Junction Conditions and Gravitational Collapse in Scalar-Tensor-Vector Gravity

This paper formulates junction conditions for Scalar-Tensor-Vector Gravity (STVG) to model gravitational collapse, demonstrating that an interior FLRW spacetime can match with an exterior Reissner-Nordström-like spacetime via a charged shell to form RN-like black holes in finite proper time.

The Gist

The Big Picture: Fixing the "Missing Mass" Mystery

Imagine you are watching a spinning carousel. If you spin it too fast, the horses should fly off. But in our universe, galaxies spin so fast that, according to our current laws of gravity, they should fly apart. Yet, they don't.

For decades, scientists have tried to solve this by inventing "Dark Matter"—an invisible, ghostly substance that acts like extra glue holding the galaxies together. But no one has ever actually seen or caught this ghost.

This paper explores a different idea proposed by J.W. Moffat, called Scalar-Tensor-Vector Gravity (STVG) or MOG. Instead of adding invisible ghosts (Dark Matter), this theory suggests that the "rules of gravity" themselves are slightly different. It proposes that gravity isn't just a simple pull; it has a few extra "knobs" (fields) that can change how strong the pull is and even add a repulsive push, much like a spring.

The Main Experiment: Crashing a Star

The authors of this paper wanted to see what happens if a massive ball of gas (a star) collapses under its own weight in this new theory. In standard physics, this process is like a balloon deflating until it becomes a tiny, dense point (a black hole).

They asked: Does this new theory of gravity allow stars to collapse into black holes, and if so, what do those black holes look like?

The Setup: Two Rooms and a Door

To study this, they had to build a mathematical model with two different "rooms":

1. The Inside Room (The Star): A collapsing ball of normal matter and "Dark Energy" (a mysterious force pushing things apart).
2. The Outside Room (The Void): The empty space surrounding the star, which they modeled after a specific type of black hole solution known as Reissner-Nordström.

The Problem: In standard physics, you can just tape these two rooms together. But in this new theory, the "knobs" (fields) inside the star don't match the "knobs" outside. If you just tape them, the universe would tear apart at the seam.

The Solution: The Charged Shell (The Doorframe)
To fix the tear, the authors introduced a special "doorframe" or shell between the inside and outside.

• Think of this shell as a thin, magical skin wrapping the collapsing star.
• This skin carries a special kind of "charge" (called STVG-charge). It's not electric charge like in a battery; it's a gravitational charge specific to this theory.
• This charge acts like a bridge, allowing the different rules of gravity inside and outside to connect smoothly without breaking the universe.

The Two Scenarios: How the Collapse Plays Out

The authors ran the simulation with two different settings for this "magic skin," leading to two different endings:

Scenario 1: The "Loose" Skin (Sub-extremal Black Hole)

In this version, the skin has a bit of flexibility. It allows for some energy to flow in and out.

• What happens: The star collapses, and the skin shrinks.
• The Result: It forms a black hole, but it's a "standard" one with two distinct boundaries (horizons).
• The Catch: As the skin gets very close to the inner boundary (the Cauchy horizon), things get messy. The energy density on the skin starts to behave strangely, almost like the skin is becoming unstable or "negative." It's like a rubber band stretching so far it starts to vibrate uncontrollably.

Scenario 2: The "Perfect" Skin (Extremal Black Hole)

In this version, the authors imposed a stricter rule: the skin must be perfectly balanced, with no "waste" energy (mathematically, its "trace" must be zero).

• What happens: The star collapses, and the skin shrinks.
• The Result: It forms an Extremal Black Hole. This is a very special, rare type of black hole where the two boundaries (horizons) merge into one. It's like a sphere where the "surface" and the "center" of the event horizon are the same.
• The Observer's View: If you were watching from far away, you would see the shell slow down as it approaches the horizon, eventually freezing and fading away, never quite crossing it from your perspective. But for the shell itself, it crosses the horizon in a finite amount of time.

