Pith. sign in

REVIEW 4 major objections 5 minor 87 references

On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims a strictly conserved, positive-definite Hamiltonian energy exists for axially symmetric linear perturbations of subextremal Kerr black holes.

desk verdict A concrete positive-energy construction for a restricted class of axisymmetric perturbations; the main theorem is as strong as its unproved zero-angular-momentum and boundary-regularity assumptions. read the letter →

arxiv 2507.08326 v1 pith:6TPTHSHP submitted 2025-07-11 gr-qc math.APmath.DG

Nishanth Gudapati This is my paper · ORCID
classification gr-qcmath.APmath.DG MSC 83C5758E2058E3035C1535Q75
keywords Kerrblackholesholestabilityergo-regionHamiltonianmechanicswavemapsPoissonequationaxialsymmetrypositive-definiteenergy
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 claims that axially symmetric linear perturbations of the exterior of subextremal Kerr black holes admit a positive-definite and strictly conserved Hamiltonian energy, despite the ergo-region obstruction that usually prevents such an energy. If true, this provides a form of linear stability within axial symmetry for the full subextremal range $|a| < M$. The energy is constructed through a dimensional reduction to a $2+1$ Einstein--wave map system whose target is the negatively curved hyperbolic plane, and its conservation is established by showing that all boundary fluxes vanish at the horizon, the axes, and spatial infinity.

What carries the argument

The key machinery is the Hamiltonian dimensional reduction of $3+1$ axially symmetric Ricci-flat spacetimes to a $2+1$ Einstein--wave map system with the negatively curved hyperbolic $2$-plane as target. The positive-definite energy density involves the wave-map curvature term $-\tfrac12 N \bar\mu_q q^{ab}_0 h_{AE} U'^{A} R^{E}_{\,BCD}\,\partial_a U^B\,\partial_b U^C U'^D$, whose sign is fixed by negative target curvature. The conservation argument uses the fact that $(N,0)^T$ lies in the kernel of the adjoint of the linearized constraint map, so the time derivative of the energy density is a pure spatial divergence; the remaining task is to show that the resulting flux terms vanish dynamically at the boundaries of the orbit space.

What would settle it

Construct an admissible axially symmetric linear perturbation with nonzero angular-momentum perturbation that satisfies the linearized constraints in harmonic gauge, transform it to the Weyl--Papapetrou gauge, and check whether the boundary condition $U'|_{\Gamma_1}=U'|_{\Gamma_2}$ still holds and whether the horizon flux of $J^b$ vanishes; if either fails, the strict conservation claimed in Theorem 8.1(2) does not extend beyond the zero-$\delta J$ class.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 8.1: for the maximal development of axially symmetric linear perturbations of the exterior of subextremal Kerr spacetimes, after a $C^\infty$ diffeomorphism from harmonic coordinates to the Weyl--Papapetrou gauge, the regularized Hamiltonian $H^{\mathrm{Reg}}$ is strictly conserved both forwards and backwards in time. Since $H^{\mathrm{Reg}}$ is positive definite, this yields a spacetime gauge-invariant, target-gauge-independent, positive-definite and conserved energy for this perturbation class, giving a form of dynamical linear stability of the Kerr exterior.

Load-bearing premise

The paper assumes, without proof, that one may set the perturbation of angular momentum to zero without effective loss of generality; the conservation argument depends on the resulting equality of twist-potential boundary data on the two axes.

Editorial extensions

If this is right

  • For admissible axially symmetric linear perturbations, the energy at one time equals the energy at any later time without the need for a bulk Morawetz estimate.
  • The construction covers the full subextremal range $|a|<M$, not just slowly rotating Kerr, because the ergo-region positivity issue is resolved geometrically rather than by smallness of angular momentum.
  • The energy provides a natural starting point for uniform boundedness and decay estimates of Maxwell and linearized Einstein perturbations within axial symmetry.
  • The target-covariant formulation extends locally to higher-dimensional black holes with toroidal spacelike symmetry, such as five-dimensional Myers--Perry black holes.
  • The result confirms the axisymmetric-stability criterion tied to positivity of canonical energy: within this symmetry class, a failure of positive definiteness would have forced exponential growth.

