Coarse-Grained Geometries

When many exact microstates are represented by one semiclassical geometry, the essential datum is the coarse-graining map. One may restrict the observable algebra, replace a pure state by a maximum-entropy state with the same selected expectation values, average over time, or average over an ensemble. These operations are not equivalent, and averaging metrics is generally not the same as solving the field equations for averaged sources. A common geometry is justified only when the retained observables and their fluctuations agree to the stated accuracy throughout a controlled semiclassical regime.

Required background. Quantum Information supplies quantum channels, maximum-entropy states, and distinguishability; Gauge Theory and String-Theory Preparation supplies constrained algebras and the holographic context for geometric descriptions.

Helpful background. Semiclassical Gravity supplies effective geometries and renormalized sources; Quantum Fields in Curved Spacetime supplies the low-point field observables retained by the coarse description.

Four different coarse-graining operations

Let $\rho_i$ be exact microstates in one energy-and-charge sector. Four maps occur commonly.

Restricted algebra. Keep only expectation values $a_\alpha=\operatorname{Tr}(\rho_i A_\alpha)$ for $A_\alpha\in\mathcal A_{\rm simple}$. States are operationally equivalent at accuracy $\epsilon$ when

$$\sup_{A\in\mathcal A_{\rm simple},\,\lVert A\rVert_\infty\le1} \left|\operatorname{Tr}A(\rho_i-\rho_j)\right|\le\epsilon.$$

Maximum entropy. Replace the exact state by the Jaynes state

$$\rho_{\rm cg} =\frac{1}{Z}\exp\!\left(-\sum_\alpha\lambda_\alpha A_\alpha\right),$$

whose multipliers reproduce the retained $a_\alpha$. Ensemble averaging. Use $\bar\rho=\sum_i p_i\rho_i$, which discards which-state information. Temporal smoothing. Average correlators over a declared time window, which suppresses phases but depends on spectral resolution and observation time.

These maps answer different questions. A restricted pure state remains pure globally, whereas $\bar\rho$ is mixed. A time average can preserve diagonal microstate data that an ensemble average removes. Every geometric claim should name which operation is performed.

For example, retaining only $\langle Z\rangle=0$ for a qubit is consistent with both pure states $|+\rangle$ and $|-\rangle$, and with the mixed state $I/2$. Maximum entropy selects $I/2$, whose entropy is $\log2$, although each original pure state has entropy zero. Measuring $X$ distinguishes the two pure states immediately. The entropy increase records the discarded observable information; it is not evidence of nonunitary evolution.

From retained data to an effective geometry

A semiclassical metric $g_{\mu\nu}^{\rm eff}$ can represent the coarse data if there is an Einstein-gravity regime in which retained one- and low-point functions agree with quantum fields on that geometry and the renormalized source satisfies

$$G_{\mu\nu}[g^{\rm eff}]+\Lambda g_{\mu\nu}^{\rm eff} =8\pi G\,\langle T_{\mu\nu}\rangle_{\rm cg}$$

to the stated order. The equation uses the site convention $(-,+,+,+)$; a higher-derivative theory requires its corresponding renormalized field equation. It does not imply $g^{\rm eff}_{\mu\nu}=\sum_i p_i g^{(i)}_{\mu\nu}$. Einstein’s equation is nonlinear, and individual exact states need not possess classical metrics at all.

Controlled fluctuations are an additional condition. Stress tensors are operator-valued distributions, so their fluctuations require specified smearing and renormalization. The connected stress two-point function drives metric fluctuations through the gravitational response. If those induced fluctuations are large at the resolution of interest, one deterministic metric is inadequate even when the mean source is smooth. The ratio of stress variance to squared mean is only a heuristic: it is undefined when the mean vanishes and is not a universal validity test. The response, scale, and state must be examined together.

Balasubramanian and collaborators distinguish typical-state correlators from a thermal density matrix in Balasubramanian et al. 2008, §§ 2–4. Their examples make the key point: agreement for low-point probes need not extend to sufficiently precise or high-point observables.

Results and open questions through 2026-08-19

This discussion reflects results available through 2026-08-19. The permissive interpretation treats a stable common set of simple correlators as sufficient for an emergent geometry. The conservative interpretation demands a derived coarse map, controlled fluctuations, and a demonstration that the geometry is not an ensemble artifact.

