Gravitational collapse of matter fields in de Sitter spacetimes

This paper investigates the spherically symmetric gravitational collapse of diverse matter fields in de Sitter spacetime by combining analytical and numerical methods to demonstrate that the quasilocal formalism of marginally trapped surfaces effectively tracks the evolution of black hole and cosmological horizons.

The Gist

Imagine the universe not as an empty void, but as a giant, expanding balloon filled with a thick, sticky fluid. Now, imagine you take a handful of that fluid and squeeze it into a tight ball. What happens? Does it just collapse into a tiny, invisible point (a black hole), or does the "stickiness" of the fluid change how fast it falls?

This paper by Akriti Garg and Ayan Chatterjee is a deep dive into exactly that scenario, but with a few cosmic twists. They are studying how matter collapses in a universe that is already expanding (called a de Sitter universe, which has a "cosmological constant" pushing things apart).

Here is the breakdown of their work using simple analogies:

1. The Setting: A Tug-of-War

Think of the universe as a giant tug-of-war.

• Gravity is the team trying to pull a ball of matter inward to crush it.
• The Cosmological Constant (dark energy) is the opposing team trying to push everything outward, expanding the universe.

The authors wanted to see what happens when a ball of matter tries to collapse in this specific environment. They looked at different types of "balls":

• Dust: Like dry sand falling through your fingers (no pressure, no stickiness).
• Perfect Fluids: Like water (has pressure).
• Viscous Fluids: Like honey or molasses (has "stickiness" or viscosity that resists flow).

2. The Problem with "Looking into the Future"

In physics, there is a concept called the Event Horizon (the point of no return for a black hole). The authors point out a weird problem with how we usually define this: it's "teleological."

The Analogy: Imagine trying to draw a line on a map to show where a flood will reach. To draw the line today, you would need to know exactly where the water will be tomorrow, next week, and next year. You can't know the future, so drawing the line based on the future is confusing for scientists trying to simulate what is happening right now.

The Solution: The authors use a tool called Marginally Trapped Tubes (MTT).

• The Analogy: Instead of guessing the future, they look at the water level right now. They track the "edge" of the water as it rises or falls in real-time.
• Why it's better: This method tells them exactly how the horizon grows as matter falls in, without needing to know the future. It's like watching a balloon inflate second-by-second rather than trying to predict its final size before you even start blowing.

3. What They Found: The "Sticky" Effect

The authors ran simulations (using math and computer models) with different types of matter. Here are their main discoveries:

• The "Honey" Effect (Viscosity): When they added "viscosity" (stickiness) to the collapsing matter, the collapse slowed down significantly.

  • Analogy: Dropping a rock in a pond (dust) makes a splash instantly. Dropping a rock in thick honey (viscous fluid) takes much longer to sink.
  • Result: The time it took for the matter to reach the center (the singularity) and the time it took for the black hole to form increased by "orders of magnitude." The universe's expansion and the fluid's stickiness acted like a brake.

• Two Horizons, One Dance: In this universe, there are two "edges" to watch:

  1. The Black Hole Horizon: As matter falls in, this horizon grows larger (like a black hole eating a meal).
  2. The Cosmological Horizon: Because the universe is expanding, there is a boundary far away where things move away too fast to be seen. As the black hole grows, this outer boundary shrinks.
  • The Dance: The authors showed that these two horizons move toward each other. Eventually, they meet at a point called the Nariai limit. It's like two people walking toward each other in a hallway until they bump into the middle.

• No Naked Singularity: A "naked singularity" is a scary concept where the center of a black hole (a point of infinite density) is exposed to the rest of the universe, breaking the laws of physics. The authors found that in all their scenarios, the "event horizon" (the protective skin) always formed before the singularity could be seen.

  • Conclusion: The universe seems to have a "cosmic censorship" rule: it always hides the messy, infinite points behind a wall of darkness.

4. The Takeaway

The paper essentially says:

1. Gravity isn't the only player: The expansion of the universe and the "stickiness" (viscosity) of matter play huge roles in how fast black holes form.
2. Viscosity matters: If the universe were filled with "thick" matter, black holes would take much, much longer to form than if the matter were "thin" dust.
3. Better Tools: Using the "real-time" tracking method (MTT) is much better for understanding how black holes grow than trying to predict the future.

In short, they took the classic idea of a collapsing star, added the complexity of an expanding universe and sticky fluids, and used a new mathematical lens to show that the process is slower, more complex, and still safely hides its dangerous center from the rest of the world.

