Symmetry inheritance, black holes and no-hair theorems Ivica Smoli - - PowerPoint PPT Presentation

▶
symmetry inheritance black holes and no hair theorems
SMART_READER_LITE
LIVE PREVIEW

Symmetry inheritance, black holes and no-hair theorems Ivica Smoli - - PowerPoint PPT Presentation

Oct 31, 2022 132 likes •1.38k views

Symmetry inheritance, black holes and no-hair theorems Ivica Smoli c University of Zagreb Symmetries Symmetries a continuous symmetry (isometry) of the spacetime metric g ab is encapsulated in a Killing vector field a , g ab =

slide-1
SLIDE 1

Symmetry inheritance, black holes and no-hair theorems

Ivica Smoli´ c

University of Zagreb

slide-2
SLIDE 2

Symmetries

slide-3
SLIDE 3

Symmetries

  • a continuous symmetry (isometry) of the spacetime metric gab

is encapsulated in a Killing vector field ξa, £ξgab = ∇

aξb + ∇ bξa = 0

slide-4
SLIDE 4

Symmetries

  • a continuous symmetry (isometry) of the spacetime metric gab

is encapsulated in a Killing vector field ξa, £ξgab = ∇

aξb + ∇ bξa = 0

  • for example,

2-dim Euclid ∂x, ∂y and x∂y − y∂x (1 + 1)-dim Minkowski ∂t, ∂x and x∂t + t∂x Kerr black hole ∂t and ∂ϕ

slide-5
SLIDE 5

Symmetry inheritance

slide-6
SLIDE 6

Symmetry inheritance

  • Suppose that some mater/gauge field ψa...

b... is present in the

spacetime, solution to a system of gravitational-mater/gauge field equations.

slide-7
SLIDE 7

Symmetry inheritance

  • Suppose that some mater/gauge field ψa...

b... is present in the

spacetime, solution to a system of gravitational-mater/gauge field equations. ...a natural question occurs :

£ξgab = 0 ⇒ £ξψa...

b... = 0 ?

slide-8
SLIDE 8

Symmetry inheritance

  • Suppose that some mater/gauge field ψa...

b... is present in the

spacetime, solution to a system of gravitational-mater/gauge field equations. ...a natural question occurs :

£ξgab = 0 ⇒ £ξψa...

b... = 0 ?

If the answer is yes, then we say that the field ψa...

b... inherits the symmetries of the metric gab

slide-9
SLIDE 9

Why bother?

slide-10
SLIDE 10
  • math. physics

phenomenology

slide-11
SLIDE 11
  • math. physics

phenomenology

classifications & uniqueness thms a quest for the black hole hair

slide-12
SLIDE 12
slide-13
SLIDE 13

Symmetry inheritance for ...

⋆ fields of various spin

slide-14
SLIDE 14

Symmetry inheritance for ...

⋆ fields of various spin ⋆ various types of couplings

slide-15
SLIDE 15

Symmetry inheritance for ...

⋆ fields of various spin ⋆ various types of couplings ⋆ various gravitational theories

slide-16
SLIDE 16

Symmetry inheritance for ...

⋆ fields of various spin ⋆ various types of couplings ⋆ various gravitational theories ⋆ different spacetime dimensions

slide-17
SLIDE 17

Desiderata

£ξψa...

b... =

some zeros and well-defined exceptions + concrete examples

s = 0 | 1/2 | 1 | · · · D = 2 | 3 | 4 | 5 | · · ·

slide-18
SLIDE 18

Plan

slide-19
SLIDE 19

Plan

  • verview

novel results

  • pen questions

1501.04967 70s → 00s 1508.03343

?

1609.04013

slide-20
SLIDE 20

⋆ leave aside the related mathematical problem of collineations

£ξgab = 0 vs £ξRabcd = 0 vs £ξRab = 0 vs £ξR = 0 vs ...

slide-21
SLIDE 21

⋆ leave aside the related mathematical problem of collineations

£ξgab = 0 vs £ξRabcd = 0 vs £ξRab = 0 vs £ξR = 0 vs ...

Katzin, Levine and Davis: J.Math.Phys. 10 (1969) 617

slide-22
SLIDE 22

Long time ago ...

slide-23
SLIDE 23

Long time ago ...

