REVIEW
Late-Time Behavior of Stellar Collapse and Explosions: II. Nonlinear Evolution
General Relativity and Quantum Cosmology · Submitted 10 Jul 1993 · 11 references
Pith's verdict: The physics holds up; the corrupted abstract is an arXiv artifact, not the authors' text, and the classifier record built on it should be ignored.
T0 · AI review
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Referee report
The paper
The paper claims
The late-time fingerprints of a black hole survive a fully nonlinear collapse simulation, and one of them appears even when no black hole forms.
The reading
What carries the argument
The engine is a characteristic (null) grid: retarded time $u$ labels outgoing light cones, $v$ labels ingoing null geodesics, and the areal radius $r(u,v)$ is itself an evolved metric function rather than a coordinate. The scalar is repackaged through an auxiliary field $psi$ with $varphi = (1/r)int_0^r psi,dr$, so that the wave equation becomes a transport equation along ingoing characteristics, $(partial_u - (bar g/2)partial_r)psi = (g-bar g)(psi-varphi)/(2r)$, while $varphi$ is recovered by radial integration along each outgoing cone. The metric functions come from quadratures, $g = exp[4pi int_0^r (partial_r varphi)^2 r,dr]$ and $bar g = (1/r)int_0^r g,dr$, so spherical symmetry removes any independent gravitational degree of freedom. Each step is a second-order Runge-Kutta update in $u$ with trapezoidal radial integrals, and the step size is chosen so that no grid point moves further in $r$ than its neighbour's spacing. The multipole test field adds a centrifugal term, $(l(l+1)/2),g,varphi_{ml}$, to the same transport equation, which is what makes the $l$-dependence of the tails and the weakly damped high-$l$ ringing accessible. The authors are candid that the code is better than first order but not second order, and that the dominant error source is the linear interpolation used to impose the $r=0$ boundary condition, which they compensate for by refining the radial grid by a factor of 16 near the origin.
Extended reading notes
Core claim
The paper's claim is that the late-time behaviour predicted by linear perturbation theory of Schwarzschild spacetime — quasinormal ringing and power-law tails — is reproduced by a full nonlinear integration of the Einstein equations coupled to a spherically symmetric scalar field, in regimes where nonlinearities genuinely matter. Measured tail exponents are close to the linear predictions: $-2.74$, $-2.63$, $-2.63$ and $-2.68$ for four Gaussian amplitudes against a predicted $-3$, and $-2.08$, $-1.98$, $-1.95$ and $-1.86$ for four static-static amplitudes against a predicted $-2$. For the test field, the least-damped ringing frequencies are matched tightly: a measured half-period of $1.811$ against a predicted $1.819$ for $l=1$ (within $0.7$ percent) and $1.105$ against $1.104$ for $l=2$ (within $0.2$ percent). The sharper half of the claim is that tails develop with the same exponents whether or not a horizon forms, and that ringing appears abruptly at the threshold of collapse for Gaussian data but fades gradually for static-static data. The authors read this as evidence that tails come from backscattering at large radius, which does not care about the small-$r$ geometry, while ringing requires a compact object and does.
Load-bearing premise
The argument assumes that predictions derived on a fixed Schwarzschild background transfer to a time-dependent, mass-losing background with some retarded-time average playing the role of $M$, and the authors say outright that it is unclear which mass to use — which is why exponent comparisons succeed while the predicted tail amplitude comes out roughly an order of magnitude too large.
Editorial extensions
If this is right
- Once a horizon has settled down, the late-time gravitational waveform of a strongly nonlinear collapse can be modelled by linear perturbation theory, so ringdown templates do not require a full nonlinear simulation for every source.
- Power-law tails should be expected in the late signal of any roughly spherical, asymptotically flat radiating spacetime, including bursts that leave no remnant behind.
- The abrupt onset of ringing at the black-hole threshold, set against the smooth continuity of the tail, gives a qualitative discriminator between 'a horizon formed' and 'it very nearly did'.
- The multipole dependence of the tail, $t^{-(2\ell+3)}$ for generic data and $t^{-(2\ell+2)}$ for initially static multipoles, survives on a time-dependent background, so the slope of a late-time signal identifies which multipole is dominating it.