Reading between the lines

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

  • The zero angular-momentum perturbation restriction is the effective boundary of the theorem: a perturbation with nonzero $\delta J$ could violate the twist-potential boundary condition $U'|_{\Gamma_1}=U'|_{\Gamma_2}$ on which the flux-vanishing argument rests, so the claimed conservation may not cover the full linearized phase space even within axial symmetry.
  • Conservation plus positivity alone does not imply decay; turning this energy into a Lyapunov functional would require coercivity estimates that control the fields near the axes and the corner $\Gamma \cap H^+$, where the energy density degenerates.
  • A numerical test could evolve a small axisymmetric perturbation in harmonic gauge, transform to Weyl--Papapetrou coordinates, and monitor $H^{\mathrm{Reg}}$ in time; if any boundary flux at the corner or horizon fails to vanish, the classification of flux terms in Section 8 would be contradicted.
Share X Bluesky LinkedIn Reddit HN

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 the existence of a positive-definite, strictly conserved Hamiltonian energy for axially symmetric linear perturbations of the exterior of subextremal Kerr spacetimes. In the first part (Sections 2–5), the author reduces the 3+1 vacuum Einstein equations under the SO(2) isometry to a 2+1 Einstein–wave map system with hyperbolic 2-plane target, linearizes it around Kerr, and constructs the energy functional H^Reg (Eqs. (4.1)–(4.2)) using linearization-stability/adjoint-kernel methods; H^Reg is shown to generate the constrained dynamics (Theorem 4.1) and to arise from a divergence-free vector density (Theorem 4.2). In the second part (Sections 6–8), the initial value problem is formulated in harmonic gauge, global regularity is asserted (Proposition 6.1), a C∞ gauge diffeomorphism to Weyl–Papapetrou coordinates is constructed (Section 7), and the boundary fluxes of the energy current are estimated and claimed to vanish at the axes, the horizon, and spatial infinity, yielding strict conservation (Theorem 8.1). The headline claim is Theorem 8.1(2): H^Reg is strictly conserved forwards and backwards in time for the maximal development of the linearized system, under the assumptions that the angular-momentum perturbation vanishes and that certain axis-regularity conditions are imposed.

Significance. If substantiated, the main theorem would supply a positive-definite, conserved energy for the axial sector of linearized gravity on Kerr for the full subextremal range, and the mechanism – negative curvature of the wave-map target combined with linearization stability – is genuinely different from the Carter–Robinson identities used in the author's earlier work; the construction also suggests extensions to Maxwell perturbations, higher-dimensional toroidal symmetries, and the Hollands–Wald canonical-energy framework. The paper's strengths are its explicit algebraic derivations (no free parameters, no fitted quantities), its transparent naming of assumptions imposed 'by fiat', and its honest discussion of the boundary-term difficulties in Section 5. The main reservations are: (i) the theorems are conditional on hypotheses (δJ=0, axis regularity, asserted spatial decay rates) whose necessity or propagation is not fully proved; (ii) the global statements (Prop. 6.1, Theorem 8.1(1)) are sketched rather than proved; and (iii) conservation without any decay statement supports only a weak notion of 'linear stability', as the paper itself hedges.

