A Finslerian notion of causal structure Omid Makhmali IMPAN, Warsaw - - PowerPoint PPT Presentation

A Finslerian notion of causal structure

Omid Makhmali

IMPAN, Warsaw

February 21, 2019

IEMath, Granada

Omid Makhmali A Finslerian notion of causal structure 1 / 1

An outline

(local) Causal structures: defjnition, motivation, and history The equivalence method: Riemannian, Finsler, conformal, causal Structure equations and local invariants Causal vs. Finsler Null Jacobi fjelds and tidal force Half-fmat indefjnite causal structures in dimension 4 Discussing Petrov type and almost Einstein condition for causal str

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Defjnitions

The relation of causal str to conformal pseudo-Riem str is intended to be an analogue of what Finsler str are to Riem str: Pseudo-Riemannian metric on Mn+1 is uniquely determined by TM ⊃ Σ2n+1 = {v ∈ TM | g(v, v) = 1} Roughly speaking, if Σx not quadratic one has a (local) Finsler metric TM ⊃ Σ2n+1 = {v ∈ TM | F(v) = 1} assuming radial transversality and non-deg of the 2nd fund form of Σx ⊂ TxM, ∀x ∈ M.

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Defjnitions

A more general notion of Finsler structures due to Bryant:

Defjnition (Bryant) A generalized pseudo-Finsler structure on Mn+1

is denoted by (M, Σ) together with an immersion ι : Σ → TM where Σ is a connected, smooth manifold of dimension 2n + 1 and ι is a radially transverse immersion satisfying The map π ◦ ι : Σ → M is a submersion with connected fjbers. In the fjbration π ◦ ι : Σ → M, the fjbers Σn

x := (π ◦ ι)−1(x) are

mapped to immersed connected hypersurfaces via ιx : Σx → TxM with non-deg 2nd fund form everywhere. (M, Σ)

locally

∼ = ( ˜ M, ˜ Σ) at x ∈ M,˜ x ∈ ˜ M if ∃ diffeo ϕ : U → ˜ U where x ∈ U ⊂ M,˜ x ∈ ˜ U ⊂ ˜ M Σ|U ϕ∗ ✲ ˜ Σ|˜

U

U µ

❄

ϕ

✲ ˜

U ˜ µ

❄

˜ x = ϕ(x) ϕ∗(Σy) = ˜ Σφ(y) ∀y ∈ U

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Defjnitions: remarks

Not every generalized Finsler structure is realizable as a Finsler metric. Note that Σ can be open and immersed as an open submanifold. There is no requirement for Σx to be compact or ι be an embedding. All classical constructions for Finsler structures, e.g., canonical connections, structure bundle and curvature will go through. For the local aspects of Finsler geometry ι can be assumed to be an embedding in a suffjciently small neighborhood of Σ. When imposing certain DEs on Finsler structures it is more natural to work in the generalized setting an then restrict to the classical setting afterwards.

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Defjnitions

The conformal class of a Pseudo-Riem metric (pseudo-conformal structures) on Mn+1 is uniquely determined by its null cones PTM ⊃ C2n = {v ∈ TM | g(v, v) = 0} Assigning a null cone at each tangent space is the main ingredient for understanding causal properties of M. Roughly speaking, if Cx not quadratic, C is a fjeld of cones, locally described by PTM ⊃ C2n = {v ∈ TM | G(v) = 0}. If the projective 2nd fund form of Cx ⊂ PTxM, ∀x ∈ M is non-degenerate one obtains a causal structure.

