An Introduction to General Relativity

Michael James Gilbert

Yaron Hadad

Contents

1. Preface 42. Notation 4

Part 1. Introduction 5

Part 2. Riemannian Geometry 93. Introduction 104. What is curvature? 125. Connections 146. Parallel Translation 207. Geodesics 218. The Riemannian Connection 229. Back to Curvature 25

Part 3. General Relativity 2710. Einstein’s Equation 2911. Linearized Gravity 2912. The Newtonian Limit 34

Bibliography 39

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4 CONTENTS

1. Preface

This term paper was written for the course Math 534 : Topology & Geometryat the University of Arizona in Spring 2008. The paper is designed as a shortintroduction to Riemannian geometry and general relativity for math or physicsstudents who are interested in the subject. As undergrads, we found it very difficultto find a rigorous and concise introduction to the general theory of relativity. Themain goal of this paper is to formulate the beautiful general theory of relativityand some of its geometric aspects, with full rigor but without getting lost in allthe mathematical details. These two goals are somewhat contradictory, and thuswe have tried to separate the treatment of the mathematical machinery needed forgeneral relativity, and the formulation of general relativity itself. So in the firstpart of the paper we discuss the main ideas of Riemannian geometry, focusing ontools that will be used in the second part, in which we discuss general relativity.

Since it is not possible to cover all the necessary knowledge within a (short?)paper, we assume that the reader is already familiar with the theory of topologicaland differential manifolds, as treated for example in Lee (Introduction to SmoothManifolds 2002). In particular, the reader should be familiar with the notions of amanifold, partitions of unity, vectors and vector fields, tangent/cotangent bundles,vector bundles, tensors and tensor fields. Furthermore, we won’t discuss classicalmechanics, electromagnetism or special relativity with all the glorious details, so aformer knowledge in those subject would make the reading more pleasant (literally).

We would like to thank Prof. Flaschka for encouraging us to work on thattopic. That was the first time in which a genuine chance to study general relativitywas given to us; Otherwise we would have both procrastinated forever...

2. Notation

Throught the paper, we will use the same notation as Lee (2002). More specif-ically, we will use the following notations unless otherwise specified:

• V,W denotes a vector field.• X,Y, Z denotes a vector.• C∞ (M) denotes smooth real-valued functions on M .• TpM is the tangent space of M at the point p.• TM and T ∗M are the tangent and cotangent bundles of M , respectively.

Furthermore, we will use Einstein’s summation notation unless otherwise specified.

Part 1

Introduction

Our story begins in Alexandria circa 300 BC, when the Greek mathematicianEuclid published his groundbreaking book The Elements. In The Elements, Euclidprovided a full treatise of the knowledge that was available on geometry at thetime. The importance of Euclid’s work is impossible to overestimate, as the bookwas not only one of the most popular books of all time (second only to the Biblein the number of editions published), but it was also an important milestone in thehistory of science, as a model of logical reasoning via a rigorous axiomatic approach.

The Elements is the first time in which a list of “self-evident” postulates werestated, and every other result was derived in a rigorous manner from these postu-lates, and thus it is the first axiomatic treatment in mathematical literature. TheElements begins by listing 23 definitions, followed by five postulates (translatedfrom the original text in Huggett (1999)as follows):

“Let the following be postulated:

(1) To draw a straight line from any point to any point.(2) To produce a finite straight line continuously in a straight line.(3) To describe a circle with any center and distance.(4) That all right angles are equal to one another.(5) That, if a straight line falling on two straight lines make the interior an-

gles on the same side less than two right angles, the two straight lines, ifproduced indefinitely, meet on that side on which are the angles less thanthe two right angles.”

Reading through the definitions and postulates of The Elements for the first time,one might wonder what is the point of stating them - “Isn’t it completely obviousthat one can draw a straight line between any two points?”

Taking this naive and myopic point of view is to miss Euclid’s greatest achieve-ment. Before The Elements, geometry was not a rigorous or systematic theory, andEuclid’s work was the first time in which all the known ’truths’ were organized ina complete treatise. Moreover, The Elements demonstrates that all of the geomet-ric knowledge that existed at that time is a consequence of five basic postulatesand definitions. Assuming the five postulates, the rest of geometry is just a mat-ter of deduction, so to some extent, those five short lines called the postulates ofEuclidean geometry encompass ALL of Euclidean geometry.

Going back to the five postulates, we notice that while the first four seemvery intuitive, the fifth postulate (see figure below) is certainly less obvious thanthe others. In fact, Euclid struggled accepting the fifth postulate as stated. Hefruitlessly tried to derive the fifth postulate from the first four, and used it as littleas possible throughout the rest of his books. From that point forward, many of theworld’s most reknowned mathematicians tried and failed in the same attempt asEuclid: deriving the notorious “parallel postulate” from the first four, thus makingthe fifth postulate the source of what can arguably be said as the most importantdebate in the history of mathematics.

7

Throughout the course of these attempts, many mathematicians tried (andsometimes even thought that they succeeded) proving the fifth postulate. Greatnames such as Ptolemy, Khayyam, Saccheri, Legendre, Lobachevski, Bolyai andGauss showed that this postulate is equivalent to many more intuitive statementsthat we usually take for granted in standard plane geometry, such as:

• The sum of the angles in every triangle is 180°,• For every line and a point that does not lie on the line, there exists a

unique line through the point that never intersects the line,• The sum of the angles is the same for every triangle,• There exists a pair of similar, but not congruent, triangles,• There exists a pair of straight lines that are at constant distance from

each other,• Two lines that are parallel to the same line are also parallel to each other,• Given two parallel lines, any line that intersects one of them also intersects

the other,• In a right-angled triangle, the square of the hypotenuse equals the sum of

the squares of the other two sides (The Pythagorean Theorem).

More than two thousand years have passed, but there was still no significantprogress on Euclid’s original question: How does the fifth postulate follow fromthe first four?

Around 1830, the Hungarian Janos Bolyai and the Russian Nikolai IvanovichLobachevsky separately found the long anticipated answer for that question: NO.They published treatises on hyperbolic geometry, known today as Bolyai-Lobachevskygeometry, in which the fifte postulate was assumed to be false. Their work was al-most completely ignored by the mathematical community until the German mathe-matician Karl Friedrich Gauss read Janos Bolyai’s work. Gauss had been intriguedby this problem, and after reading Bolyai’s work, he wrote him a letter claiming tohave derived the same results earlier.

Gauss’s role in the foundation of non-Euclidean geometry did not end here.Gauss’s student Georg Friedrich Bernhard Riemann was seeking the position of aPrivatdocent (a lecturer who receives no salary, but is voluntarily paid by studentswho attend his lectures.) To get this position, the candidate had to give an inaugu-ral lecture on a topic chosen by the faculty, from a list of three topics he propsed.The first two topics that Riemann submitted were ones for which he was clearly anexpert, and as a third topic he picked the foundations of geometry. In the history ofthe Privatdocent position, the faculty had never chosen the candidate’s third topic.Thus, Riemann had all the reasons to expect that the faculty will pick one of thefirst two topics. Contrary to all traditions, Gauss insisted to pick the third topic,as he had been interested in the foundations of geometry for years.

On June 10, 1854, the faculty of Gottingen University heard Riemann’s phe-nomenal inaugural lecture, entitled “Uber die Hypothesen, welche der Geometriezu Grunde liegen” (On the Hypothesis which lie at the foundations of Geometry.)Though he worked on it for just a few months, Riemann’s reknowned inaugurallecture marked the end of one of the biggest question marks in the history of math,that had been left open for more than two thousand years. In this lecture, not onlydid Riemann generalized Euclidean & Bolyai-Lobachevsky geometry to arbitrarydimension, he also showed that the fifth postulate is not a necessary axiom for

8

a consistent theory of geometry. Dedekind described Gauss’s reaction as (Spivak(1979)):

“Gauss sat at the lecture which surpassed all his expectations, in the greatestastonishment, and on the way back from the faculty meeting he spoke to WilhelmWeber, with the greatest appreciation, and with an excitement rare for him, aboutthe depth of the ideas presented by Riemann.”

