Metrics and Curvature

Transcription

1 Metrics and Curvature How to measure curvature? Metrics Euclidian/Minkowski Curved spaces General 4 dimensional space Cosmological principle Homogeneity and isotropy: evidence Robertson-Walker metrics Parallel transport Affine connection Torsion and curvature tensors Compatibility with the metric Interpretations of the curvature tensor Normal (Gaussian) coordinates = free falling system Deviation of geodesics Failure of parallel transport Phys Metrics 1

2 How to Measure Curvature? Cannot step out of space! We do not want to immerse our 4-dimensional space into a bigger space Intrinsic measurement of curvature For an intuitive picture, take the (ordinary) 2-sphere Sum of angles of a triangle(constructed with geodesics) 180 => Failure of Pythagoras theorem Deviation of parallel geodesics : 5th Euclid axiom not valid Failure of parallel transport along a closed loop: vector does not come back onto itself Variation of surface and volume with distance e.g. 1/r 2 law Classical tests of cosmology: Standard candle Evolution of galaxy density with distance => We need to measure distances Phys Metrics 2

3 Space = R 3 Euclidian Space Time=R Spatial Distance Cartesian coordinates Along a curve Change of coordinates s 2 = x 2 + y 2 + y 2 ds with ds 2 = dx 2 + dy 2 + dy 2 x, y, z u,v,w. Replace dx, dy,dz by dx = du + dv + dw dy =... dz =... u v w Example: Spherical coordinates x = rsin cos y = rsin sin x = rcos ds 2 = dr 2 + r 2 d 2 + r 2 sin 2 d 2 Homogeneity and isotropy of space Homogeneity = distance independent of origin Isotropy= distance independent of direction of axis = invariance under translation of origin (which can depend on time!) => velocities add! and rotation of axes => Galileo group Phys Metrics 3

4 Phys Metrics 4 (Pseudo)Metric Metrics (pseudo) scalar product of differential vectors (linear+symmetric) dx.dy = g dx dy (Summation on repeated indices) => definition of a (pseudo) length Geodesics: length along trajectory x (s )is an extremum A Euclidian space Space B ±g dx ds dx v ds ds Time independent and absolute Minkowski : Light velocity is constant Space and time are mixed in a 4-dimensional space d 2 = d d = d o with o = t τ is called the proper time= time in rest frame Lorentz transformation : keep the metric invariant B A Depends on type of curve: 0 or <0 length R 3 d 2 = g dx dx g = = [ ] 2 1 ( d 1 ) 2 + ( d 2 ) 2 + ( d 3 ) 2 c 2 1 = x 2 = y d 2 = g dx dx g = real symmetric 3 = z c 2 = c c 2

5 A Prototype of Curved Space 2 dimensional sphere The radius is R. Describe by the polar coordinate of the projection on the equatorial plane. If we imbed it in the 3 dimensional space, and adopt the same metric: Phys Metrics 5 Note: two different coordinate systems are necessary to describe the entire space 3 dimensional homogeneous and isotropic space homogeneous: no special point= invariant under translations Isotropic: no preferred direction= invariant under rotations Most general: z r d 2 = d 2 = dr 2 + r 2 d 2 + dz 2 r 2 + z 2 = R 2 z = R 2 r 2 dz = d 2 = 1 1 r2 R r 2 R 2 dr 2 + r 2 d 2 dr 2 + r 2 d 2 + r 2 sin 2 d 2 = ±1 depending on sign of curvature rdr R 2 r 2

6 General 4-Dimensional Spaces Curved space-time: 4 dimensional manifold P described locally by chart R 4 : x points on manifold P metric of proper signature (same as ) (i.e. number of positive and negative eigenvalues) d 2 = g dx dx One can change chart or frame x d 2 = g dx dx = d d = g Note that locally we can arrange that = Diagonalize Renormalize This is a free fall system: only safe way to know meaning of coordinates Cannot in general be global g oo dt 2 2 = dt ff g 11 dx 12 = dx 1 2 ff what we call time and space! There is no function x ( x ff ) ff ff v ff ff Phys Metrics 6

7 Cosmological Principe Space ( at high enough scale) is homogeneous and isotropic Evidence 1: Cosmic Microwave Background Full scale T/T =1 Full scale T/T =10-2 Full scale T/T =10-4 Subtract off galaxy Phys Metrics 7 S/N 3