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Defjnitions

More precise defjnition:

Defjnition A causal structure on Mn+1 is denoted by (M, C) together

with an immersion ι : C → PTM where C is a connected, smooth manifold of dimension 2n and ι is an immersions satisfying The map π ◦ ι : C → M is a submersion with connected fjbers. In the fjbration π ◦ ι : C → M, the fjbers Cn−1

x

:= (π ◦ ι)−1(x) are mapped to immersed connected tangentially non-degenerate projective hypersurfaces via ιx : Cx → PTxM, i.e., they have non-deg projective 2nd fund form everywhere. (M, C)

locally

∼ = ( ˜ M, ˜ C) at x ∈ M,˜ x ∈ ˜ M if ∃ diffeo ϕ : U → ˜ U where x ∈ U ⊂ M,˜ x ∈ ˜ U ⊂ ˜ M C|U ϕ∗ ✲ ˜ C|˜

U

U µ

❄

ϕ

✲ ˜

U ˜ µ

❄

˜ x = ϕ(x) ϕ∗(Cy) = ˜ Cφ(y) ∀y ∈ U

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Remarks

Locally the projective 2nd fundamental form of a hypersuface in Pn is proportional to the 2nd fundamental form of the affjne hypersuface obtained by taking an affjne chart for Pn. C2n is called the(projective) null cone bundle of the causal

• structure. We do not assume that its fjbers are convex or closed in

PTxM. Note that C can be open and be immersed as an open hypersurface in PTM. For the local aspects of causal geometry ι can be assumed to be an embedding in a suffjciently small neighborhood of C.

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Defjnitions and examples

Locally a causal structure can be expressed as C ⊃ U = {(x, [y]) ∈ PTM|L(x; y) = 0}. L : TM → R or C satisfjes      L(x; λy) = λrL(x; y) for some r [ ∂2L ∂yi∂yj ] has max rank over L = 0. L(x; y), S(x; y)L(x; y) − → same causal str (S nowhere vanishing.)

Example : L(x; y) = (y1)2 + (y2)2 + (y3)2 − (y4)2 : fmat 4D causal

structure.

Example : L(x; y) = 1

3(y2)3 + y0y3y3 − y1y2y3:

Null cones are projectively equivalent to Cayley’s cubic surface.

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Defjnitions and examples

Defjnition : (M, C) is called locally V-isotrivial if Cx ∼

= V ⊂ Pn, ∀x ∈ M (M, C) is called locally V-isotrivially fmat if (M, C)

loc

∼ = (U, U × V) where V ⊂ Pn is a projective hypersurface. i.e., locally it can be expressed as {(x; [y]) ∈ PTM | L(y) = 0} with L(y) not depending on x. Being locally V-isotrivial is the causal analogue of being locally Minkowskian in Finsler geometry.

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Motivation 1: Geometrization of DEs

This program, pioneered by Cartan and Chern, was aimed to characterize geometric structures arising from certain classes of difgerential equations. { Contact equivalence class of y′′′ = f(x, y, y′, y′′) }

locally

← → Certain foliations of J2(R, R) J2(R, R) : 2nd jet space of functions with coordinates (x, y, p, q), where p = y′, q = y′′. A contact equivalence class is an equivalence relation defjned by contact transformation: difgeomorphisms of J1(R, R) x → ¯ x = χ(x, y, y′), y → ¯ y = ϕ(x, y, y′), y′ → ¯ y′ = ψ(x, y, y′), satisfying ψ = Dϕ Dχ, D = ∂x + y′∂y + y′′∂y′ + y′′′∂y′′

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Motivation 1: Geometrization of difgerential equations

Theorem (Holland-Sparling following the works of Cartan, Chern,

Sato-Yashikawa, Newman-Kozameh, Nurowski-Godlinski,...) { contact equivalent classes

• f 3rd order ODEs

}

1−1

← → { causal structures (M3, C4) } 3-dimensional causal structures (M3, C4) J1(R, R) ∼ = K3 ✛ ρ J2(R, R) ∼ = C4 µ

✲ M3 ∼

= Space of solutions K3 : Locally defjned space of “null geodesics” which has a contact structure. M3: Locally defjned space of solutions of a 3rd order ODE. An extension of this result, with relation to Newman’s null surface formulation of General Relativity will be described later.

