ROTATING BLACK HOLES

The Geometry of Kerr Spacetime

Anusar Farooqui

Copyright

All rights reserved by the author. Commercial distribution and sale of thedocument is in violation of international copyright law. However, feel free tocopy and distribute this document free of charge.

Caveat emptor

This document contains material pertaining to the geometry of Kerr black holes.The goal of the author in creating this document was to learn the materialhimself. They are neither complete nor is there any guarantee of correctnessof the exposition. Important sections will be skimmed or skipped. For a moresubstantial treatment of the material the reader is referred to Barrett O’Neill’sexcellent text, ‘The Geometry of Kerr black holes’.

Notational remark

Throughout this document we will follow the Einstein summation conventionwithout exception. Which is, if the same index appears both as a subscript and asuperscript on one side of an equation then we suppress the summation sign, e.g.,∑ni=1 x

i ∂∂xi will appear simply as xi ∂

∂xi . This will also apply, mutatis mutandisto equations with multiple indices: no summation signs will ever appear for suchindices without an explanation.

1

1 Preliminaries

1.1 Manifolds

Differentiable Structures

A topological manifold is a topological space locally homeomorphic to Rn. Anatlas is a collection of smooth coordinate charts that cover the manifold and areglued together consistently with smooth transition maps on overlapping neigh-borhoods. An atlas endows a unique differentiable structure to a topologicalmanifold which is then called a smooth manifold. A map ϕ : M −→ N betweenmanifolds is smooth if, with coordinate charts ζ and ξ, the composition mapsζ ϕ ξ−1 are smooth.

The set of all smooth, real-valued functions f : M −→ R is a commutativering under addition and multiplication of functions denoted by F(M) or moreexplicity by C∞(M ; R). A diffeomorphism is a smooth map ϕ : M −→ N witha smooth inverse ϕ−1, in which case M and N are said to be diffeomorphic.

Tangent Vectors

A tangent vector X to M at x ∈ M is an R-linear map X : F(M) −→ R withthe Leibnizian property

X[fg] = Xfg(x) + f(x)Xg ;∀f, g ∈ F(M) (1.1.1)

where we supress x in the subscript and the brackets for clarity. The set ofall tangent vectors to a manifold M at a point x is denoted by Tx(M). Thisis a vector space under scaler multiplication by real numbers and addition offunctions.

Let x ∈ U an open subset of M and ξ : U −→ Rn be a chart with localcoordinates x1, . . . , xn. Then the partials of the coordinate expression f ξ−1

of f are tangent vectors ∂∂x1

, . . . , ∂∂xn and form a basis for the vector space

Tx(M). The disjoint union of the tangent spaces at all points of the manifold isitself a manifold called the tangent bundle TM =

∐x∈M Tx(M). The projection

π : TM −→M sends Xx 7−→ x and (x1, . . . , xn,∂∂x1

, . . . , ∂∂xn

) : π−1(U) −→ R2n

is a chart on TM .

Curves

A curve in a manifold M is a smooth map γ : I −→ M where I ⊂ R. Thetangent vector γ(s) to M at γ(s) for s ∈ I is defined by

γ(s)[f ] =d(f γ)

ds(s) ; ∀f ∈ F(M) (1.1.2)

Differential maps

Given a mapping ϕ : M −→ N , for each point x ∈M there is a linear transfor-mation dϕ|x : TxM −→ Tϕ(x)N called the differential map or the push-forward

2

defined by

ϕ∗|x[f ] ≡ dϕ(x)[f ] ≡ X[f ϕ] ; ∀X ∈ TxM and f ∈ F(M) (1.1.3)

An equivalent characterization is that dϕ preserves tangents, i.e., dϕ sendsγ(s) 7−→ (ϕ γ)′(s). One can regard it as a single map dϕ : TM −→ TN oftangent spaces.

Vector Fields

A vector field X is a cross section of TM , i.e., a smooth map X : M −→ TMsuch that π X = idM . X is smooth means that Xf is smooth wheneverf is smooth. The set X(M) of vector fields on M is a module over F(M),the ring of real-valued functions on M . X satifies the Leibnizian property (1)and is therefore a derivation†. In terms of local coordinates, we can writeX = Xi ∂

∂xi with the Einstein summation convention. This is the fundamentallink between the invariant description of X and its tensor description as an n-tuple of functions Xi ≡ X(xi).

The Lie bracket [X,Y ] of vector fields X and Y is characterized by

[X,Y ]f = X(Y f)− Y (Xf) ;∀f ∈ F(M) (1.1.4)

This operation is bilinear over R, skew-symmetric and satisfies the Jacobiidentity:

[X, [Y,Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0 (1.1.5)

Integral Curves

A vector field on a manifold has another interpretation as a differential equationwhose solutions are curves, called integral curves, that at each point have thetangent specified by the vector field. That is

γ(s) = Xγ(s) ;∀s ∈ I (1.1.6)

In local coordinates,

γ(s) = γ(s)[xi]∂

∂xi(1.1.7)

=d

ds(xi γ)

∂

∂xi(1.1.8)

Thus the vector equation becomes a linear system of ordinary differentialequations:

†implicit definition

3

d

ds(xi γ) = Xi(x1, . . . , xn) ;∀i = 1, 2, . . . , n (1.1.9)

Since X is smooth, local existence and uniqueness follows from standardresults, so that for each point x ∈ M, ∃! integral curve γx, with the largestpossible domain starting at x : γx(0) = x. And integral curves γ, β of X canmeet only if one is a reparametrization of the other, i.e., β(s) = γ(s+ const).

A vector field is complete if each of its maximal integral curves is defined onthe entire real line. Suppose X is complete. For each s ∈ R, let ϕs : M −→ Nbe the smooth mapping that sends each point x 7−→ γx(s). Clearly, ϕ0 = idMand ϕs+t = ϕs ϕt ;∀s, t ∈ R. Since it has a smooth inverse ϕ−s, each ϕs is adiffeomorphism. The collection ϕs : s ∈ R is called the flow of X.

If X is not complete its flow is still defined locally:

∀x ∈ M,∃ a neighborhood U of x and ε > 0, such that ∀y ∈ U, γy isdefined at least on (−ε, ε) so that ϕs : U −→M is well defined with the aboveproperties. In this case, it is called a local flow.

Remark 1.1.1. We will learn in the next chapter that a vector field on a semi-Riemannian manifold is Killing if its flow has the property that for each s, ϕsis not just a diffeomorphism but an isometry. Killing vector fields are extremelyimportant for the geometry of Kerr black holes.

