Cosmological Space-Times

Transcript

1 Cosmological Space-Times Lecture notes compiled by Geoff Bicknell based primarily on: Sean Carroll: An Introduction to General Relativity plus additional material 1

2 Metric of special relativity ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2 = dx 02 + dx 2 + dy 2 + dz 2 = η µν dx µ dx ν where η µν = Minkowski tensor = diag [ 1, 1, 1, 1] µ, ν = 0, 1, 2, 3 This is the metric of four-dimensional flat space time Generalised by Einstein in his 1916 General Theory of Relativity to: ds 2 = g µν dx µ dx ν Metric tensor 2

3 General relativity field equations g µν Christoffel Symbols Γ σ µν = 1 2 gσρ (g νρ,µ + g ρµ,ν g µν,ρ ) g σρ = Inverse of g µν The Christoffel symbols appear in the equations of test particles: Geodesics of space time - and also in generalised (covariant) derivatives Riemann curvature tensor: R ρ σµν = Γ ρ νσ,µ Γ ρ µσ,ν + Γ ρ µλ Γλ νσ Γ ρ νλ Γλ µσ 3

4 Tensors derived by contraction over indices Ricci tensor Ricci scalar R µν = R λ µλν R = g µν R µν Einstein tensor G µν = R µν 1 2 g µνr 4

5 Einstein s field equations Newton s Constant of gravitation G µν = Λg µν + 8πG c 4 T µν Cosmological constant Dark energy Matter tensor Matter tensor T µν =(ρc 2 + p)u µ U ν + pg µν 4-velocity of matter U µ = dxµ ds = 1 c dx µ dτ 5

6 Metric of the Universe Homogeneity and isotropy => Geometry invariant under translations and rotations => Maximally symmetric space time ds 2 = c 2 dt 2 + a 2 (t) [e 2β(r) dr 2 + r 2 dθ 2 + r 2 sin 2 dφ 2] Spatial part of metric: dσ 2 = a 2 (t) [e 2β(r) dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2] 6

7 When e 2β =1 dσ 2 = a 2 (t) [ dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2] which is the normal metric of flat space modified by the scale factor a(t) The scale factor informs us how the universe is expanding In this space-time metric the coordinates are comoving coordinates, i.e. as the Universe expands the spatial coordinates of galaxies remain constant 7

8 Space-time geometry Neighbouring world lines 3D hypersurface t = t 1 ds 2 =a 2 (t 1 ) γ ij du i du j 3D hypersurface t = t 0 ds 2 =a 2 (t 0 ) γ ij du i du j Comoving observers u i =constant Geometry of 3D hypersurfaces [e dσ 2 = a 2 (t) 2β(r) dr 2 + r 2 dω 2] 8

9 Maximally symmetric spaces (consequence of homogeneity and isotropy) Characterised by (3) R ijkl = k(γ ik γ jl γ il γ jk ) R ij = 2kγ ij Spatial metric γ ij = diag(e 2β(r),r 2,r 2 sin 2 θ) Equations for metric tensor (r (3) R 11 = e 2β β ) r = 2kγ 11 =2ke 2β (r (3) R 22 = e 2β β ) r = 2kr 2 [e ( (3) R 33 = 2β r β ) ] r 1 +1 sin 2 θ =2kr 2 sin 2 θ 9

10 Solution dω 2 = dθ 2 + sin 2 θdφ 2 e 2β 1 = 1 kr 2 [ ] dσ 2 = a 2 dr 2 (t) 1 kr 2 + r2 dω 2 Coordinate transformation r 2 = k r 2 r = k 1/2 r [ dσ 2 = a2 (t) k dr 2 ] 1 sgn(k)r 2 + r 2 dω 2 10

11 Absorb k 1/2 into a(t); k = -1, 0, 1 e 2β 1 = 1 kr 2 [ ] dσ 2 = a 2 dr 2 (t) 1 kr 2 + r2 dω 2 k = 1, 0, +1 k=0 dσ 2 = a 2 (t) [ dr 2 + r 2 dω 2] Expanding flat space 11

12 k=-1 dσ 2 = a 2 (t) [ ] dr 2 1+r 2 + r2 dω 2 New radial variable dχ = dr (1 + r 2 ) 1/2 χ = sinh 1 r r = sinh χ Metric of each 3D hypersurface dσ 2 = a 2 [ (t) dχ 2 + sinh 2 χdω 2] 12

