Self-Consistent Static and Cosmological Solutions

A state–geometry pair is self-consistent when the renormalized stress tensor evaluated in that state on that geometry sources the same geometry to the claimed order. Evaluating a small quantum source on the leading classical background can be sufficient at first order; a nonperturbative mean-field solution instead must close the full state–geometry dependence. Static and constant-curvature symmetry reductions turn this functional fixed-point problem into ordinary differential or algebraic equations, but branch choice, state regularity, conservation, boundary data, and stability still determine whether a formal root is physically controlled.

Required background. Einstein Equations supplies the sourced metric equation, and Gravitational Actions and Boundary Terms supplies the finite couplings and boundary conditions being held fixed.

Helpful background. Quantum Fields in Curved Spacetime supplies the state and renormalized stress tensor, while Black-Hole Geometry and Mechanics supplies static regularity and asymptotic conditions.

The self-consistency fixed point

For prescribed field content, renormalization conditions $c_i$, and a state-selection rule $\omega[g]$, define

$$\mathcal E_{\mu\nu}[g] =E_{\mu\nu}[g;c_i] -8\pi G\langle T_{\mu\nu}[g]\rangle_{\omega[g],\mathrm{ren}}.$$

A solution satisfies $\mathcal E_{\mu\nu}[g]=0$ together with its boundary and initial conditions. The state-selection rule is essential: “vacuum” may mean static ground state, asymptotic no-particle state, or regular horizon state, and these choices need not coexist on the same geometry.

Hu and Verdaguer define a semiclassical solution as the joint metric, field operator, and physically admissible state satisfying the renormalized equation Hu and Verdaguer 2008, § 3.1, following Eq. (3.10). Covariant conservation is a necessary residual check, but it is not sufficient for regularity or stability.

To see the perturbative distinction, write $g=g_0+\varepsilon h+O(\varepsilon^2)$ and a quantum source $\varepsilon T_1[g]$. Smooth response gives $\varepsilon T_1[g]=\varepsilon T_1[g_0]+O(\varepsilon^2)$. Thus using $T_1[g_0]$ is consistent in the equation for $h$ at first order. The justification fails if backreaction or susceptibility is large enough to invalidate that ordering. Iterating until numerical convergence is neither always required nor by itself a proof of control.

Static and constant-curvature reductions

For a static spherical ansatz, one may solve coupled radial equations for metric functions and matter variables after selecting an asymptotic state. The renormalized source must remain finite in the orthonormal frame appropriate to the claimed object. Approximations such as dimensionally reduced or Polyakov-inspired stress tensors can illuminate branches, but their central behavior and four-dimensional accuracy must be exposed. Arrechea and collaborators obtain ultracompact static examples using a regularized Polyakov source and explicitly identify the core approximation as the main limitation Arrechea et al. 2022, pp. 2–7, PDF.

For a four-dimensional de Sitter geometry, $R_{\mu\nu}=3xg_{\mu\nu}$ with $x=H^2\ge0$, a de Sitter-invariant state has stress expectation proportional to $g_{\mu\nu}$. Such a state must exist for the chosen field; for example, the massless minimally coupled scalar requires special care. A simple reduced equation can take the form

$$x=\lambda+\epsilon x^2, \qquad \lambda=\frac{\Lambda}{3}.$$

Its exact roots are

$$x_\pm=\frac{1\pm\sqrt{1-4\epsilon\lambda}}{2\epsilon}.$$

For $\epsilon,\lambda\gt 0$ and $4\epsilon\lambda\lt 1$, only $x_-$ approaches the classical value $\lambda$ as $\epsilon\to0$; $x_+\sim\epsilon^{-1}$ lies at the scale of the truncated correction. Here $[x]=[\lambda]=L^{-2}$ and $[\epsilon]=L^2$. This algebraic model tests branch selection, not a quantum state construction.

Reproducible teaching benchmark

The bounded benchmark was rerun in CPython 3.9.6 with NumPy 2.0.2 and SciPy 1.13.1, using IEEE-754 binary64 arithmetic. Choose a reference length $L_0$ and nondimensionalize $x$, $\lambda$, and $\epsilon$ by $L_0^2x$, $L_0^2\lambda$, and $\epsilon/L_0^2$; the same symbols denote these dimensionless variables below. The benchmark sets $\lambda=0.12$, $\epsilon=0.35$, brackets the weak root on $[0,(2\epsilon)^{-1}]$, and applies SciPy’s Brent method at four root tolerances.

