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On Kerr black hole formation with complete apparent horizon and a new approach toward Penrose inequality

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Pith's one-line read The paper constructs vacuum Kerr black hole formation with a complete apparent horizon and uses it to prove Penrose inequalities in the perturbative Kerr regime.

desk verdict Bold, well-architected preprint that connects short-pulse collapse to Kerr stability and proves Penrose inequalities in those spacetimes, but the load-bearing initialization step needed to invoke Klainerman–Szeftel is explicitly omitted (Remark 10), leaving the global conclusions conditional. read the letter →

arxiv 2505.11399 v1 pith:BRSBUEKT submitted 2025-05-16 gr-qc math-phmath.APmath.DGmath.MP

Xinliang An , Taoran He This is my paper · ORCID
classification gr-qcmath-phmath.APmath.DGmath.MP MSC 83C0583C5735Q7535J60
keywords KerrblackholeformationapparenthorizonmarginallyoutertrappedsurfacePenroseinequalityEinsteinvacuumequationsshort-pulsemethodstabilitycharacteristicinitialdata
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

The paper aims to close the loop of gravitational collapse: starting from admissible characteristic initial data—a short pulse of strong shear near the center joined to nearly Kerr data farther out—it constructs full solutions of the 3+1 Einstein vacuum equations in which a trapped region emerges, its boundary (the apparent horizon) forms and expands, and the spacetime settles down to a slowly rotating Kerr black hole. It then solves for the entire apparent horizon as a family of unique marginally outer trapped surfaces on incoming null cones, and shows that this horizon is smooth, locally achronal, area-nondecreasing, asymptotically null, and converging to the event horizon at late times. If the construction is correct, no symmetry assumption is needed to prove that collapse ends at Kerr, and the Penrose area–mass bound follows in these formation spacetimes and in perturbed Kerr. The essential bridge is feeding the characteristic data into the nonlinear Kerr stability theorem, which supplies the global late-time spacetime and decay estimates.

What carries the argument

The load-bearing object is the incoming null cone foliation $\tilde H_{\tilde u}$ with regular coordinate $\tilde r$ obtained by solving the eikonal equation in the perturbed Kerr spacetime, using Pretorius–Israel type coordinates as the leading model. On these null cones the crucial identity is $\operatorname{tr}\tilde\chi_0 = F(\tilde r - r_+)$ with $F \sim 1$ for all subextremal Kerr data $|a|<m$, which converts the MOTS location equation into a quasilinear elliptic PDE whose zeroth-order coefficient is negative and bounded away from zero; that negativity yields the $C^0$ bound, invertibility of the linearized operator, and the area asymptotics. The elliptic existence engine is a Leray–Schauder fixed point argument whose a priori $C^{1,\alpha}$ estimates do not require Hölder continuity of the coefficients: they use the Campanato condition, a sharp Miranda–Talenti type inequality on $S^2$, and a generalized quasiconformal method with only an $L^\beta$ bound on the source. The null comparison principle then upgrades the MOTS family to a locally achronal apparent horizon, giving the area non-decrease used in the Penrose inequality proof.

What would settle it

Run the omitted initialization check: for admissible data from Section 3 and the smooth examples in Appendix A, compute the outgoing initial layer's transition coefficients $f', \lambda'$ and the $r^p$-weighted flux of the curvature components along $H_{v_0}$, and test the claimed bound $|f'| \lesssim \epsilon_1/r'$ and the energy bound $E^{N-3}_{0,\mathrm{in}} \lesssim \epsilon_1$; a single admissible data set violating these would invalidate the bridge to the Kerr stability theorem and with it the global apparent horizon construction. Alternatively, in exact subextremal Kerr with $|a/m|<1$, evaluate $\operatorname{tr}\tilde\chi_0/(\tilde r - r_+)$ at $r=m$ for a near-extremal value: if it ever fails to be positive and bounded below on $[m,r_0]$, Proposition 4.5 and the MOTS existence argument fail.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: admissible characteristic initial data produce Kerr black hole formation solutions of the Einstein vacuum equations, each with a complete apparent horizon that originates from a spacetime center point, is spacelike in the short-pulse region, asymptotically null, and converges to the event horizon as advanced time tends to infinity. The proof has a hyperbolic part that welds scale-critical short-pulse collapse to nonlinear Kerr stability with small angular momentum through precisely constructed initial data layers and many coordinate and frame changes, and an elliptic part that solves the MOTS equation on each incoming null cone using Leray–Schauder fixed point theory together with new a priori estimates. A structural identity for exact Kerr—the outgoing null expansion has the form $\operatorname{tr}\tilde\chi_0 = F(\tilde r - r_+)$ with $F\sim 1$ and $r_+$ the horizon radius—gives the MOTS equation a negative zeroth-order term, which drives existence, uniqueness, asymptotics, and the inequality chain. The paper then proves the dynamical Penrose inequality $M_B(\tilde u) \ge m_\infty \ge \sqrt{m_\infty(m_\infty+\sqrt{m_\infty^2-a_\infty^2})/2} \ge \sqrt{A_M(\tilde u)/16\pi}$ and the spacetime Penrose inequality $m \ge \sqrt{A/16\pi}$ in the perturbed Kerr regime, with rigidity when equality holds and the MOTS has constant Gauss curvature $1/(4m^2)$.

