Eikonal, nonlocality and regular black holes

This paper investigates gravitational eikonal scattering in nonlocal $D$-dimensional theories to derive effective geometries that, when nonlinearly completed, yield singularity-free, asymptotically flat black hole solutions featuring a de Sitter core.

The Gist

The Big Picture: Smoothing Out the Rough Edges of Gravity

Imagine the universe as a giant trampoline. In standard physics (Einstein's General Relativity), if you put a heavy bowling ball on it, the fabric stretches. If you put a tiny, infinitely heavy marble on it, the fabric stretches so sharply that it tears a hole right through the center. This "tear" is what physicists call a singularity—a point where the math breaks down and the rules of the universe stop working. This happens inside black holes.

This paper asks a simple question: What if the fabric of the universe isn't perfectly smooth and sharp, but slightly fuzzy?

The authors explore a theory called Nonlocal Gravity. In this theory, gravity doesn't act at a single, sharp point. Instead, it's "smeared out" over a tiny area, like a drop of ink spreading on a wet piece of paper rather than staying as a sharp dot. This "smearing" is controlled by a mathematical tool called a form factor.

The Experiment: Bouncing Balls in Space

To test this idea, the authors didn't build a black hole in a lab (which is impossible). Instead, they used a theoretical trick called the Eikonal Approximation.

Think of this like a game of billiards, but played with invisible particles at near-light speed.

1. The Setup: Imagine two particles zooming past each other. They don't hit head-on; they just graze each other.
2. The Measurement: Because gravity pulls on them, their paths bend slightly. This bending is called the deflection angle.
3. The Twist: In standard gravity, if they get too close, the math explodes. But in this "fuzzy" gravity, the authors calculated how much the particles would bend when they get very close.

They found that because gravity is "smeared out," the pull doesn't get infinitely strong. It gets strong, but then it levels off. It's like driving a car toward a cliff: in normal physics, you just fall off the edge. In this fuzzy physics, the cliff turns into a gentle, steep hill that you can drive over without falling.

The Discovery: The "Fuzzy" Black Hole

Using the results from their particle scattering experiment, the authors tried to reconstruct what a black hole would actually look like in this fuzzy universe.

The Old Black Hole (Standard Physics):

• The Core: A point of infinite density (the singularity).
• The Horizon: A point of no return.
• The Problem: The center is a mathematical disaster.

The New Black Hole (This Paper's Theory):

• The Core: Instead of a sharp point, the center is a De Sitter core. Imagine the center of the black hole isn't a hole, but a tiny, expanding bubble of space (like a microscopic version of our whole universe). It is smooth, finite, and safe.
• The Horizon: Depending on how heavy the black hole is, it might have two horizons (like an onion with two skins) or none at all.
• The Result: A Regular Black Hole. It has all the cool features of a black hole (it traps light, it bends space) but it has no "tear" in the fabric of reality.

The "Dirty" Solution

The authors found something interesting about the shape of this new black hole. In standard physics, the math describing the black hole is "clean" and symmetrical. In this new theory, the math is "dirty."

Think of it like this:

• Clean Black Hole: A perfect sphere where the inside and outside rules match perfectly.
• Dirty Black Hole: A slightly lopsided shape where the rules for time (how fast clocks tick) and space (how far you have to walk) don't match up perfectly.

The authors realized that to make the math work and avoid the singularity, the black hole must be "dirty." It's a necessary imperfection to keep the universe from breaking.

Why Does This Matter?

1. Fixing the Singularity: It offers a way to solve the biggest problem in black hole physics: the infinite point in the middle. If this theory is right, black holes are safe, smooth objects, not mathematical nightmares.
2. Connecting to Strings: The math they used looks very similar to String Theory (where particles are tiny vibrating strings). This suggests that even if we don't have a full theory of everything yet, we can use these "fuzzy" rules to predict what happens at the smallest scales.
3. Real-World Tests: While this is theoretical, the authors suggest that if we look closely at black holes (perhaps with future telescopes or gravitational wave detectors), we might see subtle differences in how they behave compared to Einstein's predictions. For example, the way a black hole "cools down" (Hawking radiation) might stop at a certain point, leaving behind a cold, stable remnant instead of vanishing completely.

