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 Omid Makhmali A Finslerian notion of causal structure 2 / 1

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: assuming radial transversality and non-deg of the 2nd fund form of Omid Makhmali A Finslerian notion of causal structure 3 / 1 Pseudo-Riemannian metric on M n + 1 is uniquely determined by TM ⊃ Σ 2 n + 1 = { v ∈ TM | g ( v , v ) = 1 } Roughly speaking, if Σ x not quadratic one has a (local) Finsler metric TM ⊃ Σ 2 n + 1 = { v ∈ TM | F ( v ) = 1 } Σ x ⊂ T x M , ∀ x ∈ M .

Defjnitions locally A Finslerian notion of causal structure Omid Makhmali U U U M A more general notion of Finsler structures due to Bryant: U M 4 / 1 with non-deg 2nd fund form everywhere. transverse immersion satisfying Defjnition ( Bryant ) A generalized pseudo-Finsler structure on M n + 1 is denoted by ( M , Σ) together with an immersion ι : Σ → TM where Σ is a connected, smooth manifold of dimension 2 n + 1 and ι is a radially The map π ◦ ι : Σ → M is a submersion with connected fjbers. x := ( π ◦ ι ) − 1 ( x ) are In the fjbration π ◦ ι : Σ → M , the fjbers Σ n mapped to immersed connected hypersurfaces via ι x : Σ x → T x M ϕ ∗ ✲ ˜ Σ | U Σ | ˜ ∼ ( ˜ M , ˜ ( M , Σ) Σ) = ˜ x = ϕ ( x ) x ∈ ˜ at x ∈ M , ˜ ϕ ∗ (Σ y ) = ˜ µ µ ˜ Σ φ ( y ) if ∃ diffeo ϕ : U → ˜ ∀ y ∈ U ❄ ❄ ϕ x ∈ ˜ U ⊂ ˜ where x ∈ U ⊂ M , ˜ ✲ ˜

Defjnitions: remarks Not every generalized Finsler structure is realizable as a Finsler metric. embedding. All classical constructions for Finsler structures, e.g., canonical connections, structure bundle and curvature will go through. 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. Omid Makhmali A Finslerian notion of causal structure 5 / 1 Note that Σ can be open and immersed as an open submanifold. There is no requirement for Σ x to be compact or ι be an For the local aspects of Finsler geometry ι can be assumed to be an embedding in a suffjciently small neighborhood of Σ .

Defjnitions The conformal class of a Pseudo-Riem metric (pseudo-conformal Assigning a null cone at each tangent space is the main ingredient for described by non-degenerate one obtains a causal structure. Omid Makhmali A Finslerian notion of causal structure 6 / 1 structures) on M n + 1 is uniquely determined by its null cones P TM ⊃ C 2 n = { v ∈ TM | g ( v , v ) = 0 } understanding causal properties of M . Roughly speaking, if C x not quadratic, C is a fjeld of cones, locally P TM ⊃ C 2 n = { v ∈ TM | G ( v ) = 0 } . If the projective 2nd fund form of C x ⊂ P T x M , ∀ x ∈ M is

Defjnitions locally A Finslerian notion of causal structure Omid Makhmali U U U M More precise defjnition: U M 7 / 1 mapped to immersed connected tangentially non-degenerate non-deg projective 2nd fund form everywhere. x Defjnition A causal structure on M n + 1 is denoted by ( M , C ) together with an immersion ι : C → P TM where C is a connected, smooth manifold of dimension 2 n 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 C n − 1 := ( π ◦ ι ) − 1 ( x ) are projective hypersurfaces via ι x : C x → P T x M , i.e., they have ϕ ∗ ✲ ˜ C| U C| ˜ ∼ ( ˜ M , ˜ ( M , C ) = C ) ˜ x = ϕ ( x ) x ∈ ˜ at x ∈ M , ˜ ϕ ∗ ( C y ) = ˜ µ µ ˜ C φ ( y ) if ∃ diffeo ϕ : U → ˜ ∀ y ∈ U ❄ ϕ ❄ x ∈ ˜ U ⊂ ˜ ✲ ˜ where x ∈ U ⊂ M , ˜

Remarks Locally the projective 2nd fundamental form of a hypersuface in structure. We do not assume that its fjbers are convex or closed in Omid Makhmali A Finslerian notion of causal structure 8 / 1 P n is proportional to the 2nd fundamental form of the affjne hypersuface obtained by taking an affjne chart for P n . C 2 n is called the (projective) null cone bundle of the causal P T x M . Note that C can be open and be immersed as an open hypersurface in P TM . For the local aspects of causal geometry ι can be assumed to be an embedding in a suffjciently small neighborhood of C .

