The first part of this course will cover the foundational material of homogeneous big bang cosmology. There are three basic topics:

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1 The first part of this course will cover the foundational material of homogeneous big bang cosmology. There are three basic topics:. General Relativity. Cosmological Models with Idealized Matter 3. Cosmological Models with Understood Matter General Relativity References: Landau and Lifshitz, Volume : The Classical Theory of Fields Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity Misner, Thorne, and Wheeler, Gravitation. Transformations and Metrics Covariant Derivatives: Affine Structure Covariant Derivatives: Metric Invariant Measure Curvature Invariant Actions Field Equations Newtonian Limit 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,... ).. 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 dx µ = x µ dx ν x ν x µ R µ ν GL(4) x ν (Note: summation convention) (invertible real matrix)

2 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 g µν (x) so that we can define intervals ds = g µν dx µ dx ν ( Pythagoras ) For this to be invariant (g µν dx µ dx ν = g µν dx µ dx ν ), we need g µν (x ) = (R ) α µ (R ) β ν g αβ (x) Since R µ = x µ, (R ) ρ = xρ (chain rule: δ α = x α = xλ x α ) ν x ν σ x σ β x β x β x λ We can write the transformation law for g µν in matrix form: G = R G(R ) T ( From linear ) algebra, we can insure G is diagonal with ± (or 0) entries. The signature, e.g., is determined. There are residual transformations that leave this form of g µν intact. They are the Lorentz transformations! Generalizing g µν, dx µ, we define tensors of more general kinds by the transformation law T µ...µ m ν...ν n (x) T ν...ν n µ...µ m (x ) = (R ) α (R ) αm µ m R ν R νn β β n T...β n µ β α...α m (x) Example: Inverse metric g µν g µα g αν = δ ν µ Operations: Muliplication by number Tensor, of the same type can be added Outer Product Contraction T V ν W ν T ν ν µ µ µ ν...ν n = T ν...ν m µ µ...µ m µ...µ m Example: g µν dx α dx β = T µν αβ a tensor Contraction: T µν µν = ds 0 is a tensor (all components = 0). δ ν µ is a tensor.

3 . Covariant Derivatives: Affine Structure Scalar Field φ (x ) = φ(x) Vector A µ(x ) = xα x A α (x) µ ( (R ) α µa α ) Operator x α ν x ν = x ν x α Is there an invariant derivative? x α x β ν A µ = A β x ν x α x µ x α x β x β = α A β + A β } x ν x {{ µ }} x ν x {{ µ } good Add correction term: ν A µ ν A µ Γ λ νµa λ : x α where S α ν (R ) α ν = x ν. This will work if? = S α ν S β σ µ α A β Γ αβ A σ bad (hard to use not a tensor) x σ ν A µ = S α ν S β µ α A β + x µ x ν A σ Γ λ Γ λ = S α ν S β Γ σ x σ νµs σ λ µ αβ + x ν x µ νµs σ λ A σ that is, Γ λ = R λ σs α ν S β Γ σ x λ x σ νµ µ αβ + x σ x ν x µ Note that the inhomogeneous part is symmetric in µ ν. So we can assume Γ λ αβ = Γβα λ consistently. (The antisymmetric torsion part is a tensor on its own!) Given Γ, we can take covariant derivatives as ν = α T Γ λ T... Γ λ T +Γ ν α T...ν n ν...ν n ν...ν n ν...ν n λν µ...µ m µ...µ m αµ λµ...µ m αµ m µ...λ αλ T...ν n µ µ...µ m This gives a tensor. We use the Leibniz rule in products..3 Covariant Derivatives: Metric Big result: given a metric, there is a unique preferred connection Γ Demand λ g µν = 0 (and symmetry) 0 = λ g λµ g µν Γ α αν Γ α λν g αµ 0 = λµ g αν Γ α µ g νλ Γ α µν g αλ 0 = ν g λµ Γ α Γ α λν g αµ µν g αλ 3

