PH5011 General Relativity

Transcription

1 PH5011 General Relativity Martinmas 2012/2013 Dr HongSheng Zhao shortened/expanded from notes of MD

2 0 General issues 0.1 Summation convention dimension of coordinate space pairwise indices imply sum 0.2 Indices Apart from a few exceptions, upper and lower indices are to be distinguished thoroughly 2

3 To Exam Or Not To Exam 1. Basis (not examined) intro. tensor and Coordinates transformation (exam). 2. Tensor operations all examinable. 3. Mechanics classical NOT exam. 4. Mechanics in curved space NOT exam. 5. Special Rela. NOT exam. 6. General Rela. (Einstein Eq.) exam. 7. Application of GR Examinable: FRW (p1-6), Schwarzschild (p1-4), Tutorials (1,2,3). Adv. (1p, for intuition) 3

4 1 Curvilinear coordinates 1.1 Basis and coordinates location described by set of coordinates coordinate line given by for all tangent vector at basis vector related to coordinate set of basis vectors spans tangent space at infinitesimal displacement in space on variation of coordinate given by line element in general, the basis vectors 3 depend on

5 1 Curvilinear coordinates 1.1 Basis and coordinates Example A: Cartesian coordinates (I) 4

6 1 Curvilinear coordinates 1.1 Basis and coordinates Example B: Constant, non-orthogonal system (I) 5

7 1 Curvilinear coordinates 1.2 Reciprocal basis Kronecker-delta construction: orthogonality normalization for for orthogonal basis orthonormal basis for 6

8 1 Curvilinear coordinates 1.2 Reciprocal basis Special case: 3 dimensions 7

9 1 Curvilinear coordinates 1.2 Reciprocal basis Example A: Cartesian coordinates (II) 8

10 1 Curvilinear coordinates 1.2 Reciprocal basis Example B: Constant, non-orthogonal system (II) 9

11 1.3 Metric 1 Curvilinear coordinates coefficients of metric tensor ( 1.5) symmetry: as matrix 10

12 1 Curvilinear coordinates 1.3 Metric Examples A+B: Cartesian & non-orthogonal constant basis (III) 11

13 1 Curvilinear coordinates 1.3 Metric length of curve given by parametric representation of curve 12

14 1 Curvilinear coordinates 1.3 Metric Example: Length of equator in spherical coordinates use parameter along the azimuth in one only needs to consider : one full turn for and 13

15 1 Curvilinear coordinates 1.3 Metric With the reciprocal basis, one defines reciprocal components of the metric tensor which fulfill, equivalent to the condition for the inverse matrix 14

16 1 Curvilinear coordinates 1.3 Metric metric tensor orthonormality condition lowers index raises index 15

17 1 Curvilinear coordinates 1.4 Vector fields mathematics: vector field physics: vector (field) vector components defined by means of basis vectors contravariant components covariant components ( 1.6) raising/lowering indices 16

18 1.5 Tensor fields mathematics: tensor field physics: tensor (field) 1 Curvilinear coordinates tensor is multi-dimensional generalization of vector product of vector spaces behaves like a vector with respect to each of the vector spaces rank of tensor tensor of rank 0 tensor of rank 1 tensor of rank 2 tensor of rank 3... scalar vector square matrix cube 17

19 1 Curvilinear coordinates 1.5 Tensor fields basis vectors apply to each of the vector spaces contravariant components covariant components mixed components 18

20 1 Curvilinear coordinates 1.5 Tensor fields Example: Rank-2 tensor Coincidentally, with the matrix product For Cartesian coordinates: 19

21 1.6 Coordinate transformations consider different set of coordinates 1 Curvilinear coordinates (chain rule) different coordinate systems describe same locations 20

22 1 Curvilinear coordinates 1.6 Coordinate transformations vector fields covariant contravariant } tensor fields { derivatives components transform like coordinate differentials 21

23 1 Curvilinear coordinates 1.6 Coordinate transformations Proof: are covariant components of a tensor 22

24 1.7 Affine connection 1 Curvilinear coordinates in general, basis vectors depend on the coordinates derivative of basis vector written in basis affine connection (Christoffel symbol) derivative of reciprocal basis vector: 23

25 1 Curvilinear coordinates 1.7 Affine connection Example C: Spherical coordinates (IV) 24

26 1 Curvilinear coordinates 1.7 Affine connection Example C: Spherical coordinates (IV) [continued] 25

27 1 Curvilinear coordinates 1.7 Affine connection given that the Christoffel symbols can be expressed by means of the components of the metric tensor and their derivatives 26

28 1 Curvilinear coordinates 1.7 Affine connection Proof: (I) (II) (III) (II) + (III) - (I) : 27

29 2 Tensor analysis 2.1 Covariant derivative vector field both the vector components depend on the coordinates derivative: and the basis vectors define covariant derivative of a contravariant vector component as so that 28

30 2 Tensor analysis 2.1 Covariant derivative derivatives transform as can be considered the covariant components of the vector (gradient) covariant components of a vector form components of a tensor, not 29

