Potentials and linearized gravity

Transcription

1 Potentials and linearized gravity Lars Andersson Albert Einstein Institute Sanya 2016 joint work with Steffen Aksteiner, Thomas Bäckdahl, and Pieter Blue Lars Andersson (AEI) Potentials and linearized gravity Sanya 1 / 47

2 Outline 1 Introduction 2 Kerr 3 Spin geometry 4 Potentials 5 Concluding remarks Lars Andersson (AEI) Potentials and linearized gravity Sanya 2 / 47

3 Introduction The Kerr black hole solution is expected to be unique and dynamically stable. Kerr is algebraically special, of Petrov type D. This property is essential for uniqueness and stability. Spinor methods allow one to exploit the special geometry of Kerr. A key problem for stability is to prove dispersive estimates fields on Kerr. Lars Andersson (AEI) Potentials and linearized gravity Sanya 3 / 47

4 Introduction The Kerr black hole solution is expected to be unique and dynamically stable. Kerr is algebraically special, of Petrov type D. This property is essential for uniqueness and stability. Spinor methods allow one to exploit the special geometry of Kerr. A key problem for stability is to prove dispersive estimates fields on Kerr. For scalar waves this is well understood. In the case of fields with integer spin (Maxwell and linearized gravity) one encounters new difficulties: non-radiating modes lack of conservation laws For Maxwell on slowly rotating Kerr, dispersive estimates are known. For linearized gravity this is open. Lars Andersson (AEI) Potentials and linearized gravity Sanya 3 / 47

5 Introduction The radiating modes for Maxwell and linearized gravity can be represented in terms of one complex scalar (Debye) potential. On Kerr, the scalars with extreme spin weight for Maxwell and linearized gravity satisfy TME: Teukolsky Master Equation (wave equation) TSI: Teukolsky-Starobinsky Identities (integrability condition) Lars Andersson (AEI) Potentials and linearized gravity Sanya 4 / 47

6 Introduction The radiating modes for Maxwell and linearized gravity can be represented in terms of one complex scalar (Debye) potential. On Kerr, the scalars with extreme spin weight for Maxwell and linearized gravity satisfy TME: Teukolsky Master Equation (wave equation) TSI: Teukolsky-Starobinsky Identities (integrability condition) Main results in this talk A higher-order conserved energy-momentum tensor for Maxwell on Kerr A covariant form of TSI for linearized gravity on Kerr Lars Andersson (AEI) Potentials and linearized gravity Sanya 4 / 47

7 Introduction Related work Spin-0: Scalar waves: (Finster, Kamran, Smoller, & Yau, 2006),(Tataru & Tohaneanu, 2008),(L.A. & Blue, 2015) (Dafermos, Rodnianski, & Shlapentokh-Rothman, 2014) Spin- 1 2 : Dirac: (Finster, Kamran, Smoller, & Yau, 2002) Spin-1: Maxwell: Schwarzschild: (Blue, 2008),(Metcalfe, Tataru, & Tohaneanu, 2014), (L.A., Bäckdahl, & Blue, 2015a), Kerr a M: (L.A. & Blue, 2013) Spin- 3 2 : Rarita-Schwinger:? Spin-2: Linearized gravity: Schwarzschild: (Dafermos, Holzegel & Rodnianski, w.i.p.) Nonlinear stability: Axially symmetric, a = 0 (Klainerman & Szeftel, w.i.p.) Lars Andersson (AEI) Potentials and linearized gravity Sanya 5 / 47

8 Potentials Maxwell Let F ab = F ab + i F ab. Then F ab is anti-self dual, F ab = if ab and Maxwell s equation takes the form (df) abc = 0 Suppose H ab is self-dual, H ab = ih ab and (dd + d d)h ab = 0. Let F ab = dd H ab ( = d dh ab ). Then (df) abc = 0 and F ab = dd H ab = d d H ab = if ab Lars Andersson (AEI) Potentials and linearized gravity Sanya 6 / 47

9 Potentials Maxwell Let F ab = F ab + i F ab. Then F ab is anti-self dual, F ab = if ab and Maxwell s equation takes the form (df) abc = 0 Suppose H ab is self-dual, H ab = ih ab and (dd + d d)h ab = 0. Let F ab = dd H ab ( = d dh ab ). Then (df) abc = 0 and F ab = dd H ab = d d H ab = if ab Spin-1 Hertz map Hertz map: H ab F ab sends a self-dual solution of wave equation to an anti-self dual solution of the Maxwell equation. Lars Andersson (AEI) Potentials and linearized gravity Sanya 6 / 47

10 Potentials Maxwell Let F ab = F ab + i F ab. Then F ab is anti-self dual, F ab = if ab and Maxwell s equation takes the form (df) abc = 0 Suppose H ab is self-dual, H ab = ih ab and (dd + d d)h ab = 0. Let F ab = dd H ab ( = d dh ab ). Then (df) abc = 0 and F ab = dd H ab = d d H ab = if ab Spin-1 Hertz map Hertz map: H ab F ab sends a self-dual solution of wave equation to an anti-self dual solution of the Maxwell equation. Minkowski M 4 The Hertz map is surjective. Decay estimates for H ab yield decay estimates for F ab, including peeling (L.A. & Bäckdahl, T. & Joudioux, J., 2014). Lars Andersson (AEI) Potentials and linearized gravity Sanya 6 / 47

11 Potentials Maxwell Nisbet gauge (Nisbet, 1955): H ab = (dg) ab + (d W) ab Represent Hertz potential in terms of a scalar Debye potential. Ansatz H ab = χz ab where Z ab is a background self-dual 2-form, χ scalar. Lars Andersson (AEI) Potentials and linearized gravity Sanya 7 / 47

12 Potentials Maxwell Nisbet gauge (Nisbet, 1955): H ab = (dg) ab + (d W) ab Represent Hertz potential in terms of a scalar Debye potential. Ansatz H ab = χz ab where Z ab is a background self-dual 2-form, χ scalar. On an algebraically special background with Z ab aligned with principal directions, can use Nisbet gauge freedom to get separable, decoupled wave equation for χ. Debye map: χ F ab. Lars Andersson (AEI) Potentials and linearized gravity Sanya 7 / 47

13 Potentials Linearized gravity Supermetric (Sachs & Bergmann, 1958): g ab = c d M acbd where M abcd has symmetries of the Riemann tensor, M abcd = M [ab]cd = M cdab, M [abc]d = 0. If M abcd = 0, then g ab solves the linearized Einstein equation DR[g] ab = 0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 8 / 47

