting with the non-geodesic fluid flow. For a positive cosmological constant, all curvature and matter invariants remain finite at the horizon, making the crossing innocuous for test bodies, while in the Λ = 0 case zero- and first-order invariants stay finite but the second-order curvature invariant diverges, and some parallel-propagated curvature components blow up, suggesting potential deformational effects on extended bodies. The analysis combines explicit geodesic equations, autonomous-system techniques, and a careful expansion of curvature invariants (including the Kretschmann scalar and derivatives) to reveal the distinct horizon behavior governed by the asymptotics of the Hubble function H(t). The work clarifies the black hole interpretation of McVittie spacetimes and highlights how the presence or absence of a cosmological constant crucially influences horizon regularity and tidal forces.

Abstract

Yes and maybe. In contrast to the fluid particles of this perfect fluid spacetime which follow non-geodesic world-lines and escape to infinity, we prove that freely-falling test particles of McVittie spacetime can reach the black hole horizon in finite proper time. We review the relevant evidence and argue that the fate of an extended test body is less clear. More precisely: we consider expanding McVittie spacetimes with a non-negative cosmological constant. In each member of this class, we identify a region of the spacetime such that an observer following an initially-ingoing timelike geodesic crosses the black hole horizon of the spacetime in a finite amount of proper time. The curvature behaves in interesting ways along these geodesics. In the case of a positive cosmological constant, curvature scalars (of zero, first and second order in derivatives), Jacobi fields and parallel propagated (p.p.) frame components of the curvature remain finite along timelike geodesics running into the black hole horizon. For a vanishing cosmological constant, scalar curvature terms of zero and first order as well as Jacobi fields remain finite in this limit. However, scalar curvature terms of second order diverge, and we show that there are p.p.\ frame components of the curvature tensor that also diverge in this limit. We argue that this casts a doubt as to whether or not an extended test body can survive crossing the black hole horizon in this case.

Can you fall into a McVittie black hole? Will you survive?

TL;DR

This paper shows that in expanding McVittie spacetimes, freely falling timelike geodesics can reach the black hole horizon in finite proper time, contrasting with the non-geodesic fluid flow. For a positive cosmological constant, all curvature and matter invariants remain finite at the horizon, making the crossing innocuous for test bodies, while in the Λ = 0 case zero- and first-order invariants stay finite but the second-order curvature invariant diverges, and some parallel-propagated curvature components blow up, suggesting potential deformational effects on extended bodies. The analysis combines explicit geodesic equations, autonomous-system techniques, and a careful expansion of curvature invariants (including the Kretschmann scalar and derivatives) to reveal the distinct horizon behavior governed by the asymptotics of the Hubble function H(t). The work clarifies the black hole interpretation of McVittie spacetimes and highlights how the presence or absence of a cosmological constant crucially influences horizon regularity and tidal forces.

Abstract

Yes and maybe. In contrast to the fluid particles of this perfect fluid spacetime which follow non-geodesic world-lines and escape to infinity, we prove that freely-falling test particles of McVittie spacetime can reach the black hole horizon in finite proper time. We review the relevant evidence and argue that the fate of an extended test body is less clear. More precisely: we consider expanding McVittie spacetimes with a non-negative cosmological constant. In each member of this class, we identify a region of the spacetime such that an observer following an initially-ingoing timelike geodesic crosses the black hole horizon of the spacetime in a finite amount of proper time. The curvature behaves in interesting ways along these geodesics. In the case of a positive cosmological constant, curvature scalars (of zero, first and second order in derivatives), Jacobi fields and parallel propagated (p.p.) frame components of the curvature remain finite along timelike geodesics running into the black hole horizon. For a vanishing cosmological constant, scalar curvature terms of zero and first order as well as Jacobi fields remain finite in this limit. However, scalar curvature terms of second order diverge, and we show that there are p.p.\ frame components of the curvature tensor that also diverge in this limit. We argue that this casts a doubt as to whether or not an extended test body can survive crossing the black hole horizon in this case.

Paper Structure

This paper contains 15 sections, 14 theorems, 113 equations, 1 figure.

Table of Contents

  1. Introduction
  2. McVittie spacetimes as black holes
  3. Falling into a McVittie black hole: timelike geodesics
  4. Curvature at the horizon
  5. Curvature invariants.
  6. The case $H_0>0$.
  7. The case $H_0=0$.
  8. Universal behaviour of causal geodesics approaching the horizon.
  9. The background equation of state and asymptotics of $H^{(n)}$.
  10. Some key limits.
  11. Curvature scalars at the horizon
  12. Jacobi fields and tidal forces at the horizon.
  13. Conclusions
  14. Second order curvature scalars
  15. Coefficients of $\nabla_f\nabla_eR_{abcd}\nabla^f\nabla^eR^{abcd}$

Key Result

Proposition 1.1

Let $(M,g)$ be an expanding McVittie spacetime with a Big Bang background.

Figures (1)

  • Figure 1: Penrose diagram of a McVittie spacetime with $\Lambda>0$. $\mathcal{I}^+$, future null infinity, is spacelike everywhere in this case. The regular region is shaded: the unshaded region is anti-trapped. $\mathcal{H}^+$ is the black hole horizon - the future boundary of the causal past of $\mathcal{I}^+$. $t$ increases from left to right along the past singularity at $r=2m$. The left-most arrow (black) represents an ingoing radial null geodesic, along which $(t,r)\to(+\infty,r_-)$ as $\mathcal{H}^+$ is approached. The right-most arrow (blue) represents a fluid flow line: $(t,r)\to(+\infty,+\infty)$ as proper time increases without bound along these worldlines. The key question we address in this paper is whether or not timelike geodesics (represented by the central arrow (red)) can reach $\mathcal{H}^+$ in finite proper time. See kaloper2010mcvittie,lake2011more and nolan2014particle for further details.

Theorems & Definitions (14)

  • Proposition 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.1
  • Theorem 2.1
  • Lemma 2.3
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 4 more