- Tail amplitude scales as the cube of the initial data amplitude while the Bondi mass scales as the square, which is what one expects if the tail is radiation scattered once off the spacetime curvature, picking up a single factor of mass.
Reading between the lines
- If large-radius backscatter really is the mechanism, the tail should weaken or vanish when the numerical domain is truncated well inside the scattering region; that is a cheap test the authors did not run and would sharpen the causal story considerably.
- The order-of-magnitude shortfall in tail amplitude relative to the linear prediction may be an artifact of the first-order treatment at $r=0$ that the authors themselves name as their main error source; a code with a properly regular centre condition would separate the physics from the numerics.
- Because the tail crosses the collapse threshold continuously while the ringing does not, the tail amplitude could in principle serve as a mass calibration for a near-critical remnant, since it tracks radiated energy rather than horizon formation.
- The test-field equations deliberately omit the backreaction a genuine perturbation of the background scalar would produce, so the agreement reported here is a best case; the disagreement case, where the perturbation is coupled to the geometry, is left untested.
Original abstract
Problem with the figures should be corrected. Apparently a broken uuencoder was the cause.
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
Problem with the figures should be corrected. Apparently a broken uuencoder was the cause.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Problem with the figures should be corrected. | limitation 0.66 | 0.14 | 0.02 | asserted 0.96 |
| s1 | Apparently a broken uuencoder was the cause. | background 0.49 | 0.28 | 0.02 | hedged possibility 0.54 |
What was examined, and what was not
- arXiv listing abstract v2
- 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
Showing 11 of the paper's 15 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:gr-qc/9307010 by the authors.
References
How the literature uses it
35 arXiv papers cite this in 39 sentences
Papers citing it · 14 sentences without a confident label
Uses its method or tool 14 papers
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We use the light cone coordinates [equation] then Eq. ([ref]) can be written as [equation] We adopt the discretization scheme suggested by Gundlach and Price et al. [ref] [equation] where $S=(u,v),W=(u+Δu,v),E=(u,v+Δv),N=(u+Δu,v+Δv)$.
Ringing and echoes from black bounces surrounded by the string cloud · The methods · 1.00
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For the time-domain profile, we can calculate it using the numerical method suggested by Gundlach and Price et al. [ref] .
Ringdown of a black hole sourced by a Burkert-density effective anisotropic source · Numerical methods for time-domain profiles · 1.00
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These results are illustrated using the finite difference method [ref] , where the resulting time evolutions for effective potentials with different parameters $(ν_1,ν_2)$ are shown in Fig. [ref].
Late-time tail and echoes of Damour-Solodukhin wormholes · Generalization to the case of singular ingoing wave · 1.00
Compares with it 1 paper
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This behaviour is in contrast to that of asymptotically flat BH perturbations, where the quasinormal ringing gives way to a power-law cutoff [ref] , and its occurrence is related to the asymptotic nature of timelike infinity in AdS spacetimes which serves as a reflective boundary.
Echoes of Compact Objects in Scalar-Tensor Theories of Gravity · Black hole · 0.91
Cites it as background 9 papers
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This is famously represented by Price's scaling: massless fields in a non-spinning BH background decay at fixed spatial locations as $∼t^-2ℓ-3$ at very late times, where $ℓ$ is the corresponding multipole number [ref] .
Nonlinear Tails of Gravitational Waves in Schwarzschild Black Hole Ringdown · Introduction · 0.85
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Linear black hole perturbation theory (BHPT) has long provided the framework for modeling black hole ringdowns, yielding well-known results such as quasinormal modes (QNMs) and Price’s power law tails [ref] .
Dynamical nonlinear tails in Schwarzschild black hole ringdown · Introduction · 1.00
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There is an initial burst that depends on the source [ref] , a ringdown dominated by a discrete spectrum of complex frequencies known as quasinormal modes (QNMs) [ref] , and a late-time power-law tail governed by the long-range structure of the spacetime [ref] .
Parametrized beyond-Teukolsky framework in the time domain · Introduction · 0.91
Discussion
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