Gravitational scattering amplitudes from curved space

This thesis develops a framework for computing gravitational scattering amplitudes in curved backgrounds by quantizing general relativity on Schwarzschild-Tangherlini spacetime and applying worldline quantum field theory to derive Compton amplitudes that match established flat-space results.

The Gist

Imagine you are trying to understand how two massive objects, like black holes, dance around each other in the universe. For a long time, physicists have used two main ways to study this dance:

1. The "Flat Map" Approach: Pretending the universe is a perfectly flat, empty sheet of paper, and the black holes are just heavy marbles rolling on it.
2. The "Curved Map" Approach: Acknowledging that the heavy marbles actually warp the paper itself, creating a deep valley (gravity) that other objects roll into.

This thesis, written by Carl Jordan Eriksen, is a master's project that asks a very specific question: Does it matter which map we use? Specifically, can we calculate how a "ripple" in spacetime (a graviton) bounces off a black hole using the "Curved Map," and will we get the exact same answer as if we used the "Flat Map"?

The answer, surprisingly, is yes. The paper proves that both methods lead to the same result, but the "Curved Map" method offers a clever shortcut that makes the math much more organized.

Here is a breakdown of the journey, using simple analogies:

1. The Setup: The Black Hole and the Ripple

Imagine a black hole as a giant, heavy bowling ball sitting on a trampoline. It creates a deep dip. Now, imagine throwing a tiny pebble (a graviton) at it. The pebble doesn't just bounce off; it skims along the curved surface of the trampoline, gets deflected, and flies away.

In physics, this is called Compton Scattering (usually, it's light hitting an electron, but here it's a gravity-wave hitting a black hole). The goal is to calculate exactly how the pebble bounces.

2. The Two Ways to Do the Math

Method A: The Flat World (The Old Way)
In this method, physicists pretend the trampoline is perfectly flat. They treat the black hole as a source of "force" that pushes the pebble.

• The Problem: To get an accurate answer, you have to add up an infinite number of tiny, messy interactions. It's like trying to calculate the path of a ball by adding up every single tiny bump in the road, one by one. It gets incredibly complicated very fast.

Method B: The Curved World (The New Way)
In this method, the physicist starts with the trampoline already warped by the bowling ball. The "background" is already curved.

• The Advantage: Because the background is already curved, the math "knows" about the gravity immediately. You don't have to add up all those tiny bumps one by one. The curvature does the heavy lifting for you.

3. The Big Discovery: The "Magic Trick"

The author spent the thesis developing the rules (Feynman rules) for Method B. He then calculated the bounce of the pebble at two different levels of precision:

• Level 1 (1PM): A simple bounce.
• Level 2 (2PM): A more complex bounce where the pebble interacts with the gravity field twice.

The Result: When he did the math using the "Curved World" (Method B), he got the exact same answer as the "Flat World" (Method A).

Why is this cool?
Think of it like solving a maze.

• Method A is like walking the maze, hitting every wall, and counting your steps.
• Method B is like looking at the maze from a helicopter and seeing the path clearly.
  The thesis proves that looking from the helicopter (Curved) gives you the same destination as walking the maze (Flat), but the helicopter view is much more efficient for complex problems.

4. The "Ghost" Problem and the Solution

There was a tricky part. When you put a black hole on a trampoline, the math gets "divergent" (it blows up to infinity) right where the ball touches the fabric.

• The Fix: The author used a mathematical trick called "Dimensional Regularization." Imagine the trampoline isn't just 2D, but exists in a weird, slightly higher dimension where the sharp point of the ball smoothes out. This removes the infinities and leaves a clean, finite answer.

5. The "Infrared" Glitch

At the second level of precision (2PM), the math produced a "glitch"—an infinite number that shouldn't be there.

• The Analogy: Imagine you are listening to a radio. If you turn the volume up too high, you get static. In physics, this "static" is called an infrared divergence. It happens because gravity waves can travel forever and never truly stop.
• The Resolution: The author showed that this "static" isn't a mistake. It's a known feature of gravity. It turns out this infinite noise is exactly what you expect based on previous theories (Weinberg's theorem). It's like the universe whispering, "Hey, I'm infinite, so my math has infinite parts." The author proved his calculation was correct because the "noise" matched the theoretical prediction perfectly.

