Fundamental Cosmology: 4.General Relativistic Cosmology

Transcription

1 Fundamental Cosmology: 4.General Relativistic Cosmology Matter tells space how to curve. Space tells matter how to move. John Archibald Wheeler 1

2 4.1: Introduction to General Relativity Introduce the Tools of General Relativity Become familiar with the Tensor environment Look at how we can represent curved space times How is the geometry represented? How is the matter represented? Derive Einstein s famous equation Show how we arrive at the Friedmann Equations via General Relativity Examine the Geometry of the Universe Suggested Reading on General Relativity Introduction to Cosmology - Narlikar 199 Cosmology - The origin and Evolution of Cosmic StructureColes & Lucchin 1995 Principles of Modern Cosmology - Peebles 199 2

3 4.1: Introduction to General Relativity Newton v Einstein Newton: Mass tells Gravity how to make a Force Force tells mass how to accelerate Newton: Flat Euclidean Space Universal Frame of reference Einstein: Mass tells space how to curve Space tells mass how to move Einstein: Space can be curved Its all relative anyway!!

4 4.1: Introduction to General Relativity The Principle of Equivalence Recall: Equivalence Principle m g = m i A more general interpretation led Einstein to his theory of General Relativity PRINCIPLE OF EQUIVALENCE: An observer cannot distinguish between a local gravitational field and an equivalent uniform acceleration r a Implications of Principle of Equivalence Imagine 2 light beams in 2 boxes the first box being accelerated upwards the second in freefall in gravitational field For a box under acceleration Beam is bent downwards as box moves up For box in freefall photon path is bent down!! g = r a Fermat s Principle: Light travels the shortest distance between 2 points Euclid = straight line Under gravity - not straight line Space is not Euclidean! 4

5 4.2: Geometry, Metrics & Tensors The Interval (line element) in Euclidean Space: z ds ds dy dr rsinqdf dx ds dz rdq rsinq q dq r dr (dx 2 +dy 2 ) 1/2 f df y 2-D ds 2 = dx 2 + dy 2 x -D ds 2 = dx 2 + dy 2 + dz 2 ds 2-D ds 2 = dr 2 + r 2 dq 2 rdq -D ds 2 = dr 2 + r 2 dq 2 + r 2 Sin 2 qdf 2 dr 5

6 4.2: Geometry, Metrics & Tensors The Interval in Special Relativity (The Minkowski Metric) : The Laws of Physics are the same for all inertial Observers (frames of constant velocity) The speed of light, c, is a constant for all inertial Observers Events are characterized by 4 co-ordinates (t,x,y,z) Length Contraction, Time Dilation, Mass increase Space and Time are linked P The notion of SPACE-TIME ds 2 = c 2 dt 2 - (dx 2 + dy 2 + dz 2 ) ds (dx 2 +dy 2 +dz 2 ) 1/2 dt ds 2 = c 2 dt 2 - [ dr 2 + r 2 dq 2 + r 2 Sin 2 qdf 2 ] 6

7 4.2: Geometry, Metrics & Tensors The Interval in Special Relativity (The Minkowski Metric) : Causally Connected Events in Minkowski Spacetime t ds 2 = c 2 dt 2 - (dx 2 + dy 2 + dz 2 ) dx o Equation of a light ray, ds 2 =0, dt = ±c Trace out light cone from Observer in Minkowski S-T Spreading into the future Collapsing from the past ds 2 <0 ds 2 >0 ds 2 =0 ds 2 <0 o Area within light cone: ds 2 >0 Events that can affect observer in past,present,future. This is a timelike interval. Observer can be present at 2 events by selecting an appropriate speed. x ds 2 =0 ds 2 >0 o Area outside light cone: ds 2 <0 Events that are causally disconnected from observer. This is a spacelike interval. These events have no effect on observer. y 7

8 4.2: Geometry, Metrics & Tensors The Interval in General Relativity : o In General The interval is given by: ds 2 = Âg ij dx i dx j g ij is the metric tensor (Riemannian Tensor) : n i, j= 0 Tells us how to calculate the distance between 2 points in any given spacetime Components of g ij Multiplicative factors of differential displacements (dx i ) Generalized Pythagorean Theorem 8