Key Takeaways for the Everyday Reader

1. No Dark Matter Needed: This paper shows that you can explain gravitational collapse and black hole formation without needing invisible Dark Matter, provided you use this modified version of gravity (STVG).
2. The "Charge" is Key: In this theory, black holes aren't just empty pits; they are surrounded by a shell carrying a special gravitational charge that holds the different parts of the universe together.
3. Two Types of Black Holes: Depending on how the "skin" of the collapsing star behaves, the universe could end up with two different types of black holes: a standard one with two horizons, or a special "extremal" one where the horizons merge.
4. Instability Warning: The paper notes that as these collapsing objects get very close to their inner limits, they might become unstable, suggesting that nature might have strict rules about how deep these objects can go.

Summary Analogy

Imagine a collapsing star is a deflating beach ball.

• Standard Physics: The ball just shrinks until it's a tiny dot.
• This Paper's Theory: The ball is wrapped in a special, charged plastic wrap. As it shrinks, this wrap has to stretch and adjust to connect the inside of the ball to the outside universe.
  • If the wrap is a bit loose, the ball shrinks into a standard black hole, but the wrap gets shaky near the end.
  • If the wrap is perfectly tight and balanced, the ball shrinks into a unique, "perfect" black hole where the boundaries merge.

The authors successfully wrote the "instruction manual" (junction conditions) for how to attach the inside of the star to the outside universe using this special wrap, proving that black holes can indeed form in this new theory of gravity.

Technical Summary

Technical Summary: Junction Conditions and Gravitational Collapse in Scalar-Tensor-Vector Gravity

Problem Statement
Scalar-Tensor-Vector Gravity (STVG), also known as Modified Gravity (MOG), is a covariant theory proposed by J. W. Moffat to address the missing mass problem without invoking non-baryonic dark matter. While STVG successfully reproduces galaxy rotation curves and cluster data, a critical gap existed in understanding the dynamical evolution of gravitational collapse within this framework. Specifically, there was no established formalism for "gluing" an interior collapsing spacetime to an exterior black hole spacetime in STVG. Unlike General Relativity (GR), where the Oppenheimer-Snyder-Datt collapse matches a Friedmann-Lemaître-Robertson-Walker (FLRW) interior to a Schwarzschild exterior, STVG introduces additional fields (a massive vector field $A_\mu$ and scalar fields $G, \omega, \phi$) that complicate the matching conditions. The central problem addressed is formulating the correct junction conditions for STVG to determine if a gravitational collapse can proceed to form a black hole and what the nature of the resulting horizon would be.

Methodology
The authors employ a rigorous mathematical framework based on the Israel junction conditions, adapted for the specific field content of STVG.

1. Derivation of Junction Conditions: Starting from the total STVG action, the authors derive the field equations for the metric, vector, and scalar sectors. They then analyze the behavior of these fields across a timelike hypersurface $\Sigma$ (the collapsing shell).

   • Metric: They enforce the continuity of the induced metric ($[g_{\mu\nu}] = 0$) and derive the relationship between the jump in extrinsic curvature and the surface energy-momentum tensor $S_{\mu\nu}$.
   • Scalar Fields ($G, \omega, \phi$): To avoid ill-defined distributional products (e.g., $\Theta \delta$) in the equations of motion, the authors impose continuity of the scalar fields and their normal derivatives across the junction ($[G] = [\omega] = [\phi] = 0$ and $[\partial_\mu G] = [\partial_\mu \omega] = [\partial_\mu \phi] = 0$). Crucially, they demonstrate that the trace of the surface energy-momentum tensor must vanish ($S = g^{\mu\nu}S_{\mu\nu} = 0$) if the scalar field $G$ is treated as a dynamical field.
   • Vector Field ($A_\mu$): Instead of matching the potential $A_\mu$ directly, the authors match the field strength tensor $B_{\mu\nu}$. They show that a discontinuity in $B_{\mu\nu}$ across the shell necessitates a surface STVG-charge density ($Q$), which acts as the source for the exterior vector field.

