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Cited by 92Black Holes and Theoretical PhysicsRelativity and Gravitational TheoryCosmology and Gravitation Theories

Causality and hyperbolicity of Lovelock theories

Open access
Reall, Harvey S, Tanahashi, Norihiro, Way, Benson·2014·Classical and Quantum Gravity
DOI 10.1088/0264-9381/31/20/205005
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Abstract

In Lovelock theories, gravity can travel faster or slower than light. The causal structure is determined by the characteristic hypersurfaces. We generalize a recent result of Izumi to prove that any Killing horizon is a characteristic hypersurface for all gravitational degrees of freedom of a Lovelock theory. Hence gravitational signals cannot escape from the region inside such a horizon. We investigate the hyperbolicity of Lovelock theories by determining the characteristic hypersurfaces for various backgrounds. First we consider Ricci flat type N spacetimes. We show that characteristic hypersurfaces are generically all non-null and that Lovelock theories are hyperbolic in any such spacetime. Next we consider static, maximally symmetric black hole solutions of Lovelock theories. Again, characteristic surfaces are generically non-null. For some small black holes, hyperbolicity is violated near the horizon. This implies that the stability of such black holes is not a well-posed problem.

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Written from the paper’s full text.

TL;DR

The paper proves that any Killing horizon is a characteristic hypersurface for all gravitational degrees of freedom in Lovelock theories, so gravitational signals cannot escape such horizons. It then shows Lovelock theories are hyperbolic in Ricci-flat type N spacetimes, but hyperbolicity fails near the horizon of some small static maximally symmetric black holes, making linearized stability ill-posed.

Key findings

  • A Killing horizon is characteristic for all gravitational degrees of freedom in any Lovelock theory; the kernel of the principal symbol on the horizon has dimension \(d(d-3)/2\).
  • In Ricci-flat type N spacetimes, there are \(d(d-3)/2\) effective metrics \(G_{I\,ab}=g_{ab}+\omega_I \ell_a \ell_b\); characteristic hypersurfaces are null with respect to one of them, and Lovelock theory is hyperbolic in any such background. For \(d=5\), the relevant eigenvalues include \(\nu_I=\pm 2\sqrt{\Omega'_{ij}\Omega'_{ij}}\).
  • For static, maximally symmetric black holes, characteristic surfaces are null with respect to effective metrics for scalar, vector, and tensor perturbations. Large black holes have all \(c_A(r)>0\) outside the horizon and are hyperbolic; for some small black holes, one \(c_A\) vanishes at \(r_c>r_0\) and becomes negative for \(r<r_c\), violating hyperbolicity. Examples include scalar modes in \(d=5\) and tensor modes in \(d=6\) Einstein-Gauss-Bonnet theory with \(\Lambda=0\).
  • When hyperbolicity is violated, linearized “bad” modes grow as \(e^{\alpha l t}\), solutions do not depend continuously on initial data, and for generic smooth initial perturbations no corresponding linearized solution exists.

Methods

  • The paper uses a second-order, gauge-invariant principal-symbol approach: it defines the characteristic polynomial \(Q(x,\xi)=\det P(x,\xi)\), characterizes hypersurfaces by \(Q=0\), and defines hyperbolicity via Lovelock-spacelike surfaces and real roots.
  • For Ricci-flat type N spacetimes, it introduces a null basis, computes the principal symbol, diagonalizes the relevant map on traceless symmetric matrices, and derives the effective metrics.
  • For static, maximally symmetric black holes, it uses known scalar/vector/tensor master equations for linearized perturbations and their large-\(l\) behavior to read off the effective metrics \(G_A^{\mu\nu}\) and the functions \(c_A(r)\).

Limitations

  • The paper states that its study of hyperbolicity in black hole backgrounds is restricted to the black hole exterior; the interior is not investigated.
  • The paper states that the two cases studied are atypical: factorization of the characteristic polynomial into quadratic factors and the appearance of effective metrics are special to these solutions, while generic Lovelock solutions are expected to have a higher-degree normal cone.
  • The paper states that extending the Killing-horizon result to stationary black holes would require a Lovelock analogue of Hawking’s rigidity theorem, and whether hyperbolicity violation can arise dynamically remains an open question.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1088/0264-9381/31/20/205005

Cites

45 of 45 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

14 of 21 cited sources checked · 6 supported · 0 flagged · accuracy not yet measured

  • Findingcitation

    Any Ricci flat type N spacetime is a solution of Lovelock theory with Λ = 0.

