The Erlangen Program and General Relativity

Transcription

1 The Erlangen Program and General Relativity Derek K. Wise University of Erlangen Department of Mathematics & Institute for Quantum Gravity Colloquium, Utah State University January 2014

2 What is geometry?

3 What is geometry?

4 Oversimplified flow of geometric ideas Euclid 300 BC Galois 1832 Lobachevsky 1829 Klein Erlangen 1872 Cartan 1923 Ehresmann Riemann 1854 Newton 1687 Einstein general relativity 1916 Yang & Mills

5 Klein s Erlangen Program The Galois theory of geometry

6 Klein Geometry Klein geometry studies homogeneous spaces (X, G): X a manifold G a Lie group acting transitively on X Given x X, define Klein realized: H := {h G : hx = x} the stabilizer of x. X = G/H symmetries of X symmetries fixing a point.

7 Klein geometry Basic examples: Elliptic: Sn Euclidean: E n Hyperbolic: H n = SO(n + 1)/SO(n) ISO(n)/SO(n) = SO(n, 1)/SO(n) =

8 De Sitter Spacetime A more physicsy example: Write R 4,1 for R 5 equipped with Minkowski inner product : η(x, y) = x 0 y 0 + x 1 y x 4 y 4 Then de Sitter space is the unit pseudosphere, a 4d hyperboloid: X = {x R 4,1 : η(x, x) = 1} The group G = SO(4, 1) of linear isometries of R 4,1 acts transitively on de Sitter space, and the stabilizer of a point is H = SO(3, 1), so we can identify X = SO(4, 1)/SO(3, 1)

9 Figures in a Klein geometry One beauty of Klein s approach is how different spaces of features or figures in a geometry are related... Example: In 3d Euclidean geometry: Space of points is E 3 = ISO(3)/SO(3) Space of lines is L = ISO(3)/O(2) Space of regular icosahedra I = ISO(3)/A 5. of a given size is And the possible geometric relationships between features is reduced to algebraic relationships between the stabilizers.

10 Cartan Geometry A new flavor of differential geometry for each Klein geometry.

11 Cartan geometry approximation by tangent planes Euclidean Geometry allow curvature Riemannian Geometry generalize symmetry group generalize tangent space geometry Klein Geometry allow curvature Cartan Geometry arbitrary homogeneous space approximation by tangent homogeneous spaces (Adapted from diagram by R.W. Sharpe.)

12 Cartan geometry In Cartan geometry, a Cartan connection specifies a way to move a Klein geometry along a path. Example: Any 2d surface Σ R 3 has a canonical Cartan connection modeled on S 2 = SO(3)/SO(2) described by rolling S 2 on Σ without slipping or twisting. Here the Cartan connection is a certain so(3)-valued 1-form on a principal SO(2) bundle. Why? Configuration space of a ball on a surface is an SO(3) bundle. So, what s this SO(2) bundle?

13 Broken symmetry in Cartan geometry Ball has an SO(3) s worth of rotations;

14 Broken symmetry in Cartan geometry Ball has an SO(3) s worth of rotations; Hamster has an SO(2) s worth of rotations.

15 Broken symmetry in Cartan geometry Ball has an SO(3) s worth of rotations; Hamster has an SO(2) s worth of rotations. But... Hamster s motion completely determines ball s motion! Rolling without slipping breaks SO(3) symmetry to SO(2).

16 (G/H) Cartan geometry: P M A: T P g properties: Principal H bundle 1. A p : T y P g is an isomorphism y P 2. (R h ) A = Ad(h 1 ) A h H 3. A( X) = X X h ( X := vertical vector field corresponding to X h) Hamster Geometry : (SO(3)/SO(2) Cartan geometry) P M Principal SO(2) bundle of hamster configurations A: T P so(3) tiny changes in hamster configuration tiny rotations of hamsterball 1. Hamster moves ball rotates (no slipping!) 2. Hamster s direction is a gauge symmetry 3. Hamsterball does not move when the hamster does a pure rotation (no twisting!)

