Vladimir Matveev Jena (Germany) Geodesic equivalence of Einstein pseudo-riemannian metrics and projective Lichnerowicz conjecture
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1 Vladimir Matveev Jena (Germany) Geodesic equivalence of Einstein pseudo-riemannian metrics and projective Lichnerowicz conjecture Kiosak M : arxiv: and arxiv: matveev/
2 How Isaac Newton found the gravitational law? Isaac Newton in Philosophiae Naturalis Principia Mathematica suggested the following strategy to study nature: Example: Gravitational Law: r one needs to understand the mathematical model describing the situation, and then find the parameters in the model using experiments or observations. From the Newton s third law and general sense it is clear that the gravitational force between two bodies is proportional to the masses of the bodies and depends on the distance between the bodies: F = F 1 = F 2 = m 1 m 2 f (r). How to find f? Newton tried different functions for f and solved the equations of motion. He proved that for the function f (r) = G r 2 the trajectories of the planets are ellipses which was observed by Kepler. Then, he concluded that the function f (r) is indeed G r, i.e., F = F 2 1 = F 2 = G m1m2 r. 2
3 Let us apply the same method to the general relativity The model is known since Einstein: the structure of the vacuum space-time is given by a solution of the Einstein equations R ij = R n g ij. In many cases, the only objects one can observe are geodesics considered as unparameterized curves (since there is no notion of absolute time). The clock of object has nothing to do with the clock of the observer The clock of observer has nothing to do with the clock of the object
4 In many cases, the only thing one can obtain by observations are unparametrized geodesics One can obtain unparametrized geodesics by observation: Observer N 1 Information Information Observer N 2 We take 4 freely falling observers that measure the angular coordinate of the visible objects and send this information to one place. This place will have 4 functions angle(t) for every visible object which are in general case 4 coordiantes of the object.
5 In many cases, the only thing one can get by observations are unparametrised geodesics If one can not register a periodic process on the observed body, one can not get the own time of the body This situation is extremely rare
6 As it is known since Weyl and Ehlers, if one knows the light cone (i.e., the conformal structure), one can reconstruct the own time of an free falling object. The only way to get the light cone is to send the light ray to the object and to register echo -- also possible very rarely and register echo echo world line of the observer the light ray to the body observer shoots with light in objects
7 Jürgen Ehlers 1972 We reject clocks as basic tools for setting up the space-time geometry and propose... freely falling particles instead. We wish to show how the full space-time geometry can be synthesized.... Not only the measurement of length but also that of time then appears as a derived operation. The first game: we know the geodesics of a certain 4-dimensional Einstein metric of Lorenz signature; we need to find the metric. The second game: if we already found an Einstein metric of Lorenz signature such that its geodesics coincides with the observations, do we know for sure that the metric is the one we need? The first question of my talk (H. Weyl 1931): Could two 4- dimensional Einstein metrics of Lorenz signature have the same geodesics considered as unparameterized curves?
8 Main Definition and example of Lagrange 1789 Def. Two metrics (on one manifold) are geodesically equivalent if they have the same unparameterized geodesics (notation: g ḡ) f(x) X Radial projection f : S 2 R 2 takes geodesics of the sphere to geodesics of the plane, because geodesics on sphere/plane are intersection of plains containing 0 with the sphere/plane.
9 Examples of Dini 1869 Theorem (Dini 1869) The metric (X(x) Y (y))(dx 2 + dy 2 ) ( is geodesically equivalent to they have sense). 1 Y(y) 1 X(x) ) ( dx 2 X(x) + dy2 Y(y) ), (if Moreover, every two nonproportional Riemannian geodesically equivalent metrics on the surface have this form in a neighbourhood of almost every point. Levi-Civita 1896: The metrics of Dini can be generalized for every dimension. In the pseudo-riemannian case there exist other examples (the complete description in dim 2 is due to Bolsinov, Pucacco, M 2008; for other dimensions is joint project with Bolsinov)
10 Main Theorem 1 Theorem 1 (Kiosak, M 2009): 4 dim Einstein manifolds are geodesically rigid: Let (M 4,g) be a pseudo-riemannian Einstein manifold of nonconstant curvature. Then, every ḡ geodesically equivalent to g has the same Levi-Civita connection with g. It was a popular subject; partial cases of Theorem 1 were proved by Weyl 1921, Couty 1961, Petrov 1963, Ehlers , Mikes 1982, Barnes 1993, Hall (-Lonie) , Kiosak 2000, Mikes-Kiosak-Hinterleitner Remark 1. If there exists an open subset U TM such that it is sufficient big: for every x M we have T x M U (for example, the set of all time-like vectors is sufficiently big), such that every g-geodesic γ with γ(t 0 ) U is a reparametrized ḡ-geodesic, Then the metrics g and ḡ are geodesically equivalent. Remark 2. Actually, we prove more: if a metric is close in the C 6 -topology to a Einstein 4-dimensional metric of nonconstant curvature, then it admits no nontrivial geodesic equivalence
11 Main Theorem 2 Theorem 1 is very nontrivial and fails in dimension 5: there exist counterexamples. Thought the metrics in the counterexamples are not complete: Theorem 2 (Kiosak, M 2009:) n dim complete Einstein manifolds are geodesically rigid: Let (M n,g) be a complete pseudo-riemannian Einstein manifold of nonconstant curvature. Then, every complete ḡ geodesically equivalent to g has the same Levi-Civita connection with g.