The Takeaway

This paper is like a blueprint for a renovated black hole. The original blueprint (Einstein's) had a fatal flaw in the basement (the singularity). These architects (the authors) used a new tool (nonlocality/fuzziness) to redesign the basement. Now, instead of a bottomless pit, there is a smooth, expanding room. The building still looks like a black hole from the outside, but the inside is safe, stable, and free of disasters.

Technical Summary

1. Problem Statement

The paper addresses the challenge of understanding high-energy gravitational scattering in nonlocal theories of gravity, which are proposed as ultraviolet (UV) completions of General Relativity (GR) capable of resolving singularities. Specifically, the authors investigate:

• How nonlocal form factors (introduced to tame UV divergences and remove ghosts) modify the leading gravitational eikonal phase in $D$-dimensional theories.
• How to reconstruct the full nonlinear spacetime geometry (specifically black hole solutions) from linearized scattering data (the eikonal approximation) in the presence of nonlocality.
• Whether these nonlocal theories naturally lead to regular black holes (singularity-free solutions) with specific thermodynamic and geometric properties.

2. Methodology

The authors employ a multi-step approach combining perturbative scattering amplitudes with geometric reconstruction:

• Nonlocal Action: They consider a class of nonlocal gravity actions quadratic in the Ricci scalar and tensor, parametrized by an entire function form factor $e^{H(-\alpha \Box)}$, where $\alpha = \ell^2$ is the nonlocality scale. The graviton propagator is modified by a factor $e^{-H(\alpha q^2)}$.
• Eikonal Approximation: They calculate the tree-level $2 \to 2$ scattering amplitude for massless and massive scalars in the Regge limit (high energy, fixed momentum transfer). The leading eikonal phase $\delta_0(s, b)$ is extracted via a Fourier transform of the Born amplitude to impact parameter space ($b$).
• Probe Limit: For the massive case, they specialize to the "probe limit" ($m_{probe} \ll M_{source}$), where the scattering is interpreted as the geodesic motion of a light particle in the background of a heavy source.
• Geometric Reconstruction (Eikonal Compatibility):
  1. They derive the deflection angle $\Theta$ from the eikonal phase.
  2. They postulate a general ansatz for a nonlinear, static, spherically symmetric metric with two unknown functions, $g(r/\ell)$ and $h(r/\ell)$, deforming the Schwarzschild solution.
  3. They calculate the deflection angle for a probe moving in this ansatz metric.
  4. By equating the scattering-derived deflection angle with the geometric one (eikonal compatibility), they derive integral equations to solve for $g$ and $h$.
• Nonlinear Completion: To ensure regularity at the horizon (where $g_{tt}=0$), they introduce a "dirty" black hole ansatz (where $-g_{tt} \neq g_{rr}^{-1}$) involving an exponential redshift factor, ensuring the metric is regular everywhere.

3. Key Contributions and Results

A. Massless Scattering and Generalized Aichelburg-Sexl Geometries

• Eikonal Phase: For a Gaussian form factor ($H(\alpha q^2) = \alpha q^2$), the eikonal phase involves the incomplete gamma function.
• Comparison with String Theory: The authors compare their results to string theory scattering. They note that while string theory exhibits logarithmic enhancements and Regge trajectory effects, the nonlocal field theory result is a "degenerate" limit where strings are treated as rigid, non-localized objects.
• Resolution of 't Hooft Poles: In standard GR, the S-matrix exhibits singularities ('t Hooft poles) at specific values of the coupling due to small-$b$ divergences. The authors show that the nonlocal form factor acts as a regulator, rendering the S-matrix finite and removing these poles.
• Generalized Shockwaves: The eikonal phase is interpreted as the phase shift of a particle moving in a generalized Aichelburg-Sexl shockwave. The nonlocality smears the energy density of the source from a delta function to a Gaussian profile, modifying the shockwave geometry.