Load-bearing premise

The entire global construction hinges on the claim, sketched in Section 3.2.2 with one proof explicitly omitted, that the characteristic data layers can be initialized—after coordinate and frame changes—so that they satisfy the hypotheses of the Kerr stability theorem; if those transition-coefficient and curvature estimates along the outgoing initial hypersurface fail, the late-time spacetime and all subsequent MOTS and Penrose conclusions collapse.

Editorial extensions

If this is right

  • The full collapse process—trapped surface emergence, apparent horizon growth, settling to Kerr—is realized without any symmetry assumption, so the later evolution of short-pulse data is no longer a separate open problem.
  • The apparent horizon can be located, tracked, and shown to be achronal in these spacetimes; in particular its area is non-decreasing along the formation, giving a black-hole area law during collapse.
  • The dynamical Penrose inequality holds along the formation: Bondi mass is bounded below by the final mass, which is bounded below by $\sqrt{A_M(\tilde u)/16\pi}$, with the MOTS area approaching the Kerr horizon area $4\pi(r_+^2+a_\infty^2)$ at late times.
  • In the perturbed Kerr regime the spacetime Penrose inequality $m \ge \sqrt{A/16\pi}$ holds without time symmetry, with rigidity characterized by constant Gauss curvature equal to $1/(4m^2)$.
  • Once nonlinear Kerr stability is established for the full subextremal range $|a/m|<1$, the same argument extends the horizon conclusions and the Penrose inequalities to all subextremal Kerr targets; the paper already proves the needed null-expansion structure in that range.

Reading between the lines

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

  • A likely transferable residue is the quasiconformal $C^{1,\alpha}$ estimate that works with only an $L^\beta$ source bound and no Hölder coefficients; it should apply to other quasilinear elliptic equations with uniformly elliptic but rough coefficients, such as MOTS equations on slightly timelike or spacelike slices.
  • The initial-layer gluing recipe suggests a modular route for black hole formation: if a target spacetime has a proven stability theorem and explicit null coordinates, the same transition-region construction may weld short-pulse collapse to that target—e.g., charged rotating black holes or other stationary backgrounds.
  • A concrete testable extension would be to compute how tight the inequality chain is at finite time: the gaps between $A_M(\tilde u)$, its limiting value $4\pi(r_+^2+a_\infty^2)$, and the corresponding Bondi mass should be optimally controlled by the stated $\epsilon_0/\tilde u^{1+\delta_{\mathrm{dec}}}$ rates, and any loss would indicate where the elliptic estimates degrade.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper claims a proof that the 3+1 Einstein vacuum equations admit Kerr black hole formation solutions arising from admissible characteristic initial data. The strategy is to combine the scale-critical short-pulse trapped-surface formation result of An--Luk with the Klainerman--Szeftel nonlinear stability of Kerr with small angular momentum. To make this connection, the authors introduce admissible Kerr black hole formation initial data, construct initial data layers in a neighborhood of the characteristic hypersurfaces, and then pass through a chain of coordinate and null-frame transformations to the principal geodesic structures used in [56]. In the resulting spacetimes, they construct incoming null hypersurfaces by solving the eikonal equation in Pretorius--Israel type coordinates, solve a quasilinear elliptic equation to find a unique marginally outer trapped surface (MOTS) on each incoming null cone, and assemble these MOTSs into a complete apparent horizon. They further prove that this apparent horizon is smooth, locally achronal, asymptotically null, and approaches the event horizon at timelike infinity. Building on this analysis, they derive an area monotonicity law and then prove a dynamical Penrose inequality in their black-hole formation spacetimes and in Klainerman--Szeftel spacetimes, as well as a spacetime Penrose inequality, without a time-symmetry assumption, in the perturbative Kerr regime.

