1 General Relativity

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:

G� = R−1G(R−1)T

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

ν ...ν T � 1 n (x�) = (R−1)α1 · · · (R−1)αm Rν1 · · · Rνn · T β1...βn (x) µ1...µm µ1 µm β1 βn α1...αm Example: Inverse metric gµν 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 T µ1µ2...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

� T ν1...νn = ∂ T ν1...νn −Γλ T ν1...νn −. . .−Γλ T ν1...νn +Γν1 T λν2...νn 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 gλβ 1 × : Γβ = g λβ �∂ g + ∂ g − ∂ g � 2 µν 2 µ λν ν λµ λ 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 G� = R−1G(R−1)T

� 1 det g = det g µν (det R)2 µν

Write g ≡ det(gµν ); then � √ g�d4 x� = gd4 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 √ d4 x → gd4 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. Spinors! These are foundational, but relegated to the Appendix.

6 1.7 Field Equations 1. Scalar ﬁeld Λ � �� √ �1 � S = d4 x g gµν ∂ φ∂ φ − V (φ) 2 µ ν

δΛ δΛ ∂µ = δ∂µφ δφ √ µν √ � ∂µ ( gg ∂ν φ) = − gV (φ) µν � or �µ (g ∂ν φ) = −V (φ)

2. Transverse vector ﬁeld

1 � √ � √ S = − d4 x gg αγ g βδ (∂ A − ∂ A ) (∂ A − ∂ A ) − d4x gjµA 4 α β β α γ δ δ γ µ � �� coupling to current

√ δ gL �√ µγ νδ � ∂µ = −∂µ gg g (∂γ Aδ − ∂δAγ ) δ∂µAν √ µν · √ µν = −∂µ ( gF ) = − g �µF √ exercise! δ gL √ = − gjν δ∂µAν

µν ν Equation of motion: �µF = j

�√ µγ νδ � √ ν or ∂µ gg g Fγδ = gj

�√ ν � √ ν As a consistency condition we have: ∂ν gj = 0 ⇒ g �ν j 3. Longitudinal vector ﬁeld

� √ � √ 1 √ 1 √ g � Aµ� Aν = g √ ∂ ( gAµ) √ ∂ ( gAν ) µ ν g µ g ν

� 1 ≡ √ ∂ pµ∂ p ν g µ ν � � δL 1 µ σ ∂µ ν = 2∂µ √ δν ∂σp δ∂µp g

� 1 � = 2∂ √ ∂ p σ ν g σ

√1 σ If there is no source, then we have that g ∂σp = λ, which is constant.

� √ µ ν 2 � √ So g �µA �ν A → λ g gives a “dynamical” cosmological term 4. Field equation for gravity

7 √ αβ a. The hard part of ﬁnding the ﬁeld equation for gravity is varying gg Rαβ . However, √ αβ we can use the trick that gg δRαβ is a total derivative. To prove this relatively painlessly, we can adopt a system of locally geodesic coordinates � � ⇒ ∂αgβγ = 0 . Then

αβ αβ � µ µ � g δRαβ = g ∂µδΓαβ − ∂αδΓβµ

� αβ µ µβ α � = ∂µ g δΓαβ − g δΓβα µ ≡ ∂µω

Note that δΓ has no inhomogeneous terms in its transformation law - it is a tensor! So √ gαβδR = ∂ ωµ → � ωµ is now valid in any coordinate system. So ggαβ δR = αβ µ µ αβ √ µ �√ µ � g�µω = ∂µ gω is a boundary term; it does not contribute to the Euler-Lagrange equations. √ Thus with S = κ � gR we get ⎛ ⎞ R � √ � �� √ ⎜ αβ αβ ⎟ δS = κ ⎝δ g g Rαβ + gδg Rαβ ⎠

� 1√ √ = κ − gg δgαβ R + gR δgαβ 2 αβ αβ

� √ � 1 � = κ g R − g R δgαβ αβ 2 αβ

b. We have for the cosmological term

� √ δS = −Λ δ g

� √ �1 � = Λ g g δgαβ 2 αβ

c. Matter

� √ S = gΛ � � √ √ � δ gΛ δ gΛ αβ δS = αβ − ∂µ αβ δg δg δ∂µg

We deﬁne the energy-momentum tensor by √ √ √ δ gΛ δ gΛ g αβ − ∂µ αβ = Tαβ δg δ∂µg 2 We will now see that this makes sense with both examples and conservation laws. i. Examples:

8 • Scalar ﬁeld:

1 αβ 1 2 2 Λ = g ∂αφ∂β φ − m φ √ 2 2 δ gΛ 1√ = − gg Λ δgαβ 2 αβ √ g 1 �√ �1 1 �� 1√ T = − gg gµν ∂ φ∂ φ − m2 φ2 + g∂ φ∂ φ 2 αβ 2 αβ 2 µ ν 2 2 α β 1 1 T = ∂ φ∂ φ + g m 2φ2 − g gµν ∂ φ∂ φ αβ α β 2 αβ 2 αβ µ ν In ﬂat space we have: 1 1 � � T = ∂ φ∂ φ + m2 φ2 − ∂ φ∂ φ − (�φ)2 00 0 0 2 2 0 0 1 1 1 = φ˙2 + (�φ)2 + m 2φ2 =� W SB 2 2 2

δ • Maxwell ﬁeld: (now we use δgµν )

�√ αγ βδ � 1 √ αγ βδ √ βδ δ gg g Fαβ Fγδ = − gµν gg g Fαβ Fγδ + 2 gg Fµβ Fµδ 2 2 � 1� �√ � T = √ − δ gg αγg βδF F µν g 4 αβ γδ 1 = −F F g βδ + g g αγ g βδF F ⇒ µβ νδ 4 µν αβ γδ µν g Tµν = 0 (�)

In ﬂat space we have: 1 T = E2 + · 2 �B2 − E2� 00 4 1 =� �E2 + B2� 2 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 − xµ = δxµ = �µ, small, vanishing at ∞

�µ �ν � ∂x ∂x g µν (x�) = g αβ(x) ∂xα ∂xβ µν µ αν ν µβ ≈ g (x) + ∂α� g + ∂β � g �µν � µν µν α g (x ) ≈ g + ∂αg �

µν �µν µ αν ν µα µν α Thus δg = g (x) = ∂α� g + ∂α� g − ∂αg � ↑ not x�

9 Now notice that

µν αµ ν αν µ δg = g �α� + g �α� (Killing equations) ν Γαρ � �� 1 = gαµ∂ �ν + gαµ · g νβ �∂ g + ∂ g − ∂ g � �ρ α 2 α βρ ρ αβ β αρ 1 + g αν ∂ �µ + g αν · gµβ �∂ g + ∂ g − ∂ g � �ρ α 2 α βρ ρ αβ β αρ αµ ν αν µ αµ ρν ρ = g ∂α� + g ∂α� + g g ∂ρgαβ � αµ ν αν µ αβ ρ = g ∂α� + g ∂α� − ∂ρg �

µβ β µβ where the last step follows from diﬀerentiating: gαµ g = δα; so ∂λgαµg + µβ ρα µβ ρα ρβ gαµ ∂λg = 0. Multiplying by g , we get ∂λgαµg g = −∂λg . We can also write this as δgµν = �ν;µ + �µ;ν . Now we have, from the invariance of the action and symmetry of Tµν in its deﬁnition � √ µ;ν 0 = g Tµν �

But also � ν �√ µ � 0 = ∂ g Tµν � � √ µ � µ � = g � Tµν � � � √ ν √ µ;ν = g � Tµν + gTµν � � �� 0 µ ν Since this holds for an arbitrary � , we conclude that � Tµν = 0.

1.8 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 T00 = ρ significant. (For now, of course, we ignore the cosmological term.) The stationary action condition gives

� 1 � 1 κ R − g R = T or αβ 2 αβ 2 αβ 1 2κR = T − T g (g ≈ c2 � g ) αβ αβ 2 αβ 00 ij Focus on R : 00 ρ 2κR = 00 2 · ∂ 1 ∂ in R ··· terms with ΓΓ pieces higher order. Also terms with ∂x0 ∼ c ∂xi are small. Thus in γ γ R00 ≈ ∂γ Γ00 − ∂0Γ0γ ≈ 1 −→ �2g 2 00

10 and ﬁnally 2 2κ� g00 = ρ 00 To interpret this, wrte g00 = 1 + � (g = 1 − �). The action density of matter is perturbed by √ δ( gΛ) ρ ≈ − � δg00 2 This looks like the Newtonian coupling if � = 2φ. The equation

2 2 2κ� g00 = ρ ⇒ 4κ� φ = ρ This fixes κ = 1 . 16πGN � � GN M GN M Gauss 2 � φ = − ; �φ = 2 ˆr −−−→ dV � φ = 4πGN 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

1. The notion of local Lorentz invariance, vierbeins, and R recipe (Appendices 1-2). 2. The idea that we are building a model of the world and are free to try anything (Scholium 4).