In the sequel we present Riemann’s groundbreaking ideas in his inaugural lec-ture.

Part 2

Riemannian Geometry

3. Introduction

Roughly speaking, the difference between topology and geometry is that topol-ogy deals with the global nature of space, whereas geometry is concered with localproperties such as lengths, angles, areas and volumes. Thus, the first thing we needfor constructing a theory of geometry on a manifold is a notion of length. Linearalgebra tells us that an inner product gives us a definition of length and anglesbetween vectors in a vector space, thus introduces the most elemetary notions ofgeometry. Transferring this idea to manifolds, we recall that the tangent space atany point of the manifold is a vector space. This leads to the following definition:

Definition 3.1. Given a smooth manifold M , a Riemannian metric g on Mis a smooth, symmetric 2-tensor field which is positive definite at each point. Sucha pair (M, g) is called a Riemannian manifold.

It is important to note that a Riemannian metric is NOT the same as a metric(in the sense of metric spaces), albeit the former gives rise to the latter.

Given a point p ∈M , g induces an inner product on each tangent space TpM ,denoted by 〈X,Y 〉 = g (X,Y ) for X,Y ∈ TpM . Furthermore, in any smooth localcoordinates

(xi)a Riemannian metric can be written as

g = gijdxi ⊗ dxj .

Using the symmetry of g, we may write it in terms of symmetric products as

g = gijdxidxj .

A pseudo-Riemannian metric is a symmetric 2-tensor field g that is nondegen-erate at each point p ∈ M . Everything that we will do next in the contents ofRiemannian manifolds also holds for pseudo-Riemannian manifolds, and thus maybe used for the general theory of relativity in the last part of the paper.

Whenever we introduce a new definition on an abstract manifold, we hopethat the definition generalizes our good-old notions in Euclidean spaces. The nextexample shows that the standard inner product on Rn is actually a special case ofa Riemannian metric.

Example 3.2. We define the Euclidean metric g on Rn using standard coordi-nates by

g = δijdxidxj

=(dx1)2

+ · · ·+ (dxn)2

where δij is the Kronecker delta. For any two vectors v, w ∈ TpRn, we get

g (v, w) = δijviwj

=n∑i=1

viwi

= v · w.

Indeed, g is the 2-tensor field whose value at each point is the Euclidean dotproduct.

Definition 3.3. We are ready to construct some of the basic geometric notionsthat can be defined on a Riemannian manifold (M, g).

3. INTRODUCTION 11

• The length or norm of a tangent vector X ∈ TpM is defined to be

|X|g = 〈X,X〉1/2g = g (X,X)1/2

• The angle between two nonzero tangent vectors X,Y ∈ TpM is the uniqueθ ∈ [0, π] satisfying

cos θ =〈X,Y 〉g|X|g |Y |g

.

• Two tangent vectors X,Y ∈ TpM are said to be orthogonal if 〈X,Y 〉g = 0;Equivalently, if the angle between them is π

2 .• If γ : [a, b]→M is a piecewise-smooth curve, the length of γ is defined to

be

Lg (γ) =ˆ b

a

|γ′ (t)|g dt

This integral is well-defined since |γ′ (t)|g is continuous at all but finitelymany values of t, and has well-defined left and right-handed limits atthose points. Moreover, it is possible to show that the length of a curveis independent of the parameterization (as we would expect it to be).

• If (M, g) and(M, g

)are two Riemannian manifolds, a smooth map F :

M → M is called an isometry if it is a diffeomorphism that satisfiesF ∗g = g. We say that (M, g) and

(M, g

)are isometric if there exists an

isometry between them.

Roughly speaking, Riemannian geometry is the study of properties of Riemann-ian manifolds that are invariant under isometries.

3.1. The Tangent-Cotangent Isomorphism. One of the most importantproperties of a Riemannian metric, is that it provides us with a natural isomorphismbetween the tangent and cotangent bundles. Given a Riemannian manifold (M, g),we define a bundle map g : TM → T ∗M by mapping each Xp ∈ TpM to thecovector g (Xp) ∈ T ∗pM defined by:

g (Xp) (Yp) = gp (Xp, Yp)

Its action on smooth vector fields is:

g (X) (Y ) = g (X,Y )

which is linear over C∞ (M) as a function of Y , thus g (X) is a smooth covectorfield. Because g (X) is linear over C∞ (M) as a function of X, g is a smoothbundle map. Moreover, g is injective at each point, because if g (Xp) = 0 then0 = g (Xp) (Xp) = 〈Xp, Xp〉g, which implies that Xp = 0. Since the dimensions ofTpM and T ∗pM are the same, g is bijective and therefore it is a bundle isomorphism.Now, if X and Y are smooth vector fields, we can write in coordinates:

g (X) (Y ) = gijXiY j ,

which implies that the cover field g (X) has the coordinate expression

g (X) = gijXidyj .

12

So g is the bundle map whose matrix with respect to coordinate frames for TMand T ∗M is the same as the matrix of g itself. We will denote the components ofthe covector field g (X) by

Xj = gijXi

so thatg (X) = Xjdy

j .

It is common to say that g (X) is obtained from X by lowering an index.The matrix of the inverse map g−1 : T ∗pM → TpM is the inverse of (gij).

It should be denoted by(g−1

)ij , but this is cumbersome so we write(gij), thus

have the equality gijgjk = δik. No confusion arises from this since the upper indicesdistinguishes the inverse metric from the metric. Therefore if ω ∈ T ∗M is a covectorfield, g−1 (ω) has the coordinate representation

g−1 (ω) = ωi∂

∂xi

where ωi = gijωj . In that case, we say that g−1 (ω) is obtained from ω byraising an index. We will use the operations of lowering/raising the index heavilyin the sequel.

So far we have defined a Riemannian metric and discussed some of the geometricconstructions that are induced by it. Here a natural question arises: How commonare Riemannian metrics? Can we define a Riemannian metric on any manifold?The answer to that question is given by the following proposition:

Proposition 3.4. Every smooth manifold admits a Riemannian metric.

Proof. We just give a sketch of the proof. A complete proof can be find inLee (2002, 1997).

We coverM by smooth coordinate charts {Uα}, and define a Riemannian metricgα by the Euclidean metric in each coordinate chart. Then, using a smooth parti-tion of unity {ψα} subordinate to the cover {Uα}, g =

∑α ψαgα is a well-defined

Riemannian metric on g. A shorter proof that uses more machinery, is by embed-dingM in Rn for some n (using Whitney’s embedding theorem) and restricting theEuclidean metric of Rn to M . �

4. What is curvature?

Intuitively, the curvature of a curve or a surface is the way it is bent as a subsetof Euclidean space. In order to give this intuitive idea a rigorous meaning, we willstart by considering the simplest possible case: a circle or a line in R2. In thefollowing image, we see five circles intersecting at the same point. As the radius ofthe circle is bigger, the circle resembles a line more, and is thus more flat.

4. WHAT IS CURVATURE? 13

Thus, we may define the curvature of a circle or a line to be

κ =1R

where R is the radius of the circle and is ∞ in a case of a line. So far, thecurvature is 0 if and only if we have a straight line, so this definition correspondsto our intuitive notion of curvature.

In the next step of defining curvature, we would like to generalize this idea toany smooth curve in the plane. For a general smooth curve, we would expect thecurvature to vary from a point to a point, but would like this value to be 1

R inthe case of a circle or a straight line. We would like to find a unique circle whosecurvature corresponds to the curvature of our curve (at a point.) Here enters thepicture the idea of the osculating circle, first introduced by Leibniz as circulumosculans (Latin for kissing circle.)

