Telling Tails In The Presence Of A Cosmological Constant

Chris M. Chambers (Montana State); Pablo Laguna (Penn State); Patrick R. Brady (CalTech); Penn State); William Krivan (Tubingen

REVIEW

Telling Tails In The Presence Of A Cosmological Constant

Patrick R. Brady (Caltech), Chris M. Chambers (Montana State), William Krivan (Tubingen, Penn State), Pablo Laguna (Penn State)

General Relativity and Quantum Cosmology · Submitted 24 Nov 1996 · 25 references

Pith's verdict: The ℓ>0 exponential-tail result is solid and properly cross-checked; the ℓ=0 constant φ₀ is the part with an unresolved tension between linearity and the claimed (M,q,Λ) dependence.

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CitationChris M. Chambers (Montana State), Pablo Laguna (Penn State), Patrick R. Brady (CalTech), Penn State), William Krivan (Tubingen. "Telling Tails In The Presence Of A Cosmological Constant." arXiv:gr-qc/9611056 (1996). https://pith.science/paper/ZNYSCLBB

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Editor's note

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Short version: the ℓ>0 result holds up, and the ℓ=0 result is the interesting one, but the paper's own framing of φ₀ has a hole in it that a referee should press on. Also, the concluding paragraph contains a sentence that contradicts the rest of the paper — more on that below. What is new: before this, the exponential-tail claim for de Sitter black holes was an argument, not a computation. Brady et al. actually evolve the test field on both Schwarzschild–de Sitter and Reissner–Nordström–de Sitter with two independent codes (null and Cauchy) and report agreement, then back the ℓ=0 piece with a nonlinear Einstein-scalar evolution validated against the exact solution of [25] and against the known Λ=0 power-law tails of [8]. The decay rates are checked against surface gravities computed independently from Eq. (6): 2% for ℓ=1,2, 10% for ℓ=0. That is a genuine cross-check, not a self-referential fit. On the reader's worry that backreaction could alter the ℓ>0 rate: it does not land. The nonlinear source terms in Eqs. (16)–(19) are O(φ²); for ℓ>0 the field decays, so the corrections are exponentially suppressed and cannot shift the asymptotic rate. Restricting §III to spherical symmetry is the right call, not a shortcut — ℓ=0 is the only mode where backreaction can matter at late times, and the authors say so. The soft spot is ℓ=0. The paper states that the amplitude A is irrelevant because Eq. (8) is linear, then asserts that φ₀ is a function of (M,q,Λ). Those cannot both be right: for a linear equation φ₀ scales with A, so the invariant statement is about φ₀/A. Fig. 6's Λ^0.6 is a fit at a single pulse (v1=50, σ=3), reported with a correlation coefficient and no error bars. And no run tests data with zero overlap with the constant mode, so “a generic perturbation leaves a residual constant” may be kinematical rather than a dynamical tail. Separately, the concluding remarks say “for all modes except the ℓ=0 mode, it approaches a constant at late times,” which flatly contradicts Eq. (1) and everything before it. A typo, but it sits in the summary paragraph. Who should read it: anyone using these tails as initial data for the Reissner–Nordström–de Sitter Cauchy horizon problem. Worth a referee.

Referee report

Significance

If correct, this is a useful and citable result: it is, as far as I am aware, the first detailed numerical study of radiative tails on de Sitter black-hole backgrounds, and it supplies exactly the late-time horizon data that the Cauchy-horizon literature cited in §IV needs. Several features deserve explicit credit. The use of two independent linear codes that agree completely is a real robustness check, not a formality. The nonlinear code is validated against the exact solution of [25] and reproduces the Λ=0 power-law tails of [8] before being applied to the new case. The fitted decay exponent is compared against κ₁ computed independently from Eq. (6), which is a genuine cross-check rather than a self-referential step. The paper is also candid about the scope of the nonlinear check, stating that spherical symmetry restricts it to ℓ=0. The main claims are modest in scope and, where they are numerical, adequately supported.