31 2 Tensor analysis 2.1 Covariant derivative contravariant components covariant components covariant derivatives of tensor components for each { upper } { index, add lower where takes place of in or 30

32 2 Tensor analysis 2.1 Covariant derivative Covariant derivative of 2nd-rank tensor 31

33 2 Tensor analysis 2.2 Riemann tensor order of 2nd covariant derivatives of vector is not commutative, but with the Riemann (curvature) tensor (not intended to be memorized) with and 32

34 2 Tensor analysis 2.2 Riemann tensor Riemann tensor [ [ has two pairs of indices and is antisymmetric in the indices of each pair symmetric in exchanging the pairs Moreover, (1st Bianchi identity) (2nd Bianchi identity) 33

35 2 Tensor analysis 2.2 Riemann tensor Proof: The scalar product of two vectors is a scalar On the other hand (Riemann curvature tensor is antisymmetric in first two indices) 34

36 2 Tensor analysis 2.3 Einstein tensor 2nd-rank curvature tensor fulfilling must relate to Riemann tensor only a single non-vanishing contraction (up to a sign) (Ricci tensor) with next-level contraction (Ricci scalar) matches required conditions 35

37 3 Review: Classical Mechanics 3.1 Principle of stationary action Mechanical system completely described by (Lagrangian) coordinate velocity time action (Hamilton s) principle of stationary action Fermat s principle (optics) Feynman s path integral (QM) (Euler-) Lagrange equations for : 36 kinetic energy potential energy

38 3 Classical mechanics 3.1 Principle of stationary action Example: 1D harmonic oscillator (I) 37

39 4 Intro: Mech. in curved space 4.1 Principle of stationary paths stationary path between two points (e.g. path length is locally shortest) (geodesic equation, assume = s ) 47 Christoffel symbols (affine connection)

40 Path length ds = G d, stationary path means 4 Mechanics in curved space 4.1 Stationary paths Define Constant L factored out of derivatives. Write derivative as dot, if we define t = s = 48

41 4 Mechanics in curved space 4.1 Stationary paths L resembles Lagrangian for a free particle of mass m in curved space (Euler-Lagrange equations) } with and 44

42 4 Mechanics in curved space 4.1 Stationary paths based on Newton s law purely space geometry Eq. of motion along geodesics, = s, or in shorthand: 49

43 4.2 Geodesics as parallel transport 4 Mechanics in curved space i.e. = tangent unit vector to a curve is geodesic if unit tangent vector is parallelly transported moving along geodesics means to keep the same direction geodesics form straight lines (geodesic equation) 46

44 4.3 Conserved momentum p k dp k /d =0 if the metric g independent of q k 4 Mechanics in curved space (geodesic equation) if all do not depend on 50

45 5 Review: Special Relativity 5.1 Minkowski space event described by time and location reference frame defines coordinate origin and motion inertial system force-free particles move uniformly all reference frames moving uniformly with respect to an inertial system are inertial system themselves laws of physics assume the same form in all inertial systems 54

46 5 Special Relativity 5.1 Minkowski space invariance of speed of light along light rays: for all reference frames homogeneity and isotropy of space and time invariance of describes distance in four-dimensional space whereas both and depend on reference frame 55

47 5 Special Relativity 5.1 Minkowski space use 4-dimensional vectors Latin indices Greek indices flat three-dimensional space described by cartesian coordinates 56

48 5.2 Light cone or: causality and the finite speed of light defines light cone photons trace null geodesics between events inside light cone massive particles move on time-like geodesics outside light cone elsewhere, no causal connection 45 opening angle in 5 Special Relativity non-relativistic theories: instantaneous knowledge of interaction invariance of : light cone widens, all events get into causal contact categorization holds irrespective of coordinate system and reference frame 57

49 5.3 Proper time along worldline of clock with attached rest frame 5 Special Relativity proper time time shown on clock invariance of so that (moving clock observed t appears big ) 58

50 5.4 Relativistic mechanics define 4-velocity as 5 Special Relativity known: free particle moves along geodesic [ all ] (as anticipated for inertial system) 59

51 5 Special Relativity 5.4 Relativistic mechanics relativistic action ansatz: (matches invariance of ) let non-relativistic limit ( ) 60

52 conjugate momentum energy 5 Special Relativity 5.4 Relativistic mechanics with (sign in spatial part due to in metric) (relativistic Hamilton-Jacobi equation) 61

53 5.5 Energy-momentum tensor components of stress tensor force area of cross-section normal to cross-section 5 Special Relativity provides relation between the forces and the cross-sections these are exerted on for fluid in thermodynamic equilibrium: (no shear stresses) complement to energy density momentum density stress in fluid rest frame: 62 energy-momentum tensor pressure mass density

54 5 Special Relativity 5.5 Energy-momentum tensor non-relativistic limit: ( Newton s law) (continuity equation) 63

