new conservation laws for fields on the kerr spacetime · i the debye potential construction cohen,...
TRANSCRIPT
![Page 1: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/1.jpg)
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)
![Page 2: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/2.jpg)
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)
![Page 3: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/3.jpg)
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)
![Page 4: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/4.jpg)
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)
![Page 5: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/5.jpg)
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
![Page 6: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/6.jpg)
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
![Page 7: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/7.jpg)
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)
![Page 8: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/8.jpg)
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)
![Page 9: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/9.jpg)
Maxwell on Minkowski
�A = QA: symmetry of the action + Noether ; conservation laws.Apply with Q “non-geometric” symmetry.
Helicity
D : (
~E,~B) ! (
~B,�~E) Duality rotation Heaviside (1893)
H =
y12 (~A · ~B � ~C · ~E)d3r Helicity Candlin (1965)
Chirality
�~A = r⇥ @t~A Philbin (2013)
C =
y12 (~E ·r⇥ ~E +
~B ·r⇥ ~B)d3r Lipkin (1964), Tang, Cohen (2010)
![Page 10: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/10.jpg)
Maxwell on Minkowski
�A = QA: symmetry of the action + Noether ; conservation laws.Apply with Q “non-geometric” symmetry.
Helicity
D : (
~E,~B) ! (
~B,�~E) Duality rotation Heaviside (1893)
H =
y12 (~A · ~B � ~C · ~E)d3r Helicity Candlin (1965)
Chirality
�~A = r⇥ @t~A Philbin (2013)
C =
y12 (~E ·r⇥ ~E +
~B ·r⇥ ~B)d3r Lipkin (1964), Tang, Cohen (2010)
![Page 11: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/11.jpg)
Maxwell on Minkowski
�A = QA: symmetry of the action + Noether ; conservation laws.Apply with Q “non-geometric” symmetry.
Helicity
D : (
~E,~B) ! (
~B,�~E) Duality rotation Heaviside (1893)
H =
y12 (~A · ~B � ~C · ~E)d3r Helicity Candlin (1965)
Chirality
�~A = r⇥ @t~A Philbin (2013)
C =
y12 (~E ·r⇥ ~E +
~B ·r⇥ ~B)d3r Lipkin (1964), Tang, Cohen (2010)
![Page 12: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/12.jpg)
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
![Page 13: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/13.jpg)
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
![Page 14: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/14.jpg)
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
![Page 15: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/15.jpg)
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
t0
Z
⌦e dtdx . I
H �
H+
I+
I�
i0
i+
i�
![Page 16: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/16.jpg)
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
t0
Z
⌦e dtdx . I
H �
H+
I+
I�
i0
i+
i�
![Page 17: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/17.jpg)
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
t0
Z
⌦e dtdx . I
H �
H+
I+
I�
i0
i+
i�
![Page 18: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/18.jpg)
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)
![Page 19: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/19.jpg)
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)
![Page 20: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/20.jpg)
Operator identitiesAdjoint operator method and conservation laws
Potential 7! field strength
Aa
hab
�A ! = TA
⇢Fab = Fab + i(?F)ab
˙Cabcd =
˙Cabcd + i(? ˙C)abcd
Field equation
EA = 0
Operator IdentitySE = OT Conserved currents
Q
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
![Page 21: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/21.jpg)
Operator identitiesAdjoint operator method and conservation laws
Potential 7! field strength
Aa
hab
�A ! = TA
⇢Fab = Fab + i(?F)ab
˙Cabcd =
˙Cabcd + i(? ˙C)abcd
Field equation
EA = 0
Operator IdentitySE = OT Conserved currents
Q
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
![Page 22: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/22.jpg)
Hidden symmetriesIrreducible symmetry operators
Hidden symmetryof ETY = 0
Q1
TME separationanti-self dual field strength
Q2
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.
![Page 23: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/23.jpg)
Hidden symmetriesIrreducible symmetry operators
Hidden symmetryof ETY = 0
Q1
TME separationanti-self dual field strength
Q2
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.
![Page 24: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/24.jpg)
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
1 4)
$ = S†K : complex “nearly pure gauge" solution of thelinearized Einstein equations on Kerr
p
q
ð
ð0 i
i0
+ 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
![Page 25: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/25.jpg)
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
1 4)
$ = S†K : complex “nearly pure gauge" solution of thelinearized Einstein equations on Kerr
p
q
ð
ð0 i
i0
+ 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
![Page 26: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/26.jpg)
The symplectic current
Green’s identityA · EB � E†A · B = r · ⌦(A,B)
⌦(A,B) is the symplectic current
⌦⌃(A,B) =Z
⌃⌦(A,B) =
Z
⌃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).
![Page 27: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/27.jpg)
The symplectic current
Green’s identityA · EB � E†A · B = r · ⌦(A,B)
⌦(A,B) is the symplectic current
⌦⌃(A,B) =Z
⌃⌦(A,B) =
Z
⌃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).
![Page 28: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/28.jpg)
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
![Page 29: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/29.jpg)
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
![Page 30: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/30.jpg)
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)
![Page 31: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/31.jpg)
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)
![Page 32: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/32.jpg)
Conserved currents from symmetry operatorsLinearized gravity on Kerr
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)
![Page 33: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/33.jpg)
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.
![Page 34: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/34.jpg)
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.
![Page 35: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/35.jpg)
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
![Page 36: New conservation laws for fields on the Kerr spacetime · I The Debye potential construction Cohen, Kegeles (1975), Chrzanowski (1975) leads to the second type of symmetry operator](https://reader036.vdocuments.mx/reader036/viewer/2022070809/5f07a9817e708231d41e1b3c/html5/thumbnails/36.jpg)
Symbolic calculations with xActEfficient tensor/spinor computer algebra for Mathematica
The characterization of Kerr in terms of its Killing spinor has been implemented inxAct �! coordinate free symbolic calculations can be applied!
Open source, www.xAct.es
Powerful algorithms to handle and use symmetriesof tensors and spinors
xAct Packages: SymManipulator, SpinFrames,TexAct
I Irreducible decompositionsI Fundamental spinor operatorsI NP and GHP formalismsI Structured Tex output
Mathematica
xAct
Spinors
xPert
SymManipulator
xCoba
SpinFrames
Spinor variations
Linearized Gravity
TexAct
...