The first part of this course will cover the foundational material of homogeneous big bang cosmology. There are three basic topics: 1. General Relativity 2. Cosmological Models with Idealized Matter 3. Cosmological Models with Understood Matter 1 General Relativity References: • Landau and Lifshitz, Volume 2: The Classical Theory of Fields • Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity • Misner, Thorne, and Wheeler, Gravitation 1.1 Transformations and Metrics . 1 1.2 Covariant Derivatives: Affine Structure . 3 1.3 Covariant Derivatives: Metric . 3 1.4 Invariant Measure . 4 1.5 Curvature . 4 1.6 Invariant Actions . 6 1.7 Field Equations . 7 1.8 Newtonian Limit . 10 This will be a terse introduction to general relativity. It will be logically complete, and adequate for out later purposes, but a lot of good stuff is left out (astrophysical applications, tests, black holes, gravitational radiation, . ). 1.1 Transformations and Metrics We want equations that are independent of coordinates. More precisely, we want them to be invariant under “smooth” reparameterizations x�µ = x�µ(x) To do local physics we need derivatives. Of course ∂x�µ dx�µ = dxν (Note: summation convention) ∂xν ∂x�µ ≡ Rµ ∈ GL(4) (invertible real matrix) ∂xν ν 1 We want to have special relativity in small empty regions, so we must introduce more structure. The right thing to do is to introduce a symmetric, non-singular (signature + – – –), tensor field gµν (x) so that we can define intervals 2 µ ν ds = gµν dx dx (“Pythagoras”) � �µ �ν µ ν For this to be invariant (gµν dx dx = gµν dx dx ), we need � � −1 α −1 β gµν (x ) = (R ) µ(R ) ν gαβ(x) µ ∂x�µ −1 ρ ∂xρ α ∂x�α ∂xλ ∂x�α Since R ν = ∂xν , (R ) σ = ∂x�σ (chain rule: δβ = ∂x�β = ∂x�β ∂xλ ) We can write the transformation law for gµν in matrix form: � −1 −1 T G = R G(R ) From linear algebra, we can insure G� is diagonal with ±1 (or 0) entries. The signature, e.g. � 1 � −1 −1 , is determined. −1 There are residual transformations that leave this form of gµν intact. They are the Lorentz transformations! µ Generalizing gµν , dx , we define tensors of more general kinds ν1...νn Tµ1...µm (x) by the transformation law � ν1...νn � −1 α1 −1 αm ν1 νn β1...βn T (x ) = (R ) · · · (R ) R · · · R · T (x) µ1...µm µ1 µm β1 βn α1...αm µν Example: Inverse metric g µα µ g gαν = δν Operations: • Muliplication by number • Tensor, of the same type can be added • Outer Product ν1 ν2 ν1ν2 V W µ ≡ T µ • Contraction µ1ν2...νn ˜ ν2...νm Tµ 1µ2...µm = Tµ2...µm Example: α β αβ gµν dx dx = Tµν a tensor µν 2 Contraction: Tµν = ds µ “0” is a tensor (all components = 0). δν is a tensor. 