4 Subtract the first line from the sum of the second and third: λβ g Γ α µν g αλ = µ g λν + ν g λµ λ g µν β : Γ µν = g λβ ( µ g λν + ν g λµ λ g µν ) Conversely, this works! Extra term in Γ : [ ( λβ x σ ) x σ g µ x λ g σν + µ x ν g λσ x σ x σ xσ x σ ] + ν x λ g σµ + λ g σν λ ν gσµ x µ g λσ x µ x ν The boxed terms give the desired inhomogeneous terms; the others cancel..4 Invariant Measure Write g det(g µν ); then is an invariant measure. x µ d 4 x = det d 4 x x }{{ ν } Jacobian; x α x β g µν = x µ x ν g αβ = det R G = R G(R ) T det g µν = (det R) det g µν g d 4 x = gd 4 x.5 Curvature In order to make the g µν into dynamical variables, we want them to appear with derivatives in the Lagrangian. Can we form tensors such that this occurs? α Trick: ( µ ν ν µ ) A β R βµν }{{} A α }{{} involves g no derivatives This R α βµν automatically transforms as a proper tensor; the hard part is to show that the derivatives on A α all cancel so we get this form. µ A β = µ A β Γνβ λ A λ σ ( ν A β ) = µ }{{ Γ σ µ ( ν A β ) Γ µν σ A β µβ ν A σ } symm etric drop it = (Γ σ µν ν A σ + Γ ρ µ ν A β µ νλ A σ) Γ σ µβ Γ σ νρa σ Γ σ µλ ν A σ + Γ ρ νλ µ Γ σ µ νλ A σ Γ σ A σ Γ σ µβ νρa σ so R α = µ Γ α µβ + Γα ρ Γ νβ Γ α Γ ρ βµν νβ ν Γ α µρ νρ µβ 4

5 Symmetry properties of R αβγδ We can go to a frame where g µν = 0 (at point of interest). These are geodesic coordinates. x λ There we know that Γ α βγ = 0 (but not its derivatives!) Then [ ( )] R α βµν = µ g ασ ν g σβ β g νσ σ g βν (µ ν) [ ] = g ασ µ β g νβ µ σ g βν ν β g µσ ν σ g βµ = g ασ R σβγδ [ ] with R αβγδ = α δ g βγ + β γ g αδ α γ g βδ β δ g αγ (e.g. Look at the coefficient of α δ g βγ : R αβγδ = R βαγδ R αβγδ = R αβδγ R αβγδ = R γδαβ R αβγδ + R αγδβ + R αδβγ = 0 + in R αβγδ in R αγδβ 0 in R αδβγ ) Since these are tensor identities, they hold in any frame! Also notable is the Bianchi identity; It follows from: e.g.: α R µνβγ + β R µνγα + γ R µναβ = 0 [ 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 () () α [ β, γ ]A µ = α R ν µβγ A ν = α R ν µβγ A ν R ν µβγ αa ν [ β, γ ] α A µ = R ν αβγ ν A µ + R ν µβγ αa ν (3) (4) () cancels against (4), (3) will go away by the symmetry of R ν αβγ + Rν βγα + Rν γαβ generates the Bianchi identity. This identity is the gravity analogue of = 0, so () α F βγ + β F γα + γ F αβ = 0 in electromagnetism, or B = 0, E = B (existence of vector potential). t 5

6 .6 Invariant Actions Since we have an invariant measure gd 4 x (L ), we get invariant field theories by putting invariant expressions inside (L ). Given a special-relativistic invariant theory, we just need to change d 4 x gd 4 x η µν g µν µ µ to make a general-relativistic invariant theory. This is the minimal coupling procedure. Examples:. Scalar field: L = g µν µ φ ν φ V (φ). Transverse vector field σ σ F µν = µ A ν ν A µ = µ A ν Γ µν A σ ν A µ + Γ νµ A σ = µ A ν ν A µ (no need for Γ, or g) L = g αγ g βδ F αβ F 4 γδ supports gauge symmetry A µ A µ + µ χ 3. Longitudinal vector field (instructive) µ A µ = µ A µ µ + Γ µα A α Γ µ = g µσ ( µα α g µ g ασ σ g σµ + ) µα = g µσ α g σµ = g α g [To prove α g = gg µσ α g µσ, use expansion by minors and expansion for inverse matrix. Check on diagonal matrices!] A µ ( So d 4 A µ µ = g µ ga µ ). Thus x g µ = d 4 x µ ( ga µ ) is semi-trivial: it is a boundary term. d 4 x µ A µ ν A ν gives dynamics. This supports a gauge transformation ga µ ga µ + ɛ µνρσ ν Λ ρσ 4. Gravity itself Λ ρσ = Λ σρ R = g βγ R α βαγ and are invariant. The latter is non-trivial, due to the measure factor d 4 x g Cosmological term 5. Spinors! These are foundational, but relegated to the Appendix. 6