Major objections (3)

[§II B, p. 4, and Fig. 6] There is an internal tension in the treatment of the ℓ=0 asymptote. The paper states that 'the amplitude A is irrelevant since Eq. (8) is linear', and then asserts that 'the final field value, φ₀, is a function of the black hole parameters (M,q,Λ)'. For a linear homogeneous equation the late-time constant must scale linearly with the initial amplitude, so the background quantity is φ₀/A, not φ₀. As written, the statement is incorrect, and it matters because the Λ^{0.6} scaling of Fig. 6 is presented as a result. The authors should (i) report φ₀ normalized by the initial amplitude, (ii) state explicitly that the scaling is a fit at one fixed pulse (v₁=50, σ=3) and show at least one run with a different amplitude and width to confirm that φ₀/A is independent of the initial data, and (iii) give error bars or a fit residual for the Λ^{0.6} law, since a linear correlation coefficient of 0.98 on a log-log plot with no quoted uncertainties is weak support for a specific exponent. Relatedly, §II B states that 'the results are qualitatively similar for all Λ>0', yet φ₀ varies by orders of magnitude over the Λ range of Fig. 6; the sense in which the results are 'qualitatively similar' should be made precise.
[Eq. (1), Figs. 7–8] The decay law φ ∼ e^{−ℓκ₁t} is fitted, not derived, and the paper offers no argument for why the rate is exactly ℓκ₁. This is the quantitative content of the headline result, so at least a heuristic derivation should be supplied. The effective potential falls off as e^{−2κ₁r*} near the cosmological horizon and as e^{−2κ₂r*} near the black-hole horizon, and the figures show the same rate κ₁ at both horizons and at timelike infinity. It is not obvious to a reader why the black-hole-horizon tail is governed by the cosmological surface gravity κ₁ rather than by κ₂, and a short asymptotic or WKB argument explaining this asymmetry would convert a descriptive fit into a predictive statement. The paper should also state the range of ℓ tested and whether the ℓκ₁ law was verified for ℓ=3 or higher.
[§IV and Eq. (11)] The physical status of the constant mode needs to be stated explicitly, because §IV is the paper's stated motivation. Since the field is massless and minimally coupled, only gradients carry stress-energy, and the manuscript itself notes that 'there is no stress-energy associated with a constant field'. The reader is therefore left unsure whether the ℓ=0 asymptote is a physically meaningful late-time statement or a statement about the zero-frequency mode of the field. The authors should say plainly that the quantity entering the Cauchy-horizon initial data is the flux, which decays as e^{−2κ₁u} (Eq. 2 and Fig. 10), and that the residual constant is dynamically harmless. Without that sentence, the ℓ=0 result reads as a stronger claim about late-time dynamics than the analysis supports.

Minor objections (8)

[Eq. (1) and Figs. 4–8] Define t precisely. At the horizons the natural affine parameters are u and v, and the statement that the same exponential rate holds 'at the cosmological horizon, the black-hole event horizon and future timelike infinity' would be clearer if the paper stated which coordinate time is plotted in each figure and confirmed that the rate is the same when expressed in u at r₂.
[§II B] The remark that 'for sufficiently small values of Λ, a regime of power law decay followed the quasi-normal ringing' is interesting and is not shown anywhere. A sentence quantifying the Λ below which this intermediate regime is visible, or a figure, would strengthen the connection with the asymptotically flat results of [3,4].
[Fig. 3] The ℓ=0 potential well is mentioned but not characterized. Stating its depth and location, and whether it is responsible for the constant mode, would help the reader connect the potential shape to the result.
[Figs. 4, 5, 7, 8] The two constant-radius curves are not labelled with their r values. Please add the radii to the caption or legend so the reader can judge how the approach to the asymptotic value depends on position.
[§III] The initial data used for Figs. 9 and 10 (amplitude, width σ, centre r₀) are not given. Please state them, as was done for the linear runs in Eq. (10).
[Eq. (5)] In the q=0 branch the term involving κ₃ and r₃ is retained even though r₃ is negative and non-physical. A sentence explaining why the unphysical root is kept in the tortoise coordinate would prevent confusion.
[Abstract and §III] The multipole index is written as 'l' in the abstract and as 'ℓ' in the body. Please use one symbol throughout.
[Throughout] There are several typographical errors to correct: 'dependance' for 'dependence' in §II B, 'the the' in §II B, 'who's' for 'whose' in the acknowledgments, and 'prop-agating' in the abstract. The author list and affiliations also contain broken spacing and accents ('I nstitute', 'f¨ ur', 'Biˇ c´ ak').

