1. PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013

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

3. 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) To Exam Or Not To Exam 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

34. 2 Tensor analysis 33 2.2 Riemann tensor Riemann tensorhas 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)

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

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

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

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

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

40. Path length ds = G d l, stationary path means 48 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 = l

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

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

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

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

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

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

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

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

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

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

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

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

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

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

55. 6 General Relativity 64 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 → Einstein ’ s field equations only remaining issue: relation between and Newton ’ s law

56. 6 General Relativity 6.1 Principles 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 Particles move along geodesics The laws of Special Relativity apply locally for all non-accelerated (inertial) observers The curvature follows the energy-momentum tensor as described by Einstein ’ s field equations General Relativity summarized in 6 points 65

57. 66 6 General Relativity 6.2 Einstein’s field equations independence on choice of coordinates formulate theory by means of tensor fields 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 ? (energy-momentum tensor) matter is completely described by 2nd-rank tensor description of curvature by 2nd-rank tensor (Einstein tensor)

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

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

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

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

62. 71 6 General Relativity 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? can be locally transformed to are not completely arbitrary matrix witheigenvalues of corresponding to 1 time-like and 3 space-like coordinates have signs

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

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

70. Challenge: prove this

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

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

73. 7 Consequences 7.1 Relativistic Kepler problem 87 Perihelion shift of the planets in the Solar system semi-major axis a[AU] orbital period P[yr]eccentricityε perihelion shift per century Mercury ☿ 0.390.250.20643˝ Venus ♀ 0.720.620.00688.6˝ Earth ♁ 110.01673.8˝ Mars ♂ 1.51.880.09331.4˝ Jupiter ♃ 5.211.90.0480.06˝ Saturn ♄ 9.529.50.0560.01˝ Uranus ♅ 19840.0460.002˝ Neptune ♆ 301650.0100.0008˝ (essentially inversely proportional to a 5/2 )

74. 7 Consequences 7.1 Relativistic Kepler problem 86

75. 90 7 Consequences 7.2 Bending of light asymptoticstotal deflection Deflection of light by gravity (1915) α = 4GM c2ξc2ξ 1. ″ 7 measurable at Solar limb: α = bending angle

76. 92

77. 93

78. 7 Consequences 7.2 Bending of light "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." (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 British expeditions to Sobral (Brazil) and the island of Principe to observe the total Solar Eclipse of 29 May 1919 95

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

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

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

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

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

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

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