Significance. If the central claims are fully substantiated, the paper represents a substantial advance: it would extend Christodoulou's trapped-surface formation theorem to a complete, dynamical Kerr black hole formation picture, and it would provide a new hyperbolic route to the Penrose inequality outside the time-symmetric setting. The paper contains several genuinely new ingredients: the construction of admissible initial data that connects the short-pulse regime to the Kerr-stability regime, the construction of incoming null foliations in perturbed Kerr spacetimes via Pretorius--Israel type coordinates, a quasilinear elliptic existence and uniqueness theory for MOTSs based on quasiconformal estimates, and the extraction of a precise leading-order structure for the null expansion. These techniques, if valid, are likely to be influential beyond the present setting, particularly the elliptic method that does not rely on smallness of the angular momentum. The main limitation is that the bridge to the Klainerman--Szeftel theorem is not fully proved in the manuscript; the omitted initialization estimates are load-bearing, and several other key steps are asserted with varying degrees of detail.

major comments (4)
  1. [Section 3.2.2, Remark 10] The load-bearing bridge to Theorem 3.4 is not proved. The transition coefficients (3.13) and the curvature estimates for the initial data along H_{v0} with respect to the new null frame are exactly what is needed to pass from the double-null frame of the short-pulse construction to the principal geodesic frame required by [56]. Remark 10 states 'We omit the proof in the present paper,' and this is not a peripheral detail: without these initialization estimates, Proposition 3.3 does not verify the hypotheses of Theorem 3.4, and therefore Theorem 1.1, the global spacetime used in Sections 4--6, and the Penrose inequalities in Section 7 all lack support. The 'last slice argument' also needs a uniform closure in v*; the current text asserts the rp-weighted estimates close, but the initialization step is needed for that closure. The authors should provide the full proof or a precise theorem from [45, 60, 73] that covers this exact situation, including the smallness conditions matched to (3.5)--(3.6).
  2. [Section 5, transition region after Proposition 4.4] The manuscript asserts that the MOTS existence result of [3] 'extends' to the transition region F(δ) ≤ eu ≤ F(v1), but no proof or precise reference is supplied. This is not a cosmetic gap: the global optical function (glo)eu constructed in Section 4.2 is neither the short-pulse v-level foliation nor the eikonal foliation of (int)M, and the MOTS equation must be re-derived and re-solved for this intermediate foliation. Without a demonstration that the existence, uniqueness, and regularity of MOTSs remain valid across the transition region, the claimed 'complete apparent horizon' and its smooth connection between the short-pulse and Kerr-stability portions are not established. Please either provide the proof or state explicitly which theorem in [3] applies, and explain why its hypotheses are satisfied by the transition foliation.
  3. [Section 4.3, Proposition 4.8] The proof of Proposition 4.8, which yields the key leading-order identity tr˜χ0 = F·(˜r-r+) with F∼1, relies on algebraic computations summarized as 'a direct calculation' and 'it can be check readily.' In particular, the expression for λ^{-1}tr˜χ′ assembled in the proof of Lemma 4.7 contains many terms, and the positivity of the resulting factor G(r,θ) over the entire range m ≤ r ≤ r0 is essential for the MOTS equation and for the Penrose inequality. The manuscript does not display the complete algebraic verification that G has no zeros; it instead uses monotonicity and the two limits r→r+ and r→∞. Those arguments are plausible, but they are not sufficient as written because the intermediate expression is not fully derived. Please include the full computation or a verifiable derivation that shows H(r,θ) > 0 for all relevant r, θ, including the treatment of the polar angular coordinates.
  4. [Section 7.1, proof of Theorem 1.4] The spacetime Penrose inequality requires the global existence of a spacetime with complete future null infinity arising from the spacelike initial data set (Σ,g,k) in the perturbative Kerr regime. The proof cites [56] and then refers to [20, 74] in a parenthetical remark, but no combined statement of the resulting global existence theorem is given, nor is it verified that the initial data set satisfies the precise hypotheses of the cited works, such as asymptotic flatness conditions and compatibility with the characteristic data framework. The equality m = MB(-∞) ≥ MB(∞) = m∞ depends on this global development. Please provide a precise statement of the global theorem being invoked, with its hypotheses, and confirm that the initial data set (Σ,g,k) and the MOTS M0 satisfy them.
minor comments (5)
  1. [Section 3.1 vs Section 3.2.2, Remark 9] The smallness hierarchy is stated inconsistently: Definition 3.1 and the surrounding text require δA^{1/2} ≪ ϵ1 ≪ 1, while Remark 9 says the upper bound on the initial energy is not necessarily required to be small and only the characteristic length d needs to be small. Please clarify the exact smallness assumptions used in Proposition 3.3.
  2. [Theorem 1.1] The phrase 'processes a complete apparent horizon' appears to be a typo for 'possesses a complete apparent horizon'; please correct it.
  3. [Section 4, notation near (4.26) and Proposition 4.5] The notation for the frames (eµ), (1)eµ, (˜eµ), and (˜e′µ) is dense, and the same symbol f is used for transition coefficients and for the function f in the definition of the optical function. A summary table of the frames, their transition coefficients, and the corresponding coordinate systems would substantially improve readability.
  4. [Section 5.1, equation (5.2)] The expression for trχ′ is stated as a lemma from [7], but the sign conventions and the derivation of the quadratic gradient terms are not repeated. Since the signs are crucial for the elliptic structure in Proposition 5.2, please either include a short derivation or explicitly indicate the equation numbers in [7] that are being used.
  5. [Section 7, Theorem 1.4 rigidity statement] The rigidity statement is phrased both as 'if equality holds then the Gauss curvature is pointwise 1/(4m²)' and 'conversely, assuming Gauss curvature is pointwise 1/(4m²), A = 16πm².' The second implication is immediate from Gauss--Bonnet, but the first requires the transport argument along the apparent horizon to be written out with the relevant estimates; the current text states this in a few sentences and should be expanded.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are assembled from the external Kerr-stability theorem [56], prior short-pulse results, and new elliptic estimates; no prediction reduces by construction to its inputs.