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Motivation 2: Hwang-Mok program and VMRTs

Their program is to give a difgerential geometric characterization of uniruled projective manifolds using their variety of minimal rational tangents (VMRT). On a uniruled manifold M consider the scheme of rational curves with minimal degree wrt to −KM. At a general point x ∈ M, Cx =VMRT at x ={tangent directions to such curves at x}. Assume for generic x, Cx      smooth

• f codimension one
• f degree ≥ 2

     : M has causal str.

Theorem (Hwang, 2013) Causal structures arising from smooth

VMRTs are locally isotrivially fmat. This gives a generalization of results on holomorphic conformal str.

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History

Apart from the works presented so far, the notion of causal structure closely related to what I defjned can be found in Irvine Segal’s book Mathematical cosmology and extragalactic astronomy, 1976. In Hwang-Mok program a cone structure is defjned as a fjeld of cones which can have any codimension or degeneracy condition. I borrowed the term causal from the work of Holland-Sparling on scalar 3rd order ODEs. They defjned causal structure in 3D via an axiomatic approach.

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The equivalence problem: Riemannian geometry

Cartan invented a powerful machinery to solve the equivalence problem

• f geometric structures by constructing a principal bundle and
• btaining invariants.

Given a Riemannian metric g ∈ Γ(S2(T∗M)), defjne (ω0, · · · , ωn) s.t. g = (ω0)2 + · · · + (ωn)2. The choice of ωi’s are ambiguous up to an O(n + 1)-action. Now, consider the principal O(n + 1)-bundle π : F → M. There is a taugological lift of the 1-forms ωi’s to F, defjning the so-called tautological Rn+1-valued 1-form on F (aka the soldering form).

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The equivalence problem: Riemannian geometry

The structure group O(n + 1) acts transitively on the fjbers of F, i.e., the right action Rg(fx) = g−1fx where fx : TxM → Rn+1 is a coframe on M and f = (x ; fx) = (x ; ω0

x, · · · , ωn x ) ∈ F, with ωi ⊂ T∗M.

Defjne the tautological Rn+1-valued 1-form ω on pr : F → M by setting at f = (x, fx) ∈ F ωf(v) := fx(pr∗(v)) ∈ Rn+1, v ∈ TfF The 1-forms pr∗(ωi) furnish a basis of semi-basic forms on F. In practice, F|U ∼ = U × O(n + 1). Take a coframe (ω0, · · · , ωn)T. ω(x ; g) = (ω0, · · · , ωn)(x ; g) := g−1(pr∗ω0, · · · , pr∗ωn)T = g−1ωe. dω = dg−1 ∧ ω + g−1dω = −g−1dg ∧ g−1ω + g−1dω = −α ∧ ω + T(ω ∧ ω)

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The equivalence problem: Riemannian geometry

The 1-forms α are the Maurer-Cartan forms of O(n + 1). dropping the underline, we obtain dωi = −ωi

j ∧ ωj + 1

2Ti

jkωj ∧ ωk,

where ωi

j = −ωj i, Ti jk = −Ti kj, (T is called torsion.)

The 1-forms ωi

j are ambiguous up to a linear combination of ωi’s.

Pin down ωi

j (Levi-Civita conn) so that Ti jk = 0, i.e.,

ωi

j → ωi j + 1

2(Ti

jk − Tj ik)ωk + δklTk ijωl.

After one more difgerentiation it follows that dωi = −ωi

j ∧ ωj

dωi

j = −ωi k ∧ ωk j + 1

2Ri

jklωk ∧ ωl

where Ri

jkl = −Rj ikl, Ri jkl = −Ri jlk, Ri jkl + Ri klj + Ri ljk = 0.

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The equivalence problem: Conformal geometry

Defjne (ω0, · · · , ωn) so that g = (ω0)2 + · · · + (ωn)2. The choice of ωi’s are ambiguous up to a CO(n + 1)-action. Consider the principal CO(n + 1)-bundle π : F → M. Using the same absorption, it follows dωi = −(ωi

j + λδi j) ∧ ωj

However, ωi

j, λ still have ambiguity, i.e., one needs to prolong.

