Geometry of almost-product (pseudo-)riemannian manifold. manifolds and the dynamics of the observer. Aneta Wojnar

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1 Geometry of almost-product (pseudo-)riemannian manifolds and the dynamics of the observer University of Wrocªaw Barcelona Postgrad Encounters on Fundamental Physics, October 2012

2 Outline 1 Motivation

3 An observer as a moving matter (Ehlers) There are a few equivalent ways of thinking about the observer on Lorentzian manifold (with the metric of signature (, +, +, +)): ow of a relativistic uid congruence of lines (family of integral curves), arrow of time, normalized vector eld u α u α = 1, 1-dimensional time-like distribution (+ 3-dimensional space-like distribution = 3+1 decomposition).

4 Distribution Motivation Distribution (a geometrical denition) Let M be n-dimensional manifold (possibly with a pseudo -Riemannian metric and the corresponding Levi-Civita connection ). A k-distribution (k-dimensional tangent distribution) is a map D which associates a k-dimensional subspace D p T p M with the point p M: D : p D p T p M

5 Frobenius' theorems Local Frobenius theorem Every involutive distribution is completely integrable. Proof: Introduction to smooth manifolds, J.M.Lee Global Frobenius theorem Let D be an involutive k-dimensional tangent distribution on a smooth manifold M. The collection of all maximal connected integral manifolds of D forms a foliation of M. Proof: Introduction to smooth manifolds, J.M.Lee

6 Almost-product manifold Almost-product structure Let M and TM denote a smooth manifold and its tangent bundle respectively. An almost-product structure (P-structure) on M is a eld of endomorphisms of TM, i.e. ( 1 1) tensor eld P on M, such that P 2 = 1. If M is equipped with a metric g such that: g(px, PY ) = g(x, Y ), X, Y TM, then P is called the (pseudo-)riemannian almost-product structure. Let D be a distribution on (M,g) and D the distribution orthogonal to D. At every point p M, we have then T m (M) = D m D m.

7 Naveira's classication of Riemannian almost-product manifold Let D be a distribution on (M,g) and D the distribution orthogonal to D. At every point p M, we have then T m(m) = D m D m. Thus we can uniquely dene a ( 1 1 ) tensor eld P such that P 2 = 1, P D = 1, P D = 1. The distribution D is called: geodesic, minimal or umbilical if and only if D has property D 1, D 2, D 3 respectively, where: D 1 ( A P)A = 0, D 2 α(x ) = 0, D 3 g(( A P)B, X ) + g(( B P)A, X ) = 2 g(a, B)α(X ), p where X D, A, B D. Here {e a} k a=1 (k = dimd) is a local orthonormal frame of D and α(x ) = k a=1 g(( ea P)e a,x ).

8 Geometrical properties of foliations We have 3 kinds of foliations related with the following properties of integrable submanifolds: Totally geodesic foliation: N M is a totally geodesic submanifold if and only if every geodesic on N is also a geodesic on M. Condition: K = 0. Minimal surface: a surface with the smallest possible value of the area bounded by a certain curve. Condition: H = 0. Umbilical manifold: a manifold for which all points are umbilical points (for Riemannian submanifold the second fundamental form is proportional to an induced metric).

9 of minimal surfaces Helicoid Catenoid

10 of an umbilical manifold Sphere

11 Totally geodesic foliation Source: 3+1 Formalism and Bases of Numerical Relativity Eric Gourgoulhon

12 Foliations Motivation Theorem (O. Gil-Medrano) A foliation D is a totally geodesic, minimal or totally umbilical if and only if D has the property F 1, F 2, F 3 respectively, where: F i F + D i, i = 1,2,3, and F ( A P)B = ( B P)A A, B D The property F is equivalent to Frobenius' theorem, i.e. the distribution D which has this property is a maximal foliation.

13 Ehlers' decomposition (1961) u α;σ = σ αβ + ω αβ Θ h αβ u α u β Irreducible components and their physical interpretation u α = u α;β u β - acceleration vector, ω αβ = u [α;β] + u [α u β] - rotation tensor, Θ = u α ;α - scalar of expansion, σ αβ = u (α;β) + u (α u β) 1 3 Θ h αβ - shear tensor.

14 Relation between u and P Lorentzian almost-product structure Let (M, g) be the Lorentzian manifold and u α (u α u α = 1) the independent of the metric observer. The corresponding Lorentzian almost-product structure compatible with the metric g is dened P = h v, where h α β = δ α β + uα u β and v = 1 h are the orthogonal projection on 3-distribution and 1-distribution respectively. It has the following form: P α β = δ α β + 2uα u β

15 Final results: table 1

16 Final results: table 2

17 of observers Minkowski space-time The observer u µ = (1,0,0,0) belongs to the class (F 1,F 1 ), ( ) The observer u µ r =, t r 2 t 2 r 2 t,0,0 belongs to the class (F,F ), ( 2 ) 2, The observer u µ y =, x x 2 +y 2 x 2 +y,0 is from the class (F,D 2 2 ). Schwarzschild space-time The stationary observer u µ = ( ) (1 2M r ) 1 2,0,0,0 belongs to (F,F 1 ), There is no observer of the class (F 1,F 1 ) in the Schwarzschild spacetime.

18 Further questions Existance of preferred observer for a given metric. Canonical form of a metric for a given observer. Classication of structures.

19 Appendix For Further Reading For Further Reading I A.M. Naveira A classication of Riemannian almost product manifolds Rendiconti di Matematica (1983) O. Gil-Medrano Geometric properties of some classes of Riemannian almost-product manifolds Rendiconti del Circolo Matematico di Palermo (1983) A. Borowiec and M. Ferraris and M. Francaviglia and I. Volovich Almost-complex and almost-product Einstein manifolds from a variational principle Journal of Mathematical Physics (1999) J. Ehlers Contributions to the relativistic mechanics of continuous media General Relativity and Gravitation (1961) J.M. Lee Introduction to smooth manifolds Springer Verlag (2000)