Practical Stress-Tensor Calculations

A numerical stress-tensor calculation is credible only when the field, state, subtraction, boundary conditions, and convergence tests are specified together. This page compares the main computational strategies and then gives a reproducible cylinder benchmark in which two different regulators converge to the same Casimir energy, while deliberately incorrect subtractions fail visibly. The benchmark tests machinery; it is not a four-dimensional black-hole production calculation.

Required background. Stress–Energy and Conservation provides the tensor identities used as numerical residuals. Curvature supplies the local geometric subtraction terms.

Helpful background. Flat-Spacetime Quantum Field Theory provides mode normalization and vacuum sums. Partial Differential Equations and Green Functions supplies Green-function and point-splitting methods. Causality and Global Structure develops the background assumptions. Quantum Fields in Curved Spacetime provides the common Hadamard and state framework.

Choosing a numerical representation

Four families of methods recur.

• Mode-sum point splitting separates the points in time, angle, or another symmetry coordinate, subtracts the large-frequency singular part, and sums the regular remainder. It is efficient on stationary or highly symmetric backgrounds but sensitive to mode normalization and tail fitting.
• DeWitt–Schwinger or heat-kernel subtraction expands local geometric coefficients. It exposes the ultraviolet counterterms directly, but a truncated large-mass expansion is not an exact state-dependent answer.
• Adiabatic regularization subtracts a WKB expansion mode by mode, especially in homogeneous cosmology. Its adiabatic order and its relation to a Hadamard prescription must be stated.
• Numerical Green-function point splitting solves for a two-point function or Euclidean Green function and removes a local parametrix before coincidence. It handles less separable problems but demands careful control of boundary data and near-coincidence cancellation.

Modern implementations often combine them: analytic subtraction handles the singular tail, while numerics evaluate the smooth remainder. The classic point-separation construction is given in Christensen 1976, pp. 2490–2501. Practical mode-sum variants include time splitting and angular splitting developed by Levi and Ori.

Whatever the representation, report the tensor component in a declared frame. A finite Schwarzschild-coordinate component near a horizon does not by itself establish regularity in a freely falling or Kruskal frame.

A controlled cylinder benchmark

Take one real massless $c=1$ scalar oscillator sector on

$$ds^2=-dt^2+dx^2,\qquad x\sim x+L,$$

with $\hbar=1$. The noncompact scalar zero mode has no normalizable Fock vacuum, so it is explicitly excluded; the benchmark concerns the derivative-field oscillator sector. Define the dimensionless energy coefficient $C=L^2\rho$. For periodic modes, Abel and Gaussian regulators give

$$C_A(a)=2\pi\left[\sum_{n=1}^{\infty}n e^{-an}-\frac{1}{a^2}\right],$$ $$C_G(a)=2\pi\left[\sum_{n=1}^{\infty}n e^{-(an)^2}-\frac{1}{2a^2}\right].$$

The continuum counterterm differs between regulators; using the Abel counterterm in the Gaussian sum is an intentional failure case. Both correct branches approach

$$C=-\frac{\pi}{6}.$$

This universal finite-size coefficient is the $c=1$ specialization of the conformal results in Blöte, Cardy, and Nightingale 1986, pp. 742–745 and Affleck 1986, pp. 746–748.

The finite limit and the convergence order can also be derived before computing any sum. For the Abel regulator,

$$\sum_{n\ge1}ne^{-an} =\frac{1}{4\sinh^2(a/2)} =\frac1{a^2}-\frac1{12}+\frac{a^2}{240}+O(a^4).$$

For the Gaussian regulator, Euler–Maclaurin applied to $f(x)=xe^{-a^2x^2}$ gives

$$\sum_{n\ge1}ne^{-(an)^2} =\frac1{2a^2}-\frac1{12}-\frac{a^2}{120}+O(a^4).$$

Consequently $C_A=-\pi/6+\pi a^2/120+O(a^4)$ and $C_G=-\pi/6-\pi a^2/60+O(a^4)$. The two sequences approach the same limit from opposite sides, with leading errors in a ratio of $-2$. This is a stronger diagnostic than agreement of their extrapolated limits alone.

Reproducible refinement results

The reproducible run used Node.js v23.11.0, no third-party dependencies, deterministic IEEE-754 binary64 arithmetic, and compensated summation. The downloadable executable source and machine-readable JSON output contain every tolerance and all 77 checks. Use the standalone download-and-run recipe for the cylinder case:

curl -fL https://blackhole.org/code/qft-in-curved-spacetime/qft-curved-spacetime-benchmarks.mjs -o qft-curved-spacetime-benchmarks.mjs
node qft-curved-spacetime-benchmarks.mjs --case cylinder

The last column is the absolute Abel error, so the table shows convergence quantitatively rather than describing it qualitatively.

