Complexity equals Spacetime Volume (CV2.0)

• CV2.0 is a proposal that equates holographic complexity with the spacetime volume of the Wheeler–DeWitt patch normalized by pressure and ℏ, differing from volume or action proposals.
• It leverages extended thermodynamics and Noether-charge methods to connect gravitational action sectors with the thermodynamic volume of AdS black holes.
• The model exhibits late-time linear growth and intriguing features in multi-horizon and de Sitter scenarios, yet remains a conjectural candidate amidst ongoing debates.

Complexity equals Spacetime Volume, usually abbreviated CV2.0 or CV-2, is a holographic complexity proposal in which the complexity of a boundary state is identified not with the volume of a maximal spatial slice but with the spacetime volume of the Wheeler–DeWitt (WDW) patch, normalized by the AdS pressure and by $\hbar$:

$$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}.$$

With the extended-thermodynamic identification

$P=-\frac{\Lambda}{8\pi G},$

the proposal ties complexity to the cosmological-constant sector of the on-shell gravitational action and to the thermodynamic volume of AdS black holes. In its original form, CV2.0 was presented as a conjectural alternative to complexity=action (CA), motivated by late-time WDW dynamics, black-hole chemistry, and improved behavior with respect to the Lloyd bound in some charged examples (Couch et al., 2016).

1. Definition and distinction from other holographic complexity proposals

The defining idea of CV2.0 is that the primitive bulk observable is a codimension-zero quantity: the spacetime volume of the WDW patch. This differs from the original complexity=volume proposal, which uses the volume of a maximal codimension-one spatial slice, and from complexity=action, which uses the on-shell action of the WDW patch. In the notation used in the original proposal,

$$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}, \qquad \dot{\mathcal C}\sim \frac{PV}{\hbar},$$

where $V$ is the thermodynamic volume for one-horizon black holes, or the difference of thermodynamic volumes for two-horizon geometries. By contrast, CA is

$\mathcal{C}=\frac{\mathcal A}{\pi\hbar},$

with $\mathcal A$ the WDW action (Couch et al., 2016).

Two structural features distinguish CV2.0 from the original CV proposal. First, it is a covariant spacetime quantity associated with the WDW region rather than a codimension-one extremal slice. Second, because the normalization is supplied by the pressure $P$, it does not require inserting an arbitrary length scale by hand in the way the original CV proposal does. The proposal therefore isolates a specific bulk contribution—pressure times spacetime volume—rather than the full gravitational action or the maximal-slice volume (Couch et al., 2016).

The WDW patch here is the union of spacelike surfaces anchored at chosen boundary times. In one-horizon cases the relevant late-time growth is controlled by the outer horizon; in two-horizon cases the relevant quantity is an outer-minus-inner contribution. This codimension-zero character later became central to broader “complexity equals anything” frameworks, within which CV2.0 appears as a special limit rather than an isolated prescription.

2. Thermodynamic and Noether-charge basis

The original motivation for CV2.0 came from combining CA-duality with extended black-hole thermodynamics. In that framework the ADM mass is interpreted as enthalpy, the pressure is

$P=-\frac{\Lambda}{8\pi G},$

and the thermodynamic volume is

$V=\left(\frac{\partial H}{\partial P}\right)_S.$

The key observation was that the late-time growth of the WDW spacetime volume is naturally governed by this thermodynamic volume, suggesting that $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}.$$0 has a direct geometric and holographic meaning rather than being merely a formal derivative (Couch et al., 2016).

A central result was the decomposition of late-time CA growth for 4d AdS-Schwarzschild: $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}.$$1 Using the Smarr relation

$$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}.$$2

this simplifies to

$$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}.$$3

The significance of the intermediate form is that the $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}.$$4 term arises from the bulk interior part of the WDW patch, while the $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}.$$5 term comes from the singularity contribution and the $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}.$$6 term from corner terms. This singled out $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}.$$7 as a distinguished bulk-interior contribution inside the CA computation itself (Couch et al., 2016).

The Noether-charge underpinning was developed through an adaptation of the Iyer–Wald formalism with varying $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}.$$8. Defining

$$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}.$$9

one has on shell

$P=-\frac{\Lambda}{8\pi G},$0

After integrating over a hypersurface $P=-\frac{\Lambda}{8\pi G},$1, the $P=-\frac{\Lambda}{8\pi G},$2 term in the extended first law emerges from the variation of the cosmological constant. This gives thermodynamic volume a geometrical and variational meaning within gravity, and it is precisely this structure that motivates relating complexity to the $P=-\frac{\Lambda}{8\pi G},$3-controlled spacetime volume of the WDW patch (Couch et al., 2016).

For Einstein vacua the bridge becomes particularly direct. With

$P=-\frac{\Lambda}{8\pi G},$4

and

$P=-\frac{\Lambda}{8\pi G},$5

the on-shell bulk action is proportional to

$P=-\frac{\Lambda}{8\pi G},$6

so the cosmological-constant contribution is

$P=-\frac{\Lambda}{8\pi G},$7

This is the direct route from CA to CV2.0: pressure times WDW spacetime volume is already present as a special sector of the CA action (Couch et al., 2016).

