Kerr-Schild solutions in Multigravity and the Classical Double Copy

This paper derives new exact proportional and double Kerr-Schild solutions in multigravity by extending General Relativity results with constant Ricci scalars, while also investigating the associated classical double copy relations and their single and zero copy field dynamics.

The Gist

Imagine the universe as a grand orchestra. For over a century, our best understanding of how gravity works has been the "General Relativity" symphony, composed by Albert Einstein. In this symphony, gravity is a single, smooth melody played by one instrument: the graviton (a massless particle).

But what if the universe is actually a jazz ensemble with multiple instruments playing together? What if there isn't just one gravity, but many?

This paper, written by Hugo García-Compeán and Everardo Rivera-Oliva, explores a theory called Multigravity. It asks: What happens if we have a whole family of gravity fields interacting with each other?

Here is a breakdown of their findings using simple analogies.

1. The Big Idea: A Gravity Family Tree

In standard physics, gravity is like a soloist. In Multigravity, imagine a choir of $N$ singers (metrics).

• The Soloist (General Relativity): One singer, perfect pitch, massless.
• The Choir (Multigravity): Many singers. Some are heavy (massive gravitons), some are light. They are all singing together, but they are slightly out of sync with each other.
• The Problem: When you add too many singers, the harmony can get messy. Sometimes, the song becomes unstable (a "ghost" appears, which is a mathematical error that breaks the theory).
• The Solution: The authors focus on a specific type of harmony called "Proportional Solutions." Imagine all the singers in the choir are singing the exact same song, but some are just singing it slightly louder or softer (a constant volume factor). This keeps the harmony stable and avoids the "ghosts."

2. The Magic Trick: The "Double Copy"

The paper uses a fascinating mathematical shortcut called the Classical Double Copy.

• The Analogy: Think of gravity as a complex, heavy double-decker bus. Think of electromagnetism (light and electricity) as a sleek, fast motorcycle.
• The Trick: The "Double Copy" theory suggests that if you know how to build the motorcycle, you can mathematically "copy and paste" parts of it to build the double-decker bus.
  • Zero Copy: A scalar field (like a simple temperature map).
  • Single Copy: A vector field (like a magnetic field or light).
  • Double Copy: The gravity field itself.
• What the Authors Did: They took their new "Multi-Gravity" choir songs (the double-decker buses) and used the Double Copy trick to see what kind of "motorcycles" (electromagnetic fields) and "temperature maps" (scalar fields) they would produce.

3. The New Discoveries: Building New Black Holes

The authors didn't just talk about theory; they built specific solutions (mathematical models of black holes and waves) for this multi-gravity choir.

• The "Proportional" Rule: They found that if all the gravity fields are proportional (singing the same tune at different volumes), the math simplifies beautifully.
• The New Black Holes: They created a whole catalog of new black hole solutions:
  • Multi-Schwarzschild: A chain of black holes, each with its own mass, but all following the same rules.
  • Multi-Kerr: Spinning black holes, where the spin is shared, but the masses differ.
  • Multi-Waves: Ripples in spacetime (like Kundt waves or pp-waves) traveling through this multi-gravity medium.
• The Twist: In these solutions, the "cosmological constant" (the energy of empty space that makes the universe expand) isn't just a background setting. In Multigravity, it emerges naturally from the interaction between the different gravity fields. It's like the energy of the choir singing together creates a new pressure that pushes the universe apart.

4. The Results: What Do the "Copies" Look Like?

When the authors applied the Double Copy trick to their new Multi-Gravity black holes, they found something surprising:

• The Single Copy (The Motorcycle): Instead of just getting normal light (Maxwell's equations), they got massive photons (Proca fields).
  • Analogy: In our normal universe, photons (light particles) have no mass and zip around at the speed of light. In this Multi-Gravity universe, the "light" generated by the black holes has a tiny bit of weight. It moves slower and behaves differently because the "background music" (the curvature of space) gives it mass.
• The Zero Copy (The Temperature Map): Similarly, the scalar fields became massive particles.
  • Analogy: Imagine a ripple in a pond. Usually, it spreads out forever. In this theory, the ripple has a "weight" that makes it settle down faster.

