sign in sign up
<<
Home , Abstract index notation, Differentiable manifold, Normal coordinates, Tensor field

Definitions from Harvey Reall’s Course

Yichen Shi Michaelmas 2012

1 Ch.2 - and

Definition 1.1. A differentiable is a M together with a collection of subsets Oα such that 1. ∪αOα = M, i.e., the subsets Oα cover M. n 2. For each α there is a one-to-one and onto map φα : Oα → Uα where Uα is an open subset of R . φα are called charts and the set {φα} is called an . 3. If Oα and Oβ overlap, i.e., Oα ∩ Oβ 6= ∅, then

−1 n n φβ ◦ φα : φα(Oα ∩ Oβ) ⊂ Uα ⊂ R → φβ(Oα ∩ Oβ) ⊂ Uβ ⊂ R . We require that this map be smooth (infinitely differentiable).

Definition 1.2. A function f : M → R is smooth iff for any chart φ : O → U, F := f ◦ φ−1 : U → R is a smooth function.

Definition 1.3. Let λ : I → M be a smooth with λ(0) = p. The to λ at p is the Xp form the of smooth functions on M to R defined by

d Xp(f) = [f(λ(t))] . dt t=0 Proposition 1.4. The set of all tangent vectors at p forms an n-dimensional , the TpM.

µ Definition 1.5. Let {eµ, µ = 1, ...n} be a for TpM. We can expand any vector X ∈ TpM as X = X eµ. We call the numbers Xµ the components of X with respect to this basis. Definition 1.6. Let V be a real vector space. The V ∗ of V is the vector space of linear maps from V to R. ∗ ∗ Lemma 1.7. If V is n-dimensional then so is V . If {eµ, µ = 1, ..., n} is a basis for V then V has a basis µ µ µ {f , µ = 1, ..., n}, the dual basis defined by f eν = δν . Theorem 1.8. If V is finite dimensional then (V ∗)∗ is naturally isomorphic to V . The is Φ: V → (V ∗)∗ where Φ(X)(ω) = ω(X) for all ω ∈ V ∗.

∗ Definition 1.9. The dual space of TpM is denoted Tp M and called the at p. If {eµ} is a µ µ basis for TpM and {f } is the dual basis then we can expand a covector η as ηµf .

Definition 1.10. Let f : M → R be a smooth function. Define a covector (df)p by (df)p(X) = X(f) for any vector X ∈ TpM.(df)p is the of f at p.

1 Definition 1.11. A of type (r, s) at p is a

∗ ∗ T : Tp M × ... × Tp M × TpM × ... × TpM → R

∗ where there are r factors of Tp M and s factors of TpM. In other words, given r covectors and s vectors, a tensor of type (r, s) produces a .

µ Definition 1.12. Let T be a tensor of type (r, s) at p. If {eµ} is a basis for TpM with dual basis {f } then the components of T in this basis are the numbers

µ1...µr µ1 µr T ν1...νs = T (f , ..., f , eν1 , ..., eνs ).

Definition 1.13. If S is a tensor of type (p, q) and T is a tensor of type (r, s) then the outer product of S and T , denoted S ⊗ T is a tensor of type (p + r, q + s) defined by

(S ⊗ T )(ω1, ..., ωp, η1, ...ηr,X1, ..., Xq,Y1, ..., Ys) = S(ω1, ..., ωp,X1, ..., Xq)T (η1, ...ηr,Y1, ..., Ys).

Definition 1.14. A vector field is a map X which maps any point p ∈ M to a vector Xp at p. Given a vector field X and a function f we can define a new function X(f): M → R by X(f): p → Xp(f). The vector field X is smooth if this map is a smooth function for an smooth f.

Definition 1.15. A covector field is a map ω which maps any point p ∈ M to a covector ωp at p. Given a covector field and a vector field X we can define a function ω(X): M → R by ω(X): p 7→ ωp(Xp). The covector field ω is smooth for any smooth vector field X.