3. Late-time growth laws and canonical examples

In the original proposal, the basic late-time formulas are simple. For one-horizon black holes such as AdS-Schwarzschild,

$P=-\frac{\Lambda}{8\pi G},$8

For two-horizon black holes,

$P=-\frac{\Lambda}{8\pi G},$9

The geometric content is that the late-time rate of change of WDW spacetime volume equals the thermodynamic volume in the one-horizon case, or the outer-minus-inner thermodynamic-volume difference in the two-horizon case (Couch et al., 2016).

The prototype is 4d AdS-Schwarzschild. There the late-time bulk contribution from the two WDW slivers is

$$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}, \qquad \dot{\mathcal C}\sim \frac{PV}{\hbar},$$0

Thus the spacetime-volume growth of the WDW patch reproduces the thermodynamic volume $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}, \qquad \dot{\mathcal C}\sim \frac{PV}{\hbar},$$1. In the high-temperature regime $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}, \qquad \dot{\mathcal C}\sim \frac{PV}{\hbar},$$2, the standard thermodynamic quantities satisfy

$$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}, \qquad \dot{\mathcal C}\sim \frac{PV}{\hbar},$$3

so $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}, \qquad \dot{\mathcal C}\sim \frac{PV}{\hbar},$$4, $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}, \qquad \dot{\mathcal C}\sim \frac{PV}{\hbar},$$5, and $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}, \qquad \dot{\mathcal C}\sim \frac{PV}{\hbar},$$6 are parametrically of the same order, with $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}, \qquad \dot{\mathcal C}\sim \frac{PV}{\hbar},$$7 and $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}, \qquad \dot{\mathcal C}\sim \frac{PV}{\hbar},$$8 coinciding at leading order. This explains why CV2.0 and CA are not parametrically far apart in the large-black-hole regime (Couch et al., 2016).

For AdS-Reissner–Nordström in $$\mathcal{C} \sim \frac{1}{\hbar} P\,\mathrm{(Spacetime~Volume)}, \qquad \dot{\mathcal C}\sim \frac{PV}{\hbar},$$9 dimensions,

$V$0

and the WDW spacetime volume grows as

$V$1

so that

$V$2

This is a nontrivial check because the late-time WDW patch terminates on the inner horizon rather than on a singularity (Couch et al., 2016).

The same outer-minus-inner structure persists in lower-dimensional and rotating examples. For charged BTZ,

$V$3

and the charge-dependent subtraction cancels in $V$4. For rotating BTZ,

$V$5

For Kerr-AdS$V$6,

$V$7

In this rotating case, the outer-minus-inner volume difference decreases toward zero at extremality, which the original paper regarded as physically appealing from the viewpoint of vanishing late-time complexity growth (Couch et al., 2016).

4. Bounds, comparisons, and controversies

CV2.0 was partly motivated by the Lloyd bound. The uncharged version reviewed in the original paper is

$V$8

while for charged systems the proposed grand-canonical form is

$V$9

refined to

$\mathcal{C}=\frac{\mathcal A}{\pi\hbar},$0

For AdS-RN near extremality with $\mathcal{C}=\frac{\mathcal A}{\pi\hbar},$1, both CA and CV2.0 violate the refined bound because both growth rates scale linearly in $\mathcal{C}=\frac{\mathcal A}{\pi\hbar},$2, whereas the right-hand side scales quadratically. By contrast, near empty AdS with $\mathcal{C}=\frac{\mathcal A}{\pi\hbar},$3, CV2.0 satisfies the bound while CA violates it; in 4d AdS-RN the original analysis showed

$\mathcal{C}=\frac{\mathcal A}{\pi\hbar},$4

whereas the corresponding CA expression is manifestly positive (Couch et al., 2016).

The main controversy concerns the relation between thermodynamic volume and WDW spacetime volume. “On the Noether charge and the gravity duals of quantum complexity” argued that the generalized Wald–Iyer construction more naturally singles out the $\mathcal{C}=\frac{\mathcal A}{\pi\hbar},$5-dependent part of the non-derivative action, leading to a different proposal, CA-2, rather than to raw spacetime volume. In that treatment,

$\mathcal{C}=\frac{\mathcal A}{\pi\hbar},$6

but the paper emphasized that thermodynamic volume is in general not equal to the WDW spacetime volume except in special cases such as $\mathcal{C}=\frac{\mathcal A}{\pi\hbar},$7. Even so, after testing Einstein–Maxwell–dilaton examples, it concluded that CA generally violates the Lloyd bound, CA-2 sometimes improves but can still fail, whereas CV-2 always respects the Lloyd bound in the examples studied; the paper therefore stated that CV-2 might be the best holographic dual among the proposals it compared (Fan et al., 2018).