7 .7 Field Equations. Scalar field S = { ( }} ){ d 4 x g gµν µ φ ν φ V (φ) Λ δλ δλ µ δ µ φ = δφ µ ( gg µν ν φ) = gv (φ) or µ (g µν ν φ) = V (φ). Transverse vector field S = d 4 x gg αγ g βδ ( d 4 α A β β A α ) ( γ A δ δ A γ ) x gj µ A µ 4 }{{} δ gl ( gg ) µ = µγ µ g νδ ( γ A δ δ A γ ) δ µ A ν = µ ( gf µν ) = g F µν µ exercise! δ gl = gj ν δ µ A ν coupling to current 3. Longitudinal vector field g µ A µ ν A ν = Equation of motion: µ F µν = j ν ( ) µγ or µ gg g νδ F γδ = gj ν As a consistency condition we have: ν ( gj ν ) = 0 g ν j ν g g µ ( ga µ ) g ν ( ga ν ) ν µ p µ ν p g δl µ δ µ p = µ δ µ g ν σ p σ ν = ν g σp σ If there is no source, then we have that g σp σ = λ, which is constant. So g µ A µ ν A ν λ g gives a dynamical cosmological term 4. Field equation for gravity 7

8 a. The hard part of finding the field equation for gravity is varying gg αβ R αβ. However, we can use the trick that gg αβ δr αβ is a total derivative. To ( prove this relatively ) painlessly, we can adopt a system of locally geodesic coordinates α g βγ = 0. Then g αβ δr αβ = g αβ δγ µ µ αβ αδγ µ βµ = µ g αβ δγ µ µβ α αβ g δγ βα µ ω µ Note that δγ has no inhomogeneous terms in its transformation law - it is a tensor! So g αβ δr αβ = µ ( ωµ µ ω µ is now valid in any coordinate system. So gg αβ δr αβ = ω µ g µ = µ gω µ ) is a boundary term; it does not contribute to the Euler-Lagrange equations. Thus with S = κ gr we get R {}}{ δs = κ δ g g αβ R αβ + gδg αβ R αβ = κ ggαβ δg αβ R + gr αβ δg αβ = κ g Rαβ g αβ R δgαβ b. We have for the cosmological term δs = Λ δ g = Λ g gαβ δg αβ c. Matter S = gλ δ gλ δ gλ δs = δg αβ µ δ µ g αβ δg αβ We define the energy-momentum tensor by δ gλ δ gλ g = δg αβ µ δ µ g αβ T αβ We will now see that this makes sense with both examples and conservation laws. i. Examples: 8

9 Scalar field: Λ = g αβ α φ β φ m φ δ gλ ggαβ Λ δg αβ = g ( ( )) Tαβ = ggαβ g µν µ φ ν φ m φ T αβ = α φ β φ + g αβ m φ g αβ gµν µ φ ν φ In flat space we have: δ Maxwell field: (now we use δg ) µν T 00 = 0 φ 0 φ + m φ 0 φ 0 φ ( φ) φ + φ = ( φ) + m = W SB + g α φ β φ ( gg ) αγ αγ δ g βδ F αβ F γδ = gµν gg g βδ F αβ F γδ + gg βδ F µβ F µδ αγ T µν = δ gg g βδ F g 4 αβ F γδ = F µβ F νδ g βδ + g 4 µν g αγ g βδ F αβ F γδ In flat space we have: g µν T µν = 0 ( ) T 00 = E + 4 B E = E + B As an exercise, check the Poynting vector and stresses for the Maxwell field. ii. Conservation laws Preliminary: We expect the conservation of energy-momentum to be tied up with invariance under translations. So: let us translate! x µ x µ = δx µ = ɛ µ, small, vanishing at g µν (x ) = x µ x ν g αβ (x) x α x β g µν (x) + α ɛ µ g αν + β ɛ ν g µβ µν (x ) g µν + α g µν ɛ α g Thus δg µν = g µν (x) = α ɛ µ g αν + α ɛ ν g µα α g µν ɛ α not x 9