    “We note that any such spacetime is a solution of Lovelock theory (with Λ = 0) (slightly extending a result of [13]).”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [13] source

      Section II.A (p. 3) states that all Weyl type N Einstein spacetimes, which include Ricci-flat type N spacetimes, solve quadratic gravity with an appropriately chosen effective cosmological constant. Its footnote notes the Λ = 0 result for Gauss–Bonnet gravity. The paper does not establish the extension to Lovelock theory; the citing sentence itself calls that a slight extension of the cited result. (LLM, conf=0.86)

      Experimental — accuracy not yet measured
  • Findingcitation

    For certain small Lovelock black holes, one of the effective metrics may change signature near, but outside, the horizon of the black hole.

    “It turns out that, for certain small Lovelock black holes, one of these metrics may change signature near, but outside, the horizon of the black hole.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [25] source

      Section 3.2 reports that sufficiently small Lovelock black holes can have an instability-producing negative region outside the horizon. Section 3.1 also says that a negative region of T outside the horizon gives perturbations a wrong-sign kinetic term. The paper does not describe this as an effective metric changing signature, so that specific interpretation is not established by our copy. (LLM, conf=0.82)

      Experimental — accuracy not yet measured
  • Findingreasoning

    For static spherically or planar symmetric black hole solutions of Lovelock theories, there exist effective metrics G_S, G_V and G_T such that characteristic hypersurfaces are null hypersurfaces with respect to one of these metrics.

    “From this one can infer that there exist 'effective metrics' G ab S , G ab V and G ab T such that characteristic hypersurfaces are null hypersurfaces with respect to one of these metrics.”
    Quote in the paperSubstantial claim3 of 3 sources checked
    • cites [18] Dotti G, 2005, Class. Quantum Grav. source

      Section 2 (pp. L2–L3) studies tensor perturbations of spherically symmetric Einstein–Gauss–Bonnet solutions, and Section 4 (p. L6) summarizes a stability analysis. Our copy does not address effective metrics for scalar, vector, and tensor modes or say that characteristic hypersurfaces are null with respect to them (searched: effective metrics, characteristic hypersurfaces, and propagation cones). (LLM, conf=0.76)

      Experimental — accuracy not yet measured
    • cites [19] source

      Sections II–III study tensor perturbations of static Einstein–Gauss–Bonnet spacetimes, and Section VI says vector results are preliminary and the scalar case remains under investigation. Searches for effective metrics, characteristic hypersurfaces, and null propagation found no passage establishing the claimed three metrics or their relation to characteristics. (LLM, conf=0.76)

      Experimental — accuracy not yet measured
    • cites [20] source

      The Introduction and Conclusions discuss tensor, vector, and scalar perturbations of Einstein–Gauss–Bonnet spacetimes as a stability problem. Searches for “effective metric,” “characteristic,” and “null hypersurface” found no passage stating that characteristic hypersurfaces are null with respect to sector-specific effective metrics. (LLM, conf=0.76)

      Experimental — accuracy not yet measured
  • Premisecitation

    Lovelock theories admit static, maximally symmetric black hole solutions with metric of the form (5.1).

    “Lovelock theories admit static, maximally symmetric black hole solutions with metric of the form [14-17] Σ = -+ + ds f r dt f r dr r d ( ) ( ) , (5.1) 2 2 1 2 2 2”
    Quote in the paperSubstantial claim3 of 4 sources checked
    • cites [14] source

      In the section headed “S. Deser” (page unavailable), the paper derives static, spherically symmetric solutions in Schwarzschild gauge for its Einstein–Gauss–Bonnet model and describes solutions with event horizons. This supports a black-hole solution of the stated general form, but the copy does not establish the claim for Lovelock theories more broadly or for maximally symmetric horizons beyond spherical ones; the metric equation itself is not decoded in our copy. (LLM, conf=0.84)

      Experimental — accuracy not yet measured
    • cites [15] source
    • cites [16] source
    • cites [17] source
  • Findingown data

    A Killing horizon is a characteristic hypersurface for all gravitational degrees of freedom, generalising a result of [9].

    “We proved that a Killing horizon is a characteristic hypersurface for all gravitational degrees of freedom, generalising a result of [9].”
    Quote in the paperSubstantial claim1 of 1 sources checked
  • Classified as probably not a substantial claim. 1
  • May combine two claims. 28
  • The quote only partly states this claim. 2

  • Classified as probably not a substantial claim.

    Premisecitation

    Izumi proved that a Killing horizon is a characteristic hypersurface for all gravitational degrees of freedom in Einstein-Gauss-Bonnet theory.