Definition 1.16. An (r, s) tensor field is a map T which maps any point p ∈ M to an (r, s) tensor Tp at p. Given r covector fields η1, ..., ηr and s vector fields X1, ..., Xs we can define a function T (η1, ..., ηr,X1, ..., Xs): M → R by p 7→ T ((η1)p, ..., (ηr)p, (X1)p, ..., (Xs)p). The tensor field T is smooth if this function is smooth for any smooth covector field η1, ..., ηr and vector fells X1, ..., Xs. Definition 1.17. The commutator of two vector fields X and Y is the vector field [X,Y ] defined by

[X,Y ](f) = X(Y (f)) − Y (X(f)) for any smooth function f. Definition 1.18. Let X be a vector field on M and p ∈ M. An curve of X through p is a curve through p whose tangent at every point is X. If we let λ denote an integral curve of X with λ(0) = p, then in a coordinate chart, this definition reduces to the initial value problem

dxµ(t) = Xµ(x(t)), xµ(0) = xµ. dt p

2 Ch.3 - The Tensor

Definition 2.1. A at p ∈ M is a (0, 2) tensor g with the following properties: 1. It is symmetric: g(X,Y ) = g(Y,X) for all X,Y ∈ TpM, i.e., gab = gba. 2. It is non-degenerate: g(X,Y ) = 0 for all Y ∈ TpM iff X = 0. Definition 2.2. A Riemannian (Lorentzian) manifold is a pair (M, g) where M is a differentiable manifold and g is a Riemannian (Lorentzian) metric tensor field. A Lorentzian manifold is called a space-time.

Definition 2.3. Since gab is non-degenerate, it must be invertible. The inverse metric is a symmetric (2, 0) ab ab a tensor field denoted g and obeys g gbc = δc .

2 Definition 2.4. A metric determines a natural isomorphism between vectors and covectors. Given a vector a b a ab X we can define a covector Xa = gabX . Given a covector ηa we can define a vector η = g ηb. These maps are clearly inverses of each other.

Definition 2.5. On a Lorentzian manifold (M, g), a non-zero vector X ∈ TpM is timelike if g(X,X) < 0, null if g(X,X) = 0, and spacelike if g(X,X) > 0. Definition 2.6. On a , the of a vector X is |X| = pg(X,X) and the angle between two non-zero vectors X and Y is θ where cos θ = g(X,Y )/(|X||Y |). The same definitions apply to space-like vectors on a Lorentzian manifold. Definition 2.7. A curve in a Lorentzian manifold is timelike/null/spacelike if its tangent vector is everywhere timelike/null/spacelike. Definition 2.8. Let λ(u) be a timelike curve with λ(0) = p. Let Xa be the tangent vector to the curve. The proper time τ from p along the curve is defined by dτ q = −(g XaXb) , τ(0) = 0. du ab λ(u)

µ µ 2 µ ν In a coordinate chart, X = dx /du so this definition can be rewritten as dτ = −gµν x x , evaluated along the curve. Definition 2.9. If proper time τ is used to parametrise a timelike curve then the tangent to the curve is called the 4-velocity of the curve. In a coordinate basis, it has components uµ = dxµ/dτ. Definition 2.10. The Christoffel symbols are defined by 1 Γµ = gµσ(∂ g + ∂ g − ∂ g ), νρ 2 ν ρσ ρ νσ σ νρ giving the equation

d2xµ dxν dxρ + Γµ = 0, or, ∇ X = 0. dτ 2 νρ dτ dτ X

3 Ch.4 - Covariant

Definition 3.1. A ∇ on a manifold M is a map sending every pair of smooth vector fields X,Y to a smooth vector field ∇X Y , with the properties

∇fX+gY Z = f∇X Z + g∇Y Z;

∇X (Y + Z) = ∇X Y + ∇X Z;

∇X (fY ) = f∇X Y + (∇X f)Y = f∇X Y + X(f)Y. Definition 3.2. Let Y be a vector field. The covariant derivative of Y is the (1, 1) tensor field ∇Y . In a a a abstract we write (∇Y ) b as ∇bY or Y ;b.

µ Definition 3.3. In a basis {eµ} the componentsΓ νρ are defined by

µ ∇ρeν := ∇eρ eν = Γνρeµ

Definition 3.4. A connection ∇ is torsion-free if ∇a∇bf = ∇b∇af for any function f. This is equivalent ρ to Γ[µν] = 0.

3 Lemma 3.5. For a torsion-free connection, if X and Y are vector fields, then

∇X Y − ∇Y X = [X,Y ].

Proof: In coordinate basis,

µ µ ν µ ν µ ∇X Y − ∇Y X = X ∇ν Y − Y ∇ν X ν µ ν µ ρ ν µ ν µ ρ = X ∂ν Y + X Γρν Y − Y ∂ν X − Y Γρν X = Xν ∂ Y µ − Y ν ∂ Xµ + Xν Y ρΓµ ν ν [ρν] = [X,Y ]µ.

