Gravitational action with null boundaries

TL;DR

The paper delivers a complete framework for the gravitational action with boundaries that include null segments, identifying and taming ambiguities from null parametrization and null joints via affinely parametrized generators and a set of additivity rules. It provides a rigorous variational-principle treatment, derives explicit boundary terms for spacelike, timelike, and null segments, and formulates joint contributions with precise sign conventions. Applying this to Wheeler–DeWitt patches in AdS black holes, the authors recover the standard late-time result dI/dt = 2M (or dS/dt = 32πG_N M in their normalization) and extend it to charged and non-spherical horizons, while clarifying comparisons with prior work. The findings strengthen the foundations of the complexity–action conjecture by clarifying boundary-term ambiguities and demonstrating robust, symmetry-protected results under reasonable prescriptions.

Abstract

We present a complete discussion of the boundary term in the action functional of general relativity when the boundary includes null segments in addition to the more usual timelike and spacelike segments. We confirm that ambiguities appear in the contribution from a null segment, because it depends on an arbitrary choice of parametrization for the generators. We also show that similar ambiguities appear in the contribution from a codimension-two surface at which a null segment is joined to another (spacelike, timelike, or null) segment. The parametrization ambiguity can be tamed by insisting that the null generators be affinely parametrized; this forces each null contribution to the boundary action to vanish, but leaves intact the fredom to rescale the affine parameter by a constant factor on each generator. Once a choice of parametrization is made, the ambiguity in the joint contributions can be eliminated by formulating well-motivated rules that ensure the additivity of the gravitational action. Enforcing these rules, we calculate the time rate of change of the action when it is evaluated for a so-called "Wheeler-deWitt patch" of a black hole in asymptotically-anti de Sitter space. We recover a number of results cited in the literature, obtained with a less complete analysis.

Figures (13)

• Figure 1: Wheeler-deWitt patch of an eternal Schwarzschild-anti de Sitter black hole. The patch is defined by a future light cone originating inside the white-hole horizon and reaching the left boundary at time $t_{\rm L}$ and the right boundary at time $t_{\rm R}$. This light cone is joined to a past light cone converging to the future singularity.
• Figure 2: A region ${\scr V}$ of spacetime with its broken boundary $\partial {\scr V}$. The boundary consists of four spacelike segments ${\cal S}_1$, ${\cal S}_2$, ${\cal S}_3$, ${\cal S}_4$, and four null segments ${\cal N}_1$, $\bar{{\cal N}}_2$, $\bar{{\cal N}}_3$, ${\cal N}_4$. These are joined at codimension-two surfaces denoted ${\cal B}_{jk}$.
• Figure 3: Domain ${\scr V}$ bounded by a closed hypersurface $\partial {\scr V}$ consisting of a timelike segment ${\cal T}$ and two spacelike segments ${\cal S}_1$ and ${\cal S}_2$. The intersection between ${\cal T}$ and ${\cal S}_j$ is the closed two-surface ${\cal B}_j$.
• Figure 4: Joint terms in the boundary action. In panel $a$, a past boundary is broken at ${\cal B}$ into two spacelike segments of normal $n_1^\alpha$ and $n_2^\alpha$; the contribution from ${\cal B}$ to the boundary action is $-2\oint_{\cal B} \eta\, dS$. In panel $b$, a future boundary is broken into two spacelike segments; the contribution to the boundary action is $2\oint_{\cal B} \eta\, dS$. In panel $c$, a timelike boundary is broken into two timelike segments of normal $s_1^\alpha$ and $s_2^\alpha$; the contribution to the boundary action is $-2\oint_{\cal B} \eta\, dS$. In panel $d$, a timelike boundary of normal $s^\alpha$ is joined at ${\cal B}$ to a future, spacelike boundary of normal $n^\alpha$; the contribution to the boundary action is $2\oint_{\cal B} \eta\, dS$. In panel $e$, a timelike boundary is joined to a past boundary, with contribution $-2\oint_{\cal B} \eta\, dS$. In panel $f$, two spacelike boundaries are joined, with contribution $2\oint_{\cal B} \eta\, dS$. Finally, two timelike boundaries are joined in panel $g$, with contribution $2\oint_{\cal B} \eta\, dS$. In all panels the shaded region represents the interior of ${\scr V}$. The figure also shows the null vectors $k^\alpha$ and $\bar{k}^\alpha$, which are introduced in the main text.
• Figure 5: A closed hypersurface $\partial {\scr V}$ consisting of a past spacelike surface ${\cal S}_1$, a truncated past null cone ${\cal N}$, and a future spacelike surface ${\cal S}_2$.