    “He proved that a Killing horizon is a characteristic hypersurface for all gravitational degrees of freedom.”
    Quote in the paperSubstantial claim1 of 1 sources checked

  • May combine two claims.

    Premisecitation

    For bad modes, V_l is negative for r <= r_bad and admits negative energy bound states.

    “But for the bad modes, V r ( ) l is negative for ⩽ r r bad , and admits negative energy bound states [18].”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [18] Dotti G, 2005, Class. Quantum Grav. source

      Section 3 gives a negative-potential-integral criterion for negative-energy bound states and shows it holds for sufficiently small mass and large angular eigenvalue in the exceptional n = 4 case. The copy does not define r_bad or establish that the potential is negative throughout r ≤ r_bad. No page number is available for these passages in the indexed copy. (LLM, conf=0.83)

      Experimental — accuracy not yet measured

  • May combine two claims.

    Findingown data

    For a d = 5 small spherical black hole with Λ = 0, the scalar modes have V_eff < 0 near the horizon, giving the instability discovered in [20]; c_S < 0 implies the theory is not hyperbolic near the horizon.

    “For the d = 5 example, the plot for the scalar modes reveals that < V 0 eff near the horizon.This gives rise to the instability discovered in [20].However, from our perspective this is much worse than an instability: < c 0 S implies that the theory is not hyperbolic near the horizon.”
    Quote in the paperSubstantial claim1 of 1 sources checked

  • May combine two claims.

    Findingown data

    For a d = 6 small spherical black hole with Λ = 0, the tensors have V_eff < 0 near the horizon, giving the instability of [18,19]; this is a failure of hyperbolicity.

    “Similarly, for the d = 6 example the tensors have < V 0 eff near the horizon, which gives the instability of [18,19].But again, this is much worse than an instability, it is a failure of hyperbolicity of the theory.”
    Quote in the paperSubstantial claim2 of 2 sources checked
    • cites [18] Dotti G, 2005, Class. Quantum Grav. source
    • cites [19] source

      The stability analysis (p. 12) says that for a six-dimensional spherical black hole with Λ = 0, low mass brings a tensor-mode instability. Section V (pp. 10–11) establishes instability when its S-deformed potential is negative somewhere, but does not specifically state that the undeformed effective potential is negative near the horizon. The failure-of-hyperbolicity characterization is the citing authors’ own conclusion, not a finding established by this source. (LLM, conf=0.84)

      Experimental — accuracy not yet measured

  • May combine two claims.

    Findingnone given

    The ω_I are real homogeneous functions of Ω'ij of weight 1 and are independent of k_p for p > 2.

    “The ω I are real homogeneous functions of Ω ′ ij of weight 1 and independent of k p for > p 2.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given

    Generically ω_I is nonzero for all I, so the null cones of the effective metrics do not coincide with the null cone of the physical metric.

    “Generically ω ≠ 0 I for all I so the null cones of the effective metrics do not coincide with the null cone of the physical metric.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given

    For generic Ω'ij, the quantities ω_I are distinct and hence the effective metrics G_I are distinct.

    “For generic Ω ′ ij , the quantities ω I are distinct and hence so are the metrics G Iab .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    There is generically at least one positive ω_I, and the outermost cone lies outside the light cone except where they are tangent along l.

    “We will show below that there is generically at least one positive ω I and so this outermost cone lies outside the light cone (except where they are tangent along ℓ a ).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    If ξ^a is null and parallel to l^a, then ξ^a is characteristic and non-trivial elements of the kernel of P correspond to the subspace V_physical^transverse of V_physical, with dimension d(d-3)/2.

    “Finally, if ξ a is null and parallel to ℓ a then  ξ x ( , ) vanishes so the analysis is the same as for GR (section 3.2).Such ξ a is characteristic and non-trivial elements of the kernel of P correspond to the subspace V physicaltransverse of V physical , with dimension d d ( 3) 2.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    The vector l is null with respect to any of the effective metrics, so the null cones are all tangent along l.

    “However, the vector ℓ a is null with respect to any of these metrics so the null cones are all tangent along ℓ a .”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingown data

    For d = 5, the remaining eigenvalues of the operator are ν = ± Ω'_{ij} Ω'_{ij}.

    “For d = 5 we can do this explicitly with the result ν Ω Ω = ± ′ ′ = d 2 3 , 5. (4.12)I ij ij”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given

    The tensors G_I are non-degenerate with Lorentzian signature.