14 Potentials Linearized gravity Supermetric (Sachs & Bergmann, 1958): g ab = c d M acbd where M abcd has symmetries of the Riemann tensor, M abcd = M [ab]cd = M cdab, M [abc]d = 0. If M abcd = 0, then g ab solves the linearized Einstein equation DR[g] ab = 0 Spin-2 Hertz and Debye map on M 4 Hertz: M abcd Ċ abcd, Debye: χ χ M abcd Ċ abcd Decay estimates for M abcd yield decay estimates for linearized gravity, including peeling (L.A. & Bäckdahl, T. & Joudioux, J., 2014) Lars Andersson (AEI) Potentials and linearized gravity Sanya 8 / 47

15 Kerr Kerr stationary, rotating black hole (Kerr, 1963) Carter tetrad in Boyer-Lindquist coordinates (t, r, θ, φ): Let = r 2 2Mr + a 2, Σ = r 2 + a 2 cos 2 θ, l a = n a = m a = a( φ ) a 2 1/2 Σ + (a2 + r 2 )( t ) a 1/2 2 1/2 Σ + 1/2 ( r ) a, 1/2 2Σ 1/2 a( φ ) a 2 1/2 Σ + (a2 + r 2 )( t ) a 1/2 2 1/2 Σ 1/2 ( r ) a, 1/2 2Σ 1/2 ( θ) a + i csc θ( φ) a 2Σ 1/2 2Σ 1/2 + ia sin θ( t) a. 2Σ 1/2 g ab = 2(l (a n b) m (a m b) ). AF, vacuum, 2 parameters M, a Lars Andersson (AEI) Potentials and linearized gravity Sanya 9 / 47

16 Kerr Contains black hole for a M, non-degenerate for a < M 2 Killing vector fields t, φ Superradiance, trapping Algebraically special, Petrov type D Carter constant, integrability for the geodesic equations, symmetry operators,... i + H I + i 0 trapping I Lars Andersson (AEI) Potentials and linearized gravity Sanya 10 / 47

17 Kerr Contains black hole for a M, non-degenerate for a < M 2 Killing vector fields t, φ Superradiance, trapping Algebraically special, Petrov type D Carter constant, integrability for the geodesic equations, symmetry operators,... i + H I + Black hole stability problem For data close to exterior Kerr data, show that the maximal globally hyperbolic Cauchy extension is asymptotic to exterior Kerr need dispersive estimates for fields on Kerr trapping I i 0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 10 / 47

18 Non-radiating modes Maxwell on Kerr Conserved charges S 2 F ab dσ ab, can be non-zero due to topology. S 2 ( F) ab dσ ab dispersion to non-radiating Coloumb solution Lars Andersson (AEI) Potentials and linearized gravity Sanya 11 / 47

19 Non-radiating modes Maxwell on Kerr Conserved charges S 2 F ab dσ ab, can be non-zero due to topology. S 2 ( F) ab dσ ab dispersion to non-radiating Coloumb solution Conjecture A Maxwell field with vanishing charges can be represented in terms of a Hertz/Debye potential. Lars Andersson (AEI) Potentials and linearized gravity Sanya 11 / 47

20 Non-radiating modes Linearized gravity on Kerr ξ a = ( t ) a, η a = ( φ ) a Killing fields, Conserved charges (Komar integrals) for linearized gravity: Ṁ = 1 (d 4π ξ) ab dσ ab, J = 1 (d η) ab dσ ab, S 2 8π S 2 can be nonzero due to topology dispersion to non-radiating mode Lars Andersson (AEI) Potentials and linearized gravity Sanya 12 / 47

21 Non-radiating modes Linearized gravity on Kerr ξ a = ( t ) a, η a = ( φ ) a Killing fields, Conserved charges (Komar integrals) for linearized gravity: Ṁ = 1 (d 4π ξ) ab dσ ab, J = 1 (d η) ab dσ ab, S 2 8π S 2 can be nonzero due to topology dispersion to non-radiating mode Conjecture A linearized gravitational field with vanishing charges can be represented (modulo gauge) in terms of a Hertz/Debye potential. Lars Andersson (AEI) Potentials and linearized gravity Sanya 12 / 47

22 Lorentzian spin geometry in 4D 2-spinors Spin group SL(2,C) = SO(3, 1) 0 acts on C 2,C 2 Correspondence C 4 C 2 C 2 : x AA = σ AA ax a x AA = 1 ( x 0 + x 3 x 1 + ix 2 ) 2 x 1 ix 2 x 0 x 3 Lars Andersson (AEI) Potentials and linearized gravity Sanya 13 / 47

23 Lorentzian spin geometry in 4D 2-spinors Spin group SL(2,C) = SO(3, 1) 0 acts on C 2,C 2 Correspondence C 4 C 2 C 2 : x AA = σ AA ax a x AA = 1 ( x 0 + x 3 x 1 + ix 2 ) 2 x 1 ix 2 x 0 x 3 spin tensors: φ AB FA B F spin dyad: o A, ι A, o A ι A = 1 null tetrad: l a = o A ō A, n a = ι A ῑ A, m = o A ῑ A, m a = ō A ι A, l a n a = 1 = m a m a n a la m a, m a Lars Andersson (AEI) Potentials and linearized gravity Sanya 13 / 47

24 Lorentzian spin geometry in 4D 2-spinors spin metric: ɛ AB = ɛ [AB] area element in C 2 : g ab = ɛ AB ɛ A B ɛ AB used to raise and lower indices on spinors: κ B = κ A ɛ AB, κ A = ɛ AB κ B Lars Andersson (AEI) Potentials and linearized gravity Sanya 14 / 47

25 Lorentzian spin geometry in 4D 2-spinors spin metric: ɛ AB = ɛ [AB] area element in C 2 : g ab = ɛ AB ɛ A B ɛ AB used to raise and lower indices on spinors: κ B = κ A ɛ AB, κ A = ɛ AB κ B symmetric spinors φ A DA D = φ (A D)(A D ) irreps of SL(2,C) Lars Andersson (AEI) Potentials and linearized gravity Sanya 14 / 47

26 Lorentzian spin geometry in 4D Spinor-tensor correspondence Each spinor can be written as a sum of symmetric spinors and ɛ AB. Example: χ AB = χ (AB) ɛ ABɛ CD χ CD Each tensor can be written as a sum of symmetric spinors and ɛ AB. Example: F ab = φ AB ɛ A B + φ A B ɛ AB 2-form Lars Andersson (AEI) Potentials and linearized gravity Sanya 15 / 47

27 Lorentzian spin geometry in 4D Spinor-tensor correspondence Each spinor can be written as a sum of symmetric spinors and ɛ AB. Example: χ AB = χ (AB) ɛ ABɛ CD χ CD Each tensor can be written as a sum of symmetric spinors and ɛ AB. Example: F ab = φ AB ɛ A B + φ A B ɛ AB 2-form spin connection: AA lift of Levi-Civita AA ɛ BC = 0 Curvature R abcd curvature spinors Weyl: C abcd = Ψ ABCD ɛ A B ɛ C D + c.c., traceless Ricci: S ab = Φ ABA B, Ricci scalar: R = 1 24 Λ Lars Andersson (AEI) Potentials and linearized gravity Sanya 15 / 47

28 Lorentzian spin geometry in 4D Fundamental operators D, C, C, T Projection of AA φ B F B F on irreducible representations gives 4 fundamental operators D, C, C, T analogues of Stein-Weiss operators. Examples: (D 2,2 φ) A A = BB φ ABA B (C 2,2 φ) ABC A = (A B φ BC) A B (C 2,2 φ) A A B C = B(A φ AB B C ) (T 2,2 φ) ABC A B C = (A (A φ BC) B C ) All expressions reduce to combinations of these operators, and can be simplified by applying commutation rules. Lars Andersson (AEI) Potentials and linearized gravity Sanya 16 / 47

29 Lorentzian spin geometry in 4D Fundamental operators D, C, C, T Projection of AA φ B F B F on irreducible representations gives 4 fundamental operators D, C, C, T analogues of Stein-Weiss operators. Examples: (D 2,2 φ) A A = BB φ ABA B (C 2,2 φ) ABC A = (A B φ BC) A B (C 2,2 φ) A A B C = B(A φ AB B C ) (T 2,2 φ) ABC A B C = (A (A φ BC) B C ) All expressions reduce to combinations of these operators, and can be simplified by applying commutation rules. Schematically φ = Tφ + ɛcφ + ɛc φ + ɛ ɛdφ Lars Andersson (AEI) Potentials and linearized gravity Sanya 16 / 47

30 Lorentzian spin geometry in 4D ker C = massless fields Massless spin-s fields (C 2s,0 φ) A B F A A φ AB F = 0 Buchdahl constraint: spin-s equation for s 3/2 test fields has restricted solutions, Ψ (A DEF φ B C)DEF = 0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 17 / 47

31 Lorentzian spin geometry in 4D ker C = massless fields Massless spin-s fields (C 2s,0 φ) A B F A A φ AB F = 0 Buchdahl constraint: spin-s equation for s 3/2 test fields has restricted solutions, Ψ (A DEF φ B C)DEF = 0 s = 1 Maxwell: F ab = φ AB ɛ A B + c.c. (C 2,0 φ) A A = 0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 17 / 47

32 Lorentzian spin geometry in 4D ker C = massless fields Massless spin-s fields (C 2s,0 φ) A B F A A φ AB F = 0 Buchdahl constraint: spin-s equation for s 3/2 test fields has restricted solutions, Ψ (A DEF φ B C)DEF = 0 s = 1 Maxwell: F ab = φ AB ɛ A B + c.c. (C 2,0 φ) A A = 0 s = 2 Weyl field: C abcd = Ψ ABCD ɛ A B ɛ C D + c.c. vacuum Einstein: R ab = 0 (C 4,0 Ψ) A ABC = 0 Linearized gravity: (C 4,0 Ψ) A ABC = j A ABC Lars Andersson (AEI) Potentials and linearized gravity Sanya 17 / 47

33 Scalars Newman-Penrose Normalization o A ι A = 1 perserved by rescaling o A λo A, ι A λ 1 ι A dyad components are weighted scalars ϕ λ p λq ϕ type {p, q} NP-scalars φ i = φ A1 A i A i+1 A 2s ι A1 ι A i o A i+1 o A 2s Lars Andersson (AEI) Potentials and linearized gravity Sanya 18 / 47

34 Scalars Newman-Penrose Normalization o A ι A = 1 perserved by rescaling o A λo A, ι A λ 1 ι A dyad components are weighted scalars ϕ λ p λq ϕ type {p, q} NP-scalars φ i = φ A1 A i A i+1 A 2s ι A1 ι A i o A i+1 o A 2s q Maxwell φ AB φ 0, φ 1, φ 2 Weyl Ψ ABCD Ψ 0,, Ψ 4 φ 2 Ψ 0 φ 1 φ 0 p Ψ 4 Ψ 3 Ψ 2 Ψ 1 Lars Andersson (AEI) Potentials and linearized gravity Sanya 18 / 47

35 Scalars Christodoulou-Klainerman/Ionescu-Klainerman type notation Tetrad m a, m a, n a, l a e a 1, ea 2, la, l a with l a l a = 2 (C-K), m a, m a, l a, l a with l a l a = 1 (I-K) Spin weight C-K/I-K Definition NP +2 α(m, m) C(l, m, l, m) Ψ 0 ρ (I-K) 0 C( m, l, m, l) Ψ 2 ρ + iσ (C-K) -2 α( m, m) C(l, m, l, m) Ψ 4 Lars Andersson (AEI) Potentials and linearized gravity Sanya 19 / 47

36 Petrov classification Decomposition into principal spinors: Ψ ABCD = α (A β Bγ Cδ D) I = {1, 1, 1, 1} general II = {2, 1, 1} Ψ ABCD = α (A α Bβ Cγ D) D = {2, 2} Ψ ABCD = α (A α Bβ Cβ D) III = {3, 1} Ψ ABCD = α (A α Bα Cβ D) N = {4} Ψ ABCD = α Aα Bα Cα D O = { } conformally flat I D O II N III Lars Andersson (AEI) Potentials and linearized gravity Sanya 20 / 47

37 Petrov classification Decomposition into principal spinors: Ψ ABCD = α (A β Bγ Cδ D) I = {1, 1, 1, 1} general II = {2, 1, 1} Ψ ABCD = α (A α Bβ Cγ D) I II D = {2, 2} Ψ ABCD = α (A α Bβ Cβ D) III = {3, 1} Ψ ABCD = α (A α Bα Cβ D) N = {4} Ψ ABCD = α Aα Bα Cα D O = { } conformally flat D O N III Kerr is Petrov type D Ψ ABCD = 6Ψ 2o (A o Bι Cι D), if o A, ι A principal. For Kerr in Boyer-Lindquist coordinates: Ψ 2 = M (r ia cos θ) 3 Extreme linearized Weyl scalars Ψ 0, Ψ 4 are gauge invariant. Lars Andersson (AEI) Potentials and linearized gravity Sanya 20 / 47