6. Why Does This Matter?

We are entering a new era of astronomy. We have detectors (like LISA) that will listen to the "chirps" of black holes colliding. To understand these sounds, we need incredibly precise maps of how gravity works.

• The Takeaway: This thesis gives physicists a new, powerful tool. Instead of struggling with messy, flat-space math, they can use the "Curved Space" approach. It organizes the chaos, resums infinite series of interactions into neat packages, and makes it easier to predict what happens when black holes collide.

In a nutshell: The author built a new set of mathematical glasses. When you look at gravity through these glasses (the curved background), the picture is just as clear as looking through the old glasses (flat space), but the view is much less cluttered, making it easier to solve the universe's hardest puzzles.

Technical Summary

1. Problem Statement

The thesis addresses the computation of classical gravitational scattering amplitudes, specifically the Compton amplitude (the scattering of a graviton off a massive compact object, modeled as a black hole or point particle). While recent advancements in the post-Minkowskian (PM) expansion have utilized quantum field theory (QFT) tools to compute classical observables, these calculations are traditionally performed in flat space (Minkowski background) with a perturbative expansion in Newton's constant $G$.

The central problem investigated is whether one can perform these computations directly in a curved background (specifically the Schwarzschild–Tangherlini metric) and recover the same results. This is motivated by the study of extreme mass-ratio inspirals (EMRIs), where the self-force expansion naturally involves an expansion around a curved background. The thesis aims to:

1. Develop the formalism for quantizing gravity on an arbitrary curved background.
2. Derive Feynman rules for gravity in both flat and curved (Schwarzschild–Tangherlini) backgrounds.
3. Compute the 1st (1PM) and 2nd (2PM) post-Minkowskian contributions to the Compton amplitude using Worldline Quantum Field Theory (WQFT).
4. Verify that the curved-space computation yields results identical to the flat-space computation, thereby validating the curved-background approach for classical gravity.

2. Methodology

A. Theoretical Framework

• Worldline Quantum Field Theory (WQFT): The author employs WQFT, a formalism where the classical limit is taken at the level of the action. In this approach, compact objects are treated as point particles moving on a worldline $x^\mu(\tau)$, and the classical limit corresponds to retaining only tree-level diagrams in the worldline expansion.
• Background Field Method: The metric is decomposed as $g_{\mu\nu} = \bar{g}_{\mu\nu} + \kappa h_{\mu\nu}$, where $\bar{g}_{\mu\nu}$ is the background metric (either Minkowski $\eta_{\mu\nu}$ or Schwarzschild–Tangherlini) and $h_{\mu\nu}$ is the fluctuating graviton field.
• Dimensional Regularization: Calculations are performed in $d$ dimensions to regulate infrared and ultraviolet divergences.

B. Derivation of Feynman Rules

1. Flat Space: Standard weak-field expansion is used, deriving the de Donder propagator and the infinite tower of graviton self-interaction vertices from the Einstein-Hilbert action.
2. Curved Space (Schwarzschild–Tangherlini):
   • The author derives the $d$-dimensional Schwarzschild–Tangherlini solution in isotropic coordinates.
   • The Einstein-Hilbert action is expanded around this curved background.
   • Key Insight: The interaction between the graviton and the background is encoded in a two-point vertex (a "fat" graviton line connecting to the worldline). This vertex is expanded in powers of $G$ (or $\Lambda_d$), generating an infinite tower of effective vertices.
   • Cancellation Mechanism: A crucial technical result is the demonstration that the "source rule" (the vertex where the worldline emits a graviton in flat space) cancels exactly with the linear term in the Einstein-Hilbert action when expanded around the curved background. This eliminates diagrams containing self-energy loops on the worldline.
   • Resummation: The curved-space two-point vertex effectively resums an infinite set of "potential" graviton diagrams found in the flat-space expansion.

