Complexity of Formation and Time Growth

Complexity of formation is a matched subtraction between a thermofield-double preparation and two reference preparations; time growth is the derivative of a separately normalized proposal. The subtraction can remove leading UV divergences, but only when induced cutoffs, boundary sources, null normalizations, and reference states match. Similar late growth in CV and CA is a qualitative cross-check, not evidence that the proposals or boundary tasks are equivalent.

Required background. Complexity Equals Volume Proposals and Complexity Equals Action Proposals define the two bulk functionals.

Helpful background. Mixed-State, Purification, and Formation Complexity fixes the boundary task. Spectral Statistics, Form Factors, and Late-Time Evidence prevents a classical growth law from being extended to exact late times.

The matched formation subtraction

For a two-sided thermofield double at inverse temperature $\beta$, define proposal by proposal

$$\Delta\mathcal C_X(\beta) =\mathcal C_X(|\mathrm{TFD}_\beta\rangle) -\mathcal C_X(|0\rangle_L) -\mathcal C_X(|0\rangle_R), \qquad X\in\{V,A\}.$$

The black-hole and vacuum regulators must induce the same boundary metric and spatial volume. In CV this means matching the Fefferman–Graham cutoff before subtracting volumes. In CA it also means identical WDW anchoring, null-normal convention, null counterterm scale, and vacuum energy convention. A coordinate cutoff $r=r_{\max}$ used independently in the two geometries generally does not satisfy this requirement.

For high-temperature AdS black holes, both proposals often yield a formation cost proportional to thermal entropy, with proposal- and dimension-dependent coefficients. In boundary dimension two, special cancellations make the CA formation result temperature independent in the standard setup Chapman, Marrochio, and Myers 2017. These are outputs of the specified bulk prescriptions, not universal circuit theorems.

Growth and conserved quantities

Time-reflection symmetry gives

$$\left.\frac{\mathrm d\mathcal C_X}{\mathrm dt}\right|_{t=0}=0$$

for the unperturbed symmetric TFD. At late classical times, neutral two-derivative black holes give

$$\frac{\mathrm d\mathcal C_A}{\mathrm dt}\to\frac{2M}{\pi}, \qquad \frac{\mathrm d\mathcal C_V}{\mathrm dt} \to\frac{\Omega_{d-1}r_m^{d-1}\sqrt{-f(r_m)}}{G_N\ell_V}.$$

The CA expression is tied to the chosen action normalization; the CV expression retains $\ell_V$. Charge and rotation replace the simple mass dependence by combinations of conserved potentials and horizon data. Classical linear growth cannot continue through recurrence times of a finite-entropy quantum system without modification.

First application

For a neutral BTZ black hole, match the boundary circle and Fefferman–Graham cutoff to two copies of global AdS$_3$. Evaluate $V_{\mathrm{BTZ}}-2V_{\mathrm{AdS}}$ and $I_{\mathrm{WDW}}^{\mathrm{BTZ}}-2I_{\mathrm{WDW}}^{\mathrm{AdS}}$ at $t=0$. Then evolve both boundaries symmetrically and extract the late rates. CA gives $2M/\pi$; CV gives $\pi r_+^2/(G_3L\ell_V)$.

Report each formation value and rate with its own normalization. The common facts—finite matched formation difference and late linear growth—survive. Their coefficients do not define a proposal-independent complexity.

Adversarial control

Change the vacuum matching by holding a coordinate cutoff rather than the induced metric fixed; the formation difference shifts or diverges. Next move to a charged or rotating family while keeping the neutral normalization. Inner horizons and chemical potentials change the CA result, while $r_m$ and the arbitrary $\ell_V$ control CV. Qualitative linear growth remains possible even when the two functions disagree at intermediate times or in their parameter dependence.

Regime, evidence ceiling, and handoff

The calculation assumes a classical saddle with $G_N/L^{d-1}\ll1$, controlled curvature and $\alpha'$ corrections, a declared compactification, and times below the regime where finite-$N$ discreteness and recurrences dominate. The subtraction does not cancel every finite counterterm or reference-state ambiguity.

The evidence ceiling is a proposal-specific finite difference and time diagnostic. Continue to Shockwaves, Switchbacks, and Scrambling Diagnostics for perturbed growth and Divergences, Counterterms, and Scheme Dependence for the allowed scheme shifts.

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. (2016), “Complexity, Action, and Black Holes,” Physical Review D 93, 086006. DOI; arXiv:1512.04993.
• Chapman, S., Marrochio, H., and Myers, R. C. (2017), “Complexity of Formation in Holography,” Journal of High Energy Physics 2017(01), 062. DOI; arXiv:1610.08063.
• Stanford, D., and Susskind, L. (2014), “Complexity and Shock Wave Geometries,” Physical Review D 90, 126007. DOI; arXiv:1406.2678.