Heat and Wave kernel expansions for stationary spacetimes

Alexander Strohmaier; Steve Zelditch

arxiv: 2308.12148 · v3 · submitted 2023-08-23 · 🧮 math.SP · gr-qc· math-ph· math.AP· math.MP

Heat and Wave kernel expansions for stationary spacetimes

Alexander Strohmaier , Steve Zelditch This is my paper

Pith reviewed 2026-05-24 08:05 UTC · model grok-4.3

classification 🧮 math.SP gr-qcmath-phmath.APmath.MP

keywords stationary spacetimeswave-trace expansionheat kernel coefficientsspectral geometryzeta functionsnull-geodesicsultra-static spacetimesscalar curvature

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The pith

The second non-zero term in the wave-trace expansion on stationary spacetimes is computed as the direct analogue of the second heat kernel coefficient of the Laplace operator.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an explicit formula for the second coefficient in the wave-trace expansion of the generator of time translations for the wave equation on stationary spacetimes. This extends classical spectral geometry results from Riemannian manifolds, where the same generator reduces to the square root of the Laplacian, to the broader stationary setting. The formula is involved in general but simplifies exactly to the scalar-curvature term when the spacetime is ultra-static. The computation is tied to prior results on the Weyl law and the relation of the expansion to the geometry of null-geodesics, and it connects the wave-trace coefficients to heat-kernel coefficients and zeta-function residues.

Core claim

If the spacetime is spatially compact the spectrum is discrete and admits a wave-trace expansion at time zero. The second non-zero term in this expansion is an analogue in the category of stationary spacetimes of the second heat kernel coefficient of the Laplace operator. The general formula reduces to the usual term involving the scalar curvature when specialised to ultra-static spacetimes.

What carries the argument

The wave-trace expansion at time zero of the generator of time-translations on the solution space of the wave equation, whose second coefficient encodes the geometric invariant analogous to scalar curvature.

If this is right

The residues of associated zeta functions are determined by the same geometric data that appear in the second wave-trace coefficient.
Spectral analysis of the time-translation generator yields geometric invariants for any spatially compact stationary spacetime, not only the ultra-static subclass.
The wave-trace formula remains linked to the geometry of the space of null-geodesics even after the second coefficient is extracted.
Heat-kernel methods on the spatial slices can be compared term-by-term with the wave-trace coefficients via the ultra-static reduction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same expansion technique may supply curvature-type invariants for stationary metrics that arise in general relativity but are not ultra-static.
Comparison of the derived coefficient against known examples such as rotating black-hole exteriors could test whether the formula captures frame-dragging effects.
If the coefficient can be expressed in terms of local curvature scalars plus lower-order terms, it would parallel the classical heat-kernel expansion more closely than the general expression suggests.

Load-bearing premise

The spacetime must be spatially compact so that the spectrum is discrete and admits a wave-trace expansion at time zero.

What would settle it

Explicit computation of the second wave-trace coefficient for a concrete non-ultra-static stationary metric whose spectrum can be determined independently, followed by direct comparison with the general formula.

read the original abstract

The generator of time-translations on the solution space of the wave equation on stationary spacetimes specialises to the square root of the Laplacian on Riemannian manifolds when the spacetime is ultrastatic. Its spectral analysis therefore constitutes a generalization of classical spectral geometry. If the spacetime is spatially compact the spectrum is discrete and admits a wave-trace expansion at time zero. A Weyl law for the eigenvalues and a wave-trace formula was shown in a previous paper and related to the geometry of the space of null-geodesics. In this paper we investigate the relation to heat kernel coefficients and residues of zeta functions in this context and compute the second non-zero term in the wave-trace expansion. This second coefficient is an analogue in the category of stationary spacetimes of the second heat kernel coefficient of the Laplace operator. The general formula is quite involved but reduces to the usual term involving the scalar curvature when specialised to ultra-static spacetimes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper computes the second non-zero coefficient in the wave-trace expansion for stationary spacetimes and verifies the reduction to the scalar curvature term in the ultrastatic case.