38 Type D Theorem (Walker & Penrose, 1970) Assume that (M, g ab ) is vacuum, Petrov type D. Let κ AB = 2κ 1 o (A ι B), κ 1 Ψ 1/3 2 Then κ AB is a Killing spinor of valence 2, (T 2,0 κ) A ABC A (Aκ BC) = 0 ξ AA = (C 2,0 κ) AA is a Killing vector field: (aξ b) = 0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 21 / 47

39 Type D Theorem (Walker & Penrose, 1970) Assume that (M, g ab ) is vacuum, Petrov type D. Let κ AB = 2κ 1 o (A ι B), κ 1 Ψ 1/3 2 Then κ AB is a Killing spinor of valence 2, Kerr (T 2,0 κ) A ABC A (Aκ BC) = 0 ξ AA = (C 2,0 κ) AA is a Killing vector field: (aξ b) = 0 ξ AA = ξ AA (i.e. ξ a ( t ) a ) Y ab = i(κ AB ɛ A B κ A B ɛ AB) is a Killing-Yano tensor: (a Y b)c = 0. K ab = Y ac Y c b is a Killing tensor: (a K bc) = 0 k = K ab γ a γ b is conserved along geodesics Carter constant Lars Andersson (AEI) Potentials and linearized gravity Sanya 21 / 47

40 Killing spinors Killing spinors propagate causally: Theorem (Bäckdahl, T. and Valiente Kroon, J. A., 2011) Assume that (M, g ab ) is vacuum and contains a Cauchy surface with valence 2 Killing spinor data. Then (M, g ab ) admits a valence 2 Killing spinor. Lars Andersson (AEI) Potentials and linearized gravity Sanya 22 / 47

41 Killing spinors Killing spinors propagate causally: Theorem (Bäckdahl, T. and Valiente Kroon, J. A., 2011) Assume that (M, g ab ) is vacuum and contains a Cauchy surface with valence 2 Killing spinor data. Then (M, g ab ) admits a valence 2 Killing spinor. Kerr can be characterized in terms of Killing spinor data: Theorem (L.A., Bäckdahl, & Blue, 2015b) If (M, g ab ) is vacuum, and contains a Cauchy surface with valence 2 Killing spinor data which is 1 asymptotically flat 2 contains a MOTS (apparent horizon) then (M, g ab ) is Kerr. Lars Andersson (AEI) Potentials and linearized gravity Sanya 22 / 47

42 Twistors and tractors κ A Killing spinor: (T 1,0 κ) A AB = 0 A A κ B = ɛ AB π A A A π B = 0 where π A = 1 2 (C 1,0 κ)a. Lars Andersson (AEI) Potentials and linearized gravity Sanya 23 / 47

43 Twistors and tractors κ A Killing spinor: (T 1,0 κ) A AB = 0 where π A = 1 2 (C 1,0 κ)a. A A κ B = ɛ AB π A A A π B = 0 τ = (κ A, π A ) is Penrose s twistor: w.r.t. connection parallel: šτ = 0 š = ( ) 0 ɛ 0 0 We can generate a complete table to derivatives for κ A! Lars Andersson (AEI) Potentials and linearized gravity Sanya 23 / 47

44 Twistors and tractors Assume (M, g ab ) vacuum. Let κ AB be a Killing spinor, (T 2,0 κ) A ABC = 0. ξ a = (C 2,0 κ) AA is Killing. Assume generalized Kerr-NUT condition: ξ a real κ = ξɛ ξ = κ Ψɛ + κψ ɛ (schematically) Lars Andersson (AEI) Potentials and linearized gravity Sanya 24 / 47

45 Twistors and tractors Assume (M, g ab ) vacuum. Let κ AB be a Killing spinor, (T 2,0 κ) A ABC = 0. ξ a = (C 2,0 κ) AA is Killing. Assume generalized Kerr-NUT condition: ξ a real κ = ξɛ ξ = κ Ψɛ + κψ ɛ (schematically) τ = (κ, κ, ξ) is parallel w.r.t. a modified connection šτ = 0 tractor (Bailey et al., 1994) We can generate a complete table to derivatives for κ AB! Lars Andersson (AEI) Potentials and linearized gravity Sanya 24 / 47

46 Encoding Kerr The geometry of Kerr can be characterized in terms of the Killing spinor κ AB and its derivatives. This makes it possible to perform efficient coordinate-free symbolic calculations Spinor formalism for linearized gravity (Bäckdahl & Valiente Kroon, 2015) xact, SymManipulator, SpinFrames Lars Andersson (AEI) Potentials and linearized gravity Sanya 25 / 47

47 Encoding Kerr Diagonal wave operator Recall κ AB = 2κ 1 o (A ι B) where κ 1 Ψ 1/3 2 ( r ia cos θ in BL coordinates). Define extended fundamental operators D k,l,m, C k,l,m, C k,l,m, T k,l,m as projections of AA m AA log(κ 1 ) Lars Andersson (AEI) Potentials and linearized gravity Sanya 26 / 47

48 Encoding Kerr Diagonal wave operator Recall κ AB = 2κ 1 o (A ι B) where κ 1 Ψ 1/3 2 ( r ia cos θ in BL coordinates). Define extended fundamental operators D k,l,m, C k,l,m, C k,l,m, T k,l,m as projections of AA m AA log(κ 1 ) Lemma (Diagonal wave operator) The operator ϕ A F (C 2s 1,0, 2s C 2s,0 ϕ) A F is diagonal on spin-s fields. Lars Andersson (AEI) Potentials and linearized gravity Sanya 26 / 47

49 Encoding Kerr TME and TSI The first order Maxwell and linearized Bianchi systems imply higher order integrability conditions TME and TSI. Maxwell: { (C 2,0 φ) AA = 0 (C 1,1, 2 C 2,0 φ) AB = 0 (Spin-1 TME) κ AB (T 1,1, 4 C 2,0 φ) ABA B = 0 (Spin-1 TSI) Lars Andersson (AEI) Potentials and linearized gravity Sanya 27 / 47