full rationale

The paper's derivation chain is not circular in any of the seven flagged senses. Theorem 1.1 concatenates two independent ingredients: the scale-critical short-pulse existence result of An–Luk [9] for the early collapse, and Klainerman–Szeftel's Kerr stability theorem [56] for the late-time convergence. The paper prescribes 'admissible Kerr black hole formation initial data' whose post-short-pulse part is required to satisfy the decay/energy bounds of [56]; this is an explicit hypothesis, not a conclusion smuggled in. The verification that the constructed initial-data layers satisfy [56]'s conditions is summarized in Proposition 3.3 and Section 3.3, with the caveat that Remark 10 omits the initialization estimates for curvature components on H_{v0}; that is a completeness gap in the present version, not a circular reduction. The MOTS existence and uniqueness arguments (Section 5) are new and self-contained, using the uniformly elliptic quasilinear equation (5.4) derived from the null-expansion formula (5.2), and the leading-order structure trχ = F·(r̃−r+)+O(ε0/u^{1+δ_dec}) is proved in the exact Kerr spacetime in Section 4.3 rather than assumed. The area-increasing law in Section 6.3 follows from the achronality established by verifying the null comparison principle of [7] in Proposition 6.4, i.e., the cited criterion is applied only after its hypotheses are proved in this paper. Finally, the Penrose inequalities in Section 7 rest on the chain A≤A_M(ũ)≤A_∞≤16πm_∞^2≤16πM_B(ũ)^2≤16πm^2, where each inequality comes from the proved area monotonicity, the asymptotic area computation in Lemma 7.1, the Bondi-mass loss formula, and the convergence to Kerr from [56]; no fitted parameter is renamed as a prediction, and no equation is used that is equivalent by construction to the target inequality. The self-citations to [7], [9] and [4] are prior mathematical results used as tools, not unverified premises carrying the argument by themselves.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No new fields, particles, forces, or conserved quantities are postulated. The 'Pretorius-Israel type coordinates' and the optical function ~u are new constructions within standard GR, not new physical entities. The smallness constants are technical hierarchy parameters, not fitted values: the admissible initial data satisfy open conditions and no constant is tuned to match a target inequality.