Let γ : [a, b] → R2 be a smooth curve, and let p = γ (t) be a point on thecurve. There are many circles tangent to γ at p, namely, all those circles that havea parametric representation whose velocity vector at p is the same as that of γ.Among these parametrized circles, there is exactly one whose acceleration vectorat p is the same as that of γ (when taking an arc length parameterization); This isthe osculating circle.

Since we already defined the curvature of a circle to be κ = 1R , we define the

curvature of the curve γ at the point p = (x0, y0) to be 1R , where R is the radius

of the osculating circle. An arc length circle that intersects p at t = 0 has thefollowing parameterization:

c (t) =(R cos

(t

R

)− x0 −R,R sin

(t

R

)− y0

).

The acceleration vector is:

c (t) = − 1R

(cos(t

R

), sin

(t

R

)).

Setting ‖c (t) ‖ = ‖γ (t) ‖, we see that 1R = ‖γ (t) ‖ and that the solution is

unique, i.e. there is exactly one osculating circle. Therefore, the curvature of the

14

osculating circle is 1R = ‖γ (t) ‖, thus we define the curvature of any smooth curve

to beκ (t) = ‖γ (t) ‖

Obviously from this definition, a circle has a constant curvature κ ≡ 1R , and a

straight line has a curvature zero, as its acceleration is zero. Defining a curvature onan abstract manifold is a bit more involved, and will be discussed in the followingsections.

5. Connections

5.1. What are connections? In Euclidean spaces, a curve γ (t) is a straightline if and only if its Euclidean acceleration γ (t) is identically zero. Moreover, as wehave seen, the curvature of a curve on the plane can be defined using the accelerationof the curve, which again involves a second-order differentiation. Eventually, wewould like to generalize these two notions to define geodesics on a Riemannianmanifold which correspond to straight lines in Euclidean spaces, and to define anotion of curvature of a manifold.

Given a curve γ : [a, b] → M , we know that the velocity vector γ (t) has acoordinate-independent meaning for each t ∈ M , and its expression in any coor-dinate system matches the usual notion of velocity of a curve in Rn. Unfortu-nately, unlike the velocity vector, the acceleration vector has no such coordinate-invariant interpretation. For example, the parametrized circle in the plane given by(x (t) , y (t)) = (cos t, sin t) has the acceleration vector (x (t) , y (t)) = (− cos t,− sin t).But in polar coordinates, the same curve is described by (r (t) , θ (t)) = (1, t), inwhich the acceleration vector is

(r (t) , θ (t)

)= (0, 0). The acceleration vector is

not coordinate-independent.

The problem is that the acceleration is defined as the derivative of the velocityvector γ (t), which involves writing a difference quotient of γ (t+ h) and γ (t) - butthis two vectors live in different vector spaces!

In order to make sense of this difference quotient, we need to find a way toidentify vectors that live in different tangent spaces, or more intuitively, a smoothway to “connect” nearby tangent spaces. In order to do that, we will add some more

5. CONNECTIONS 15

structure to the manifold, namely a connection, which will be a rule for computingdirectional derivatives of vector fields.

For concreteness, let us consider the case of Euclidean space again. In Rn, thereis a notion of taking the derivative of a vector field Y in the direction of the vectorfield X, namely,

∇XY := (X · ∇)Y

where ∇ is the gradient. We see that it has the following properties:(1) Linearity over C∞ (Rn) in X - For all f, g ∈ C∞ (Rn) we have:

∇f1X1+f2X2Y = f1∇X1Y + f2∇X2Y

(2) Linearity over R in Y - For all c1, c2 ∈ R we have:

∇X (c1Y1 + c2Y2) = c1∇XY1 + c2∇XY2

(3) Leibniz rule - For all f ∈ C∞ (Rn) we have

∇X (fY ) = (X · ∇) (fY )= (X · ∇f)Y + f (X · ∇)Y

= (X · ∇f)Y + f∇XY

In order to define a similar notion on a general manifold, we will use the propertiesof the ∇ operator as the defining properties of a connection on a manifold. Thisapproach is in the same spirit as the way we define, for example, tangent vectors ona manifold as derivations: since we cannot define the object directly, we will defineit using some characterizing properties that do make sense on an abstract manifold.It turns out to be easiest to define a connection as a way of differentiating sectionsof vector bundles, and we will later adapt this definition to the case of vector fieldsalong curves. Keeping in mind the above motivation, we are led to the followingdefinition of a connection:

Definition 5.1. Let π : E → M be a vector bundle over a manifold M , andlet E (M) denote the space of smooth sections of E. A connection in E is a map

∇ : T (M)× E (M)→ E (M)

written (X,Y ) 7→ ∇XY that satisfies the following properties:(1) Linearity over C∞ (M) in X - For all f, g ∈ C∞ (M) we have:

∇f1X1+f2X2Y = f1∇X1Y + f2∇X2Y

(2) Linearity over R in Y - For all c1, c2 ∈ R we have:

∇X (c1Y1 + c2Y2) = c1∇XY1 + c2∇XY2

(3) The Leibniz rule - For all f ∈ C∞ (M) we have

∇X (fY ) = (Xf)Y + f∇XY

∇XY is called the covariant derivative of Y in the direction of X.Thus a connection gives us a way of differentiating sections of vector bundles.

Next we adapt this definition to the case of vector fields, keeping in mind the generaldefinition to which we will return quite soon.

16

Definition 5.2. A (linear) connection on M is a connection in TM , i.e. it isa map

∇ : T (M)× T (M)→ T (M)satisfying properties (1)-(3) in the definition of a connection above.

Next we examine the structure of a connection in term of components. Let{Ei} be a local frame for TM on an open subset U ⊆ M . ∇Ei

Ej is a vector fieldon M , so we may expand it in terms of this frame:

∇EiEj = ΓkijEk

for some Γkij ∈ C∞ (M). This defines n3 functions Γkij on U , called the Christof-fel symbols of ∇ with respect to the frame {Ei}. Notice that the Christoffel symbolsare not the components of a tensor field, since they do not (in general) satisfy thetensor transformations law. The importance of the Christoffel symbols is that theycompletely determine the action of ∇ on U , as the following lemma proves:

Lemma 5.3. Let ∇ be a connection, let X,Y ∈ T (U) and let {Ei} be a localframe for TM . Then:

∇XY =(XY k +XiY jΓkij

)Ek.

Proof. We compute

∇XY = ∇X(Y jEj

)=

(XY j

)Ej + Y j∇XEj

= XY jEj + Y j∇XiEiEj

= XY jEj +XiY j∇EiEj

= XY kEk +XiY jΓkijEk

The second line follows from the Leibniz rule, the fourth line follows from thelinearity condition and the last line follows from renaming the dummy index j inthe first term and from applying the definition of the Christoffel symbols to thesecond term. �

Examples:• The Euclidean connection on Rn is defined by:

∇X(Y j∂j

)=(XY j

)∂j .

In other words, ∇X is just the vector field whose components are theordinary directional derivatives of the components of Y in the directionof X. It is easy to see that ∇ is indeed a connection, and its Christoffelsymbols in standard coordinates are:

Γkij =(∇∂i∂j

) (xk)

=(∂iδ

lj

)∂lx

k

= 0

Where the last equality holds since the partial derivatives of the constantδlj are zero.

• The same idea can be extended to any manifold that is covered by a singlecoordinate chart:

5. CONNECTIONS 17

Lemma 5.4. If M is a manifold covered by a single coordinate chart, then thereis a one-to-one correspondence between connections on M and choices of n3 smoothfunctions

{Γkij}on M , by the rule:

∇XY =(Xi∂iY

k +XiY jΓkij)∂k

Proof. On one hand, the rule

∇XY =(Xi∂iY

k +XiY jΓkij)∂k

is equivalent to ∇XY =(XY k +XiY jΓkij

)Ek when Ei = ∂i is a coordinate

frame. Thus for every connection, the functions{

Γkij}satisfy the above rule. On

the other hand, given{

Γkij}, the above rule is clearly C∞ (M) in X and linear over

R in Y . The Leibniz rule can also be verified by direct computation. �

The above example motivates the following proposition.