By the end of the Golden era

  • f GR people began to won-

der what are the symmetry inheritance properties of the various physical fields. A thorough search was con- ducted ...

slide-24
SLIDE 24

Strategies

slide-25
SLIDE 25

Strategies

  • short-sighted: do the analysis for a concrete EOM, concrete

isometry and in adopted coordinates

slide-26
SLIDE 26

Strategies

  • short-sighted: do the analysis for a concrete EOM, concrete

isometry and in adopted coordinates

  • ...a bit beter: do the analysis for a general isometry, but

concrete gravitational EOM, using some peculiar features of the particular field equations

slide-27
SLIDE 27

Strategies

  • short-sighted: do the analysis for a concrete EOM, concrete

isometry and in adopted coordinates

  • ...a bit beter: do the analysis for a general isometry, but

concrete gravitational EOM, using some peculiar features of the particular field equations Can we be even more general?

slide-28
SLIDE 28

A general strategy

slide-29
SLIDE 29

A general strategy

  • Gravitational field equation

Eab = 8πTab

slide-30
SLIDE 30

A general strategy

  • Gravitational field equation

Eab = 8πTab

  • For any Killing vector field ξa,

£ξRabcd = 0 , £ξǫab... = 0 , £ξ∇

a = ∇ a £ξ

slide-31
SLIDE 31

A general strategy

  • Gravitational field equation

Eab = 8πTab

  • For any Killing vector field ξa,

£ξRabcd = 0 , £ξǫab... = 0 , £ξ∇

a = ∇ a £ξ

  • Thus, £ξEab = 0 and the problem is reduced to

£ξTab = 0

slide-32
SLIDE 32

Electromagnetic Field

slide-33
SLIDE 33

EM Symm.Inh. in a Nutshell

(1 + 1) £ξFab = 0 (1 + 2) £ξFab = 0 CDPS ’16 (1 + 3) £ξFab = f ∗F ab MW ’75 / WY ’76 D ≥ 5 £ξFab = ??

slide-34
SLIDE 34

A warm up: (1+1)-dim EM field

slide-35
SLIDE 35

A warm up: (1+1)-dim EM field

Fab = f ǫab

slide-36
SLIDE 36

A warm up: (1+1)-dim EM field

Fab = f ǫab 0 = d F ✓ 0 = d ∗F = −d f ⇒ f = const.

slide-37
SLIDE 37

A warm up: (1+1)-dim EM field

Fab = f ǫab 0 = d F ✓ 0 = d ∗F = −d f ⇒ f = const. £ξFab = (£ξf ) ǫab + f (£ξǫab) = 0

slide-38
SLIDE 38

Classical result: (1+3)-dim EM field

slide-39
SLIDE 39

Classical result: (1+3)-dim EM field

  • first results via Rainich-Misner-Wheeler formalism

[Woolley 1973; Michalski and Wainwright 1975]

slide-40
SLIDE 40

Classical result: (1+3)-dim EM field

  • first results via Rainich-Misner-Wheeler formalism

[Woolley 1973; Michalski and Wainwright 1975]

  • generalization via basis of vectors

[Wainwright and Yaremovicz 1976]

slide-41
SLIDE 41

Classical result: (1+3)-dim EM field

  • first results via Rainich-Misner-Wheeler formalism

[Woolley 1973; Michalski and Wainwright 1975]

  • generalization via basis of vectors

[Wainwright and Yaremovicz 1976]

  • simplified proof via spinors [Tod 2007]

FABA′B′ = φAB ǫA′B′ + φA′B′ ǫAB , TABA′B′ = 1 2π φAB φA′B′

slide-42
SLIDE 42

Classical result: (1+3)-dim EM field

  • first results via Rainich-Misner-Wheeler formalism

[Woolley 1973; Michalski and Wainwright 1975]

  • generalization via basis of vectors

[Wainwright and Yaremovicz 1976]

  • simplified proof via spinors [Tod 2007]

FABA′B′ = φAB ǫA′B′ + φA′B′ ǫAB , TABA′B′ = 1 2π φAB φA′B′

  • “master equation” £ξTABA′B′ = 0 implies £ξφAB = ia φAB for

some real function a and

£ξFab = f ∗F ab

slide-43
SLIDE 43
  • f is constant if Fab is non-null

(we say that Fab is null if Fab F ab = Fab ∗F ab = 0)

slide-44
SLIDE 44
  • f is constant if Fab is non-null

(we say that Fab is null if Fab F ab = Fab ∗F ab = 0)