assumptions (4)
  • domain assumption Nonlinear stability of Kerr with small angular momentum (Theorem 3.4, [56]) holds and is applicable to the constructed initial data layers.
    The entire late-time spacetime, including decay estimates (4.2) and the final mass and angular momentum, is imported from [56]. The paper verifies the required initial conditions only through sketched arguments, with one initialization step explicitly omitted in Remark 10.
  • domain assumption Scale-critical short-pulse existence and estimates from An-Luk [9] and An [3,4] hold in the short-pulse region P, including the early apparent horizon.
    The hyperbolic estimates and the MOTS in the early region are taken from these prior works without reproof, and they form the starting point of the collapse construction.
  • standard math Standard 2D elliptic regularity tools (Leray-Schauder fixed point, Campanato condition, Miranda-Talenti inequality, quasiconformal C^{1,alpha} estimates) are valid as invoked.
    Section 5 builds on these tools; the paper generalizes the quasiconformal estimate to finite L^beta, which is a new technical lemma that is only partially proved.
  • standard math Null structure equations and the Raychaudhuri equation apply to the constructed null frames in vacuum.
    Used in Section 4.3 to establish monotonicity of tr~chi' and in Section 5 to derive the MOTS equation.

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Pith. "Pith review of On Kerr black hole formation with complete apparent horizon and a new approach toward Penrose inequality." pith.science (2026). https://pith.science/paper/BRSBUEKT

@misc{pith2026250511399,
  author       = {Pith},
  title        = {Pith review of: On Kerr black hole formation with complete apparent horizon and a new approach toward Penrose inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BRSBUEKT}},
  note         = {Machine review of arXiv:2505.11399}
}
read the original abstract

Arising from admissible extended scale-critical short-pulse initial data, we show that 3+1 dimensional Einstein vacuum equations admit dynamical Kerr black hole formation solutions. Our hyperbolic arguments combine the scale-critical gravitational-collapse result by An--Luk with the recent breakthrough by Klainerman--Szeftel on proving nonlinear Kerr stability with small angular momentum, which requires us to perform various specific coordinate changes and frame transformations. Furthermore, allowing large spacetime angular momentum, with new elliptic arguments and precise leading order calculations, we also solve the apparent horizon in Kerr black hole formation spacetimes (including Klainerman--Szeftel's Kerr stability spacetimes) and conduct an exploration, detailing the emergence, evolution, asymptotics and final state of the apparent horizon. Building on our analysis, without time symmetric assumption, we then put forward a new mathematical framework and prove both the dynamical Penrose inequality and the spacetime Penrose inequality in our black-hole formation spacetimes and in the perturbative regime of subextremal Kerr black holes. Collectively, without assuming any symmetry, we extend Christodoulou's celebrated trapped surface formation theorem to a black hole formation result.

Figures

Figures reproduced from arXiv: 2505.11399 by the authors.