8 Cosmological Principle 2 Evidence 2: Structure at large scale e.g. Las Campanas redshift survey astro-ph/ The Universe on Very Large Scales: A View from the Las Campanas Redshift Survey D.L. Tucker (FNAL), H. Lin (Steward Obs.), S. Shectman (OCIW) Phys Metrics 8

9 Robertson-Walker Metric Theorem in Differential Geometry / General Relativity Most general 4-dimensional homogeneous and isotropic negative metric space can be parametrized such that the metric is d 2 = dt 2 a2 (t) 1 c 2 1 with positive (closed) or negative (open). Proof: - g oi terms can be cancelled by clock synchronization - curvature can only depend of time. Take time out by considering comoving coordinates r2 R 2 d phys = a( t)d com dr 2 + r 2 d 2 + r 2 sin 2 d 2 a(t) Phys Metrics 9 a(t) is called the scale parameter. Hubble constant : v = d a phys = t a t d a phys H = ( t) a( t) Not constant! t

10 t2 + t 2 t2 t t1 Redshift Consider light rays emitted at comoving point r, and time t 1 and received by the observer at the origin and time t 2. t1+ t1 Geodesics passing through the origin are straight: The null square line element is then: Integrating t2 t 1 dt a( t) dt a( t) = 1 c d 2 = dt 2 a2 (t) 1 c 2 1 r 2 dr 2 = 0 1 R 2 dr (dr < 0!) 1 r2 R = dr = independent of t 1 r c 1 r2 R 2 In particular, a signal emitted at time t 1 t 1 will be received at time t 2 t 2 such that r (comoving coordinate) t2 dt = t 1 a( t) t2+ t2 t 1 + t 1 dt a( t) Redshift or for small time intervals t 2 t 1 = 0 a t 2 a t 1 Phys Metrics 10 in particular light emitted at wave length λ 1 is observed at wave length λ z = a t 2 a t 1 2 = a t 2 a t 1 1 or defining z = <= Dilatation of space ratio of scale parameters!

11 Distances 1 Proper distance of object at redshift z For photon emitted d p Comoving distance dr 1 r2 R 2 = c dt a t r( z ) dr t o dt a o da z dz ( z) = a o 0 1 r = ca 2 o = ca t e a( t) o a a = ca a o e 2 0 a R 2 a a z We need to know evolution of a(t) a o r = a o R sin sinh where the sin or sinh has to be used depending whether ε>0 or <0 d p ( z) a o R Phys Metrics 11

12 Distances 2 Apparent angle of object of size x. Angular distance Goto to comoving coordinates with origin at observer r = x = x a e a e r d A = x = a er = a or 1+ z Apparent luminosity. Luminosity distance Assume that the object has absolute luminosity L over 4π Two effects: Solid angle subtended by the telescope of area A simplest is to go to comoving system with the origin at time of emission Ω e = A a o 2 / r 2 Phys Metrics 12 Power P received is decreased by the redshift of the photons and by the decrease of number by unit time 2 factors of 1+z l = P A = L 4πa 2 o r z 1/ 2 L Luminisoity distance d L = 4πl 2 = a o r( 1 + z) = ( 1 + z) 2 d A

13 Parallel Transport World lines= trajectories = curves on the manifold { } : x Tangent space at one point: Space of velocities: u = dx d Parallel transport Let us move along a curve with velocity u(τ) Variation of quantities along curve? Phys Metrics 13 Starting from r u x r r u V u For scalar no problem df ( x) d r f x u But for vectors (in tangent spaces) tangent spaces are not the same! we have to specify how the bases vary = velocity and vectorv r ( x) = V ( x) r D V r Dx u V r ; e { } = u V = u f ( x) r e + u V = dx d e x f ( x) r e { = u V Γ e r r e + u V Γ e r Affine connection

14 Parallel Transport 2 A vector is parallel transported along u when r r u V = 0 Geodesics = autoparallel u r r u d 2 x d 2 = 0 Advantage Explicitly covariant : Covariant derivative By definition, a vector U µ in a space tangent to a manifold (surface) changes under the change of coordinates x µ to ξ ρ (dx µ is a vector!): U' = U DU Dx Phys Metrics 14 = U D U + Γ U = DU = U D Dx Note the Γ are not tensors Γ = Γ + 2 x Therefore they can be transformed to zero= Free falling frame! + Γ U