  • further constraints in the presence of the black hole
slide-45
SLIDE 45
  • f is constant if Fab is non-null

(we say that Fab is null if Fab F ab = Fab ∗F ab = 0)

  • further constraints in the presence of the black hole

⋆ an example of symmetry noninheriting EM field [Tariq and Tupper 1975; Michalski and Wainwright 1975]

ds2 = 1 (2r)2 (dr2 + dz2) + r2dϕ2 − (dt − 2z dϕ)2 F = − √ 8 cos α r dr ∧ (dt − 2z dϕ) + sin α dz ∧ dϕ

  • α = −2 ln r + α0

ξ = r ∂ ∂r + z ∂ ∂z − ϕ ∂ ∂ϕ , £ξF = −2 ∗F

slide-46
SLIDE 46

A recent result: (1+2)-dim EM field

[M. Cvitan, P. Dominis Prester and I.Sm.: CQG 33 (2016) 077001]

slide-47
SLIDE 47

A recent result: (1+2)-dim EM field

[M. Cvitan, P. Dominis Prester and I.Sm.: CQG 33 (2016) 077001]

  • introduce auxiliary “electric” and “magnetic” fields

N = ξaξa , Ea = ξbFab , B = ξa ∗F a

slide-48
SLIDE 48

A recent result: (1+2)-dim EM field

[M. Cvitan, P. Dominis Prester and I.Sm.: CQG 33 (2016) 077001]

  • introduce auxiliary “electric” and “magnetic” fields

N = ξaξa , Ea = ξbFab , B = ξa ∗F a

  • the key observation

8πTabξaξb = EaEa + B2 Ea £ξEa + B £ξB = 0 4π ∗ (ξ ∧ T(ξ))a = −BEa B £ξEa + (£ξB)Ea = 0

slide-49
SLIDE 49

A recent result: (1+2)-dim EM field

[M. Cvitan, P. Dominis Prester and I.Sm.: CQG 33 (2016) 077001]

  • introduce auxiliary “electric” and “magnetic” fields

N = ξaξa , Ea = ξbFab , B = ξa ∗F a

  • the key observation

8πTabξaξb = EaEa + B2 Ea £ξEa + B £ξB = 0 4π ∗ (ξ ∧ T(ξ))a = −BEa B £ξEa + (£ξB)Ea = 0

£ξFab = 0

slide-50
SLIDE 50

Fermions

slide-51
SLIDE 51

Spin-1/2 fields

slide-52
SLIDE 52

Spin-1/2 fields

  • C.A. Kolassis for the Einstein-Weyl EOM
slide-53
SLIDE 53

Spin-1/2 fields

  • C.A. Kolassis for the Einstein-Weyl EOM

J.Math.Phys. 23 (9) 1982

slide-54
SLIDE 54

Spin-1/2 fields

  • C.A. Kolassis for the Einstein-Weyl EOM

J.Math.Phys. 23 (9) 1982 Phys.Let. 95 A, 1983

slide-55
SLIDE 55

Spin-1/2 fields

  • C.A. Kolassis for the Einstein-Weyl EOM

J.Math.Phys. 23 (9) 1982 Phys.Let. 95 A, 1983

(a) if ℓa = νAνA′ is collinear with one of the principal null directions of the Weyl tensor £ξνA = is νA with real constant s

slide-56
SLIDE 56

Spin-1/2 fields

  • C.A. Kolassis for the Einstein-Weyl EOM

J.Math.Phys. 23 (9) 1982 Phys.Let. 95 A, 1983

(a) if ℓa = νAνA′ is collinear with one of the principal null directions of the Weyl tensor £ξνA = is νA with real constant s (b) ...otherwise £ξνA = f νA

slide-57
SLIDE 57

Scalar Fields

slide-58
SLIDE 58

Real scalar field

slide-59
SLIDE 59

Real scalar field

  • minimally coupled, canonical

[Hoenselaers 1978; I.Sm. 2015]

Tab = (∇

aφ)(∇ bφ) + (X − V(φ))gab ,

X ≡ −1 2 (∇cφ)(∇

cφ)