12 Proof of main Theorems: reformulaition as PDE Folklor (Levi-Civita 1896): Two symmetric affine connections Γ and Γ have the same (unparameterized) geodesics, iff there exists a 1-form φ such that Γ i jk = Γi jk + φ jδk i + φ kδj i ( ) Then, all metrics geodesically equivalent to a connection Γ are solutions of [ the system ] k ḡij ḡ ia Γ a kj ḡ jaγ a ki φ i ḡ kj φ j ḡ ki 2φ k ḡ ij = 0 on unknowns ḡ ij and φ i. The system is nonlinear and contains an artificial unknown φ i Remark. One can actually find the artificial unknown φ in the terms of g and ḡ: indeed, contracting ( ) with respect to i and j, we obtain Γ a ai = Γ a ai + (n + 1)φ i. Using that for the Levi-Civita connection Γ of( a metric ) g we have Γ a ai = 1 log( det(g) ) 2 x i, we get φ i = 1 det(ḡ) 2(n+1) x i log det(g) = φ,i for the ( ) function φ : M R given by φ := 1 2(n+1) log det(ḡ) det(g).
13 One more important observation: As we explained above, the following two equivalent statements are definitions of geodesic equivalence: Γ i jk = Γi jk + φ jδ i k + φ kδ i j ( ) for every parametrized geodesic γ(τ) of Γ, there exists a function τ(t) such that the curve γ(τ(t)) is a parametrized geodesic of Γ. Natural questions: How φ and τ(t) are related? ( ( Answer (Levi-Civita 1896): φ a γ a = 1 d 2 dt log dτ )) dt.
14 Naive trick that works: [ We consider the PDE-system ] k ḡij ḡ ia Γ a kj ḡ jaγ a ki φ i ḡ kj φ j ḡ ki 2φ k ḡ ij = 0 ( ). We think that Γ and φ are known and therefore the PDE is a system of n 2 (n+1) 2 of LINEAR PDE of first order on the n(n+1) 2 unknown functions ḡ ij. As we explained above, every ḡ geodesically equivalent to g is a solution of such PDE-system for a certain φ. Now we differentiate ( ) two times w.r.t. all variables and consider the derivatives as new unknowns: Initial equations after one differentiation after two differentiations Number of unknowns Number of equations n(n+1) + n2 (n+1) 2 2 n 2 (n+1) 2 + n2 (n+1) n3 (n+1) 2 We see that (for n 3) the total number of unknowns, n(n+1) 2 + n2 (n+1) 2 + n2 (n+1) n2 (n+1) 2 (n+2) 12 = n(n+1)2 (n+2)(n+3) + n2 (n+1) 2 (n+2) 12 + n3 (n+1) of our linear system of algebraic equations, is less than the number of equations, n2 (n+1) 2 + n3 (n+1) 2 + n3 (n+1) 2 4 = n2 (n+1) 2 (n+2) 4. Then, the coefficients of our system (which are expressions in Γ, φ i, and their first and second derivatives) must satisfy certain algebraic equations.
15 It is a nontrivial linear algebra to find these algebraic equations; those we will use will be formulated in the next Theorem A, which will be used in the proof of Main Theorem 2, and Theorem B, which will be used in the proof of Main Theorem 1. Theorem A Let g be Einstein; n = dim(m) 3. Suppose ḡ is geodesically equivalent to g. Then, the corresponding φ satisfy the following system of PDE. φ i,jk = R n(n+1) (2g ijφ k + g ik φ j + g jk φ i ) + 2(φ k φ i,j + φ i φ j,k + φ j φ k,i ) 4φ i φ j φ k ( ) Corollary Suppose φ satisfy the PDE ( ). Then, along every light-line geodesics γ(t) the following ODE holds:... φ = 6 φ φ 4( φ) 3. Proof: contracting ( ) with γ i γ j γ k and using g ij γ i γ j = 0 we obtain the desired equality.