Proposition 5.5. Every manifold admits a connection.

Proof. Let {Uα} be a cover of M by coordinate charts, and let {ϕα} bea partition of unity subordinate to {Uα}. The preceding lemma guarantees theexistence of a connection ∇α on each of the Uα’s, we would like to patch theseconnections together using the partition of unity. We define

∇XY =∑α

ϕα∇αXY.

It is obvious that this expression is smooth, linear over C∞ (M) in X and linearover R in Y . We check the Leibniz rule by direct computation:

∇X (fY ) =∑α

ϕα∇αX (fY )

=∑α

ϕα ((Xf)Y + f∇αXY )

=

(∑α

ϕα

)(Xf)Y + f

∑α

ϕα∇αXY

= (Xf)Y + f∇XY.

�

5.2. Covariant Derivatives of Tensor Fields. From now on, we will as-sume M is a manifold with a (linear) connection ∇. A connection was defined asa map on vector fields which lives on the manifold M . Actually, any connectionautomatically induces connections on all tensor bundles over M , and thus givesus a way to compute covariant derivatives of any tensor field. In fact, this is theway in which many physics books (e.g. Robert Wald’s “General Relativity”) definecovariant derivatives in the first place.

Lemma 5.6. Let ∇ be a connection on M . There is a unique connection ineach tensor bundle T kl M , also denoted by ∇, that satisfies:

(1) Consistency: ∇ agrees with the given connection on TM . Moreover, onT 0M , ∇ is given by ordinary differentiation of functions:

∇Xf = Xf

18

(2) The Leibniz rule:

∇X (F ⊗G) = (∇XF )⊗G+ F ⊗ (∇XG)

(3) Commutativity with contraction: If ”tr” denotes the trace with respect toany pair of indices,

∇X (trY ) = tr (∇XY )

(4) Product rule with respect to natural pairing: If ω is a covector field and Yis a vector field, then

∇X 〈ω, Y 〉 = 〈∇Xω, Y 〉+ 〈ω,∇XY 〉(5) For any F ∈ T kl (M), vector fields Yi, and 1-forms ωj,

(∇XF )(ω1, . . . , ωl, Y1, . . . , Yk

)= X

(F(ω1, . . . , ωl, Y1, . . . , Yk

))−

l∑j=1

F(ω1, . . . ,∇Xωj , . . . , ωl, Y1, . . . , Yk

)−

k∑i=1

F(ω1, . . . , ωl, Y1, . . . ,∇XYi, . . . , Yk

)Proof. Could be found in Lee (1997). �

Notice that since the covariant derivative ∇XY of a tensor field Y is linear overC∞ (M) in X, it can be used to construct another tensor field as follows.

Lemma 5.7. If ∇ is a (linear) connection on M and F ∈ T kl (M), then themap

∇F : T 1 (M)× · · · × T 1 (M)× T (M)× · · · × T (M)→ C∞ (M)

defined by

∇F(ω1, . . . , ωl, Y1, . . . , Yk, X

)= ∇XF

(ω1, . . . , ωl, Y1, . . . , Yk

)is a

(k+1l

)-tensor field.

The tensor field ∇F is called the total covariant derivative of F .

Example 5.8. If f ∈ C∞ (M) is a smooth function on M , then ∇f ∈ T 1 (M)is defined by ∇f (Y ) = ∇Y f = Y f . Recalling the definition of the differential, wesee that ∇f is just the differential df .

We write the components of a total covariant derivative using a semicolonto separate indices resulting from differentiation from the preceding indices. Forexample, if Y is a vector field written in components as Y = Y i∂i, the componentsof the

(11

)-tensor field ∇Y are written Y i;j . Hence ∇Y is given by

∇Y = Y i;j∂i ⊗ dxj

and by definition,

Y i;j = ∇Y(dxi, ∂j

)= ∇∂j

Y(dxi)

= ∇∂jY i

= ∂jYi + Y kΓijk

5. CONNECTIONS 19

More generally, the next lemma gives us a formula for the components of co-variant derivatives of all tensor fields.

Lemma 5.9. Let ∇ be a (linear) connection. The components of the total co-variant derivative of a

(kl

)-tensor field F with respect to a coordinate system are

given by

F j1...jli1...ik;m = ∂mFj1...jli1...ik

+l∑

s=1

F j1...p...jli1...ikΓjsmp −

k∑s=1

F j1...jli1...p...ikΓpmis

Proof. This lemma is left as an exercise to those readers who love indices. �

5.3. Vector Fields along Curves. Finally, we can address the question thatmotivated the original definition of connections and use them in order to make senseof directional derivatives of a vector field along a curve. But before, a couple ofdefinitions that will make the following discussion more rigorous.

Definition 5.10. A curve is a smooth map γ : I → M , where I ⊆ R is someinterval. Please notice that many books use this definition for a parameterizedsmooth curve. We will sometime refer to the image of the curve γ (I) as “thecurve”.

Definition 5.11. Given a curve γ : I →M , a vector field along γ is a smoothmap V : I → TM such that V (t) = Tγ(t)M for every t ∈ I. That is, for everypoint on the curve, the vector V (t) “lives on” the tangent space to that point.

The set of all vector fields along a curve γ is denoted by T (γ). Clearly, T (γ)is a vector space over R under pointwise addition.

A vector field along γ is called extendable if there exists a vector field V on aneighborhood of the curve γ (i.e. the image of γ) that is related to V by V (t) = Vγ(t)

for all t ∈ I. I.e. V |γ= V .

Example 5.12. Given a curve γ, its velocity vector γ defined by the push-forward γ∗

(ddt

), or by its action on functions by

γ (t) f =d

dt(f ◦ γ) (t)

is a vector field along the curve γ, since γ (t) ∈ Tγ(t)M for each t, and if γ (t) =(γ1 (t) , . . . , γn (t)

)then the coordinate expression γ (t) = γi (t) ∂i shows that it is

smooth.

The following lemma shows how we may use connections in order to make senseof directional derivatives of vector fields along curves.

Lemma 5.13. Let ∇ be a (linear) connection on M . For each curve γ : I →M ,∇ determines a unique operator

Dt : T (γ)→ T (γ)

satisfying:(1) Linearity - for all a, b ∈ R we have:

Dt (aV + bW ) = aDtV + bDtW

(2) Leibniz rule - For all f ∈ C∞ (I)

Dt (fV ) = fV + fDtV

20

(3) Consistency with the connection: If V is extendable, then for any connec-tion V of V ,

DtV (t) = ∇γ(t)VFor any V ∈ T (γ) , DtV is called the covariant derivative of V along γ.

Proof. Uniqueness: Suppose Dt is such an operator. For any t0 ∈ I, thevalue of DtV at t0 depends only on the values of V in any interval (t0 − ε, t0 + ε)containing t0. Let

(xj)be coordinates near γ (t0), and write V (t) = V j (t) ∂j near

t0. By the properties of Dt,

DtV (t0) = Dt

(V j (t0) ∂j

)= V j (t0) ∂j + V j (t0)Dt∂j

= V j (t0) ∂j + V j (t0)∇γ(t0)∂j

=(V k (t0) + V j (t0) γi (t0) Γkij (γ (t0))

)∂k

where the second line follows from the second property and the third line followsfrom the third property. In the following, we will refer to this formula has theacceleration formula (this is not a common terminology). This shows that such anoperator is unique, if it exists.

