Circular orbits in spherically symmetric spacetimes and BSW effect with nonzero force

This paper develops a general framework for analyzing circular particle orbits under arbitrary external forces in static spherically symmetric spacetimes, revealing that such forces can induce new stable orbit branches and near-horizon trajectories in extremal black holes, thereby enabling high-energy collisions with behavior analogous to the BSW effect in rotating black holes.

The Gist

Imagine you are watching a cosmic dance around a black hole. Usually, we think of particles (like tiny stars or dust) as free-spirited dancers, gliding along the smooth, invisible tracks of gravity called "geodesics." They don't need to push or pull; they just follow the curve of space.

But in this paper, the authors ask a "What if?" question: What if the dancers aren't free? What if they are being pushed, pulled, or held by an invisible hand (an external force) that keeps them on a specific circular path?

Here is the story of their findings, broken down into simple concepts.

1. The Cosmic Tether

In the real world, things like accretion disks (swirling clouds of gas around black holes) or self-gravity can act like that "invisible hand." The authors created a mathematical toolkit to figure out:

• How hard do you have to push a particle to keep it in a perfect circle?
• Is that circle stable, or will the particle fly off or crash into the black hole?
• How does this change if the black hole is "charged" (like a giant magnet) versus just a normal heavy one?

Think of it like a child on a merry-go-round. If the child just sits there, they might fly off if it spins too fast. But if a parent holds their hand (the force), they can stay on. The paper calculates exactly how hard that parent needs to hold on.

2. The "Impossible" Zone Near the Edge

Black holes have an edge called the Event Horizon. Once you cross it, you can't get out.

• Normal Black Holes: If you try to orbit a normal black hole right at the edge, the force required to keep you there becomes infinite. It's like trying to hold a rope that is being pulled by a hurricane; eventually, the rope snaps. You simply cannot have a stable circular orbit right at the edge of a normal black hole without infinite effort.
• Extremal Black Holes: These are special, "super-charged" or "maximally spinning" black holes. The authors found that for these special black holes, the force required to stay in a circle near the edge is finite. It's like the hurricane calms down just enough that a strong person could hold the rope. This opens up a whole new playground of orbits that didn't exist before.

3. The "Sweet Spot" (ISCO)

Astronomers love to find the Innermost Stable Circular Orbit (ISCO). Think of this as the "last safe seat" at a concert before you fall off the stage.

• Without any external force, this seat is at a specific distance.
• With a force: The authors discovered that if you apply a steady push or pull, you can create new seats. Sometimes, you can even find two or three different stable spots where a particle can orbit, depending on how hard you push. It's like adding a safety net that allows you to sit closer to the edge than before.

4. The Cosmic Particle Smash (The BSW Effect)

The most exciting part of the paper is about High-Energy Collisions.
Imagine two particles smashing into each other near a black hole. If they hit just right, the energy released can be astronomical—potentially creating a "particle accelerator" more powerful than anything we can build on Earth. This is known as the BSW effect.

The authors looked at what happens when these particles are forced into circular orbits:

• Scenario A (The Glide): One particle is stuck in a stable circle, and another crashes into it. The energy is huge, but not the most huge.
• Scenario B (The Plunge): A particle is forced into a circle, but then it gets nudged and falls straight toward the black hole's edge. When it hits another particle right at the edge, the energy explosion is massive.

The Big Surprise: They found that for these "forced" orbits, the energy explosion behaves exactly the same way as it does for rotating black holes (which are usually thought to be the only ones capable of this).

• The Metaphor: It's as if the external force (the invisible hand pushing the particle) is a "fake rotation." Even though the black hole isn't spinning, the force makes the physics act as if it were spinning, allowing for these incredible energy collisions.

Summary

This paper is like a new instruction manual for a cosmic playground. It tells us:

1. Forces matter: If you push particles, you can create stable orbits that gravity alone couldn't support.
2. Special black holes are special: Near "extremal" black holes, you can orbit right at the edge without needing infinite strength.
3. Forces mimic spin: By pushing particles, you can trick the universe into thinking the black hole is spinning, allowing for particle collisions with energies that could theoretically be infinite.

In short, the authors showed that by adding a little "push" to the cosmic dance, we can unlock new, high-energy secrets hidden right at the edge of the universe's most mysterious objects.

Technical Summary

1. Problem Statement

The paper addresses a significant gap in the theoretical understanding of the Ba˜nados-Silk-West (BSW) effect, which describes the possibility of arbitrarily high-energy particle collisions near black hole horizons. Previous studies (including the authors' own prior work) largely focused on particles falling radially toward a black hole or free particles moving on geodesics.

The specific problems addressed in this work are:

• Circular Orbits with External Forces: How do circular orbits behave when particles are subject to an unspecified, non-zero external force (acceleration) in static, spherically symmetric spacetimes?
• Stability and ISCO: How does the presence of a force alter the definition and location of the Innermost Stable Circular Orbit (ISCO)?
• Near-Horizon Behavior: Do circular orbits exist near the horizons of extremal and near-extremal black holes when a force is present, given that they typically do not exist for free particles in non-extremal cases?
• High-Energy Collisions: Can the BSW effect be realized via collisions involving these forced circular orbits, and how does the collision energy depend on the black hole's surface gravity ($\kappa$)?