After prolongation, the structure equations is obtained dωi = −(ωi

j + λδi j) ∧ ωj

dωi

j = −ωi k ∧ ωk j + πj ∧ ωi − πi ∧ ωj + 1

2Wi

jklωk ∧ ωl,

dλ = πi ∧ ωi, dπi = (ωi

j + λδi j) ∧ πj + 1

2Wijkωi ∧ ωj

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Curvature decomposition

The quantities Wi

jkl, Wijk are called the Weyl tensor and the

Cotton-York tensor of the conformal structure satisfying Wi

jkl = −Wj ikl, Wi jkl = −Wi jlk, Wi jkl + Wi klj + Wi ljk = 0, Wi jil = 0

Wijk = −Wilk, Wijk + Wjki + Wkij = 0. The Cotton-York tensor satisfjes Wjkl =

1 n−3 ∂ ∂ωi Wi jkl.

The symmetries of Wi

jkl is like Ri jkl and is trace-free: Wi jil = 0.

In fact, the Riemann tensor as an O(n + 1)-module decomposes as Ri

jkl = Wi jkl − δi lujk + δj luik + δi kujl − δj kuil,

where uij is the Rho tensor uij = 1 n − 2 ( Rij − R 2(n − 1)δij )

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The equivalence problem: Finsler geometry

As originally done by Cartan and Chern, Hilbert form η0 is the essential notion in Finsler geometry: If the indicatrix bundle is given by F(v) = 1 then η0 = ∂xiFdyi. η0 can be defjned more abstractly: at u = (x ; v) ∈ Σ η0

u = Ann(TvΣx),

η0

u(v) = 1.

The non-deg of the indicatrices implies η0 is a contact 1-form: η0 ∧ (dη0)n ̸= 0. The geodesic fmow is the Reeb vector fjeld of η0. Moreover, dη0 = −ζi ∧ ηi, The 1-forms (η1, · · · , ηn) are semi-basic wrt π : Σ → M. (η0, · · · , ζn) give a coframe on Σ ambiguous up to certain matrix group.

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The equivalence problem: Finsler geometry

After a few steps, one can obtain a canonical coframe (ηi, ζa, ψa

b) on a

principal O(n)-bundle satisfying structure equations dη0 = −ζi ∧ ηi, dηi = δijζj ∧ η0 − ψij ∧ ηj − Iijkζj ∧ ηk, dζi = ψji ∧ ζj + R0i0jη0 ∧ ηj + 1

2R0ijkηj ∧ ηk + Jijkζj ∧ ηk

dψij = −ψik ∧ ψk

j + Rij0kη0 ∧ ηk + 1

2Rijklηk ∧ ηl + Pijklζk ∧ ηl The fmag curvature, i.e. the symm tensor R = R0i0jωj ◦ ωj, is a fund inv. The cubic form I = Iijkθi ◦ θj ◦ θk is the other fund inv and when restricted to an indicatrix it coincides with its centroaffjne cubic form. If I = 0, then J = 0 and the structure equations are the same as a Riemannian metric. The vanishing I and R implies that the metric is fmat.

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The equivalence problem: causal geometry

At (x; [y]) ∈ C, with µ : C2n → Mn+1, µ−1(x) = Cn−1

x

defjne Vn−1 := µ−1

∗ (0) ⊂ J n := µ−1 ∗ (ˆ

y) ⊂ P2n−1 := µ−1

∗ (

TyCx) ⊂ T(x;[y])C Let ω0 = Ann(P) := Ann(T[y]Cx) {ω0, · · · , ωn−1} = Ann(J ) {ω0, · · · , ωn} = Ann(V), with {ω0, · · · , ωn, θ1, · · · , θn−1} being a coframe on C. ω0 is called projective Hilbert form.