Existence: If γ (I) is contained in a single chart, we define DtV by the the aboveformula, i.e. we define DtV to be:

DtV (t0) =(V k (t0) + V j (t0) γi (t0) Γkij (γ (t0))

)∂k

It is easy to verify that it satisfies the requisites properties. If γ (I) is notcontained in a single coordinate chart, we can cover γ (I) with coordinate chartsand define DtV by the same formula in each chart. The uniqueness of the operatorimplies that the various definitions agree whenever two or mort charts overlap.Therefore there exists a unique such operator. �

6. Parallel Translation

Definition 6.1. A vector field V along a curve γ is said to be parallel along γwith respect to the (linear) connection ∇ if DtV ≡ 0.

A vector field V on M is said to be parallel if it is parallel along every curve,or equivalently, if its total covariant derivative ∇V vanishes identically.

We are finally ready to come back to the purpose for which we defined con-nections in the first place: to identify tangent vectors that live in different tangentspaces. The following proposition shows that given a vector, we may parallel trans-port it along any given curve.

Theorem 6.2. (Parallel Transport) Given a curve γ : I → M , t0 ∈ I, and avector V0 ∈ Tγ(t0)M , there exists a unique parallel vector field V along γ such thatV (t0) = V0. This vector field is called the parallel translate of V0 along γ.

Proof. If γ (I) is contained in a single coordinate chart, by the accelerationformula, V is parallel along γ if and only if

V k (t) = −V j (t) γi (t) Γkij (γ (t))

for k = 1, 2, . . . , n. This is a linear system of ODEs for(V 1 (t) , . . . , V n (t)

), and

therefore there exists a unique solution on all I with initial condition V (t0) = V0.

7. GEODESICS 21

If γ (I) is not covered by a single chart, let β denote the supremum of all b > t0 forwhich there is a unique parallel translate on [t0, b].β > b, since for b small enough,γ ([t0, b]) is contained in a single chart, so the above argument applies. Thereforea unique parallel translate V exists on [t0, β). If β ∈ I is an interior point, wetake a coordinate chart on an open set containing γ (β − δ, β + δ) for some positiveδ. Then by the above argument again, there exists a unique parallel vector fieldV satisfying the initial condition V

(β − δ

2

)= V

(β − δ

2

). By uniqueness, V = V

on their common domain, and therefore V is an extension of V past β, which is acontradiction. A similar argument works to the left side of t0.

If γ : I →M is a curve and t0, t1 ∈ I, parallel translation defines an operator

Pt0,t1 : Tγ(t0)M → Tγ(t1)M

mapping V0 7→ V (t1), where V is the parallel translation of V0 along γ. It isimportant to note that this operator depends on the curve we pick, and we willlater use this fact to define curvature on a general manifold. Moreover, Pt0,t1 is alinear isomorphism between Tγ(t0)M and Tγ(t1)M , and can be used to recover thecovariant derivative by the following formula, that we won’t prove here:

DtV (t0) = limt→t0

P−1t0,tV (t)− V (t0)

t− t0�

7. Geodesics

Using the notion of parallel transport we may define acceleration and geodesicsof curves.

Definition 7.1. The acceleration of a curve γ is the vector field Dtγ along γ.

Definition 7.2. A curve γ is called a geodesic (with respect to the (linear)connection ∇) if its acceleration is zero, i.e. Dtγ ≡ 0. Equivalently, a geodesic is acurve whose velocity vector field is parallel along the curve itself.

Example 7.3. In Euclidean space (Rn, g), a curve γ : I → Rn is a geodesic ifand only if (using the acceleration formula) for any t0 ∈ I

0 = Dtγ

= γj (t0) ∂j

which holds if and only if γ ≡ 0, that is, when γ is a straight line with constantspeed parameterization. Thus geodesics are a generalization of our familiar notionof straight lines in Euclidean spaces.

Here it is also important to note that a straight line (in Euclidean space) witha different parameterization may not be a geodesic, even though the images of thetwo are the same.

Theorem 7.4. (Existence and Uniqueness of Geodesics) Let M be a manifoldwith a (linear) connection. For any p ∈ M , V ∈ TpM , and t0 ∈ R there existsan interval I ⊂ R containing t0 and a geodesic γ : I → M satisfying γ (t0) = p,γ (t0) = V .

22

Proof. Let(xi)be coordinates on some neighborhood U of p. From the

acceleration formula, a curve γ : I → U is a geodesic if and only if its componentfunctions γ (t) =

(x1 (t) , . . . , xn (t)

)satisfy the geodesic equation

xk (t) + xi (t) xj (t) Γkij (x (t)) = 0

This is a second-order system of ordinary differential equations for the functionsxi (t) subject to the initial conditions γ (0) = p and γ (0) = V . Thus, by theexistence and uniqueness for second-order ODEs, there exists a unique local solutionto the geodesic equation. To prove the uniqueness claim of the theorem, supposeγ1, γ2 : I →M are two geodesics with γ1 (t0) = γ2 (t0) and γ1 (t0) = γ2 (t0). By thelocal uniqueness we have just proved, they agree on some neighborhood of t0. Letβ be the supremum of numbers b such that they agree on [t0, b]. If β ∈ I, then bythe smoothness of the curves, γ1 (β) = γ2 (β) and γ1 (β) = γ2 (β). By applying thelocal uniqueness in a neighborhood of β, γ1 and γ2 agree on a neighborhood of β,which is a contradiction to the fact that β is a supremum. By applying the sameargument to the left of t0, we conclude that they agree on all of I.

Using a similar argument to the one we used for the parallel transport theorem,the uniqueness statement in the preceeding argument shows that for any p ∈ Mand V ∈ TpM , there is a unique maximal geodesic γ : I → M with γ (0) = p andγ (0) = V . This maximal geodesic is called the geodesic with initial point p andinitial velocity V , and will be denoted γV . We won’t specify the initial point p inthe notation. �

8. The Riemannian Connection

In the beginning of the chapter, we defined a Riemannian manifold as a manifoldin which every tangent space is equipped with an inner product, thus allowing usto measure lengths of curves, angles and many other geometric properties of thespace. Later, we discussed the notion of curvature, and noted that in order togeneralize this notion from the plane to an abstract manifold, we had to introducean additional structure on the manifold, the connection. Connections allow us totalk about parallel transports and geodesics, and thus generalizing the familiarnotion of a “straight line” in Euclidean geometry.

In the next part of the paper, we would like to use the notion of a geodesic todescribe the motion of point particles under the influence of gravity. The problemis that on a given manifold, there are many different ways to define a connection,and thus obtaining many different notions of geodesics. In a classical theory such asgeneral relativity, a particle travels in a unique path, so it is evident that we needa way to single out a natural connection that actually describes the motion of theparticle in the gravitational field. Here the Riemannian metric comes back into thepicture: we would hope that the natural connection reflects the properties of themetric. Namely, that our connection will correspond naturally to the Riemannianmetric g.

In order to make the above discussion more concrete, we return to our favoriteexample: Euclidean space. The Euclidean connection on Rn has one very sub-stantial property that relates it to the Euclidean metric: It satisfies the productrule

∇X 〈Y,Z〉 =⟨∇XY,Z

⟩+⟨Y,∇XZ

⟩

8. THE RIEMANNIAN CONNECTION 23

This compatibility property still makes sense on an abstract Riemannian man-ifold, and as we will soon see, will be one of two conditions that will allow us toobtain a unique connection on any Riemannian manifold. Thus, we use this definingproperty of the connection as our “natural” connection.