15 Affine Connections What is second order covariant derivative with respect to d 2 x d 2 = d d dx d = dx d D Dx dx d = component of uu The laws of motion of free particles will be generalized u u = 0 d 2 x + Γ dx dx = 0 d 2 d d Geodesics! An example: accelerated frame in Newtonian Mechanics e.g. if origin is at R and instantaneous rotation is ω: ma r rot = F r r m d2 R r dt 2 m r 2m r v r rot 3 m r r ( r ) inertial Coriolis Centrifugal we had to take into account derivatives of the base vectors Phys Metrics 15

16 Torsion and Curvature Tensors Phys Metrics 16 Definitions Torsion Curvature Properties Obviously tensors T( X,Y) = X Y Y X [X,Y] where [X,Y] = ( X i i Y j Y i i X j )e j R( X,Y)Z = X ( Y Z) Y ( X Z) [ X,Y ] Z T( X,Y) = T( Y, X) R( X,Y ) = R( Y, X) T( fx,gy) = fgt( X,Y) R( fx,gy) ( hz) = fghr X,Y Z Note that the last property comes from the commutator term in the definitions = Γ k k ij Γ ji T e i,e j e k T = 0 Γ ij k = Γ ji k In general relativity we will consider only torsion free connections! It is long but straightforward to show that R = Γ v, Γ, + Γ Γ Γ Γ with R( e,e )e = R e

17 Compatibility with Metrics How does g µv transforms? g U V should be a scalar In order to have D g U V = g U V Dx with the usual chain differentiation rule, we should have Dg = g g Γ g Γ Dx General transformation of covariant tensor! Compatibility with metrics Parallel transported metric should be the same as the original metric (the scalar product should be invariant) u r g = 0 r u Dg Dx = 0 Phys Metrics 17

18 Compatibility with Metrics 2 Unique (for torsion free connections) g With zero torsion, this is equivalent to = 1 2 g g + g g Γ g Γ = 0 with ( g ) = ( g ) 1 g Proof: Using circular permutations of the invariance of g under parallel transport g v g g Γ g Γ = 0 Adding first two and subtracting last one,and using symmetries (T =0!) we get the resul g Γ g Γ = 0 Phys Metrics 18 The affine connection is given in terms of the first derivative of the metric tensor Geodesics Autoparallels are also extrema of proper time. Proof: cf Weinberg 3.4

19 Normal (Gaussian) Coordinates Locally we can arrange that the s vanish In the neighborhood of a point p on the manifold every point can be reached with a geodesic of appropriate initial velocity <= solution of second order equation u u = 0 Phys Metrics 19 Label points along the geodesic of initial velocity u with time τ: x = u Because the equation is linear, we can choose a base for the u s, and construct a coordinate system (a local chart): x = u Exponential mapping, normal or Gaussian coordinates, in GR free falling or inertial frames in d 2 x + Γ dx dx = 0 d 2 x = 0 Γ u u = 0 d 2 d d d 2 Γ + Γ v = 0 or if the torsion is zero Γ = 0 Hence the curvature of space is not encoded in the s We have to go to one more order in differentiating the g. Curvature tensor!

20 Geodesic Deviation n Geodesic deviation equation (no torsion) Let us consider a family of geodesics,n. Call u the velocity along the geodesics and η their distance : u = (,n) n = (,n) n Theorem: u u = R( u, )u j = j or 2 u u = R u u Phys Metrics 20 Proof 1) [ u, ]= 0 Go to coordinates u = [ u, ] = 2 x n n 2 x = n 2) u u = 0 geodesics 2 3) u u ( u )= u ( u) zero torsion add zero terms u u [ u, ] u 2 u =Ru, u = 0 because of commutation of derivatives

21 v p u Phys Metrics 21 Interpretation of Curvature Tensor R=Riemann curvature tensor constructed from metric, and its first and second derivatives: Γ s are not sufficient because they can be locally transformed away! Complex! n 2 ( n 2 1) 12 n=2: 1 n=3: in each direction, distance and angle:3x2=6 n=4: 20! Similarly explains failure of parallel transport Curvature prevents parallel transport to be independent of the path q Similar demonstration Consider a family of curves u = (,n) =,n with = 0 at p n 1) as before [ u, ]= 0 2) vector v is parallel transported u v = 0 independent components and start in well defined position at p : v = 0 at p 3) if curvature vanishes u v (,n) (no more geodesics) going from point p to point q = u v = 0 Derivative with respect to =0 at origin + its derivative with respect to u = 0 =>stays 0 Independent of path! 4) If parallel displacement is independent of path u v u v =R(u, )v=0