New conservation laws for fields on the Kerr spacetime

Lars Andersson

Albert Einstein Institute

BHI Conference, May 2017

joint work with Steffen Aksteiner (AEI) and Thomas Bäckdahl (Chalmers)

Background

Stability of KerrProof of stability of Kerr requires polynomial decay in time for perturbations ;Energy and Morawetz (integrated energy) currents must be constructed.

stability for Maxwell on Kerr: (L.A. & Blue arXiv:1310.2664)linear stability of Schwarzschild: (Dafermos & Holzegel & RodnianskiarXiv:1601.06467)linear stability of Kerr: open

EMRI, self-force:Currents corresponding to Carter constant: Flanagan& Grant (2017)

Classification problemFind all conserved currents for fields on Kerr (Maxwell, linearized gravity)

Conserved currents for Maxwell on Kerr (L.A. & Bäckdahl & BluearXiv:1504.02069)Conserved currents for linearized gravity on Kerr (this talk)

Background

Symmetries of Minkowski

10-dimensional space of Killing vectors: ⌫a, r(a⌫b) = 0: so(1, 3)� 1+3

15-dimensional space of conformal Killing vectors: c(1, 3)

(T⌫)ab := r(a⌫b) �14rc⌫cgab = 0

Hidden symmetries20-dimensional space of conformal Killing-Yano tensors Yab = Y[ab]:

(TY)abc := r(aYb)c +13 gabrdYc

d � 13 g(a|c|rdYb)d = 0

$ valence (2, 0) Killing spinor AB, (T)A0ABC = 0

Symmetries of spacetime ; conservation laws (Lie, Noether)Hidden symmetries ; "hidden" conservation laws

Symmetries of Minkowski

10-dimensional space of Killing vectors: ⌫a, r(a⌫b) = 0: so(1, 3)� 1+3

15-dimensional space of conformal Killing vectors: c(1, 3)

(T⌫)ab := r(a⌫b) �14rc⌫cgab = 0

Hidden symmetries20-dimensional space of conformal Killing-Yano tensors Yab = Y[ab]:

(TY)abc := r(aYb)c +13 gabrdYc

d � 13 g(a|c|rdYb)d = 0

$ valence (2, 0) Killing spinor AB, (T)A0ABC = 0

Symmetries of spacetime ; conservation laws (Lie, Noether)Hidden symmetries ; "hidden" conservation laws

Maxwell on MinkowskiSymmetries

Classical: c(1, 3)⌦ u(1) — 16 dim., conformal symmetries, duality rotation Heaviside(1892), Bateman (1909)

Modern: c(1, 3)⌦ u(2)⌦ u(2) — 23 dim.,

Includes "non-geometric" symmetries, acting on jet space.

Conserved currents"Classical":

I 15 stress Ja = Tab⌫b, ⌫a 2 c(1, 3)

I 1 Helicity $ duality rotation u(1)

"Modern": 84 Zilch + 378 odd w.r.t. duality reflection

(up to 1st order in derivatives of field strength, defined in terms of CKY, not equivalentto any "classical" current)

Maxwell on MinkowskiSymmetries

Classical: c(1, 3)⌦ u(1) — 16 dim., conformal symmetries, duality rotation Heaviside(1892), Bateman (1909)

Modern: c(1, 3)⌦ u(2)⌦ u(2) — 23 dim.,

Includes "non-geometric" symmetries, acting on jet space.

Conserved currents"Classical":

I 15 stress Ja = Tab⌫b, ⌫a 2 c(1, 3)

I 1 Helicity $ duality rotation u(1)

"Modern": 84 Zilch + 378 odd w.r.t. duality reflection

(up to 1st order in derivatives of field strength, defined in terms of CKY, not equivalentto any "classical" current)

Maxwell on Minkowski

�A = QA: symmetry of the action + Noether ; conservation laws.Apply with Q “non-geometric” symmetry.

Helicity

~E,~B) ! (

~B,�~E) Duality rotation Heaviside (1893)

y12 (~A · ~B � ~C · ~E)d3r Helicity Candlin (1965)

Chirality

�~A = r⇥ @t~A Philbin (2013)

y12 (~E ·r⇥ ~E +

~B ·r⇥ ~B)d3r Lipkin (1964), Tang, Cohen (2010)

Helicity

~E,~B) ! (

Chirality

�~A = r⇥ @t~A Philbin (2013)

y12 (~E ·r⇥ ~E +

Helicity

~E,~B) ! (

Chirality

�~A = r⇥ @t~A Philbin (2013)

y12 (~E ·r⇥ ~E +

Symmetries of Kerr

2-dimensional space of Killing vectors:@t $ energy e, @� $ azimuthal angular momentum lzKerr is algebraically special, Petrov type D