One-forms

The cotangent space to M at x, is the dual of the tangent space whose elementsare called covectors:

T ∗xM ≡ (TxM)∗ ≡ ωx : TxM −→ R, ωx linear (1.1.10)

The disjoint union of cotangent spaces is called the cotangent bundle, T ∗M ≡∐x T∗xM . A section of the cotangent bundle is called a one-form. That is, a

1-form ω is a covector field, i.e., it assigns to each x ∈M a covector ωx ∈ T ∗xM .The space of all 1-forms or covector fields on M , X∗(M) is a module over F(M)under pointwise addition and multiplication by real valued functions on M .

A vector field X ∈ X(M) and a one form ω ∈ X∗(M) combine naturallyto give a real valued function ωX ∈ F(M). Each function f ∈ F(M) gives riseto a 1-form, its differential df . The cotangent space has a basis of coordinateone-forms dx1, . . . , dxn, dual to ∂

∂x1 , . . . ,∂∂xn and each one-form can be ex-

pressed as ω = ω( ∂∂xi )dx

i. In particular, computing in local coordinates, werecover the multilinear calculus formula for the total differential of a function:df = ∂f

∂xi dxi.

Remark 1.1.2. We the following transformation rules under change of local co-ordinates:

4

∂

∂xi=

∂yj

∂xi∂

∂yj(1.1.11)

dxi =∂xi

∂yjdyj (1.1.12)

Bundles

A (left) action of a Lie group G on a manifold F : G 11 F , is a smooth mapG× F −→ F, (g, x) 7−→ gx such that

g(h(x)) = gh(x) ;∀g, h ∈ G and x ∈ F

and with the identity e ∈ G

ex = x ;∀x ∈ F

For a fixed g ∈ G, the map λg : F −→ F, x 7−→ gx is a differomorphismwith the smooth inverse λg−1 . The action of a Lie group is said to be effectiveif

λg = idM =⇒ g = e (1.1.13)

In Kerr applications, the circle group acts effectively on the unit 2-sphere,S1 11 S2 by rotation around the z-axis as follows. For α ∈ R, we have

Rα(x, y, z) = (x cosα− y sinα, x sinα+ y cosα, z) (1.1.14)

A smooth fiber bundle consists of a smooth map π : E −→ B, a Lie groupacting on a manifold: G 11 F , and a collection Φ of smooth mappings φ suchthat

1. Local product condition: Each φ : π−1(U) −→ U ×F is a diffeomorphismfor some open neighborhood U ⊂ B such that π(φ−1(b, x)) = b ;∀x ∈F and b ∈ U , i.e., the following diagram commutes

π−1(U)

φ // U × F

Proj.

yyttttttttttt

U

2. Overlap condition: If φ, ψ ∈ Φ have intersecting domains U ∩ V in B,there is a smooth map γ : U ∩ V −→ G such that

φ(b, x) = ψ(b, λγ(b)(x)) ;∀b ∈ U ∩ V and x ∈ F

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Terminology: π is the projection, E is the total manifold, B is the base, F is thefiber, G is the structure group and Φ is the set of bundle coordinate systems φ.For brevity, one often writes

G,,

F // Eπ // B

That is, Structure group 11 Fiber // Total manifold // Base.A vector bundle of rank k over M , is a bundle F −→ E −→ M with fiber

F = Rk with the general linear group as the structure group, in which case wehave the local trivialization†

π−1(U)

Φ // U ×Rk

Proj.

yyssssssssss

U

and the restriction of Φ to π−1(x) is the linear isomorphism π−1(x) ∼= x×Rk ∼=Rk. In the case of vector bundles, we usually denote the bundle by E −→M orsimply by E. For instance, the tangent bundle TM −→ M is a vector bundleas is the cotangent bundle T ∗M −→M .If there exists a map Φ such that

E

π

Φ // M ×Rk

Proj.

wwwww

wwww

M

then Φ is called a global trivialization and E is said to be a trivial bundle.

1.2 Tensors

The Invariant approach

Let s and r be nonnegative integers, not both zero. A tensor field of type (r, s)on M is an F-multilinear function

A : X∗ × · · · × X∗︸ ︷︷ ︸r

×X× · · · × X︸ ︷︷ ︸s

−→ F

with the convention that a tensor field of type (0, 0) is just a function f ∈ F.Thus, for every choice of r covector fields (1-forms) ω1, . . . , ωr and s vectorfields X1, . . . , Xs, A(ω1, . . . , ωr, X1, . . . , Xs) is a real valued function and Ais linear over F in each argument separately. In particular, a function f can befactored out of each slot.

†Definition

6

Let Trs denote the set of all type (r, s) tensor fields on M . Clearly, T01 =

X∗. Now X is the dual of X∗, indeed, we have X 3 X : X∗ −→ F such thatX(ω) = ω(X) ,∀ω. So we also have T1

0 u X. Only tensors of the same typecan be added but arbitrary tensors can be multiplied using the tensor product.For example, if A is of type (1, 1) and B is of type (2, 3) the A⊗B is given by

(A⊗B)(θ, ω,X, Y, Z) = A(θ,X)B(ω, Y, Z) (1.2.1)

for all covector fields θ, ω and vector fields X,Y, Z on M .The tensor product is F-bilinear and associative but may not be commuta-

tive. However, if A is covariant, that is, of type (0, s) and B is contravariant,that is, of type (r, 0), then A and B commute.

The Classical approach

Let x1, . . . , xn be local coordinates for a chart on a neighborhood U ⊂ M .Then, on U , the components of A ∈ Trs are the real valued functions computedas follows:

Ai1,...,irj1,...,js= A(dxi1 , . . . , dxir ,

∂

∂xj1, . . . ,

∂

∂xjs) (1.2.2)

The superscripts are contravariant indices and the subscripts are covariant in-dices. The operations given in the previous section can be easily computed inlocal coordinates. For example, if A is a tensor field of type (1, 1) and we aregiven ω = ωidx

i and X = Xi ∂∂xi then

A(ω,X) = AijωiXj (1.2.3)

where we have applied the Einstein summation convention to multi-indices forthe first time and supressed

∑i,j . The components of a sum of (r, s) tensors are

the sum of components. The components of a tensor product are the productof the components in tensorial sense: if A is of type (1, 1) and B is type (1, 2)then A⊗B has components:

(A⊗B)ijk`m = AikBj`m (1.2.4)

Remark 1.2.1. Its useful to define tensors invariantly, since then we don’t haveto check that they are well defined. But they are more explicit and easier tocompute in local coordinates. Also, there is the built in error detection featurecalled index balance: that the same unsummed subscripts and superscripts mustappear on each side of a tensor equation.