read the letter

The main result is the explicit computation of that second coefficient, which was not in the prior literature on the Weyl law and wave-trace formula. The paper frames the time-translation generator as a direct generalization of the square root of the Laplacian and ties the expansion to null-geodesic geometry and zeta residues. The reduction to the usual scalar curvature term when the spacetime is ultrastatic serves as a consistency check that the generalization is set up correctly. This is the concrete new piece the work delivers. The setup is clean and the spatial compactness assumption is stated upfront as the condition needed for a discrete spectrum and the t=0 expansion. The formula is described as involved, which limits immediate applicability, but the check against the classical case is a useful verification. No internal inconsistency appears in the claims. The derivation steps are not visible from the abstract, so the technical strength rests on the full text, but the overall structure follows from the earlier wave-trace result without obvious circularity. This is for specialists working on spectral geometry for Lorentzian or stationary metrics and related questions in mathematical GR. A reader already using wave-trace or heat-kernel methods on manifolds would find the new coefficient and the geometric relations worth consulting. It deserves peer review because it supplies a specific new term in an established framework.

Referee Report

0 major / 2 minor

Summary. The manuscript computes the second non-zero coefficient in the wave-trace expansion at t=0 for the generator of time translations on spatially compact stationary spacetimes. Building on a prior Weyl law and wave-trace formula, the authors derive an explicit (though involved) expression for this coefficient and show that it reduces to the standard scalar-curvature term when the spacetime is ultra-static, thereby establishing it as the stationary analogue of the second heat-kernel coefficient a2 of the Laplace-Beltrami operator.

Significance. If the derivation holds, the work supplies a concrete geometric invariant linking the spectrum of the time-translation generator to curvature quantities on stationary backgrounds, extending classical spectral geometry. The explicit reduction to the ultra-static case serves as a non-trivial consistency check. The parameter-free character of the reduction and the direct tie to the geometry of null geodesics are strengths.

minor comments (2)

The abstract states that the general formula is 'quite involved'; a compact display of the final expression (perhaps as a displayed equation in the introduction or §3) would help readers locate the main result without searching the body.
Notation for the wave-trace coefficients and the generator of time translations should be compared explicitly with the corresponding objects in the cited previous paper to avoid any ambiguity in the reduction step.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of its contributions, and recommendation for minor revision. The referee correctly notes the derivation of the second coefficient in the wave-trace expansion, its relation to the heat kernel coefficient a2, and the consistency check in the ultrastatic case.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained with external verification

full rationale

The paper cites prior work by the same authors only for the existence of the wave-trace expansion and Weyl law as a starting framework. The central new result is the explicit computation of the second non-zero coefficient, which is then shown to reduce to the known scalar-curvature term under ultra-static specialization. This reduction serves as an external benchmark rather than a self-referential fit. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the new coefficient to prior inputs by construction appear in the provided material. The derivation therefore retains independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on spatial compactness (domain assumption) and on the wave-trace formula established in the authors' previous paper; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The spacetime is stationary and spatially compact, yielding a discrete spectrum that admits a wave-trace expansion at time zero.
Stated explicitly in the abstract as the condition under which the expansion exists.

pith-pipeline@v0.9.0 · 5693 in / 1198 out tokens · 36152 ms · 2026-05-24T08:05:54.499108+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The second coefficient is an analogue in the category of stationary spacetimes of the second heat kernel coefficient of the Laplace operator. The general formula is quite involved but reduces to the usual term involving the scalar curvature when specialised to ultra-static spacetimes.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

c2(x) = 2 π^{-n/2} Γ(n/2 -1)/(4 Γ(n-3)) × ˜c2(x) with ˜c2 containing (1/6 scal - W) N |Z|^{-n+2} and curvature terms

