Weak null singularity for the Einstein-Euler system

Referee: Proposition 2.3: the assertion that the singular Christoffels {trχ̄, χ̂, ω̄} cancel in deriving ∂_u(v, log τ) on H̄_0 is given in one sentence; please display the cancellation component-by-component for ∂_u v_A, ∂_u v³, ∂_u v⁴, ∂_u L, confirming no residual term proportional to a bounded p'-function times trχ̄ or χ̂ survives.

Authors: We agree this point deserves an explicit display and will expand Proposition 2.3 accordingly. The mechanism is the following. In the equations (2.1)–(2.2) the only places where ψ_H = {trχ̄, χ̂, ω̄} could enter ∂_u acting on (v,L) are through the covariant derivatives D_α v_β of components transverse to e_3, since by our frame choice e_3 = ∂_u + b^A ∂_A and ω̄ = 0 identically. Schematically: (i) D_A v_B = e_A v_B − Γ^C_{AB} v_C + ½ χ_{AB} v_4 + ½ χ̄_{AB} v_3 contributes χ̄_{AB} v³; (ii) D_A v_4 = e_A v_4 + ½ χ_{AB} v^B contributes only the regular χ; (iii) D_3 v_α involves no ψ_H other than ω̄ = 0. The χ̄_{AB} v³ term in D_A v_B feeds, via (2.1), into the equation for ∂_u v_A as +½ p' γ^{AB} χ̄_{AB} v³ = ½ p' trχ̄ v³, and via the v_α p' v^ν D_ν L term it appears as −v_A p' v³ (½ trχ̄). These two contributions cancel. The χ̄ contribution to ∂_u v³ from −2 χ̄_A^B v^B v_4-style products similarly recombines into trχ̄ pieces that cancel against the v³ trχ̄ explicitly displayed in (2.6). The traceless χ̂ never appears in isolation because v^A enters only through D_A acting on the *trace* combination v^4 v^4 + v^A v_A, which by the normalization v^4 = (2v³)⁻¹(γ^{AB} v_A v_B + 1) reduces to a trace contraction. We will add this explicit component-by-component computation as a lemma in Section 2.3. revision: yes

Referee: The schematic ∇_3 Z^i r =_s ε⁻¹ Σ Z^j r + Σ Z^j Γ in (5.2) and its decisive use in Proposition 5.1: please trace one representative case — e.g. (ε∇_4) ∇_3 v³ — exhibiting where ε⁻¹ enters, and confirm that higher commutator order produces no further negative powers of ε.

Authors: The ε⁻¹ originates exclusively from the rescaling built into Z = {∇_A, ε∇_4}. Concretely, the Euler equation written along ∇_3 in (2.2) has the schematic form ∇_3 r = (∇_A r) + Γ r (the ∇_4 r term is absent in the form solved for ∇_3, since the system is solved for the transverse derivative). Commuting ε∇_4 we obtain (ε∇_4)∇_3 r = ε(∇_4 ∇_A r) + ε(∇_4 Γ) r + ε Γ ∇_4 r. The last term is rewritten as Γ · (ε∇_4) r, which is regular. The first term, after [∇_4, ∇_A] commutation, contributes ε ∇_A ∇_4 r + ε ψ ∇_4 r = ∇_A (ε∇_4 r) + ψ (ε∇_4 r), again regular. The ε⁻¹ appears only when we re-express a bare ∇_4 r (not (ε∇_4) r) using a Z derivative, i.e. ∇_4 r = ε⁻¹ · Z r. This happens once per commutation level, and only on the source side of the equation; consequently at order i the worst factor is ε⁻¹ · Z^{i+1} r as written. No iterated ε⁻ᵏ with k > 1 appears, because each subsequent commutation reproduces the same algebraic structure (the ∇_4 derivative of the source is absorbed into Z). In Proposition 5.1 the ε⁻¹ is then paired with f(u) and integrated: ‖f(u) ε⁻¹‖_{L²_u} = ε⁻¹ ‖f‖_{L²_u} ≤ ε⁻¹ · ε² = ε under (1.5). We will add this trace-through as a remark following (5.2). revision: yes

Referee: In the (β,α) energy estimate the cubic commutator contribution is bounded by f(u)⁻² C(∆_1)(1+R+Õ_{4,2}); since f⁻² is only in L¹_u with norm controlled by ε⁴ (not uniformly), please state which norm of f⁻¹ is used and verify the Grönwall constant remains finite as u → u*.

