Non-uniqueness of the shockwave dynamics in effective loop quantum gravity

Francesco Fazzini

REVIEW 2 major objections 5 minor 1 cited by

Non-uniqueness of the shockwave dynamics in effective loop quantum gravity

T0 review · 2 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper shows that extending effective loop quantum gravity dust collapse beyond shell-crossing singularities is non-unique: an infinite family labeled by n gives shockwave speeds and lifetimes scaling as $M^{n+2}$, so the familiar…

desk verdict Fazzini explicitly constructs an infinite family of weak extensions of the shockwave model, showing the M^2 lifetime is not unique among integral forms, but the physical significance is limited by the absence of a selection principle. read the letter →

arxiv 2502.03003 v1 pith:BH3TAUUT submitted 2025-02-05 gr-qc

Francesco Fazzini This is my paper · ORCID

classification gr-qc MSC 83C5783C7535L6583C45 PACS 04.60.Pp04.70.Dy

keywords loopquantumgravityeffectivedustcollapseshell-crossingsingularitiesweaksolutionsconservationlawsshockwavesblackholelifetimenon-uniqueness

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether the shockwave model for effective loop quantum gravity dust collapse is uniquely determined once the equations cease to hold at shell-crossing singularities. It answers no. Starting from the same PDE, the author constructs an infinite family of conservation laws by multiplying by $r^{3n/2}\sin^n(B/r^2)$ and rearranging; these laws agree with the original equation on smooth solutions but produce different weak solutions after discontinuities form. Applying the Rankine-Hugoniot condition, the shock velocity and the black hole lifetime depend on $n$, with lifetimes growing as $M^{n+2}$ for even $n$ and as odd power laws for odd $n$. A sympathetic reader should care because observable predictions such as black hole explosion times and the fate of information depend on a choice of integral form that the effective theory does not single out.

What carries the argument

The central object is the infinite class of conservation laws (8), with conserved density (9) and flux (10), parameterized by $n$. It is generated from the original PDE (3) by multiplication by $r^{3n/2}\sin^n(B/r^2)$ and rearrangement using an identity that expresses $\sin^n$ as a finite sum of cosines. This class does the argument's work because all members coincide on smooth solutions but have different weak solutions after characteristic crossing; the Rankine-Hugoniot condition (23) then converts each choice of $n$ into a distinct shock trajectory and lifetime.

What would settle it

A numerical simulation of the integral form of (8) for two values of $n$ (say $n=0$ and $n=1$) that produced identical shock trajectories and identical lifetimes would contradict the claimed non-uniqueness, as would a derivation from a more fundamental quantum theory that singled out exactly one conservation law.

Watch

Extended reading notes

Core claim

The central claim is that the shockwave extension of effective loop quantum gravity dust collapse is not unique. Equation (3), the PDE governing marginally bound collapse, is equivalent to each member of the class (8) while the fields remain smooth, because (8) is obtained by multiplying (3) by $r^{3n/2}\sin^n(B/r^2)$ and rewriting the result as a conservation law using a trigonometric identity. Once characteristics cross and a shell-crossing singularity forms, however, the weak solutions of these conservation laws differ, and their shock speeds obey different Rankine-Hugoniot conditions. The paper derives explicit shock velocities and shows that the black hole lifetime scales as $T_{\mathrm{BH}}\propto R_S^{n+2}$ for even $n$, with odd-power lifetimes for odd $n$; the $n=0$ case reproduces the known $M^2$ lifetime, now revealed as one element of an infinite family. The author also notes that the class can likely be extended to $n<0$, presumably giving shorter lifetimes.

Load-bearing premise

The argument assumes every member of the constructed family (8) is a physically legitimate weak form of the same collapse model, so that no principle selects one integral form over the others; if only the original form is physical, the non-uniqueness collapses.