Definition 3.6. Let M be a manifold with a metric g. There exists a unique torsion-free connection ∇ such that the metric is covariantly constant, i.e., ∇g = 0 or gab;c = 0. This is the Levi-Civita connection. Definition 3.7. Let M be a manifold with a connection ∇. An affinity parametrised geodesic is an integral curve of a vector field X satisfying ∇X X = 0.

Theorem 3.8. Let M be a manifold with a connection ∇. Let p ∈ M and Xp ∈ TpM. Then there exists a uniquely parametrised geodesic through p with tangent vector Xp at p.

Definition 3.9. Let M be a manifold with a connection ∇. Let p ∈ M. The exponential map from TpM to M is defined as the map which sends Xp to the point unit affine parameter distance along the geodesic through p with tangent Xp at p.

Definition 3.10. Let {eµ} be a basis for TpM. coordinates at p are defined in a neighbourhood of µ a p as follows. Pick q near p. Then the coordinates of q are X where X is the element of TpM that maps to q under the exponential map.

µ µ Lemma 3.11. Γ(νρ)(p) = 0 in at p. Hence for a torsion-free connection, Γνρ(p) = 0 in normal coordinates at p. µ µ Proof: Affinely parametrised through p are given in normal coordinates by X (t) = tXp , hence the µ ν ρ µ ν ρ geodesic equation reduces to Γνρ(X(t))Xp Xp = 0. At t = 0, we have Γνρ(p)Xp Xp = 0. The antisymmetric µ part of Γ vanishes when contracted with the Xp’s and we are left with Γ(νρ)(p) = 0. Lemma 3.12. On a manifold with a metric, if the Levi-Civita connection is used to define normal coordinates at p then ∂ρgµν = 0 at p. σ Proof: At p,Γνρ = 0. Hence σ 0 = 2gµσΓνρ = ∂ν gµρ + ∂ρgνµ − ∂µgνρ.

Now if we symmetrise µ, ν, then the last two terms cancel and we have ∂ρgµν = 0.

Lemma 3.13. On a manifold with metric one can choose normal coordinates at p so that ∂ρgµν (p) = 0, and gµν (p) = ηµν (Lorentzian case); gµν (p) = δµν (Riemannian case). Definition 3.14. In a Lorentzian manifold a local inertial frame at p is a set of normal coordinates at p with the above properties.

4 Ch.6 - Curvature

Definition 4.1. Let Xa be the tangent to a curve. A tensor field T is parallelly transported along the curve if ∇X T = 0.

4 a a b c d Definition 4.2. The R bcd of a connection ∇ is defined by R bcdZ X Y = (R(X,Y )Z)a, where X,Y,Z are vector fields and R(X,Y )Z is the vector field

R(X,Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ]Z. In coordinate basis, µ µ µ τ µ τ µ R νρσ = ∂ρΓνσ − ∂σΓνρ + ΓνσΓτρ − ΓνρΓτσ. Definition 4.3. The tensor is the (0, 2) tensor defined by

c Rab = R acb. Proposition 4.4. If ∇ is torsion-free then a R [bcd] = 0. µ µ Proof: Let p ∈ M and choose normal coordinates at p. Vanishing torsion implies Γ νρ(p) = 0. So R νρσ(p) = µ µ µ ∂ρΓνσ − ∂σΓνρ. Antisymmetrising on ρνσ gives R [νρσ] = 0. But if components of a tensor vanish in one basis then they vanish in any basis, so our result is not limited to normal coordinates. On top, p is an a arbitrary point, so our result is also not limited to the point p. Hence R [bcd] = 0. Proposition 4.5 (Bianchi identity). If ∇ is torsion-free then

a R b[cd;e] = 0.

µ Proof: Let p ∈ M and choose normal coordinates at p. Vanishing torsion implies Γνρ(p) = 0. Now, µ µ µ R νρσ;τ = ∂τ ∂ρΓνσ − ∂τ ∂σΓνρ. µ a Antisymmetrising on ρτσ gives R ν[ρσ;τ] = 0, and by same reasons as above, we have R b[cd;e] = 0. Definition 4.6. Let M be a manifold with a connection ∇.A 1-parameter family of geodesics is a map γ : I × I0 → M where I and I0 both are open intervals in R, such that 1. for fixed s, γ(s, t) is a geodesic with affine parameter t (so s is the parameter that labels the geodesic); 2. the map (s, t) 7→ γ(s, t) is smooth and one-to-one with a smooth inverse. This implies that the family of geodesics forms a 2-dimensional Σ ⊂ M. Definition 4.7 (geodesic deviation equation). If ∇ has vanishing torsion then

∇T ∇T S = R(T,S)T. Or in , c b a a b c d T ∇c(T ∇bS ) = R bcdT T S .

