Universality of Gravity from Entanglement (1405.2933v1)

Abstract: The entanglement "first law" in conformal field theories relates the entanglement entropy for a ball-shaped region to an integral over the same region involving the expectation value of the CFT stress-energy tensor, for infinitesimal perturbations to the CFT vacuum state. In recent work, this was exploited at leading order in $N$ in the context of large N holographic CFTs to show that any geometry dual to a perturbed CFT state must satisfy Einstein's equations linearized about pure AdS. In this note, we investigate the implications of the leading 1/N correction to the exact CFT result. We show that these corrections give rise to the source term for the gravitational equations: for semiclassical bulk states, the expectation value of the bulk stress-energy tensor appears as a source in the linearized equations. In particular, the CFT first law leads to Newton's Law of gravitation and the fact that all sources of stress-energy source the gravitational field. In our derivation, this universality of gravity comes directly from the universality of entanglement (the fact that all degrees of freedom in a subsystem contribute to entanglement entropy).

• The paper demonstrates that variations in entanglement entropy in CFTs lead directly to linearized gravitational dynamics via the first law of entanglement.
• It employs a holographic framework incorporating leading 1/N corrections, showing that bulk stress-energy acts as a source in gravitational equations.
• The study underscores the universal role of quantum entanglement in shaping spacetime geometry and generating gravitational forces.

Universality of Gravity from Entanglement

The paper authored by Brian Swingle and Mark Van Raamsdonk titled "Universality of Gravity from Entanglement" presents a sophisticated exploration of the connection between quantum entanglement and gravitational dynamics in the context of large $N$ conformal field theories (CFTs) and their AdS/CFT correspondence. This analysis is grounded in the conceptual framework where certain entanglement properties in CFTs mirror gravitational dynamics in higher-dimensional spacetime, specifically via the linearized Einstein equations.

Core Contributions

The authors leverage the entanglement "first law" for conformal field theories, which links the variation in entanglement entropy for a ball-shaped region to the expectation value of the stress-energy tensor, for states infinitesimally perturbed from the CFT vacuum. This foundation is crucial for demonstrating that any geometry dual to a perturbed CFT must satisfy the linearized Einstein equations. The novelty in this work is extended to include leading $1/N$ corrections that introduce the source term for the gravitational equations. At this order, the expectation value of the bulk stress-energy tensor appears as a source, thereby connecting the universality of gravity to the universality of entanglement across all degrees of freedom contributing to entanglement entropy.

Methodology

The approach is based on translating the CFT relation over to the gravitational theory through the holographic dictionary. The holographic entanglement entropy beyond leading order is employed in the analysis. The entanglement entropy $S_A$ for any region $A$ in the CFT can be expressed as the sum of the bulk extremal surface area and a bulk entanglement term of quantum fields on that surface. This setup allows the authors to probe deeper into the gravitational dynamics associated with semiclassical bulk states.

Results Summary

• The paper shows how the variation in entanglement entropy is linked with linearized gravitational dynamics, introducing bulk field contributions that serve as sources in the linearized equations.
• By incorporating $1/N$ corrections, the authors provide evidence for the source terms in Newtonian gravity, inferring that bulk stress-energy universally sources gravitational fields.
• These results suggest that modifications to the holographic entanglement entropy at subleading order directly correlate to gravitational source dynamics in dual descriptions.

Theoretical Implications

This work robustly affirms the essential role of quantum entanglement in the emergent gravitational dynamics within holographic settings, underscoring that the interplay between quantum field perturbations and gravitational geometry operates universally. The derivation of gravitational equations from entanglement constraints is particularly compelling for formulating a consistent theory where spacetime geometry and its dynamics emerge from underlying quantum entanglement phenomena.

Future Directions

This research opens multiple avenues for future investigation:

• Extending these assumptions and results to different dimensions and more complex geometries.
• Exploring potential generalizations beyond AdS/CFT correspondence which align with more extensive holographic principles.
• Delving into the holographic renormalization and multiscale entanglement renormalization ansatz (MERA) framework for solidifying the bulk entanglement contributions and boundary theories.
• Addressing the non-locality in potential gravitational theories, ensuring that such principles hold across broader quantum gravitational regimes.

In sum, Swingle and Van Raamsdonk provide a rigorous theoretical analysis that bolsters the understanding of gravity as an emergent force, fundamentally influenced by quantum informational principles in the form of entangled CFT degrees of freedom. The paper not only reinforces the theoretical foundations of holographic duality but also highlights a crucial intersection of quantum field theories and gravitational dynamics.