9 4.2: Geometry, Metrics & Tensors The Interval in General Relativity : ds 2 = n Âg ij dx i dx j i, j= 0 Euclidean Metric dx 1 = x dx 2 = y dx = z ds 2 = dx 2 + dy 2 + dz 2 (n = 2) Dimensions g i, j = d i, j d i, j = 0 ( i j) g i, j = È Í Í Î Í Minkowski Metric ds 2 = cdt 2 - (dx 2 + dy 2 + dz 2 ) (n = ) 4 Dimensions dx 0 = t dx 2 = y dx 1 = x dx = z È Í g i, j = Í Í Í Î

10 4.2: Geometry, Metrics & Tensors A little bit about Tensors - 1 o Consider a matrix M=(m ij ) ds 2 = n Âg ij dx i dx j i, j= 0 o Tensor ~ arbitray number of indicies (rank) Scalar = zeroth rank Tensor Vector = 1st rank Tensor Matrix = 2nd rank Tensor : matrix is a tensor of type (1,1), i.e. m i j o Tensors can be categorized depending on certain transformation rules: Covariant Tensor: indices are low Contravariant Tensor: indices are high Tensor can be mixed rank (made from covariant and contravariant indices) Euclidean D space: Covariant = Contravariant = Cartesian Tensors m i jk mij k 10

11 4.2: Geometry, Metrics & Tensors A little bit about Tensors - 2 o Covariant Tensor Transformation: Consider Scalar quantity f. Scalar invariant under coordinate transformations f(x i ) = f(x' i ) e.g. f(x 1, x 2, x ) = f'(x' 1, x' 2, x' ) Taking the gradient f = f x ˆ 1 + f x ˆ 2 + f x ˆ x 1 x 2 x this normal has components A i = f x i Also have the relation Which should not change under coordinate transformation, so f' x' i = f x j x j x' i So for, A i = f x i A' i = 'f x' i A' i = f' x' i = f x' i A j = f x j etc A' i = 'f x' i = x j x' i f x j = x j x' i A j A' i = a i, j A j a i, j = x j x' i Generalize for 2nd rank Tensors, Covariant 2nd rank tensors are animals that transform as :- A' ij = x k x' i x l x' j A kl 11

12 4.2: Geometry, Metrics & Tensors A little bit about Tensors - o Contravariant Tensor Transformation: Consider a curve in space, parameterized by some value, r Coordinates of any point along the curve will be given by: x i (r) Tangent to the curve at any point is given by: A i = dx i dr The tangent to a curve should not change under coordinate transformation, so A' i = dx'i dr Also have the relation dx' i dr = x'i x j dx j dr So for, A i = dx i dr A' i = dx'i dr A j = dx j dr etc A' i = dx'i dr = x'i x j dx j dr = x'i x j A j A' i = a i, j A j a i, j = x'i x j Generalize for 2nd rank Tensors, Contravariant 2nd rank tensors are animals that transform as :- A' ij = x' i x k x' j x l A kl 12

13 4.2: Geometry, Metrics & Tensors More about Tensors - 4 o Index Gymnastics o Raising and Lowering indices: Transform covariant to contravariant form by multiplication by metric tensor, g i,j g ij A i = A j g ij A i = A j o Einstein Summation: Repeated indices (in sub and superscript) are implicitly summed over Âa i a i = a i a i i  k  i,k A ik A ik B k = A ik B k B i B k = A ik B i B k Einstein c.1916: "I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice..." n Thus, the metric ds 2 = Âg ij dx i dx j May be written as ds 2 = g ij dx i dx j i, j= 0 1

14 4.2: Geometry, Metrics & Tensors More about Tensors - 5 o Tensor Manipulation o Tensor Contraction: Set unlike indices equal and sum according to the Einstein summation convention. Contraction reduces the tensor rank by 2. A i B j For a second-rank tensor this is equivalent to the Scalar Product i = j A i B j = A 0 B 0 + A 1 B 1 + A 2 B 2 + A B Minkowski Spacetime (Special Relativity) g ij A i A j < 0 g ij A i A j > 0 g ij A i A j = 0 spacelike timelike null 14

15 4.2: Geometry, Metrics & Tensors More about Tensors - 6 o Tensor Calculus o The Covariant Derivative of a Tensor: Derivatives of a Scalar transform as a vector A i = f x i How about derivative of a Tensor?? Does A i x j transform as a Tensor? Lets see.. Take our definition of Covariant Tensor A' i = x j x' i A j Problem A' i x' m = x j x' i x n x' m A j x n + 2 x j x' m x' i A j Implies transformation coefficients vary with position Correction Factor: G ij k such that for, da i = G ij k A k dx j Correction Factor: A i x j Christoffel Symbols Defines parallel vectors at neighbouring points Parallelism Property: affine connection of S-T Christoffel Symbols = fn(spacetime) Redefine the covariant derivative of a tensor as; A i; j A i - G k x j ij A k A i, j - G k ij A k, normal derivative ; covariant derivative 15