Editorial extensions

If this is right

Black hole lifetimes in the shockwave model are not a single prediction: they range over an infinite family of power laws, with $M^{n+2}$ for even $n$ and odd powers for odd $n$ (for example, $M^3$ for $n=1$).
Astrophysical searches for exploding black holes become a way to constrain $n$: choices with $n<0$ or with small odd $n$ predict many more, or differently timed, explosions than the $M^2$ case.
Whether the information paradox is avoided depends on $n$: only choices whose lifetimes are shorter than the Page time allow information to escape, while longer-lifetime members may behave like black-to-white hole transitions or leave an almost classical evaporating black hole.
The analytic approximations used for the $n=0$ shock, including the exterior vacuum solution and the Rankine-Hugoniot velocity, carry over directly to every $n$, so the family can be probed numerically by simulating (8) for different values of $n$.
The ambiguity could be resolved only by a physical selection principle, such as a more fundamental quantization that singles out one conservation law or variables that never become discontinuous.

Reading between the lines

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

An editorial extension: the same algebraic ambiguity applies to any nonlinear PDE rewritten as a conservation law by multiplication with a smooth function, so this non-uniqueness is probably generic to weak extensions of nonconservative equations, not special to loop quantum gravity dust collapse.
The paper leaves open $n<0$; a natural testable extension is to construct those shock solutions numerically and check whether superluminal motion or anti-trapped regions appear, which would change the astrophysical signatures.
If future observations measure a population of black hole explosions, the effective lifetime exponent could be fit to $n$, effectively converting the ambiguity into an observable parameter of the model.
If a fundamental theory such as group field theory dynamically selects one of the conservation laws, the non-uniqueness would be resolved; this is a concrete use of the paper's ambiguity rather than a flaw.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Fazzini does something useful: he takes the standard integral-form extension used in the 'quantum black hole' shockwave model and shows explicitly that the same smooth PDE can be rewritten as an infinite family of conservation laws, parameterized by n, which give different shock speeds and therefore different black hole lifetimes. The n=0 member recovers the known M^2 result, so the claim that the lifetime prediction is not robust is concrete and verifiable. The algebraic trick (multiply by sin^n(B/r^2) and repackage) is valid on smooth solutions, and the Rankine-Hugoniot calculation is worked out for even and odd n. The paper is also honest: Sec. VI admits the transformation does not apply discontinuously and that no mathematical prescription selects one form.

The soft spots are mostly typos and gaps. Eq (3) as printed has a sqrt(B) that should be B, and the n=0 lifetime in the text is missing a factor 3 compared to Eq (28). Both are fixable. The monotonicity/invertibility of v(B,r) is asserted rather than proven, which matters because the whole conservation-law form rests on it. The approximations for the odd-n denominator are made quickly, though they look plausible.

The bigger issue is conceptual. The paper's non-uniqueness is real at the level of integral forms, but it does not establish that all n>0 forms are admissible weak formulations of the same effective LQG dynamics. Multiplying the PDE by sin^n and rearranging is an identity on smooth solutions, but across a shock it gives a different equation. The paper says there's no mathematical prescription, but that doesn't mean the physics doesn't select n=0 through junction conditions, the underlying quantization, or conservation of the actual charges of the theory. So the result is best read as a warning: the integral-form approach itself has an ambiguity. It doesn't prove that the M^2 lifetime is one of infinitely many physical possibilities. The abstract slightly overstates this.

Who should read it: anyone using the shockwave model to make claims about black hole lifetimes or the information paradox. It's a useful caution and it will force the n=0 advocates to say why their conservation form is the right one.

Recommendation: send it out for review. The core construction is correct and the typos are easy to fix, but the admissibility question deserves a referee discussion.

Referee Report

2 major / 5 minor

Summary. This paper considers spherically symmetric effective dust collapse in a polymerized loop quantum gravity model. Starting from the evolution equation (1) for the connection component b, the author rewrites it as (3), then constructs an infinite family of conservation laws (8) parameterized by n∈N by multiplying by sin^n(B/r^2) and rearranging (Eqs. (5)-(10)). The paper shows via Rankine-Hugoniot analysis (Sec. V) that the shock velocity after shell-crossing depends on n, leading to black hole lifetimes scaling as M^{n+2} for even n and as odd powers of M for odd n. The n=0 case is claimed to reproduce the earlier shockwave result of [8]. The paper concludes that the extension of the spacetime beyond shell-crossing singularities is non-unique, with no mathematical prescription to choose among the forms.