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The equivalence problem: causal geometry

If fjbers Cx are tangentially non-degenerate, then for suitable θa’s dω0 = −2ϕ0 ∧ ω0 − θa ∧ ωa Let 0 ≤ i, j, k ≤ n, 1 ≤ a, b, c ≤ n − 1. The 1-forms (ωi, θa) give a 1-adapted coframe on C. ω0 ∧ (dω0)n−1 ̸= 0 ⇒ rank(ω0) = 2n − 1. Hence, ω0 induces a quasi-contact structure on C2n. ∃ line bundle ℓ ⊂ TC : ω0(ℓ) = 0, ℓ ⌟ dω0 = 0. The curves tangent to ℓ are called the characteristic curves of (M, C). Locally,

∂ ∂ωn is a section of ℓ.

Their projection to M give an analogue of null geodesics.

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The equivalence problem: causal geometry

After several coframe adaptation and a prolongation, a canonical coframing on certain principal bundle is obtained. Fundamental invariants are: A totally symmetric trace-free cubic form F := Fabcθa ◦ θb ◦ θc which pulls back to the Fubini cubic form of the fjbers. A symmetric trace-free bilinear form W := Wnanbωa ◦ ωb. In conformal geometry, at (x; [y]) ∈ C Wanbn ∝ Waibjyiyj. The vanishing of the fundamental invariants implied that the causal structure is the fmat conformal structure.

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Causal vs. Finsler

First structure equation for Causal structures: dω0 = − 2ϕ0 ∧ ω0 − θa ∧ ωa, dωa = − γa ∧ ω0 − ϕab ∧ ωb − (ϕ0 + ϕn) ∧ ωa − εabθb ∧ ωn − Kab θb ∧ ω0 − Fabc θb ∧ ωc, dωn = − γa ∧ ωa − 2ϕn ∧ ωn − La θa ∧ ω0 dθa = − πa ∧ ω0 − πn ∧ ωa + ϕba ∧ θb − (ϕ0 − ϕn) ∧ θa + Wanbn ωb ∧ ωn + 1

2Wanbc ωb ∧ ωc − fabc θb ∧ ωc.

First structure equations for Finsler structures: dη0 = −ζi ∧ ηi, dηi = δijζj ∧ η0 − ψij ∧ ηj − Iijkζj ∧ ηk, dζi = ψji ∧ ζj + R0i0jη0 ∧ ηj + 1

2R0ijkηj ∧ ηk + Jijkζj ∧ ηk.

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Causal vs. Finsler

Finsler Causal Indicatix bdle Σ2n+1 → Mn+1 (Proj.) null cone bdle C2n → Mn+1

• Loc. expressed as F = 1
• Loc. expressed as L = 0

Hilbert form η0 = ∂F

∂yi dxi

Pojective Hilbert form ω0 = ∂L

∂yi dxi

η0 : contact form on Σ2n+1 ω0 : quasi-contact form on C2n dη0 = −ζ1 ∧ η1 − · · · − ζn ∧ ηn, dω0 = −θ1 ∧ ω1 − · · · − θn−1 ∧ ωn−1 η0 ∧ (dη0)n ̸= 0 −2ϕ0 ∧ ω0, ω0 ∧ (dω0)n−1 ̸= 0 Geodesics: integral curves of Null geodesics: integral curves of the Reeb vector fjeld the characteristic line fjeld η0(u) = 1, dη0(u, .) = 0 ω0(v) = 0, dω0(v, .) = 0 Σx ⊂ TxM is Legendrian Cx ⊂ PTxM are quasi-Legendrian Σn

x = Ker{ηi}

Cn−1

x

= Ker{ωi} g = (η0)2 + δijηiηj [g] = [2ω0ωn + εabωaωb] is well-def on Σ (osc. quadric) is well-def on C (osc. quadric)