Definition 8.1. Given a Riemannian manifold (M, g), a (linear) connection∇ is said to be compatible with g, if it satisfies the following product rule for allvector fields X,Y, Z:

∇X 〈Y, Z〉 = 〈∇XY,Z〉+ 〈Y,∇XZ〉In fact, this definition is even more “natural” than how it appears at first glance,

as the following lemma shows:

Lemma 8.2. Let ∇ be a (linear) connection on a Riemannian manifold (M, g).Then the following conditions are equivalent:

(1) ∇ is compatible with g.(2) ∇g ≡ 0.(3) For any vector fields V,W along any curve γ,

d

dt〈V,W 〉 = 〈DtV,W 〉+ 〈V,DtW 〉

(4) If V,W are parallel vector fields along a curve γ, then 〈V,W 〉 is constant.(5) Parallel translation Pt0,t1 : Tγ(t0)M → Tγ(t1)M is an isometry for each

t0, t1.Compatability with the metric is not enough to determine a unique connection,so we would have to restrict the connection even more, in order to get a uniqueone. This involves the torsion tensor of the connection, which is the

(21

)-tensor field

τ : T (M)× T (M)→ T (M) defined by

τ (X,Y ) = ∇XY −∇YX − [X,Y ]

Definition 8.3. A linear connection ∇ is said to be symmetric, or torsion-free,if its torsion vanishes identically, that is, if

∇XY −∇YX ≡ [X,Y ]

The reason it is called symmetric, is that a linear connection ∇ is symmetricif and only if its Christoffel symbols are symmetric with respect to the first twoindices, i.e.

Γkij = Γkji.

Theorem 8.4. (Fundamental Lemma of Riemannian Geometry) Let (M, g) bea Riemannian manifold. There exists a unique (linear) connection ∇ on M that iscompatible with g and symmetric.

This connection is called the Riemannian connection or the Levi-Civita con-nection of g.

Proof. (Need to be rewritten)Let ∂a be the ordinary derivative operator with respect to a particular co-

ordinate system. Then, according to the above equation, we know that for anyconnection ∇, we should have

∇agbc = ∂agbc − Γdabgdc − Γdacgbd

24

Thus, in order for ∇agbc = 0, we must have

∂agbc = Γdabgdc + Γdacgbd,

and lowering indices, we can rewrite this equation as

Γcab + Γbac = ∂agbc.

and then, if we substitute the indices, we also get the equations

Γcba + Γabc = ∂bgac

Γbca + Γacb = ∂cgab

and then, adding the first two above equations together and subtracting the thirdyields:

Γcab + Γbac + Γcba + Γabc − Γbca − Γacb = ∂agbc + ∂bgac − ∂cgaband since in General Relativity we are always assuming that ∇ is torsion free, thisimplies that Γ satisfies Γabc = Γacb, so that in our situation, we have Γcab = Γcba,Γbac = Γbca, and Γabc = Γacb, so that the above equation simplifies to somethingvery simple:

2Γcab = ∂agbc + ∂bgac − ∂cgab,

and then, dividing both sides of the equation by 2, and operating on both sides ofthe equation by the inverse metric, gab, we get

gcdΓdab =12gcd (∂agbd + ∂bgad − ∂dgab) ,

this technique is used to raise the index on Γcab, so that gcdΓdab = Γcab, and theabove equation becomes

Γcab =12gcd (∂agbd + ∂bgad − ∂dgab) .

The proof is essentially complete, for this shows that given any spacetime (M, g),finding a derivative operator ∇ satisfying ∇g = 0 reduces to taking partial deriva-tives of the metric tensor in any local coordinate system {xi} given by

Γkij =12gkl(∂gjl∂xi

+∂gil∂xj− ∂gij∂xl

),

Thus, given any metric, such a derivative operator exists, and is manifestly unique,since it must satisfy the above equation in any local coordinate system, whichcompletes the proof. �

Lemma 8.5. All Riemannian geodesics are constant speed curves.

Proof. Let γ be a Riemannian geodesic. Since γ is parallel along γ, its length|γ| = 〈γ, γ〉1/2 is constant by the fourth part of the lemma about the compatibilityof a connection. �

9. BACK TO CURVATURE 25

9. Back to Curvature

One of Riemann’s major contributions in 1854 was to show that near any pointp of a Riemannian manifold, there are normal coordinates centered at p, i.e. chartsunder which the preimage of every straight line through the origin is a geodesic of(M, g) through p. Then, he defined a distance function ds (p, q) to be the infimumof the lengths of all curves from p to q, and showed that in normal coordinatescentered at some point p ∈ M , the squared distance function (ds)2 takes the form(here we don’t employ the summation convention):

ds2 = dx21 + · · ·+ dx2

n +112

∑i,j,k,l

Rijkl (p) (xidxj − xjdxi) (xkdxl − xldxk) +

o(x2

1 + · · ·+ x2n

).

Therefore the Riemannian distance function is Euclidean at the zeroth and firstorders, and the correction to the second order is given by a tensor Rijkl. This ledRiemann to the following definition of the curvature tensor :

Definition 9.1. If (M, g) is a Riemannian manifold, the (Riemann) curvatureendomorphism is the map

R : T (M)× T (M)× T (M)→ T (M)

defined byR (X,Y )Z =

([∇x,∇Y ]−∇[X,Y ]

)Z

The (Riemann) curvature tensor is the covariant 4-tensor field obtained bylowering the last index of the curvature endomorophism. In coordinates it is givenby:

Rabcd = Rijkldxi ⊗ dxj ⊗ dxk ⊗ dxl

Definition 9.2. A Riemannian manifold is said to be flat if it is locally isomet-ric to Euclidean space, that is, if every point has a neightborhood that is isometricto an open set in Rn with its Euclidean metric.

Theorem 9.3. A Riemannian manifold is flat if and only if its curvature tensorvanishes identically.

Proof. The proof can be found in Lee (1997). �

In the next part, we will apply our geometric constructions of Riemannian man-ifolds to the general theory of relativity. As any physical theory, general relativitybecomes the most valuable when we use it to make predictions about the physicalnature of our universe. Thus a big portion of the next part will be devoted tocomputations, and in order to simplify them we next state some of the importantproperties of the curvature tensor.

Proposition 9.4. (Symmetries of the Curvature Tensor) The components ofthe curvature tensors have the following symmetries:

• Rijkl = −Rjikl• Rijkl = −Rijlk• Rijkl = Rklij• Rijkl +Rjkil +Rkijl = 0

26

There is another important identity that is satisfied by the covariant derivatives ofthe curvature tensor, called the second Bianchi identity (or by the modern name,the differential Bianchi identity). This identity played an important role in thederivation of Einstein’s equation and we will discuss its importance in the nextpart of the paper.

Proposition 9.5. (Differential Bianchi Identity) The components of the cur-vature tensor satisfy the following identity:

Rijkl;m +Rijlm;k +Rijmk;l = 0

or, in a coordinate-free expression:

∇R (X,Y, Z, V,W ) +∇R (X,Y, V,W,Z) +∇R (X,Y,W,Z, V ) = 0

The different symmetries of the curvature tensor shows that it contains someredundant information. Thus it is often useful to construct simpler tensors thatsummarizes some of the information contained in the curvature tensor. The mostimportant such tensor in the case of general relativity is the Ricci curvature tensor,which is the covariant 2-tensor field obtained by taking the trace of the curvatureendomorphism on its first and last indices. In math books, the Ricci tensor iscommonly written as Rc, but since we employ the abstract index notation, noconfusion will arise if we use R for the Ricci tensor as well, since the indices willdifferentiate between the two.

Definition 9.6. The Ricci curvature tensor, is the 2-tensor:

Rij = Rkkij = gkmRkijm

The scalar curvature is the function S defined as the the trace of the Ricci tensor:

S = Rii = gijRij

Proposition 9.7. The Ricci curvature is a symmetric 2-tensor field.

