The 77th Compton lecture series Frustrating geometry: Geometry and incompatibility shaping the physical world

Transcription

1 The 77th Compton lecture series Frustrating geometry: Geometry and incompatibility shaping the physical world Lecture 6: Space-time geometry General relativity and some topology Efi Efrati Simons postdoctoral fellow James Franck Institute The University of Chicago 1

2 Outline: Geometry of space-time in special relativity Geometry of space-time in general relativity Swimming in space Some Topology 2

3 Geometry of Space-time Special relativity: The Lorenz metric, length contraction, time dilation Does this lead to a non-euclidean space-time structure? No! 3

4 Geometry of special relativity The ingredients of standard (Gallilean) relativity: The laws of motion are the same in all inertial frames (non accelarating frames). V =V+U The ingredients of special relativity: All the laws of physics are the same in all inertial frames. The laws of Electromagnetism depend on a parameter, c, The speed of light. It too must be constant at all frames. c =? c+u 4

5 Geometry of special relativity The ingredients of standard (Gallilean) relativity: The laws of motion are the same in all inertial frames (non accelarating frames). V =V+U The ingredients of special relativity: All the laws of physics are the same in all inertial frames. The laws of Electromagnetism depend on a parameter, c, The speed of light. It too must be constant at all frames. c =? c+u Einstein (1949): The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events... The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws... 5

6 Geometry of special relativity When transitioning between inertial frames, time and space mix. Loss of the notion of simultaneity! The proper time interval -c 2 dτ 2 =- c 2 dt 2 +dl 2= ds 2 6

7 Geometry of special relativity The proper time interval -c 2 dτ 2 =- c 2 dt 2 +dl 2= ds 2 The pole and barn paradox is due to the lack of simultaneity 7

8 Geometry of special relativity The proper time interval -c 2 dτ 2 =- c 2 dt 2 +dl 2= ds 2 For the shirt to fit it must match the true rest length 0.9 c 8

9 Geometry of special relativity 1. The notion of simultaneity is lost 2. Space and time mix when going between inertial frames 3. Space is Riemannianly flat (Minkowsky metric)

10 Geometry of general relativity The principal of general relativity claims that an observer cannot distinguish the effects of gravity from those of acceleration. Each frame is associated with a different acceleration. As velocity changes we go from using one inertial frame to another: Similarly to gluing together pieces of flat geometry

11 Geometry of general relativity Einstein s equation relates the geometry of space to the matter and energy through the energy momentum tensor

12 Geometry of general relativity Einstein s equation relates the geometry of space to the matter and energy through the energy momentum tensor Cosmological constant Curvature Energy momentum tensor

13 Geometry of general relativity Schwarzschild solution of Einstein equation to black holes Flamm s paraboloid

14 Geometry of general relativity Schwarzschild solution of Einstein equation to black holes Not to be confused with gravity well Flamm s paraboloid

15 Swimming through space J. Wisdom (2003) Say you find yourself near the horizon of a black hole with no propulsion. Can you swim your way out? 15

16 Swimming through space J. Wisdom (2003) Say you find yourself near the horizon of a black hole with no propulsion. Can you swim your way out? 16

17 Baron Münchhausen Baron Münchhausen s maneuver The Surprising Adventures of Baron Munchausen by Rudolf Erich Raspe Having sunk with his horse in a swamp he pulls himself out by his own hair... This is physically impossible because an isolated object cannot change the momentum of its center of mass. 17

18 Baron Münchhausen Baron Münchhausen s maneuver In the language of physics: 1. An isolated object experiencing no external forces (ignoring gravity) is displaced. 2. In the absence of external forces a body cannot change its momentum. Are these two mutually contradicting? 18

19 Baron Münchhausen Cats do it... When a cat is dropped from rest with its legs facing up it manages to rotate and point its legs down without any external torque! 19

20 Baron Münchhausen Cats do it... A free floating cat can rotate through a cyclic set of motions (swim in circle...) 20

21 Baron Münchhausen Non-commutativity in curved geometry Parallel transport is non-commutative in curved space 21

22 Baron Münchhausen The shape of the universe Gravitational lensing 22

23 Topology

24 Topology Topology is the mathematical study of the properties that are preserved under deformations (not allowing for self intersections or the introduction of cuts). Quantifies the connectedness of the pieces of an object.

25 Topology Topology is the mathematical study of the properties that are preserved under deformations (not allowing for self intersections or the introduction of cuts). Quantifies the connectedness of an object. Topologically equivalent 25

26 Topology Topology is the mathematical study of the properties that are preserved under deformations (not allowing for self intersections or the introduction of cuts). Quantifies the connectedness of an object. Topologically equivalent Topologically not equivalent 26

27 Topological invariants A volume in three dimensions can be associated with the closed surface consisting its boundary 27

28 Topological invariants The genus of a closed surface = the number of handles g = 0 g = 1 g = 2 g = 3 28

29 Euler s characteristic Euler s Characteristic χ = Vertices - Edges + Faces

30 Euler s characteristic Euler s Characteristic χ = Vertices - Edges + Faces =1

31 Euler s characteristic Euler s Characteristic χ = Vertices - Edges + Faces =1

32 Euler s characteristic Euler s Characteristic χ = Vertices - Edges + Faces Does not change when sides or edges deform Does not change when edges are added or eliminated Δ Vertices = -1 Δ Edges = -1 Δ Faces = 0 32 Δ Vertices = 0 Δ Edges = -1 Δ Faces = -1

33 Euler s characteristic Euler s Characteristic χ = Vertices - Edges + Faces Does not change when sides or edges deform Does not change when edges are added or eliminated = = =2 33

34 Euler s characteristic Euler s Characteristic χ = Vertices - Edges + Faces Does not change when sides or edges deform Does not change when edges are added or eliminated = =0 (-4)-(-4)+(-2)=-2 Combining the two we loose four vertices, four edges, and two faces 2+2-2=2 (a sphere fused to another sphere is a sphere) 2+0-2=0 (a torus fused to a sphere is a torus) 34

35 Euler s characteristic The genus of a closed surface = the number of handles g = 0 g = 1 g = 2 g = 3 χ=2 χ=0 χ=-2 χ=-4 χ=2-2g 35

36 The Gaussian curvature and Gauss Bonnet theorem The Gauss-Bonnet theorem relates the sum over area of the Gaussian curvature to the turning rate of the boundary. For a geodesic triangle it reads K da = 3 θ i π i=1 ence of a specific configurati Given a closed surface we can triangulate it with geodesics. On a smooth surface the sum of angles at every vertex is 360 (2 π) 36

37 The Gaussian curvature and Gauss Bonnet theorem The Gauss-Bonnet theorem relates the sum over area of the Gaussian curvature to the turning rate of the boundary. K da = 3 θ i π = 2π #Vertices π #Triangles = 2π(V E + F) = 2πχ i=1 3 Given a closed surface we can triangulate it with geodesics. #Edges = 3 2 #Triangles ence of a specific configur On a smooth surface the sum of angles at every vertex is 360 (2 π) 37

38 Geometry and Topology K da = 2πχ Geometry Topology 38

39 Next Week Guest lecturer: Dr. Dustin Kleckner 39