Regulator $a$	Abel cutoff	$C_A$	Gaussian cutoff	$C_G$	Abel absolute error
0.400	100	-0.519436432722	16	-0.532314803627	0.004162342876
0.200	200	-0.522553237941	32	-0.525713401767	0.001045537657
0.100	400	-0.523337080063	64	-0.524123625421	0.000261695535
0.050	800	-0.523533332244	127	-0.523729753277	0.000065443354
0.025	1600	-0.523582413543	253	-0.523631505393	0.000016362055

Both errors scale as $a^2$; the observed refined orders are $1.9999$ for Abel and $2.0006$ for Gaussian. Richardson extrapolation of the last two Abel values gives $-0.523598773976446$, only $1.62\times10^{-9}$ from $-\pi/6$. The explicit Abel loop agrees with its closed hyperbolic-function sum within $4.55\times10^{-13}$, providing an independent implementation benchmark rather than another extrapolation of the same data.

State, boundary, and tensor checks

Changing to antiperiodic modes changes the exact coefficient to $+\pi/12$; the regulated sequence approaches it at second order. Adding one quantum in the lowest nonzero periodic mode shifts $C$ by the exact finite state difference $2\pi$; a quantum in mode $n$ shifts it by $2\pi n$. These differences need no continuum subtraction. The tests discriminate a correct boundary phase and state normalization from an implementation that merely reproduces one vacuum number.

For the static cylinder result, the computed tensor has zero flux, vanishing homogeneous conservation residual, and the expected traceless oscillator-sector form. The script tests all three. It also detects two failures: omitting the Abel continuum term makes the result diverge like the inverse regulator squared, and using the wrong Gaussian counterterm produces the same nonconvergent growth. A result is rejected when any declared tolerance fails.

For a curved four-dimensional calculation the analogous checklist is longer:

1. demonstrate mode-tail convergence under at least three justified refinements;
2. compare the subtracted sum with an analytic limit or independent method;
3. verify $\nabla^a\langle T_{ab}\rangle$ and the expected trace;
4. test boundary conditions and state regularity in a nonsingular frame;
5. vary numerical precision when large singular pieces nearly cancel.

Failure modes and scope

An unresolved tail can look like a finite counterterm; a wrong boundary phase can look like a different state; and subtracting two large approximations can lose the small physical remainder to cancellation. Nonuniform convergence is especially dangerous near horizons or coincidence. Conservation may also be satisfied accidentally by a symmetric ansatz while individual components carry a normalization error, so it is not an independent benchmark by itself.

The cylinder calculation has no curvature, horizon, or four-dimensional angular momentum barrier. It validates regulator subtraction, convergence estimation, state differences, boundary phases, trace, and conservation in a setting with an exact answer. It does not validate a production code’s geometric coefficients or black-hole state construction.

Exercise

Using the last two rows, estimate the observed order of the Abel error and explain what would happen to that estimate if the $1/a^2$ subtraction were omitted.

Solution

Halving $a$ changes the error from $6.5443354\times10^{-5}$ to $1.6362055\times10^{-5}$. Therefore $p=\log_2(e_{\mathrm{coarse}}/e_{\mathrm{fine}})\simeq1.9999$, consistent with quadratic regulator error. Without the continuum subtraction, the leading sum grows as $1/a^2$; halving $a$ would increase rather than decrease the magnitude by about a factor of four. An apparent negative convergence order would expose the missing counterterm.

References

• Affleck, Ian. “Universal Term in the Free Energy at a Critical Point and the Conformal Anomaly.” Physical Review Letters 56 (1986): 746–748. DOI.
• Anderson, Paul R., William A. Hiscock, and David A. Samuel. “Stress-Energy Tensor of Quantized Scalar Fields in Static Spherically Symmetric Spacetimes.” Physical Review D 51 (1995): 4337–4358. DOI.
• Blöte, Henk W. J., John L. Cardy, and M. P. Nightingale. “Conformal Invariance, the Central Charge, and Universal Finite-Size Amplitudes at Criticality.” Physical Review Letters 56 (1986): 742–745. DOI.
• Christensen, Stephen M. “Vacuum Expectation Value of the Stress Tensor in an Arbitrary Curved Background: The Covariant Point-Separation Method.” Physical Review D 14 (1976): 2490–2501. DOI.
• Levi, Adam, and Amos Ori. “Mode-Sum Regularization of $\langle\phi^2\rangle$ in the Angular-Splitting Method.” Physical Review D 94 (2016): 044054. DOI.
• Levi, Adam, and Amos Ori. “Pragmatic Mode-Sum Regularization Method for Semiclassical Black-Hole Spacetimes.” Physical Review D 91 (2015): 104028. DOI. Open PDF.