16 Collecting what we know: Proof of Main Theorem 2 Theorem 2(Kiosak, M 2009) Let (M n,g) be a complete pseudo-riemannian Einstein manifold of nonconstant curvature. Then, every complete ḡ geodesically equivalent to g has the same Levi-Civita connection with g. We already know: 1. The reparameterisation τ(t) that makes g geodesics from ḡ geodesics satisfy ( ( )) φ = φ k γ k = 1 d 2 dt log dτ(t) dt = 1 d 2 dt (log ( τ)) Along every light-line geodesics γ(t), we have φ = 6 φ φ 4( φ) If both metrics are complete, the mapping τ is a diffeomorphism of R. Analyse of the conditions (1,2) gives us an ODE on τ(t) whose solution is τ(t) = t dξ t 0 C 2ξ 2 +C 1ξ+C 0 + const; in view of the condition (3), τ(t) const implying by (1) that φ const 1 implying that the metrics are affine equivalent as we claim in Main Theorem 2 (provided g is indefinite.) The proof for Riemannian metrics with R 0 is similar. For R > 0 it requires a result of Tanno/Gallot 1978/79 and will not be given here.
17 The same trick works in another problem Theorem 3. Let g be a complete pseudo-riemannian Einstein metric of indefinite signature (i.e., for no constant c the metric c g is Riemannian) on a connected manifold M n>2. Assume the metric ψ 2 g is also Einstein. Then, the function ψ is a constant. Remark. Theorem fails for Riemannian metrics Möbius transformations of the standard round sphere and the stereographic map of the punctured sphere to the Euclidean space are conformal nonhomothetic mappings. One can construct other examples on warped Riemannian manifolds (Kühnel 1988) Proof. An easy exercise (Brinkmann 1925) ist that the Ricci curvatures R ij and R ij of g and ḡ = ψ 2 g = e 2φ g are related by R ij = R ij + ( φ (n 2) φ 2 )g ij + n 2 ψ i j ψ ( ) We take a light-line geodesic γ(t) and contract ( ) with γ i γ j : we obtain d 2 dt ψ(γ(t)) = 0 implying ψ(γ(t)) = const 2 1 t + const. Since ψ(γ(t)) is defined for all t R and equals zero at no point, const 1 = 0 implying ψ const,
18 The same trick works in one more problem Theorem (Alekseevsky, Cortes, Leitner, Galaev 2007) Let g be a light-line-complete pseudo-riemannian metric of indefinite signature on a closed n dimensional manifold M n. Then, the corresponding cone ij g ijdx i dx j ) is not decomposable. ( ˆM n+1 = R >0 M n,ĝ = dx0 2 + x2 0 Proof. By Gallot 1979, ( ˆM,ĝ) is decomposable if and only if λ : M R such that λ const and k j i λ = i λ g jk + j λ g ik + 2 k λ g ij ( ) We take a light-line geodesic γ(s) and contract ( ) with γ i γ j γ k. The right-hand side vanishes and we obtain 0 = γ i γ j γ k k j i λ = d3 ds 3 λ(γ(s)). Thus, along γ we have λ(γ(t)) = const 2 s 2 + const 1 s + const 0. Since the manifold is closed, the function λ is bounded, and const 2 = const 1 = 0, implying λ = const 0 May be there are other applications of the same trick? Remark. Ten days ago Pierre Mounoud in arxiv: combined our results with those of Gallot and A-C-L-G proved that the completeness assumption in Theorem above could be removed by the price of allowing the cone ( ˆM,ĝ) to be flat.