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Causal vs. Finsler

Finsler Causal Cartan’s conn on Σ

• reg. norm. Cartan conn on C

Parabolic geometry of type (Bn−1, P12), (Dn, P12), n ≥ 4 (D3, P123), (B2, P12) Essential invariants Essential invariants (Harmonic) Iijk : centro-affjne invariant of Σx Fabc : Fubini cubic form of Cx Ri0j0 : Flag curvature Wanbn : Weyl shadow fmag curvature Iijk = 0 ⇒ Riem. geom. on M Fabc = 0 ⇒ Conformal pseudo-Riem. geom. on M Ri0j0 = 0 ⇒ β-int Segre Wanbn = 0 ⇒ β-int Lie contact str on K (space of geod) str on K (space of null geod)

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Some remarks

Locally isotrivially fmat ⇐ ⇒ Wanbn = 0,

∂ ∂ωi Fabc = 0.

One could have started with the dual cones C∗ ⊂ PT∗M, via a Legendre transformation in which case ω0 is pulled-back to the natural contant 1-form on PT∗M. The submaximal 4D causal structure that is not conformal is unique and isotrivially fmat. Its null cones are Cayley’s cubic surface and its Lie algebra of infjnitesimal symmetries is 8D.

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Some remarks: causal structure from a system of PDEs

Following K. Yamaguchi, one can show for n + 1 ≥ 4:

Theorem :

{ causal str.

• n (Mn+1, C2n)

}

1−1

← →            contact classes of certain systems of n(n−1)

2

− 1 2nd order PDEs

• f finite type

           J2(Rn−1, R) ∼ =LG(n − 1, TK) ✛

⊃ C2n

J1(Rn−1, R) ∼ = K2n−1

❄ ✛

ρ Mn+1 µ

✲

K2n−1 : Locally defjned space of “null geodesics” which has a contact structure. LG(n − 1, TK) : bundle of Lagrangian-Grassmannians of TK. Mn+1: Locally defjned space of solutions of the system of PDEs.

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Null Jacobi fjelds and tidal force

Characteristic curves for causal structures are defjned topologically and do not arise from a variational problem. The variational problem of char curves is given by the triple (I, ωn; ω0), where I = {ω0, · · · , ωn−1, θ1, · · · , θn−1} Given a char curve: γ : I = (a, b) → M, we have γ∗(I) = 0, γ∗(ωn) ̸= 0 and we consider the functional Φ : U(I) → R Φ([γ]) := ∫

I

γ∗ω0. where U(I) := {γ : I → R|γ∗I = 0}/parametrization.

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Null Jacobi fjelds and tidal force

Consider a variation Γ : (−ϵ, ϵ) → U(I). Parametrize the curves of Γ(s)

• Γ : (−ϵ, ϵ) → M where

Γ(s, t) = (Γ(s))(t) As a result, Γ∗(s0, t)I = 0 for any fjxed s0. Note that TγU(I) := { ∂ Γ ∂s (0, t) : Γ : (−ϵ, ϵ) → U(I) is a compact variation of γ } . The variational equations of a characteristic curve can be thought of as the fjrst order approximation of TγU(I). If η ∈ I then L ∂

∂s

• Γ∗(s, t)η ≡ 0 modulo {ds}.

Setting s = 0, the expression above reads as γ∗(ˆ J ⌟ dη + d(ˆ J ⌟ η)) = 0, where ˆ J(t) = Γ∗(0, t) ∂

∂s

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Null Jacobi fjelds and tidal force

The tangential component of the vector fjeld ˆ J(t) along γ(t) has no efgect on the variational equations. Hence

∂ ∂ωn -component of ˆ

J will be dropped. ˆ J along the curve γ(t) is ˆ J(t) = Va(t) ∂ ∂ωa + Va(t) ∂ ∂θa . The vector fjeld J = π∗ˆ J is called a null Jacobi fjeld along the null geodesics π(γ(t)) ⊂ M. The variational equations for ˆ J(t) can be written as ˆ J ⌟ dω0 + d(ˆ J ⌟ ω0) = 0, ˆ J ⌟ dωa + d(ˆ J ⌟ ωa) = 0, mod I. ˆ J ⌟ dθa + d(ˆ J ⌟ θa) = 0,

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Null Jacobi curves and tidal force

The fjrst equation has no new information. The last two equations give dVa + Vbϕab − Vaωn ≡ 0, dVa + Vbϕab + WanbnVbωn ≡ 0, mod I. Defjne the covariant derivative of ˆ J along characteristic curves as Dvˆ J ≡ v ⌟ ( dVa ∂ ∂ωa + Vbϕab ∂ ∂ωa ) , mod ∂ ∂θ1 , · · · , ∂ ∂θn−1 , where v is the characteristic vector fjeld satisfying ωn(v) = 1.