Proof: Vanishing torsion gives ∇T S − ∇ST = [T,S] = 0. Hence

∇T ∇T S = ∇T ∇ST = ∇S∇T T + R(T,S)T = R(T,S)T

since ∇T T = 0 as T is tangent to affinely parametrised geodesics. Proposition 4.8. The Riemann tensor satisfies

Rabcd = Rcdab,R(ab)cd = 0. Proof: The second follows from the first and the antisymmetry of the Riemann tensor. For the first, introduce µ normal coordinates at p, so ∂µgνρ(p) = 0, and Γνρ(p) = 0. Then at p, ν νσ νσ νσ 0 = ∂µδρ = ∂µ(g gσρ) = gσρ∂µg ⇒ ∂µg = 0. 1 ⇒ ∂ Γτ = gτµ(g + g − g ). ρ νσ 2 µν,σρ µσ,νρ νσ,µρ 1 R = (g + g − g − g ) = R . µνρσ 2 µσ,νρ νρ,µσ νσ,µρ µρ,νσ ρσµν This is a tensor equation so is valid not only in normal coordinates, and p is arbitrary. Hence Rabcd = Rcdab.

5 Proposition 4.9. The Ricci tensor is symmetric, i.e.,

Rab = Rba.

Proof: cd cd c Rab = g Rcadb = g Rdbca = R bca = Rba. Definition 4.10. The Ricci is ab R = g Rab. Definition 4.11. The is the symmetric (0, 2) tensor 1 G = R − Rg . ab ab 2 ab Proposition 4.12. The Einstein tensor satisfies the contracted Bianchi identity 1 ∇aG = ∇aR − ∇ R = 0. ab ab 2 b Proof: From the Bianchi identity Rabcd;e + Rabec;d + Rabde;c = 0. Contract a with c gives ac Rbd;e − Rbe;d + g Rabde;c = 0. Contract b with d gives bd bd ac R;e − g Rbe;d + g g Rabde;c = 0. bd ac ac bd But g g Rabde;c = −g Rae;c = −g Rbe;d. Hence, 1 ∇ R − 2∇bR = 0 ⇒ ∇aR − ∇ R = 0. e be ab 2 b

Theorem 4.13 (Lovelock). Let Hab be a such that 1. in any coordinate chart, at any point, Hµν is a function of gµν , gµν,ρ, gµν,ρσ; a 2. ∇ Hab = 0; 3. either space-time is four-dimensional or Hµν depends linearly on gµν,ρσ. Then there exist constants α, β such that Hab = αGab + βgab. Hence we have the Einstein equation

Gab + Λgab = 8πGTab.

5 Ch.7 - Diffeomorphisms and

Definition 5.1. Let M,N be differentiable manifolds of m, n. A function φ : M → N is smooth −1 iff ψA ◦ φ ◦ ψα is smooth for all charts ψα of M and all charts ψA of N. Note that this is a map from a subset of Rm to a subset of Rn. Definition 5.2. Let φ : M → N and f : N → R be smooth functions. The pull-back of f by φ is the function φ∗(f): M → R defined by φ∗(f) = f ◦ φ, i.e., φ∗(f)(p) = f(φ(p)).

Definition 5.3. Let φ : M → N be smooth. Let p ∈ M and X ∈ TpM. The push-forward of X with respect to φ is the vector φ∗(X) ∈ Tφ(p)N defined as follows. Let λ be a smooth curve in M passing through p with tangent X at p. Then φ∗(X) is the tangent vector to the curve φ ◦ λ in N at the point φ(p).

6 Lemma 5.4. Let f : N → R. Then ∗ (φ∗(X))(f) = X(φ (f)). Proof: Let λ(0) = p wlog. Then

d d (φ (X))(f) = [(f ◦ (φ ◦ λ)(t))] = [((f ◦ φ) ◦ λ)(t)] = X(φ∗(f)). ∗ dt t=0 dt t=0

∗ ∗ Definition 5.5. Let φ : M → N be smooth. Let p ∈ M and η ∈ Tp M defined by (φ (η))(X) = η(φ∗(X)) for any X ∈ TpM.

Lemma 5.6. Let f : N → R. Then φ∗(df) = d(φ∗(f)).

Proof: Let X ∈ TpM. Then

∗ ∗ ∗ (φ (df))(X) = df(φ∗(X)) = (φ∗(X))(f) = X(φ (f)) = d(φ (f))(X).