Frame Fields

A frame field E1, . . . , En, is an alternative basis for the tangent space TxMfor each x ∈ U ⊂M , where each basis element can be expressed in terms of thecoordinate frame ∂

∂x1 , . . . ,∂∂xn as follows

7

Ei = Eji∂

∂xj(1.2.5)

The coframe field consists of the dual 1-forms ω1, . . . , ωn defined implicitlyby

ωi(Ej) = δij (1.2.6)

For any vector field X, we have the following duality formula

X = ωi(X)Ei (1.2.7)

The components of a tensor field can be expressed relative to a frame field,replacing the coordinate frames in (16) by the new basis vectors. One advantageof using generalized frame fields is that we can often define them over largerdomains. For instance, these is a globally defined frame field on a torus while aminimum of 4 charts are required to construct an atlas.

Change of coordinates is replaced by change of frame fields. Given a framefield Ei with dual ωi and a new frame field Fj with dual θj on thesame domain U . There are unique n×n-matrix valued functions a, b on U suchthat

Fj = akjEk and θi = bimωm (1.2.8)

By duality

bikakj = δij (1.2.9)

Note that in the special case of coordinate frames, a and b are just the Jacobian

matrices ∂yi

∂xj and ∂xi

∂yj . Akin to the coordinate frames case, we have a tensortransformation rule for change of frame fields that determines a unique tensorin the new frame field. For example, if A is a (1, 1) tensor with components Aijrelative to Ei, then its components in the new frame field Fi are given by

A(θi, Fj) = A(bi`ω`, akjEk) = bi`a

kjA

`k (1.2.10)

Clearly, the tensor A is independent of the choice of frame field.

Contraction

This is a generalization of the concept of the trace of a linear operator to tensors.For a choice of a contravariant index and a covariant index of the componentsof A, take the trace. For example, is A is of type (2, 3) then choosing thefirst contravariant superscript index and the second covariantsubscript index, theresulting tensor C1

2A has components A`ij`k and is called a contraction of A. Ingeneral, contraction reduces type from (r, s) to (r − 1, s− 1).

8

Tensors at a point

Let x ∈ M . Then an (r, s) tensor at the point x ∈ M is an R-multilinearfunction

A : T ∗x (M)× · · · × T ∗x (M)︸ ︷︷ ︸r

×Tx(M)× · · · × Tx(M)︸ ︷︷ ︸s

−→ R (1.2.11)

A tensor field on M determines at each point x ∈ M , a tensor Ax called itsvalue at x.

One can further replace, Tx(M) by an finite dimensional real vector space Vand T ∗x (M) by V ∗ and define (r, s) tensors as before. In these terms, an innerproduct g on V is a symmetric, positive definite (0, 2) tensor on V . g is scalarproduct if, instead of positive definiteness we just require it to be nondegenerate:g(v, w) = 0 ,∀w ∈ V =⇒ v = 0.

1.3 Differential Geometry

Metric Tensors

A metric tensor on a manifold M is a smooth covariant (0, 2) tensor thatassigns to each point on the manifold, a scalar product on the tangent space,M 3 x 7−→ gx ∈ TxM . Then (M, g) is called a semi-Riemannian manifold. Forvector fields V,W on M , g(V,W ) is a real valued function, sometimes denotedby 〈V,W 〉.

If g is positive definite (the scaler products gx are inner products), then gis a Riemannian metric and (M, g) is a Riemannian manifold. If each gx is aLorentz scaler product (see section 1.5), then g is a Lorentz metric and (M, g)is called a Lorentz manifold.

On a chart U with local coordinates x1, . . . , xn, the components of themetric tensor are

gij = g(∂

∂xi,∂

∂xj) = 〈 ∂

∂xi,∂

∂xj〉 (1.3.1)

Hence, we can write g = gijdxi⊗dxj . The metric tensor is often replaced by

the line element ds2 ≡ q, which assigns to each point x the associated quadraticform qx of gx. So the value of ds2 on a vector field V is g(V, V ) = 〈V, V 〉. Themetric tensor can be obtained from the line element by polarization. Locally,we can express the line element as

ds2 = gijdxidxj (1.3.2)

Indeed, given a vector field V , we have

9

ds2(V ) = 〈V, V 〉

= 〈V i ∂

∂xi, V j

∂

∂xj〉

= V iV j〈 ∂∂xi

,∂

∂xj〉

= gijViV j

= gijdxi(V )dxj(V )

= (gijdxidxj)(V )

Key Examples

For 0 ≤ ν ≤ n, let Rnν be the manifold Rn furnished with the line element

ds2 = εi(dxi)2, where

ε =

−1 ,∀i ≤ ν+1 ,∀i > ν

Rnν is called a semi-Euclidean n-space of index ν. When ν = 0 this is just or-

dinary Euclidean space, which is obviously a Riemannian manifold. And whenν = 1 and n ≥ 2, it is Minkowski n-space, a Lorentz manifold.

The standard metric on Rn is denoted by gstd. The pullback of gstd un-der the local coordinate maps endows every smooth manifold with a cannonicalRiemannian metric also denoted by gstd. Locally, this is clear. One only needsto check for the consistency on the overlap to demonstrate this very importantexistence result.

Isometries

A diffeomorphism φ : M −→ N of semi-Riemannian manifolds is an isometry ifit is preserves scarlar products:

〈dφ(v), dφ(w)〉 = 〈v, w〉 for all tangent vectors v, w ∈M (1.3.3)

Then M and N are isometric and are geometrically indistinguishable. Supposeds2N = gijdy

idyj on V ⊂ N and ds2M = gijdx

idxj on U ⊂ M . Then, we havethe following useful criterion in terms of local coordinates.

The mapping φ : U −→ V is an isometry on U if ds2N pulls back to ds2

M :

φ∗(ds2N ) = ds2

M (1.3.4)

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The Levi-Civita Connection

1.3.1 The Fundamental Theorem of semi-Riemannian Geometry. Ona semi-Riemannian manifold there exists a unique function ∇ : X × X −→ Xsuch that

1. ∇V (W ) is F-linear in V

2. ∇V (W ) is R-linear in W

3. ∇V (fW ) = V [f ]W + f∇V (W ) ;∀f ∈ F

4. [V,W ] = ∇V (W )−∇W (V )

5. X〈V,W 〉 = 〈∇XV,W 〉+ 〈V,∇XW 〉

∇ is called the Levi-Civita connection of M . For vector fields V and W ,∇V (W ) is the covariant derivative of W with respect to V .