major comments (4)
  1. [Section 2, p. 14] The statement 'without effective loss of generality, we shall assume that the perturbation of the angular-momentum is zero' is load-bearing but unproved. Angular-momentum conservation for the vacuum axisymmetric problem implies only that δJ is constant in time, not that it vanishes; a perturbation along the Kerr family itself has δJ≠0. Section 4 (p. 28) uses this assumption to impose the twist boundary data U'^A|Γ1=U'^A|Γ2, and the Γ-flux estimates in Section 8 inherit it. Since general axially symmetric perturbations of Kerr need not preserve J, the theorem as stated (and the abstract's 'axially symmetric linear perturbations') overstates the class covered. Either the reduction to δJ=0 must be proved, or the theorem must be restated explicitly for the δJ=0 sector with the δJ≠0 case analyzed separately.
  2. [Section 4, p. 28; Section 6, p. 40] The axis regularity conditions (|Φ|'=0, ∂nU'^A=0, p'_A=∂tp'_A=∂np'_A=0, ∂nν'=0) are imposed 'by fiat' on the initial hypersurface and are used in Section 8 to kill every flux term at the axes, yet the paper does not prove that these conditions are preserved by the maximal development in harmonic gauge. The assertion in Section 6 ('we have ∂nA=0, for a scalar, and A⊥=0, ∂nA∥=0 ... globally in time') does not supply the required argument: propagation of boundary conditions for quotient fields is not a consequence of the hyperbolicity of the interior system; it requires a separate boundary/corner analysis that is not given. Without such an argument, the vanishing of the Γ-fluxes – and hence the strict conservation claim – rests on unverified hypotheses.
  3. [Proposition 6.1, Section 6; Theorem 8.1(1), Section 8] Proposition 6.1 asserts that the harmonic-gauge development is 'globally regular' including at the axes, infinity, and the corners Γ∩H+, and Theorem 8.1(1) asserts the existence of a C∞ diffeomorphism from harmonic coordinates to the Weyl–Papapetrou gauge. Both statements are supported only by sketches. The gauge-vector construction in Section 7 solves Poisson equations for Y^R/R and Y^θ with Dirichlet data at H+ and regularity conditions on Γ (pp. 46–47), but the compatibility of the overdetermined system, the regularity at the corners, and the global nature of the resulting diffeomorphism are asserted rather than proved. Since the conservation proof in Section 8 applies only to solutions obtained through this diffeomorphism, Theorem 8.1(2) inherits this gap.
  4. [Section 8, pp. 58–60] The flux-vanishing estimates rely on asymptotic behaviors stated without derivation, e.g., 'the behaviour of the canonical pair (γ′,p′) near the future horizon is 4∂Rγ′=(1−R+²/R²) and Np′/μq=O(1−R+²/R²)', and at spatial infinity 'γ′=1/R, ∂Rγ′=O(1/R²)', 'Np′/μq=O(1/R³)', '∂Rω=O(1/R³)'. These rates are not derived from the linearized Einstein system in harmonic gauge nor from the elliptic estimates of Section 7; they are assumed. A rigorous proof of strict conservation requires either a derivation of these pointwise rates from the initial data and the evolution equations, or the addition of such rates as explicit hypotheses of Theorem 8.1. As it stands, the conclusion that every flux integral vanishes is not established.
minor comments (5)
  1. [Section 6, p. 39] The statement 'The number of independent degrees of freedom, modulo the gauge degrees of freedom, is 6' is misleading; the standard count for linearized gravitational perturbations is 2 propagating degrees of freedom (10 components minus 4 gauge minus 4 constraints). The sentence should be corrected or clarified.
  2. [Section 8, p. 60] The displayed limit 'lim_{θ→0,π} ∂θγ′ = cot θ' cannot hold for a smooth perturbed field γ′; presumably this is a typo for the background ∂θγ. Please correct.
  3. [Sections 5 and 8] The two displayed definitions of the flux vector density (J^b)^Reg (p. 32 and p. 56) do not agree in indices and bracketing; these expressions should be reconciled so that the flux estimates can be checked against a single formula.
  4. [Theorem 4.1 proof, pp. 29–30] The first variation computation contains unexplained factors (N^{-1} versus N) and an apparent use of the momentum constraint inside Dp'_A·H^Reg; a cleaner derivation of the two Hamilton equations would aid verification.
  5. [Proposition 7.4, pp. 52–54] The equivalence of the Dirichlet (ν′=2γ′ on Γ) and Neumann (∂nν′=0 on Γ) data at the axes is asserted but not proved; since the O(1/R) decay of ν′ feeds into the J3–J5 flux estimates in Section 8, this equivalence needs a proof or an explicit reference.

Circularity Check

0 steps flagged · score 2.0 of 10

No constructional circularity: the energy is an explicit divergence computation; the δJ=0 and axis-regularity assumptions limit scope but are inputs, not outputs.

full rationale

The central claim is not equivalent to its inputs by construction. H^Reg in (4.1)-(4.2) is defined directly from perturbed phase-space variables; positivity is manifest from the target curvature and the τ'=0 gauge choice, and conservation is obtained by showing ∂_t e^Reg is a spatial divergence (Theorem 4.2), with each flux term in J^b estimated at Γ, H+, and i0 in Section 8. No parameter is fitted to data and then renamed a prediction. The main caveats are scope conditions rather than circular reductions: Section 2 states 'without effective loss of generality, we shall assume that the perturbation of the angular-momentum is zero', and Section 4 says 'We impose the regularity conditions on the axis of initial hypersurface Σ by fiat'; these are used to make axis fluxes vanish, so if they are genuine restrictions the theorem's advertised generality is unsupported. Likewise, Proposition 6.1 and the C∞-diffeomorphism in Theorem 8.1(1) are asserted with a sketch and the paper notes it is a merger of the author's preprints [35, 36]. These are completeness and provenance concerns, but they do not exhibit an equation that reduces to its own input, so the circularity score is low.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The central claim rests on a chain of geometric and analytic assumptions: the Weyl-Papapetrou representation, zero angular-momentum perturbation, axis regularity by fiat, harmonic-to-WP gauge passage, and the maximal gauge condition. None of these are fitted parameters; they are structural inputs that the conservation proof depends on.