5. Why Does This Matter?

You might ask, "Why do we care about a choir of gravity fields?"

1. Dark Matter & Dark Energy: The universe is mostly made of invisible stuff (Dark Matter) and is expanding faster than expected (Dark Energy). Standard gravity struggles to explain this without adding "magic." Multigravity offers a new way to explain these phenomena using the interaction between multiple gravity fields, rather than invisible magic dust.
2. Cosmology: The "massive photons" and "massive scalars" the authors found are exactly the kinds of particles cosmologists look for in theories about the early universe (like inflation) or the "Dark Sector" (dark photons).
3. Mathematical Unity: It proves that the "Double Copy" trick works even in these complex, multi-field theories. It suggests a deep, hidden connection between the forces of nature that holds true even when gravity gets complicated.

Summary

The authors took a complex theory where multiple gravity fields interact, found a way to keep them stable (the "Proportional" rule), built a library of new black holes and waves, and then used a mathematical magic trick (Double Copy) to show that these heavy gravity objects are secretly related to massive light and massive particles.

It's like discovering that a heavy, complex symphony is actually just a remix of a simple pop song, but with a twist: the instruments in the remix are slightly heavier than usual. This could help us understand the invisible forces shaping our universe.

Technical Summary

1. Problem Statement

The paper addresses two primary challenges in modern theoretical physics:

1. Exact Solutions in Multigravity: While massive gravity and bigravity (bimetric gravity) have been extensively studied, finding exact solutions in Multigravity (theories with $N$ interacting spin-2 fields) remains difficult. Existing solutions were often derived using the vielbein formalism or were limited to specific cases. There is a need for a systematic derivation of exact solutions using the metric formalism, particularly those that can be extended to the "Classical Double Copy" framework.
2. Extension of the Classical Double Copy: The Classical Double Copy conjecture posits a mapping between solutions in gauge theory (spin-1) and gravity (spin-2), and further to scalar theory (spin-0). While established for General Relativity (GR) and Bigravity, its applicability to Multigravity (with $N > 2$ metrics) and the nature of the underlying gauge theories (specifically multi-vector and multi-scalar theories) had not been fully explored.

2. Methodology

The authors employ a systematic approach combining metric formalism, ansatz construction, and field-theoretic mapping:

• Theoretical Framework: They utilize the ghost-free Multigravity action (generalizing dRGT massive gravity and Hassan-Rosen bigravity) involving $N$ dynamical metrics $g_{\mu\nu}^j$ interacting with nearest neighbors. The equations of motion are derived via variation of the action.
• Proportional Kerr-Schild Ansatz: The core methodology involves assuming a proportional Kerr-Schild structure for all $N$ metrics:
  $$(g_j)_{\mu\nu} = C_j^2 \left[ \bar{g}_{\mu\nu} + 2S_j l_\mu l_\nu \right]$$
  where $\bar{g}_{\mu\nu}$ is a background metric, $l_\mu$ is a null vector, $S_j$ is a scalar function, and $C_j$ are constant conformal factors. This ansatz linearizes the inverse metric and the interaction matrix $\gamma$, simplifying the highly non-linear field equations.
• Circularity Theorem: To derive rotating solutions (Kerr-like), the authors extend the Circularity Theorem to the multigravity context. This imposes constraints on the scalar functions $S_j$ and the interaction parameters, ensuring the existence of commuting Killing vectors.
• Double Kerr-Schild Extension: The analysis is extended to Double Kerr-Schild ansatzes ($g_{\mu\nu} = \bar{g}_{\mu\nu} + 2S l_\mu l_\nu + 2Q f_\mu f_\nu$) to capture more complex geometries like Taub-NUT and Plebański-Demiański metrics.
• Classical Double Copy Mapping: The authors apply the standard double copy prescription:
  • Single Copy (Spin-1): Map the gravitational perturbation $\phi k_\mu k_\nu$ to a gauge field $A_\mu = \phi k_\mu$.
  • Zero Copy (Spin-0): Map the scalar component $\phi$ to a scalar field.
    They analyze the resulting equations of motion to identify the underlying gauge and scalar field theories.