In local coordinates, ∇ is described by Christoffel symbols, real-valued func-tions Γkij such that ∇ ∂

∂xi= Γkij

∂∂xk

for all i, j. Then,

Γkij =1

2g`k[∂gi`

∂xj− ∂gij

∂xl+∂gj`

∂xi

](1.3.5)

For a curve γ in M with γ 6= 0 and a vector field Y , the covariant derivativeof Y along γ is defined by Y ′ = ∇γY which can be written locally as

DY

dt≡ Y ′ =

[dY k

ds+ Γkij

dY i

ds

dxj

ds

]∂

∂xk(1.3.6)

where in a slight abuse of notation we have written xi in place of xi γ.

If Y ′ = 0, the vector field is said to be parallel for parameter values sand t, Y (t) is said to be obtained by parallel transport along γ. The tangent γof γ is a vector field on γ, and the covariant derivative of γ is the accelerationγ ≡ Dγ

dt ≡ ∇γ γ of the curve.

Geodesics

A curve γ is a geodesic if its acceleration is zero: ∇γ γ = 0. Straight linesin Euclidean space generalize to geodesics. In local coordinates, a geodesic ischaraterized by the geodesic equations:

xk + Γkij xixj = 0 for all k = 1, . . . , n (1.3.7)

11

Where we write xi for xi γ and supress the argument for γ(·). One usesstandard results from ODE theory to obtain the following result.

1.3.2 Local Existence and Uniqueness for Geodesics. Given a point x ∈M and a tangent vector X ∈ TxM , there exists a unique geodesic γ : I −→ Mpassing through x with tangent vector X.

If every geodesic can be extended over the entire real line R then M is saidto be geodesically complete. A reprametrization β(s) = γ(h(s)) of a geodesic γis a also a geodesic only if it is affine, i.e., h(s) = as + b. A pregeodesic is acurve that admits a reparametrization as a geodesic.

Here is a useful reformulation of the geodesic equations. Let L : TM −→ Rbe a smooth function on the tangent bundle. A curve γ on M is extremal for Lif there is an atlas for TM such that the Euler-Lagrange equations hold:

d

ds

[∂L

∂xi(γ)

]=∂L

∂xi,∀i = 1, . . . , n (1.3.8)

For a semi-Riemannian manifold M , let L : TM −→ R, ν 7−→ 12 〈ν, ν〉.

In local coordinates, L = 12gij(x

1, . . . , xn)xixj . The resulting Euler-Lagrangeequations are

d

ds

[gklxl

]=

1

2

∂gij∂xk

xixj ,∀k = 1, . . . , n (1.3.9)

These equations are equivalent to the geodesic equations. This formulationwill be extremely useful in computing geodesics in Kerr spacetimes.

Type-changing

On a semi-Riemannian manifold, the metric tensor can be used to change anytensor of type (r, s) to that of type (r′, s′) with r+s = r′+s′. They are regardedas being equivalent. It suffices to show how to change a tensor of type (r, s) tothat of type (r − 1, s+ 1) and (r + 1, s− 1). Let A be a (r, s) tensor field withcomponents

Ai1,...,irj1,...,js

relative to a frame field Ei. We change A to type (r − 1, s + 1) by loweringan index to obtain the components for the new tensor as follows:

Ai1,...,ν,...,irj1,...,ν,...,js= gνεA

i1,...,ε,...,irj1,.........,js

(1.3.10)

for a choice of contravariant index ν ∈ i1, . . . , ir, where gνε ≡ 〈Eν , Eε〉 arethe components of the metric tensor relative to the frame field. Similarly, wechange A to type (r + 1, s− 1) by raising an index

Ai1,...,ν,...,irj1,...,ν,...,js= gνεAi1,.........,irj1,...,ε,...,js

(1.3.11)

where gνε is the inverse of the matrix gνε.

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Lemma 1.3.1. Let Ei be an othonormal frame field, so gij = δijεj, withεj = ±1. The effect of raising or lowering an index i on the components of atensor is just multiplication by εi.

On a semi-Riemannian manifold, the contraction operation can be extendedto two indices of the same variance. One needs only to raise or lower one ofthem and use the natural contraction.

Curvature

The Riemannian curvature tensor of a semi-Riemannian manifold M is thefunction R : X× X× X −→ X given by

RXY Z = ∇X(∇Y Z)−∇Y (∇XZ)−∇[X,Y ]Z (1.3.12)

If the Lie bracket of vector fields X and Y is zero then the Riemannian curvaturetensor measures the failure of ∇X and ∇Y to commute.

1.3.3 Symmetries of the curvature tensor. For all vector fields X,Y, Z,W

1. RXY Z = −RY XZ

2. 〈RXY Z,W 〉 = −〈RXYW,Z〉

3. RXY Z +RY ZX +RZXY = 0

4. 〈RXY Z,W 〉 = 〈RZWX,Y 〉

In terms of the frame field Ei, the components of the curvature tensor are

Rijk` = ωi(REkE`Ej) (1.3.13)

Using the duality formula X = ωi(X)Ei one obtains

R ∂

∂xk∂

∂x`

(∂

∂xj

)= Rijk`

∂

∂xi(1.3.14)

Due to the symmetries of the curvature tensor, it has only one nonzerocontraction modulo the sign called the Ricci curvature tensor denoted by Ric.By convention, the sign is specified by the following lemma:

Lemma 1.3.2. In terms of an arbitrary frame field, the components of Ric are

Rij = Ric(Ei, Ej) = Rkikj (1.3.15)

In the orthonormal case, Ric(X,Y ) = εk〈RXEkEk, Y 〉.

13

Further contracting Ric one obtains the scaler curvature S = gijRij . Thesectional curvature K, a real valued function on the set of all 2-planes Π tangentto M is defined as follows. For a basis X,Y of Π, we have

K(Π) =−〈RXYX,Y 〉

〈X,X〉〈X,X〉 − 〈X,Y 〉2(1.3.16)

If S is a surface and x ∈ S, the sectional curvature K is just the classicalGaussian curvature of S at x.

The Riemannian curvature tensor R has the following symmetry properties withthe covariant derivative:

(∇ξR)XY Z = ∇ξ(RXY Z)−R∇ξ(X)Y Z −RX∇ξ(Y )Z −RXY∇ξ(Z) (1.3.17)

and the Bianchi curvature identities

(∇XR)Y Z + (∇YR)ZX + (∇ZR)XY = 0 (1.3.18)

The various forms of curvature are geometric invariants, i.e., they are preservedby isometries.

Killing Vector Fields

Let X be a vector field on a semi-Riemannian manifold M with flow ϕs. Ifeach ϕs is not just a diffeomorphism but an isometry then X is called a Killingvector field or sometimes an infinitisimal isometry.

