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Linear Response to Black Hole Deformations

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Stefan Hohenegger

General Relativity and Quantum Cosmology · Submitted 13 Oct 2026 · 73 references

Pith's verdict: Useful framework paper: universal linear-response coefficients for deformed Schwarzschild observables, with one load-bearing global-validity assumption that is asserted rather than proved.

T0 · AI review

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arXiv ID2610.18521 (version 1)
Pith IDKMX7BSK6, Pith's permanent record of this paper and its review
CitationStefan Hohenegger. "Linear Response to Black Hole Deformations." arXiv:2610.18521 (2026). https://pith.science/paper/KMX7BSK6
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The paper

The paper claims

The paper claims every observable outside a slightly deformed static black hole reduces to its Schwarzschild value plus universal coefficients times derivatives and nested integrals of the Einstein-tensor eigenvalue.

The reading

This paper proposes a linear-response treatment of black hole deformations: the leading corrections to any observable outside a static, spherically symmetric black hole horizon are written as a universal functional of a single function $\mathcal{E}$, an eigenvalue of the Einstein tensor that vanishes for Schwarzschild and is interpreted as an external energy-momentum source. The central claim is that every such observable takes the form $O[\mathcal{E}](t) \sim O[0,0](t) + u_C(t)\,C + \sum_n u_n(t)\,I_n[\mathcal{E}](t)$, where the coefficients $u_n$ are fixed by the Schwarzschild geometry alone and the $I_n$ are derivatives, the function itself, or nested integrals of $\mathcal{E}$. If correct, measurements of the black hole shadow, photon-sphere quantities, the ISCO, the Hawking temperature, and scalar quasi-normal mode frequencies become invertible constraints on $\mathcal{E}$, giving a direct and model-uniform bridge between observations and effective physics beyond general relativity. The formalism is illustrated on three known deformed geometries and reproduces their exact numerical observables to within a few percent for small deformation parameters.

What carries the argument

The load-bearing object is the linear-response decomposition (2.15) together with the momenta $I_n[\mathcal{E}]$ of (2.16), which are derivatives, the function itself, or nested integrals of the deformation function. The decomposition rests on two functional identities, (2.11) and (2.12), imported from the author's earlier series-based effective-description paper [36], which express the metric function and the radial coordinate as linear functionals of $\mathcal{E}$; the paper maintains that these relations hold globally outside the horizon whenever $\mathcal{E}$ and all its momenta remain small. The choice of $t$ as the parametrisation is secondary: a change of coordinates merely shifts the coefficients $u_n$ in a way computable entirely within the Schwarzschild geometry.

Extended reading notes

Core claim

The discovery this paper argues for is that the response of a deformed black hole factorises: the observable-specific part is universal, and the model-dependent part is carried entirely by $\mathcal{E}$ and its momenta $I_n[\mathcal{E}]$, defined as ordinary derivatives for $n<0$, the function itself for $n=0$, and $n$-fold nested integrals weighted by $(3t+4)$ for $n>0$. Using as coordinate the normalised proper time $t$ of a radially infalling observer to reach the horizon, explicit coefficient functions are worked out for the surface gravity, photon-sphere position, shadow radius, circular-orbit frequency, Lyapunov exponent, ISCO position and angular momentum, and the scalar quasi-normal mode spectrum computed by third-order WKB. Because the coefficients depend only on undeformed Schwarzschild data, the same expression describes any small deformation in this class, and reversing the logic turns each observable into an integro-differential equation for $\mathcal{E}$. In the eikonal regime the quasi-normal mode spectrum becomes a quasi-continuous family of such equations probing a region just inside the photon sphere.

Load-bearing premise

The whole expansion stands or falls on the claim that the near-horizon linearised relations (2.11) and (2.12) remain valid as functional identities at every point outside the horizon whenever $\mathcal{E}$ and all its derivatives and nested integrals stay small; the paper states this on the strength of those smallness conditions and supports it numerically for three small-parameter models, but gives no convergence or analytic-continuation argument that would justify the global extension.

