PH5011 General Relativity Martinmas 2012/2013 Dr HongSheng Zhao shortened/expanded from notes of MD [email protected] 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 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 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 depend on 3 1 Curvilinear coordinates 1.1 Basis and coordinates Example A: Cartesian coordinates (I) 4 1 Curvilinear coordinates 1.1 Basis and coordinates Example B: Constant, non-orthogonal system (I) 5 1 Curvilinear coordinates 1.2 Reciprocal basis Kronecker-delta construction: orthogonality for normalization for orthogonal basis for orthonormal basis 6 1 Curvilinear coordinates 1.2 Reciprocal basis Special case: 3 dimensions 7 1 Curvilinear coordinates 1.2 Reciprocal basis Example A: Cartesian coordinates (II) ⤿ 8 1 Curvilinear coordinates 1.2 Reciprocal basis Example B: Constant, non-orthogonal system (II) ⤿ 9 1 Curvilinear coordinates 1.3 Metric ⤿ coefficients of metric tensor (→ 1.5) symmetry: as matrix 10 1 Curvilinear coordinates 1.3 Metric Examples A+B: Cartesian & non-orthogonal constant basis (III) ⤿ ⤿ 11 1 Curvilinear coordinates 1.3 Metric length of curve given by parametric representation of curve 12 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 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 1 Curvilinear coordinates 1.3 Metric metric tensor orthonormality condition ⤿ “lowers index” “raises index” 15 1 Curvilinear coordinates 1.4 Vector fields mathematics: vector field physics: vector (field) vector components defined by means of basis vectors contravariant components (→ 1.6) covariant components “raising/lowering indices” 16 1 Curvilinear coordinates 1.5 Tensor fields mathematics: tensor field physics: tensor (field) tensor is multi-dimensional generalization of vector product of vector spaces behaves like a vector with respect to each of the vector spaces tensor of rank 0 scalar tensor of rank 1 vector rank of tensor tensor of rank 2 square matrix tensor of rank 3 cube ........ 17 1 Curvilinear coordinates 1.5 Tensor fields basis vectors apply to each of the vector spaces ⤿ contravariant components covariant components mixed components 18 1 Curvilinear coordinates 1.5 Tensor fields Example: Rank-2 tensor ⤿ Coincidentally, with the matrix product For Cartesian coordinates: 19 1 Curvilinear coordinates 1.6 Coordinate transformations consider different set of coordinates (chain rule) different coordinate systems describe same locations ⤿ ⤿ 20 1 Curvilinear coordinates 1.6 Coordinate transformations vector fields ⤿ covariant derivatives components transform like coordinate contravariant } {differentials tensor fields ⤿ 21 1 Curvilinear coordinates 1.6 Coordinate transformations Proof: are covariant components of a tensor ⤿ 22 1 Curvilinear coordinates 1.7 Affine connection 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 1 Curvilinear coordinates 1.7 Affine connection Example C: Spherical coordinates (IV) ⤿ ⤿ 24 1 Curvilinear coordinates 1.7 Affine connection Example C: Spherical coordinates (IV) [continued] ⤿ ⤿ ⤿ 25 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 1 Curvilinear coordinates 1.7 Affine connection Proof: ⤿ (I) (II) (III) (II) + (III) - (I) : ⤿ 27 2 Tensor analysis 2.1 Covariant derivative vector field both the vector components and the basis vectors depend on the coordinates derivative: define covariant derivative of a contravariant vector component as so that 28 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 2 Tensor analysis 2.1 Covariant derivative contravariant components covariant components covariant derivatives of tensor components upper for each index , add {lower } { where takes place of in or 30 2 Tensor analysis 2.1 Covariant derivative Covariant derivative of 2nd-rank tensor ⤿ 31 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 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 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 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 3 Review: Classical Mechanics 3.1 Principle of stationary action Mechanical system completely described by coordinate (Lagrangian) velocity time action (Hamilton’s) principle of stationary action Fermat’s principle (optics) Feynman’s path integral (QM) ⤿ (Euler-) Lagrange equations kinetic energy for : potential energy 36 3 Classical mechanics 3.1 Principle of stationary action Example: 1D harmonic oscillator (I) ⤿ ⤿ ⤿ 37 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 ) Christoffel symbols (affine connection) 47 4 Mechanics in curved space 4.1 Stationary paths Path length ds = G d , stationary path means ⤿ Define ⤿ ⤿ Constant L factored out of derivatives. Write derivative as dot, if we define t = s = 48 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 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 4 Mechanics in curved space 4.2 Geodesics as parallel transport = tangent unit vector to a curve is geodesic if unit tangent vector is parallelly transported i.e. • moving along geodesics means to keep the same direction • geodesics form “straight lines” (geodesic equation) 46 4 Mechanics in curved space 4.3 Conserved momentum pk dpk/d=0 if the metric g independent of qk (geodesic equation) if all do not depend on ⤿ 50 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 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 5 Special Relativity 5.1 Minkowski space use 4-dimensional vectors Latin indices Greek indices flat three-dimensional space described by cartesian coordinates ⤿ 56 5 Special Relativity 5.2 Light cone or: “causality and the finite speed of light” defines light cone 45° opening angle in photons trace null geodesics between events inside light cone massive particles move on time-like geodesics outside light cone ‘elsewhere’, no causal connection non-relativistic theories: instantaneous knowledge of interaction : light cone widens, all events get into causal contact invariance of ⤿ categorization holds irrespective of coordinate system and reference frame 57 5 Special Relativity 5.3 Proper time along worldline of clock with attached rest frame ⤿ proper time time shown on clock invariance of ⤿ ⤿ so that (moving clock observed “t” appears big ) 58 5 Special Relativity 5.4 Relativistic mechanics define 4-velocity as ⤿ ⤿ known: free particle moves along geodesic ⤿ [ all ] ⤿ (as anticipated for inertial system) 59 5 Special Relativity 5.4 Relativistic mechanics relativistic action ansatz: (matches invariance of ) let" ⤿ non-relativistic limit ( ) 60 5 Special Relativity 5.4 Relativistic mechanics conjugate momentum energy !" with (sign in spatial part due to in metric) !" (relativistic Hamilton-Jacobi equation) 61 5