How to deal with conformal and pure scale-invariant theories of gravity in d dimensions?

This paper formulates conformally and scale-invariant theories of gravity in $d$ dimensions, presenting an elegant framework that reveals how these higher-dimensional theories exhibit fundamentally different properties compared to their four-dimensional counterparts.

The Gist

Imagine you are an architect designing a building. For centuries, you've been perfecting a design for a 4-story building (our universe as we usually think of it). You know exactly how the stairs, elevators, and rooms work. You know which beams are strong and which ones might collapse.

Now, you decide to build a 10-story skyscraper (a universe with more dimensions). You might think, "It's just a taller version of the same building, right? I'll just stack more floors on top."

This paper by Anamaria Hell and Dieter Lüst is about what happens when you try to build that skyscraper using two very specific, fancy types of architectural rules: Scale-Invariant and Conformal gravity.

Here is the breakdown of their findings in simple terms:

1. The Two Types of "Magic" Rules

In physics, gravity usually follows Einstein's rules (General Relativity). But these authors are looking at "fancier" versions of gravity that have special symmetries:

• Scale-Invariant Gravity (The "Zoom" Rule): Imagine a photo. If you zoom in or out, the picture looks the same. In this theory, if you stretch the entire universe like a rubber sheet, the laws of physics don't change.

  • The 4D Version: In our 4D world, this theory is actually quite boring. In empty space, nothing moves. It's like a silent movie where no actors are on stage.
  • The High-D Version: When they tried to stretch this to 5, 6, or more dimensions, they found it behaves almost the same way. It's still mostly silent. It's a "pure" theory that stays stable.

• Conformal Gravity (The "Shape" Rule): Imagine a piece of clay. You can squish it, stretch it, and twist it, but as long as you don't tear it, the angles between lines stay the same. This theory only cares about angles, not distances.

  • The 4D Version: This is already a bit tricky. It has some "ghosts" (unstable parts that shouldn't exist physically), but they are mostly hidden in the "tensor" (gravitational wave) sector.
  • The High-D Version: This is where the paper gets exciting. When they built the 5D skyscraper with these rules, the building didn't just get taller; it started shaking violently.

2. The Problem: "Ostrogradsky Ghosts"

In physics, a "ghost" isn't a spooky spirit. It's a mathematical error that represents an unstable energy state. Think of it like a ball balanced on the very tip of a sharp mountain peak. If you nudge it even slightly, it doesn't roll down gently; it flies off into infinity, destroying the system.

• In 4 Dimensions: The "ghosts" in Conformal Gravity were like a few loose screws in the elevator shaft. Annoying, but manageable.
• In 5+ Dimensions: The authors found that the ghosts multiplied. Suddenly, the stairs (scalar modes) and the walls (vector modes) were also made of unstable, exploding material.
  • The Analogy: In 4D, you had one unstable room. In 5D, the entire building—floors, walls, and elevators—is made of unstable material. The theory becomes much more "monstrous" and difficult to control.

3. The Solution: Changing the "Frame of Reference"

The paper introduces a clever trick to understand these messy theories. They call it changing the "Frame."

• The Metaphor: Imagine you are looking at a tangled ball of yarn. It looks impossible to untangle. But if you put on a special pair of glasses (change the frame), the yarn suddenly looks like a straight, neat string.
• How they did it: They took the complex, messy equations of these high-dimensional theories and used a mathematical "lens" to rewrite them.
  • For the Scale-Invariant theory, they turned it into a standard gravity theory plus a simple "scalar field" (like a new type of particle).
  • For the Conformal theory, they turned it into a theory involving the "Weyl tensor" (a measure of shape distortion) plus a cosmological constant (like dark energy).

This "Einstein Frame" made it much easier to count the particles and see exactly where the ghosts were hiding.

4. The Big Conclusion

The main takeaway of the paper is a warning to physicists: Do not assume that what works in 4 dimensions works in higher dimensions.

• The "Pure" Scale Theory: It's a good citizen. It behaves nicely whether you are in 4D or 100D. It stays quiet and stable.
• The Conformal Theory: It's a troublemaker. In 4D, it's a manageable headache. In 5D and above, it becomes a chaotic nightmare with way too many unstable "ghosts" and new, weird particles that don't exist in our current universe.

Summary

If you are trying to build a theory of everything that includes extra dimensions, you can't just copy-paste the rules from our 4D universe.

• Some rules (Scale-Invariance) are robust and survive the upgrade.
• Other rules (Conformal Invariance) break down and become full of instability (ghosts) when you add more dimensions.

The authors have provided a new "mathematical toolkit" (the alternative frame) to help physicists analyze these high-dimensional monsters without getting lost in the complexity.

Technical Summary

1. Problem Statement

The paper addresses the fundamental question of how gravitational theories behave when extended from four dimensions ($d=4$) to higher dimensions ($d > 4$). Specifically, it focuses on two classes of higher-derivative gravity theories:

1. Pure Scale-Invariant Gravity: Actions containing only powers of the Ricci scalar ($R$) such that the action is invariant under constant metric rescaling ($g_{\mu\nu} \to F g_{\mu\nu}$).
2. Conformal Gravity: Actions constructed from the Weyl tensor ($W_{\mu\nu\rho\sigma}$) that are invariant under local conformal transformations ($g_{\mu\nu} \to F(x) g_{\mu\nu}$).