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Null Jacobi fjelds and tidal force

it follows that DvDvˆ J ≡ −WanbnVb ∂ ∂ωa , mod ∂ ∂θ1 , · · · , ∂ ∂θn−1 ≡ −

sf

Wˆ J, mod ∂ ∂θ1 , · · · , ∂ ∂θn−1 Note that Wanbn = ωa(W( ∂

∂ωb , v)v. Hence, defjning

J′ = π∗(Dvˆ J), the null Jacobi equation can be expressed as J′′ + W(J, v)v = 0, where J = π∗ˆ J and W(J, v)v denotes π∗ ( W(ˆ J, v)v ) . Following this line of argument, and defjning expansion, vorticity and shear in terms of null Jacobi tensor, one can obtain the Raychaudhuri equation.

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Analogue of self-duality in 4D

Penrose’s twistor theory: In conformal structure of signature (2,2), null cones are doubly ruled by 1-parameter family of α-planes and β-planes. β-surface: a surface whose tangent space is a β-plane everywhere. selfduality ⇐ ⇒ β−integrability. i.e., ∃ a 3-parameter family of surfaces (β-surfaces) such that at each point and through each β-plane at that point, there passes a unique surface tangent to that β-plane. self-duality (Wasd = 0) is usually defjned by the hodge star operator W = Wsd ⊕ Wasd. For a 4D causal structure of indefjnite signature [g] (null cones are indefjnite proj surf) the essential invariants are W+, W− and F+, F−. F+ = F− = 0 ⇒ W+ generates Wsd and W− generates Wasd. F−F+ = 0 ⇒ the null cones are ruled. F−, W− = 0 ⇒ ∃ a 3-parameter family of surfaces s.t. along each ruling-plane at every point there passes a unique surface.

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Double fjbrations

For 4D indefjnite self-dual causal structure: T3 ✛ F−, W− = 0 C6 F−, F+ = 0

✲ M4

path geom ✛ causal

✲ conformal

T3 is the space of β-surfaces. If F+ = 0, then (C, M) gives a self-dual conformal strucure and T is equipped with a torsion-free path geometry The torsion of the path geometry is given by F+ and its derivatives. If F−, W−, W+ = 0 then T has a projective str.

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4D causal

Theorem

indefjnite selfdual causal on M4 ⇐ ⇒ path geom. on T3

Remarks : A path geometry in 3D can be expressed locally in terms of

a pair of ODEs under point equivalence relation z′′

1 = F1(t, z1, z2, z′ 1, z′ 2),

z′′

2 = F2(t, z1, z2, z′ 1, z′ 2)

Theorem : The submaximal indefjnite 4D causal structure that does

not descend to a pseudo-conformal structure has 8-dimensional Lie algebra of infjnitesimal symmetries and is locally isotrivially fmat whose null cone is Cayley’s cubic scroll. It corresponds to the following pair of ODEs: z′′

1 = z2,

z′′

2 = 0.

In terms of affjne coordinates for PTM : y3 = y1y2 + 1 3(y1)3.

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Future directions

Find a generalization of the almost Einstein condition arising from BGG operators in parabolic geometry. Find an analogue of principal null directions (planes) for 4D causal structures. Is there an analogue of Goldberg-Sachs theorem in the causal setting? When is a 4D causal structure locally equivalent to the twisted product of two Finsler metrics?

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Thank you for your attention!

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