1.3.4 Proposition. The following are equivalent:

1. X is a Killing vector field

2. X〈Y, Z〉 = 〈[X,Y ], Z〉+ 〈Y, [X,Z]〉

3. 〈∇YX,Z〉 = −〈∇ZX,Y 〉

We have the following important conservation lemma:

Lemma 1.3.3. If X is a Killing vector field on M and γ is a geodesic, thenthe scalar product 〈X, γ〉 is constant along γ.

Proof. Since γ = 0, dds 〈X, γ〉 = 〈∇γX, γ〉 = 0 by part 3 of the previous proposi-tion.

If M is connected, then any isometry ϕ of M is completely determined byits differential map dϕx at a single point x ∈ M . So a Killing vector field iscompletely determined by its values on a arbitrarily small neighborhood of asingle point.

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1.4 Extending Manifolds

Coordinate Extension

Let U be a semi-Riemannian manifold that we want to extend. Find a diffeo-morphism φ : U −→ V ⊂ M a smooth or analytic manifold. Choose a metricg0 on V that makes φ an isometry. Extend the tensor field g0 to a larger setW ⊃ V such that g|V = g0 and g is still a metric tensor on W .

Let (U, ξ) be a coordinate chart with local coordinates x1, . . . , xn andlet gij be the components of a metric tensor g on U . So ξ is a diffeomorphism ofU onto its image ξ(U) ⊂ Rn. Assign ξ(U) the metric that makes ξ an isometry.The components gij have the same expressions as before but with x1, . . . , xnnow considered as the natural Cartesian coordinates of Rn. Now it we maybe able to extend the functions gij to an open set containing ξ(U) and in theanalytic case such extensions are unique.

Gluing Topological Spaces

Given two semi-Riemannian manifolds M and N , together with an isometry µfrom U ⊂ M to V ⊂ N . There is a natural way to glue them together alongU ≈ V producing a new manifold Q. We call M,N, µ the gluing data and µthe matching map.

Consider M and N as topological spaces. Define on M tN , the equivalencerelation ∼ defined by

x ∼ y! [x = y or x = µ(y) or y = µ(x)]

Then Q = M tµ N ≡ M t N/ ∼. The natural injections ı : M → Q and : N → Q combine to give the natural projection π : M t N Q for a quo-tient space. In particular, a subset S ⊂ Q is open ⇐⇒ both ı−1(S) ⊂ M and−1(S) ⊂ N are open. Equivalently, a map ϕ : Q −→ X is continuous ⇐⇒both ϕ ı and ϕ are continuous.

The natural injections ı and are homeomorphisms onto their respectiveimages. If ϕM : M −→ P and ϕN : N −→ P agree on M ∩N then they definewell a map ϕ : Q −→ P . And if ϕM and ϕN are continuous, and supposeϕ|M = ϕM and ϕ|M = ϕM , then ϕ is continous as well.

Note that even if M and N are Hausdorff, Q need not be. As a counterex-ample, consider M = N = R and U = V = t : t < 0 with µ the identity map.

We say that the gluing data M,N, µ : U −→ V safisfies the Hausdorffcondition if there does not exist any sequence xn in U such that both

limxn ∈M − U and limµ(xn) ∈ N − V

If the gluing data satisfies the Hausdorff condition and ifM andN are Hausdorff,then so is Q.

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Gluing Semi-Riemannian Manifolds

For smooth manifolds it suffices to assign Q the atlas consisting of all chartsof M ≈ ı(M) and N ≈ (N). Its straightforward to check that they agree onthe overlaps and the transition maps are smooth. Moreover, the same holds foranalytic manifolds as well.

Furthermore, on semi-Riemannian manifolds with the matching map anisometry the metric tensors agree on M ∩ N and hence we obtain a metrictensor on Q. We will use this such extensions often in the construction ofmaximal extensions of Kerr spacetimes.

1.5 Lorentz Vector Spaces

Let V be a vector space over the reals of dimension n. A scalar product g on V isa symmetric, nondegenerate bilinear form on V . Vectors v and w are orthogonalif g(v, w) ≡ 〈v, w〉 = 0 and we write v ⊥ w. g is nondegenerate asserts that zerois the only vector orthogonal to all the vectors, i.e., v ⊥ V =⇒ v = 0. A usefulcriterion for nondegeneracy is that one, and hence every, basis v1, . . . , vn forV the matrix gij = g(vi, vj) is invertible.

(V, g) is called a scalar product space. If g is positive definite, it is an innerproduct and V is then called an inner product space. The norm is defined as|v| ≡ |〈v, v〉| 12 . If |u| = 1 then u is called a unit vector and a set of orthogonalunit vectors is called orthonormal. Every scalar product space has an orthonor-mal basis. Indeed, any orthonormal set can be enlarged to an orthonormal basis.

Associated to an orthonormal basis e1, . . . , en are n numbers εi = ± suchthat 〈ei, ej〉 = δij εj . Modulo the order, every orthonormal set has the samesigns. (ε1, . . . , εn) is called the signature of V , usually listing the negative signsfirst. The number ν of negative signs is called the index of V .

A Lorentz vector space is a scalar product space of dimension n ≥ 2 withindex ν = 1. A vector v in a Lorentz vector space V is

¶ spacelike if 〈v, v〉 > 0 or v = 0¶ timelike if 〈v, v〉 < 0 and v 6= 0¶ null if 〈v, v〉 = 0

The type that a vector falls into is called its causal character. A vector or-thogonal to a timelike vector z is spacelike. Thus, z⊥ is an inner product spaceand V = Rz ⊕ z⊥. For v ∈ V , writing v = au+ x for a timelike vector u and aspacelike vector x we obtain

〈v, v〉 = −a2 + |x|2 (1.5.1)

Since a cone is a subset closed under multiplication by positive scalars, the setΛ of all null vectors of V is called the nullcone.

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The nullcone Λ has two components Λ+ and Λ− with Λ− = −Λ+. Eachcomponent is a cone diffeomorphic to R+ × Sn−2. Λ+ and Λ− are also callednullcones and there is no invariant way to distinguish between them. For n ≥ 3,Sn−2 is connected, hence the disjoint sets Λ+ and Λ− are actually the connectedcomponents of the nullcone. Since S0 = ±1 when n = 2, Λ+ and Λ− are notconnected.

The set I of all timelike vectors in V is called the timecone. It has two compo-nents I+ and I−. Each is an open convex cone with I− = −I+. The boundaryof the timecone is the nullcone along with the zero vector: I± = Λ± ∪ 0.

Two timelike vectors z and w are in the same timecone ⇐⇒ 〈z, w〉 < 0. Avector that is not spacelike is called causal. A subspace W of V is

¶ spacelike if g|W is positive definite¶ timelike if g|W is nondegenerate and has index 1¶ null if g|W is degenerate

and its type is called the causal character of W . Note that 0 is spacelike.W is spacelike ⇐⇒ it is an inner product space. For n ≥ 2, W is timelike⇐⇒ W is a Lorentz vector space. In a Lorentz vector space, orthogonal nullvectors are collinear.