⇒ 0 = £ξV(φ) = V ′(φ) £ξφ

slide-60
SLIDE 60

Real scalar field

  • minimally coupled, canonical

[Hoenselaers 1978; I.Sm. 2015]

Tab = (∇

aφ)(∇ bφ) + (X − V(φ))gab ,

X ≡ −1 2 (∇cφ)(∇

cφ)

⇒ 0 = £ξV(φ) = V ′(φ) £ξφ

slide-61
SLIDE 61

Real scalar field

  • minimally coupled, canonical

[Hoenselaers 1978; I.Sm. 2015]

Tab = (∇

aφ)(∇ bφ) + (X − V(φ))gab ,

X ≡ −1 2 (∇cφ)(∇

cφ)

⇒ 0 = £ξV(φ) = V ′(φ) £ξφ V ′(φ) = 0 £ξφ = 0 V ′(φ) = 0 £ξφ = a = const. and if ξa has compact orbits then a = 0.

slide-62
SLIDE 62
  • an example of time dependent real scalar field in a stationary

spacetime: M. Wyman, Phys. Rev. D 24 (1981) 839 ds2 = −eν(r)dt2 + eλ(r)dr2 + r2(dθ2 + sin2 θ dϕ2) φ(t) = γ t , γ = const.

slide-63
SLIDE 63
  • an example of time dependent real scalar field in a stationary

spacetime: M. Wyman, Phys. Rev. D 24 (1981) 839 ds2 = −eν(r)dt2 + eλ(r)dr2 + r2(dθ2 + sin2 θ dϕ2) φ(t) = γ t , γ = const.

  • two solutions: a simpler one with eν = 8πγ2r2 and eλ = 2, and

the second one in a form a Taylor series.

slide-64
SLIDE 64
  • “k-essence” theories

a generic model for the inflationary evolution Tab = p,X (∇

aφ)(∇ bφ) + p gab ,

p = p(φ, X)

slide-65
SLIDE 65
  • “k-essence” theories

a generic model for the inflationary evolution Tab = p,X (∇

aφ)(∇ bφ) + p gab ,

p = p(φ, X)

  • lemma [I.Sm. 2015] £ξp = £ξ(Xp,X) = 0
slide-66
SLIDE 66
  • “k-essence” theories

a generic model for the inflationary evolution Tab = p,X (∇

aφ)(∇ bφ) + p gab ,

p = p(φ, X)

  • lemma [I.Sm. 2015] £ξp = £ξ(Xp,X) = 0

Xp,X = 0 ⇒ £ξ£ξφ = 0 along the orbit (for admissible Tab)

slide-67
SLIDE 67
  • “k-essence” theories

a generic model for the inflationary evolution Tab = p,X (∇

aφ)(∇ bφ) + p gab ,

p = p(φ, X)

  • lemma [I.Sm. 2015] £ξp = £ξ(Xp,X) = 0

Xp,X = 0 ⇒ £ξ£ξφ = 0 along the orbit (for admissible Tab) Xp,X = 0 ⇒ £ξφ is a solution to p,φ(£ξφ)2 + 2Xp,X £ξ£ξφ = 0 which is either identically zero or doesn’t have any zeros along the orbit of ξa

slide-68
SLIDE 68

Addendum: Ideal fluid

slide-69
SLIDE 69

Addendum: Ideal fluid

  • [Hoenselaers 1978]

Tab = (ρ + p) uaub + p gab

slide-70
SLIDE 70

Addendum: Ideal fluid

  • [Hoenselaers 1978]

Tab = (ρ + p) uaub + p gab £ξTab = 0 ⇒ £ξρ = £ξp = 0 = £ξua

slide-71
SLIDE 71

Complex scalar field

slide-72
SLIDE 72

Complex scalar field

  • energy-momentum tensor

Tab = ∇

(aφ∇ b)φ∗ − 1

2

  • ∇cφ∇

cφ∗ + V(φ∗φ)

  • gab
slide-73
SLIDE 73

Complex scalar field

  • energy-momentum tensor

Tab = ∇

(aφ∇ b)φ∗ − 1

2

  • ∇cφ∇

cφ∗ + V(φ∗φ)

  • gab
  • e.g. in polar form φ = Aeiα :

Tab = ∇

aA ∇ bA + A2 ∇ aα ∇ bα + T + V(A2)

D − 2 gab

slide-74
SLIDE 74
  • subcase #1: symmetry inheriting amplitude, £ξA = 0

→ £ξα is a constant !

slide-75
SLIDE 75
  • subcase #1: symmetry inheriting amplitude, £ξA = 0

→ £ξα is a constant !

