Roche limit and stellar disruption in the Simpson--Visser spacetime

This paper investigates the tidal forces and Roche limits for various stellar objects within the Simpson--Visser spacetime, comparing static and infalling observers to determine how the black bounce regularization affects tidal disruption and the observability of these events for astrophysical black holes like M87* and Sgr A*.

The Gist

Imagine the universe is filled with cosmic vacuum cleaners known as black holes. Usually, we think of them as having a terrifying center called a "singularity," a point of infinite density where the laws of physics break down. But what if that center isn't a broken point, but rather a smooth tunnel? This is the idea behind the Simpson–Visser spacetime, a theoretical model explored in this paper.

Think of a standard black hole like a funnel that gets narrower and narrower until it pinches off into a sharp, impossible point. The Simpson–Visser model is like a funnel that narrows down to a smooth, round tunnel (called a "throat") and then opens up again on the other side. It's a "black bounce" because instead of crushing everything into a singularity, the universe "bounces" the path back out.

Here is what the authors discovered about how stars behave near these cosmic tunnels, explained simply:

1. The Cosmic Stretching Machine (Tidal Forces)

When a star gets close to a black hole, the gravity on the side of the star closest to the hole is much stronger than the gravity on the far side. This difference acts like a giant cosmic hand pulling the star apart. This is called tidal force.

• The Analogy: Imagine holding a piece of taffy. If you pull the ends, it stretches. If you pull hard enough, it snaps. The point where it snaps is the Roche limit.
• The Discovery: In a normal black hole, this stretching gets infinitely strong as you get closer to the center. But in the Simpson–Visser model, because the center is a smooth tunnel, the stretching force doesn't go to infinity. In fact, it can actually flip! Instead of just stretching the star, the gravity can start to squeeze it sideways, like a gentle hug, before potentially stretching it again.

2. The Observer Effect: Standing Still vs. Falling In

The paper points out a fascinating difference depending on how you are watching the star.

• The Static Observer: Imagine a camera hovering in space, using powerful rockets to stay in one spot. From this view, the forces look one way.
• The Falling Observer: Now imagine a camera falling freely into the hole, like a skydiver.
• The Twist: For a normal black hole, both cameras see the same stretching. But for this "bouncing" black hole, the falling camera sees something different. The "squeeze" (transverse force) depends on how fast the camera is falling. The faster you fall, the further out from the center this "squeeze" effect starts happening. It's like the speed of your fall changes the shape of the gravity field you experience.

3. The "Roche Limit" Game

The authors calculated the Roche limit (the "snap point") for three types of stars:

• Neutron Stars: These are incredibly dense, like a sugar cube weighing a billion tons. They are tough.
• White Dwarfs: Dense, but not as tough as neutron stars.
• Sun-like Stars: Big, fluffy, and easy to tear apart.

The Big Finding:
The "smooth tunnel" parameter (let's call it the "bounciness" of the hole) acts like a shield.

• If the black hole is "bouncy" enough (has a large tunnel), the tidal forces become so weak that they can't tear the star apart at all. The star might fall right through the event horizon and into the tunnel without ever getting ripped to shreds.
• For massive black holes (like the ones at the centers of galaxies, M87* and Sgr A*), the authors found that if the "bounciness" is high, the star gets swallowed whole before it has a chance to break apart. The disruption happens inside the "horizon" (the point of no return), making it invisible to the outside universe.

4. The Dynamic Dance (The Affine Model)

To make their math more realistic, the authors didn't just treat stars as rigid balls. They used a model that treats the star like a blob of jelly.

• What happened: As the "jelly star" fell toward the tunnel, it didn't just stretch into a long noodle (spaghettification).
• The Surprise: Because of the unique geometry of the tunnel, the star would get squeezed sideways, and then, as it got very close to the tunnel, it would actually rebound and stretch sideways. It's as if the star was being squeezed by a hand, and then suddenly the hand let go and pulled it apart in a different direction.
• The Result: For stars falling into these "bouncy" black holes, the "jelly" often survived the trip intact, or at least didn't get torn apart as violently as it would near a standard black hole.

Summary

This paper suggests that if black holes are actually these "bouncing" tunnels rather than singular points, they are much gentler on falling stars.

• Standard Black Holes: Rip stars apart violently outside the event horizon (if the hole isn't too massive).
• Simpson–Visser "Bouncy" Black Holes: Can act as a protective shield. They can weaken the tearing forces so much that stars might fall inside the black hole without ever being ripped apart, or they might get stretched in weird, sideways ways that we don't see in normal black holes.

The authors conclude that by watching how stars get torn apart (or don't get torn apart) near black holes, we might be able to tell if these "bouncy" tunnels actually exist in our universe.

Technical Summary

Technical Summary: Roche Limit and Stellar Disruption in the Simpson–Visser Spacetime

Problem Statement
The detection of black hole shadows and gravitational waves has reinforced the theoretical prediction of black holes, yet alternative compact objects, such as regular black holes and wormholes, can mimic these signatures in certain regimes. A critical method for distinguishing between standard singular black holes and regular alternatives lies in the behavior of matter in their vicinity, specifically through tidal forces. While tidal forces have been extensively analyzed for standard black holes (e.g., Schwarzschild, Reissner–Nordström), the Simpson–Visser spacetime presents a unique "black bounce" topology where the central singularity is replaced by a wormhole throat at $r=a$. This work investigates how this topological shift alters tidal force profiles, the resulting Roche limit (the radius of tidal disruption), and the observability of stellar disruption events for neutron stars, white dwarfs, and Sun-like stars.