Let W be a null subspace of V . Then, there is a nonzero vector d ∈ W ,unique upto multiplication by nonzero scalars, such that d ⊥W . In particular,d is null. Moreover, every vector in W−Rd is spacelike. For a spacelike subspaceW , extending an orthonormal basis to one for V shows that W⊥ is timelike andvice-versa so that V = W ⊕W⊥. It follows that W is null =⇒ W⊥ is null aswell.

1.6 General Relativity

Let S be a 3-dimensional Riemannian manifold called Space. A particle in spaceis a curve γ : I −→ S where we think of t ∈ I as the time parameter. The classicNewtonian case is a mass m located at the origin in S = R3, and particles ofvarious masses m m move in S according to Newton’s laws of motion andthe gravitational law.

Now consider the Riemannian manifold S×R with R as the time axis. Callthis manifold spacetime. Each point (x, t) is called an event. A particle is acurve t 7−→ (γ(x), t) in spacetime. Now replace the Riemannian line elementdσ2 + dt2 by the Lorentz line element dσ2 − dt2. With S = R3 and setting m= 0 gives us Minkowski spacetime. A particle moving solely under the influenceof the central mass is said to be free falling. The key idea of relativity is thatfreely falling particles are geodesics in spacetime.

Spacetimes

Let M be a Lorentz manifold, whose points are called events. We time-orientM by selecting continuously, for the tangent space TxM at each point x ∈ M

17

one of two components of the timecone I calling it the future timecone or simplythe futurecone. The others are called past timecones.

So, a spacetime† is a connected time-oriented 4-dimensional Lorentz manifold.A timelike tangent vector is z is future-pointing if it is in a future timecone,otherwise its past-pointing. A nullcone in the boundary of each future timeconeis a future nullcone, and its vectors are future-pointing. A non-spacelike curveor vector field is future-pointing if all its tangent vectors are future-pointing.

Einstein’s insight was that physical invariants of gravitational fields are pre-cisely the geometric invariants of spacetime. Therefore, no concepts expressedin terms of coordinates are of physical significance unless they are independentof the coordinate descriptions.

Particles

A material particle α in a spacetime M is a future-pointing, timelike curve.The proper time τ of α is its arc-length function, and its mass is m = |α| > 0.Light or electromagnetic radiation has a special place is relativity. A photonor lightlike particle γ in a spacetime M is a future-pointing null geodesic. Alightlike particle has no proper time since |γ| = 0. A material particle is freelyfalling if it is a geodesic. Moreover, it has a constant mass m since, for a geodesic〈α, α〉 = −m2 is constant.

The tangent vector γ = ddsα of a material particle is called energy-momentum

4-vector and denoted by p. Reparametrizing α by its proper time τ and denotingit by α we have

p =dα

ds= m

dα

dssince m =

∣∣∣dαds

∣∣∣ =dτ

ds

The unit vector dαds is called the 4-velocity of the particle. In an abuse of notation

we often write just α for α when the parametrization is clear from the context.For a lightlike particle γ, the energy-momentum 4-vector is just p = γ.

Curvature

In special relativity, the curvature vanishes and spacetime is flat. A vaccumspacetime has no sources of gravity, which is equivalent, via Einstein’s equation,to the vanishing of Ricci curvature. Such a spacetime is called Ricci flat. Thus,a model of gravitational field of a single star must be Ricci flat away from thesource of gravity. Of course, the Riemannian curvature need not be zero. WhenRicci curvature does not vanish, it describes the motion of the source of gravity.Such models are useful in cosmology.

†Henceforth, this will serve as the definition for spacetime.

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Stationary Observers

An observer in spacetime is simply a material particle parametrized by propertime. Observers can send and receive messages and keep track of their propertime. In special relativity, a freely falling observer can impose his proper timeand space on the entire Minkowski spacetime but in general relativity, we needan entire family of observers to get analogous results.

A observer field U on a spacetime M is a future-pointing unit vector field.Each integral curve of U is an observer called a U-observer. An observer fieldU is stationary if there exists a smooth function f > 0 on M such that fU is aKilling vector field. If U is also hypersurface orthogonal, i.e. U⊥ is integrable,then U is static.

A spacetime is absolutely stationary if it has a unique stationary observerfield, in which case the U -observers are said to be at rest. If U is static, theintegral manifolds of U⊥ are 3-dimensional spacelike, submanifolds that are iso-metric under the flow, and hence constitute a common space for the U -observers.The gravitational field outside a single star is stationary if the physical prop-erties of the star are not changing. It it is also not rotating, the spacetime isstatic.Suppose we have local coordinates ξ = (x0, x1, x2, x3), where the coordinatevector field ∂

∂x0 is timelike future-pointing and span ∂∂x1 ,

∂∂x2 ,

∂∂x3 is spacelike.

Then, the coordinate slices x0 = constant are 3-dimensional Riemannian man-ifolds. Moreover, we have the following

Lemma 1.6.1. If (x0, x1, x2, x3) are local coordinates such that ∂∂x0 is timelike

future-pointing and suppose∂gij∂x0 = 0 for all i, j. Then, the observer field U =

∂∂x0 /| ∂∂x0 | is stationary. Conversely, for a stationary U such a coordinate systemcan be found at every point. If in addition g0i = 0 for i > 0, then the observerfield U is static and conversely.

Proof.∂gij∂x0 = 0 for all i, j =⇒ ∂

∂x0 is a Killing vector field. Then, fU = ∂∂x0

with f = | ∂∂x0 |. Conversely, let f > 0 be a function such that fU is Killing. SincefU is nonvanishing, at each point there is a coordinate system (x0, x1, x2, x3)

such that ∂∂x0 = fU . But ∂

∂x0 Killing =⇒ ∂gij∂x0 = 0 for all i, j.

Instantaneous Observers

An instantaneous observer at an event p ∈ M , is a future pointing timelikeunit vector u ∈ TpM . u has the instantaneous information common to everyordinary observer α whose 4-velocity is u as it passes through p. In particular,u knows TpM ∼= Ru⊕ u⊥ ∼= Ru⊕R3. There is a natural way for u to measurethe speed of any particle as it passes through p at proper time τ0. If α is amaterial particle, then α(τ0) = au + x, where x ∈ u⊥. Note that a is theinstantaneous rate at which u’s time, t, is increasing relative to α’s time τ , sowe write a = dt/dτ . Similarly, |x| is the rate at which the arc length σ in u⊥

is increasing relative to τ , so |v| = dσ/dτ . Thus, u measures the speed of α atevent p as

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dσ

sτ=dσ/dτ

dτ/dt=|x|a< 1 (1.6.1)

Where the last inquality holds sinceα is timelike =⇒ 0 > 〈α, α〉|p = −a2 + |x|2.