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The paper

The paper claims

On black-hole backgrounds with a positive cosmological constant, a massless scalar field's late-time tail decays exponentially rather than as a power law, and the monopole settles to a nonzero constant instead of vanishing.

The reading

This paper asks what happens to the radiation that lingers after a black hole settles down when the surrounding universe has a positive cosmological constant. Evolving a massless scalar field on fixed Schwarzschild–de Sitter and Reissner–Nordström–de Sitter backgrounds, the authors find that the late-time tail is not the power law familiar from asymptotically flat space but an exponential decay, $|varphi| sim e^{-ell kappa_1 t}$ for $ell > 0$, with the rate set by the surface gravity of the cosmological horizon. The monopole is the exception: instead of decaying to zero it settles to a constant value that is the same at the cosmological horizon, at the black hole horizon, and at future timelike infinity. A separate nonlinear integration of the coupled Einstein–scalar equations, restricted to spherical symmetry and therefore to the $ell=0$ mode, reproduces that constant and shows the field gradient falling as $e^{-2kappa_1 u}$. The motivation is Cauchy horizon stability: if perturbations at the event horizon die exponentially rather than as a slow power law, the case that inner horizons of charged black holes in de Sitter space remain stable gets stronger.

What carries the argument

The argument turns on the shape of the effective potential in the tortoise coordinate $r_*$. For a metric function $f(r)$ with a cosmological horizon at $r_1$ and a black hole horizon at $r_2$, the potential $V_ell(r)$ is localized near $r_* = 0$ and falls off exponentially as $r_* to pm infty$, so the wave equation $Psi_{,uv} = -tfrac{1}{4}V_ell(r)Psi$ has exponential rather than power-law barriers on both sides. That exponential suppression is what converts Price's power-law tail into an exponential one. The $ell=0$ case adds a potential well after the barrier, and it is this feature that lets a constant mode be transmitted to both horizons. The supporting apparatus is two independent numerical evolutions of the linear equation on a null grid and a Cauchy code, plus a nonlinear characteristic integration of the coupled Einstein–scalar equations in the Christodoulou variables $h$ and $bar{h}$.

Extended reading notes

Core claim

The central claim is that the radiative tail of a massless, minimally coupled scalar field on a black hole–de Sitter background is qualitatively different from the asymptotically flat case. For every multipole with $ell > 0$ the field falls off as $e^{-ell kappa_1 t}$ at late times, where $kappa_1$ is the surface gravity of the cosmological horizon, rather than as Price's inverse power law in $t$. For $ell = 0$ the field does not decay at all: a generic initial pulse leaves behind a residual constant, $varphi|_{ell=0} simeq varphi_0 + varphi_1(r) e^{-2kappa_1 t}$, and the same value is reached on the cosmological horizon, on the black hole event horizon, and on surfaces of constant radius approaching timelike infinity. The authors show this with two independent linear codes, a null evolution scheme and a Cauchy scheme, and then check the $ell=0$ case against a nonlinear evolution of a self-gravitating scalar field in spherical symmetry, which they report agrees with the linear result. They also fit the dependence of the residual constant on the cosmological constant, finding $varphi_0 sim Lambda^{0.6}$ for $q=0$, $M=1$.

Load-bearing premise

The exponential-tail result for $ell > 0$ rests entirely on linear test-field evolution on a fixed background, since the only nonlinear check is spherically symmetric and therefore sees only the $ell=0$ mode, a restriction the authors state explicitly.