10 Now notice that δg µν = g αµ α ɛ ν + g αν α ɛ µ (Killing equations) {}}{ = g αµ α ɛ ν + g αµ g νβ ɛ ρ α g βρ + ρ g αβ β g αρ where the last step follows from differentiating: g αµ g µβ β = δ α ; so λ g αµ g µβ + g αµ λ g µβ = 0. Multiplying by g ρα, we get λ g αµ g µβ g ρα = λ g ρβ. We can also write this as δg µν = ɛ ν;µ + ɛ µ;ν. Now we have, from the invariance of the action and symmetry of T µν in its definition 0 = g Tµν ɛ µ;ν Γ ν αρ + g αν α ɛ µ + g αν g µβ α g βρ + ρ g αβ β g αρ ɛ ρ = g αµ α ɛ ν + g αν α ɛ µ + g αµ g ρν ρ g αβ ɛ ρ = g αµ α ɛ ν + g αν α ɛ µ ρ g αβ ɛ ρ But also 0 = = = ν ( g Tµν ɛ µ ) g ( Tµν ɛ µ ) µ g ν T µν + gtµν ɛ µ;ν }{{} 0 Since this holds for an arbitrary ɛ µ, we conclude that ν T µν = 0..8 Newtonian Limit We should be able to identify Newtonian gravity (and fix the coupling constant) by looking at the situation with nearly flat space and only T 00 = ρ significant. (For now, of course, we ignore the cosmological term.) The stationary action condition gives κ R αβ g αβ R = T αβ or κr αβ = T αβ T gαβ (g 00 c g ij ) Focus on R 00 : ρ κr 00 = in R terms with ΓΓ pieces higher order. Also terms with x 0 R 00 γ Γ γ 00 0Γ γ g 00 0γ c x i are small. Thus in 0

11 and finally κ g 00 = ρ To interpret this, wrte g 00 = + ɛ (g 00 = ɛ). The action density of matter is perturbed by δ( gλ) ρ δg 00 ɛ This looks like the Newtonian coupling if ɛ = φ. The equation κ g 00 = ρ 4κ φ = ρ This fixes κ = 6πG N. ( G ) N M G N M Gauss φ = ; φ = ˆr dv φ = 4πG N M r r }{{} M = 4κ Note on the appendices and scholia: These are not necessarily self-contained, specifically, they refer to facts about quantum field theory and the standard model that are not assumed elsewhere in the course. Don t worry if not everything is clear (or even meaningful) to you at this stage. Ask me if you re curious! Central material. The notion of local Lorentz invariance, vierbeins, and R recipe (Appendices -).. The idea that we are building a model of the world and are free to try anything (Scholium 4). Appendix : Spinors and Local Lorentz Invariance Spinor fields play a very important role in fundamental physics, so we must learn how to treat them in general relativity. The essential thing is to define γ a matrices. They transform under Lorentz transformations (that are in SO(3, ) not GL(4)). Indeed the defining relation γ a γ b + γ b γ a = η ab refers to the flat-space Minkowski metric and the transformation law γ a = S (Λ)γ a S(Λ) = Λ a b γb requires an S that does not exist in general. (Λ GL(4).) So we postulate local Lorentz invariance under transformations of this form with Λ(x) a function of x, Λ(x) SO(3, ). To connect this structure to the metric we introduce a vierbein e a µ (x) such that η ab e µ a (x)e ν b (x) = g µν (x) g µν e a µ(x)e b ν (x) = η ab ( square root of the metric) ( moving frame ) Now, for example, we can form the Dirac equation (γ a e a µ D µ + m) ψ = 0