For a lightlike particle γ, since γ is null we have 0 = 〈γ, γ〉 = −a2 + |x|2, andtherefore dσ/ds = 1. So instantaneous observers always measure the speed oflight to be c = 1.

Now let u and v be two instantaneous observers at an event p ∈M . u writesu = 1 · u + 0 and dσ/ds = 0 so he is at rest. Then writing v = a · u + x heconcludes that v is moving since u 6= v =⇒ x 6= 0. But v writes v = 1 · v + 0and u = b · v + y, concluding that he is at rest and it is u that is moving. Nowa = dtu/dtv, but −1 = 〈u, v〉 = −a2 + |x|2 so a = (1+ |x|2)1/2 > 1. So accordingto u, his time runs faster than v’s. Since u sees himself at rest and v moving,he claims: Moving clocks run slower. v measures dtv/dtu > 1, so his time runsfaster than u’s, but he agrees with the slogan. Their disagreements derive fromtheir different decompositions of TpM into space and time.

Special Relativity

Special relativity is general relativity of a spacetime isometric to Minkowskispacetime R4

1, where the subscript denotes the index. For an arbitrary spacetimeM , special relativity obtains in each tangent space TpM ≈ R4

1.Let α be a material particle with energy momentum vector p = dα/ds =

m dα/dτ , and u be an infinitisimal observer at some event p = α(τ0). Let α bethe projection of α onto the space u⊥.

TpM = Ru+ u⊥ =⇒ α(τ) = t(τ)u+ α(τ)

where t is the observers time and τ is the proper time of the particle. Hence,

dα

dτ=dt

dτu+

dα

dτ

and since this is a timelike unit vector we get

−1 = −( dtdτ

)2

+∣∣∣dαdτ

∣∣∣2Now v = |dα/dt| = dσ/dt is the Newtonian speed of α as measured by u andby chain rule |dα/dτ | = v dt/dτ . Thus, dt/dτ = (1− v2)−1/2. Consequently,

p = mdα

dτ=

m√1− v2

u+m√

1− v2

dα

dt

In the binomial expansion

E = m(1− v2)−1/2 = m+ 12mv

2 +O(v4)

the second term 12mv

2 is kinetic energy, and Einstein declared this scalar Eto be the energy of the particle measured by u, with rest mass m = E|v=0 asmerely one form of energy. This is the famous formula E = mc2.

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1.7 Submanifolds

Smooth Submanifolds

A manifold P is a submanifold of a manifold M if it is a topological subspace†

with a smooth inclusion map : P → M whose push-forward, ∗ is one-to-one. If P has dimension one less than M it is called a hypersurface. We saythat submanifold is closed if it is a closed set of M . Every smooth manifoldis diffeomorphic to a submanifold of some Euclidean space. The following is auseful way to get submanifolds:

Lemma 1.7.1. Let f be a smooth real valued function on M . If the push-forward f∗ = df is nonvanishing on a level set f = const, then the level setis a closed hypersurface of M .

As an example, consider the function f(x) =∑i(x

i)2 on Rn. For r > 0, thelevel set f = r is a closed hypersurface, the (n− 1)-sphere of radius r.

Remark 1.7.1. Let S2 be the unit sphere in R3. Let ϑ denote colatitude and ϕlongitude. We treat ϕ as circular so that the coordinate system ϑ, ϕ covers allof the sphere except the poles (0, 0, ±1). ϕ is undefined at the poles, but itscoordinate vector field ∂

∂ϕ is well defined and smooth on the entire sphere and

is zero at the poles. Defining ϑ(0, 0,+1) = 0 and ϑ(0, 0,−1) = π extends ϑ tothe entire sphere, with 0 ≤ ϑ ≤ π. At the poles, ϑ is only continuous but sinϑand cosϑ are analytic on all of S2.

Foliations

A distribution Π of dimension k on a smooth manifold M is a smooth field ofk-planes on M . That is, Π assigns to each point p ∈ M a k-dimensional sub-space Πp of TpM , and locally Π has a basis of k smooth vector fields.

Thus, a distribution Π is a subbundle of the tangent bundle TM . A subman-ifold P of M such that TpP = Πp for all p ∈ P is called an integral submanifoldof Π. If there is an integral manifold of Π through every point x ∈ M , then Πis said to be integrable.

A vector field V is said to be in a distribution Π if Vp ∈ Πp for all p ∈ M ,i.e., if it is a section of the subbundle Π.

1.7.1 Frobenius’ Theorem. A distribution

Π on M is integrable ⇐⇒ X,Y ∈ Π =⇒ [X,Y ] ∈ Π (1.7.1)

Through each point of M there is a unique largest connected integral mani-fold of Π. The collection of all such maximal integral manifolds on Π is calleda foliation of dimension k = dim(Π). For example, on Rn − 0 the spheresf = rr>0 constitute a (n−1)-dimensional foilation, with Π consisting of theirtangent spaces.

†So it has the relative topology.

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Every one-dimensional distribution is integrable since the bracket conditionis trivial. Such foliations are called congruences. Every nonvanishing vector fielddetermines a unique congruence on the manifold. For a given one dimensionaldistribution Π, such a vector field need not exist; if it does, Π is said to beorientable. Then Π supplies a direction at each point p ∈ M , i.e., an orientedtangent line Πp ∈ TpM .

Hypersurfaces

If a curve α meets a hypersurface S at α(s0) ∈ S the meeting is transverse ifα(s0) is not tangent to S. For a closed hypersurface S in M , a neighborhood Nof S is tubular if there is a line bundle π : E −→ S over S and a diffeomorphismφ : E −→ N such that φ(0p) = p, ∀p ∈ S. Then, N is foliated by curves thatcut transversally across S.

A hypersurface is said to be two-sided if there exists a continuous vectorfield Z of M on S such that Z is never tangent to S, and is therefore nowherevanishing. Let N be a tubular neighborhood of S. Suppose S is two sided,then, N − S has two connected components. Otherwise, S is called one-sidedand N − S is connected. A closed hypersurface S in M separates M if M − Sis not connected.

Lemma 1.7.2. Let S be a closed connected hypersurface of M. If S is two-sided,then, M − S has at most two components. On the hand, if S is one-sided thenM − S is connected.