assumptions (6)
  • domain assumption Kerr metric in subextremal range admits the Weyl-Papapetrou representation with the stated orbit space and boundary structure.
    Invoked throughout Sections 1, 5, and 7; the coordinate representation of Kerr is standard, but the global orbit-space picture, including axes and horizon as boundary, is a nontrivial geometric input.
  • ad hoc to paper The perturbation of angular momentum vanishes.
    Section 2 states 'without effective loss of generality, we shall assume that the perturbation of the angular-momentum is zero.' This is used for the twist-potential boundary data and flux vanishing; no proof that δJ=0 is WLOG is given.
  • ad hoc to paper Axis regularity conditions are imposed by fiat and propagate for all times.
    Section 4 says 'We impose the regularity conditions on the axis of initial hypersurface Σ by fiat'; Section 5 uses these conditions to lift fields to the original Σ and to obtain decay rates. The propagation is not proved independently.
  • domain assumption Global existence in harmonic gauge and existence of a C∞ diffeomorphism to Weyl-Papapetrou gauge.
    Proposition 6.1 asserts global regularity and global hyperbolicity; Theorem 8.1 assumes the C∞ diffeomorphism. The proof is a sketch based on standard hyperbolic theory and does not construct the diffeomorphism.
  • domain assumption The 2+1 maximal gauge condition τ'=0 is preserved and compatible with all admissible initial data.
    H^Reg is manifestly positive only when τ'=0; Theorems 4.1 and 8.1 restrict to CH' ∩ CH'a ∩ Cτ'. Compatibility of this gauge with the harmonic-to-WP transformation is assumed.
  • standard math Standard elliptic theory and method of images yield O(1/R) decay for the Poisson problems in the orbit space.
    Section 7 uses representation formula (7.5), reflection-antisymmetric images, and separated solutions to derive decay rates; these are standard tools, but their application here assumes the stated reflection and boundary conditions.
invented entities (1)
  • H^Reg, the regularized Hamiltonian energy functional
    purpose: Provides the candidate positive-definite conserved energy for axially symmetric linear perturbations of Kerr.
    The functional is constructed inside the paper via divergence identities and the maximal gauge condition; no falsifiable prediction outside the paper is supplied. Its positivity is largely built in by construction, not inferred from external data.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes." pith.science (2026). https://pith.science/paper/6TPTHSHP

@misc{pith2026250708326,
  author       = {Pith},
  title        = {Pith review of: On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6TPTHSHP}},
  note         = {Machine review of arXiv:2507.08326}
}
read the original abstract

The lack of a positive-definite and conserved energy is a serious obstacle in the black hole stability problem. In this work, we will show that there exists a positive-definite and conserved Hamiltonian energy for axially symmetric linear perturbations of the exterior of Kerr black hole spacetimes. In the first part, based on the Hamiltonian dimensional reduction of 3+1 axially symmetric, Ricci-flat Lorentzian spacetimes to a 2+1 Einstein-wave map system with the negatively curved hyperbolic 2-plane target, we construct a positive-definite, spacetime gauge-invariant energy functional for linear axially symmetric perturbations in the exterior of Kerr black holes, in a manner that is also gauge-independent on the target manifold. In the construction of the positive-definite energy, various dynamical terms at the boundary of the orbit space occur critically. In the second part, after setting up the initial value problem in harmonic coordinates, we prove that the positive energy for the axially symmetric linear perturbative theory of Kerr black holes is strictly conserved in time, by establishing that all the boundary terms dynamically vanish for all times. This result implies a form of dynamical linear stability of the exterior of Kerr black hole spacetimes.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

87 extracted references · 63 canonical work pages

  1. [61]

    Moncrief V., Gudapati N., A positive-definite energy functional for the axisymmetric perturbations of Kerr– Newman black holes, arXiv:2105.12632

    work page Pith review arXiv

  2. [1]

    Stability for linearized gravity on the Kerr spacetime

    Andersson L., B¨ ackdahl T., Blue P., Ma S., Stability for linearized gravity on the Kerr spacetime, Ann. PDE 11 (2025), 11, 161 pages, arXiv:1903.03859

    work page Pith review arXiv 2025

  3. [2]

    Andersson L., Blue P., Hidden symmetries and decay for the wave equation on the Kerr spacetime, Ann. of Math. 182 (2015), 787–853, arXiv:0908.2265

    arXiv 2015

  4. [3]

    Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior

    Andersson L., Blue P., Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior, J. Hyperbolic Differ. Equ. 12 (2015), 689–743, arXiv:1310.2664

    work page Pith review arXiv 2015

  5. [4]

    Global Regularity for the 2+1 Dimensional Equivariant Einstein-Wave Map System

    Andersson L., Gudapati N., Szeftel J., Global regularity for the 2 + 1 dimensional equivariant Einstein-wave map system, Ann. PDE 3 (2017), 13, 142 pages, arXiv:1501.00616

    work page Pith review arXiv 2017

  6. [5]

    Arnowitt R., Deser S., Misner C.W., The dynamics of general relativity, in Gravitation: An Introduction to Current Research, Wiley, New York, 1962, 227–265, arXiv:gr-qc/0405109

    arXiv 1962

  7. [6]

    Ashtekar A., Varadarajan M., Striking property of the gravitational Hamiltonian, Phys. Rev. D 50 (1994), 4944–4956, arXiv:gr-qc/9406040

    work page Pith review arXiv 1994

  8. [7]

    Decay of the Maxwell field on the Schwarzschild manifold

    Blue P., Decay of the Maxwell field on the Schwarzschild manifold, J. Hyperbolic Differ. Equ. 5 (2008), 807–856, arXiv:0710.4102

    work page Pith review arXiv 2008

Show all 87 references
  1. [8]

    Physics 7 (1959), 466–483

    Brill D.R., On the positive definite mass of the Bondi–Weber–Wheeler time-symmetric gravitational waves, Ann. Physics 7 (1959), 466–483

    1959

  2. [9]

    Thesis, University of New England, 1983

    Bunting G., Proof of the uniqueness conjecture for black holes, Ph.D. Thesis, University of New England, 1983

    1983

  3. [10]

    Carter B., Axisymmetric black hole has only two degrees of freedom, Phys. Rev. Lett. 26 (1971), 331–333

    1971

  4. [11]

    Chandrasekhar S., The mathematical theory of black holes, Internat. Ser. Monogr. Phys. , Vol. 69, The Clarendon Press, New York, 1983

    1983

  5. [12]

    Choquet-Bruhat Y., Geroch R., Global aspects of the Cauchy problem in general relativity, Comm. Math. Phys. 14 (1969), 329–335

    1969

  6. [13]

    Choquet-Bruhat Y., Moncrief V., Existence theorem for solutions of Einstein’s equations with 1 parameter spacelike isometry groups, in Quantization, Nonlinear Partial Differential Equations, and Operator Algebra (Cambridge, MA, 1994), Proc. Sympos. Pure Math. , Vol. 59, Americ...

    1994

  7. [14]

    Henri Poincar´ e2 (2001), 1007–1064, arXiv:gr-qc/0112049

    Choquet-Bruhat Y., Moncrief V., Future global in time Einsteinian spacetimes with U(1) isometry group, Ann. Henri Poincar´ e2 (2001), 1007–1064, arXiv:gr-qc/0112049

    2001 arXiv

  8. [15]

    Pure Appl

    Christodoulou D., Tahvildar-Zadeh A.S., On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math. 46 (1993), 1041–1091

    1993

  9. [16]

    PDE 5 (2019), 2, 118 pages, arXiv:1711.07944

    Dafermos M., Holzegel G., Rodnianski I., Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: The case |a| ≪M , Ann. PDE 5 (2019), 2, 118 pages, arXiv:1711.07944. 70 N. Gudapati

    2019 arXiv

  10. [17]

    222 (2019), 1–214, arXiv:1601.06467

    Dafermos M., Holzegel G., Rodnianski I., The linear stability of the Schwarzschild solution to gravitational perturbations, Acta Math. 222 (2019), 1–214, arXiv:1601.06467

    2019 arXiv

  11. [18]

    Dafermos M., Holzegel G., Rodnianski I., Taylor M., The non-linear stability of the Schwarzschild family of black holes, arXiv:2104.08222

    arXiv

  12. [19]

    Dafermos M., Rodnianski I., A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds, Invent. Math. 185 (2011), 467–559, arXiv:0805.4309

    2011 arXiv

  13. [20]

    Dafermos M., Rodnianski I., Shlapentokh-Rothman Y., Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M, Ann. of Math. 183 (2016), 787–913, arXiv:1402.7034

    2016 arXiv

  14. [21]

    Dain S., Axisymmetric evolution of Einstein equations and mass conservation, Classical Quantum Gravity 25 (2008), 145021, 18 pages, arXiv:0804.2679

    2008 arXiv

  15. [22]

