Semiclassical Causal Limits and Quantum-Gravity Questions

Semiclassical gravity assumes one differentiable manifold with one mean Lorentzian metric, sourced by quantum matter and sometimes supplemented by controlled stochastic perturbations. It can compute causal response and probabilities for relational probes near that mean. It cannot define an exact causal order when inequivalent geometries or topologies occur in coherent quantum superposition and no single background dominates.

Required background. Quantum Information supplies mixtures, coherent branches, and operational probabilities.

Helpful background. Quantum Fields in Curved Spacetime supplies microcausality on a fixed geometry; Einstein Equations supplies the mean metric; Black-Hole Geometry and Mechanics supplies horizons; Semiclassical Gravity supplies mean and stochastic validity; and Causality and Global Structure supplies causal relations and topology.

The one-geometry closure

The mean equation has the form

$$E_{ab}[\bar g]=8\pi G\, \langle\hat T_{ab}[\bar g]\rangle_\rho.$$

It presupposes that matter operators, the state $\rho$, renormalization, and retarded propagation are all defined on the same background $\bar g$. Stochastic gravity adds a covariance $\langle h_{ab}h_{cd}\rangle$ while requiring $h$ to remain a perturbation. A full distribution of smeared observables additionally requires higher moments or a justified Gaussian approximation. Hu and Verdaguer state this hierarchy and its large-$N$ interpretation in Hu and Verdaguer 2008, §§ 3.2–3.3, PDF.

Within that closure one can ask whether a retarded kernel has past support, whether a causal horizon of $\bar g$ is stable, how a wavepacket arrival time is broadened, and whether quantum stress prevents a proposed geometry from solving the constraints. Those are genuine semiclassical causal questions.

Why the mean metric may not encode causal order

Suppose a quantum state has approximately decohered geometric branches $g_i$ with weights $p_i$. For two relationally specified events $A$ and $B$, and after fixing a common relational identification of the branches, an operational causal probability is

$$P(B\in J^+(A)) =\sum_i p_i\, \mathbf 1\!\left[B\in J^+_{g_i}(A)\right]$$

for a classical mixture, with a measurement model required when branches interfere. In general this is not equal to the indicator computed from a componentwise mean metric $\bar g=\sum_i p_i g_i$ in that same relational gauge. Without such an identification, even the metric average is undefined; with one, the map from a metric to its causal cone is still nonlinear.

For a local example, put both events in a convex normal neighborhood in each branch and fix their future ordering when they are causally related. In common squared-length units, their geodesic squared intervals may be $s_1^2=-4$ and $s_2^2=+2$. With weights $0.7$ and $0.3$, the averaged number is $-2.2$, while the probability of the timelike branch is only $0.7$. This average of squared intervals need not equal the squared interval computed from the averaged metric, since the connecting geodesic also depends on the metric. A null average can likewise arise from timelike and spacelike branches without either branch being null. Interval signs give this simple classification only locally; global causal connectivity cannot in general be inferred from one geodesic’s length.

Topology, locality, and exact observables

A perturbation $h_{ab}$ on one manifold cannot describe creation of a handle, disappearance of a component, or a sum over inequivalent topologies. Classical topology-change results already show that a smooth Lorentzian interpolation with strong causal properties faces sharp obstructions under stated hypotheses Geroch 1967, pp. 782–786. Deciding what replaces those hypotheses quantum mechanically belongs to Quantum Gravity.

Locality also changes character. On a fixed background, spacelike separated matter operators commute. In a diffeomorphism-invariant quantum theory, an observable must be gravitationally dressed or relationally anchored, and that dressing can extend to infinity or another reference system. Donnelly and Giddings explain why such observables obstruct an exact tensor factorization into compact spatial subsystems Donnelly and Giddings 2016, §§ II–V, PDF. Semiclassical algebras remain useful approximations when the dressing and metric uncertainty are perturbative; they are not exact microscopic locality data.

Testing the limits of a mean causal geometry

Pass a causal question to Quantum Gravity when any of the following is essential to its answer:

• no single manifold and topology supports all relevant branches;
• the rms metric fluctuation of the relational observable is comparable to its mean separation from the light cone;
• interference between geometries, rather than a classical stochastic mixture, changes the outcome;
• singularity resolution or topology change selects whether worldlines continue;
• defining the observable requires nonperturbative gravitational dressing or an exact factorization; or
• response frequencies, curvature, or localization reach the EFT cutoff.

A comparison with a microscopic theory should retain the last controlled mean geometry, its covariance and smearing, the quantum state, renormalized couplings, constraint and response residuals, causal probabilities for named probes, topology assumptions, and the first failed inequality. These quantities can be matched between descriptions without assuming the microscopic theory’s causal ontology.

Checks and limits

If every branch gives the same causal relation and metric differences are perturbatively small, $P$ approaches zero or one and the mean-metric description is recovered. In a flat-space narrow-state limit, ordinary microcausality returns. Conversely, a large coordinate variance does not establish a failure of mean geometry: first replace it with a gauge-invariant smeared observable. Gauge artifacts do not become quantum geometry merely because their distribution is broad.

Exercise

Two decohered branches have probabilities $0.7$ and $0.3$. For the same relational events within a convex normal neighborhood in each branch, $s_1^2=-4$ and $s_2^2=+2$ in common squared-length units. Use the same future ordering in the timelike branch. Compute the mean squared interval and the probability that the events are causally related. Why are these different summaries?

Solution

The mean is

$$\langle s^2\rangle=0.7(-4)+0.3(2)=-2.2.$$

With signature $(-,+,+,+)$ it is negative, but the causal-relation probability is only $0.7$, because the second branch is spacelike. Averaging the interval and asking a branchwise yes-or-no causal question are nonlinear operations that do not commute. A microscopic treatment is needed if interference between the branches matters.

References

• Donnelly, W., and Giddings, S. B. (2016). “Observables, Gravitational Dressing, and Obstructions to Locality and Subsystems.” Physical Review D 94, 104038. DOI. Open PDF.
• Geroch, R. (1967). “Topology in General Relativity.” Journal of Mathematical Physics 8, 782–786. DOI.
• Hu, B. L., and Verdaguer, E. (2008). “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11, 3. DOI. Open PDF.