Authors: Thank you — the bound is correct but the typesetting on that line is misleading and we will rewrite it. The factor f(u)⁻² there is *not* the integrand for the Grönwall step; it appears as the Hölder pairing ‖f⁻¹‖_{L²_u} · ‖f · (something)‖_{L²_u L²_ū L²(S)}, and we then absorb ‖f⁻¹‖²_{L²_u} ≤ ε⁴ from (1.5). Specifically, in the cubic term Σ Z^{i_1} ψ_H^{i_2} Z^{i_3} ψ_H Z^{i_4} α we place the most singular factor in L²_u with weight f, the second singular factor in L²_ū L^∞_u with weight f, and the curvature factor in L^∞_u L²_ū L²(S); the two weights f combine to f², which we then split as f² = (f⁻¹)⁻² and apply Cauchy–Schwarz against ‖f⁻¹‖_{L²_u}² ≤ ε⁴. The Grönwall integration that closes the (β,α) energy is in the ū direction (via ∇_3 transport) and uses only the trχ̄ multiplier, which is in L¹_ū L^∞_u L^∞(S) with norm ≤ ‖f⁻²‖_{L¹_ū} ≤ ε² < ∞. Hence the Grönwall constant is exp(C ε²), finite up to u → u*. We will rewrite the displayed line to make this Hölder pairing explicit rather than leaving the f⁻² out front. revision: yes

Referee: Theorem 2 requires smallness in both u* and ū* (Remark 1.2); please indicate whether the obstruction to fixed ū* is in the Einstein sector or the fluid sector, and which estimate fails.

Authors: We thank the referee for raising this; we did not want to expand on it in the present paper but are happy to add a discussion. The obstruction is in the Einstein sector, not the fluid sector. In the fluid energy estimate (Proposition 5.4) the prefactor (1 − (p')^{1/3}) > 0 is genuine and does not require smallness in ū*; the slower characteristic speed produces a true positive coercivity gap, and one can absorb the cross terms using only smallness in u*. The Einstein-side obstruction is the same one as in [19, Luk]: the trχ-multiplier in the (K, σ̌, β) energy estimate generates a Grönwall factor whose exponent is controlled by ‖trχ‖_{L¹_ū L^∞_u L^∞(S)}, which we currently bound only by Cε. Removing the smallness in ū* would require reorganizing this Grönwall as in [19] using the renormalized quantity trχ + (Ω-correction). We expect this can be done following [19] and have added a sentence to Remark 1.2 indicating that the obstruction is purely on the geometric side. revision: partial

Referee: Corollary 1.1 / Section 6: the existence of the (1.6)-class data is deferred and uses an extra ad hoc hypothesis (2.7) on ∂_u ∂_u Ψ. Please make explicit that (2.7) is compatible with the singular hierarchy assumed elsewhere, and that the resulting data still belong to Theorem 2's class; the openness of (1.6) deserves a stand-alone verification.

Authors: We will add a short stand-alone subsection at the end of Section 2.3 verifying that (2.7) is compatible with all assumptions of Theorem 2. Concretely: (2.7) requires |∂_u ∂_ū Ψ| ~ u⁻¹ f(ū)⁻², together with |(∂̸, ε∂_u)^i ∂_u ∂_ū Ψ| ≲ u⁻¹ f(ū)⁻² for i ≤ N. After integration in u this gives |∂_ū Ψ|(u, ū) ≲ log(1/u) f(ū)⁻², which is consistent with the bound |∂_ū Ψ| ~ f(ū)⁻² imposed earlier (the log(1/u) factor is harmless because u ≤ ε² and the bootstrap norms have an ε^{1/2} room). The L²-based norms used in Theorem 2's data hypotheses involve f(ū) Z^i χ̄ in L^∞_u L²(S_{0,ū}), and Z^4 χ̄ ~ ∂_u ∂_ū Ψ in L^∞_u L²(S_{0,ū}) is bounded by ‖u⁻¹‖_{L^∞_u} · f(ū)⁻² which is compatible (∇_4 commutators include ε∂_u, so the u⁻¹ is multiplied by ε to give ε/u ≤ 1 on the bootstrap region). Concerning openness of (1.6): the lower bound is a strict inequality on a compact set in (u,θ); since the map (initial Ψ, Φ) → α(·, 0) is continuous in the C¹ data topology by (A.11), a small perturbation of Ψ within the data class preserves (1.6). We will display this in a self-contained statement. revision: yes