    Dain S., de Austria I.G., On the linear stability of the extreme Kerr black hole under axially symmetric perturbations, Classical Quantum Gravity 31 (2014), 195009, 30 pages

    2014

  16. [23]

    Emparan R., Reall H.S., Black holes in higher dimensions, Living Rev. Relativ. 11 (2008), 6, 87 pages, arXiv:0801.3471

    2008 arXiv

  17. [24]

    Finster F., Kamran N., Smoller J., Yau S.T., An integral spectral representation of the propagator for the wave equation in the Kerr geometry, Comm. Math. Phys. 260 (2005), 257–298, arXiv:gr-qc/0310024

    2005 arXiv

  18. [25]

    Finster F., Kamran N., Smoller J., Yau S.T., Decay of solutions of the wave equation in the Kerr geometry, Comm. Math. Phys. 264 (2006), 465–503, arXiv:gr-qc/0504047

    2006 arXiv

  19. [26]

    Decay of solutions of the wave equation in the Kerr geometry

    Finster F., Kamran N., Smoller J., Yau S.T., Erratum to “Decay of solutions of the wave equation in the Kerr geometry”, Comm. Math. Phys. 280 (2008), 563–573

    2008

  20. [27]

    Finster F., Kamran N., Smoller J., Yau S.T., A rigorous treatment of energy extraction from a rotating black hole, Comm. Math. Phys. 287 (2009), 829–847, arXiv:gr-qc/0701018

    2009 arXiv

  21. [28]

    Fischer A.E., Marsden J.E., Deformations of the scalar curvature, Duke Math. J. 42 (1975), 519–547

    1975

  22. [29]

    Fischer A.E., Marsden J.E., Moncrief V., The structure of the space of solutions of Einstein’s equations. I. One Killing field, Ann. Inst. H. Poincar´ e Sect. A (N.S.) 33 (1980), 147–194

    1980

  23. [30]

    Geroch R., A method for generating solutions of Einstein’s equations, J. Math. Phys. 12 (1971), 918–924

    1971

  24. [31]

    Giorgi E., Klainerman S., Szeftel J., A general formalism for the stability of Kerr, arXiv:2002.02740

    2002 arXiv

  25. [32]

    Giorgi E., Klainerman S., Szeftel J., Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes, Pure Appl. Math. Q. 20 (2024), 2865–3849, arXiv:2205.14808

    2024 arXiv

  26. [33]

    Thesis, Freie Uni- versit¨ at Berlin, 2013, arXiv:1311.4495

    Gudapati N., The Cauchy problem for energy critical self-gravitating wave maps, Ph.D. Thesis, Freie Uni- versit¨ at Berlin, 2013, arXiv:1311.4495

    2013 arXiv

  27. [34]

    Gudapati N., A note on the dimensional reduction of axisymmetric spacetimes, arXiv:1702.07950

    arXiv

  28. [35]

    Gudapati N., Axially symmetric perturbations of Kerr black holes I: A gauge-invariant construction of ADM energy, arXiv:1904.09670

    1904 arXiv

  29. [36]

    Gudapati N., Axially symmetric perturbations of Kerr black holes II: Boundary behaviour of the dynamics in the orbit space, arXiv:2109.02131

    arXiv

  30. [37]

    Gudapati N., A positive-definite energy functional for axially symmetric Maxwell’s equations on Kerr– de Sitter black hole spacetimes, C. R. Math. Acad. Sci. Soc. R. Can. 40 (2018), 39–54, arXiv:1710.11294

    2018 arXiv

  31. [38]

    Relativity Gravi- tation 50 (2018), 93, 25 pages, arXiv:1703.06331

    Gudapati N., On 3 + 1 Lorentzian Einstein manifolds with one rotational isometry, Gen. Relativity Gravi- tation 50 (2018), 93, 25 pages, arXiv:1703.06331

    2018 arXiv

  32. [39]

    Gudapati N., A conserved energy for axially symmetric Newman–Penrose–Maxwell scalars on Kerr black holes, Proc. A. 475 (2019), 20180686, 18 pages, arXiv:1806.09162

    2019 arXiv

  33. [40]

    H¨ afner D., Hintz P., Vasy A., Linear stability of slowly rotating Kerr black holes,Invent. Math. 223 (2021), 1227–1406, arXiv:1906.00860

    2021 arXiv

  34. [41]

    Hollands S., Ishibashi A., Black hole uniqueness theorems in higher dimensional spacetimes, Classical Quan- tum Gravity 29 (2012), 163001, 47 pages, arXiv:1206.1164