50 Encoding Kerr TME and TSI The first order Maxwell and linearized Bianchi systems imply higher order integrability conditions TME and TSI. Maxwell: { (C 2,0 φ) AA = 0 (C 1,1, 2 C 2,0 φ) AB = 0 (Spin-1 TME) κ AB (T 1,1, 4 C 2,0 φ) ABA B = 0 (Spin-1 TSI) Equivalent system The spin-1 TME/TSI system is equivalent to Maxwell modulo non-radiating modes (Coll et al., 1987). A similar situation is expected for linearized gravity. Diagonal wave operator TME for linearized gravity: (C 4,0 Ψ) ABCA = j ABCA (C 4,1, 4 C 4,0 Ψ) ABCD = z ABCD Spin-2 TME for Ψ 0, Ψ 4 Lars Andersson (AEI) Potentials and linearized gravity Sanya 27 / 47

51 GHP Symmetries: (bar) : l a l a, n a n a, m a m a, m a m a, {p, q} {q, p}, (prime) : l a n a, n a l a, m a m a, m a m a,{p, q} { p, q} Θ a lift of a to weighted scalars of type {p, q} Lars Andersson (AEI) Potentials and linearized gravity Sanya 28 / 47

52 GHP Symmetries: (bar) : l a l a, n a n a, m a m a, m a m a, {p, q} {q, p}, (prime) : l a n a, n a l a, m a m a, m a m a,{p, q} { p, q} Θ a lift of a to weighted scalars of type {p, q} GHP operators þ = l a Θ a, þ = n a Θ a, ð = m a Θ a, ð = m a Θ a weighted operators: q ð þ p þ ð Lars Andersson (AEI) Potentials and linearized gravity Sanya 28 / 47

53 GHP Type D: Only non-zero connection coefficients are ρ = m a ð l a, τ = m a þ l a and primes important simplifications in Maxwell, linearized Bianchi Lars Andersson (AEI) Potentials and linearized gravity Sanya 29 / 47

54 GHP Type D: Only non-zero connection coefficients are ρ = m a ð l a, τ = m a þ l a and primes important simplifications in Maxwell, linearized Bianchi Maxwell: and prime versions. (þ 2ρ)φ 1 (ð τ )φ 0 = 0, (ð 2τ )φ 1 (þ ρ)φ 2 = 0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 29 / 47

55 GHP Type D: Only non-zero connection coefficients are ρ = m a ð l a, τ = m a þ l a and primes important simplifications in Maxwell, linearized Bianchi Maxwell: (þ 2ρ)φ 1 (ð τ )φ 0 = 0, (ð 2τ )φ 1 (þ ρ)φ 2 = 0 and prime versions. Consistent weights restrict the possible relations: q þ ð þ ð p φ 2 ð φ þ 1 ð φ þ 0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 29 / 47

56 TME and TSI The first order Maxwell/linearized Bianchi systems imply wave equations Teukolsky Master equation (TME) integrability conditions Teukolsky-Starobinsky Identities (TSI) Maxwell TME and TSI: TME for φ 0 follows by applying a derivative to the Maxwell equations 1 and 2, relating φ 0 and φ 1, similarly for φ 2 and linearized gravity. q þ 1 ð p φ 2 φ 1 ð 2 φ þ 0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 30 / 47

57 TME and TSI The first order Maxwell/linearized Bianchi systems imply wave equations Teukolsky Master equation (TME) integrability conditions Teukolsky-Starobinsky Identities (TSI) Maxwell TME and TSI: TME for φ 0 follows by applying a derivative to the Maxwell equations 1 and 2, relating φ 0 and φ 1, similarly for φ 2 and linearized gravity. q 1 ð p φ 2 φ 1 2 þ φ 0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 30 / 47

58 TME and TSI The first order Maxwell/linearized Bianchi systems imply wave equations Teukolsky Master equation (TME) integrability conditions Teukolsky-Starobinsky Identities (TSI) Maxwell TME and TSI: TME for φ 0 follows by applying a derivative to the Maxwell equations 1 and 2, relating φ 0 and φ 1, similarly for φ 2 and linearized gravity. TSI are relations between derivatives of φ i. There are three TSI for Maxwell. φ 2 þ þ q φ 1 ð ð φ 0 p Lars Andersson (AEI) Potentials and linearized gravity Sanya 30 / 47

59 TME and TSI The first order Maxwell/linearized Bianchi systems imply wave equations Teukolsky Master equation (TME) integrability conditions Teukolsky-Starobinsky Identities (TSI) Maxwell TME and TSI: TME for φ 0 follows by applying a derivative to the Maxwell equations 1 and 2, relating φ 0 and φ 1, similarly for φ 2 and linearized gravity. TSI are relations between derivatives of φ i. There are three TSI for Maxwell. φ 2 ð þ q φ 1 þ ð φ 0 p Lars Andersson (AEI) Potentials and linearized gravity Sanya 30 / 47

60 TME and TSI The first order Maxwell/linearized Bianchi systems imply wave equations Teukolsky Master equation (TME) integrability conditions Teukolsky-Starobinsky Identities (TSI) Maxwell TME and TSI: TME for φ 0 follows by applying a derivative to the Maxwell equations 1 and 2, relating φ 0 and φ 1, similarly for φ 2 and linearized gravity. TSI are relations between derivatives of φ i. There are three TSI for Maxwell. q φ 2 φ 1 φ þ 0 ð ð þ p Lars Andersson (AEI) Potentials and linearized gravity Sanya 30 / 47

61 TME and TSI The first order Maxwell/linearized Bianchi systems imply wave equations Teukolsky Master equation (TME) integrability conditions Teukolsky-Starobinsky Identities (TSI) Maxwell TME and TSI: TME for φ 0 follows by applying a derivative to the Maxwell equations 1 and 2, relating φ 0 and φ 1, similarly for φ 2 and linearized gravity. TSI are relations between derivatives of φ i. There are three TSI for Maxwell. Analogously, there are five TSI for linearized gravity. q φ 2 φ 1 φ þ 0 ð ð þ p Lars Andersson (AEI) Potentials and linearized gravity Sanya 30 / 47

62 TME and TSI Coordinate form of TME The extreme Maxwell and linearized Weyl scalars solve TME TME is the wave equation [ T sψ (s) = r r 1 { 2 (r 2 + a 2 ) t + a φ (r M)s} 4sr t + 1 sin θ θ sin θ θ + 1 { } ] 2 a sin 2 θ sin 2 t + φ + is cos θ 4sia cos θ t ψ (s) = θ Lars Andersson (AEI) Potentials and linearized gravity Sanya 31 / 47