Technical Summary

Technical Summary: Gravitational Collapse of Matter Fields in de Sitter Spacetimes

Problem Statement
The paper addresses the gravitational collapse of spherically symmetric matter fields within a de Sitter universe (spacetime with a positive cosmological constant, $\Lambda > 0$). While the formation of spacetime singularities and horizons is well-studied in asymptotically flat spacetimes (e.g., the Oppenheimer-Snyder-Datt model), the authors note that systematic analytical progress for collapse in de Sitter spacetimes remains limited, particularly regarding complex matter fields. Existing literature largely focuses on homogeneous dust or specific inhomogeneous dust models. There is a lack of detailed studies incorporating fluids with tangential and radial pressures, as well as bulk and shear viscosities, within the de Sitter framework. Furthermore, the paper seeks to clarify the role of the positive cosmological constant in altering the dynamics of horizon formation and the time scales of collapse.

Methodology
The authors employ a quasi-local formalism based on the Einstein field equations for a general spherically symmetric metric. The study utilizes the following methodological components:

1. Metric and Matter Formalism: The spacetime is described by a general spherically symmetric metric with functions $\alpha(r,t)$, $\beta(r,t)$, and the areal radius $R(r,t)$. The energy-momentum tensor ($T_{\mu\nu}$) is generalized to include a wide variety of matter fields: dust, perfect fluids, fluids with tangential ($p_t$) and radial ($p_r$) pressures, and fluids with shear ($\eta$) and bulk ($\zeta$) viscosities.
2. Quasi-Local Horizons (MTT): Instead of relying on the teleological Event Horizon (EH), which requires knowledge of the entire future null infinity, the authors utilize the Marginally Trapped Tube (MTT) formalism. They define horizons as the time evolution of marginally trapped surfaces (where the expansion of outgoing null geodesics vanishes, $\theta_{(\ell)} = 0$). The signature of the MTT (spacelike, timelike, or null) is determined by the constant $C$, derived from the energy-momentum tensor and the area of the surface.
3. Analytical and Numerical Integration: The authors solve the coupled Einstein equations and Bianchi identities. For various matter configurations, they derive exact analytical solutions for the shell dynamics ($R(r,t)$) using parametric methods involving Weierstrass elliptic functions ($P, \zeta, \sigma$) and hypergeometric functions. These analytical results are supplemented by numerical techniques to track the evolution of density profiles and horizon locations under different initial conditions.
4. Initial Conditions: The study imposes regular initial data on velocity and density profiles, ensuring the absence of shell-crossing singularities and initial trapped surfaces. Both marginally bound ($k=0$) and gravitationally bound ($k>0$) configurations are analyzed.

Key Contributions and Results

• Generalized Matter Collapse: The paper extends the formalism of gravitational collapse to include fluids with tangential pressure, radial pressure, and both shear and bulk viscosities in a de Sitter background. It provides exact analytical solutions for the time evolution of collapsing shells for these complex matter types.
• Horizon Evolution via MTT: The study demonstrates that the MTT formalism effectively tracks the simultaneous evolution of the black hole horizon ($r_{BH}$) and the cosmological horizon ($r_{CH}$).
  • As matter collapses, the black hole horizon grows (spacelike signature) while the cosmological horizon shrinks (timelike signature).
  • When matter stops falling, both horizons become null (Isolated Horizon phase).
  • In all cases studied, the black hole and cosmological horizons eventually merge at the Nariai limit ($r = 1/\sqrt{\Lambda}$).
• Impact of Viscosity and Pressure: A primary finding is that the inclusion of tangential pressure and, more significantly, viscosity, drastically alters the collapse dynamics.
  • Viscous forces generate anisotropies that slow down the gravitational collapse.
  • The time required for matter shells to reach the central singularity and the time required for horizon formation are increased by several orders of magnitude compared to dust models.
  • The time to reach the Nariai limit is also significantly delayed in viscous fluids.
• Singularity Formation: For the initial conditions chosen (regular density and velocity profiles), the collapse leads to the formation of a central spacetime singularity. The study confirms that the singularity remains hidden behind the black hole horizon, supporting the cosmic censorship conjecture in these specific de Sitter scenarios.
• Analytical Solutions: The authors provide explicit closed-form solutions for the time-radius relationship ($t(R)$) for dust, pressure-supported fluids, and viscous fluids, utilizing special functions (Weierstrass elliptic functions, hypergeometric functions) to describe the shell dynamics.

Significance and Claims
The paper claims that the quasi-local formalism of trapped surfaces (MTT) provides an ideal, non-teleological framework for studying the dynamics of horizons in gravitational collapse, particularly in the presence of a cosmological constant. It asserts that this approach successfully captures the transition between dynamical horizons (during collapse) and isolated horizons (in equilibrium).

The authors conclude that the presence of a positive cosmological constant and complex fluid properties (specifically viscosity) are critical factors that cannot be neglected when modeling the formation of compact objects. The significant delay in collapse timescales caused by viscosity suggests that the formation of black holes in a viscous de Sitter universe requires more careful consideration than previously modeled by simple dust approximations. The work serves as a step toward a more general understanding of gravitational collapse beyond spherically symmetric dust models, though the authors acknowledge that their exact solutions rely on specific assumptions regarding equations of state and that numerical methods would be necessary for cases where exact analytical solutions are not available.