    “The tensors G I ab are non-degenerate with Lorentzian signature.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    For the same d = 7 black hole at intermediate r, 0 < c_S < c_V < c_T < 1, so the tensor metric still determines causality but all modes propagate slower than light.

    “At intermediate r we have < < < < c c c 0 1 S V T so it is still the tensor metric that determines causality but now all modes propagate slower than light 13 .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    G_Aab is Lorentzian if c_A(r) > 0, degenerate if c_A(r) = 0, and Lorentzian with 'mostly minus' signature if c_A(r) < 0.

    “Note that G A ab is Lorentzian if > c r ( ) 0 A , degenerate if = c r ( ) 0 A , and Lorentzian with 'mostly minus' signature if < c r ( ) 0 A .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For the same d = 7 black hole near the horizon, 0 < c_T < c_V < c_S < 1, so the scalar effective metric determines causality, with all modes subluminal.

    “Near to the horizon, we have < < < < c c c 0 1 T V S and so it is now the effective metric for the scalars that determines causality, with all modes subluminal.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingcitation

    These solutions grow exponentially in time and are regular on the future horizon.

    “These grow exponentially in time and are regular on the future horizon 12 .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For a small spherical black hole with d = 7 and Λ = 0, at large r, c_S < c_V < c_T < 1, so the tensor effective metric determines causality and tensor modes propagate faster than light.

    “Figure 2 shows the behaviour of the effective potentials for a small spherical (κ = 1) black hole with d = 7 and Λ = 0. From the plot we see that < < < < c c c 0 1 S V T for large r.Hence at large r, the effective metric that determines causality is that of the tensors, and these modes propagate faster than light.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    At large r, c_A → 1, so all effective metrics approach the physical metric asymptotically and have Lorentzian signature.

    “At large r, → c 1 A , so all the effective metrics approach the physical metric asymptotically and hence have Lorentzian signature.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    In the Gauss-Bonnet case α_3 = 0, not all of these can be negative due to the bound (5.16) on αΛ; the scalars are negative in d = 5 for -3 < αΛ < -3/2, and the tensors are negative in d = 6 for -5 < αΛ < -2.

    “In the Gauss-Bonnet case α = ˜0 3 for example, not all of these can be negative due to the bound (5.16) on α Λ ˜2 .The scalars are negative in d = 5 for α Λ -< < -3 ˜3 2 2 , and the tensors are negative in d = 6 for α Λ -< < -5 ˜2 6 2 .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given

    Near the horizon, the effective potential goes as V_A^eff = c_A(r_0) f'(r_0)(r - r_0)/2, with f'(r_0) > 0.

    “Near the horizon, the effective potential goes as = ′ - V c r f r r r r ( ) ( ) 2 ( ), (5.19) A eff 0 0 0 2 0 with ′ > f r ( ) 0 0 .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    All small Lovelock black holes with d < P + 1/3 break hyperbolicity, and the bad modes can show up in any scalar, tensor or vector sector.

    “We have thus shown that all small Lovelock black holes with < + d P 1 3 break hyperbolicty, and the 'bad' modes can show up in any (scalar, tensor or vector) sector.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Solutions exist for arbitrarily small horizon radius r_0.

  • The quote only partly states this claim.

    Interpretationreasoning

    The effective metric with the most positive ω_I, rather than the physical metric, determines causal properties of gravitational propagation in this background.

    “Therefore it is this effective metric, rather than the physical metric, which determines causal properties of gravitational propagation in this background.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The non-characteristic condition might be violated after a finite time, in which case one cannot evolve further.

    “But the non-characteristic condition might be violated after a finite time, in which case one cannot evolve further.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The initial value problem and stability for small perturbations near such black holes is ill-posed.

    “This implies that the stability of such black holes is not a well-posed problem.”
    • Also stated in: § 1. Introduction · plain
      “Hence the initial value problem is ill-posed: time evolution does not make sense here.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Asking about stability presupposes that the equations admit a well-posed initial value formalism for initial data in a neighbourhood of the black hole initial data.

  • May combine two claims.

    Interpretationreasoning

    For solutions with all c_A positive, there exist Lovelock-spacelike hypersurfaces (e.g., surfaces of constant t), and hence Lovelock theory is hyperbolic in these backgrounds.

    “For these solutions with all c A positive, it is clear that we can find Lovelock-spacelike hypersurfaces, which are hypersurfaces that are spacelike with respect to all of the G Aab .For example, surfaces of constant t have this property.Hence Lovelock theory is hyperbolic in these backgrounds.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    For these small black holes, surfaces of constant t are spacelike for good modes but timelike in r < r_c for bad modes, implying the problem of studying stability by specifying an initial perturbation on a surface of constant t and evolving in time is ill-posed.

