REVIEW
Late Time Tail of Wave Propagation on Curved Spacetime
General Relativity and Quantum Cosmology · Submitted 30 Oct 1994 · 12 references
Pith's verdict: The general tail formula and the exceptional-case analysis are real and useful; the weak point is that the Born-approximation justification is plausible rather than proved, and the letter format hides the l≠0 coefficient.
T0 · AI review
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Referee report
The paper
The paper claims
The late-time tail of a wave on curved spacetime is not always a power law: for a potential that falls off as $x^{-2}$-plus-tail, the decay is $t^{-(2l+2)}$ generically but picks up a $t^{-(2l+3)}$-style exception when the falloff exponent,
The reading
What carries the argument
The mechanism is a branch cut in the frequency-domain Green's function. Writing the retarded Green's function as $G = f(ω,y)g(ω,x)/W(ω)$, with $f$ fixed by the boundary condition at the origin and $g$ by outgoing waves at large $x$, the contour of the inverse Fourier transform is pushed into the lower half-plane. Zeros of the Wronskian $W$ give quasinormal modes, which decay exponentially and do not control late times; the large-arc contribution vanishes after a time of order $x$; what remains is a cut on the negative imaginary $ω$ axis, extending to $ω=0$, generated by the slow falloff of $V(x)$ at large radius. A first Born approximation applied only to the potential tail, while treating the centrifugal barrier exactly with Hankel functions, evaluates the discontinuity across this cut and yields the coefficients $C(l,α)$ and $F(α) = 2(2x_0)^{α-2}Γ(2l+α)/Γ(α)$. The authors argue the Born step is exact at large $t$ because it is used only to compute an $x$-independent Wronskian. The heuristic picture behind it is simple: the wave runs out to a radius of order $t/2$, scatters off the weak far field, and returns, so the tail tracks $V(t/2)$ up to the centrifugal suppression.
Extended reading notes
Core claim
The paper establishes that for a wave equation $[∂_t^2 - ∂_x^2 + V(x)]φ = 0$ with a potential that behaves at large $x$ as a centrifugal barrier $l(l+1)/x^2$ plus a remainder $V(x) ∼ x^{-α}(log x)^β$ with $α > 2$ and $β = 0$ or $1$, the late-time tail is generically $φ ∼ t^{-(2l+α)}(log t)^β$. The time dependence comes only from the asymptotic form of the potential; the amplitude depends on local geometry through the functions $f(0,x)$ and $g_0$, and hence on whether a horizon exists. The exception is the load-bearing surprise: when $α$ is an odd integer less than $2l+3$, the coefficient $C(l,α) = ∏_{j=0}^{l-1}(α-2j-3)/(α+1+2j)$ vanishes, the leading tail disappears, and the next contribution from multiple scattering goes as $t^{-(2l+2α-2)}$ for $β=0$, or as a pure power $t^{-(2l+α)}$ with no logarithm for $β=1$. Since Schwarzschild has $α = 3$, it belongs to this exceptional set, which explains the known pure-power result. Numerical evolutions of Gaussian initial data agree with the analytic curves to within plotting resolution, and the authors note that varying the potential parameters continuously can change the late-time exponent discontinuously.
Load-bearing premise
The first Born approximation, applied only to the far tail of the potential while the centrifugal barrier is treated exactly, is assumed to capture the leading behaviour near the tip of the branch cut at $ω = 0$, with all higher-order scatterings contributing only to subleading terms.
Editorial extensions
If this is right
- For any static, spherically symmetric background, the late-time decay exponent can be predicted from the large-radius falloff of the effective potential alone, without solving the full wave equation.
- A logarithmic factor in the leading late-time tail is a genuine possibility, not an artefact, and should be looked for in backgrounds whose potentials carry a logarithmic tail.
- Because the amplitude depends on local geometry and the exponent does not, the presence or absence of a horizon changes how loud the tail is but not how fast it fades.
- The Schwarzschild pure-power result is reclassified as a cancellation rather than a generic feature, so extrapolating it to other geometries is unjustified.
- The discontinuous jump in the decay exponent as $α$ crosses an odd integer gives a sharp, checkable signature in numerical evolution.
Reading between the lines
- The cancellation condition $α$ an odd integer below $2l+3$ is a statement about a product of simple factors, so one could test it directly by scanning $α$ in a fixed-$l$ numerical evolution and watching for the exponent to switch without passing through intermediate values.