C. Computation of the Amplitude

• Kinematics: The scattering of a graviton with momentum $p$ off a black hole with velocity $v$ is analyzed. The momentum transfer is $q = p_2 - p_1$, with $q \cdot v = 0$.
• Integration-by-Parts (IBP): The 2PM amplitude involves loop integrals (specifically "fan" integrals in the curved expansion). The author uses IBP identities (implemented via the LiteRed package) to reduce the complex integrand to a basis of Master Integrals.
• Master Integrals: Three master integrals are identified: a cut bubble, a cut triangle, and a cut box. These are evaluated using:
  • Analytic continuation for the bubble.
  • Feynman parameters for the triangle.
  • The method of differential equations and the method of regions for the box integral.

3. Key Contributions

1. Formalism for Curved Background Quantization: The thesis provides a rigorous derivation of Feynman rules for gravity on an arbitrary background, specifically specializing to the Schwarzschild–Tangherlini metric. It clarifies the relationship between the curved background expansion and the standard weak-field expansion.
2. Equivalence of Flat and Curved Computations: The author explicitly demonstrates that the 1PM and 2PM Compton amplitudes computed in the curved background are identical to those computed in flat space. This validates the use of curved backgrounds for classical amplitude calculations.
3. New 2PM Result: The thesis presents the full 2PM Compton amplitude in $d$ dimensions and specifically in four dimensions. While parts of the 2PM amplitude appeared in previous literature, the complete result derived here is new.
4. Infrared Structure Analysis: The 2PM amplitude is shown to possess an infrared (IR) divergence proportional to the 1PM amplitude. The author verifies that this divergence matches the expected behavior from Weinberg's theorem on the exponentiation of IR divergences in theories with massless external states.
5. Cross-Checks:
   • The result is checked against the known 1PM amplitude.
   • The scalar limit (extracting the coefficient of $(\varepsilon_1 \cdot \varepsilon_2)^2$) is compared to the scattering of massless scalars off massive scalars, showing agreement in the functional form (up to a sign and a numerical factor in the imaginary part, which the author discusses as a potential normalization or convention issue).

4. Results

• 1PM Amplitude: Successfully reproduced using both flat and curved rules, confirming the consistency of the formalism.
• 2PM Amplitude:
  • Expressed as a linear combination of three master integrals ($\mathcal{K}_1, \mathcal{K}_2, \mathcal{K}_3$) with gauge-invariant coefficients.
  • The result contains:
    • An IR pole ($1/\epsilon$) proportional to the 1PM amplitude.
    • Logarithmic terms ($\log(-q^2)$ and $\log(\omega^2)$).
    • A non-analytic term proportional to $1/|q|$.
    • Rational terms.
  • The imaginary part of the amplitude satisfies the expected unitarity cuts.
• Infrared Divergence: The divergence is shown to be of the form $\mathcal{M}_{2PM}^{IR} \propto \frac{1}{\epsilon} \mathcal{M}_{1PM}$, consistent with the exponentiation of soft graviton emissions.

5. Significance

• Validation of Curved Background Methods: This work proves that expanding around a curved background is a viable and consistent method for computing classical gravitational scattering amplitudes. This is significant for EMRI physics, where the self-force expansion is naturally formulated in curved space.
• Resummation of Potential Diagrams: The curved expansion effectively resums an infinite series of potential graviton exchanges into a single vertex. This suggests that curved background methods could simplify higher-order PM calculations by reducing the number of diagrams, although the complexity of the vertex itself increases.
• Foundation for Higher Orders: By establishing the formalism and computing the 2PM amplitude, this thesis lays the groundwork for computing higher-order PM corrections (3PM, 4PM) and investigating tidal effects (via the decomposition of the Compton amplitude into spherical harmonics).
• Spin and Kerr Metrics: The author suggests that extending this formalism to the Kerr metric (using Kerr-Schild coordinates) could allow for exact-in-spin calculations, which are crucial for modeling spinning black holes in gravitational wave astronomy.

In summary, the thesis successfully bridges the gap between curved-space effective field theory and flat-space scattering amplitudes, providing a robust new tool for calculating classical gravitational observables relevant to the LISA mission and extreme mass-ratio systems.