Definition 5.7. A map φ : M → N is a diffeomorphism iff it is one-to-one, onto, smooth, and has a smooth inverse. Definition 5.8. Let φ : M → N be a diffeomorphism and T a tensor of type (r, s) on M. Then the push-forward of T is a tensor φ∗(T ) of type (r, s) on N defined by

∗ ∗ −1 −1 φ∗(T )(η1, ..., ηr,X1, ..., Xs) = T (φ (η1), ..., φ (ηr), (φ )∗(X1), ..., (φ )∗(Xs)).

∗ −1 Note that pull-back can be defined in a similar way, with the result φ = (φ )∗. Definition 5.9. Let φ : M → N be a diffeomorphism. Let ∇ be a covariant derivative on M. The push-forward of ∇ is a covariant derivative ∇˜ on N defined by

˜ ∗ ∇X T = φ∗(∇φ∗(X)(φ (T ))) where X is a vector field and T a tensor field on N. Definition 5.10. A diffeomorphism φ : M → M is a transformation of a tensor field T iff φ∗(T ) = T everywhere. A symmetry transformation of the metric tensor is called an isometry.

Definition 5.11. Let X be a vector field on a manifold M. Let φt be the map which sends a point p ∈ M to the point parameter distance t along the integral curve of X through p. It can be shown that φt is a diffeomorphism. Definition 5.12. The Lie derivative of a tensor field T with respect to a vector field X at p is

((φ−t)∗T )p − Tp (LX T )p = lim . t→0 t

Definition 5.13. If φt is a symmetry transformation of T for all t then LX T = 0. If φt are a 1-parameter of isometries then LX g = 0, i.e., ∇aXb + ∇bXa = 0. This is Killing’s equation and solutions are called Killing vector fields. Proof (LX g = 0 ⇔ ∇aXb + ∇bXa = 0):

(LX g)(Y,Z) = LX (g(Y,Z)) − g(LX Y,Z) − g(Y, LX Z).

µ ν ρ µ ν ρ µ ρ µ ν µ ρ ν ρ ν ⇒ (LX g)µν Y Z = X ∂ρ(gµν Y Z ) − gµν (X ∂ρY − Y ∂ρX )Z − gµν Y (X ∂ρZ − Z ∂ρX ) ρ µ ν ρ µ ν ν ρ µ µ ρ ν = X ∂ρ(gµν Y Z ) − gµν (X ∂ρ(Y Z ) + gµν (Z Y ∂ρX + Y Z ∂ρX ) ρ ρ ρ µ ν = (X ∂ρgµν + gρν ∂µX + gµρ∂ν X )Y Z .

7 ρ ⇒ (LX g)µν = X ∂ρgµν + ∂µXν + ∂ν Xµ.

Now, in normal coordinates, ∂ρgµν = 0 and ∂ ⇔ ∇ since Γ = 0.

⇒ (LX g)µν = ∇µXν + ∇ν Xµ.

This is a tensor equation so is valid not only in normal coordinates, and p is arbitrary. Hence

(LX g)ab = ∇aXb + ∇bXa.

Lemma 5.14. Let Xa be a Killing vector field and let V a be tangent to an affinely parametrised geodesic. a Then XaV is constant along the geodesic. Proof: a a b b a ∇V (XaV ) = V V ∇bXa + XaV ∇bV = 0

since the first term vanishes by the antisymmetry of ∇bXa and the second vanishes since V is parallely transported along itself.

6 Ch.9 - Differential forms

Definition 6.1. Let M be a differentiable manifold. A p-form on M is an antisymmetric (0, p) tensor field on M. Definition 6.2. The wedge product of a p-form X and a q-form Y is the (p + q)-form X ∧ Y defined by

(p + q)! (X ∧ Y ) = X Y . a1...apb1...bq p!q! [a1...ap b1...bq ]

Definition 6.3. The of a p-form X is the (p + 1)-form dX defined in a coordinate basis by

(dX)µ1...µp+1 = (p + 1)∂[µ1 Xµ2...µp+1]. Definition 6.4. X is closed if dX = 0 everywhere. X is exact if there exists a (p − 1)-form Y such that X = dY everywhere. Exact implies closed, but the converse is true only locally. Lemma 6.5 (Poincar´e). If X is a closed p-form, p ≥ 1, then for any r ∈ M, there exists a neighbourhood O of r and a (p − 1)-form Y such that X = dY in O.

7 Others

Definition 7.1. The energy-momentum tensor is defined in general relativity as 2 δS T ab = √ matter . −g δgab

8