    2012 arXiv

  35. [42]

    Hollands S., Wald R.M., Stability of black holes and black branes, Comm. Math. Phys. 321 (2013), 629–680, arXiv:1201.0463

    2013 arXiv

  36. [43]

    On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 71

    Holzegel G., Conservation laws and flux bounds for gravitational perturbations of the Schwarzschild metric, Classical Quantum Gravity 33 (2016), 205004, 28 pages. On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 71

    2016

  37. [44]

    Differential Geom

    Hung P.-K., Keller J., Wang M.-T., Linear stability of Schwarzschild spacetime: Decay of metric coefficients, J. Differential Geom. 116 (2020), 481–541, arXiv:1702.02843

    2020 arXiv

  38. [45]

    PDE 1 (2015), 1, 78 pages, arXiv:1412.5679

    Ionescu A.D., Klainerman S., On the global stability of the wave-map equation in Kerr spaces with small angular momentum, Ann. PDE 1 (2015), 1, 78 pages, arXiv:1412.5679

    2015 arXiv

  39. [46]

    Klainerman S., Szeftel J., Kerr stability for small angular momentum, Pure Appl. Math. Q. 19 (2023), 791–1678, arXiv:2104.11857

    2023 arXiv

  40. [47]

    Krieger J., Schlag W., Concentration compactness for critical wave maps, EMS Monogr. Math. , European Mathematical Society (EMS), Z¨ urich, 2012

    2012

  41. [48]

    Henri Poincar´ e 21 (2020), 815–863, arXiv:1705.06621

    Ma S., Uniform energy bound and Morawetz estimate for extreme components of spin fields in the ex- terior of a slowly rotating Kerr black hole I: Maxwell field, Ann. Henri Poincar´ e 21 (2020), 815–863, arXiv:1705.06621

    2020 arXiv

  42. [49]

    Ma S., Uniform energy bound and Morawetz estimate for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole II: Linearized gravity, Comm. Math. Phys. 377 (2020), 2489–2551, arXiv:1708.07385

    2020 arXiv

  43. [50]

    Mazur P.O., Black hole uniqueness theorems, arXiv:hep-th/0101012

    arXiv

  44. [51]

    Misner C.W., Harmonic maps as models for physical theories, Phys. Rev. D 18 (1978), 4510–4524

    1978

  45. [52]

    Misner C.W., Thorne K.S., Wheeler J.A., Gravitation, W. H. Freeman and Company, San Francisco, CA, 1973

    1973

  46. [53]

    Moncrief V., Gravitational perturbations of spherically symmetric systems. I. The exterior problem, Ann. Physics 88 (1974), 323–342

    1974

  47. [54]

    Moncrief V., Odd-parity stability of a Reissner–Nordstr¨ om black hole, Phys. Rev. D. 9 (1974), 2707–2709

    1974

  48. [55]

    Moncrief V., Stability of Reissner–Nordstr¨ om black holes, Phys. Rev. D. 10 (1974), 1057–1059

    1974

  49. [56]

    Moncrief V., Gauge invariant perturbations of Reissner–Nordstr¨ om black holes, Phys. Rev. D. 12 (1975), 1526–1537

    1975

  50. [57]

    Moncrief V., Spacetime symmetries and linearization stability of the Einstein equations. I, J. Math. Phys. 16 (1975), 493–498

    1975

  51. [58]

    Physics 167 (1986), 118–142

    Moncrief V., Reduction of Einstein’s equations for vacuum space-times with spacelike U(1) isometry groups, Ann. Physics 167 (1986), 118–142

    1986

  52. [59]

    Moncrief V., Reduction of the Einstein–Maxwell and Einstein–Maxwell–Higgs equations for cosmological spacetimes with spacelike U(1) isometry groups, Classical Quantum Gravity 7 (1990), 329–352

    1990

  53. [60]

    Moncrief V., Relativistic Teichm¨ uller theory: a Hamilton–Jacobi approach to 2 + 1-dimensional Einstein gravity, in Surveys in Differential Geometry. Vol. XII. Geometric Flows, Surv. Differ. Geom. , Vol. 12, International Press, Somerville, MA, 2008, 203–249

    2008

  54. [62]

    Prabhu K., Wald R.M., Black hole instabilities and exponential growth, Comm. Math. Phys. 340 (2015), 253–290, arXiv:1501.02522

    2015 arXiv

  55. [63]

    Regge T., Wheeler J.A., Stability of a Schwarzschild singularity, Phys. Rev. 108 (1957), 1063–1069

    1957

  56. [64]

    Rinne O., Stewart J.M., A strongly hyperbolic and regular reduction of Einstein’s equations for axisymmetric spacetimes, Classical Quantum Gravity 22 (2005), 1143–1166, arXiv:gr-qc/0502037

    2005 arXiv

  57. [65]

    Robinson D.C., Classification of black holes with electromagnetic fields, Phys. Rev. D 10 (1974), 458–460

    1974

  58. [66]

    Schoen R., Yau S.T., On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), 45–76

    1979

  59. [67]

    II, Comm

    Schoen R., Yau S.T., Proof of the positive mass theorem. II, Comm. Math. Phys. 79 (1981), 231–260

    1981

  60. [68]

    Henri Poincar´ e14 (2013), 1747–1773, arXiv:1209.0019

    Schoen R., Zhou X., Convexity of reduced energy and mass angular momentum inequalities, Ann. Henri Poincar´ e14 (2013), 1747–1773, arXiv:1209.0019

    2013 arXiv

  61. [69]

    Pure Appl

    Shatah J., Tahvildar-Zadeh A., Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds, Comm. Pure Appl. Math. 45 (1992), 947–971

    1992

  62. [70]

    Pure Appl

    Shatah J., Tahvildar-Zadeh A.S., On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math. 47 (1994), 719–754. 72 N. Gudapati

    1994

  63. [71]

    Sterbenz J., Tataru D., Energy dispersed large data wave maps in 2 + 1 dimensions, Comm. Math. Phys. 298 (2010), 139–230, arXiv:0906.3384

    2010 arXiv

  64. [72]

    Sterbenz J., Tataru D., Regularity of wave-maps in dimension 2 + 1, Comm. Math. Phys. 298 (2010), 231–264, arXiv:0907.3148

    2010 arXiv

  65. [73]

    Struwe M., Radially symmetric wave maps from (1+2)-dimensional Minkowski space to the sphere, Math. Z. 242 (2002), 407–414

    2002

  66. [74]

    Pure Appl

    Struwe M., Equivariant wave maps in two space dimensions, Comm. Pure Appl. Math. 56 (2003), 815–823

    2003

  67. [75]

    Synge J.L., On the deviation of geodesics and null-geodesics, particularly in relation to the properties of spaces of constant curvature and indefinite line-element, Ann. of Math. 35 (1934), 705–713

    1934

  68. [76]

    Tao T., Global regularity of wave maps. I. Small critical Sobolev norm in high dimension, Int. Math. Res. Not. (2001), 299–328, arXiv:math.AP/0010068

    2001

  69. [77]

    Tao T., Global regularity of wave maps. II. Small energy in two dimensions, Comm. Math. Phys. 224 (2001), 443–544, arXiv:math.AP/0011173

    2001

  70. [78]

    Tao T., Global regularity of wave maps. III. Large energy from R1+2 to hyperbolic spaces, arXiv:0805.4666

    arXiv

  71. [80]

    Tao T., Global regularity of wave maps. V. Large data local wellposedness and perturbation theory in the energy class, arXiv:0806.3592

    arXiv

  72. [81]

    Tao T., Global regularity of wave maps. VI. Abstract theory of minimal-energy blowup solutions, arXiv:0906.2833

    arXiv

  73. [82]

    Tao T., Global regularity of wave maps. VII. Control of delocalised or dispersed solutions, arXiv:0908.0776

    arXiv

  74. [83]

    Tataru D., Tohaneanu M., A local energy estimate on Kerr black hole backgrounds, Int. Math. Res. Not. 2011 (2011), 248–292, arXiv:0810.5766

    2011 arXiv

  75. [84]

    Teukolsky S.A., The Kerr metric, Classical Quantum Gravity 32 (2015), 124006, 32 pages, arXiv:1410.2130

    2015 arXiv

  76. [85]

    Thorne K.S., Energy of infinitely long, cylindrically symmetric systems in general relativity, Phys. Rev. 138 (1965), B251–B266

    1965

  77. [86]

    Witten E., A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), 381–402

    1981

  78. [87]

    Zerilli F.J., Effective potential for even parity Regge–Wheeler gravitational perturbation equations, Phys. Rev. Lett. 24 (1970), 737–738

    1970

  79. [88]

    Zerilli F.J., Perturbation analysis for gravitational and electromagnetic radiation in a Reissner–Nordstr¨ om geometry, Phys. Rev. D. 9 (1974), 860–868

    1974

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.