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

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Berger, M.; Gauduchon, P.; Mazet, E., Le spectre d'une variété riemannienne. (French) Lecture Notes in Mathematics, Vol. 194 Springer-Verlag, Berlin-New York 1971

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J.J.Duistermaat and V.Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Inv.Math. 24 (1975), 39-80

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Off diagonal short time asymptotics for fundamental solutions of diffusion equations

Kannai, Y. Off diagonal short time asymptotics for fundamental solutions of diffusion equations. Comm. Partial Differential Equations 2 (1977), no. 8, 781-830

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& Ralston, J

Majda, A. & Ralston, J. An analogue of Weyl's theorem for unbounded domains. I, II, III Duke Math. J. . 45, no. 1, 183-196, no.3, 513–536, 46, no. 4, 725–731 (1978, 1979)

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Forward scattering by a convex obstacle

Melrose, R. Forward scattering by a convex obstacle. Comm. Pure Appl. Math. . 33, 461-499 (1980)

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Riesz, L'int\'egrale de Riemann-Liouville et le probleme de Cauchy

M. Riesz, L'int\'egrale de Riemann-Liouville et le probleme de Cauchy. Acta Math. 81 (1949). 1--223

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Strohmaier and S

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[39]

Strohmaier and S

A. Strohmaier and S. Zelditch, Spectral asymptotics on stationary space-times. Reviews in Mathematical Physics, 2060007, 2020

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[40]

Local decay of waves on asymptotically flat stationary space-times

Tataru, D. Local decay of waves on asymptotically flat stationary space-times. Amer. J. Math. . 135, 361-401 (2013)

work page 2013
[41]

On the Fourier transforms of retarded Lorentz-invariant functions

Trione, S. On the Fourier transforms of retarded Lorentz-invariant functions. J. Math. Anal. Appl. . 84, 73-112 (1981)

work page 1981
[42]

R. M. Wald, General Relativity. Chicago: University of Chicago Press (1984)

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[43]

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Eigenfunctions of the Laplacian on a Riemannian manifold

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[1] [1]

Atiyah, M.; Bott, R.; Patodi, V. K. Errata to: "On the heat equation and the index theorem'' (Invent. Math. 19 (1973), 279-330). Invent. Math. 28 (1975), 277-280

work page 1973

[2] [2]

& Kath, I

Baum, H. & Kath, I. Normally hyperbolic operators, the Huygens property and conformal geometry. Annals Of Global Analysis And Geometry . 14 pp. 315-371 (1996)

work page 1996

[3] [3]

On the wave equation on a compact Riemannian manifold without conjugate points

B\'erard, Pierre H. On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155 (1977), no. 3, 249-276

work page 1977

[4] [4]

(French) Lecture Notes in Mathematics, Vol

Berger, M.; Gauduchon, P.; Mazet, E., Le spectre d'une variété riemannienne. (French) Lecture Notes in Mathematics, Vol. 194 Springer-Verlag, Berlin-New York 1971

work page 1971

[5] [5]

Einstein manifolds

Besse, A. Einstein manifolds. (Springer-Verlag, Berlin,2008), Reprint of the 1987 edition

work page 2008

[6] [6]

ar, N. Ginoux and F. Pf\

C. B\"ar, N. Ginoux and F. Pf\"affle, Wave equations on Lorentzian manifolds and quantization. ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Z\"urich, 2007

work page 2007

[7] [7]

B\"ar, A.Strohmaier, Local Index Theory for Lorentzian Manifolds, arXiv: 2012.01364

C. B\"ar, A.Strohmaier, Local Index Theory for Lorentzian Manifolds, arXiv: 2012.01364

work page arXiv 2012

[8] [8]

& Häfner, D

Bony, J. & Häfner, D. Decay and non-decay of the local energy for the wave equation on the de Sitter-Schwarzschild metric. Comm. Math. Phys. . 282, 697-719 (2008)

work page 2008

[9] [9]