16 4.2: Geometry, Metrics & Tensors Riemann Geometry higher rank fl extra Christoffel Symbol Covariant Differentiation of Metric Tensor g ik;l g ik,l - G il p g pk - G kl p g ip 1 General Relativity formulated in the non-euclidean Riemann Geometry Simplifications i G kl = G lk i g ik;l = 0 1 i 40 linear equations - ONE Unique Solution for Christoffel symbol G kl SPECIAL RELATIVITY ds 2 = c 2 dt 2 - (dx 2 + dy 2 + dz 2 ) ds 2 = g ij dx i dx j Ú path Ú ds = constant d ds = 0 path = gim ( 2 g + g - g mk,l ml,k kl,m) Shortest distance between 2 points is a straight line. But lines are not straight (because of the metric tensor) Gravity is a property of Spacetime, which may be curved Path of a free particle is a geodesic where d 2 x i ds 2 + G i kl dx k ds dx l ds = 0 16

17 4.2: Geometry, Metrics & Tensors The Riemann Tensor Recall correction factor for the transportation of components of parallel vectors between neighbouring points da i = G ik l A l dx k Such that we needed to define the Covariant derivative A i;k A i x - G l k ika l A i,k - G l ik A l To transport a vector without the result being dependent on the path, Require a vector A i (x k ) A such that - i x = G l k ika 1 l x n 1 2 A i x n x = k x n ( G l ik A l ) = G ik x n l A + G l A l l ik x = Ê G ik Á n Ë x n m + G ik l G ln m ˆ A m Interchange differentiation w.r.t x n & x k and use A i,nk =A i,kn Necessary condition for 1 0 = G m ik x n Can show: A i;nk - A i;kn R i m kn A m - G m in x k Where RHS=LHS=Tensor + G l ikg m ln - G l in G m m lk R i kn R i m kn is the Riemann Tensor Defines geometry of spacetime (=0 for flat spacetime). Has 256 components but reduces to 20 due to symmetries. 17

18 4.2: Geometry, Metrics & Tensors The Einstein Tensor Lowering the second index of the Riemann Tensor, define j g ij R klm = R iklm Contracting the Riemann Tensor gives the Ricci Tensor describing the curvature R kl = g im R iklm R m klm Contracting the Ricci Tensor gives the Scalar Curvature (Ricci Scalar) R = g kl R kl R i m kn is the Riemann Tensor Define the Einstein Tensor as G ik = R kl g ik R Symmetry properties of R iklm fi R iklm;n + R iknl;m + R ikmn;l = 0 Bianchi Identities Multyplying by g im g kn and using above relations Ê R ik g ˆ Á Ë ikr ;k 0 or The Einstein Tensor has ZERO divergence 18

19 4.: Einstein s s Theory of Gravity The Energy Momentum Tensor Define Energy Momentum Tensor T ik T ikï Ì Ó T 00 T i0 The flux of the i momentum component across a surface of constant x k Conservation of Energy and momentum - T k i;k =0 T ik Density of mass /unit vol. Density of j th component of momentum/unit vol. k-component of flux of j component of momentum o Non relativistic particles (Dust) r o = r o (x) - Proper density of the flow u i = dx i /dt - 4 velocity of the flow In comoving rest frame of N dust particles of number density, n Consider Units Density, r = mass/vol Momentum, p = energy/vol In general, T = p ƒ u = mnu ƒ u = ru ƒ u Vol T ik = T 00 = Âm N c 2 = nmc 2 = r o c 2 N Dust approximation World lines almost parallel Speeds non-relativistic Looks like classical Kinetic energy In any general Lorentz frame T ik = ru i u k 19

20 4.: Einstein s s Theory of Gravity The Energy Momentum Tensor o Relativistic particles (inc. photons, neutrinos) Ê Particle 4 Momentum, p i = E c, p r ˆ Á Ë E 2 = c 2 r p 2 + m 2 c 4 ª r p 2 c 2 Relativistic approximation Particles frequently colliding Pressure Speeds relativistic T 00 = Vol  E N = e e - energy density N T 11 = T 22 = T 22 = Vol  N r p 2 c 2 E ª 1 e 1/ since random directions o Perfect Fluid General Particle 4 Momentum, In rest frame of particle Vol T 00 =  gmc 2 ª rc 2 T 11 = T 22 = T 22 = N r p 0 = gmc 2 r p a = gmv r (a =1,2,) Vol  N g r p 2 c 2 ª P Looks like gas law Fluid approximation Particles colliding - Pressure Speeds nonrelativistic For any reference frame with fluid 4 velocity = u T ik = (P + rc 2 )u i u k - g ik P Pressure is important 20

