electromagnetic fields with vanishing scalar invariants ... · electromagnetic fields with...

Electromagnetic fields with vanishing scalar invariants1

Marcello Ortaggio

Institute of MathematicsAcademy of Sciences of the Czech Republic

Trento – June 6th, 2016

1Joint work with V. Pravda, Class. Quantum Grav. 33 (2016) 115010

Electromagnetic fields with vanishing scalar invariants

Contents

1 Null electromagnetic fields

2 Null gravitational fields

3 Main result: VSI p-forms

4 Universal solutions

Electromagnetic fields with vanishing scalar invariants

Null electromagnetic fields

Null electromagnetic fields

Definition [Synge (Silberstein, Bateman, Rainich, Ruse, . . . )]

1 FabFab = 0 (⇔ E2 −B2 = 0)

2 Fab∗F ab = 0 (⇔ ~E · ~B = 0)

Electromagnetic fields with vanishing scalar invariants

Null electromagnetic fields

Null electromagnetic fields

Definition [Synge (Silberstein, Bateman, Rainich, Ruse, . . . )]

1 FabFab = 0 (⇔ E2 −B2 = 0)

2 Fab∗F ab = 0 (⇔ ~E · ~B = 0)

intrinsically Lorentzian property

Electromagnetic fields with vanishing scalar invariants

Null electromagnetic fields

Null electromagnetic fields

Definition [Synge (Silberstein, Bateman, Rainich, Ruse, . . . )]

1 FabFab = 0 (⇔ E2 −B2 = 0)

2 Fab∗F ab = 0 (⇔ ~E · ~B = 0)

intrinsically Lorentzian property

plane waves

Electromagnetic fields with vanishing scalar invariants

Null electromagnetic fields

Null electromagnetic fields

Definition [Synge (Silberstein, Bateman, Rainich, Ruse, . . . )]

1 FabFab = 0 (⇔ E2 −B2 = 0)

2 Fab∗F ab = 0 (⇔ ~E · ~B = 0)

intrinsically Lorentzian property

plane waves

radiating systems: asymptotically F = Nr + . . .

Electromagnetic fields with vanishing scalar invariants

Null electromagnetic fields

Null electromagnetic fields

Definition [Synge (Silberstein, Bateman, Rainich, Ruse, . . . )]

1 FabFab = 0 (⇔ E2 −B2 = 0)

2 Fab∗F ab = 0 (⇔ ~E · ~B = 0)

intrinsically Lorentzian property

plane waves

radiating systems: asymptotically F = Nr + . . .

“ǫ-property”

Electromagnetic fields with vanishing scalar invariants

Null electromagnetic fields

Null electromagnetic fields

Definition [Synge (Silberstein, Bateman, Rainich, Ruse, . . . )]

1 FabFab = 0 (⇔ E2 −B2 = 0)

2 Fab∗F ab = 0 (⇔ ~E · ~B = 0)

intrinsically Lorentzian property

plane waves

radiating systems: asymptotically F = Nr + . . .

“ǫ-property”

solutions of Maxwell ⇒ also solve Born-Infeld’s and any NLE![Schrodinger’35,’43]

Electromagnetic fields with vanishing scalar invariants

Null gravitational fields

Null gravitational fields

All Riemann scalar invariants I(Riem) vanish:

R = 0, RabRab = 0, RabcdR

abcd = 0, . . .

Electromagnetic fields with vanishing scalar invariants

Null gravitational fields

Null gravitational fields

All Riemann scalar invariants I(Riem) vanish:

R = 0, RabRab = 0, RabcdR

abcd = 0, . . .

equivalent to “Riemann type” III (N, O)

Electromagnetic fields with vanishing scalar invariants

Null gravitational fields

Null gravitational fields

All Riemann scalar invariants I(Riem) vanish:

R = 0, RabRab = 0, RabcdR

abcd = 0, . . .

equivalent to “Riemann type” III (N, O)

in HD: Einstein+type N ⇒ also solve

quadratic (⊃Gauss-Bonnet): Gab + Λ0gab + (Riem)2 +∇2Ric = 0

Lovelock gravity: Gab + Λ0gab +∑

k=2

(Riem)k = 0

[Pravdova-Pravda’08, Malek-Pravda’11, Reall-Tanahashi-Way’14]

Electromagnetic fields with vanishing scalar invariants

Null gravitational fields

Null gravitational fields

All Riemann scalar invariants I(Riem) vanish:

R = 0, RabRab = 0, RabcdR

abcd = 0, . . .

equivalent to “Riemann type” III (N, O)

in HD: Einstein+type N ⇒ also solve

quadratic (⊃Gauss-Bonnet): Gab + Λ0gab + (Riem)2 +∇2Ric = 0

Lovelock gravity: Gab + Λ0gab +∑

k=2

(Riem)k = 0

[Pravdova-Pravda’08, Malek-Pravda’11, Reall-Tanahashi-Way’14]

certain type N pp -waves: solve any L(Riem,∇Riem, . . .)[Guven’87, Amati-Klimcık’89, Horowitz-Steif’90]