& Gilkey, P

Branson, T. & Gilkey, P. The asymptotics of the Laplacian on a manifold with boundary. Comm. Partial Differential Equations . 15, 245-272 (1990)

work page 1990

[10] [10]

Scattered plane waves, spectral asymptotics and trace formulae in exterior problems

Buslaev, V. Scattered plane waves, spectral asymptotics and trace formulae in exterior problems. Dokl. Akad. Nauk SSSR . pp. 999-1002 (1971)

work page 1971

[11] [11]

Déterminant relatif et la fonction Xi

Carron, G. Déterminant relatif et la fonction Xi. Amer. J. Math. . 124, 307-352 (2002)

work page 2002

[12] [12]

Colin de Verdi\`ere, Y., Spectre conjoint d'op\'erateurs pseudo-diff\'erentiels qui commutent. I. Le cas non int\'egrable. Duke Math. J. 46 (1979), no. 1, 169-182

work page 1979

[13] [13]

DeWitt, B., Quantum Field theory on a curved spacetime, Phys. Reps. 19 (6) (1975), 295-357

work page 1975

[14] [14]

& Soffer, A

Donninger, R., Schlag, W. & Soffer, A. On pointwise decay of linear waves on a Schwarzschild black hole background. Comm. Math. Phys. . 309, 51-86 (2012)

work page 2012

[15] [15]

auser Classics. Birkh\

J.J. Duistermaat, Fourier integral operators . Reprint of the 1996 edition. Modern Birkh\"auser Classics. Birkh\"auser/Springer, New York, 2011

work page 1996

[16] [16]

24 (1975), 39-80

J.J.Duistermaat and V.Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Inv.Math. 24 (1975), 39-80

work page 1975

[17] [17]

and H\"ormander, L.,Fourier integral operators

Duistermaat, J.J. and H\"ormander, L.,Fourier integral operators. II. Acta Math. 128 (1972), no. 3-4, 183-269

work page 1972

[18] [18]

& Yau, S

Finster, F., Kamran, N., Smoller, J. & Yau, S. Decay of solutions of the wave equation in the Kerr geometry. Comm. Math. Phys. . 264, 465-503 (2006)

work page 2006

[19] [19]

Friedlander, F. G. The wave equation on a curved space-time. Cambridge Monographs on Mathematical Physics, No. 2. Cambridge University Press, Cambridge-New York-Melbourne, 1975

work page 1975

[20] [20]

Geodesics in stationary spacetimes and classical Lagrangian systems

Germinario, A. Geodesics in stationary spacetimes and classical Lagrangian systems. J. Differential Equations . 232, 253-276 (2007)

work page 2007

[21] [21]

& Werner, M

Gibbons, G., Herdeiro, C., Warnick, C. & Werner, M. Stationary metrics and optical Zermelo-Randers-Finsler geometry. Phys. Rev. D . 79, 044022, 21 (2009)

work page 2009

[22] [22]

and Uribe A.,

Guillemin V. and Uribe A.,

work page

[23] [23]

With appendices by V

G\"unther P., Huygens' principle and hyperbolic equations. With appendices by V. Wunsch. Perspectives in Mathematics, 5. Academic Press, Inc., Boston, MA, 1988

work page 1988

[24] [24]

C., Periodic Orbits and Classical Quantization Conditions, J

Gutzwiller M. C., Periodic Orbits and Classical Quantization Conditions, J. Math. Phys. 12 (1971), 343–358]

work page 1971

[25] [25]

, Lectures on Cauchy's problem in linear partial differential equations

Hadamard, J. , Lectures on Cauchy's problem in linear partial differential equations . Dover Publications, New York, 1953

work page 1953

[26] [26]

Acta Math

Hadamard, J., Th\'eorie des Equations aux d\'eriv\'ees partielles lin\'eaires hyperboliques et du probl\`eme de Cauchy. Acta Math. 31 (1908), no. 1, 333--380

work page 1908

[27] [27]

