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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
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Electromagnetic fields with vanishing scalar invariants
Contents
1 Null electromagnetic fields
2 Null gravitational fields
3 Main result: VSI p-forms
4 Universal solutions
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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)
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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
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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
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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 + . . .
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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”
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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]
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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, . . .
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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)
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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]
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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]
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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]
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Electromagnetic fields with vanishing scalar invariants
1 what about VSI Maxwell fields?
I(F,∇F, . . .) = 0
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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, . . . ]
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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 .
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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”.
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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)
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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]
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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]
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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
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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).
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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
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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)
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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]
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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, . . .
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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]
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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]
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Electromagnetic fields with vanishing scalar invariants
Universal solutions
Universal solutions
Can some VSI F solve any theory L = L(F,∇F, . . .)?
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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]
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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]
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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
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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]
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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]