Proof. This follows directly from the symmetries of the Riemann curvaturetensor. �

Part 3

General Relativity

While Riemann was working on non-Euclidean geometry, he tried to applyhis ideas to the connection between electricity, magnetism, light and graviation.Unfortunately, his investigations ended up incomplete due to his early age. Halfa century later in 1916, Albert Einstein generalized his original work on specialrelativity to systems in which gravity presents. This work, which was the firstapplication of Riemann’s ideas of curvature and non-Euclidean geometry to physics,is commonly known as the general theory of relativity, and it will be the topic ofthe following sections.

Interestingly enough, though the fifth postulate was the source of such a longdebate stretching over 2,000 years; the general theory of relativity and some ofthe experiments that were conducted due to it showed that in fact, our universe isnon-Euclidean, and thus falsified a very long wrong belief.

9.0.1. Notation & Units. Throughout the remainder of this paper, we will forthe most part employ the abstract index notation that is popularly used in mostphysics books that use differential geometry. Thus, if T is an arbitrary tensorof rank (k, l), we will write T = T a1···ak

b1···bl, which, although closely resembling the

component notation for the components of T in an arbitrary basis, is actuallydesigned specifically to tell us that T is, in fact, a tensor of type (k, l), by countingthe number of upper indices and lower indices. Then, if va is an arbitrary vector, i.e.(1, 0)-tensor, we denote by T a1···ak

b1···bj ···blvbj the tensor of type (k, l−1) that is obtained

by putting v in the k + jth slot of the tensor T. Similarly, if ωa is a covector, i.e.(0, 1)-tensor, we denote T a1···aj ···ak

b1···blωaj to be the tensor of type (k − 1, l) that is

obtained by putting ω into the jth slot of the tensor T. Note that although thisnotation looks like component notation (and for all practical purposes it is), it isdefined in such a way that the symbols T a1···ak

b1···blhave coordinate free meaning, and

are thus true geometric tensor expressions.This also gives us a simple relationship between the expressions for general

tensors, and the expressions for tensors in an arbitrary local coordinate patch. Butwe need to distinguish between the two, since there might be times when we canchoose a patch such that the coordinate expressions become simple. So from thispoint forward, we make the distinction that coordinate free tensors will be writtenwith the Latin indices a, b, c, d, while local coordinate expressions will be writtenwith the indices i, j, k, l, . . . . This is the standard convention in Wald, and indeedin most physics books. Thus, if (

(xi), U) is a local coordinate patch then the tensor

T a1···ak

b1···blwill have coordinate expression

T i1···ikj1···jl∂

∂xj1⊗ · · · ⊗ ∂

∂xjk⊗ dxj1 ⊗ · · · ⊗ dxjl

Since this notation is quite cumbersome, we will mostly describe tensors in localcoordinates by their individual components.

We must make note that in general relativity, we will be concerned with space-times (M, gab), which are neighborhoods of spacetime in with the metric gab anda coordinate patch where this metric is valid throughout. Thus, for our purposes,there will be no inherent problems from using the abstract index / coordinate nota-tion, since everything will be taking place in particular spacetimes and coordinateneighborhoods.

Furthermore, we use natural units, where G = c = 1, to simplify some of theexpressions.

11. LINEARIZED GRAVITY 29

10. Einstein’s Equation

The contents of general relativity may be summarized as follows: Spacetimeis a manifold M on which there is a Lorentz metric gab. The curvature of gab isrelated to the matter disctribution in spacetime by Einstein’s equation:

Gab = 8πTab

where Gab ≡ Rab − 12Rgab; Rab is the Ricci curvature tensor and R is the scalar

tensor. Continuous matter distributions in general relativity are described by thesymmeric tensor Tab called the stress-energy tensor. For an observer with 4-velocityva, the component Tabvavb is interpreted as the enrgy density, i.e., the mass-enrgyper unit volume, as measured by this observer.

A derivation of Einstein’s equation and some of the physical interpretations ofit can be found in Robert Wald’s “General Relativity” book.

11. Linearized Gravity

Now that we have introduced the theory of general relativity, and given a briefoverview of the geometry involved in the physical theory, we will conclude the paperwith a brief application to show how Einstein’s equation might be used in practice,in order to solve real life physical problems.

We begin by assuming that we are dealing with a spacetime (M, gab), in whichthe metric tensor is very close to the metric that describes a flat spacetime, ηab.We write

gab = ηab + γab,

where γab is a small pertubation from the original flat metric ηab. There is nomathematically rigorous way to define “small” in this context, since the metric gabhas a signature that is not postive definite, so in many cases it may suffice to assumethat the absolute value of the entries in the matrix γab are all much less than 1. Amore concrete way of looking at the pertubation γab, is to consider a one parameterfamily of spacetimes (M, gab(λ)), which satisfy the conditions that (i) gab dependssmoothly on λ, and (ii) gab(0) = ηab. Then, an arbitrary infinitesmal pertubationfrom ηab is given by dgab

dλ |λ=0. Thus, we can write

γab =dgabdλ|λ=0,

which will provide us with a mathematically precise and concrete way of talkingabout the purtubation γab. The purpose of this procedure has to do with the factthat finding exact solutions to Einstein’s equations is hard. But since we alreadyhave exact solutions to the equation in the case of the flat metric ηab, then if weperturb ηab by a small amount, and only keep the linear terms in the pertubationin our equations, then it becomes much simpler to find solutions that approximateregions of spacetime that are not entirely flat.

Throughout the calculations, we will raise the indices with ηab instead of withgab, so that the pertubation term γab is not hidden in a raised or lowered index. Wewill derive the linearized Einstein’s equation for such a pertubation, by throwingaway all the terms that arise which are not linear in the term γab. Since we areassuming that γab is “small,” solving this linearized form of the equation will giveus good approximate solutions to the actual equation.

30

Letting ∂a denote the ordinary derivative operator associated with the flatmetric, ηab, we can compute the Christoffel symbols by

Γcab =12gcd (∂agbd + ∂bgad − ∂dgab) ,

where gcd denotes the inverse metric. Then, since gab = ηab + γab, we havegab = ηab − γab, so that

Γcab =12ηcd (∂agbd + ∂bgad − ∂dgab) ,

since the term γcd(∂agbd+∂bgad−∂dgab) contains no linear terms in γab. Then,expanding gab = ηab + γab in the above equation, we have

Γcab =12ηcd (∂aηbd + ∂aγbd + ∂bηad + ∂bγad − ∂dηab − ∂dγab)

and since ∂a is the ordinary derivative operator associated with ηab, each termof the form ∂aηbc = 0, so we have

Γcab =12ηcd (∂aγbd + ∂bγad − ∂dγab) .

Now that we have computed the Christoffel Symbols, we will calculate the Riccitensor Rab to linear order, using the formula

Rab = ∂cΓcab − ∂aΓccb + ΓdabΓcdc − ΓdcbΓ

cda,

but since each of the terms ΓdabΓcdc, and ΓdcbΓ

cda will be nonlinear in γab, to linear

order, we have

R(1)ab = ∂cΓcab − ∂aΓccb,

and substituting our formula for Γcab, we have

R(1)ab = ∂c

(12ηcd(∂aγbd + ∂bγad − ∂dγab)

)− ∂a

(12ηcd(∂cγbd + ∂bγcd − ∂dγcb

)=

12ηcd∂c∂aγbd +

12ηcd∂c∂bγad −

12ηcd∂c∂dγab −

12ηcd∂a∂cγbd −

12ηcd∂a∂bγcd −

12ηcd∂a∂dγcb

=12

((∂d∂aγbd + ∂d∂bγad)− ∂d∂dγab − ∂a∂dγbd − ∂a∂cγdd + ∂a∂cγcb)

=12

(∂c∂aγbc + ∂c∂bγac)−12∂c∂cγab −

12∂a∂bγ

= ∂c∂(aγb)c −12∂c∂cγab −

12∂a∂bγ,

and thus, we have the formula for the first order Ricci tensor, R(1)ab = ∂c∂(aγb)c−

12∂

c∂cγab− 12∂a∂bγ, where γ = γaa is the trace of γab, and on the second to last line

we used the fact that γcb = γbc. Now, we calculate R = Raa:

Raa = ∂c∂(aγa)c −12∂c∂cγ

aa −

12∂a∂aγ

=12∂c∂aγac +

12∂c∂aγac −

12∂a∂aγ −

12∂a∂aγ

= ∂b∂aγab − ∂a∂aγ,

11. LINEARIZED GRAVITY 31

Then we note that Rgab = Rηab, since the higher order terms of γab drop outof the sum Rgab, and we calculate G(1)

ab = R(1)ab −

12R

(1)ηab, or

G(1)ab = ∂c∂(aγb)c −

12∂c∂cγab −

12∂a∂bγ −

12ηab(∂c∂dγcd − ∂c∂cγ).