63 TME and TSI Coordinate form of TME The extreme Maxwell and linearized Weyl scalars solve TME TME is the wave equation [ T sψ (s) = r r 1 { 2 (r 2 + a 2 ) t + a φ (r M)s} 4sr t + 1 sin θ θ sin θ θ + 1 { } ] 2 a sin 2 θ sin 2 t + φ + is cos θ 4sia cos θ t ψ (s) = θ TME admits commuting symmetry operators T s = R s + Q s, where [R s, Q s ] = 0 separation of variables For s 0, TME is not formally self-adjoint no real conserved currents Lars Andersson (AEI) Potentials and linearized gravity Sanya 31 / 47

64 TME and TSI Debye map The Hertz/Debye map yields a solution to Maxwell/Linearized gravity from a solution of TME. q φ 2 φ 1 φ 0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 32 / 47

65 TME and TSI Debye map The Hertz/Debye map yields a solution to Maxwell/Linearized gravity from a solution of TME. q Example Let χ 0, χ 2 solve TME. Then χ A B = 2κ2 1 χ 2 o A o B, are Hertz potentials for Maxwell on Kerr. φ 2 ð φ 1 ð þ φ 0 ð χ 2 þ Lars Andersson (AEI) Potentials and linearized gravity Sanya 32 / 47

66 TME and TSI Debye map The Hertz/Debye map yields a solution to Maxwell/Linearized gravity from a solution of TME. q Example Let χ 0, χ 2 solve TME. Then þ χ 0 ð χ A B = 2κ2 1 χ 2 o A o B, χ A B = 2κ2 1 χ 0 ι A ι B, þ ð ð are Hertz potentials for Maxwell on Kerr. φ 2 φ 1 φ 0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 32 / 47

67 TME and TSI Debye map The Hertz/Debye map yields a solution to Maxwell/Linearized gravity from a solution of TME. q TSI from Debye If χ 0, χ 2 are extreme scalars of the same Maxwell field, then the Debye maps agree. This is a manifestation of the TSI. Analogous statements hold for linearized gravity. þ φ 2 ð χ 0 þ ð φ 1 ð ð ð þ φ 0 ð χ 2 þ Lars Andersson (AEI) Potentials and linearized gravity Sanya 32 / 47

68 TME and TSI Wald s method Field equation: E(f ) = 0 Decoupled equation O(χ) = 0 SE(f ) = O(χ) = OT (f ) S: decoupling T : projection Suppose O (ψ) = 0 SE = OT E S = T O E S ψ = 0 If E = ±E then S ψ solves E(S ψ) = 0 Debye map: ψ S ψ Lars Andersson (AEI) Potentials and linearized gravity Sanya 33 / 47

69 Maxwell conservation laws TME/TSI on Kerr Lemma Assume that φ AB solves the Maxwell equation (C 2,0 φ) AA = 0 on Kerr. Let Θ AB = 2κ C (A φ B)C. Then Θ AB has components (Θ i ) = ( 2κ 1 φ 0, 0, 2κ 1 φ 2 ) In particular, Θ AB depends only on the extreme components of φ AB. Let η AA = (C 2,0 Θ) AA. (C 1,1 η) AB = 2 3 L ξφ AB (TME) (C 1,1 η) A B = 0 (TSI) (D 1,1 η) = 0 where ξ a = ( t ) a. Lars Andersson (AEI) Potentials and linearized gravity Sanya 34 / 47

70 Maxwell conservation laws Higher order conserved stress-energy Theorem (L.A., Bäckdahl, & Blue, 2014, Aksteiner, L.A., & Bäckdahl, 2016b) 1 The Maxwell TSI is the Euler-Lagrange equation for S = η AA η AA (1) 2 The action (1) has energy-momentum tensor V ab = η AB η A B + η BA η B A L ξφ A B Θ AB L ξφ AB ΘA B 3 V ab is is conserved, a V ab = 0. Lars Andersson (AEI) Potentials and linearized gravity Sanya 35 / 47

71 Maxwell conservation laws Higher order conserved stress-energy Remark V ab is quadratic in first derivatives of the Maxwell field, in contrast to the standard Maxwell energy-momentum tensor T ab = φ AB φa B. The leading order term η AB η A B + η BA η B A in V ab satisfies the dominant energy condtion. The non-radiating Coloumb field on Kerr is up to a normalization φ Coloumb AB = 2κ 2 1 o (Aι B), with components (φ Coloumb i ) = (0, κ 2 1, 0), and is therefore cancelled by V ab. This allows one to prove dispersive estimates for Maxwell using V ab. Lars Andersson (AEI) Potentials and linearized gravity Sanya 36 / 47

72 TSI for linearized gravity Spinor formulation Given a solution to the linearized vacuum Einstein equations δg ab, define G ABA B = δg (AB)(A B ) Let φ ABCD = 1 2 (C 3,1C 2,2 G) ABCD Then φ ABCD is a modified linearized Weyl spinor. The extreme linearized Weyl scalars φ 0 = Ψ 0, φ 4 = Ψ 4 are gauge invariant. Linearized Bianchi (C 4,0 φ) A ABC = j A ABC Lars Andersson (AEI) Potentials and linearized gravity Sanya 37 / 47

73 TSI for linearized gravity Potential Let κ 1 = κ AB o A ι B Ψ 1/3 2 and define the spinor φ ABCD to have components ( φ 0, φ 1, φ 2, φ 3, φ 4 ) = (κ 4 1 Ψ 0, 0, 0, 0, κ 4 1 Ψ 4 ) φ ABCD has pure spin-2, a rescaling, and a sign flip. Let M ABA B = (C 3,1 C 4,0,4 φ) ABA B Then M ab is a complex solution of the linearized Einstein equation. The linearized metric M ab is generated using φ ABCD as Hertz potential. Lars Andersson (AEI) Potentials and linearized gravity Sanya 38 / 47

74 TSI for linearized gravity Spin-2 TSI on Kerr Lemma There is a complex vector field A a so that where ξ a = ( t ) a. M ab = M 27 (L ξδg) ab + L A g ab Lars Andersson (AEI) Potentials and linearized gravity Sanya 39 / 47

75 TSI for linearized gravity Spin-2 TSI on Kerr Lemma There is a complex vector field A a so that where ξ a = ( t ) a. From the above we get M ab = M 27 (L ξδg) ab + L A g ab Theorem (Aksteiner, L.A., & Bäckdahl, 2016a) (C 1,3 C 2,2 C 3,1 C 4,0,4 φ) A B C D = M 27 (L ξφ) A B C D + (L AΨ) A B C D This is the TSI for linearized gravity. Lars Andersson (AEI) Potentials and linearized gravity Sanya 39 / 47