Electromagnetic fields with vanishing scalar invariants

Null gravitational fields

Null gravitational fields

All Riemann scalar invariants I(Riem) vanish:

R = 0, RabRab = 0, RabcdR

abcd = 0, . . .

equivalent to “Riemann type” III (N, O)

in HD: Einstein+type N ⇒ also solve

quadratic (⊃Gauss-Bonnet): Gab + Λ0gab + (Riem)2 +∇2Ric = 0

Lovelock gravity: Gab + Λ0gab +∑

k=2

(Riem)k = 0

[Pravdova-Pravda’08, Malek-Pravda’11, Reall-Tanahashi-Way’14]

certain type N pp -waves: solve any L(Riem,∇Riem, . . .)[Guven’87, Amati-Klimcık’89, Horowitz-Steif’90]

these are in fact VSI spacetime: any I(Riem,∇Riem, . . .) = 0![Pravda-Pravdova-Coley-Milson’02, Coley-Milson-Pravda-Pravdova’04]

Electromagnetic fields with vanishing scalar invariants

1 what about VSI Maxwell fields?

I(F,∇F, . . .) = 0

Electromagnetic fields with vanishing scalar invariants

1 what about VSI Maxwell fields?

I(F,∇F, . . .) = 0

2 also solutions of higher-derivative NLE?

L = L(F,∇F, . . .)

old idea [Bopp’40, Podolsky’42]

effective theories motivated by string theorye.g. [Andreev-Tseytlin’88, Thorlacius’98, Chemissany-Kallosh-Ortin’12, . . . ]

Electromagnetic fields with vanishing scalar invariants

1 what about VSI Maxwell fields?

I(F,∇F, . . .) = 0

2 also solutions of higher-derivative NLE?

L = L(F,∇F, . . .)

old idea [Bopp’40, Podolsky’42]

effective theories motivated by string theorye.g. [Andreev-Tseytlin’88, Thorlacius’98, Chemissany-Kallosh-Ortin’12, . . . ]

We studied this in arbitrary dimension n for a p-form F .

Electromagnetic fields with vanishing scalar invariants

Main result: VSI p-forms

Main result: VSI p-forms

Theorem ([M.O.-Pravda’15])

The following two conditions are equivalent:

1 a p-form field F is VSI in a spacetime gab

2 (a) F = ℓ ∧ ω, with ℓaℓa = 0 = ℓa1ωa1...ap−1

(i.e., VSI0)(b) £ℓF = 0(c) gab is “degenerate Kundt”.

Electromagnetic fields with vanishing scalar invariants

Main result: VSI p-forms

Main result: VSI p-forms

Theorem ([M.O.-Pravda’15])

The following two conditions are equivalent:

1 a p-form field F is VSI in a spacetime gab

2 (a) F = ℓ ∧ ω, with ℓaℓa = 0 = ℓa1ωa1...ap−1

(i.e., VSI0)(b) £ℓF = 0(c) gab is “degenerate Kundt”.

Maxwell equations not employedhowever, dF = 0 with (2a) ⇒ (2b)

Electromagnetic fields with vanishing scalar invariants

Main result: VSI p-forms

Main result: VSI p-forms

Theorem ([M.O.-Pravda’15])

The following two conditions are equivalent:

1 a p-form field F is VSI in a spacetime gab

2 (a) F = ℓ ∧ ω, with ℓaℓa = 0 = ℓa1ωa1...ap−1

(i.e., VSI0)(b) £ℓF = 0(c) gab is “degenerate Kundt”.

Maxwell equations not employedhowever, dF = 0 with (2a) ⇒ (2b)(2c) ⇔ ℓ is geodesic, shearfree, twistfree, expansionfree andall ∇kRiem are “multiply aligned” with ℓ

(includes all Einstein-Kundt, Minkowski, (A)dS, . . . )[Coley-Hervik-Pelavas’09, Coley-Hervik-Papadopoulos-Pelavas’09]

Electromagnetic fields with vanishing scalar invariants

Main result: VSI p-forms

Main result: VSI p-forms

Theorem ([M.O.-Pravda’15])

The following two conditions are equivalent:

1 a p-form field F is VSI in a spacetime gab

2 (a) F = ℓ ∧ ω, with ℓaℓa = 0 = ℓa1ωa1...ap−1

(i.e., VSI0)(b) £ℓF = 0(c) gab is “degenerate Kundt”.

