On the Ghost Problem of Conformal Gravity (2306.13714v2)

Abstract: We study the metric perturbations around the de Sitter and Minkowski backgrounds in Conformal Gravity. We confirm the presence of ghosts in both cases. In the de Sitter case, by applying the Maldacena boundary conditions - the Neumann boundary condition and the positive-frequency mode condition - to the metric, we show that one cannot recover a general solution for the perturbations. In turn, alongside the Neumann boundary condition, we derive an additional condition with which the perturbations of conformal gravity and dS perturbations of Einstein gravity with cosmological constant coincide. We further show that the Neumann boundary condition does not lead to a general solution in Minkowski space. Conversely, we derive the alternative boundary conditions, with which we attain an agreement between the perturbations of conformal and Einstein gravity in full generality, thus removing the ghost of conformal gravity.

• The paper demonstrates that novel boundary conditions effectively resolve ghost instabilities in Conformal Gravity, ensuring compatibility with Einstein gravity's tensor modes.
• It shows that applying boundary conditions akin to Maldacena's method eliminates unwanted vector contributions, leaving only the expected de Sitter tensor perturbations.
• The findings imply that breaking conformal invariance selectively can pave the way for improved quantum gravity models and more accurate cosmological predictions.

Insights on the Ghost Problem of Conformal Gravity

The paper "On the Ghost Problem of Conformal Gravity" addresses key mathematical challenges and theoretical questions in the field of gravitation by focusing on conformal gravity and its ghost solution problem. Conformal Gravity (CG) is an extension of Einstein's theory of general relativity that includes higher derivative terms making it power-counting renormalizable. However, it suffers from ghost-like degrees of freedom which render the theory unstable, both classically and quantum mechanically. The authors, Anamaria Hell, Dieter Lüst, and George Zoupanos, explore boundary conditions that resolve the ghost issues in perturbations of CG, making it observationally indistinguishable from Einstein gravity under specific spatial conditions.

Main Contributions

• Ghosts in Conformal Gravity: The authors begin by reconfirming the presence of ghosts in perturbations around both de Sitter (dS) and Minkowski backgrounds. Ghosts are negative-norm states that lead to non-unitary evolution in a quantum context. For CG, this pathology shows up through additional tensor degree modes that persist without the intervention of additional constraints.
• Boundary Conditions Approach: Building on work by Maldacena, the authors apply boundary conditions to resolve these ghost issues. The paper confirms that the Maldacena boundary conditions lead to a particular solution where the vector degrees vanish and leave a specific remnant of tensor modes that align with the effective degrees of freedom expected in dS space.
• New Boundary Conditions for Full Solution Recovery: The paper's primary contribution lies in presenting new boundary conditions that allow for the recovery of all tensor perturbations that coincide with those expected in Einstein gravity. These boundary conditions carefully break the conformal invariance in CG and select the background uniquely as the dS universe. This approach effectively mitigates the adverse effects of the ghost modes.
• Minkowski Perturbations Without Neumann Condition: The paper examines CG perturbations in flat (Minkowski) spacetime where conventional methods restrict to non-general solutions. The authors propose alternative boundary conditions to derive a full set of Minkowski-compatible solutions, completely removing the pathological ghost solutions.

Theoretical and Practical Implications

The implications of this paper are notable both theoretically and practically:

• Theoretical Alignment with Einstein Gravity: By solving the ghost problem, the studied conformal gravity can effectively behave as Einstein gravity under certain conditions, thus theoretically validating the model within the well-tested sphere of known gravitation physics while allowing for new predictions at high energies.
• Framework for Other Higher-Order Theories: The techniques and results could potentially pave the way for similar problem-solving strategies in other higher-order gravitational theories that face similar challenges of ghost or additional degrees of freedom.
• Implications for Quantum Gravity and Cosmology: Since CG is considered a candidate for UV completion of gravity, these findings might assist in refinements in cosmological models, especially in early universe scenarios influenced by quantum gravity effects.

In conclusion, the paper meticulously engages with the mathematical aspects of conformally invariant gravity models, providing a roadmap for resolving perturbative issues in one of the more conservative extensions to the standard model of cosmology. The developments here constitute a precise mathematical treatment that aligns conformal gravity with the observable universe while setting a new stage for further investigations into the quantum realms of gravity. Future research may expand on how these boundary conditions might be experimentally realized or further generalized to a wider class of spacetimes.