  • subcase #2: symmetry inheriting phase, £ξα = 0,

(£ξA)2 + N D − 2 V(A2) = λ

slide-76
SLIDE 76
  • subcase #1: symmetry inheriting amplitude, £ξA = 0

→ £ξα is a constant !

  • subcase #2: symmetry inheriting phase, £ξα = 0,

(£ξA)2 + N D − 2 V(A2) = λ ⋆ for V = µ2A2, the only symmetry noninheriting amplitude A which is bounded or periodic along the orbits of ξa is A ∼ sin(√κ(x − x0)) but N = const. > 0 and ξa is hypersurface orthogonal

slide-77
SLIDE 77

Black Hole Hair

slide-78
SLIDE 78

What is black hole hair?

slide-79
SLIDE 79

What is black hole hair?

  • the term was coined by J.A. Wheeler and R. Ruffini, Introducing

the black hole, Physics Today 24 (1971) 30

slide-80
SLIDE 80

What is black hole hair?

  • the term was coined by J.A. Wheeler and R. Ruffini, Introducing

the black hole, Physics Today 24 (1971) 30

  • roughly, a broad definition:

any non-gravitational field in a black hole spacetime

slide-81
SLIDE 81

What is black hole hair?

  • the term was coined by J.A. Wheeler and R. Ruffini, Introducing

the black hole, Physics Today 24 (1971) 30

  • roughly, a broad definition:

any non-gravitational field in a black hole spacetime

  • more refined definition:

any non-gravitational field in a black hole spacetime contributing to the conserved “charges” associated to the black hole, apart from the total mass M, the angular momentum J, the electric charge Q and the magnetic charge P (see also: primary/secondary hair distinction)

slide-82
SLIDE 82

No-hair theorems

slide-83
SLIDE 83

No-hair theorems

  • Bekenstein, PRL 28 (1971) 452
slide-84
SLIDE 84

No-hair theorems

  • Bekenstein, PRL 28 (1971) 452

The absence of the scalar black hole hair is always proven under some particular assumptions about the scalar field φ,

slide-85
SLIDE 85

No-hair theorems

  • Bekenstein, PRL 28 (1971) 452

The absence of the scalar black hole hair is always proven under some particular assumptions about the scalar field φ, (a) a choice of the scalar field coupling to gravity,

slide-86
SLIDE 86

No-hair theorems

  • Bekenstein, PRL 28 (1971) 452

The absence of the scalar black hole hair is always proven under some particular assumptions about the scalar field φ, (a) a choice of the scalar field coupling to gravity, (b) an energy condition,

slide-87
SLIDE 87

No-hair theorems

  • Bekenstein, PRL 28 (1971) 452

The absence of the scalar black hole hair is always proven under some particular assumptions about the scalar field φ, (a) a choice of the scalar field coupling to gravity, (b) an energy condition, (c) details about the “asymptotics”

slide-88
SLIDE 88

No-hair theorems

  • Bekenstein, PRL 28 (1971) 452

The absence of the scalar black hole hair is always proven under some particular assumptions about the scalar field φ, (a) a choice of the scalar field coupling to gravity, (b) an energy condition, (c) details about the “asymptotics” (d) the assumption that the scalar field φ inherits the spacetime symmetries

slide-89
SLIDE 89

Symmetry noninheriting scalar black hole hair

slide-90
SLIDE 90

Symmetry noninheriting scalar black hole hair

  • Herdeiro and Radu, PRL 112 (2014) 221101
slide-91
SLIDE 91

Symmetry noninheriting scalar black hole hair

  • Herdeiro and Radu, PRL 112 (2014) 221101

numerical stationary axially symmetric solution of the Einstein-Klein-Gordon EOM, with the complex scalar field φ = A(r, θ) ei(mϕ−ωt) with ω = ΩHm

slide-92
SLIDE 92

Symmetry noninheriting scalar black hole hair

  • Herdeiro and Radu, PRL 112 (2014) 221101

numerical stationary axially symmetric solution of the Einstein-Klein-Gordon EOM, with the complex scalar field φ = A(r, θ) ei(mϕ−ωt) with ω = ΩHm

  • Are there any other hairy black hole solutions based on

symmetry noninheritance? What are the constraints on the existence of the sni scalar black hole hair?

slide-93
SLIDE 93
  • n any Killing horizon H[ξ] we have R(ξ, ξ) = 0
slide-94
SLIDE 94
  • n any Killing horizon H[ξ] we have R(ξ, ξ) = 0
  • thus, for the Einstein-KG, T(ξ, ξ) = 0 on H[ξ]
slide-95
SLIDE 95
  • n any Killing horizon H[ξ] we have R(ξ, ξ) = 0
  • thus, for the Einstein-KG, T(ξ, ξ) = 0 on H[ξ]

→ implications [I.Sm. 2015]

slide-96
SLIDE 96
  • n any Killing horizon H[ξ] we have R(ξ, ξ) = 0
  • thus, for the Einstein-KG, T(ξ, ξ) = 0 on H[ξ]

→ implications [I.Sm. 2015] ⋆ real canonical scalar field, £ξφ = 0 (no sni BH hair!)

slide-97
SLIDE 97
  • n any Killing horizon H[ξ] we have R(ξ, ξ) = 0
  • thus, for the Einstein-KG, T(ξ, ξ) = 0 on H[ξ]

→ implications [I.Sm. 2015] ⋆ real canonical scalar field, £ξφ = 0 (no sni BH hair!) ⋆ complex scalar field with symmetry inheriting amplitude: a constraint for H[χ] with χa = ka + ΩHma £kα + ΩH £mα = 0

slide-98
SLIDE 98
  • n any Killing horizon H[ξ] we have R(ξ, ξ) = 0
  • thus, for the Einstein-KG, T(ξ, ξ) = 0 on H[ξ]

→ implications [I.Sm. 2015] ⋆ real canonical scalar field, £ξφ = 0 (no sni BH hair!) ⋆ complex scalar field with symmetry inheriting amplitude: a constraint for H[χ] with χa = ka + ΩHma £kα + ΩH £mα = 0 ⋆ complex scalar field with symmetry inheriting phase: no sni BH hair (via Vishveshwara-Carter tm)

slide-99
SLIDE 99

Hair constraints beyond Einstein

[I.Sm. arXiv:1609.04013]

slide-100
SLIDE 100

Hair constraints beyond Einstein

[I.Sm. arXiv:1609.04013]

  • idea: use the Frobenius’ theorem (diff. geom.)
slide-101
SLIDE 101

Hair constraints beyond Einstein

[I.Sm. arXiv:1609.04013]

  • idea: use the Frobenius’ theorem (diff. geom.)

⋆ integrable iff involute, [X(i), X(j)]a ∈ ∆

slide-102
SLIDE 102

Hair constraints beyond Einstein

[I.Sm. arXiv:1609.04013]

  • idea: use the Frobenius’ theorem (diff. geom.)

⋆ integrable iff involute, [X(i), X(j)]a ∈ ∆ ⋆ orthogonally-transitive iff X(1) ∧ . . . ∧ X(n) ∧ dX(i) = 0

slide-103
SLIDE 103

static k ∧ dk = 0 Schwarzschild circular k ∧ m ∧ dk = = k ∧ m ∧ dm = 0 Kerr

slide-104
SLIDE 104

static k ∧ dk = 0 Schwarzschild circular k ∧ m ∧ dk = = k ∧ m ∧ dm = 0 Kerr

  • static → Ricci static,

k ∧ R(k) = 0 , Rti = 0

slide-105
SLIDE 105

static k ∧ dk = 0 Schwarzschild circular k ∧ m ∧ dk = = k ∧ m ∧ dm = 0 Kerr

  • static → Ricci static,

k ∧ R(k) = 0 , Rti = 0

  • circular → Ricci circular,

k ∧ m ∧ R(k) = k ∧ m ∧ R(m) = 0 , Rti = Rϕi = 0

slide-106
SLIDE 106
  • generalization:

a spacetime with commuting Killing vectors {ξa

(1), . . . , ξa (n)}, s.t.

ξ(1) ∧ . . . ∧ ξ(n) ∧ dξ(i) = 0

slide-107
SLIDE 107
  • generalization:

a spacetime with commuting Killing vectors {ξa

(1), . . . , ξa (n)}, s.t.