2. Construction of Collapse Models: The authors construct two simplified models of spherical gravitational collapse:

   • Interior: A closed FLRW spacetime containing baryonic matter (dust) and dark energy, with all scalar fields frozen and the vector field component $A_0$ set to zero.
   • Exterior: A static, spherically symmetric Reissner-Nordström (RN)-like spacetime in STVG, characterized by a mass $M$ and an STVG charge $Q$, with a non-zero $A_0(r)$ component.
   • Matching: The two regions are joined by a thin shell carrying the STVG charge.

3. Numerical and Analytical Solutions: The authors solve the resulting differential equations for the shell's radius $R(\tau)$ as a function of proper time $\tau$ under two distinct scenarios regarding the scalar field $G$.

Key Contributions and Results

1. Formulation of STVG Junction Conditions: The paper provides the first complete set of junction conditions for STVG. A key finding is that the discontinuity in the STVG vector field strength tensor requires a charged shell, and the nature of the scalar fields imposes strict constraints on the shell's energy-momentum tensor.

2. Two Distinct Collapse Scenarios:

   • Scenario A (Truncated Theory, $S \neq 0$): The authors consider a simplified version where the scalar field $G$ is treated as a non-dynamical constant, effectively disregarding its equation of motion. In this case, the trace of the shell's energy-momentum tensor need not vanish. The results show that the shell collapses and approaches the Cauchy horizon of the RN-like spacetime. As the shell nears this horizon, the shell's energy density rapidly decreases and becomes negative, indicating an instability. For an asymptotic observer, the shell never crosses the outer horizon.
   • Scenario B (Full Theory, $S = 0$): In the more rigorous treatment where $G$ remains a scalar field, the junction conditions enforce a traceless energy-momentum tensor ($S=0$). This corresponds to a conformal shell (e.g., dominated by photon-like interactions). In this scenario, the collapse leads to an extremal RN-like black hole (where the inner and outer horizons coincide, $Q^2 = GM$). The shell crosses the Cauchy horizon in finite proper time. While the system exhibits bounce-like behavior near the horizon in the analytical approximation, the authors note that the shell enters the horizon and does not return, consistent with the expectation that the Cauchy horizon acts as a limit of predictability.

3. Nature of the Resulting Black Hole: The study confirms that gravitational collapse in STVG does not necessarily produce a Schwarzschild black hole. Instead, the end state is an RN-like black hole characterized by the STVG charge. This implies that observational signatures, such as gravitational lensing or black hole shadows, would differ from standard GR predictions, even if the object is electrically neutral.

Significance and Claims
The paper claims to establish the theoretical foundation for studying dynamical processes in STVG, specifically gravitational collapse. By deriving the junction conditions, the authors demonstrate that the theory is consistent with the formation of black holes from collapsing matter.

The significance lies in:

• Theoretical Consistency: Proving that STVG allows for a smooth matching of spacetimes via a charged shell, validating the theory's applicability to strong-field regimes.
• Distinctive Features: Highlighting that the requirement for a traceless energy-momentum tensor (in the full scalar-field treatment) is a unique feature of STVG that distinguishes it from standard GR collapse scenarios.
• Observational Implications: The authors suggest that if STVG is correct, astrophysical black holes formed via collapse should exhibit RN-like properties (modified horizons and shadows) rather than Schwarzschild properties. This provides a potential observational test to distinguish STVG black holes from standard GR black holes, provided the electric charge can be ruled out by other means.

The authors remain modest regarding the complexity of the full theory, noting that their models rely on "frozen" scalar degrees of freedom and simplified interaction terms. They acknowledge that the instabilities near the Cauchy horizon and the full interaction Lagrangian between matter and the vector field require further dedicated investigation. However, within the scope of these simplified toy models, the paper successfully demonstrates a viable mechanism for gravitational collapse leading to STVG black hole formation.