Renormalized Stress–Energy Near Horizons

Near a horizon, a renormalized stress tensor must be tested in regular coordinates and tied by conservation to the asymptotic flux. A dependency-free benchmark makes that logic explicit for one conformal scalar in an intrinsic $1+1$ Schwarzschild spacetime: the Unruh state carries positive Killing energy outward at infinity, negative ingoing energy at the future horizon, and a finite Kruskal-frame outgoing component. This is a controlled model, not a spherical reduction of the four-dimensional scalar.

Helpful background. Flat-Spacetime Quantum Field Theory provides null stress components. Partial Differential Equations and Green Functions supplies coordinate and residual checks. Causality and Global Structure develops future-horizon regularity. Quantum Fields in Curved Spacetime provides renormalized stress tensors. Black-Hole Geometry and Mechanics supplies Schwarzschild surface gravity. Black-Hole Perturbation Theory provides flux and horizon boundary conventions.

Anomaly and conservation solution

Write $\mu=GM$ for the Schwarzschild gravitational length. The executable chooses $G=1$ and $M=1$, hence $\mu=1$; the analytic formulas below retain $\mu$. Use

$$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)} =-f(r)\,du\,dv, \qquad f(r)=1-\frac{2\mu}{r},$$

where $u=t-r_*$, $v=t+r_*$, and $dr_*/dr=f^{-1}$. With the site curvature convention,

$$R=-f''(r), \qquad T^a{}_a=\frac{R}{24\pi}.$$

Conservation and the trace anomaly determine the stationary stress tensor up to two state constants:

$$T_{uu}=H(r)+t_u,\qquad T_{vv}=H(r)+t_v,\qquad T_{uv}=Q(r),$$ $$H(r)=\frac{2ff''-f'^2}{192\pi}, \qquad Q(r)=\frac{ff''}{96\pi}.$$

Indeed $2g^{uv}T_{uv}=-4Q/f=-f''/(24\pi)=R/(24\pi)$. The complete anomaly-and-conservation strategy originates in Christensen and Fulling 1977, pp. 2088–2104. In higher dimensions, angular pressure and additional state data prevent this compact formula from being a complete solution.

State constants and Killing flux

For Schwarzschild $\kappa=1/(4\mu)$ and define

$$F=\frac{\kappa^2}{48\pi} =\frac{1}{768\pi\mu^2}.$$

The three stationary choices are

State	$t_u$	$t_v$	Asymptotic outward flux
Boulware	$0$	$0$	$0$
Hartle–Hawking	$F$	$F$	$0$
Unruh	$F$	$0$	$F$

With the orientation used here, the conserved outward Killing current is

$$J^r=T_{uu}-T_{vv}=t_u-t_v.$$

At infinity the Unruh state therefore has $T_{uu}\to F$ and $T_{vv}\to0$. At the horizon, $H\to-F$, so $T_{uu}\to0$ while $T_{vv}\to-F$. The same positive conserved current is thus positive outgoing energy at infinity and negative ingoing Killing energy across the future horizon. The physical interpretation of this two-dimensional result and its relation to evaporation were established in Davies, Fulling, and Unruh 1976, pp. 2720–2723.

Future-horizon regularity

Let $U=-\kappa^{-1}e^{-\kappa u}$ be regular across the future horizon. Since

$$T_{UU}=\left(\frac{du}{dU}\right)^2T_{uu},$$

finite $T_{UU}$ requires the Unruh cancellation to leave $T_{uu}=O(f^2)$, not merely a finite Schwarzschild component. Factoring the near-horizon expression analytically avoids subtracting two nearly equal binary64 numbers. The exact limit is

$$\lim_{r\to2\mu^+}\frac{\mu^2T_{uu}^{\mathrm U}}{f^2} =\frac{1}{128\pi}.$$

To see the cancellation before numerical evaluation, define $y=2\mu/r$, so $f=1-y$. Direct substitution gives

$$\frac{T_{uu}^{\mathrm U}}{F} =1-4y^3+3y^4 =(1-y)^2(1+2y+3y^2).$$

Hence $\mu^2T_{uu}^{\mathrm U}/f^2=(1+2y+3y^2)/(768\pi)$, which is manifestly finite as $y\to1$. In Kruskal coordinates $U=-\kappa^{-1}e^{-\kappa u}$ and $V=\kappa^{-1}e^{\kappa v}$, $f$ is proportional to $-UV$ near a horizon. The $O(f^2)$ criterion therefore applies to a future-horizon crossing at finite nonzero $V$; the bifurcation surface requires both null components to be regular. Unruh has no such two-horizon extension, while Hartle–Hawking does.