12 But D µ needs discussion. We want invariance under local Lorentz transformations. This requires (exercise!) D µ S(Λ(x)) = S(Λ(x))D µ a typical gauge invariance. We solve this problem as usual by introducing a gauge potential ω ab (x) so(3, ) µ where so(3, ) is the Lie algebra corresponding to SO(3, ), and thus ω µ ab (x) = ω µ ba (x), and (in matrix notation) ω µ (x) = Λ ω µ (x)λ Λ µ Λ and writing D µ = µ + ω µ τ [ ] where the τ matrices provide the appropriate representation of the symmetry, e.g. σ ab i = 4 γ a, γ b for spin or the identity for spin. To avoid introducing additional structure (c.f. Scholium 4) we demand leading to a unique determination D µ e a = µ e a ν Γ α a α + ω a b e b = 0 () ν µν e µ ν ν a µ c = e µ e ν + e ν e a Γ α c c α µν ω a Appendix : Moving Frames Method and Recipe The discussion of Appendix is not as profound as it should be: ω ab µ should be more primitive than Γ. This is worth pursuing, since it leads to a beautiful analogy and useful formulae. We can eliminate Γ from the defining relation () for ω by antisymmetrizing in µ ν. Thus µ e a ν ν e a µ = ω µ ac e cν ω ν ac e cµ To solve for ω we go through a slight rigamarole, reminiscent of what we did to get Γ from g = 0 +e µ e a aρ ν ν e a µ = ω ac µ e aρ e cν ωac ν e aρ e cµ e a a ω ac ω ac aµ ν e ρ ρ e ν = e ν aµ e cρ + ρ e aµ e cν +e aν ρ e a µ µ e a ρ = ω ac ρ e aν e cµ ω ac µ e aν e cρ e aρ µ e a ν e aρ ν e a µ e aµ ν e a ρ + e aµ ρ e a ν + e aν ρ e a µ e aν µ e a ρ = ω µ ac e aρ e cν using ω ac = ω ca. Multiplying both sides by e eρ e fν, we get µ µ A B C C B A {}}{{}}{{}}{{}}{{}}{{}}{ ω µ ef = e fν µ e e ν e fν ν e a µ e aµ e ρe e νf ν e a ρ + e aµ e eρ e fν ρ e a ν + e ρe ρ e f µ e ρe µ e f ρ or e fν ω µ ef = µ e e ν ν e e µ + e aµ e eρ ρ e a ν (e f)

13 Now we can construct a curvature by differentiating (say) a space-time scalar, which is a local Lorentz vector field (D µ D ν D ν D µ )φ a a = R b φ b This leads to a a a a c a c R µν b = µ ω ν b ν ω µ b + ω µ c ω ν b ω ν c ω µ b Now you will be delighted (but not too surprised) to learn that this gauge curvature is intimately related to the Riemann curvature we had before; indeed R µναβ = R µν a b e aα e β b This actually gives the most powerful recipe for computing R. The technique of introducing frames a e µ (x) to make the geometry locally flat was developed by E. Cartan and is called the moving frame method. It is usually presented in very obscure ways. Scholium : Structure and Redundancy Allowing a very general framework and demanding symmetry is an alternative to finding a canonical form that solves the symmetry. Thus we consider general coordinate transformations, but postulate a metric to avoid solving for local Lorentz frames and then pasting them together. Vierbeins or moving frames make this much explicit. Fixing down to a specific frame is gauge fixing in the usual sense. Scholium : Why General Relativity? Ordinary spin- gauge fields are in danger of producing wrong-metric particles or ghosts. This is because covariant quantization conditions (commutation relatives (?) ) for the different polarizations: [a µ, a ν ] = g µν if normal for the space-like pieces are abnormal for the time-like and vice versa. Gauge symmetry allows me to show the wrong-metric excitations don t couple (e.g. one can choose A 0 = 0 gauge). Similarly, general convariance/local Lorentz symmetry are required for consistent quantum theories of spin. Scholium 3: General Relatvitiy vs. Standard Model In the Standard Model symmetry and weak cutoff dependence (renormalizability) greatly restrict the possible couplings. One requries a linear manifold of fields (i.e. φ (x) + φ (x) is allowed if φ (x), φ (x) are). In General Relativity there is still symmetry, and minimal coupling leads to weak cutoff dependence below the Planck scale. However, the fields manifold is not linear (g µν + gµν may not be invertible, hence not allowed) and there is a dimensional fundamental coupling. It looks like an effective theory thus, with spontaneous symmetry breaking, i.e. e a =?, like the σ model. µν µ 3

14 Scholium 4: Gauge and Dilaton Extensions The standard assumption of Riemannian geometry is α g µν = 0. However, we might want to relax this to incorporation additional symmetry. Some important physical ideas have arisen (or have natural interpretations) along this line. Weyl wanted to unify electromagnetism with gravity. He postulated and the symmetry g α g µν = s α g µν µν (x) = λ(x)g µν (x) s α(x) = s α + α λ This was the historical origin of gauge invariance! He wanted to identify s α with the electromagnetic potential A α. That has problems, but the ideas are profound. Symmetry of the type g µν (x) = λ(x)g µν (x) arises in modern conformal field theory and string theory. It is called Weyl symmetry. Given the importance of massless particles, and the idea that massive fundamental particles can acquire mass by spontaneous symmetry breaking, this idea of a scale symmetry retains considerable appeal. A closely related variant is φ-fields of this sort are called dilatons. α g µν = α φ g µν g µν = λg µν φ = φ + λ 4