17 Cartan Geometry In the SO(3)/SO(2) example, there is an SO(2)-invariant splitting: so(3) = so(2) R 2 ( ) ( ) ( 0 u a 0 u u 0 b = u 0 + a b 0 0 a b tiny rotation of a sphere tiny rotation fixing chosen basepoint a b ) tiny translation moving basepoint This breaks the Cartan connection into two piecies: an SO(2) connection and a soldering form.

18 Example: de Sitter Cartan Geometry Analogously, for a Cartan geometry modeled on de Sitter space, we have an so(4, 1)-valued 1-form A living on a principal SO(3, 1) bundle, so: so(4, 1) = so(3, 1) R 3,1 (as reps of SO(3, 1)) A = ω + e Lorentz soldering connection form These are exactly the physical fields used in general relativity.

19 Example: de Sitter Cartan Geometry Analogously, for a Cartan geometry modeled on de Sitter space, we have an so(4, 1)-valued 1-form A living on a principal SO(3, 1) bundle, so: so(4, 1) = so(3, 1) R 3,1 (as reps of SO(3, 1)) A = ω + e Lorentz soldering connection form These are exactly the physical fields used in general relativity. = rolling de Sitter space along physical spacetime : (See gr-qc/ )

20 Geometry of Relativity

21 Special relavitivity In special relativity, if two observers use the same coordinates on space at time t = 0, but are in relative motion with velocity v = tanh φ in the x direction: then their measures of time and x-distance are related by a hyperbolic rotation: ( ) ( ) ( ) t cosh φ sinh φ t x = sinh φ cosh φ x

22 Spacetime Minkowski invented spacetime in 1908 as a geometric backdrop for Einstein s special theory of relativity (1905). As topological spaces, spacetime = space time at least locally, but this split is observer-dependent!

23 Spacetime Minkowski invented spacetime in 1908 as a geometric backdrop for Einstein s special theory of relativity (1905). As topological spaces, spacetime = space time at least locally, but this split is observer-dependent! Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

24 Spacetime Minkowski was right. Spacetime was key to constructing general relativity Einstein s theory of gravity (1916).

25 Spacetime Minkowski was right. Spacetime was key to constructing general relativity Einstein s theory of gravity (1916). But what if Minkowski hadn t discovered spacetime? Using only the idea of observers, could we still invent general relativity?

26 Observer space The observer space of de Sitter spacetime the space of all unit future-timelike vectors is a homogeneous space So... O = SO(4, 1)/SO(3)

27 Observer space The observer space of de Sitter spacetime the space of all unit future-timelike vectors is a homogeneous space So... O = SO(4, 1)/SO(3) Definition: An observer space geometry is a Cartan geometry modeled on SO(4, 1)/SO(3).

28 Observer space As representations of SO(3), the Lie algebra so(4, 1) splits into four irreducible pieces, each with nice geometric interpretation in terms of observers: so(4, 1) = so(3) (R 3 R 3 R) rotations boosts (velocity changes) spatial translations time translations But a general observer space has not only no coherent notion of space and time individually (as in general relativity) but no coherent notion of spacetime either.

29 Observer space Theorem [S. Gielen, DKW]: The unit future tangent bundle of any Lorentzian spacetime has a canonical observer space geometry. Conversely, an observer space geometry arises in this way whenever certain integrability conditions on the Cartan connection hold.

30 Observer space Theorem [S. Gielen, DKW]: The unit future tangent bundle of any Lorentzian spacetime has a canonical observer space geometry. Conversely, an observer space geometry arises in this way whenever certain integrability conditions on the Cartan connection hold. In fact, we can write down equations for an observer space geometry which imply: 1. spacetime exists as a quotient of observer space 2. Einstein s equations of general relativity hold on this quotient spacetime. And we can derive these equations from a Lagrangian on observer space.