13 k=+1 dσ 2 = a 2 (t) [ ] dr 2 1 r 2 + r2 dω 2 New radial variable dχ = dr 2 (1 r 2 ) 1/2 χ = sin 1 r r = sin χ Metric of each 3D hypersurface dσ 2 = a 2 (t) [ dχ 2 + sin 2 χdω 2] 13

14 Summary of 3D metrics dσ 2 = a 2 (t) [ dχ 2 + S 2 (χ)dω 2] S(χ) = χ k =0 S(χ) = sinh(χ) k = 1 S(χ) = sin χ k = +1 What geometry do these metrics represent? k=0 => Metric of an expanding flat space 14

15 k=+1 Consider a 3-sphere embedded in a 4-dimensional Euclidean space (not space-time) Let the equation of the sphere in (w,x,y,z) space be: w 2 + x 2 + y 2 + z 2 = a 2 The metric of the 4-dimensional space is: dσ 2 = dw 2 + dx 2 + dy 2 + dz 2 15

16 Metric of the surface of the 3-sphere Consider the following set of spherical polars in 4-space; these provide a parametric description of the surface of the 3-sphere which has radius a(t). There are 3 angular parameters. w = a cos χ z = a sin χ cos θ x = a sin χ sin θ cos φ y = a sin χ sin θ sin φ We now determine the metric of the surface of the sphere by determining the differentials of the coordinates w, x, y and z. 16

17 Euclidean metric restricted to 3-sphere These are the differentials of w,x,y,z in terms of the polar angles dw = a sin χdχ dz = a cos χ cos θ dχ a sin χ sin θ dθ dx = a cos χ sin θ cos φ dχ + a sin χ cos θ cos φ dθ a sin χ sin θ sin φ dφ dx = a cos χ sin θ sin φ dχ + a sin χ cos θ sin φ dθ + a sin χ sin θ cos φ dφ This gives: dw 2 + dx 2 + dy 2 + dz 2 = a 2 (t) [ dχ 2 + sin 2 χ(dθ 2 + sin 2 θdφ 2 ) ] which is the spatial part of the space-time metric 17

18 Conclusions for k=+1: 1. The 3-space of this metric can be thought of a as a 3-sphere of radius a(t) embedded in a 4 dimensional Euclidean space 2. The 3-sphere is expanding 3. Since a 3-sphere is closed the k=1 metric represents a closed Universe 18

19 Consider section y=0: Embedding diagram y = a sin χ sin θ sin φ =0 w φ = 0 or π φ = 0 section a x w = a cos χ z = a sin χ cos θ x = a sin χ sin θ φ = π section w = a cos χ z Embedding of a 3-sphere in a 4-dimensional Euclidean space z = a sin χ cos θ x = a sin χ sin θ 19

20 The case k = -1 Consider the equation of a 3-hyperboloid embedded in a 4- dimensional Euclidean space: w 2 x 2 y 2 z 2 = a 2 We can parametrically express this in terms of hyperspherical polars w = a cosh χ z = a sinh χ cos θ x = a sinh χ sin θ cos φ y = a sinh χ sin θ sin φ 20

21 Differentials: dw = a sinh χ dχ dz = a cosh χ cos θ dχ a sinh χ sin θdθ dx = a cosh χ sin θ cos φdχ + a sinh χ cos θ cos φ dθ a sinh χ sin θ sin φ dy = a cosh χ sin θ sin φdχ + a sinh χ cos θ sin φ dθ + a sinh χ sin θ cos φ Metric restricted to 3-hyperboloid dσ 2 = dw 2 + dx 2 + dy 2 + dz 2 = a 2 (t) [ dχ 2 + sinh 2 χ(dθ 2 + sin 2 θdφ 2 ) ] 21

22 Embedding: y = 0 section y =0 sin φ =0 φ = 0 or π φ=π θ χ θ=0 w θ χ φ=0 a x Embedding of a 3-dimensional space of negative curvature in a 4-dimensional Minkowskian space z 22

23 Summary The metric of the expanding Universe can be expressed in one of the 3 following ways: ds 2 = c 2 dt 2 + a 2 (t) [ dχ 2 + χ 2 dω 2] k=0 Infinite flat Universe ds 2 = c 2 dt 2 + a 2 (t) [ dχ 2 + sin 2 χdω 2] k=1 Finite closed Universe ds 2 = c 2 dt 2 + a 2 (t) [ dχ 2 + sinh 2 χdω 2] k=-1 Infinite, open Universe 23