Submanifolds of Lorentz Manifolds

A submanifold P of M is spacelike, timelike or null provided each tangent spaceTxP of P is spacelike, timelike or null respectively, as a vector subspace of TxM .A spacelike submanifold is intrinsically a Riemannian manifold and a timelikeone is a Lorentz manifold. On the other hand, for a null submanifold P , therestriction of g to P is degenerate, so there is not much to say about this caseeven though they show up all the time in GR.

A submanifold N is nontimelike if it has no timelike tangent vectors. If atimelike curve α meets N then it must cut transversally. Now, every nontimelikehypersurface S is two-sided, for being time orientable, M admits a globallydefined timelike vector field Z that cannot be tangent to S.

Proposition 1.7.2. Let S be a closed connected nontimelike hypersurface in aspacetime M. If N is a tubular neighborhood of S, then the two components ofN − S can be denoted by N+ and N− in such a way that any future-pointingtimelike curve α with α(s0) ∈ S crosses S transversally from N− to N+.

Corollary 1.7.3. Let S be a closed connected nontimelike hypersurface in aspacetime M. If S separates M, then M − S has exactly two components and

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these can be denoted C+ and C− so that any future pointing timelike curve αthat meets S tranversally crosses S from C− to C+.

A nonvanishing vector field X is hypersurface-orthogonal if X⊥ is integrable.Then its integral manifolds are hypersurface normal to X. For example, theradial vector field X = xi ∂

∂xi in Rn− 0 has this property since X is everywhereorthogonal to the spheres x : |x| = const.

Totally Geodesic Submanifolds

A submanifold P of a semi-Riemannian manifold M is totally geodesic if vectorfields X, Y are tangent to P , then the covariant derivative ∇XY is also tangentto P . Examples are hyperplanes in Rn and great spheres in Sn.

Proposition 1.7.4. Let P be a totally geodesic submanifold of M. If a geodesicγ of M meets P at a point γ(s0) where γ(s0) is tangent to P, then γ(s) remainsin P for s near s0.

Of course, γ need not be contained in P globally. However, if the submanifoldis closed there is no escape:

Corollary 1.7.5. If a geodesic γ : I −→ M is tangent to a closed, totallygeodesic submanifold P at a single point of I, then γ lies entirely in P.

The following is a prime source for closed, totally geodesic submanifolds.

Theorem 1.7.6. The following two are always closed, totally geodesic subman-ifolds of M :

1. The set of fixed points of an isometry φ : M −→M .

2. The set of zeros of a Killing vector field.

Remark 1.7.2. A connected, totally geodesic submanifold of a Lorentz manifoldhas causal character since parallel transport preserves character.

Remark 1.7.3. The failure of a submanifold to be totally geodesic can be mea-sured by its shape tensor or the second fundamental form S. For X,Y ∈ X(M)tangent to P , define S(X,Y ) = nor∇XY , the component normal to P †.

Consequently,

P is totally geodesic ⇐⇒ S = 0

1.8 Cartan Computations

Let E1, . . . , En be an orthonormal frame field on an open neighborhood ofa semi-Riemannian manifold M , with orthonormal coframe field ω1, . . . , ωn.Such coframe fields can be directly recognized from the line element ds2 asfollows.

†It is easy to check that S is a tensor, i.e., it is well defined and F(P )-bilinear.

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Lemma 1.8.1. One forms ω1, . . . , ωn are a coframe field on M and ε1, . . . , εnis the signature of M if and only if

ds2 = εi(ωi)2 (1.8.1)

The connection forms for a frame field E1, . . . , En are the one-formsωij1≤i,j≤n such that for all tangent vectors Xx

ωij(X) = ωi(∇XEj) ,∀i, j (1.8.2)

Hence, by the duality formula

∇XEj = ω`jE` (1.8.3)

In the Cartan approach, connection forms replace Christoffel symbols. Thematrix (ωij(X)) tells us how the frame is changing in the direction X.

Lemma: The First Structural Equation 1.8.2.

dωi = −ωi` ∧ ω` (1.8.4)

Lemma 1.8.3.ωij = −εiεjωji (1.8.5)

In particular, ωii = 0

Consequently, all connection forms are determined by ωiji<j . Further-more, for a coframe field ω1, . . . , ωn, the connection forms ωij1≤i,j≤n arethe unique one-forms that satisfy equations 1.8.4 and 1.8.5.

For tangent vectors X,Y let Ωij(X,Y ) be the matrix of curvature operatorRXY relative to the frame Ei. Explicitly,

RXY (Ej) = Ωij(X,Y )Ei ,∀j (1.8.6)

So Ωij1≤i,j≤n are 2-forms called the curvature forms of the frame fields.

Theorem: The Second Structural Equation 1.8.4.

Ωij = dωij + ωi` ∧ ω`j ,∀i, j (1.8.7)

As with the connection forms, lowering an index of Ω gives Ωij = εiΩij ,

which by symmetry of curvature is skew-symmetric in i and j. Hence,

Corollary 1.8.5.Ωij = −εiεjΩji (1.8.8)

In particular, Ωii = 0.

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The above results imply that considered as matrix valued forms Ωij and ωijtake their values in the appropriate Lie algebra†.

The components of the curvature tensor R with respect to the frame fieldare

Rijk` = Ωij(Ek, E`) (1.8.9)

Hence, Rijk` = εiRijk`. The sectional curvatures of the frame field 2-planes

are

K(Πij) = K(Ei, Ej) = εiεjRijij = εjRijij (1.8.10)

and the curvature invariant κ can be written as κ = εiεjεkε`(Rijk`)2.

Remark 1.8.1. If M is a Riemannian manifold (ν = 0) with frame field Ei,raising and lowering indices does not require a sign change. Hence, the matrices(ωij)1≤i,j≤n and (Ωij)1≤i,j≤n are skew-symmetric.

Remark 1.8.2. If M is a Lorentz manifold, then a sign change is required forthe timelike index. The skew-symmetry of ωij leads to

ωi0 = ω0i (1.8.11)

butωij = −ωji for i, j > 0 (1.8.12)

and similarly for Ωij .

Corollary 1.8.6. Let u, v be orthogonal coordinates in a semi-Riemannian sur-face S. Write 〈 ∂∂u ,

∂∂u 〉 = E = ε1e

2, 〈 ∂∂v ,∂∂v 〉 = G = εeg

2 where ε1 and ε2 are±1, and e, g > 0. The Gaussian curvature of S is given by

KS = − 1

eg

[ε1

(evg

)v

+ ε2

(gue

)u

](1.8.13)

†For Minkowski spacetime M = Rn1 this is possibly the Lie algebra o1(n) of the Lorentz

group O1(n) which is the group of isometries of M.

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