55 6 General Relativity 6.1 Principles experiments cannot distinguish between: virtual forces present in non-inertial frames true forces gravitation can be described by space-time metric gravitation becomes property of space-time with particles moving on geodesics local free-falling frame is an inertial frame, where free particles are on straight lines and only remaining issue: relation between and Newton s law Einstein s field equations 64

56 6 General Relativity 6.1 Principles General Relativity summarized in 6 points The laws of physics are the same for all observers, irrespective of their motion Physical laws take the same covariant form in all coordinate systems We live in a 4-dimensional curved metric space-time The curvature follows the energy-momentum tensor as described by Einstein s field equations The laws of Special Relativity apply locally for all non-accelerated (inertial) observers Particles move along geodesics 65

57 6.2 Einstein s field equations independence on choice of coordinates formulate theory by means of tensor fields 6 General Relativity matter is completely described by 2nd-rank tensor (energy-momentum tensor) description of curvature by 2nd-rank tensor (Einstein tensor)? if non-relativistic limit reproduces Newton s law, this is not necessarily the only possible theory, but the most simple one that conforms to the principles 66

58 6 General Relativity 6.2 Einstein s field equations non-relativistic limit (, ): dominating Einstein s field equations: 67

59 6 General Relativity 6.2 Einstein s field equations with Newton: 68

60 [note: Einstein s orignal sign convention for the Ricci tensor differs from ours] 69 6 General Relativity 6.2 Einstein s field equations

61 6.3 Cosmological constant modified Einstein tensor 6 General Relativity also fulfills measurements suggest Solar neighbourhood baryonic matter in the Universe negligible correction, unless huge length scales are considered theories modifying the law of gravity provide alternative models 70 effective repulsion (dark) vacuum energy??

62 6.4 Time and distance Laws of physics described by tensors do not depend on coordinates coordinates do not have immediate physical meaning What is the time and distance? are not completely arbitrary 6 General Relativity can be locally transformed to eigenvalues of matrix with have signs corresponding to 1 time-like and 3 space-like coordinates 71

63 6 General Relativity 6.4 Time and distance time interval given by between two events at the same location proper time in general, the relation between the proper time interval depends on the location and cannot define spatial distance by means of for neighbouring events at the same time 72

64 6 General Relativity 6.5 Synchronisation 6.5 Synchronisation (e.g. FRW cosmology) if global synchronisation possible (with regard to time coordinate, but measured depends on location) coordinate transformation can always provide (at cost of time-dependent ) (synchronized reference frame) everywhere coordinate line of (i.e. ) is geodesic 76

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70 Challenge: prove this

71 7 GR Applications 7. Satellites: GPS orbit Earth ~ stars orbit BH Beepers on sat. are Doppler/Gravitational-shifted, time delayed 110

72 GPS satellites perform two orbits per sidereal day, 7 Consequences 7. Satellite navigation Doppler shift (transverse motion) per day gravitational potential per day in total, GPS clock appears to run faster by per day GPS clocks are shipped with factory offset to compensate 112

73 7 Consequences 7.1 Relativistic Kepler problem Perihelion shift of the planets in the Solar system semi-major axis a [AU] orbital period P [yr] eccentricity ε perihelion shift per century Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune (essentially inversely proportional to a 5/2 ) 87

74 86 7 Consequences 7.1 Relativistic Kepler problem

75 Deflection of light by gravity (1915) 7 Consequences 7.2 Bending of light asymptotics total deflection bending angle α = 4GM c 2 ξ measurable at Solar limb: α =

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78 7 Consequences 7.2 Bending of light The British expeditions to Sobral (Brazil) and the island of Principe to observe the total Solar Eclipse of 29 May 1919 (Sir) Arthur Stanley Eddington Negative of one of the photographic plates taken by the British expedition to Sobral (Brazil) during the total Solar Eclipse of 29 May 1919 The Royal Society "The present eclipse expeditions may for the first time demonstrate theweight of light; or they may confirm Einstein's weird theory of non-euclidean space; orthey may lead to a result of yet more far-reaching consequences -- no deflection." "The generalized relativity theory is a most profound theory of Nature,embracing almost all the phenomena of physics." 95

79 7 Consequences 7.2 Bending of light Notes about gravitational microlensing dated to 1912 on two pages of Einstein s scratch notebook 99

80 7 Consequences 7.2 Bending of light Images by a gravitational lens 6 I side view η ξ I+ with (angular Einstein radius) (two images) 96

81 bending of light of stars due to intervening foreground stars within the Milky Way 7 Consequences 7.2 Bending of light images cannot be resolved image distortion leads to observable transient brightening (animation by Daniel Kubas, ESO) 98

82 7 Consequences 7.2 Bending of light The chance is one in a million! B. Paczyński 1986, ApJ 304, 1 100

83 First reported microlensing event 7 Consequences 7.2 Bending of light MACHO LMC#1 Nature 365, 621 (October 1993) 101

84 7 Consequences 7.2 Bending of light Astronomy & Geophysics Vol. 47 (June 2006) 102

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89 A Sample of Advanced Material: Geodesics around Black Hole Metric