Maxwell equations not employedhowever, dF = 0 with (2a) ⇒ (2b)(2c) ⇔ ℓ is geodesic, shearfree, twistfree, expansionfree andall ∇kRiem are “multiply aligned” with ℓ

(includes all Einstein-Kundt, Minkowski, (A)dS, . . . )[Coley-Hervik-Pelavas’09, Coley-Hervik-Papadopoulos-Pelavas’09]

proof based on “algebraic VSI theorem” [Hervik’11] andboost-weight classification [Milson-Coley-Pravda-Pravdova’05]

Electromagnetic fields with vanishing scalar invariants

Main result: VSI p-forms

Adapted coordinates (ℓ = ∂r, ℓadxa = du)

ds2 = 2du[dr +H(u, r, x)du+Wα(u, r, x)dxα] + gαβ(u, x)dx

αdxβ

F = 1(p−1)!fα1...αp−1

(u, x)du ∧ dxα1 ∧ . . . ∧ dxαp−1

Electromagnetic fields with vanishing scalar invariants

Main result: VSI p-forms

Adapted coordinates (ℓ = ∂r, ℓadxa = du)

ds2 = 2du[dr +H(u, r, x)du+Wα(u, r, x)dxα] + gαβ(u, x)dx

αdxβ

F = 1(p−1)!fα1...αp−1

(u, x)du ∧ dxα1 ∧ . . . ∧ dxαp−1

with Wα(u, r, x) = rW (1)α (u, x) +W (0)

α (u, x),

H(u, r, x) = r2H(2)(u, x) + rH(1)(u, x) +H(0)(u, x).

Electromagnetic fields with vanishing scalar invariants

Main result: VSI p-forms

Adapted coordinates (ℓ = ∂r, ℓadxa = du)

ds2 = 2du[dr +H(u, r, x)du+Wα(u, r, x)dxα] + gαβ(u, x)dx

αdxβ

F = 1(p−1)!fα1...αp−1

(u, x)du ∧ dxα1 ∧ . . . ∧ dxαp−1

with Wα(u, r, x) = rW (1)α (u, x) +W (0)

α (u, x),

H(u, r, x) = r2H(2)(u, x) + rH(1)(u, x) +H(0)(u, x).

Maxwell equations: f harmonic in gαβ

Electromagnetic fields with vanishing scalar invariants

Main result: VSI p-forms

Adapted coordinates (ℓ = ∂r, ℓadxa = du)

ds2 = 2du[dr +H(u, r, x)du+Wα(u, r, x)dxα] + gαβ(u, x)dx

αdxβ

F = 1(p−1)!fα1...αp−1

(u, x)du ∧ dxα1 ∧ . . . ∧ dxαp−1

with Wα(u, r, x) = rW (1)α (u, x) +W (0)

α (u, x),

H(u, r, x) = r2H(2)(u, x) + rH(1)(u, x) +H(0)(u, x).

Maxwell equations: f harmonic in gαβ

Chern-Simons term F ∧ F ∧ . . . vanishes identically(except when linear n = 2p− 1)

Electromagnetic fields with vanishing scalar invariants

Main result: VSI p-forms

Adapted coordinates (ℓ = ∂r, ℓadxa = du)

ds2 = 2du[dr +H(u, r, x)du+Wα(u, r, x)dxα] + gαβ(u, x)dx

αdxβ

F = 1(p−1)!fα1...αp−1

(u, x)du ∧ dxα1 ∧ . . . ∧ dxαp−1

with Wα(u, r, x) = rW (1)α (u, x) +W (0)

α (u, x),

H(u, r, x) = r2H(2)(u, x) + rH(1)(u, x) +H(0)(u, x).

Maxwell equations: f harmonic in gαβ

Chern-Simons term F ∧ F ∧ . . . vanishes identically(except when linear n = 2p− 1)

constraints on W(1)α , W

(0)α , H(2), H(1), H(0), gαβ if

backreaction included [Podolsky-Zofka’09]

Electromagnetic fields with vanishing scalar invariants

Main result: VSI p-forms

Adapted coordinates (ℓ = ∂r, ℓadxa = du)

ds2 = 2du[dr +H(u, r, x)du+Wα(u, r, x)dxα] + gαβ(u, x)dx

αdxβ

F = 1(p−1)!fα1...αp−1

(u, x)du ∧ dxα1 ∧ . . . ∧ dxαp−1

with Wα(u, r, x) = rW (1)α (u, x) +W (0)

α (u, x),

H(u, r, x) = r2H(2)(u, x) + rH(1)(u, x) +H(0)(u, x).

Maxwell equations: f harmonic in gαβ

Chern-Simons term F ∧ F ∧ . . . vanishes identically(except when linear n = 2p− 1)

constraints on W(1)α , W

(0)α , H(2), H(1), H(0), gαβ if

backreaction included [Podolsky-Zofka’09]

waves in Minkowski, (A)dS, Nariai, gyratons, . . .

