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Electromagnetic fields with vanishing scalar invariants 1 Marcello Ortaggio Institute of Mathematics Academy of Sciences of the Czech Republic Trento – June 6th, 2016 1 Joint work with V. Pravda, Class. Quantum Grav. 33 (2016) 115010

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Page 1: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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

Page 2: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

Electromagnetic fields with vanishing scalar invariants

Contents

1 Null electromagnetic fields

2 Null gravitational fields

3 Main result: VSI p-forms

4 Universal solutions

Page 3: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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)

Page 4: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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

Page 5: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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

Page 6: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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 + . . .

Page 7: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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”

Page 8: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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]

Page 9: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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, . . .

Page 10: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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)

Page 11: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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]

Page 12: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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]

Page 13: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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]

Page 14: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

Electromagnetic fields with vanishing scalar invariants

1 what about VSI Maxwell fields?

I(F,∇F, . . .) = 0

Page 15: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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, . . . ]

Page 16: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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 .

Page 17: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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”.

Page 18: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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)

Page 19: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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]

Page 20: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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]

Page 21: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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

Page 22: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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).

Page 23: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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

Page 24: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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)

Page 25: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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]

Page 26: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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, . . .

Page 27: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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]

Page 28: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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]

Page 29: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

Electromagnetic fields with vanishing scalar invariants

Universal solutions

Universal solutions

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

Page 30: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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]

Page 31: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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]

Page 32: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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

Page 33: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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]

Page 34: Electromagnetic fields with vanishing scalar invariants ... · Electromagnetic fields with vanishing scalar invariants1 Marcello Ortaggio Institute of Mathematics Academy of Sciences

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]