Methodology
The study employs a two-pronged analytical and numerical approach within the Simpson–Visser metric, which describes a spacetime containing a wormhole throat hidden behind an event horizon (for $a \leq 2M$).

1. Tidal Force Derivation: The authors derive the tidal tensor components using the geodesic deviation equation. They analyze two distinct observer frames:

   • Static Observer: A tetrad basis fixed in space.
   • Radially Infalling Observer: A tetrad basis attached to a particle falling freely into the black hole.
     The components are projected onto orthonormal tetrads to ensure physical interpretability.

2. Roche Limit Calculation (Analytical): Using a first-order Newtonian approximation, the Roche radius ($r_{Roche}$) is determined by equating the radial component of the tidal force ($K_1$) with the star's internal gravitational binding force ($M_\star/R_\star^3$). This algebraic approach assumes the star remains a rigid, incompressible sphere until disruption. The authors analyze the roots of the resulting fifth-order polynomial to determine if disruption occurs outside the event horizon ($r_{Roche} > r_h$).

3. Dynamical Evolution (Numerical): To address the limitations of the rigid-sphere approximation, the authors implement the Affine Model (Carter and Luminet). This treats the star as a compressible, self-gravitating fluid ellipsoid. The evolution of the principal semi-axes ($R_1, R_2, R_3$) is governed by coupled ordinary differential equations incorporating the full tidal tensor (including energy-dependent transverse components), self-gravity, and internal pressure. This allows for the tracking of continuous deformation and the identification of instability thresholds.

Key Contributions and Results

• Observer Dependence of Tidal Forces: A primary finding is that, unlike in Schwarzschild or Reissner–Nordström spacetimes, the tidal tensor in the Simpson–Visser geometry is not invariant under a change from a static to a radially infalling frame. While the radial component ($K_1$) remains identical for both observers, the angular (transverse) components ($K_2, K_3$) acquire an explicit dependence on the particle's conserved energy $E$. For infalling observers, the radius at which transverse forces vanish or reach extrema shifts outward as energy increases.

• Sign Inversion and Regularization Effects: The regularization parameter $a$ (the throat radius) significantly modifies the tidal field.

  • The radial component $K_1$ can vanish and change sign (from stretching to compression) at $r = \sqrt{3/2}a$.
  • The transverse component $K_2$ vanishes at the throat ($r=a$) for static observers but shifts for infalling observers.
  • Crucially, increasing $a$ suppresses the intensity of tidal forces. For sufficiently large $a$, the tidal forces may become too weak to overcome the star's self-gravity, preventing disruption entirely.

• Roche Limit and Observability:

  • Neutron Stars: For typical black hole masses ($\sim 10 M_\odot$), the Roche radius is often hidden inside the event horizon. The regularization parameter $a$ further reduces the Roche radius; if $a$ exceeds a critical value ($a_{crit}$), no real solution for the Roche radius exists, implying the star is swallowed intact.
  • White Dwarfs: While more susceptible to disruption than neutron stars, white dwarfs orbiting supermassive black holes (e.g., M87*, Sgr A*) generally have their Roche radii hidden behind the horizon. For M87*, disruption is only possible if $a \lesssim 10^{-4}M$.
  • Sun-like Stars: These stars exhibit the most observable disruption potential. For Sgr A*, the Roche radius can remain outside the event horizon even for relatively large values of $a$ (up to $\sim 10M$), allowing for observable tidal disruption events (TDEs). However, for M87*, the disruption radius is typically hidden unless $a$ is extremely small ($\lesssim 10^{-2}M$).

• Dynamical Deformation (Affine Model): The numerical simulations confirm the analytical trends but reveal richer dynamics:

  • Geometric Shielding: Increasing $a$ drastically dampens radial stretching ($R_1$), often preventing the star from reaching the disruption threshold ($R_1/R_\star \gtrsim 2$).
  • Transverse Rebound: A striking signature of the Simpson–Visser geometry is the behavior of the transverse axis ($R_2$). Unlike the monotonic compression seen in Schwarzschild "spaghettification," the transverse axis in the regularized geometry can undergo a "rebound," stretching beyond its initial radius ($R_2/R_\star > 1$) near the throat. This indicates a transition from compression to transverse stretching, a direct consequence of the modified tidal tensor.
  • Mass Dependence: Increasing the black hole mass reduces the amplitude of tidal deformation at a given dimensionless radius, pushing the disruption point deeper into the gravitational well or hiding it behind the horizon.

Significance and Claims
The paper claims that the Simpson–Visser spacetime introduces a fundamentally different physical paradigm for tidal interactions compared to standard regular black holes. While standard models typically replace the singularity with a de Sitter-like core, the black bounce topology (wormhole throat) leads to unique signatures, including the observer-dependent transverse tidal forces and the potential for transverse stretching (rebound) rather than just compression.

The authors assert that the regularization parameter $a$ acts as a "geometric shield," capable of suppressing tidal disruption entirely or hiding it behind the event horizon. This suggests that the absence of observed TDEs for certain stellar types or black hole masses could, in principle, constrain the parameter $a$ of the underlying spacetime geometry. The work highlights that while the radial disruption threshold is robust across observer frames, the full dynamical evolution of a star is sensitive to the kinematic state of the infall and the topological structure of the spacetime.

The study concludes that while the simplified Roche limit provides a useful boundary for observability, the Affine Model is necessary to capture the continuous dynamical nature of disruption, particularly the non-trivial transverse dynamics unique to black bounce spacetimes. Future work is proposed to extend these findings to rotating (Kerr-like) Simpson–Visser spacetimes.