Editorial extensions

If this is right

Late-time tails on black-hole–de Sitter backgrounds are exponential, not power law, with a decay rate fixed by the cosmological horizon's surface gravity and the multipole order, $e^{-ell kappa_1 t}$.
The $ell=0$ mode leaves a nonzero constant field at the cosmological horizon, the black hole event horizon, and future timelike infinity; the authors note there is no stress-energy associated with a constant field.
Although the field itself does not decay for $ell=0$, its gradient does, falling as $e^{-2kappa_1 u}$ with the fitted rate agreeing with $kappa_1$ to within about 12 percent, so the field's stress-energy still dies off exponentially.
The residual constant scales with the cosmological constant as $varphi_0 sim Lambda^{0.6}$ for $q=0$, $M=1$, a least-squares fit with correlation coefficient $0.98$.
Because horizon fluxes decay exponentially rather than as a power law, linear perturbation studies suggesting that Cauchy horizons inside charged black holes in de Sitter space are stable gain support, which bears on the letter of cosmic censorship.

Reading between the lines

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

If the flux falling on the inner horizon decays exponentially, mass inflation inside a Reissner–Nordström–de Sitter hole should saturate at a finite blueshift instead of running away; a nonlinear integration of the interior with these tails as initial data would turn that expectation into a test.
The residual constant $varphi_0$ is a new datum left over from the initial pulse, so the asymptotic state retains a memory of how the field started. The paper's runs all use a single Gaussian, so whether $varphi_0$ is truly independent of pulse shape is an open check rather than an established fact.
The $Lambda^{0.6}$ scaling is a fitted exponent with no derivation attached; a matched-asymptotics calculation combining the potential well near the black hole horizon with the exponential barriers should produce an exponent, and a value other than $0.6$ would put the fit under pressure.
Because the decay rate is $ell kappa_1$, a measured late-time slope in a de Sitter universe would read off the cosmological horizon's surface gravity directly, without needing the black hole mass separately.

Original abstract

We study the evolution of massless scalar waves propagating on spherically symmetric spacetimes with a non-zero cosmological constant. Considering test fields on both Schwarzschild-de Sitter and Reissner-Nordstrom-de Sitter backgrounds, we demonstrate the existence of exponentially decaying tails at late times. Interestingly the l=0 mode asymptotes to a non-zero value, contrasting the asymptotically flat situation. We also compare these results, for l=0, with a numerical integration of the Einstein-Scalar field equations, finding good agreement between the two. Finally, the significance of these results to the study of the Cauchy horizon stability in black hole-de Sitter spacetimes is discussed.

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.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.14 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.90 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We study the evolution of massless scalar waves propagating on spherically symmetric spacetimes with a non-zero cosmological constant. Considering test fields on both Schwarzschild-de Sitter and Reissner-Nordstrom-de Sitter backgrounds, we demonstrate the existence of exponentially decaying tails at late times. Interestingly the l=0 mode asymptotes to a non-zero value, contrasting the asymptotically flat situation. We also compare these results, for l=0, with a numerical integration of the Einstein-Scalar field equations, finding good agreement between the two. Finally, the significance of these results to the study of the Cauchy horizon stability in black hole-de Sitter spacetimes is discussed.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the evolution of massless scalar waves propagating on spherically symmetric spacetimes with a non-zero cosmological constant.	contribution 0.84	0.87	0.04	asserted 0.78
s1	Considering test fields on both Schwarzschild-de Sitter and Reissner-Nordstrom-de Sitter backgrounds, we demonstrate the existence of exponentially decaying tails at late times.	result 0.90	0.95	0.75	asserted 1.00
s2	Interestingly the l=0 mode asymptotes to a non-zero value, contrasting the asymptotically flat situation.	result 0.98	0.86	0.22	asserted 1.00
s3	We also compare these results, for l=0, with a numerical integration of the Einstein-Scalar field equations, finding good agreement between the two.	method 0.54	0.96	0.07	asserted 1.00
s4	Finally, the significance of these results to the study of the Cauchy horizon stability in black hole-de Sitter spacetimes is discussed.	interpretation 0.97	0.61	0.11	unclear 0.59

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.65.