Electromagnetic fields with vanishing scalar invariants

Main result: VSI p-forms

Examples of VSI tensors:

2[M.O.-Pravda’15]3[Pravda-Pravdova-Coley-Milson’02, Coley-Milson-Pravda-Pravdova’04,

Pelavas-Coley-Milson-Pravda-Pravdova’05]

Electromagnetic fields with vanishing scalar invariants

Main result: VSI p-forms

Examples of VSI tensors:

vector ℓ p-form F 2 Riem 3

VSI0 null N III (N,O)

VSI1 Kundt N, £ℓF = 0, Kundt N, κ = 0, σΨ4=ρΦ22

VSI2 Kundt, Riem II N, £ℓF = 0, Kundt, Riem II III, Kundt

VSI3 degKundt N, £ℓF = 0, degKundt “

... “ “ “

VSI “ “ “

2[M.O.-Pravda’15]3[Pravda-Pravdova-Coley-Milson’02, Coley-Milson-Pravda-Pravdova’04,

Pelavas-Coley-Milson-Pravda-Pravdova’05]

Electromagnetic fields with vanishing scalar invariants

Universal solutions

Universal solutions

Can some VSI F solve any theory L = L(F,∇F, . . .)?

Electromagnetic fields with vanishing scalar invariants

Universal solutions

Universal solutions

Can some VSI F solve any theory L = L(F,∇F, . . .)?

arbitrary gravity theories L = L(Riem,∇Riem, . . .): solved by“universal” (Einstein) metrics [Coley-Gibbons-Hervik-Pope’08,

Hervik-Pravda-Pravdova’14, Hervik-Malek-Pravda-Pravdova’15]

Electromagnetic fields with vanishing scalar invariants

Universal solutions

Universal solutions

Can some VSI F solve any theory L = L(F,∇F, . . .)?

arbitrary gravity theories L = L(Riem,∇Riem, . . .): solved by“universal” (Einstein) metrics [Coley-Gibbons-Hervik-Pope’08,

Hervik-Pravda-Pravdova’14, Hervik-Malek-Pravda-Pravdova’15]

Born-Infeld NLE: solved by any null Maxwell field

F[ab,c] = 0,

(

F ab −���G∗F ab

√

1 +��F −��G2

)

;b

= 0

(F ≡ 12FabF

ab, G ≡ 14Fab

∗F ab) [Schrodinger’35,’43]

Electromagnetic fields with vanishing scalar invariants

Universal solutions

Universal solutions

Can some VSI F solve any theory L = L(F,∇F, . . .)?

arbitrary gravity theories L = L(Riem,∇Riem, . . .): solved by“universal” (Einstein) metrics [Coley-Gibbons-Hervik-Pope’08,

Hervik-Pravda-Pravdova’14, Hervik-Malek-Pravda-Pravdova’15]

Born-Infeld NLE: solved by any null Maxwell field

F[ab,c] = 0,

(

F ab −���G∗F ab

√

1 +��F −��G2

)

;b

= 0

(F ≡ 12FabF

ab, G ≡ 14Fab

∗F ab) [Schrodinger’35,’43]

arbitrary L = L(F,∇F, . . .): any VSI F in type III Kundt

Electromagnetic fields with vanishing scalar invariants

Universal solutions

Universal solutions

Can some VSI F solve any theory L = L(F,∇F, . . .)?

arbitrary gravity theories L = L(Riem,∇Riem, . . .): solved by“universal” (Einstein) metrics [Coley-Gibbons-Hervik-Pope’08,

Hervik-Pravda-Pravdova’14, Hervik-Malek-Pravda-Pravdova’15]

Born-Infeld NLE: solved by any null Maxwell field

F[ab,c] = 0,

(

F ab −���G∗F ab

√

1 +��F −��G2

)

;b

= 0

(F ≡ 12FabF

ab, G ≡ 14Fab

∗F ab) [Schrodinger’35,’43]

arbitrary L = L(F,∇F, . . .): any VSI F in type III Kundt

coupling to gravity also possible: further restrictionscf. also [Guven’87, Horowitz-Steif’90, Coley’02]

NLE in [Kichenassamy’59, Kremer-Kichenassamy’60, Peres’60]

Electromagnetic fields with vanishing scalar invariants

Universal solutions

An Einstein-Maxwell universal example (n = 4, p = 2):

ds2 = 2du[

dr + 12

(

xr − xex − 2κ0exc2(u)

)

du]

+ ex(dx2 + e2udy2)

F = ex/2c(u)du ∧

(

− cosyeu

2dx+ eu sin

yeu

2dy

)

ℓ = ∂r is Kundt and recurrent (ℓa;b = fℓaℓb)

F is VSI

Petrov type III

obtained by “charging” a vacuum metric of Petrov [Petrov’62]