This distinguishes Hartle–Hawking and Unruh future regularity from Boulware divergence. Candelas’s four-dimensional state comparison and regular stress analysis are given in Candelas 1980, §§II–IV, pp. 2185–2202; the benchmark here remains intrinsically two dimensional.

Reproducing the convergence checks

The reproducible run used Node.js v23.11.0, no third-party packages, deterministic binary64 operations, and analytic factorization near the horizon. The downloadable executable and machine-readable JSON output record 40 horizon checks. Use the standalone download-and-run recipe:

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 horizon

Algorithm. The program evaluates the closed anomaly-and-conservation solution for each state, uses a factored expression for the Unruh horizon cancellation, and refines the dimensionless coordinate offset by factors of two. Independently, it applies a centered finite-difference derivative to the conservation equation at a fixed exterior radius and refines that step by factors of two. No fitted parameters, random input, interpolation, or external data enter.

Analytic benchmark. Symbolic expansion gives the exact regularity value $1/(128\pi)$, while the exact conserved current is $F$ and the exact trace is $R/(24\pi)$. These known coefficients test the coordinate limit, flux sign, and anomaly normalization independently of the finite-difference residual. Agreement of one component alone cannot make all three pass.

The last column below decreases under six coordinate refinements.

Offset refinement $(r-2\mu)/\mu$	$\mu^2T_{uu}/f^2$	absolute error
0.20000	0.002195642185819	0.000291153800
0.10000	0.002331723673771	0.000155072312
0.05000	0.002406664249207	0.000080131737
0.02500	0.002446050585952	0.000040745400
0.01250	0.002466249370828	0.000020546615
0.00625	0.002476478682175	0.000010317304

The observed order approaches one, as expected for evaluation at a finite offset from a smooth limit. A separate centered-difference calculation of the conservation equation at $r=4\mu$ converges at second order: the residual falls from $2.84\times10^{-6}$ at step $0.4\mu$ to $2.53\times10^{-9}$ at step $0.0125\mu$.

Independent checks reproduce the anomaly to $8.67\times10^{-19}$, conserve $J^r$ exactly in the analytic formulas, recover $T_{uu}/F=0.99996848$ at $r=100\mu$, and detect the Boulware Kruskal divergence. Deliberate reversal of the anomaly or Killing-current sign fails its tolerance.

Failure modes and scope

Directly evaluating $H+F$ near the horizon loses precision through cancellation; factorization and precision variation diagnose it. A finite $T_{uu}$ is not a regularity check until transformed. A wrong state constant can preserve the trace because the anomaly is state independent, while a wrong current sign can preserve conservation but reverse the physical flux. State, trace, conservation, and regularity tests are therefore independent.

The model has no angular barrier, four-dimensional trace coefficients, or spherical-reduction dilaton. It cannot provide a quantitative four-dimensional luminosity. Its purpose is to verify signs, state constants, horizon coordinates, convergence, and the exact connection between positive asymptotic flux and negative horizon Killing energy.

A production calculation would additionally need a specified four-dimensional field, angular mode sum, Hadamard subtraction, finite counterterms, numerical tail estimate, and regular tetrad. Reusing the two-dimensional coefficient as a four-dimensional luminosity would be a dimensional and physical failure, even though both problems share conservation and horizon-redshift logic.

Exercise

At the future horizon, suppose $H=-F$ and a state has $t_u=0$, $t_v=0$. Compare its outgoing Kruskal behavior with the Unruh choice $t_u=F$, $t_v=0$.

Solution

With both constants zero, $T_{uu}\to-F$, so multiplication by $(du/dU)^2$, which diverges like $U^{-2}$, makes $T_{UU}$ singular. This is the Boulware behavior. In the Unruh state the added $F$ cancels the horizon constant in $H$, and the remaining expression begins at order $f^2$; because $f$ is proportional to $U$ along a regular crossing, $T_{UU}$ stays finite. The comparison shows why matching only the asymptotic trace cannot determine horizon regularity.

References

• Candelas, Philip. “Vacuum Polarization in Schwarzschild Spacetime.” Physical Review D 21 (1980): 2185–2202. DOI.
• Christensen, Stephen M., and Stephen A. Fulling. “Trace Anomalies and the Hawking Effect.” Physical Review D 15 (1977): 2088–2104. DOI.
• Davies, Paul C. W., Stephen A. Fulling, and William G. Unruh. “Energy-Momentum Tensor Near an Evaporating Black Hole.” Physical Review D 13 (1976): 2720–2723. DOI.