3. Key Contributions

• Metric Formalism Derivation: The paper derives exact multigravity solutions directly using the metric formalism, contrasting with previous works that relied on the vielbein formalism.
• General Construction Algorithm: They establish a general algorithm (Eq. 3.56) showing that any GR solution with a constant Ricci scalar (Einstein space) can be "lifted" to a proportional multigravity solution, provided specific constraints on the interaction parameters ($P_0, P_1, P_2$) and conformal factors are met.
• Identification of Underlying Gauge Theories: A significant theoretical contribution is the identification of the specific field theories underlying the double copy in multigravity:
  • Single Copy: Corresponds to a quadratic $U(1)^N$ Proca theory (interacting massive vector fields).
  • Zero Copy: Corresponds to an interacting multi-scalar theory with $SO(3)^N$ symmetry.
• Mass Generation Mechanism: The paper demonstrates that the "mass" of the single-copy (Proca) and zero-copy (Klein-Gordon) fields is not arbitrary but is induced by the background Ricci scalar curvature of the multigravity solution ($m^2 \propto -R/6$).

4. Key Results

The authors successfully derived and classified a wide array of exact solutions in multigravity:

A. Vacuum Solutions (Proportional Kerr-Schild):

• Multi-Kerr & Multi-Kerr-AdS: A chain of $N$ rotating black holes with independent masses but shared angular momentum and background cosmological constant.
• Multi-Schwarzschild & Multi-Schwarzschild-AdS: Static spherically symmetric solutions.
• Wave Solutions: New solutions including Multi-Kundt waves, Multi-pp-waves, and Multi-Siklos-AdS waves, where wave profiles can differ between sectors while sharing a common AdS background.

B. Coupled Matter Solutions:

• Multi-Reissner-Nordström & Multi-Kerr-Newman: Solutions incorporating electric charge, where each sector has independent mass and charge parameters.
• Multi-Reissner-Nordström-AdS & Multi-Kerr-Newman-AdS: Rotating, charged black holes in an AdS background.

C. Double Kerr-Schild Solutions:

• Multi-Taub-NUT: Solutions involving NUT charges.
• Multi-Plebański-Demiański: The most general type D solution, encompassing mass, NUT charge, electric/magnetic charges, and acceleration.

D. Double Copy Results:

• For vacuum solutions (e.g., Multi-Schwarzschild), the single copy yields massless Maxwell fields (photons), and the zero copy yields massless scalars.
• For AdS solutions (e.g., Multi-Schwarzschild-AdS), the background curvature induces a mass term, resulting in Proca fields (massive photons) and massive scalars.
• For charged solutions, the double copy maps the stress-energy tensor of the source to the current density of the gauge field.

5. Significance

• Unification of Gravity and Gauge Theory in Multigravity: The work proves that the Classical Double Copy is robust enough to extend to theories with multiple interacting gravitons, not just single or double metrics.
• Cosmological Applications: By identifying the single copy as a $U(1)^N$ Proca theory and the zero copy as an $SO(3)^N$ scalar theory, the paper provides a theoretical bridge to dark photon sectors and multifield inflation models, suggesting that multigravity solutions could have direct phenomenological implications in cosmology.
• Stability and Instability: The paper notes that while proportional solutions are stable in certain regimes, non-proportional solutions may exhibit instabilities. The double copy framework offers a new tool to potentially analyze these instabilities by mapping them to the dynamics of the single/zero copy fields.
• Methodological Advancement: The derivation of these solutions in the metric formalism provides a more accessible route for future researchers to explore multigravity without the complexity of the vielbein formalism.

In conclusion, this paper significantly expands the landscape of exact solutions in multigravity and solidifies the theoretical foundation of the Classical Double Copy in multimetric theories, linking complex gravitational dynamics to well-understood multi-field gauge and scalar theories.