Defjnitions and examples Locally a causal structure can be expressed as A Finslerian notion of causal structure Omid Makhmali Null cones are projectively equivalent to Cayley’s cubic surface. structure. 9 / 1 C ⊃ U = { ( x , [ y ]) ∈ P TM | L ( x ; y ) = 0 } . L : TM → R or C satisfjes   L ( x ; λ y ) = λ r L ( x ; y ) for some r  [ ∂ 2 L ]  has max rank over L = 0 .  ∂ y i ∂ y j L ( x ; y ) , S ( x ; y ) L ( x ; y ) − → same causal str ( S nowhere vanishing.) Example : L ( x ; y ) = ( y 1 ) 2 + ( y 2 ) 2 + ( y 3 ) 2 − ( y 4 ) 2 : fmat 4D causal 3 ( y 2 ) 3 + y 0 y 3 y 3 − y 1 y 2 y 3 : Example : L ( x ; y ) = 1

Defjnitions and examples loc i.e., locally it can be expressed as Being locally V -isotrivial is the causal analogue of being locally Minkowskian in Finsler geometry. Omid Makhmali A Finslerian notion of causal structure 10 / 1 Defjnition : ( M , C ) is called locally V-isotrivial if C x ∼ = V ⊂ P n , ∀ x ∈ M ∼ ( M , C ) is called locally V-isotrivially fmat if ( M , C ) = ( U , U × V ) where V ⊂ P n is a projective hypersurface. { ( x ; [ y ]) ∈ P TM | L ( y ) = 0 } with L ( y ) not depending on x .

Motivation 1: Geometrization of DEs locally A Finslerian notion of causal structure Omid Makhmali satisfying This program, pioneered by Cartan and Chern, was aimed to A contact equivalence class is an equivalence relation defjned by contact 11 / 1 Contact equivalence class of characterize geometric structures arising from certain classes of difgerential equations. { } ← → Certain foliations of J 2 ( R , R ) y ′′′ = f ( x , y , y ′ , y ′′ ) J 2 ( R , R ) : 2nd jet space of functions with coordinates ( x , y , p , q ) , where p = y ′ , q = y ′′ . transformation : difgeomorphisms of J 1 ( R , R ) y ′ �→ ¯ y ′ = ψ ( x , y , y ′ ) , x = χ ( x , y , y ′ ) , y = ϕ ( x , y , y ′ ) , x �→ ¯ y �→ ¯ ψ = D ϕ D = ∂ x + y ′ ∂ y + y ′′ ∂ y ′ + y ′′′ ∂ y ′′ D χ,

Motivation 1: Geometrization of difgerential equations causal structures A Finslerian notion of causal structure Omid Makhmali formulation of General Relativity will be described later. An extension of this result, with relation to Newman’s null surface M 3 : Locally defjned space of solutions of a 3rd order ODE. structure. Theorem (Holland-Sparling following the works of Cartan, Chern, 12 / 1 Sato-Yashikawa, Newman-Kozameh, Nurowski-Godlinski,...) contact equivalent classes of 3rd order ODEs { } { } 1 − 1 ← → ( M 3 , C 4 ) 3-dimensional causal structures ( M 3 , C 4 ) = K 3 ✛ ρ = C 4 µ ✲ M 3 ∼ J 1 ( R , R ) ∼ J 2 ( R , R ) ∼ = Space of solutions K 3 : Locally defjned space of “null geodesics” which has a contact

Motivation 2: Hwang-Mok program and VMRTs Their program is to give a difgerential geometric characterization of A Finslerian notion of causal structure Omid Makhmali This gives a generalization of results on holomorphic conformal str. VMRTs are locally isotrivially fmat. Theorem (Hwang, 2013) Causal structures arising from smooth of codimension one smooth 13 / 1 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 − K M . At a general point x ∈ M , C x = VMRT at x ={tangent directions to such curves at x }.       Assume for generic x , C x : M has causal str.     of degree ≥ 2

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. Omid Makhmali A Finslerian notion of causal structure 14 / 1

The equivalence problem: Riemannian geometry Cartan invented a powerful machinery to solve the equivalence problem of geometric structures by constructing a principal bundle and obtaining invariants. form). Omid Makhmali A Finslerian notion of causal structure 15 / 1 Given a Riemannian metric g ∈ Γ( S 2 ( T ∗ M )) , defjne ( ω 0 , · · · , ω n ) s.t. g = ( ω 0 ) 2 + · · · + ( ω n ) 2 . The choice of ω i ’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 ω i ’s to F , defjning the so-called tautological R n + 1 -valued 1-form on F (aka the soldering