31 Observer space as a universal geometry But observer space applies just as well to other theories of space and time. All of these give observer space geometries: Galilean relativity Lorentz-violating theories (e.g. Hořava-Lifshitz gravity ) Finsler spacetimes Theories with no spacetime (e.g. relative locality ) 3d conformal geometry (related to Shape dynamics ) Twistor theory... So observer space provides a universal geometric description of a wide class of physically interesting theories.

32 Observer space as a universal geometry But observer space applies just as well to other theories of space and time. All of these give observer space geometries: Galilean relativity Lorentz-violating theories (e.g. Hořava-Lifshitz gravity ) Finsler spacetimes Theories with no spacetime (e.g. relative locality ) 3d conformal geometry (related to Shape dynamics ) Twistor theory... So observer space provides a universal geometric description of a wide class of physically interesting theories. Many such applications come from taking the Erlangen Program seriously...

33 The many faces of de Sitter observer space Sometimes a space of features in one Klein geometry is isomorphic to a space of features in a different Klein geometry. These are all isomorphic as SO(4, 1)-spaces: The unit-timelike tangent bundle of de Sitter spacetime. The space of all transverse 3-planes in the tautological bundle over the conformal 3-sphere. The unit tangent bundle of 4d hyperbolic space. The space of 2-spheres in the space of light-rays in de Sitter spacetime (related to twistor theory).. In particular, they are all isomorphic to SO(4, 1)/SO(3).

34 The many faces of de Sitter observer space Sometimes a space of features in one Klein geometry is isomorphic to a space of features in a different Klein geometry. These are all isomorphic as SO(4, 1)-spaces: The unit-timelike tangent bundle of de Sitter spacetime. The space of all transverse 3-planes in the tautological bundle over the conformal 3-sphere. The unit tangent bundle of 4d hyperbolic space. The space of 2-spheres in the space of light-rays in de Sitter spacetime (related to twistor theory).. In particular, they are all isomorphic to SO(4, 1)/SO(3). = Any of them serves as a model of de Sitter observer space.

35 The many faces of de Sitter observer space Sometimes a space of features in one Klein geometry is isomorphic to a space of features in a different Klein geometry. These are all isomorphic as SO(4, 1)-spaces: The unit-timelike tangent bundle of de Sitter spacetime. The space of all transverse 3-planes in the tautological bundle over the conformal 3-sphere. The unit tangent bundle of 4d hyperbolic space. The space of 2-spheres in the space of light-rays in de Sitter spacetime (related to twistor theory).. In particular, they are all isomorphic to SO(4, 1)/SO(3). = Any of them serves as a model of de Sitter observer space. But each one suggests a different type of deformation of the de Sitter observer space, via Cartan geometry!

36 Conformal picture of an observer For example, SO(4, 1)/SO(3) is also the space of ways to decompactify the conformal 3-sphere and turn it into a Euclidean vector space: 0 S 3 the unit 2- sphere A point in this space amounts to a choice of origin, point at infinity, and unit sphere. How does this work?...

37 Mapping observer space to conformal space Here s a hint: Think of the conformal sphere as the projective null cone P (C) in R 4,1. Then: 2d subspace of R 4,1 containing the observer s geodesic. same 2d subspace contains a pair of points in P (C). (For the full proof, see arxiv: )

38 Two Faces of Janus Like the Roman god Janus, a single geometric entity in Kleinian geometry can have two distinct faces: (Illustration by Marc Ngui)

39 Observer space from conformal space Using this idea, we can build observer space geometries from any 3d conformal geometry. For details see

40 Summary & Conclusions Taking the Erlangen Program seriously in general relativity, and using Cartan geometry, we find: Spacetime in general relativity is superfluous: it can be derived under certain conditions on observer space A unified geometric framework that includes popular extensions/modifications of general relativity Relationships between modifications of general relativity A relationship between canonical and covariant approaches to gravity. For more, see this paper: S. Gielen, DKW, Lifting general relativity to observer space, J. Math. Phys. 54, (2013), arxiv: and also these: DKW, arxiv: , arxiv: