CourseStudent8

logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up GR course 8 Metric tensor Bin Chen School of Physics Peking University March 25, 2010 Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Jj 1 Metric tensor 2 Geometry of manifolds 3 Riemann Normal coordinates 4 Geometry from metric: warm up Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Metric tensor A symmetric (0, 2) tensor; Denoted by gˆ or in components gµν ; Nondegenerate: |g| = det(gµν) 6= 0; Its inverse is also symmetric, but as a component of (2, 0) tensor: gµνgνσ = δµσ . (1) Just as its flat space counterpart, gµν and g µν could be used to raise and lower indices in curved spacetime; Define inner product between vectors or 1-forms; Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Importance of metric tensor supplies a notion of “past” and “future”;(Causality) allows the computation of path length and proper time;(Geometry) determines the “shortest distance” between two points (and therefore the motion of test particles); (Affine connection) replaces the Newtonian gravitational field φ; (Gravitational field) provides a notion of locally inertial frames and therefore a sense of “no rotation”; (RNC or LIF) replaces the traditional Euclidean three-dimensional dot product of Newtonian mechanics; and so on.; Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Local geometry Consider two infinitesimally separated points P and Q, in a coordinate system P : xµ, Q : xµ + dxµ; (2) The “distance” or “interval” between P and Q is ds2 = f(xa, dxa); (3) It determines the local geometry; Example: 2d Finsler geometry ds2 = (dξ4 + dζ4)1/2. (4) Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Riemann geometry In (Pseudo-)Riemann geometry, ds2 = gµνdxµdxν ; (5) From P to Q, we have a infinitesimal vector dsˆ: dsˆ = dxµeˆµ (6) where dxµ is the components, not 1-form; The interval: ds2 = dsˆ · dsˆ = (eˆµ · eˆν)dxµdxν = gµνdxµdxν (7) where we have used the fact that eˆµ · eˆν = gˆ(eˆµ, eˆν) = gσρθˆσ(eˆµ)θˆρ(eˆν) = gσρδσµδ ρ ν = gµν (8) Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Example: R3 In Cartessian coordinates ds2 = (dx)2 + (dy)2 + (dz)2 . (9) In spherical coordinates we have x = r sin θ cosφ y = r sin θ sinφ z = r cos θ , (10) which leads directly to ds2 = dr2 + r2dθ2 + r2 sin2 θdφ2 . (11) The space is the same, though the metric looks different; Since gµν , it has n(n+1) 2 independent components in n-dim. space; Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Canonical form of the metric tensor In this form the metric components become gµν = diag (−1,−1, . . . ,−1,+1,+1, . . . ,+1, 0, 0, . . . , 0) , where “diag” means a diagonal matrix with the given elements.; s is the number of +1’s in the canonical form, and t is the number of −1’s; s− t is the signature of the metric; For nondegenerate metric, s+ t = n, the dim. of spactime; If t = 0, the metric is called Euclidean or Riemannian; If t = 1, it is called Lorentzian or pseudo-Riemannian; If a metric is continuous, the rank and signature of the metric tensor field are the same at every point; Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up RNC Q: can we always put the metric into the canonical form? A: It is always possible to do so at some point p ∈M ; Even better: ∂σgµν = 0; But not ∂σ∂ρgµν ≡ 0; Such coordinates are known as Riemann normal coordinates(RNC), and the associated basis vectors constitute a local Lorentz frame.; In RNC, the metric at p looks like that of flat space “to first order.” This is consistent with “small enough regions of spacetime look like flat (Minkowski) space.”; Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Sketch of proof: I Transformation law: gµ′ν′ = ∂xµ ∂xµ′ ∂xν ∂xν′ gµν , (12) Taylor expansion: xµ = ( ∂xµ ∂xµ′ ) p xµ ′ + 1 2 ( ∂2xµ ∂xµ ′ 1∂xµ ′ 2 ) p xµ ′ 1xµ ′ 2+ 1 6 ( ∂3xµ ∂xµ ′ 1∂xµ ′ 2∂xµ ′ 3 ) p xµ ′ 1xµ ′ 2xµ ′ 3+· · · , (13) where xµ ′ is the sought-after coordinates. For simplicity, we have set xµ(p), xµ ′ (p) = 0. Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Sketch of proof: II To 2nd order,( g′ ) p + ( ∂′g′ ) p x′ + ( ∂′∂′g′ ) p x′x′ = ( ∂x ∂x′ ∂x ∂x′ g ) p + ( ∂x ∂x′ ∂2x ∂x′∂x′ g + ∂x ∂x′ ∂x ∂x′ ∂′g ) p x′ + ( ∂x ∂x′ ∂3x ∂x′∂x′∂x′ g + ∂2x ∂x′∂x′ ∂2x ∂x′∂x′ g + ∂x ∂x′ ∂2x ∂x′∂x′ ∂′g + ∂x ∂x′ ∂x ∂x′ ∂′∂′g ) p x′x′ . (14) Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Sketch of proof: III zero-th order: L.H.S. g′ has 10 d.o.f., R.H.S. are determined by the matrix (∂xµ/∂xµ ′ )p, 4× 4 = 16 d.o.f., therefore there are enough d.o.f. to set gµ′ν′(p) into canonical form. The extra 6 d.o.f. actually correspond to the generators of Lorentz group in local chart; 1st-order: L.H.S ∂′σg′µν has 4× 10 = 40 d.o.f., while on R.H.S. additional freedom to choose (∂2xµ/∂xµ ′ 1∂xµ ′ 2)p, it’s 4× 10 (since µ′1, µ′2 are symmetric) so we have exactly enough d.o.f. to put ∂σgµν = 0; 2nd-order: L.H.S., ∂′ρ∂′σg′µν , 10× 10 = 100, while on R.H.S. (∂3xµ/∂xµ ′ 1∂xµ ′ 2∂xµ ′ 3)p has 4× 20 = 80 d.o.f. (since µ′1, µ′2, µ′3 are symmetric 6×5×4 3×2×1 = 20) so there are not enough d.o.f.. Actually, 20 d.o.f. characterize the deviation from flatness, in Riemann tensor;Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Intrinsic and extrinsic geometry Local geometry is an intrinsic property; Consider a flat sheet of paper, ds2 = dx2 + dy2 and a bug on it; If this sheet is then rolled up into a cylinder, the bug would not be able to detect any differences in the geometrical properties of the surface; The surface can simply be unrolled back to a flat surface w/o buckling, tearing or otherwise distorting it; More precisely, the metric of a cylinder is ds2 = dz2 + a2dφ2. (15) Let x = z, y = aφ, the metric change to the one of flat space; Therefore, the surface of a cylinder is not intrinsically curved; Its curvature is extrinsic; Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Another example: S2 It cannot be formed from a flat sheet of paper w/o tearing or deformation; Its intrinsic geometry is different; Its metric is ds2 = a2(dθ2 + sin2 θdφ2), (16) which cannot be transformed to Euclidean ds2 = dx2 + dy2 over the whole surface by any coord. transf. It has nonvanishing curvature ∝ 1/a; The summation of angles of a triangle is greater than pi; Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Example of non-Euclidean geometry Two sphere: x2 + y2 + z2 = a2 so we have dz = −xdx+ydyz . The metric on S2: ds2 = dx2 + dy2 + (xdx+ ydy)2 a2 − x2 − y2 . (17) Point A: x = y = 0; Near A: ds2 = dx2 + dy2; Spherical coordinates: x = ρ cosφ, y = ρ sinφ. Metric: ds2 = a 2dρ2 a2−ρ2 + ρ 2dφ2. Singularity? ρ = a, or √ x2 + y2 = a, which corresponds to the equator of the sphere; However, nothing wrong with the equator. It’s a harmless coord. singularity; Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Length and Area Length: the length between two points is LAB = ∫ B A ds = ∫ B A √ gµνdxµdxν . (18) It’s up to the path connecting A and B. For a path parametrized by xµ(λ), the length is LAB = ∫ λB λA √ gµν dxµ dλ dxν dλ dλ. (19) Area: for simplicity assuming that gµν(x) = 0, for µ 6= ν, then the metric is ds2 = g11(dx1)2 + · · ·+ gNN (dxN )2. The proper length is just √ g11dx 1, √ g22dx 2, so the area element is dA = √ |g11g22|dx1dx2. (20) Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up Example: S2 Let the metric be ds2 = a 2dρ2 a2−ρ2 + ρ 2dφ2. Namely gρρ = a 2 a2−ρ2 , gφφ = ρ 2. Consider a circle defined by ρ = R. (21) 1 From the pole to the perimeter, the length: D = ∫ R 0 a√ a2 − ρ2dρ = a sin −1(R/a). (22) 2 Circumference: C = ∫ 2pi 0 Rdφ = 2piR. 3 Area of the cap: A = ∫ 2pi 0 ∫ R 0 a√ a2 − ρ2 ρdρdφ = 2pia 2 ( 1− ( 1− R 2 a2 )1/2) . (23) Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up S2: cont. In terms of D, C = 2pia sin(D/a), A = 2pia2(1− cos(D/a)). (24) When D increases, C and A increase correspondingly until the point when D = pia2 , which is just the equator. A subtlety: Atot = 2A(R = a) = 4pia2, (25) which is the area of the sphere of radius a. Bin Chen GR course 8 logo JJJjjj Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up S2: cont. Euclidean geometry: CircumferenceRadius = C R = 2pi,∑ (Interior angles of a triangle) = pi. On S2, 1 ∑ (Interior angles of a triangle) = pi + Aa2 ; 2 C D = 2pi sin(D/a) D/a , which turns to 2pi as D << a. Bin Chen GR course 8 Ìá¸Ù Metric tensor Geometry of manifolds Riemann Normal coordinates Geometry from metric: warm up