Can You Fall into a McVittie Black Hole and Survive? ✛
Context
McVittie black holes are solutions to Einstein's equations that embed the Schwarzschild field into an expanding universe. The question of whether they can serve as models of black holes has long remained open. Early work by Sussman and others showed that an event horizon exists, but fluid particles do not fall into the hole. Recent claims of infinite curvatures at the horizon required verification. This work fills the gap: it proves that freely falling observers reach the horizon and analyzes the behavior of curvature along their trajectories.
Methods
The author considers a class of expanding McVittie spacetimes with a nonnegative cosmological constant and a Big Bang in the past. To analyze trajectories, the equations of timelike geodesics are reduced to an autonomous system. Using monotonicity lemmas and a theorem on exiting a compact set, it is proven that geodesics reach the horizon in finite proper time. Then, scalar curvature invariants of zeroth, first, and second order, as well as curvature components in a parallel-propagated frame and Jacobi fields, are computed.
Results
It is proven that there exist timelike geodesics that reach the horizon of a McVittie black hole in finite proper time. For a positive cosmological constant, all scalar curvature invariants (zeroth, first, and second order), Jacobi fields, and curvature components in a parallel-propagated frame are finite at the horizon. For a zero cosmological constant, zeroth- and first-order scalars are finite, but the second-order scalar diverges as γ (the gamma factor). Additionally, some curvature components in the parallel-propagated frame, such as R0i0i, diverge. This indicates a possible singularity, although Jacobi fields remain finite.
Implications
This work refines the interpretation of McVittie spacetimes as models of black holes. Unlike previous studies that focused on null geodesics, it is shown here for the first time that massive particles can also fall into the hole. For Λ=0, a second-order divergence is found, which challenges the standard conclusion about the survival of extended bodies based on the finiteness of Jacobi fields. This highlights the importance of analyzing higher derivatives of curvature.
Future development
The topic could develop toward studying the behavior of extended bodies with multipole moments, as in the Dixon formalism. Second-order divergences may affect the equations of motion for a body with an octupole moment. Also promising is the investigation of the interaction of test fields (scalar, fluid) with the horizon, which could clarify the physical effects of a weak singularity.
Impact
The results are important for relativistic astrophysics and cosmology, especially when modeling black holes in an expanding universe.
Next steps
The next step could be the calculation of the motion of extended bodies with higher multipoles and the analysis of test fields near the horizon.
Key open problems
This work is connected to the fundamental problem of determining the fate of an observer crossing the horizon in cosmological black holes. The question of the physical realizability of such horizons for Λ=0 remains open due to higher-order divergences.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
The study shows that freely falling particles reach the McVittie horizon in finite proper time, unlike fluid particles, which escape to infinity. With a positive cosmological constant, all curvatures remain finite at the horizon, while with a zero cosmological constant, second-order components diverge, casting doubt on the survival of extended bodies.
Where its strength is
The strength of the work is that it demonstrates for the first time that massive particles can reach the horizon and reveals a second-order divergence at Λ=0, which refines the physical interpretation of the McVittie metric.
Where it could go next
- It might be interesting to compare the behavior of geodesics in the McVittie metric with a similar analysis in other cosmological black holes, for example, in models where the horizon is replaced by a discontinuity; our archive contains an analysis of such a scenario nearby.based on: Black Holes: A Rip Instead of an Event Horizon arXiv:2510.04165
- The question of the survival of extended bodies can be considered from the perspective of horizon stability and the possible transition of a black hole into a white hole; nearby there is a work on what needs to be 'broken' for such a transition.based on: A Black Hole Turns White: What Has to Break arXiv:2606.22404
- Since at Λ=0 divergent curvature components arise, it may be useful to look at alternative approaches to describing the internal structure, for example, the model of a black hole as a cloud of pieces of space.based on: A Black Hole as a Cloud of Spacetime Bits arXiv:2605.18922
Worth trying
- Check whether the finiteness of invariants is preserved when rotation or charge is added to the McVittie metric.
- Investigate the behavior of Jacobi fields not only along radial but also along non-radial geodesics.
- Compare the second-order divergence with singularity criteria in other expanding solutions.
Why this may matter
If the conclusions about the second-order divergence are confirmed, this could change our understanding of which objects in an expanding universe should be considered black holes, and could affect models of accretion and evaporation.
Nearby works in our archive
- Black Holes: A Rip Instead of an Event Horizonsimilarity 0.58
- How Gravity Rewires Electricitysimilarity 0.56
- How to Complete the Edge of Spacetimesimilarity 0.52
- A Black Hole as a Cloud of Spacetime Bitssimilarity 0.52
- Black Holes Survived the Universe's Contractionsimilarity 0.52
- A Black Hole Turns White: What Has to Breaksimilarity 0.51
The archive is sparse around this work: the closest review is only 0.58, against a usual 0.64. It looks like few have drilled nearby.
Cosmic fluid particles in the McVittie model do not fall into the black hole but fly off to infinity, while freely falling observers reach the horizon in finite time.
Key numbers
- finite proper time to horizon: depends on initial conditions but is always finite
- limit of Kretschmann scalar for Λ=0: 3/(4m^4)
- first-order limit for Λ=0: κ²/(16m^6)
- limit of K=Hγ^{1/2}: 1/(2m)
- horizon for Λ>0: r_- between 2m and 3m