Coarse-graining evidence	Result	Scope or limitation
Canonical typicality in Goldstein et al. 2006, pp. 1–3	Most pure states in a large constrained subspace agree on a small subsystem’s reduced state.	It assumes a factorization and measure and does not construct a metric.
Typical versus thermal correlators in Balasubramanian et al. 2008, §§ 2–4	Low-point observables can agree while sufficiently detailed probes distinguish a pure microstate from a thermal ensemble.	The resolving threshold depends on the model, operator complexity, and entropy.
ETH evidence summarized in D’Alessio et al. 2016, §§ 2–4 and 6	Simple observables in many chaotic eigenstates exhibit smooth microcanonical behavior and suppressed fluctuations.	Integrable, localized, scarred, or symmetry-mixed sectors are adverse cases; gravitational ETH remains an extrapolation.
Known smooth microstate sectors in Bena et al. 2022, §§ 2–8	Exact string backgrounds can share asymptotic charges while differing at horizon scales.	Known solutions may be atypical and show that equal asymptotic data do not imply one common interior geometry.

The controlled results support restricted-observable agreement. It remains unresolved whether a single semiclassical interior is the correct coarse-grained representative of typical black-hole microstates, whether an ensemble or single-theory average is involved, and which high-complexity observables first invalidate it.

Adverse checks and noncommuting averages

Consider a coherent superposition of two semiclassical branches with macroscopically different metrics. Averaging the metric components can produce a tensor that solves neither branch’s field equations, while a measurement of branch-sensitive correlations reveals large variance. This is a counterexample to defining the coarse geometry as a componentwise mean.

As another check, choose two ensembles with identical one-point functions but different connected two-point functions. A geometry inferred only from the mean stress tensor cannot distinguish them, yet their fluctuations source different corrections. Enlarging $\mathcal A_{\rm simple}$ therefore tests the stability of the proposed common geometry. The description is credible only within the largest algebra and resolution for which it remains accurate.

The order of operations matters as well. Projecting to a fixed charge sector and then maximizing entropy generally differs from maximizing over all charges and conditioning afterward. Time averaging before taking a nonlinear geometric functional can differ from taking that functional in each state and then averaging. A page must expose these choices rather than call all of them coarse graining.

Generic quantum-information theory, stable thermodynamics, and explicit string or holographic constructions remain separate. Here the conclusion is a criterion: name the state measure, retained algebra, averaging operation, resolution, fluctuation bound, and semiclassical equation before assigning a common geometry to many microstates.

The conclusions could change after a microscopic single-theory construction derives the coarse map and its fluctuation bounds, or a high-complexity observable distinguishes states at a parametrically different scale, or a controlled calculation shows that ensemble and state-by-state geometric prescriptions agree or disagree in the claimed black-hole regime.

Exercises

Two states have the same $\langle T_{\mu\nu}\rangle$ but variances differing by a factor of $10^6$ at the detector’s resolution. Can the same deterministic semiclassical geometry be justified from the mean alone?

Solution

No. The equal mean can support the same leading sourced equation, but the very different variances imply different stress fluctuations and potentially different metric fluctuations or higher-point responses. Specify the smeared, renormalized stress observable and propagate its connected fluctuations to the metric at the detector resolution. A variance ratio alone does not decide validity: both responses might be negligible, both large, or only one large. The one-point data cannot settle that question.

References

• Balasubramanian, Vijay, Bartłomiej Czech, Veronika E. Hubeny, Klaus Larjo, Mukund Rangamani, and Joan Simón. 2008. “Typicality versus Thermality: An Analytic Distinction.” General Relativity and Gravitation 40, 1863–1890. arXiv:hep-th/0701122.
• Bena, Iosif, Emil J. Martinec, Samir D. Mathur, and Nicholas P. Warner. 2022. “Fuzzballs and Microstate Geometries: Black-Hole Structure in String Theory.” arXiv:2204.13113.
• D’Alessio, Luca, Yariv Kafri, Anatoli Polkovnikov, and Marcos Rigol. 2016. “From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics.” Advances in Physics 65, 239–362. arXiv:1509.06411.
• Goldstein, Sheldon, Joel L. Lebowitz, Roderich Tumulka, and Nino Zanghì. 2006. “Canonical Typicality.” Physical Review Letters 96, 050403. arXiv:cond-mat/0511091.