Complexity Equals Action Proposals

The complexity-equals-action proposal assigns the on-shell gravitational action of the Wheeler–DeWitt patch to a boundary complexity, conventionally $\mathcal C_A=I_{\mathrm{WDW}}/\pi$ in units with $\hbar=1$. The number is defined only after including every bulk, non-null boundary, null boundary, joint, and null-counterterm contribution with fixed normalizations and regulators. CA is not CV in different notation and is not a unique circuit task.

Required background. Boundary Complexity Inputs: Tasks, Reference States, and Gate Sets fixes the boundary target. Two-Sided Black Holes and Thermofield-Double States supplies the WDW geometry.

Helpful background. Field Variations and Boundary Terms and Boundaries, Surface Counterterms, and Boundary Stress supply action principles and renormalization.

The complete WDW action

For Einstein gravity, a regulated patch $\mathcal W$ requires

$$\begin{aligned} I_{\mathcal W}={}&\frac{1}{16\pi G_N} \int_{\mathcal W}\!\mathrm d^{d+1}x\sqrt{\lvert g\rvert}\,(R-2\Lambda)\\ &+\frac{1}{8\pi G_N}\int_{\partial\mathcal W_{\mathrm{non-null}}} \!\mathrm d^dx\sqrt{|h|}\,K +I_{\mathrm{null}}+I_{\mathrm{joint}}+I_{\mathrm{ct}}. \end{aligned}$$

For a null generator $k^\mu=(\partial/\partial\lambda)^\mu$, $I_{\mathrm{null}}$ contains its nonaffinity $k^\nu\nabla_\nu k^\mu=\kappa k^\mu$; affine parametrization sets $\kappa=0$ but leaves constant rescalings. Null joints contain logarithms such as $a=\log|k\cdot\bar k/2|$. The reparametrization counterterm is

$$I_{\mathrm{ct}} =\frac{1}{8\pi G_N} \int\!\mathrm d\lambda\,\mathrm d^{d-1}\theta\, \sqrt\gamma\,\Theta\log(\ell_{\mathrm{ct}}\Theta), \qquad \Theta=\partial_\lambda\log\sqrt\gamma.$$

Lehner et al. derive the null and joint terms and their normalization dependence Lehner et al. 2016. The arbitrary counterterm scale $\ell_{\mathrm{ct}}$ is a residual scheme datum.

Null-normal rescaling check

Under $k\to e^c k$, an adjacent null joint shifts by a term proportional to

$$\Delta I_{\mathrm{joint}} =\frac{c}{8\pi G_N}\int\!\mathrm d^{d-1}\theta\sqrt\sigma.$$

At the same time $\lambda\to e^{-c}\lambda$ and $\Theta\to e^c\Theta$, so the logarithm in $I_{\mathrm{ct}}$ produces the compensating endpoint shift. Omitting $I_{\mathrm{ct}}$ therefore makes the result depend on an unphysical normalization of null generators. Including it restores reparametrization invariance but introduces $\ell_{\mathrm{ct}}$.

First application

For a two-sided neutral AdS–Schwarzschild black hole, anchor the WDW patch at $t_L=t_R=t/2$, regulate each asymptotic end at the same Fefferman–Graham cutoff, choose affine null generators, and include singularity, joint, and counterterm contributions. At late times the moving joints approach the horizons and the classical two-derivative calculation gives

$$\frac{\mathrm dI_{\mathrm{WDW}}}{\mathrm dt}\longrightarrow2M, \qquad \frac{\mathrm d\mathcal C_A}{\mathrm dt}\longrightarrow\frac{2M}{\pi}.$$

This is the principal neutral-black-hole result of Brown et al. Brown et al. 2016a, Brown et al. 2016b. A reproducible calculation lists the null normals, patch corners, regulator relation, counterterm scale, action normalization, and ground-state energy convention.

Adversarial control

Rescale one null normal while omitting $I_{\mathrm{ct}}$; the joint action and finite answer move. Restore the counterterm and vary $\ell_{\mathrm{ct}}$; some transient and finite terms still change. Higher-derivative gravity, charge, rotation, or quantum backreaction alters the late result and can make the neutral formula inapplicable.

The control shows that agreement with $2M/\pi$ is conditional on the classical action, state family, normalization, and complete boundary prescription.

Regime, evidence ceiling, and handoff

CA is controlled when the WDW patch lies within a valid semiclassical EFT, $G_N/L^{d-1}\ll1$, $\alpha'/L^2\ll1$, and all KK and higher-derivative terms in the action are consistently included or bounded. Because the patch can touch a singularity, small exterior quantum corrections need not imply a small action correction.

The evidence ceiling is a conjectural geometric functional with detailed black-hole tests. Continue to Proposed Complexity Bounds and Their Counterexamples for the failure of universal growth claims and to Divergences, Counterterms, and Scheme Dependence for scheme control.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Brown, A. R., Roberts, D. A., Susskind, L., Swingle, B., and Zhao, Y. (2016a), “Holographic Complexity Equals Bulk Action?” Physical Review Letters 116, 191301. DOI; arXiv:1509.07876.
• Brown, A. R., Roberts, D. A., Susskind, L., Swingle, B., and Zhao, Y. (2016b), “Complexity, Action, and Black Holes,” Physical Review D 93, 086006. DOI; arXiv:1512.04993.
• Lehner, L., Myers, R. C., Poisson, E., and Sorkin, R. D. (2016), “Gravitational Action with Null Boundaries,” Physical Review D 94, 084046. DOI; arXiv:1609.00207.