21 4.: Einstein s s Theory of Gravity The Einstein Equation After trial and error!! Einstein proposed his famous equation Energy Momentum Tensor (matter in the Universe) Einstein Tensor (Geometry of the Universe) T ik = (P + rc 2 )u i u k - g ik P G ik = R kl g ik R G ik = k T ik = R kl g ikr Classical limit must reduce to Poisson s eqn. for gravitational potential 2 f= -4pGr k= 8pG c 4 Basically: Relativistic Poisson Equation G ik = R kl g ik R = 8pG c 4 T ik 21

22 4.: Einstein s s Theory of Gravity The Einstein Equation Matter tells space how to curve. Space tells matter how to move. Fabric of the Spacetime continuum and the energy of the matter within it are interwoven G ik = R kl g ik R = 8pG c 4 T ik Einstein inserted the Cosmological Constant term L Does not give a static solution!! G ik = R kl g ik R - Lg ik = 8pG c 4 T ik The Biggest Mistake of My Life 22

23 4.: Einstein s s Theory of Gravity Solutions of the metric ds 2 = g i, j dx i dx j Cosmological Principle: q  i, j= 0 Isotropy g a,b = d a,b, only take a=b terms q Homogeneity g 0,b = g 0,a = 0 just spatial coords ds 2 =  a,b =1 g a,b dx a dx b The Metric: -ds 2 = g 00 (dx 0 ) 2 + g ab dx a dx b = g 00 (dx 0 ) 2 + dl 2 Jim Muth Proper time interval: dt = -g 00 dx 0 c fi ds 2 = c 2 dt 2 - dl 2 How about g a,b dx a dx b? embed our curved, isotropic D space in hypothetical 4-space. our -space is a hypersurface defined by in 4 space. length element becomes: (x 1 ) 2 + (x 2 ) 2 + (x ) 2 + (x 4 ) 2 = a 2 dl 2 = (dx 1 ) 2 + (dx 2 ) 2 + (dx ) 2 + (dx 4 ) 2 Constrain dl to lie in -Space (eliminate x 4 ) dl 2 = (dx 1 ) 2 + (dx 2 ) 2 + (dx ) 2 + x1 dx 1 + x 2 dx 2 + x dx a 2 - (x 1 ) 2 - (x 2 ) 2 - (x ) 2 2

24 4.: Einstein s s Theory of Gravity Solutions of the metric x 1 = rsinq cosf x 2 = rsinq sinf x = rcosq Introduce spherical polar identities in Space dl 2 = dr 2 + r 2 sin 2 qdf 2 + r 2 dq 2 + r2 dr 2 a 2 - r 2 = dr 2 1- r 2 a 2 + r2 (dq 2 + sin 2 qdf 2 ) I. Multiply spatial part by arbitrary function of time R(t) : wont affect isotropy and homogeneity because only a f(t) II. Absorb elemental length into R(t) r becomes dimensionless III. Re-write a -2 = k ds Ê dr 2 ds 2 = c 2 dt 2 - R 2 (t) 1- kr + ˆ Á 2 r2 (dq 2 + sin 2 qdf 2 ) Ë The Robertson-Walker Metric r, q, f are co-moving coordinates and don t changed with time - They are SCALED by R(t) t is the cosmological proper time or cosmic time - measured by observer who sees universe expanding around him The co-ordinate, r, can be set such that k = -1, 0, +1 24

25 4.: Einstein s s Theory of Gravity The Robertson-Walker Metric and the Geometry of the Universe Ê dr 2 ds 2 = c 2 dt 2 - R 2 (t) 1- kr + ˆ Á 2 r2 (dq 2 + sin 2 qdf 2 ) The Robertson-Walker Metric Ë k ~ defines the curvature of space time k = 0 : Flat Space k = -1 : Hyperbolic Space k = +1 : Spherical Space 25

26 4.: Einstein s s Theory of Gravity The Robertson-Walker Metric and the Geometry of the Universe k = 0 : Flat Space k = -1 : Hyperbolic Space k = +1 : Spherical Space 26