- Since the amplitude but not the exponent depends on local geometry, the tail could serve as a probe that separates near-horizon physics from far-field physics in a single observation: measure the exponent to learn the asymptotic falloff, measure the prefactor to learn about the interior.
- The paper restricts itself to $α > 2$ and defers $α ≤ 2$; that regime is where the falloff competes with the centrifugal term and where the Born-based argument is most likely to need repair, so the deferred case is the natural next test of the mechanism.
- The authors state that results for null infinity will be given elsewhere; the time-like-infinity results here do not automatically transfer, and a reader should treat the far-field asymptotics as an open companion question.
Original abstract
The late time behavior of waves propagating on a general curved spacetime is studied. The late time tail is not necessarily an inverse power of time. Our work extends, places in context, and provides understanding for the known results for the Schwarzschild spacetime. Analytic and numerical results are in excellent agreement.
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.
The abstract, sentence by sentence
The late time behavior of waves propagating on a general curved spacetime is studied. The late time tail is not necessarily an inverse power of time. Our work extends, places in context, and provides understanding for the known results for the Schwarzschild spacetime. Analytic and numerical results are in excellent agreement.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The late time behavior of waves propagating on a general curved spacetime is studied. | contribution 0.71 | 0.75 | 0.07 | asserted 0.83 |
| s1 | The late time tail is not necessarily an inverse power of time. | result 0.99 | 0.61 | 0.61 | hedged possibility 0.56 |
| s2 | Our work extends, places in context, and provides understanding for the known results for the Schwarzschild spacetime. | contribution 0.93 | 0.86 | 0.10 | asserted 0.99 |
| s3 | Analytic and numerical results are in excellent agreement. | result 0.94 | 0.86 | 0.08 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:gr-qc/9410044 by the authors.
References
How the literature uses it
38 arXiv papers cite this in 68 sentences
Papers citing it · 46 sentences without a confident label
Contests it 1 paper
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Following Ref. [ref] , the tail should be modified in this case (see Table [ref]), however our analysis suggests that this does not happen.
On the universality of late-time ringdown tail · Comparison with previous work · contest 0.81
Builds on it 2 papers
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For the late-time tail, we generalize the treatment in the seminal paper by Ching et al. [ref] to the cases where both the ingoing and outgoing waveforms possess singularities.
Late-time tail and echoes of Damour-Solodukhin wormholes · Introduction · 1.00
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One can extend the analysis to $δV = x^-ρlogx$ [ref] . ]
Hushing black holes: tails in dynamical spacetimes · Introduction · 0.84
Uses its method or tool 2 papers
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In what follows, we first outline the strategy to evaluate the asymptotical properties of late-time tail, closely following Refs. [ref] .
Late-time tail and echoes of Damour-Solodukhin wormholes · Late-time tail in Damour-Solodukhin type wormholes · 0.97
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The analytical method of [ref] shows that the Price's law obtained by a finite-distant observer is determined by the asymptotic behavior of the effective potential near infinity.
The (in)stability of quasinormal modes of Boulware-Deser-Wheeler black hole in the hyperboloidal framework · The quasinormal modes · 0.86
Compares with it 2 papers
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Consequently, both Ref. [ref] and our analysis consistently predict a \(t^-2l-3\) decay in this regime.
On the universality of late-time ringdown tail · Comparison with previous work · 0.91
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Notably, contrary to QNMs for the black holes in asymptotically flat spacetimes, where [ref] revealed the existence of power-law tails, those of AdS-Schwarzschild black holes do not exhibit such tails as shown by [ref] .
The Photon Sphere and the AdS/CFT Correspondence · Introduction · 0.83
Cites it as background 13 papers
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The small-$ν$ BCMs are known to give rise to a power-law tail decay at `late' times of an initial perturbation [ref] .
Spectroscopy of the Schwarzschild Black Hole at Arbitrary Frequencies · Introduction · 0.80
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Of direct interest for the work presented here, several authors have employed the Green's function approach in exploring the tail using a variety of techniques to calculate the Green's functions [ref] .
Scalar and vector tail radiation from the interior of the lightcone · Introduction · 0.83
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To accurately model the ringdown waveforms of BHs, significant efforts have been dedicated to refining our understanding of the QNM spectra and Price tails [ref] .
Nonlinear Tails of Gravitational Waves in Schwarzschild Black Hole Ringdown · Introduction · 0.85
Discussion
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