2 1.2 Covariant Derivatives: Affine Structure � � • Scalar Field φ (x ) = φ(x) � � ∂xα � −1 α � • Vector Aµ(x ) = ∂x�µ Aα(x) (R ) µAα � ∂ ∂xα ∂ • Operator ∂ν ≡ ∂x�ν = ∂x�ν ∂xα Is there an invariant derivative? ∂xα ∂ � ∂xβ � ∂� A� = A ν µ ∂x�ν ∂xα ∂x�µ β ∂xα ∂xβ ∂2xβ = ∂ A + A ∂x�ν ∂x�µ α β ∂x�ν ∂x�µ β � �� � � �� � good bad (hard to use ­ not a tensor) λ Add correction term: �ν Aµ ≡ ∂ν Aµ − ΓνµAλ: 2 σ ∂ x � �� A� = Sα Sβ ∂ A + A − Γ λ Sσ A ν µ ν µ α β ∂x�µ∂x�ν σ νµ λ σ ? α β � σ � = S ν S µ ∂αAβ − ΓαβAσ α −1 α ∂xα where S ν ≡ (R ) ν = ∂x�ν . This will work if 2 σ � ∂ x Γ λ Sσ = Sα Sβ Γσ + νµ λ ν µ αβ ∂x�ν∂x�µ �λ 2 σ � ∂x ∂ x that is, Γ λ = Rλ Sα Sβ Γσ + νµ σ ν µ αβ ∂xσ ∂x�ν∂x�µ λ λ Note that the inhomogeneous part is symmetric in µ ↔ ν. So we can assume Γαβ = Γβα consistently. (The antisymmetric “torsion” part is a tensor on its own!) Given Γ, we can take covariant derivatives as ν1...νn ν1...νn λ ν1...νn λ ν1...νn ν1 λν2...νn � T = ∂ T −Γ T −. .−Γ T +Γ T α µ1...µm α µ1...µm αµ1 λµ2...µm αµm µ1...λ αλ µ1µ2...µm This gives a tensor. We use the Leibniz rule in products. 1.3 Covariant Derivatives: Metric Big result: given a metric, there is a unique preferred connection Γ Demand �λgµν = 0 (and symmetry) α α 0 = ∂λgµν − Γλµgαν − Γλνgαµ � �� �� α � �� α 0 = ∂µg − Γλµgαν −� Γµν g νλ �� αλ �� α � � α 0 = ∂ν gλµ − Γλν gαµ − Γµν gαλ 3 Subtract the first line from the sum of the second and third: α 2Γµν gαλ = ∂µgλν + ∂ν gλµ − ∂λgµν λβ g β 1 λβ � � × : Γ = g ∂µg + ∂ν g − ∂λg 2 µν 2 λν λµ µν Conversely, this works! Extra term in Γ�: 1 �� ∂xσ ��� � ∂xσ � g λβ ∂� ��� g + ∂� g 2 ��µ ∂x�λ σν µ ∂x�ν λσ � σ � � � σ � � σ � � � σ � � � ∂x �� � ∂x � ∂x �� � ∂x �� � + ∂ �� g + ∂ g − ∂ �� g − ∂ �� g ��ν ∂x�λ σµ ν ∂x�µ λσ ��λ ∂x�µ σν ��λ ∂x�ν σµ The boxed terms give the desired inhomogeneous terms; the others cancel. 1.4 Invariant Measure �∂x�µ � d4 x� = det d4 x ∂xν � �� � Jacobian; = det R α β � ∂x ∂x g = g µν ∂x�µ ∂x�ν αβ � −1 − T G = R G(R 1) � 1 det gµν = det gµν (det R)2 Write g ≡ det(gµν ); then � 4 � √ 4 g�d x = gd x is an invariant measure. 1.5 Curvature In order to make the gµν into dynamical variables, we want them to appear with derivatives in the Lagrangian. Can we form tensors such that this occurs? Trick: (� � − � � ) A ≡ − Rα · A µ ν ν µ β βµν α � �� � ���� involves g no derivatives α This R βµν automatically transforms as a proper tensor; the “hard” part is to show that the derivatives on Aα all cancel so we get this form. λ �µAβ = ∂µAβ − Γνβ Aλ σ ��� σ �µ(�ν Aβ ) = ∂µ(�ν Aβ ) − �Γµν��σAβ −Γµβ �ν Aσ � �� � symm etric ⇒ drop it �� σ σ ρ σ = �∂µ�∂ν Aβ − ∂µ (ΓνλAσ) − Γµν ∂ν Aσ + Γµβ ΓνρAσ σ σ σ ρ σ − ∂µΓνλAσ − Γνλ∂µAσ − Γµλ∂ν Aσ + Γµβ ΓνρAσ α α α α ρ α ρ so R βµν = ∂µΓνβ − ∂ν Γµβ + ΓµρΓνβ − ΓνρΓµβ 4 Symmetry properties of Rαβγδ ∂gµν We can go to a frame where ∂xλ = 0 (at point of interest). These are “geodesic coordinates”. α There we know that Γβγ = 0 (but not its derivatives!) Then 1 Rα = �∂ �g ασ �∂ g − ∂ g − ∂ g � − (µ ↔ ν)�� βµν 2 µ ν σβ β νσ σ βν 1 = g ασ �∂ ∂ g − ∂ ∂ g − ∂ ∂ g − ∂ ∂ g � 2 µ β νβ µ σ βν ν β µσ ν σ βµ ασ = g Rσβγδ 1 � � with Rαβγδ = 2 ∂α∂δgβγ + ∂β∂γ gαδ − ∂α∂γ gβδ − ∂β∂δgαγ ⇒ Rαβγδ = −Rβαγδ Rαβγδ = −Rαβδγ Rαβγδ = Rγδαβ Rαβγδ + Rαγδβ + Rαδβγ = 0 (e.g. Look at the coefficient of ∂α∂δgβγ : +1 in Rαβγδ −1 in Rαγδβ 0 in Rαδβγ ) Since these are tensor identities, they hold in any frame! Also notable is the Bianchi identity; �αRµνβγ + �β Rµνγα + �γ Rµναβ = 0 It follows from: [�α, [�β, �γ ]] + [�β , [�γ , �α]] + [�γ , [�α, �β]] = �α�β�γ − �α�γ �β − �β �γ �α + �γ �β �α + �β�γ �α − �β �α�γ − �γ �α�β + �α�γ �β + �γ �α�β − �γ �β �α − �α�β�γ + �β �α�γ = 0 e.g.: (1) (2) � ν � ν ν �α[�β, �γ ]Aµ = −�α R µβγ Aν = −�αR µβγ Aν − R µβγ �αAν ν ν −[�β , �γ ]�αAµ = R αβγ �ν Aµ + R µβγ �αAν (3) (4) ν ν ν (2) cancels against (4), (3) will go away by the symmetry of R αβγ + R βγα + R γαβ = 0, so (1) generates the Bianchi identity. This identity is the gravity analogue of ∂αFβγ + ∂βFγα + ∂γ Fαβ = 0 ∂B in electromagnetism, or � · B = 0, � × E = − ∂t (existence of vector potential). 5 1.6 Invariant Actions � √ 4 Since we have an invariant measure gd x (L ), we get invariant field theories by putting invariant expressions inside (L ). Given a special-relativistic invariant theory, we just need to change 4 √ 4 d x → gd x ηµν → gµν ∂µ → �µ to make a general-relativistic invariant theory. This is the “minimal coupling” procedure. Examples: 1 µν 1. Scalar field: L = 2 g ∂µφ ∂ν φ − V (φ) 2. Transverse vector field σ σ Fµν = �µAν − �ν Aµ = ∂µAν − ΓµνAσ − ∂ν Aµ + ΓνµAσ = ∂µAν − ∂ν Aµ (no need for Γ, or ∂g) 1 L = − g αγ g βδF F 4 αβ γδ supports gauge symmetry Aµ → Aµ + ∂µχ 3. Longitudinal vector field (instructive) µ µ µ α �µA = ∂µA + ΓµαA 1 �� Γµ = gµσ �∂ g + ∂ g ��−� ∂ �g � µα 2 α σµ �µ�ασ σ µα 1 = gµσ∂ g 2 α σµ 1 √ = √ ∂ g g α µσ [To prove ∂αg = gg ∂αgµσ , use expansion by minors and expansion for inverse matrix. Check on diagonal matrices!] √ √ √ µ √1 � µ � � 4 µ � 4 µ So �µA = g ∂µ gA . Thus d x g �µA = d x ∂µ( ga ) is semi-trivial: it is a boundary term. � 4 µ ν d x�µA �ν A gives dynamics. This supports a gauge transformation √ µ √ µ µνρσ gA → gA + � ∂ν Λρσ Λρσ = −Λσρ 4. Gravity itself βγ α R = g R βαγ and 1 are invariant. The latter is non-trivial, due to the measure factor √ � d4x g “Cosmological term” 5.