27 4.: Einstein s s Theory of Gravity The Friedmann Equations in General Relativistic Cosmology Solve Einstein s equation : G ik = R kl g ik R = 8pG c 4 T ik Need: metric g ik energy tensor T ik Metric is given by R-W metric: Co-ordinates Ê dr 2 ds 2 = c 2 dt 2 - S 2 (t) 1- kr + ˆ Á 2 r2 (dq 2 + sin 2 qdf 2 ) Ë x 0 = ct x 1 = r x 2 = q x = f Use S so don t confuse with Tensors Non-zero g ik Non-zero G i kl g 00 =1 g 11 = - S 2 1- kr 2 g 22 = -S 2 r 2 g = -S 2 r 2 sin 2 q g ik = 1 g ik g 00 =1 -g = - Srsinq 1- kr 2 G 1 01 = G 2 02 = G 0 = S cs G 11 0 = S S c(1- kr 2 ) G 0 22 = S S r 2 c G 0 = S S r 2 sin 2 q c G 1 kr 11 = G 1 1- kr 2 22 = -r(1- kr 2 ) G 1 = -r(1- kr 2 )sin 2 q G 12 2 = G 1 = 1 r G 2 = -sinq cosq G 2 = cotq 27

28 4.: Einstein s s Theory of Gravity The Friedmann Equations in General Relativistic Cosmology Non-zero components of Ricci Tensor R ik R 0 0 = S R 1 c 2 1 = R 2 2 = R = 1 Ê S Á S c 2 S + 2S 2 Á Ë The Ricci Scalar R R = 6 Ê S c 2 S + S ˆ Á 2 + kc 2 Á S 2 Ë G 0 0 = R R 2 = - Ê ˆ Á S 2 + kc 2 c 2 Á S 2 Ë Non-zero components of Einstein Tensor G i k G 1 1 = G 2 2 = G R R 2 = - 1 Ê Á c 2 Á Ë 2 S S + S 2 + kc 2 S 2 + 2kc 2 S 2 ˆ ˆ Sub fl Einstein S 2 + kc 2 2 S S 2 S + S 2 = 8pG c T kc 2 S 2 = 8pG c T = 8pG c T = 8pG c T 2 28

29 4.: Einstein s s Theory of Gravity The Friedmann Equations in General Relativistic Cosmology Non-zero components of Energy Tensor T i k 1 2 S S + S 2 + kc 2 S 2 = 8pG c T = 8pG c T = 8pG c T 2 2 S 2 + kc 2 S 2 Assume Dust: P = 0 e = rc 2 = 8pG c T d(rs ) ds = e Ê S = 0 fi r = r o ˆ o Á Ë S =-P Ê fi T 0 0 = r o c 2 Á Ë 2 d(es ) dt ds S o S ˆ, T 1 1 = 0 + PS 2 = 0 actually implied by T k i;k =0 1 2 S S + S 2 + kc 2 S 2 = 0 2 S 2 + kc 2 S 2 = 8pGr o c 2 Ê Á Ë S o S ˆ 29

30 4.: Einstein s s Theory of Gravity The Friedmann Equations in General Relativistic Cosmology 2 S S + S 2 + kc 2 S 2 + kc 2 = 0 = 8pGr Ê o S o ˆ Ê R 1 2 Á r = r o o Á S 2 S 2 c 2 Ë S Ë R ˆ Return S(t)=R(t) notation Finally 2 R 2 = 8pGr R 2 - kc 2 Ê Á Ë + LR2 ˆ R = - 4pGr R Ê Á Ë + LR ˆ Our old friends the Friedmann Equations 0

31 4.4: Summary The Friedmann Equations in General Relativistic Cosmology We have come along way today!!! G ik = R kl g ik R = 8pG c 4 T ik Deriving the necessary components of The Einstein Field Equation Spacetime and the Energy within it are symbiotic The Einstein equation describes this relationship Ê dr 2 ds 2 = c 2 dt 2 - R 2 (t) 1- kr + ˆ Á 2 r2 (dq 2 + sin 2 qdf 2 ) Ë The Robertson-Walker Metric defines the geometry of the Universe R = - 4pGr R R 2 Ê Á Ë + LR = 8pGr R 2 - kc 2 ˆ Ê Á Ë + LR2 ˆ The Friedmann Equations describe the evolution of the Universe THESE WILL BE OUR TOOLBOX FOR OUR COSMOLOGICAL STUDIES 1

32 4.4: Summary Fundamental Cosmology 4. General Relativistic Cosmology Fundamental Cosmology 5. The Equation of state & the Evolution of the Universe 2