    “For example, surfaces of constant t are spacelike for the good modes but timelike in < r r c for the bad modes.This implies that the problem of studying stability by specifying an initial perturbation on a surface of constant t and evolving in time is ill-posed.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    If c_A(r) > 1 then the outermost graviton cone lies outside the light cone, so gravity travels faster than light.

    “If > c r ( ) 1 A then the outermost graviton cone lies outside the light cone: gravity travels faster than light.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    For large black holes, the effective metrics G_Aab are smooth and Lorentzian, and their null cones determine causality of Lovelock theory in this spacetime.

    “The effective metrics G Aab are smooth and Lorentzian.The characteristic surfaces are the null hypersurfaces of these effective metrics, so the null cones of G Aab determine causality of Lovelock theory in this spacetime.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    For small black holes, one of the effective metrics can change signature near the horizon, implying Lovelock theory is not hyperbolic in such backgrounds.

    “However, for small black holes, one of the effective metrics can change signature near the horizon.This implies that Lovelock theory is not hyperbolic in such backgrounds.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation

    The factorization of the characteristic polynomial into quadratic factors (and hence the appearance of effective metrics) is a consequence of the special properties of these solutions; for most solutions of Lovelock theories, one would not expect this factorization.

    “We should emphasize that the two cases investigated here are atypical.The factorization of the characteristic polynomial into quadratic factors (and hence the appearance of effective metrics) is a consequence of the special properties of these solutions.For most solutions of Lovelock theories, one would not expect this factorization.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    Since the event horizon of a static black hole is expected to be a Killing horizon, this shows that no signal can escape from the interior of such a black hole.

    “On general grounds, the event horizon of a static black hole is expected to be a Killing horizon, so this shows that no signal can escape from the interior of such a black hole.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The event horizon of a static black hole is a Killing horizon.
31 of 31

  • The authors state this as an assumption.

    The event horizon of a stationary black hole is a Killing horizon.

    “If one assumes that the event horizon of a stationary black hole is a Killing horizon then it follows that gravitational signals cannot escape from the black hole interior.”

  • The authors state this as an assumption.

    Asking about stability presupposes that the equations admit a well-posed initial value formalism for initial data in a neighbourhood of the black hole initial data.

    “Asking about stability presupposes that the equations admit a well-posed initial value formalism for initial data in a neighbourhood of the black hole initial data.”

  • The authors state this as an assumption.

    The event horizon of a static black hole is a Killing horizon.

    “On general grounds, the event horizon of a static black hole is expected to be a Killing horizon”
    “This assumption is justified for a static black hole subject to some reasonable global assumptions[27].”

  • The authors state this as an assumption.

    Solutions exist for arbitrarily small horizon radius r_0.

    “assuming solutions exist for arbitrarily small r 0”
4 of 4

Cites

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Research works printed but not linked: 1

  1. Aragone C, 1987, SILARG 6, 60

Cited by

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Related

Authors
Reall, H. S., Tanahashi, N., & Way, B.
Affiliations
  • University of Cambridge (GB)
  • Kavli Institute for the Physics and Mathematics of the Universe (JP)
Corresponding author
Way, Benson
ORCID
Venue
Classical and Quantum Gravity
Publisher
IOP Publishing
Published
2014-10-03
Volume
31
Issue
20
Pages
205005
ISSN
  • 0264-9381 (ISSN-L)
  • 1361-6382
DOI
10.1088/0264-9381/31/20/205005
OpenAlex
W2004231062
Semantic Scholar
f01f7603b9017c327cb809d433dd2bd230d8bded
Cited by
92
References
41
Citations by year
2026: 1 · 2025: 8 · 2024: 5 · 2023: 8 · 2022: 6 · 2021: 4 · 2020: 6 · 2019: 7 · 2018: 12 · 2017: 13 · 2016: 12 · 2015: 8 · 2014: 2
FWCI
5.14
Type
journal article
Language
en
Open access
Licence
Locations
  • Classical and Quantum Gravity (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Researc)
  • European Research Council (ERC-2011-StG 279363-HiDGR)
  • Ministry of Education, Culture, Sports, Science, and Technology (WPI Initiative)
Topics
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
Keywords
causality, hyperbolicity, characteristic hypersurfaces, Killing horizons, type N spacetimes, static black holes, black hole stability, well-posedness

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