Root tolerance	Weak-root result $x$	Root error	Equation residual	Iterations
1.0000000e-3	1.2549059e-1	2.3208436e-5	2.1169539e-5	5
1.0000000e-6	1.2551385e-1	4.8998356e-8	4.4693376e-8	6
1.0000000e-9	1.2551380e-1	4.3640092e-13	3.9800628e-13	7
1.0000000e-12	1.2551380e-1	1.2634338e-13	1.1529579e-13	7

The independent quadratic formula gives

$$x_-=0.125513799887, \qquad x_+=2.731629057256.$$

The decreasing error and residual verify convergence at the tested tolerances; these rows do not establish a binary64 roundoff floor. The high root is printed as a cutoff diagnostic rather than accepted as a physical solution. Download the MIT-licensed benchmark, then use the standalone download-and-run recipe:

curl -fL https://blackhole.org/code/semiclassical-gravity/semiclassical-gravity-benchmarks.py -o semiclassical-gravity-benchmarks.py
python3 semiclassical-gravity-benchmarks.py --case self-consistent-static-cosmological-solutions

This is a teaching benchmark for nonlinear root selection. It does not compute a four-dimensional mode sum, construct a Hadamard state, or establish a physical de Sitter solution, and it is not a production solver.

The branch criterion is physical input; the bracket implements it numerically by isolating the root continuously connected to $x=\lambda$. Running an unconstrained Newton iteration could converge to the high root while reporting an equally small residual. The quadratic formula is independent of the Brent iteration and checks both roots. For very small $\epsilon\lambda$, evaluate the weak root as $x_-=2\lambda/[1+\sqrt{1-4\epsilon\lambda}]$ to avoid subtracting nearly equal numbers. Repeating the calculation at four tolerances tests numerical convergence; it does not test model convergence under addition of an $O(\epsilon^2x^3)$ term. That second variation is required before interpreting the algebraic root as an effective-theory prediction.

Physical checks beyond a root

A physical calculation needs more than $\mathcal E=0$:

• State regularity: the state is Hadamard and regular in every region included in the claim.
• Conservation: $\nabla^\mu\langle T_{\mu\nu}\rangle=0$ and the tensor equation residual converges.
• Boundary data: asymptotics, center or horizon behavior, and finite couplings are held fixed across refinements.
• Branch control: the solution connects continuously to a declared weak-backreaction limit.
• Stability: causal gauge-invariant perturbations remain bounded below the cutoff.

The benchmark catches a wrong algebraic branch or failed solver tolerance, but not an inconsistent state, a hidden finite-counterterm change, or an instability. A static source approximation can also be accurate outside a small core and fail precisely where regularity is being claimed.

Failure modes and scope

Brent iteration fails if the bracket does not straddle the desired root or the discriminant becomes negative. A root may converge numerically yet violate $\epsilon x\ll1$, making the derivative truncation uncontrolled. In a differential problem, stiffness, coordinate singularities, and constraint drift add further failure modes. Independent state construction and response analysis are therefore indispensable.

Fixed-background stress calculations remain in Quantum Fields in Curved Spacetime. This page addresses their joint backreaction, not microscopic quantum-gravity completion or fine-grained information recovery. Symmetry-reduced examples demonstrate possibility within their assumptions; they do not establish a universal static or cosmological solution class.

Exercise

For $x=\lambda+\epsilon x^2$ with $\lambda,\epsilon\gt 0$, show that the weak root has the expansion $x_-=\lambda+\epsilon\lambda^2+O(\epsilon^2)$ and give a condition for its perturbative control.

Solution

Expanding $\sqrt{1-4\epsilon\lambda}=1-2\epsilon\lambda-2\epsilon^2\lambda^2+O(\epsilon^3)$ in $x_-=[1-\sqrt{1-4\epsilon\lambda}]/(2\epsilon)$ gives $x_-=\lambda+\epsilon\lambda^2+O(\epsilon^2)$. The relative correction is $\epsilon\lambda$, so a necessary algebraic control condition is $\epsilon\lambda\ll1$, together with a positive discriminant. A small residual alone cannot establish that condition because the high root also solves the equation exactly.

References

• Arrechea, J., Barceló, C., Carballo-Rubio, R., and Garay, L. J. 2022. “Semiclassical Relativistic Stars.” Scientific Reports 12, 15958. Open PDF.
• Hu, B. L., and Verdaguer, E. 2008. “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11, 3. DOI.