& Vasy, A

Häfner, D., Hintz, P. & Vasy, A. Linear stability of slowly rotating Kerr black holes. Invent. Math. . 223, 1227-1406 (2021)

work page 2021

[28] [28]

A sharp version of Price's law for wave decay on asymptotically flat spacetimes

Hintz, P. A sharp version of Price's law for wave decay on asymptotically flat spacetimes. Comm. Math. Phys. . 389, 491-542 (2022)

work page 2022

[29] [29]

H\"ormander, L., Theory of Linear Partial Differential Operators I-IV , Springer-Verlag, New York (1985)

work page 1985

[30] [30]

& Kato, T

Jensen, A. & Kato, T. Asymptotic behavior of the scattering phase for exterior domains. Comm. Partial Differential Equations . 3, 1165-1195 (1978)

work page 1978

[31] [31]

Off diagonal short time asymptotics for fundamental solutions of diffusion equations

Kannai, Y. Off diagonal short time asymptotics for fundamental solutions of diffusion equations. Comm. Partial Differential Equations 2 (1977), no. 8, 781-830

work page 1977

[32] [32]

& Ralston, J

Majda, A. & Ralston, J. An analogue of Weyl's theorem for unbounded domains. I, II, III Duke Math. J. . 45, no. 1, 183-196, no.3, 513–536, 46, no. 4, 725–731 (1978, 1979)

work page 1978

[33] [33]

Forward scattering by a convex obstacle

Melrose, R. Forward scattering by a convex obstacle. Comm. Pure Appl. Math. . 33, 461-499 (1980)

work page 1980

[34] [34]

Riesz, L'int\'egrale de Riemann-Liouville et le probleme de Cauchy

M. Riesz, L'int\'egrale de Riemann-Liouville et le probleme de Cauchy. Acta Math. 81 (1949). 1--223

work page 1949

[35] [35]

Riesz, A geometric solution of the wave equation in space-time of even dimension

M. Riesz, A geometric solution of the wave equation in space-time of even dimension. Comm. Pure Appl. Math. 13 (1960), 329--351

work page 1960

[36] [36]

Bonn (2023)

Ronge, L., Extracting Hadamard Coefficients from Green's Operators , (arXiv:2303.09976), PhD-thesis, U. Bonn (2023)

work page arXiv 2023

[37] [37]

The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation

Schoen, R. The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Comm. Pure Appl. Math. . 41 (1988), 317-392

work page 1988

[38] [38]

Strohmaier and S

A. Strohmaier and S. Zelditch, Gutzwiller trace formula for stationary spacetimes. (arXiv:1808.08425). Advances in Mathematics 376 (2019), 107434

work page arXiv 2019

[39] [39]

Strohmaier and S

A. Strohmaier and S. Zelditch, Spectral asymptotics on stationary space-times. Reviews in Mathematical Physics, 2060007, 2020

work page 2020

[40] [40]

Local decay of waves on asymptotically flat stationary space-times

Tataru, D. Local decay of waves on asymptotically flat stationary space-times. Amer. J. Math. . 135, 361-401 (2013)

work page 2013

[41] [41]

On the Fourier transforms of retarded Lorentz-invariant functions

Trione, S. On the Fourier transforms of retarded Lorentz-invariant functions. J. Math. Anal. Appl. . 84, 73-112 (1981)

work page 1981

[42] [42]

R. M. Wald, General Relativity. Chicago: University of Chicago Press (1984)

work page 1984

[43] [43]

R. M. Wald, Quantum field theory in curved spacetime and black hole thermodynamics . Chicago Lectures in Physics. University of Chicago Press, Chicago, IL, 1994

work page 1994

[44] [44]

Eigenfunctions of the Laplacian on a Riemannian manifold

Zelditch, S. Eigenfunctions of the Laplacian on a Riemannian manifold. (Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI,2017)

work page 2017