) (Walker-Penrose) One irreducible conformalKilling-Yano tensor Yab (and its dual (?Y)ab):“one hidden symmetry”raYab = 0

) Kab = YacYcb is Killing r(aKbc) = 0

)Geodesics: Carter constant k = Kab�̇a�̇b,integrabilityFields: separability, decoupling, symmetryoperators

Symmetries of Kerr

Black hole stability

Dynamical stability requiresI energy loss via dispersionI cancellation (null condition)

High frequency wave packets can trackorbiting null geodesics for a long time:

Trapping is an obstacle to dispersion

Orbiting null geodesics fill an open region inspacetime

; Must exploit hidden symmetries related to theCarter constant in order to prove dispersion

Integrated Energy EstimateZ t1

⌦e dtdx . I

Teukolsky and Teukolsky-Starobinsky(Teukolsky, Press-Teukolsky, Starobinsky-Churilov)

Field equationsMaxwell equations: (EA)a = rcrcAa �rcraAc = 0

Linearized gravity:(Eh)ab = rcrchab +rarbhc

c � 2rcr(ahb)c � gab(rcrchdd �rcrdhcd) = 0

) decoupled, separable integrability conditions

Teukolsky Master Equations (TME):

Teukolsky-Starobinsky Identities (TSI): = e�i!teim�S(✓)R(r)

Teukolsky and Teukolsky-Starobinsky(Teukolsky, Press-Teukolsky, Starobinsky-Churilov)

Field equationsMaxwell equations: (EA)a = rcrcAa �rcraAc = 0

Linearized gravity:(Eh)ab = rcrchab +rarbhc

c � 2rcr(ahb)c � gab(rcrchdd �rcrdhcd) = 0

) decoupled, separable integrability conditions

Teukolsky Master Equations (TME):

Teukolsky-Starobinsky Identities (TSI): = e�i!teim�S(✓)R(r)

Operator identitiesAdjoint operator method and conservation laws

Potential 7! field strength

�A ! = TA

⇢Fab = Fab + i(?F)ab

˙Cabcd =

˙Cabcd + i(? ˙C)abcd

Field equation

EA = 0

Operator IdentitySE = OT Conserved currents

Symmetry operator

Green’s identity

Assume self-adjointness: E†= E, O†

= OAdjoint identity: ES†

= T†O ) E†S† = 0 if O = 0

) S†: ker O ! ker E Debye map

Q = S†T : ker E ! ker E Symmetry operator

Operator identitiesAdjoint operator method and conservation laws

Potential 7! field strength

�A ! = TA

⇢Fab = Fab + i(?F)ab

˙Cabcd =

˙Cabcd + i(? ˙C)abcd

Field equation

EA = 0

Operator IdentitySE = OT Conserved currents

Symmetry operator

Green’s identity

Assume self-adjointness: E†= E, O†

= OAdjoint identity: ES†

= T†O ) E†S† = 0 if O = 0

) S†: ker O ! ker E Debye map

Q = S†T : ker E ! ker E Symmetry operator

Hidden symmetriesIrreducible symmetry operators

Hidden symmetryof ETY = 0

TME separationanti-self dual field strength

Debye mapself dual field strength

TSI: bSE =

bOT TME: SE = OT

MaxwellI Symmetry operators up to order 2 have been classified for general spacetimes

Kalnins, McLenaghan, Williams (1992), L.A., Bäckdahl, Blue (2014).Linearized gravity

I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leadsto the second type of symmetry operator Q2 of order 4.

I The first type of symmetry operator Q1 of order 6 was found in Aksteiner,Bäckdahl(2016) arXiv:1609.04584.

Hidden symmetriesIrreducible symmetry operators

Hidden symmetryof ETY = 0

TME separationanti-self dual field strength

Debye mapself dual field strength

TSI: bSE =

bOT TME: SE = OT

MaxwellI Symmetry operators up to order 2 have been classified for general spacetimes

Kalnins, McLenaghan, Williams (1992), L.A., Bäckdahl, Blue (2014).Linearized gravity

I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leadsto the second type of symmetry operator Q2 of order 4.

I The first type of symmetry operator Q1 of order 6 was found in Aksteiner,Bäckdahl(2016) arXiv:1609.04584.