Significance. The algebraic construction is explicit and the lifetime scalings are concrete and falsifiable within the model. The paper is commendably transparent about the limitation that the transformation (5) does not apply to discontinuous solutions and that no selection rule is known. If the non-uniqueness is physically genuine, it would have strong implications for black hole lifetime predictions in effective LQG and for observational searches. However, the physical significance depends crucially on whether the n>0 conservation laws are admissible weak formulations of the same gravitational theory; this is not established.

major comments (2)

[Sec. II, Eqs. (5)-(10); Sec. VI] The central claim that the shockwave dynamics of effective loop quantum gravity is non-unique rests on the unproved assumption that each member of the family (8) is a physically admissible weak formulation of the same underlying model. The derivation multiplies the smooth PDE (3) by sin^n(B/r^2) and uses the chain-rule identity (7) to obtain a new conservation law; these manipulations are valid for C^1 solutions but not distributionally across a shock, so the weak solutions of (8) are not weak solutions of (3). The paper acknowledges this in Sec. VI ('the transformation (5) does not apply to discontinuous solutions'). To conclude that the spacetime extension is non-unique, the author must show that the n>0 forms conserve the same physical charges or derive from the same geometric junction conditions; otherwise the different lifetimes (27)-(34) are features of the chosen bookkeeping, not of the gravitational theory.
[Sec. V, Eqs. (23)-(34)] The lifetime formulas are obtained by applying the Rankine-Hugoniot condition (23) to the conservation law (8) for each n. The paper does not verify that the resulting shock is the entropy-admissible weak solution for n>0; the statement in Sec. II that 'the effective shockwave dynamics described before satisfies such conditions' is made for the n=0 case only and is not demonstrated. Without an entropy condition, a scalar conservation law generally admits multiple weak solutions, and the selected shock speed may differ from the RH speed. The comparison of lifetimes across n is therefore a comparison of one particular weak solution for each equation; to establish that the integral form itself predicts a different dynamics, the author should either prove the entropy condition for all n or state the additional selection criterion.

minor comments (5)

[Eq. (3)] The argument of the sine is printed as sqrt(B)/r^2; from the definition B = r sqrt(Delta) b one obtains sin^2(B/r^2), not sin^2(sqrt(B)/r^2). This typo should be corrected.
[After Eq. (28)] The text says that for n=0 the lifetime reduces to T_BH ~ 2 pi R_S^2/(gamma sqrt(Delta)), but Eq. (28) yields T_BH ~ 2 pi R_S^2/(3 gamma sqrt(Delta)). The factor 3 discrepancy should be resolved.
[Sec. II, Eq. (9)] The claim that Eq. (9) is monotonic and hence invertible for B/r^2 in [-pi,0] is asserted without proof. Since the subsequent calculations do not use the inverse, either provide a proof or explicitly state that invertibility is not needed.
[Sec. IV] The sentence contains a typo: 'Ti is however a crucial step' should read 'This is however a crucial step'.
[Sec. II] The statement that the shockwave dynamics 'can be easily verified' to satisfy entropy conditions should be expanded, as this is a non-trivial claim.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the non-uniqueness claim is demonstrated by explicit construction of distinct weak formulations, and the self-citations used are auxiliary rather than load-bearing.

full rationale

The paper's central claim is that different integral forms of the effective dust-collapse equation lead to different shockwave dynamics and black-hole lifetimes. This is established by an explicit mathematical construction, not by fitting or by importing the conclusion from prior work. Starting from the smooth PDE (3), the author multiplies it by r^(3n/2) sin^n(B/r^2), rewrites the result using trigonometric identities, and obtains the family of conservation laws (8)-(10). On smooth solutions this family is algebraically equivalent to (3), but its weak solutions differ after characteristic crossing, as computed through the Rankine-Hugoniot condition (23) in equations (24)-(34). No parameter is fitted to produce the n-dependent lifetimes; the n-dependence follows directly from the constructed flux and conserved density. The paper does cite the author's own prior work: [32] for the shell-crossing theorem and [8] for the n=0 post-bounce profile (22) and for the benchmark M^2 lifetime. These citations are auxiliary inputs, not the source of the non-uniqueness: the SCS theorem is proved in Section III, and the n=0 profile is used only as the common pre-shock state from which the different Rankine-Hugoniot speeds are evaluated. The paper's own admission in Section VI that 'the transformation (5) does not apply to discontinuous solutions' and that 'there is no mathematical prescription to start with (3) instead of (8)' is an honest statement of an interpretational limitation, not evidence that the derivation is circular. The strongest available criticism is that the physical admissibility of the n>0 weak formulations is an assumption, but that is a correctness or modeling concern, not a circular-reasoning defect. Therefore no circular step is identifiable and the score is minimal.

