- The paper shows that null boundary contributions in gravitational action involve ambiguities that are resolved using affine parametrization.
- It rigorously computes the on-shell action for Wheeler-DeWitt patches in AdS black hole spacetimes, verifying that the action change scales with the black hole’s mass.
- The study refines the method for including null boundaries in general relativity, enhancing theoretical tools for AdS/CFT correspondence and quantum gravity research.
Gravitational Action with Null Boundaries: A Detailed Analysis
The paper by Lehner, Myers, Poisson, and Sorkin offers a comprehensive examination of the boundary term in the action functional of general relativity when the boundary of the spacetime region includes null segments. This is an extension of the traditional context where boundaries are typically timelike or spacelike, each accompanied by their own contributions to the gravitational action. The authors address the less explored area where null boundaries are involved.
The key challenge presented by null boundaries is the introduction of ambiguities in the gravitational action, primarily due to the arbitrary parametrization of the null generators. The authors show that the contribution from a null segment is sensitive to this choice, leading to an inherent ambiguity. Moreover, similar ambiguities appear in joint contributions where a null segment joins with another boundary segment. The authors propose resolving these ambiguities by demanding the affine parametrization of null generators, effectively nullifying their contributions to the action. Nevertheless, even then, there remains a permissible rescaling freedom of the affine parameter by a constant factor, which can affect the resulting joint contributions.
After establishing these foundations, the paper rigorously calculates the on-shell gravitational action for a Wheeler-deWitt patch in an anti-de Sitter black hole spacetime. This task has direct relevance to hypotheses in the framework of the AdS/CFT correspondence, notably the "complexity equals action" conjecture, which relates the growth of black hole interiors to computational complexity in quantum field theories. For Schwarzschild-AdS black holes, the authors recover a result consistent with previous literature, showing that the rate of change of the action asymptotically approaches a simple expression proportional to the black hole’s mass.
The implications of this work are twofold. Theoretically, it proposes well-defined rules for computing gravitational action in the presence of null boundaries and contributes to more precise interpretations of AdS/CFT correspondence. Practically, it advocates for a broader application of these rules in scenarios involving general spacetimes with null segments — a common scenario in theoretical explorations of black holes. Future research may extend these findings to dynamic scenarios, thereby testing the robustness of the proposed boundary terms under time-dependent conditions.
In conclusion, this paper advances our understanding of gravitational action in the context of general relativity with null boundaries. It enriches the toolkit available to researchers working with complex spacetime geometries and opens avenues for future explorations in quantum gravity and holographic principles.