B. Massive Scattering and Probe Limit

• Deflection Angle: In the probe limit, the deflection angle for a massive particle is derived. For the Gaussian case, it is proportional to the incomplete gamma function, showing a suppression of the deflection at small impact parameters ($b \ll \ell$).
• Linearized Potential: This corresponds to a smeared Newtonian potential $\Phi(r) \propto \frac{1}{r} \text{Erf}(r/2\ell)$, which is regular at the origin.

C. Reconstruction of Nonlinear Regular Black Holes

• The "Dirty" Metric: The authors demonstrate that a "clean" metric (where $-g_{tt} = g_{rr}^{-1}$) cannot simultaneously satisfy eikonal compatibility and regularity at the horizon for generic nonlocal form factors. They propose a "dirty" metric ansatz:
  $$ds^2 = -e^{-\Sigma} F(r) dt^2 + F(r)^{-1} dr^2 + r^2 d\Omega^2$$
  where $\Sigma$ is a non-vanishing exponential factor.
• Explicit Solution: For the Gaussian form factor, they derive the explicit metric functions:
  • $F(r)$ involves the incomplete gamma function $\gamma(\frac{D-1}{2}, r^2/4\ell^2)$, ensuring the mass function $M(r)$ is Gaussian.
  • The exponential factor $\Sigma$ ensures that $g_{tt}$ and $g_{rr}$ share the same zeros, removing curvature singularities at the horizon.
• Core Structure: The resulting spacetime describes a regular black hole with a de Sitter (dS) core near $r=0$, replacing the curvature singularity of the Schwarzschild solution.
• Horizon Structure: The solution admits three phases depending on the mass parameter $\mu = (R_s/\ell)^{D-3}$:
  1. Black Hole: Two horizons (inner and outer) for $\mu > \mu_c$.
  2. Extremal Black Hole: One degenerate horizon at $\mu = \mu_c$.
  3. Gravastar-like Object: No horizons for $\mu < \mu_c$.

D. Thermodynamics and Geometry

• Hawking Temperature: The temperature $T(r_H)$ is non-monotonic. As the black hole evaporates and $r_H$ approaches the extremal radius $r_0$, the temperature drops to zero, suggesting the formation of a cold remnant rather than a singularity.
• Redshift: The gravitational redshift is finite at the center ($r=0$) for horizonless objects, unlike the divergence in standard GR.

E. Infrared Form Factors

The paper briefly analyzes non-analytic form factors (infrared modifications). While these can reproduce Yukawa-like potentials, the authors find that simple nonlinear completions of these models still suffer from singularity issues at the origin, suggesting that UV-regularizing (analytic) form factors are necessary for singularity resolution.

4. Significance

• Bridging Scattering and Geometry: The work provides a rigorous method to reconstruct nonlinear spacetime geometries from perturbative scattering amplitudes in nonlocal theories, a task previously considered intractable due to the complexity of the full field equations.
• Singularity Resolution: It offers a concrete, physically motivated model for a singularity-free black hole derived directly from first principles (nonlocality), supporting the hypothesis that nonlocality acts as a natural UV regulator.
• Phenomenology: The derived solutions predict specific deviations from GR in black hole imaging (Event Horizon Telescope), orbital dynamics, and gravitational wave signals (ringdown modes), providing potential observational signatures for nonlocal gravity.
• Thermodynamic Stability: The prediction of a cold remnant offers a potential resolution to the black hole information paradox by avoiding the final singularity and infinite temperature phase.

In summary, the paper establishes a consistent framework where nonlocal gravity modifies high-energy scattering to produce regular, singularity-free black hole geometries with de Sitter cores, validated by the consistency between scattering amplitudes and geodesic motion.