Assumptions & free parameters 1 free parameters · 6 assumptions · 2 invented entities

The central claim rests on the choice of effective LQG model, on the admissibility of each constructed integral form as a weak extension, and on several unproved mathematical steps (monotonicity, asymptotic dominance). The free parameter n is not fitted to data but it is precisely the source of the ambiguity, so it must be recorded.

free parameters (1)

n = any non-negative integer (n >= 0)
The index defining the infinite family of equivalent conservation laws. It is not fitted to data but is a free choice; the black hole lifetime depends on n, e.g., T_BH proportional to M^{n+2} for even n.

assumptions (6)

domain assumption Equation (3) is the correct effective PDE for marginally bound dust collapse in the selected LQG scheme.
The paper begins from Eq (3). As printed, Eq (3) does not follow from Eq (1) under the stated change of variables, so the paper relies on Eq (3) being the accepted equation from prior literature. This is load-bearing because every subsequent construction starts from Eq (3).
ad hoc to paper The weak solutions of the constructed conservation laws (8) are physically admissible continuations beyond shell-crossing singularities.
The paper states there is no mathematical prescription to choose among integral forms. The non-uniqueness conclusion assumes each form in the class is an equally valid extension; if only the original integral form is physical, the ambiguity would be an artifact of the rewriting.
standard math v(B,r) in Eq (9) is monotonic and invertible on B/r^2 in [-pi,0] for every n.
The inversion is needed to write Eq (5) as a conservation law partial_t v + partial_r f = 0. The paper asserts this can be easily shown but supplies no proof, and the claim is not self-evident because v involves a sum of oscillatory terms.
domain assumption The post-bounce interior and exterior solution (22) is the same for all n in the class (8).
The paper evaluates the flux jump across the shock using (22), which was established for the n=0 shock solution. It argues weak solutions agree with the differential solution away from shocks, but the detailed matching of the shock to the exterior vacuum for arbitrary n is assumed.
ad hoc to paper For n even, all terms in the shock-velocity denominator except k=n/2 are negligible for L^3 >> gamma^2 Delta R_S.
The lifetime formulas (27) and (28) depend on this asymptotic dominance. The paper does not bound the neglected terms, and the approximation may break near the horizon, where the shock slows and L approaches R_S.
domain assumption The black hole lifetime is equal to the time for the shock to reach the outer horizon at L = R_S.
The paper defines t_f as the time when L(t_f)=R_S and calls this the black hole lifetime. This is a modeling choice inherited from prior work [8], not a derived consequence.

invented entities (2)

Infinite class of weak solutions parameterized by n
purpose: Show that the shockwave extension beyond shell-crossing singularities is non-unique; each n gives a different shock speed and black hole lifetime.
This family is the paper's main mathematical output. It is not an independently observed physical entity; it is a set of alternative continuations with no external evidence distinguishing them.
Shockwave discontinuity
purpose: Propagating surface of discontinuity in extrinsic curvature that extends spacetime beyond shell-crossing singularity.
Inherited from prior work [8,33]; the paper gives no new independent evidence for its existence, and the present result emphasizes that its dynamics are ambiguous.

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Cite this review

Pith. "Pith review of Non-uniqueness of the shockwave dynamics in effective loop quantum gravity." pith.science (2026). https://pith.science/paper/BH3TAUUT

@misc{pith2026250203003,
  author       = {Pith},
  title        = {Pith review of: Non-uniqueness of the shockwave dynamics in effective loop quantum gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BH3TAUUT}},
  note         = {Machine review of arXiv:2502.03003}
}

read the original abstract

Spherically symmetric effective dust collapse inspired by effective loop quantum cosmology predicts a bounce when the stellar energy density becomes planckian, which in turn inevitably leads to shell-crossing singularity formation. An extension of the spacetime beyond such singularities is possible through weak solutions of the equations of motion in integral form, leading to the shockwave model. In this work, we show explicitly that such an extension is not unique, and that relevant features like the black hole life-time strongly depend on the choice of the integral form of the equation of motion.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Dust shell in effective loop quantum black hole model
gr-qc 2025-06 conditional novelty 6.0 of 10

In a polymerized loop-quantum-gravity black hole model, a collapsing dust shell bounces and, for sufficiently heavy shells, follows a spacelike trajectory through the horizon, implying a finite horizon lifetime and a ...