ξ(1) ∧ . . . ∧ ξ(n) ∧ dξ(i) = 0

  • the class of gravitational tensors Eab such that

ξ(1) ∧ . . . ∧ ξ(n) ∧ E(ξ(i)) = 0

slide-108
SLIDE 108
  • generalization:

a spacetime with commuting Killing vectors {ξa

(1), . . . , ξa (n)}, s.t.

ξ(1) ∧ . . . ∧ ξ(n) ∧ dξ(i) = 0

  • the class of gravitational tensors Eab such that

ξ(1) ∧ . . . ∧ ξ(n) ∧ E(ξ(i)) = 0 . . . ⇒ T(χ, χ) = 0

  • n

H[χ]

slide-109
SLIDE 109

Open Qestions

slide-110
SLIDE 110

Voids in the table

slide-111
SLIDE 111

Voids in the table

  • non-minimally coupled real scalar fields

[I.Sm. 2015] → conformal symmetry inheritance, £ξgab = ψgab

slide-112
SLIDE 112

Voids in the table

  • non-minimally coupled real scalar fields

[I.Sm. 2015] → conformal symmetry inheritance, £ξgab = ψgab

  • complex scalar field

symmetry inheriting phase with general potential V = V(A2), the field with sni both A and α ;

slide-113
SLIDE 113

Voids in the table

  • non-minimally coupled real scalar fields

[I.Sm. 2015] → conformal symmetry inheritance, £ξgab = ψgab

  • complex scalar field

symmetry inheriting phase with general potential V = V(A2), the field with sni both A and α ;

  • Weyl fermions with the general gravitational EOM;

massive, Dirac fermions

slide-114
SLIDE 114

Voids in the table

  • non-minimally coupled real scalar fields

[I.Sm. 2015] → conformal symmetry inheritance, £ξgab = ψgab

  • complex scalar field

symmetry inheriting phase with general potential V = V(A2), the field with sni both A and α ;

  • Weyl fermions with the general gravitational EOM;

massive, Dirac fermions

  • EM field for D ≥ 5
slide-115
SLIDE 115

Symmetry inheritance spliting

slide-116
SLIDE 116

Symmetry inheritance spliting

  • What if we have two or more mater fields in the spacetime?
slide-117
SLIDE 117

Symmetry inheritance spliting

  • What if we have two or more mater fields in the spacetime?
  • For example, Tab = T (1)

ab + T (2) ab ; under which conditions

£ξTab = 0 can be split into £ξT (1)

ab = 0 = £ξT (2) ab

?

slide-118
SLIDE 118

Symmetry inheritance spliting

  • What if we have two or more mater fields in the spacetime?
  • For example, Tab = T (1)

ab + T (2) ab ; under which conditions

£ξTab = 0 can be split into £ξT (1)

ab = 0 = £ξT (2) ab

?

  • Wainwright and Yaremovicz [Gen.Rel.Grav. 7 (1976) 345–359

and 595–608] treat the EM field + charged ideal fluid

slide-119
SLIDE 119

Approximate symmetry inheritance

slide-120
SLIDE 120

Approximate symmetry inheritance

  • the definition of the approximate symmetries brings in certain

ambiguities ...

slide-121
SLIDE 121

Approximate symmetry inheritance

  • the definition of the approximate symmetries brings in certain

ambiguities ... ⋆ conformal Killing vector field, £ξgab = ψgab

slide-122
SLIDE 122

Approximate symmetry inheritance

  • the definition of the approximate symmetries brings in certain

ambiguities ... ⋆ conformal Killing vector field, £ξgab = ψgab ⋆ Matzner J.Math.Phys. 9 (1968) 1657, ∇b∇

(aξb) + λξa = 0;

→ Krisch and Glass arXiv:1508.04614

slide-123
SLIDE 123

Approximate symmetry inheritance

  • the definition of the approximate symmetries brings in certain

ambiguities ... ⋆ conformal Killing vector field, £ξgab = ψgab ⋆ Matzner J.Math.Phys. 9 (1968) 1657, ∇b∇

(aξb) + λξa = 0;

→ Krisch and Glass arXiv:1508.04614

  • we need a systematic approach ...
slide-124
SLIDE 124

Thank you for your atention!