While these theories are well-studied in $d=4$ (where they are candidates for UV-complete, renormalizable gravity), their properties in higher dimensions are less understood. The central problem is determining whether the degrees of freedom (DoF), stability (presence of ghosts), and propagation characteristics of these theories remain consistent with their 4D counterparts or undergo radical changes in $d$ dimensions.

2. Methodology

The authors employ a multi-step analytical approach to investigate the spectrum of these theories:

• Action Formulation in $d$ Dimensions:

  • They define the general form of scale-invariant actions as $S \sim \int d^dx \sqrt{-g} (R)^{d/2}$ and conformal actions as $S \sim \int d^dx \sqrt{-g} (W^2)^{d/4}$.
  • They analyze the behavior of these actions in Minkowski spacetime (flat background) by perturbing the metric ($g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$) and decomposing perturbations into scalar, vector, and tensor modes.

• Frame Transformation (The "Trick"):

  • Recognizing that the original actions are non-polynomial and difficult to analyze in curved backgrounds, the authors utilize a frame transformation technique.
  • For Scale-Invariant Gravity: They introduce an auxiliary scalar field $\phi$ constrained to the Ricci scalar ($R$) and perform a conformal transformation to the Einstein Frame. This converts the non-linear $R^{d/2}$ action into a linear Einstein-Hilbert term plus a scalar field with a specific potential.
  • For Conformal Gravity: They introduce an auxiliary scalar field $\phi$ constrained to the Weyl tensor squared ($W^2$) and transform to an Alternative Frame. This rewrites the action in terms of the Weyl tensor squared in the new metric plus a cosmological constant term.

• Spectral Analysis in Anisotropic Backgrounds:

  • To avoid singularities where the Weyl tensor vanishes (conformally flat backgrounds), the authors analyze the theory in a 5-dimensional anisotropic spacetime ($ds^2 = -dt^2 + a^2(t)\delta_{ij}dx^idx^j + b^2(t)dl^2$).
  • They perform a Scalar-Vector-Tensor (SVT) decomposition of metric perturbations in this background, reduce higher-order derivatives to second order, and solve for constraints to identify the physical propagating modes.

3. Key Contributions

• Generalization of Actions: The paper provides the explicit formulation of pure scale-invariant and conformal gravity actions in arbitrary $d$ dimensions.
• Elegant Frame Formalism: It introduces a robust method to handle non-polynomial higher-derivative gravities by mapping them to equivalent theories in a "dual" frame (Einstein-like frame for scale-invariant, Weyl-squared frame for conformal). This simplifies the identification of DoF and the nature of the fields.
• Dimensional Dependence of Ghosts: The work demonstrates that the stability and DoF content of conformal gravity are highly sensitive to dimensionality, contradicting the naive expectation that higher-dimensional extensions are trivial.

4. Results

A. Pure Scale-Invariant Gravity

• Flat Spacetime: In $d$ dimensions, the theory propagates no modes in Minkowski space, similar to the 4D case. The scalar modes are non-propagating and lead to constraints that set the action to zero at the leading order.
• Curved Spacetime: In the Einstein frame, the theory describes a graviton and a healthy scalar field. This property holds for both $d=4$ and $d>4$, provided the background curvature is non-zero ($R \neq 0$).

B. Conformal Gravity in $d > 4$ (Specifically $d=5$)

The results for conformal gravity differ drastically from the 4D case:

• Scalar Sector:
  • 4D: Contains no propagating scalar modes.
  • 5D: Contains three propagating scalar modes, one of which is a ghost (Ostrogradsky instability).
• Vector Sector:
  • 4D: Contains one healthy propagating vector mode (2 DoF).
  • 5D: Contains three types of propagating vector modes (6 DoF total), one of which is a ghost (2 ghost DoF).
• Tensor Sector:
  • Remains similar to 4D, containing one healthy and one ghost tensor mode (4 DoF total).
• Overall Spectrum:
  • The 5D theory possesses a significantly larger and more pathological spectrum than the 4D theory. The number of ghost modes increases, and healthy modes appear in sectors (scalars/vectors) that were previously empty in 4D.

5. Significance

• Instability in Higher Dimensions: The paper concludes that while "pure" scale-invariant gravity maintains its desirable properties (ghost-free in flat space, simple spectrum in curved space) across dimensions, conformal gravity becomes significantly more unstable in $d > 4$. The proliferation of Ostrogradsky ghosts in scalar and vector sectors suggests that these theories are likely pathological in higher dimensions unless specific mechanisms (like boundary conditions or specific background choices) are invoked to cancel them.
• Methodological Utility: The proposed frame transformation technique offers a powerful tool for analyzing complex higher-derivative theories, allowing physicists to bypass the difficulties of non-polynomial actions and directly infer the physical content (DoF and stability).
• Implications for UV Completion: Since higher-derivative gravities are often proposed as UV completions of General Relativity, the finding that conformal gravity loses its "clean" 4D properties in higher dimensions challenges the viability of using these specific theories as fundamental descriptions of gravity in string theory or extra-dimensional models.

In summary, the authors demonstrate that extending conformal and scale-invariant gravities to $d$ dimensions is not a trivial generalization; the physical content and stability of the theories change fundamentally, with conformal gravity exhibiting a dramatic increase in pathological degrees of freedom in five dimensions.