Reference graph

Works this paper leans on

59 extracted references · 25 canonical work pages · cited by 1 Pith paper

[8]

This comes from having required that the interior is almost homogeneous

for an analysis of the cosmological solution). This comes from having required that the interior is almost homogeneous. Since outside the shock the weak solutions of the in- tegral form of any equation of the class (

work page
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(26) Let’s look ﬁrst at the denominator computed in the inte- rior ( − term)

Shock velocity and black hole lifetime for n even In the case of n even, it is important noticing that the k = n/ 2 term in the series is actually given by: ak=n/ 2 =(−1) n 2 ( n n/ 2 ) lim k→n/ 2 sin [ (n − 2k) B L2 − πn 2 ] n − 2k = = ( n n/ 2 ) B L2 . (26) Let’s look ﬁrst at the denominator computed in the inte- rior ( − term). Since Bint(r, t ) = −πL ...

work page
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(5) In order to write the ﬁrst term in the required form (4), we use the following identity: sinn ( B r2 ) = n∑ k=0 (−1)k 2n (n k ) cos [ (n − 2k)B r2 − π 2 n ]

by r 3n 2 sinn(B r2 ) (with n ∈ N), and rearrange the last term, yielding sinn ( B r2 ) ∂t(r 3n 2 B)+ 1 (n + 2)γ √ ∆ ∂r [ r3 sin2 ( B r2 )] n 2 +1 = 0 . (5) In order to write the ﬁrst term in the required form (4), we use the following identity: sinn ( B r2 ) = n∑ k=0 (−1)k 2n (n k ) cos [ (n − 2k)B r2 − π 2 n ] . (6) It is then straightforward to check t...

work page
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(4) To achieve this goal, we multiply (

in r [8]), is to multiply ( 3) by functions f (B(r, t ), r ), and then rearrange the result to regain a conservation law of kind ∂tv(r, t ) + ∂rf (v(r, t ), r ) = 0 . (4) To achieve this goal, we multiply (

work page
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The analytic solution of the previous equation reads r(R, t ) = [2 Gm(R)] 1 3 { [t − α (R)]2 + ∆ } 1 3 , (12) where α (R) is ﬁxed by the initial condition

takes in LTB coordinates the following form [5]: ( ˙r r )2 = 2Gm(R) r ( 1 − 2Gm(R)γ 2∆ r3 ) , (11) where r(R, t ) is the areal radius of the shell R at time t, and m(R) is the (conserved) gravitational mass within [0, R ]. The analytic solution of the previous equation reads r(R, t ) = [2 Gm(R)] 1 3 { [t − α (R)]2 + ∆ } 1 3 , (12) where α (R) is ﬁxed by t...

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in the form of non-linear conservation laws. It is important to highlight that these equations pro- duce precisely the same dynamics as ( 3) until character- istics cross (or in general relativistic language SCS arise) and the ﬁelds become multi-valued. However, as explic- itly shown in the subsequent section, once characteristics cross, the weak solution...

work page
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and ( 8), allows to safely assume that characteristic crossings arise only in the post-bounce dynamics for decreasing initial density pro- ﬁles inhomogeneous enough. 4 IV. GENERAL FEATURES OF THE WEAK SOLUTIONS As previously stated, the solutions to equations ( 3) and (8) are identical until characteristics cross (i.e., a shell- crossing singularity arise...

work page
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(or ( 11) in LTB coordinates), until shell-crossing sin- gularities arise. At this point, while the diﬀerential form of the equations is no longer due to multi-valued solu- tions, the weak solution continues its evolution, treating the characteristic crossing as a discontinuity in the ﬁeld variables. Notably, as shown in [8] and validated nu- merically, t...