This leaves a persistent ambiguity. On one hand, CV2.0 isolates a geometrically distinguished and technically simple sector of CA. On the other, more refined Noether-charge analyses suggest that pressure times spacetime volume is not always the unique thermodynamic object naturally selected by the bulk variational structure. The result is that CV2.0 remains a motivated but non-unique codimension-zero candidate.

5. Codimension-zero reformulations and the “Complexity Equals Anything” program

Later work embedded CV2.0 into a broader family of codimension-zero observables. In “Complexity Equals Anything II,” one first selects a bulk region $\mathcal{C}=\frac{\mathcal A}{\pi\hbar},$8 bounded by two hypersurfaces $\mathcal{C}=\frac{\mathcal A}{\pi\hbar},$9 by extremizing

$\mathcal A$0

and then evaluates another observable on that extremized region. In the simple constant-functional model,

$\mathcal A$1

extremization yields constant-mean-curvature slices,

$\mathcal A$2

Taking $\mathcal A$3 with $\mathcal A$4 fixed drives the boundaries to null hypersurfaces and turns $\mathcal A$5 into the WDW patch, giving

$\mathcal A$6

This explicitly reproduces the usual CV2.0 prescription as a limit of a larger codimension-zero construction (Belin et al., 2022).

Within that framework, CV2.0 inherits two universal properties established for the codimension-zero family in the thermofield-double state: late-time linear growth and the switchback effect. For the planar AdS black hole, the WDW spacetime-volume specialization obeys

$\mathcal A$7

The same paper also used the Peierls construction to show that variations of codimension-zero observables are encoded in the gravitational symplectic form, giving the spacetime-volume observable a clean phase-space interpretation (Belin et al., 2022).

The review “Complexity Equals (Almost) Anything” sharpened the conceptual lesson: CV, CA, and CV2.0 are not isolated prescriptions but special cases inside a vast “extremize-then-evaluate” family of bulk observables. In that view,

$\mathcal A$8

is a legitimate codimension-zero member of a broad universality class, but it is not uniquely selected by the coarse tests of late-time linear growth and switchback behavior (Myers et al., 2024).

A further development concerns multi-horizon geometries. In “Complexity equals anything for multi-horizon black holes,” the codimension-zero sector—including spacetime-volume-type prescriptions—was argued to be especially well suited to black holes with several trapped regions, because it can probe all spacetime regions where the blackening factor $\mathcal A$9 and can distinguish slices asymptoting to a Cauchy horizon from those ending on a singularity. This does not single out CV2.0 uniquely, but it enlarges the class of interiors for which codimension-zero observables appear structurally advantageous (Jiang et al., 12 Jun 2025).

6. Extensions beyond AdS and current status

CV2.0 has also been adapted to de Sitter settings, though in forms that differ from the original AdS pressure-based normalization. In “Holographic Complexity in dS$P$0,” the proposal was implemented directly as

$P$1

for a WDW patch anchored on equal-time slices of stretched horizons in de Sitter. The paper found a de Sitter-specific phenomenon: hyperfast growth ending at a finite critical time $P$2, with

$P$3

and, after introducing a cutoff $P$4, a crossover to linear late-time growth

$P$5

This is a direct de Sitter implementation of a spacetime-volume complexity, though not one formulated through AdS black-hole chemistry (Jørstad et al., 2022).

A later Schwarzschild–de Sitter study introduced a timelike de Sitter adaptation,

$P$6

with $P$7 an imaginary length scale inherited from timelike CV. In both static-patch holography and dS/CFT, the asymptotic growth law was

$P$8

so CV2.0 again grew linearly, whereas the corresponding CA growth vanished because the regularized action remained finite. This work did not use a pressure formulation, which marks a significant departure from the original AdS thermodynamic interpretation (Fang et al., 2 Jun 2026).

Indirect evidence for the broader relevance of thermodynamic volume also comes from horizonless AdS$P$9 solitons. There, a nontrivial thermodynamic volume appears in a Smarr relation and first law, and for large solitons the CV and CA complexities of formation both scale as

$P=-\frac{\Lambda}{8\pi G},$0

This does not constitute a CV2.0 computation, but it suggests that thermodynamic volume can remain complexity-relevant outside black-hole interiors (Andrews et al., 2019).

The present status of CV2.0 is therefore sharply defined. It is a codimension-zero proposal with a clear geometric core, a strong thermodynamic motivation, and substantial support from explicit AdS and de Sitter examples. At the same time, the literature repeatedly emphasizes that it is a conjecture rather than a theorem, that no general proof identifies thermodynamic volume with late-time WDW spacetime-volume growth in full generality, that near-extremal charged cases remain problematic, and that the relation to a unique microscopic boundary notion of circuit complexity remains open. Current work accordingly treats CV2.0 as a serious and technically natural holographic complexity candidate, but not as a uniquely established one.