Editorial extensions

If this is right

  • Measured shadow radii, ISCO locations, and quasi-normal mode frequencies each constrain specific momenta $I_n[\mathcal{E}]$ evaluated at specific exterior locations, so a handful of observations yield approximate integro-differential equations for $\mathcal{E}$.
  • The coefficient functions $u_n(t)$ are model-independent, so any theory whose deformation produces the same $\mathcal{E}$ gives the same linearised observables, making comparisons among models and with data uniform.
  • The eikonal relation $\omega \sim \ell\,M\,\omega_c - (n+\tfrac12)|\lambda|$ is recovered, and the linear response extends it to large but finite $\ell$, where the spectrum traces a quasi-continuous family of points just inside the photon sphere.
  • Because a change of parametrisation shifts only the coefficients, observables computed in different coordinate systems agree to linear order once the change of variables is evaluated in the undeformed geometry.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A practical consequence the paper leaves implicit is a data-analysis strategy: fit the single function $\mathcal{E}(r)$ to combined shadow, ringdown, and ISCO observations rather than fitting separate per-model parameters, since the linear structure would make this a linear inverse problem.
  • The coefficient functions $u_n$ play the role of a response kernel and the nested-integral momenta resemble the convolution structure of statistical linear response theory; making that analogy precise could yield a model-independent criterion for when the linear truncation is trustworthy.
  • A proof that every sufficiently regular small $\mathcal{E}$ outside the horizon yields a metric satisfying (2.11) and (2.12) to first order would turn the numerical evidence into an existence theorem; the paper supplies the evidence but not that argument.
  • Extending the formalism to a second Einstein-tensor eigenvalue or to rotating geometries, which the paper lists as future work, would test whether one scalar function remains a sufficient descriptor of deformations.
Original abstract

Modifications of gravity beyond (vacuum) General Relativity have a subtle impact on the physics around black holes. Describing these effects across different observables is, however, a priori strongly model dependent. While effective descriptions of the geometry allow to efficiently capture deviations from benchmark predictions, they sometimes lack a clear passage towards (effective) new physics. Inspired by linear response theory, in this work I therefore propose a formalism that allows to calculate leading corrections of observables relative to the Schwarzschild black hole anywhere in space-time (outside of the horizon): the order of corrections is fixed through an eigenvalue $\mathcal{E}$ of the Einstein tensor, which is identified with an external energy momentum tensor that generates the deformation of the geometry. In this ways, $\mathcal{E}$ can be linked to effective physics models. Focusing on spherically symmetric and static black holes, I illustrate the approach at the hand of a number of observables in the vicinity of the horizon, such as characteristic quantities of particles moving around the black hole (e.g. the black hole shadow or the radius of the innermost stable circular orbit) or the spectrum of scalar quasi-normal modes. Reversing the logic, I argue that the knowledge of such observables provides non-trivial constraints on $\mathcal{E}$, which in turn grant insights into the effective physics in the vicinity of the black hole.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-14: 1 request, 5,771 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.42 · human participants 0.01 · controlled comparison 0.41 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.57 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Modifications of gravity beyond (vacuum) General Relativity have a subtle impact on the physics around black holes. Describing these effects across different observables is, however, a priori strongly model dependent. While effective descriptions of the geometry allow to efficiently capture deviations from benchmark predictions, they sometimes lack a clear passage towards (effective) new physics. Inspired by linear response theory, in this work I therefore propose a formalism that allows to calculate leading corrections of observables relative to the Schwarzschild black hole anywhere in space-time (outside of the horizon): the order of corrections is fixed through an eigenvalue $\mathcal{E}$ of the Einstein tensor, which is identified with an external energy momentum tensor that generates the deformation of the geometry. In this ways, $\mathcal{E}$ can be linked to effective physics models. Focusing on spherically symmetric and static black holes, I illustrate the approach at the hand of a number of observables in the vicinity of the horizon, such as characteristic quantities of particles moving around the black hole (e.g. the black hole shadow or the radius of the innermost stable circular orbit) or the spectrum of scalar quasi-normal modes. Reversing the logic, I argue that the knowledge of such observables provides non-trivial constraints on $\mathcal{E}$, which in turn grant insights into the effective physics in the vicinity of the black hole.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Modifications of gravity beyond (vacuum) General Relativity have a subtle impact on the physics around black holes. background 1.00 0.05 0.03 asserted 0.97
s1 Describing these effects across different observables is, however, a priori strongly model dependent. gap 0.50 0.05 0.02 asserted 0.97
s2 While effective descriptions of the geometry allow to efficiently capture deviations from benchmark predictions, they sometimes lack a clear passage towards (effective) new physics. gap 0.57 0.05 0.03 hedged possibility 0.80
s3 Inspired by linear response theory, in this work I therefore propose a formalism that allows to calculate leading corrections of observables relative to the Schwarzschild black hole anywhere in space-time (outside of the horizon): the order of corrections is fixed through an eigenvalue $\mathcal{E}$ of the Einstein tensor, which is identified with an external energy momentum tensor that generates the deformation of the geometry. contribution 0.69 0.90 0.39 asserted 0.89
s4 In this ways, $\mathcal{E}$ can be linked to effective physics models. interpretation 0.80 0.44 0.09 hedged possibility 0.70
s5 Focusing on spherically symmetric and static black holes, I illustrate the approach at the hand of a number of observables in the vicinity of the horizon, such as characteristic quantities of particles moving around the black hole (e.g. the black hole shadow or the radius of the innermost stable circular orbit) or the spectrum of scalar quasi-normal modes. contribution 0.57 0.91 0.08 asserted 1.00
s6 Reversing the logic, I argue that the knowledge of such observables provides non-trivial constraints on $\mathcal{E}$, which in turn grant insights into the effective physics in the vicinity of the black hole. interpretation 0.96 0.66 0.54 asserted 0.70