76 TSI for linearized gravity GHP form In GHP notation, the extreme TSI relations take the simple form q Ψ 4 φ 2 φ 0 Ψ0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 40 / 47

77 TSI for linearized gravity GHP form In GHP notation, the extreme TSI relations take the simple form q ð ð (κ 2 1φ 0 ) = þ þ(κ 2 1φ 2 ) Ψ 4 φ 2 φ 0 Ψ0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 40 / 47

78 TSI for linearized gravity GHP form In GHP notation, the extreme TSI relations take the simple form q ð ð (κ 2 1φ 0 ) = þ þ(κ 2 1φ 2 ) þ þ (κ 2 1φ 0 ) = ð ð(κ 2 1φ 2 ) Ψ 4 φ 2 φ 0 Ψ0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 40 / 47

79 TSI for linearized gravity GHP form In GHP notation, the extreme TSI relations take the simple form q ð ð (κ 2 1φ 0 ) = þ þ(κ 2 1φ 2 ) þ þ (κ 2 1φ 0 ) = ð ð(κ 2 1φ 2 ) ð ð ð ð (κ 4 1 Ψ 0 ) = þ þ þ þ(κ 4 1 Ψ 4 )+ M 27 (L ξ Ψ) 0 Ψ 4 φ 2 φ 0 Ψ0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 40 / 47

80 TSI for linearized gravity GHP form In GHP notation, the extreme TSI relations take the simple form q ð ð (κ 2 1φ 0 ) = þ þ(κ 2 1φ 2 ) þ þ (κ 2 1φ 0 ) = ð ð(κ 2 1φ 2 ) ð ð ð ð (κ 4 1 Ψ 0 ) = þ þ þ þ(κ 4 1 Ψ 4 )+ M 27 (L ξ Ψ) 0 þ þ þ þ (κ 4 1 Ψ 0 ) = ð ð ð ð(κ 4 1 Ψ 4 ) M 27 (L ξ Ψ) 4 Ψ 4 φ 2 φ 0 Ψ0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 40 / 47

81 TSI for linearized gravity GHP form In GHP notation, the extreme TSI relations take the simple form q ð ð (κ 2 1φ 0 ) = þ þ(κ 2 1φ 2 ) þ þ (κ 2 1φ 0 ) = ð ð(κ 2 1φ 2 ) ð ð ð ð (κ 4 1 Ψ 0 ) = þ þ þ þ(κ 4 1 Ψ 4 )+ M 27 (L ξ Ψ) 0 þ þ þ þ (κ 4 1 Ψ 0 ) = ð ð ð ð(κ 4 1 Ψ 4 ) M 27 (L ξ Ψ) 4 Ψ 4 φ 2 φ 0 Ψ0 The spin-2 case has the important new feature L ξ Ψ, cf. (Whiting & Price, 2005) Lars Andersson (AEI) Potentials and linearized gravity Sanya 40 / 47

82 The middle scalars on Schwarzschild The middle scalars φ 1, Ψ 2 satisfy wave equations ( a a + 2Ψ 2 )(κ 1 φ 1 ) = 0 (Fackerell-Ipser) ( a a + 8Ψ 2 )(κ 2 1I Ψ 2 ) = 0 (Regge-Wheeler) q p Ψ 4 φ 2 φ 0 Ψ0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 41 / 47

83 The middle scalars on Schwarzschild The middle scalars φ 1, Ψ 2 satisfy wave equations ( a a + 2Ψ 2 )(κ 1 φ 1 ) = 0 (Fackerell-Ipser) ( a a + 8Ψ 2 )(κ 2 1I Ψ 2 ) = 0 (Regge-Wheeler) They are related to the extreme scalars by Debye maps: ϕ 1 = þ ð (κ 2 1φ 0 ) q p Ψ 4 φ 2 φ 0 Ψ0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 41 / 47

84 The middle scalars on Schwarzschild The middle scalars φ 1, Ψ 2 satisfy wave equations ( a a + 2Ψ 2 )(κ 1 φ 1 ) = 0 (Fackerell-Ipser) ( a a + 8Ψ 2 )(κ 2 1I Ψ 2 ) = 0 (Regge-Wheeler) They are related to the extreme scalars by Debye maps: ϕ 1 = þ ð (κ 2 1φ 0 ) ϕ 1 = þ ð(κ 2 1φ 2 ) q p Ψ 4 φ 2 φ 0 Ψ0 Lars Andersson (AEI) Potentials and linearized gravity Sanya 41 / 47

85 The middle scalars on Schwarzschild The middle scalars φ 1, Ψ 2 satisfy wave equations ( a a + 2Ψ 2 )(κ 1 φ 1 ) = 0 (Fackerell-Ipser) ( a a + 8Ψ 2 )(κ 2 1I Ψ 2 ) = 0 (Regge-Wheeler) They are related to the extreme scalars by Debye maps: q ϕ 1 = þ ð (κ 2 1φ 0 ) ϕ 1 = þ ð(κ 2 1φ 2 ) ψ 2 = þ þ ð ð (κ 4 1 Ψ 0 ) Ψ 4 φ 2 φ 0 Ψ0 p Lars Andersson (AEI) Potentials and linearized gravity Sanya 41 / 47

86 The middle scalars on Schwarzschild The middle scalars φ 1, Ψ 2 satisfy wave equations ( a a + 2Ψ 2 )(κ 1 φ 1 ) = 0 (Fackerell-Ipser) ( a a + 8Ψ 2 )(κ 2 1I Ψ 2 ) = 0 (Regge-Wheeler) They are related to the extreme scalars by Debye maps: q ϕ 1 = þ ð (κ 2 1φ 0 ) ϕ 1 = þ ð(κ 2 1φ 2 ) ψ 2 = þ þ ð ð (κ 4 1 Ψ 0 ) ψ 2 = þ þ ð ð(κ 4 1Ψ 4 ) Ψ 4 φ 2 φ 0 Ψ0 p Lars Andersson (AEI) Potentials and linearized gravity Sanya 41 / 47