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sources read

arXiv listing abstract v1

not examined

full text
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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.** FIG. 1. A conformal diagram representing the

**Figure 2.** FIG. 2. A conformal diagram showing the Reiss

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References

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2 later papers by other authors are listed below: 1 built on it and 1 compared its own results with it.

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Late time tails and nonlinear memories in asymptotically de Sitter spacetimes (2021) · 0.99 · paraphrased: the licence does not allow quoting

Compares with it 1 of 6 papers with this label

Reported that its own observed oscillatory profile contrasts with what happens for Λ=0 and with the expectation created by this paper's study of sufficiently sub-extremal black holes with 0<M^2Λ≪1.

Strong cosmic censorship: The nonlinear story (2018) · 0.92 · paraphrased: the licence does not allow quoting

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41 arXiv papers cite this in 66 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 44 sentences without a confident label

Contests it 1 paper

On the contrary to [ref] we always find exponential tails (either oscillatory or not) at asymptotic times, which are consistent with a general observation made in [ref] .

Two Regimes of Asymptotic Fall-off of a Massive Scalar Field in the Schwarzschild-de Sitter Spacetime · Remarks on two recent works · contest 0.82

Builds on it 1 paper

This complements and extends the results of BCKL [ref] and BCLP [ref] .

Late time tails and nonlinear memories in asymptotically de Sitter spacetimes · Introduction and Setup · 0.99

Uses its method or tool 2 papers

The direct integration schemes used in [ref] can still be successfully employed.

Wormholes in de Sitter branes · Higher multipole modes ($l>0$) · 1.00
First, we use the finite difference method proposed in Ref. [ref] to reexamine the neutral massless scalar field perturbations in RN-dS spacetime, which was first reported in Ref. [ref] and later confirmed in Ref. [ref] .

Stability of Reissner-Nordstr\"om black hole in de Sitter background under charged scalar perturbation · Neutral scalar perturbation · 0.86

Compares with it 6 papers

More surprisingly, we also clearly observe an oscillatory profile; this might seem odd at first, since it is in contrast with what happens for $Λ=0$ and with the expectation created by the study of sufficiently sub-extremal BHs with $0<M^2Λ≪1$ [ref] .

Strong cosmic censorship: The nonlinear story · Results · 0.92
In addition, we may compare equations [ref] and [ref] to the results obtained by BCLK [ref] and BCLP [ref] , where they dealt with a spherically symmetric black hole in de Sitter spacetime.

Late time tails and nonlinear memories in asymptotically de Sitter spacetimes · Pseudo-Frequency Space Method · 0.97
These results show very good agreement with those observed in Refs. [ref] , which gives us confidence in the numerical method we employed.

Stability of Reissner-Nordstr\"om black hole in de Sitter background under charged scalar perturbation · Neutral scalar perturbation · 0.93

Cites it as background 8 papers

It was shown long ago that in asymptotically de Sitter spacetimes, the asymptotic decay law for massless scalar and gravitational fields is purely exponential (non-oscillatory) [ref] .

Two Regimes of Asymptotic Fall-off of a Massive Scalar Field in the Schwarzschild-de Sitter Spacetime · Introduction · 0.84
The corresponding falloff properties of radiative fields in cosmological backgrounds is less well understood, although there are a number of numerical studies, such as Brady, Chambers, Krivan and Laguna (BCKL) [ref] and Brady, Chambers, Laarakkers and Poisson (BCLP) [ref] on exact Schwarzschild-de Sitter and Reissner-Nordström-de Sitter backgrounds.

Late time tails and nonlinear memories in asymptotically de Sitter spacetimes · Introduction and Setup · 0.86
The effects of classical perturbations on the stability of inner horizon has been studied extensively [ref] .

Quantum aspects of charged black holes in de-Sitter space · Introduction · 1.00

Discussion

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