What kind of paper the classifier reports this is: method or tool 0.98 · evidence from proof or derivation 0.94.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Schematic overview over important space-time positions at which observables dis
Figure 2. Schematic drawing of a (mas￾sive) particle following a radial, infalling geodesic (red line) in the black hole geom￾etry. The dashed blue line represents the event horizon. parametrisation of (a patch of) M. A priori, any such parametrisation is viable, however, in order to describe physically meaningful invariants (i.e. observables) within our framework, we require it to be observer independent. Moreover, we prefer a choice of coordinates that can be introduced in a simple invariant fashion, that is related to (phys￾ically) measurable quantities. Following [36] we have chosen as su
Figure 3. Schematic overview of two parametrisations: the energy density is (up to symmetries
Figure 4. Energy density E as a func￾tion of t for the black holes in table 1. The deformation parameters γ have been chosen such that E(t = 0) agrees for all three black holes, i.e. concretely γ = (0.01, 0.015, 0.029) for the three models. shall consider a number examples for E. We have summarised these functions in
Figure 5. Left panel: the radial coordinate r0 = r(t)
Figure 6. Left panel: metric function f0 = f(t)
Figure 7. Left panel: Ricci scalar for three examples introduced in Section 2.5 (for the same
Figure 8. Left panel: Kretschmann scalar M4 K0 = M4 K(t)
Figure 9. Hawking Temperature TH as a function of γ for the three models in￾troduced in Section 2.5. We have nu￾merically calculated the exact expression (dashed line), the linear response (2.17) with the exact momenta In[E] calculated numerically (heavy solid line) and the lin￾ear response (2.17) with the momenta In[E] as in Section B (thin solid line). Hawking temperature, which is defined in terms of the surface gravity κ = 1 2M df dr |r=rH as follows M TH = Mκ 2π ∼ 1 8π + u (TH) C C + u (TH) 0 I0[E](0), (3.7) with u (TH) C = − 1 16π , and u (TH) 0 = − 1 2π . (3
Figure 10. Position of the photon sphere tps (left panel) and Lyapunov exponent λ (right panel) as a function of γ for the three models introduced in Section 2.5. We have numerically calculated the exact expression (dashed line), the linear response (2.17) with the exact momenta In[E] calculated numerically (heavy solid line) and the linear response (2.17) with the momenta In[E] as in Section B (thin solid line). The Lyapunov exponent is schematically plotted in the right panel of
Figure 11. Position of the ISCO tISCO (3.21) left panel) and (square of the) critical value of the angular momentum L 2 ISCO (3.22) (right panel) as a function of γ for the three models introduced in Section 2.5. We have numerically calculated the exact expression (dashed line), the linear response (2.17) with the exact momenta In[E] calculated numerically (heavy solid line) and the linear response (2.17) with the momenta In[E] as in Section B (thin solid line). 4.1 Potential We start with the Klein-Gordon equation for a massless scalar field Φ in the background of the black hole metric (2.3)…
Figure 12. Plot of the potential Vϕ for the Hayward (left panel), the Simpson-Visser I (middle panel) and Simpson-Visser II (right panel) black hole, for different values of ℓ: the solid lines represent the analytic (linear) approximations (4.5) (using the momenta given in (B.1), (B.2) and (B.3) respectively), while the dashed lines represent numerical computations for the exact potential in (4.3). The plots use the same numerical values of γ as in
Figure 13. Real (left panel) and imaginary (right panel) part of the scalar QNM for the (un
Figure 14. Real (left panel) and imaginary part (right panel) of the linear correction
Figure 15. tmax,0 as a function of ℓ (for convenience, we are using a logarithmic scale). The points indicate quantised values of ℓ (with the value of ℓ indicated), while the solid blue line shows (4.10) for continuous ℓ. The dashed black line indicates the asymptotic value tps,0 = √ 6 − 4 3 , which is the classical position of the photon sphere. Notice that for ℓ ≫ 1, the points become dense, thus justifying to consider (4.10) as a continuous function. Im(u (ω) 0 (t)) = 5(n + 1 2 ) (47n(n + 1) − 122) 324√ 2 (tps,0 − t) + o(tps,0 − t), Im(u (ω) −1 (t)) = − 9(n + 1 2 ) 4 √ 2
Figure 16. The momenta In=0,1,2[E](t) for Model 1 (left panel), Model 2 (middle panel) and Model 3 (right panel) of

Figures reproduced from arXiv:2610.18521 by the authors.

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