87 The middle scalars on Schwarzschild The middle scalars φ 1, Ψ 2 satisfy wave equations ( a a + 2Ψ 2 )(κ 1 φ 1 ) = 0 (Fackerell-Ipser) ( a a + 8Ψ 2 )(κ 2 1I Ψ 2 ) = 0 (Regge-Wheeler) They are related to the extreme scalars by Debye maps: ϕ 1 = þ ð (κ 2 1φ 0 ) ϕ 1 = þ ð(κ 2 1φ 2 ) ψ 2 = þ þ ð ð (κ 4 1 Ψ 0 ) ψ 2 = þ þ ð ð(κ 4 1Ψ 4 ) Ψ 4 Dispersive estimates hold for F-I and R-W. After integration, this yields estimates for the extreme scalars solving TME. Lars Andersson (AEI) Potentials and linearized gravity Sanya 41 / 47 φ 2 q φ 0 Ψ0 p

88 Concluding remarks A complete understanding of symmetries and conservation laws for fields on the Kerr spacetime will be significant for the black hole stability problem. Lars Andersson (AEI) Potentials and linearized gravity Sanya 42 / 47

89 Concluding remarks A complete understanding of symmetries and conservation laws for fields on the Kerr spacetime will be significant for the black hole stability problem. The complexity of the problem of analyzing symmetry operators and conservation laws for linearized gravity in a covariant manner is significantly higher than for Maxwell fields. Lars Andersson (AEI) Potentials and linearized gravity Sanya 42 / 47

90 Concluding remarks A complete understanding of symmetries and conservation laws for fields on the Kerr spacetime will be significant for the black hole stability problem. The complexity of the problem of analyzing symmetry operators and conservation laws for linearized gravity in a covariant manner is significantly higher than for Maxwell fields. In spite of much classical work, the topic of symmetries and conservation laws for higher spin fields on black hole backgrounds appears to be far from exhaused. Lars Andersson (AEI) Potentials and linearized gravity Sanya 42 / 47

91 Concluding remarks A complete understanding of symmetries and conservation laws for fields on the Kerr spacetime will be significant for the black hole stability problem. The complexity of the problem of analyzing symmetry operators and conservation laws for linearized gravity in a covariant manner is significantly higher than for Maxwell fields. In spite of much classical work, the topic of symmetries and conservation laws for higher spin fields on black hole backgrounds appears to be far from exhaused. Thank You Lars Andersson (AEI) Potentials and linearized gravity Sanya 42 / 47

92 Bibliography I Andersson, L., Aksteiner, S., & Bäckdahl, T. (2016a). Spin 2 TSI. (w.i.p.) Andersson, L., Aksteiner, S., & Bäckdahl, T. (2016b). Variational principles and spinning fields. (w.i.p.) Andersson, L., Bäckdahl, T., & Blue, P. (2014, December). A new tensorial conservation law for Maxwell fields on the Kerr background. (arxiv.org: ) Andersson, L., Bäckdahl, T., & Blue, P. (2015a, January). Decay of solutions to the Maxwell equation on the Schwarzschild background. (arxiv.org: ) Andersson, L., Bäckdahl, T., & Blue, P. (2015b, April). Spin geometry and conservation laws in the Kerr spacetime. In L. Bieri & S.-T. Yau (Eds.), One hundred years of general relativity (pp ). Boston: International Press. Lars Andersson (AEI) Potentials and linearized gravity Sanya 43 / 47

93 Bibliography II Andersson, L., Bäckdahl, T., & Joudioux, J. (2014, October). Hertz Potentials and Asymptotic Properties of Massless Fields. Communications in Mathematical Physics, 331, doi: /s x Andersson, L., & Blue, P. (2013, October). Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior. (arxiv.org: ) Andersson, L., & Blue, P. (2015). Hidden symmetries and decay for the wave equation on the Kerr spacetime. Ann. of Math. (2), 182(3), (arxiv.org: ) Bäckdahl, T., & Valiente Kroon, J. A. (2011, June). The non-kerrness of domains of outer communication of black holes and exteriors of stars. Royal Society of London Proceedings Series A, 467, doi: /rspa Lars Andersson (AEI) Potentials and linearized gravity Sanya 44 / 47

94 Bibliography III Bäckdahl, T., & Valiente Kroon, J. A. (2015, May). A formalism for the calculus of variations with spinors. (arxiv.org: ) Bailey, T., Eastwood, M., & Gover, A. (1994, 12). Thomas s structure bundle for conformal, projective and related structures. Rocky Mountain J. Math., 24(4), Retrieved from Blue, P. (2008). Decay of the Maxwell field on the Schwarzschild manifold. J. Hyperbolic Differ. Equ., 5(4), Coll, B., Fayos, F., & Ferrando, J. J. (1987, May). On the electromagnetic field and the Teukolsky-Press relations in arbitrary space-times. Journal of Mathematical Physics, 28, Dafermos, M., Rodnianski, I., & Shlapentokh-Rothman, Y. (2014, February). Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case a < M. (arxiv.org: ) Lars Andersson (AEI) Potentials and linearized gravity Sanya 45 / 47

95 Bibliography IV Finster, F., Kamran, N., Smoller, J., & Yau, S.-T. (2002). Decay rates and probability estimates for massive Dirac particles in the Kerr-Newman black hole geometry. Comm. Math. Phys., 230(2), Retrieved from doi: /s Finster, F., Kamran, N., Smoller, J., & Yau, S.-T. (2006, June). Decay of Solutions of the Wave Equation in the Kerr Geometry. Communications in Mathematical Physics, 264, doi: /s Kerr, R. P. (1963, September). Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics. Physical Review Letters, 11, doi: /PhysRevLett Metcalfe, J., Tataru, D., & Tohaneanu, M. (2014, November). Pointwise decay for the Maxwell field on black hole space-times. (arxiv.org: ) Lars Andersson (AEI) Potentials and linearized gravity Sanya 46 / 47

96 Bibliography V Nisbet, A. (1955, August). Hertzian Electromagnetic Potentials and Associated Gauge Transformations. Proceedings of the Royal Society of London Series A, 231, doi: /rspa Sachs, R., & Bergmann, P. G. (1958, October). Structure of Particles in Linearized Gravitational Theory. Physical Review, 112, doi: /PhysRev Tataru, D., & Tohaneanu, M. (2008, October). Local energy estimate on Kerr black hole backgrounds. (arxiv.org: ) Walker, M., & Penrose, R. (1970, December). On quadratic first integrals of the geodesic equations for type { 22} spacetimes. Communications in Mathematical Physics, 18, doi: /BF Whiting, B. F., & Price, L. R. (2005, August). Metric reconstruction from Weyl scalars. Classical and Quantum Gravity, 22, S589-S604. doi: / /22/15/003 Lars Andersson (AEI) Potentials and linearized gravity Sanya 47 / 47