work page

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Since the same holds also for the pre- bounce dynamics [8], and this range is the one in which the function v(B, r ) is invertible, the equations (

must agree with ( 22), we can safely assume that the B/r 2 ﬁeld dur- ing the shock dynamics takes values in [ −π, 0] for such weak solutions. Since the same holds also for the pre- bounce dynamics [8], and this range is the one in which the function v(B, r ) is invertible, the...
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Even if ﬁnding the inverse of ( 9) for generic n is a non-trivial task, it is outside the aim of this work since not necessary to derive the analytic shockwave dy- namics

can be cast in the form ( 4), and there is no invertibility issue for any n ∈ N. Even if ﬁnding the inverse of ( 9) for generic n is a non-trivial task, it is outside the aim of this work since not necessary to derive the analytic shockwave dy- namics. Ti is however a crucial ...
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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [8]

This comes from having required that the interior is almost homogeneous

for an analysis of the cosmological solution). This comes from having required that the interior is almost homogeneous. Since outside the shock the weak solutions of the in- tegral form of any equation of the class (

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[2] [1]

(26) Let’s look ﬁrst at the denominator computed in the inte- rior ( − term)

Shock velocity and black hole lifetime for n even In the case of n even, it is important noticing that the k = n/ 2 term in the series is actually given by: ak=n/ 2 =(−1) n 2 ( n n/ 2 ) lim k→n/ 2 sin [ (n − 2k) B L2 − πn 2 ] n − 2k = = ( n n/ 2 ) B L2 . (26) Let’s look ﬁrst at the denominator computed in the inte- rior ( − term). Since Bint(r, t ) = −πL ...

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[3] [2]

(5) In order to write the ﬁrst term in the required form (4), we use the following identity: sinn ( B r2 ) = n∑ k=0 (−1)k 2n (n k ) cos [ (n − 2k)B r2 − π 2 n ]

by r 3n 2 sinn(B r2 ) (with n ∈ N), and rearrange the last term, yielding sinn ( B r2 ) ∂t(r 3n 2 B)+ 1 (n + 2)γ √ ∆ ∂r [ r3 sin2 ( B r2 )] n 2 +1 = 0 . (5) In order to write the ﬁrst term in the required form (4), we use the following identity: sinn ( B r2 ) = n∑ k=0 (−1)k 2n (n k ) cos [ (n − 2k)B r2 − π 2 n ] . (6) It is then straightforward to check t...

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[4] [3]

(4) To achieve this goal, we multiply (

in r [8]), is to multiply ( 3) by functions f (B(r, t ), r ), and then rearrange the result to regain a conservation law of kind ∂tv(r, t ) + ∂rf (v(r, t ), r ) = 0 . (4) To achieve this goal, we multiply (

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[5] [4]

The analytic solution of the previous equation reads r(R, t ) = [2 Gm(R)] 1 3 { [t − α (R)]2 + ∆ } 1 3 , (12) where α (R) is ﬁxed by the initial condition

takes in LTB coordinates the following form [5]: ( ˙r r )2 = 2Gm(R) r ( 1 − 2Gm(R)γ 2∆ r3 ) , (11) where r(R, t ) is the areal radius of the shell R at time t, and m(R) is the (conserved) gravitational mass within [0, R ]. The analytic solution of the previous equation reads r(R, t ) = [2 Gm(R)] 1 3 { [t − α (R)]2 + ∆ } 1 3 , (12) where α (R) is ﬁxed by t...

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[6] [5]

in the form of non-linear conservation laws. It is important to highlight that these equations pro- duce precisely the same dynamics as ( 3) until character- istics cross (or in general relativistic language SCS arise) and the ﬁelds become multi-valued. However, as explic- itly shown in the subsequent section, once characteristics cross, the weak solution...

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[7] [6]

and ( 8), allows to safely assume that characteristic crossings arise only in the post-bounce dynamics for decreasing initial density pro- ﬁles inhomogeneous enough. 4 IV. GENERAL FEATURES OF THE WEAK SOLUTIONS As previously stated, the solutions to equations ( 3) and (8) are identical until characteristics cross (i.e., a shell- crossing singularity arise...

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[8] [7]

(or ( 11) in LTB coordinates), until shell-crossing sin- gularities arise. At this point, while the diﬀerential form of the equations is no longer due to multi-valued solu- tions, the weak solution continues its evolution, treating the characteristic crossing as a discontinuity in the ﬁeld variables. Notably, as shown in [8] and validated nu- merically, t...

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