2. Methodology

The authors developed a general formalism for particle motion in static, spherically symmetric spacetimes with metric:
$$ds^2 = -N^2(r)dt^2 + \frac{dr^2}{A(r)} + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$

Key Methodological Steps:

1. Equations of Motion: They derived the equations of motion for a particle of mass $m$ under an external force $F^\mu$. They introduced an acceleration vector $f^\mu = F^\mu/m$ and utilized the normalization condition $u^\mu u_\mu = -1$ to derive the effective potential $U_{eff}$.
2. Circular Orbit Conditions: For a circular orbit at radius $r_c$, they imposed:
   • $U_{eff}(r_c) = 0$ (Radial velocity is zero).
   • $U'_{eff}(r_c) = 0$ (Radial acceleration is zero, defining the force required to maintain the orbit).
   • They derived an explicit formula for the required radial acceleration $a_c$ (and thus the force $f$) to maintain a circular orbit with angular momentum $L$.
3. Stability Analysis: Stability is determined by the sign of the second derivative of the effective potential, $U''_{eff}(r_c)$. The condition for the ISCO is $U''_{eff} = 0$. They derived a general stability criterion involving the derivative of the external force ($f'$) and the derivative of the kinematic acceleration ($a'$).
4. Near-Horizon Expansion: They performed Taylor expansions of the metric functions $N^2$ and $A$ near the horizon ($r \to r_h$) to analyze the behavior of acceleration for non-extremal ($p=1$), extremal ($p \ge 2$), and ultra-extremal horizons.
5. Specific Metrics: The formalism was applied to:
   • Schwarzschild Metric: Assuming a constant external force.
   • Reissner-Nordström (RN) Metric: Analyzing non-extremal, extremal, and near-extremal cases for uncharged particles.
6. Collision Scenarios: They calculated the center-of-mass energy ($E_{c.m.}$) for two scenarios:
   • O-scenario: A particle on a circular orbit (ISCO) collides with a usual infalling particle.
   • H-scenario: A particle plunges from a circular orbit onto the horizon to collide with another particle.

3. Key Contributions and Results

A. General Formalism for Forced Circular Orbits

• The authors derived a machinery to calculate the specific external force required to keep a particle on a circular orbit at any radius $r$.
• They established that for a constant force, the number of possible circular orbits depends on the angular momentum $L$.
  • For low angular momentum, there is a single orbit for a given force.
  • For high angular momentum, the force-radius relationship becomes non-monotonic, leading to the existence of multiple branches of solutions (up to 3 roots) for a single force value. This introduces new orbital branches absent in geodesic motion.

B. Schwarzschild Metric Results

• Force Dependence: For a constant force, the number of circular orbits changes based on the dimensionless angular momentum $b = L/r_s$.
  • If $b < b_c \approx 1.64$, the force-radius function is monotonic (1 root).
  • If $b > b_c$, the function has a local minimum and maximum, allowing for 1, 2, or 3 orbits depending on the force magnitude.
• Stability: The stability diagram shows that the ISCO shifts depending on the force. The boundary of stability intersects the zero-force curve exactly at the standard geodesic ISCO.

C. Reissner-Nordström (RN) Metric Results

• Non-Extremal Case: The behavior is qualitatively similar to Schwarzschild, but the presence of charge ($Q$) modifies the required force and the number of roots.
• Extremal Case ($Q=M$):
  • Finite Acceleration: Unlike non-extremal black holes where acceleration diverges at the horizon, for extremal black holes, the acceleration required to maintain a circular orbit remains finite at the horizon.
  • Fake Trajectories: A formal solution exists exactly on the horizon, but it is "fake" (spacelike or lightlike) for massive particles. However, near-horizon circular trajectories ($r = r_h + \epsilon$) do exist for finite forces.
• Near-Extremal Case:
  • The authors proved that near-horizon circular orbits exist for near-extremal RN black holes.
  • ISCO Scaling: The distance of the ISCO from the horizon, $u_c = r_{ISCO} - r_h$, scales with the surface gravity $\kappa$ as:
    $$u_c \propto \kappa^{2/3}$$
  • This scaling is identical to that found for rotating (Kerr) black holes, suggesting a universality in near-horizon physics regardless of the source of the "effective" rotation (spin vs. external force).

D. High-Energy Collisions (BSW Effect)

The paper investigates the center-of-mass energy ($E_{c.m.}$) of collisions near the horizon for near-extremal RN black holes:

• O-Scenario (Collision at ISCO): One particle is on the ISCO, the other is infalling.
  • Result: $\gamma \sim \kappa^{-2/3}$.
  • Energy grows as $\kappa \to 0$, but slower than the H-scenario.
• H-Scenario (Collision at Horizon): A particle falls from the ISCO to the horizon.
  • Result: $\gamma \sim \kappa^{-1}$.
  • This scenario is more efficient, yielding higher energies.
• Significance: The presence of an external force mimics the effects of black hole rotation. Just as rotation allows for high-energy collisions in Kerr black holes, an external force allows for the BSW effect in static, spherically symmetric spacetimes (which normally forbid such collisions for free particles).

4. Significance and Conclusion

• Theoretical Advancement: The work extends the BSW effect theory from free-falling particles to accelerated observers, filling a gap in the literature regarding circular trajectories.
• Astrophysical Relevance: Circular orbits are highly relevant for accretion disks. The findings suggest that if particles in an accretion disk are subject to non-gravitational forces (e.g., magnetic fields, radiation pressure, or self-force), high-energy collisions could occur near the horizons of static black holes, which was previously thought impossible without rotation.
• Universality: The discovery that the ISCO radius scales as $\kappa^{2/3}$ and collision energies scale as $\kappa^{-1}$ (H-scenario) for forced static black holes mirrors the results for rotating black holes. This implies that the "rotation" necessary for the BSW effect can be effectively substituted by an external force in static spacetimes.
• Methodological Utility: The derived formulas for stability and force requirements provide a toolkit for analyzing particle dynamics in various modified gravity theories or environments with non-trivial matter distributions.

In summary, the paper demonstrates that external forces can stabilize circular orbits near black hole horizons and enable the BSW effect in static spacetimes, effectively acting as a substitute for the rotational frame-dragging found in Kerr black holes.