Hidden symmetriesFirst type symmetry operator for linearized gravity on Kerr

TSI $ Debye construction of “pure gauge potential”

K: sign-flipped spin-2 projection

( 0, 1, · · · , 4) ! (41 0, 0, 0, 0,�4

$ = S†K : complex “nearly pure gauge" solution of thelinearized Einstein equations on Kerr

+ 0� 4

$ = LAg � M27L@t�g ) covariant TSI

TSI Operator identitybSE = bOT � bLT, bO† = bO, bL† = bL

) Q1 = R(bS†T) spin-2 first type symmetry operator

Proofs use covariant spinor methods

arXiv:1601.06084

arXiv:1609.04584

Hidden symmetriesFirst type symmetry operator for linearized gravity on Kerr

TSI $ Debye construction of “pure gauge potential”

K: sign-flipped spin-2 projection

( 0, 1, · · · , 4) ! (41 0, 0, 0, 0,�4

$ = S†K : complex “nearly pure gauge" solution of thelinearized Einstein equations on Kerr

+ 0� 4

$ = LAg � M27L@t�g ) covariant TSI

TSI Operator identitybSE = bOT � bLT, bO† = bO, bL† = bL

) Q1 = R(bS†T) spin-2 first type symmetry operator

Proofs use covariant spinor methods

arXiv:1601.06084

arXiv:1609.04584

The symplectic current

Green’s identityA · EB � E†A · B = r · ⌦(A,B)

⌦(A,B) is the symplectic current

⌦⌃(A,B) =Z

⌃⌦(A,B) =

⌃A ·rB � B ·rA

If A 7! QA is a symmetry generator, then

⌦⌃(A,QA)

is the Hamiltonian for Q.Canonical energy:

Ecan,⌃(A) = ⌦⌃(A, @tA).

The symplectic current

Green’s identityA · EB � E†A · B = r · ⌦(A,B)

⌦(A,B) is the symplectic current

⌦⌃(A,B) =Z

⌃⌦(A,B) =

⌃A ·rB � B ·rA

If A 7! QA is a symmetry generator, then

⌦⌃(A,QA)

is the Hamiltonian for Q.Canonical energy:

Ecan,⌃(A) = ⌦⌃(A, @tA).

Conserved currents from symmetry operatorsMaxwell on Kerr

Q1,Q2 yield conserved quantities ( = TA)

⌦⌃(Q1A,A) =Z

bS† · , “Stress” + i “Zilch”

⌦⌃(Q2A,A) =Z

S† · , “D-odd”

There is a conserved symmetric tensor

V = rZrZ � 12 (rZ ·rZ)g + (L@t )Z + (L@t )Z

with Z = Y , such that

Ecan,⌃(A,Q1A) =Z

Vab(@t)anb

Conserved currents from symmetry operatorsMaxwell on Kerr

Q1,Q2 yield conserved quantities ( = TA)

⌦⌃(Q1A,A) =Z

bS† · , “Stress” + i “Zilch”

⌦⌃(Q2A,A) =Z

S† · , “D-odd”

There is a conserved symmetric tensor

V = rZrZ � 12 (rZ ·rZ)g + (L@t )Z + (L@t )Z

with Z = Y , such that

Ecan,⌃(A,Q1A) =Z

Vab(@t)anb

Conserved currents from symmetry operatorsLinearized gravity on Kerr

Conserved quantities from symmetry operatorsQ1,Q2 yield (irreducible, higher order) conserved quantities

⌦⌃(Q1�g, �g) order 7⌦⌃(Q2�g, �g) order 5

Currents representing Carter constantWave packet �gab ⇠ �abei�/✏, ✏ ! 0Linearized gravity Grant, Flanagan (2017)

⌦⌃(Q2�g,Q2�g) ⇠ k4(e2

R � e2L)

Linearized gravity (conjectured):

⌦⌃(Q1�g, �g) ⇠ k3(e2

R � e2L)

Conserved quantities from symmetry operatorsQ1,Q2 yield (irreducible, higher order) conserved quantities

⌦⌃(Q1�g, �g) order 7⌦⌃(Q2�g, �g) order 5

Currents representing Carter constantWave packet �gab ⇠ �abei�/✏, ✏ ! 0Linearized gravity Grant, Flanagan (2017)

⌦⌃(Q2�g,Q2�g) ⇠ k4(e2

R � e2L)

Linearized gravity (conjectured):

⌦⌃(Q1�g, �g) ⇠ k3(e2

R � e2L)

Higher order energy from DebyeEcan(S† ) yields a conserved canonical energy for TME.Prabhu, & Wald (2017):

I For linearized gravity on Schwarzschild

0 Ecan(S† 0)

I Ecan(S† 0) ⌘ “Regge-Wheeler" energy used in (Dafermos & Holzegel &Rodnianski arXiv:1601.06467)

Concluding remarksR. Penrose. “Chandrasekhar, Black Holes, and Singularities”. In: Journal ofAstrophysics and Astronomy 17 (1996), pp. 213–232

In spite of much classical work, the topic of symmetries and conservation lawsfor higher spin fields on black hole backgrounds appears to be far from exhaused.

Thank You

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