19 Proof of Theorem 1: We need Theorem B Theorem 1 Let (M 4,g) be a pseudo-riemannian Einstein manifold of nonconstant curvature. Then, every ḡ geodesically equivalent to g has the same Levi-Civita connection with g. Theorem B Let g be Einstein; n = dim(m) 3. Then, every ḡ geodesically equivalent to g and not affine equivalent to g satisfies W i akmḡaj + W j akmḡia = 0 ( ) where W is the projective Weyl tensor. Moreover, W has type N in Petrov classification. If W 0, the condition ( ) bears no information but if W 0, the metric has constant curvature. In dimension 4, in lorenz signature, ( ) is a very strong linear algebra restriction to the metric: Lemma (Hall-(Lonie) 2009) If W 0, has Petrov type N, and ḡ satisfies ( ), then ḡ ij = ρ(x) g ij + v i v j for a certain v. Proof of Theorem 1. The last formula is an Ansatz for ḡ, substituting it in [ k ḡij ḡ ia Γ a kj ḡ ja Γ a ki] φi ḡ kj φ j ḡ ki 2φ k ḡ ij = 0 we obtain v = 0,
20 One can use similar methods for another interesting result Def. The degree of mobility of a metric g is the dimension of the space of the metrics geodesically equivalent to g. It is known that the degree of mobility of the metric of constant curvature is (n+2)(n+1) 2. Theorem 4 (Kiosak, M 2009) Let g be a complete Riemannian or pseudo-riemannian metric on a connected M n 3. Assume that for every constant c 0 the metric c g is not the Riemannian metric of constant curvature +1. If the degree of mobility of the metric is 3, then every complete metric ḡ geodesically equivalent to g has the same Levi-Civita connection with g. In other words, if we have a complete metrics with the degree of mobility 3, then every complete metric geodesically equivalent to it has the same Levi-Civita connection with g.
21 Theorem 4 is interesting because of Lichnerowicz-Obata-Solodovnikov conjecture Question: Schouten 1924: List all complete metrics admitting complete projective vector field Conjectured Answer: Lichnerowicz-Obata-Solodovnikov (50th): Let a complete manifold (of dim 2) admit a complete projective vector field. Then, the manifold is covered by the round sphere, or the vector field is affine.
22 History of L-O-S conjecture: Most results were concentrated in the Riemannian case, since it hard to prove global results about the pseudo-riemannian metrics France (Lichnerowicz) Couty (1961) proved the conjecture assuming that g is Einstein or Kähler Japan (Yano, Obata, Tanno) Yamauchi (1974) proved the conjecture assuming that the scalar curvature is constant Proved in the Riemannian case in 2007 M Soviet Union (Raschewskii) Solodovnikov (1956) proved the conjecture assuming that all objects are real analytic and that n 3. Partial results on projective transformations of pseudo-riemannian metrics: Barnes 1993, Hall Corollary 1 from Theorem 4: L-O-S conjecture is true also in the pseudo-riemannian case under the additional assumption that the degree of mobility is 3 and dim(m) 3. Corollary 2 from Theorem 4: For (M n 3,g) such that const g is not the Riemannian manifold of constant positive curvature we have dim(proj(m, g)) dim(hom(m, g)) 1, where Proj is the Lie group of projective transformations, and Hom is the group of homotheties.
23 Let us mimic the Riemannian proof in the pseudo-riemannian situation Two cases in the Proof of L-O-S Conjecture under the assumption that the degree of mobility =2 Another group of methods works; still to be done in the pseudo-riemannian case. Joint project with Bolsinov under the assumption that the degree of mobility >2: Done by Theorem 4 (Corollary 1)
24 Proof of Theorem 4 We follow the main lines of the proof of Theorems 1,2: we consider the PDE-system [ ] k ḡij ḡ ia Γ a kj ḡ ja Γ a ki φi ḡ kj φ j ḡ ki 2φ k ḡ ij = 0 ( ). We again differentiate ( ) two times w.r.t. all variables and consider the derivatives of ḡ ij as new unknowns: Number of unknowns Number of equations Initial equations after one differentiation after two differentiations + n2 (n+1) 2 + n2 (n+1) 2 (n+2) 4 12 n(n+1) + n2 (n+1) 2 2 n 2 (n+1) 2 + n3 (n+1) 2 + n3 (n+1) 2 4 We get a huge system of linear equations on the unknowns; the number of equations is bigger than the number of equations Theorem C If the system has three independent solutions, then φ i,jk = B(2g ij φ k + g ik φ j + g jk φ i ) + 2(φ k φ i,j + φ i φ j,k + φ j φ k,i ) 4φ i φ j φ k ( ) for a certain function B. Corollary Suppose φ satisfy the PDE ( ). Then, along every... light-line geodesics γ(t) the following ODE holds: φ = 6 φ φ 4( φ) 3. This is the same equation as in proof of Thm 2!! Proof: contracting ( ) with γ i γ j γ k and using g ij γ i γ j = 0 we obtain the desired equality. Continuing as in proof of Theorem 2, we prove Theorem 4.
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