Figure 1
Figure 1. Process of Kerr Black Hole Formation We achieve our goal by the following steps: We first extend [9, 4] and introduce the admissible Kerr black hole formation initial data along an outgoing null hypersurface, such that, after the short-pulse region, the solved spacetime metric near the initial outgoing null hypersurface and along a later incoming null hypersurface enters the perturbative Kerr regime, and we further … view at source ↗
Figure 2
Figure 2. A New Approach Toward Penrose Inequality that for the MOTS Mue, which is a spacelike section of the apparent horizon, its area is non-decreasing for all ue. Given the desired nonlinear Kerr stability conclusions for the full subextremal range, we further verify that the induced metric along AH eventually stabilizes to that of the horizon for a stationary Kerr black hole with the final mass m∞ and the final angular m… view at source ↗
Figure 3
Figure 3. Spacetime Penrose Inequality 2 It is usually referred as the Bondi mass loss [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Admissible Characteristic Initial Data Specifically, define Su,v to be the intersected 2-sphere for level sets of u and v in double null coordinates. • Along Hu0 ∩ {u0 ≤ v ≤ δ} with 0 < δ ≪ 1, we prescribe the scale-critical short-pulse data that are consistent with [9…
Figure 5
Figure 5. Figure 5: Proof of Spacetime Penrose Inequality • Given the initial data triplet (Σ, g, k), we first apply the result of Kerr stability [56],14 and obtain the global existence of solutions to the EVEs (1.1). This solution (M, g) features a complete future null infinity I + and c…
Figure 6
Figure 6. Figure 6: Construction of Incoming Null Cones within (int)M Applying Appendix B, we have that the explicit expression of ue0 reads ue0 = u − f(r) + r∗(r, θ) = u + f∗(r, θ) with f ′ (r) = r 2 + a 2 ∞ ∆ . Here f∗(r, θ) is a bounded function and r∗(r, θ) satisfies (4.5) ∂rr∗ = q (r…
Figure 7
Figure 7. Figure 7: Transition Region between P and M Also recall that in Section 3 we have derived two coordinate systems for (int)M0, i.e., (u, v, θ 1 , θ 2 ) and (u, r, θ, φ). Let ((pul) eµ) and ((int) eµ) be the corresponding null frames, respectively. Now we consider the transition r…
Figure 8
Figure 8. Figure 8: Relation of Transition Coefficients between Different Null Frames By (3.35) in Section 3.3.2, we already have |d ≤N−5 ( (int) f ′ , (int) f ′ )| ≲ ϵ1 = ϵ 2 0 , |d ≤N−5 log (int)λ ′ | ≲ 1 in (int)L0. 35Recall that in Section 3.3.2 we define the transition coefficients f…
Figure 9
Figure 9. Figure 9: , we then obtain the existence and estimates of (glo)ue as stated in below proposition. A Te He ue1 Hv1 Hδ (tran)ue = const Hδ+d Hu0 [PITH_FULL_IMAGE:figures/full_fig_p050_9.png]
Figure 10
Figure 10. Figure 10: Picture for Proof of Proposition 4.8 Recall that from (4.55) we have e˜ ′ 4 (˜r) = κg(r, θ)(r − r+) · κe− 1 2 κue g 2(r − r−) · Σ |q| 2 > 0. Together with (4.61), we infer that tr ˜χ ′ [PITH_FULL_IMAGE:figures/full_fig_p055_10.png]
Figure 11
Figure 11. Figure 11: Verification of Null Comparison Principle for MOTS Mue along He ue Since f < 0, we deduce L(R ′ , ue) ≥ L(R, ue) = 0, which implies 0 ≤ L[R ′ − R] =L(R ′ ) − L(R) = Z 1 0 ∂RL(tR1 + (1 − t)R2)dt[R ′ − R] =A ij (R ′ , R)Dij (R ′ − R) + B i (R ′ , R)Di(R ′ − R) + C(R ′ ,…
Figure 12
Figure 12. Figure 12: Dynamical Penrose Inequality Note that here we have the below explicit algebraic calculation: (7.5) r 2 +,∞ + a 2 ∞ = (m∞ + p m2∞ − a 2∞) 2 + a 2 ∞ = 2m2 ∞ + 2m∞ p m2∞ − a 2∞ ≤ 4m2 ∞. This yields m∞ ≥ vuut m∞  m∞ + p m2∞ − a 2∞  2 = q r 2 +,∞ + a 2∞ 2 . Meanwhile, i…
Figure 13
Figure 13. Figure 13: Future Development of Initial Data Set (Σ, g, k) Step 2. We then consider the MOTS along Σ. Employing the barrier argument as in [11, 31], we infer that there exists a smooth MOTS M0 on Σ near r = r+. As portrayed in [PITH_FULL_IMAGE:figures/full_fig_p077_13.png]
Figure 14
Figure 14. Figure 14: Proof of the Dynamical Penrose Inequality solved outgoing optical function ue and Se ue, (ext)ue denotes the intersection of Heue and the constant (ext)ue hypersurface. Applying the estimates (C.14) established in the proof of Proposition C.2, we deduce52 (7.8) |mH(ue…

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