Appendix 1: 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 γa matrices. They transform under Lorentz transformations (that are in SO(3, 1) not GL(4)). Indeed the defining relation

γaγb + γbγa = 2ηab

refers to the ﬂat-space Minkowski metric and the transformation law

�a −1 a a b γ = S (Λ)γ S(Λ) = Λ bγ 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 a of x, Λ(x) ∈ SO(3, 1). To connect this structure to the metric we introduce a vierbein eµ(x) such that

a b ηab eµ(x)eν(x) = gµν (x) (“square root” of the metric) µν a b ab g eµ (x)eν (x) = η (“moving frame”) Now, for example, we can form the Dirac equation

a µ (γ ea Dµ + m) ψ = 0

11 But Dµ needs discussion. We want invariance under local Lorentz transformations. This requires (exercise!) DµS(Λ(x)) = S(Λ(x))Dµ a typical gauge invariance. We solve this problem “as usual” by introducing a gauge potential

ab ωµ (x) ∈ so(3, 1)

ab ba where so(3, 1) is the Lie algebra corresponding to SO(3, 1), and thus ωµ (x) = −ωµ (x), and (in matrix notation) � −1 −1 ωµ(x) = Λ ωµ(x)Λ − Λ ∂µΛ and writing Dµ = �µ + ωµ · τ

ab i � a b � where the τ matrices provide the appropriate representation of the symmetry, e.g. σ = 4 γ , γ 1 for spin 2 or the identity for spin 1. To avoid introducing additional structure (c.f. Scholium 4) we demand

a a α a a b Dµeν = ∂µeν − Γµν eα + ω µ b eν = 0 (1)

leading to a unique determination

a ν a ν a α ω µ c = −ec ∂µeν + ec eα Γµν

Appendix 2: Moving Frames Method and Recipe ab The discussion of Appendix 1 is not as profound as it should be: ωµ 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 (1) for ω by antisymmetrizing in µ ↔ ν. Thus

a a ac ac ∂µeν − ∂ν eµ = ωµ ecν − ων ecµ

To solve for ω we go through a slight rigamarole, reminiscent of what we did to get Γ from �g = 0

� a a � ac ac �� +eaρ ∂µeν − ∂ν eµ = ωµ eaρecν − �ων�eaρecµ �� a a � ac �� ac �� −eaµ ∂ν eρ − ∂ρeν = −�ων� eaµecρ + �ωρ�eaµecν �� a a� ac �� ac +eaν ∂ρeµ − ∂µeρ = �ωρ�eaν ecµ − ωµ eaν ecρ

a a a a a a ac eaρ∂µeν − eaρ∂ν eµ − eaµ∂ν eρ + eaµ∂ρeν + eaν ∂ρeµ − eaν ∂µeρ = 2ωµ eaρecν

ac ca eρ fν using ωµ = −ωµ . Multiplying both sides by e e , we get

A B C C B A

ef �fν�� e� �fν�� a� � ρe�� νf a� � eρ�� fν a� �ρe�� f� �ρe�� f� 2ωµ = e ∂µeν − e ∂ν eµ − eaµe e ∂ν eρ + eaµe e ∂ρeν + e ∂ρeµ − e ∂µeρ

or efν ωef = �∂ ee − ∂ ee + e eeρ ∂ ea − (e ↔ f) � µ 2 µ ν ν µ aµ ρ ν

12 Now we can construct a curvature by diﬀerentiating (say) a space-time scalar, which is a local Lorentz vector ﬁeld a a b (DµDν − Dν Dµ)φ = Rµν b φ This leads to a a a a c a c Rµν b = ∂µων b − ∂ν ωµ b + ωµ cων b − ων cωµ 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

a b Rµναβ = Rµν b eaαeβ

This actually gives the most powerful recipe for computing R. The technique of introducing frames a eµ(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 1: Structure and Redundancy Allowing a very general framework and demanding symmetry is an alternative to finding a canon ical 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 2: Why General Relativity? Ordinary spin-1 gauge ﬁelds are in danger of producing wrong-metric particles or “ghosts”. This is because covariant quantization conditions (commutation relatives (?) ) for the diﬀerent polar izations: † [aµ, aν ] = −gµν 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 A0 = 0 gauge). Similarly, general convariance/local Lorentz symmetry are required for consistent quantum theories of spin 2.

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. φ1(x) + φ2(x) is allowed if φ1(x), φ2(x) are). In General Relativity there is still symmetry, and minimal coupling leads to weak cutoff de 1 2 pendence below the Planck scale. However, the fields manifold is not linear (gµν + gµν may not be invertible, hence not allowed) and there is a dimensional fundamental coupling. It looks like an a effective theory thus, with spontaneous symmetry breaking, i.e. eµ = �?�, like the σ−model.

13 Scholium 4: Gauge and Dilaton Extensions µν The standard assumption of Riemannian geometry is �αg = 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

�αgµν = sαgµν

and the symmetry

� gµν (x) = λ(x)gµν (x) � sα(x) = sα + ∂αλ

This was the historical origin of gauge invariance! He wanted to identify sα with the electromagnetic potential Aα. That has problems, but the ideas are profound. � Symmetry of the type gµν (x) = λ(x)gµν (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

�αgµν = ∂αφ gµν � gµν = λgµν φ� = φ + λ

φ-ﬁelds of this sort are called “dilatons”.