So that we now have an expression for the Einstein tensor, Gab, to first orderapproximation in γab. Then, if we make the substitution γab = γab− 1

2ηabγ, we cangreatly simplify the above expression. Be warned, the expression first appears tobe pretty nasty, but simplifies pretty nicely. We have

G(1)ab =

12∂c∂a

(γbc +

12ηbcγ

)+

12∂c∂b

(γac +

12ηacγ

)−

12∂c∂c

(γab +

12ηabγ

)− 1

2∂a∂bγ −

12ηab

(∂c∂d

(γcd +

12ηcdγ

)− ∂c∂cγ

)=

12∂c∂aγbc +

14ηbc∂

c∂aγ +12∂c∂bγac +

14ηac∂

c∂bγ −12∂c∂cγab −

14ηab∂

c∂cγ −12∂a∂bγ −

12ηab∂

c∂dγcd −14ηabηcd∂

c∂dγ +12ηab∂

c∂cγ

=12∂c∂aγbc +

12∂c∂bγac −

12∂c∂cγab −

12ηab∂

c∂dγcd +14ηbc∂

c∂aγ +

14ηac∂

c∂bγ −14ηab∂

c∂cγ −12∂a∂bγ −

14ηab∂

c∂cγ +12ηab∂

c∂cγ

= ∂c∂(aγb)c −12∂c∂cγac −

12ηab∂

c∂dγcd +14∂b∂aγ +

14∂a∂bγ −

12∂a∂bγ

= ∂c∂(aγb)c −12∂c∂cγac −

12ηab∂

c∂dγcd,

and therefore, in terms of γab, the linearized Einstein’s equation G(1)ab = 8πTab

becomes

∂c∂(aγb)c −12∂c∂cγac −

12ηab∂

c∂dγcd = 8πTab,

where Tab is the stress-energy tensor. Now, we can simplify this expression evenmore, and that is the overall goal of this section, but to do so we need to introducea concept of Gauge freedom. Gauge freedom is the freedom we have to make achange of coordinates while keeping the same physical solutions to the equationsthat arise as a result of the change of coordinates.

This discussion about gauge freedom will begin by studying the effects of dif-feomorphisms on physical solutions. In general, if φ : M → N is a diffeomorphismbetween smooth manifolds, then M and N are in particular homeomorphic. Thusit is not hard to see that if a physical theory describes nature in terms of a space-time manifold M and tensor fields, {T (α)}α∈A, where α is just an arbitrary index(not meant to signify tensor type or tensor components) and if φ : M → N is adiffeomorphism, then any solutions (M,T (α)) and (N,φ∗T (α)) will have physicallyidentical properties, where φ∗ denotes the pushforward of φ, which is ok to talkabout since we are assuming that φ is a diffeomorphism. Now, we present a defi-nition and a proposition that will allow us to calculate the Gauge freedom of thepertubation γab.

32

Definition 11.1. A one-parameter group of diffeomorphisms φλ is a C∞ mapfrom R×M →M such that for any λ ∈ R, φλ : M →M is a diffeomorphism andfor all λ1, λ2 ∈ R, we have φλ1 ◦ φλ2 = φλ1+λ2 .

We observe that the relationship φλ1 ◦ φλ2 = φλ1+λ2 implies that φ0 is theidentity mapM →M , since φ0◦φλ = φλ for all λ ∈ R, and similarly for φλ◦φ0.Wewill now show that each one-parameter group of diffeomorphisms can be associatedwith a vector field V a ∈ TM. To see this, observe that given a one parameter groupof diffeomorphisms φλ, for a for a given p ∈ M , φλ (p) : R → M is a curve whichpasses through p at λ = 0. Define Vp to be tangent to this curve at λ = 0. ThenV a can be thought of as an infinitesimal generator of these diffeomorphisms.

Conversely, suppose that V a is a vector field on M. Then, there is a uniquefamily of integral curves corresponding to V a. Thus, we define φλ (p) to be thepoint lying at the parameter value λ along the integral curve of V a starting at p.Then, by the properties of integral curves, φλ will be a one-parameter group ofdiffeomorphisms, so long as we only restrict our attention to parameter values λallowed by the solution to the family of integral curves.

Thus, we can see that there is a one-to-one correspondance between one-parameter groups of diffeomorphisms, and vector fields V a ∈ TM. Having seenthis correspondance, we now return to our discussion about gauge freedom. Bythe above remark about general diffeomorphisms, we see that in our spacetime(M, gab) , if φ : M → M is any diffeomorphism, then (M, gab) and (M,φ∗gab)will have physically identical solutions, so that gab and φ∗gab represent the samespace-time geometry.

Now, recall that in the beginning of this section, we defined the pertubationγab in terms of a one parameter family of spacetimes (M, gab (λ)) . Thus, if φλ isan arbitrary one-parameter group of diffeomorphisms, we see that (M, gab (λ)) and(M, (φλ)∗ gab (λ)) represent the same one-parameter family of spacetimes. Recallthen, that at the beginning of the second, we defined the pertubation γab mathe-matically as

γab =dgabdλ|λ=0 .

But since (φλ)∗ gab represents the same space time as gab, it is immediate thatif we define γ′ab as

γ′ab =d (φλ)∗ gab

dλ,

that γab and γ′ab represent the same physical pertubation. Now, the motivationfor why we have defined the pertubation γab in such a way should become moreclear. Let V a be the vector field associated with the one parameter group φλ. Then,by the way we have defined γab and γ′ab, it follows that

γ′ab − γab = −LV ηab,

since gab (0) = ηab, where LV ηab represents the Lie derivative of ηab with respectto V a. Then, for any connection ∇, by the properties of the Lie derivative, we seethat

LV ηab = V c∇cηab + ηcb∇aV c + ηac∇bV c,

11. LINEARIZED GRAVITY 33

and thus, for the ordinary derivative operator ∂a associated to ηab (∂cηab = 0),it follows that

LV ηab = ηcb∂aVc + ηac∂bV

c

= ∂aVb + ∂bVa.

Therefore, we obtain the result that γ′ab = γab − ∂aV b − ∂bV a. And it followsthat if we make the change of coordinate

γab → γ′ab,

orγab → γab − ∂aVb − ∂bVa

for any vector field V a (since any vector field has a unique family of integralcurves), it follows that this coordinate change will preserve the physical solutions,and thus the guage freedom of linearized gravity is given by this precise coordinatechange.

Now, since this coordinate change will preserve physical solutions for any vectorfield V a that we choose on the manifold, then we can use this fact to even furthersimplify the Einstein equation under the assumptions of linearized gravity. It mightbe difficult to see the motivation behind what we are going to do, but anybody thathas taken a course in electrostatics might see this as something very familiar. Whatwe will do is find a specific vector field V a that satisfies the differential equation

∂b∂bVa = ∂bγab,

Then let us observe what happens to the Einstein equation under the gaugetransormation given above. First we observe that we have

γ′ab = γab − ∂aVb − ∂bVa,

and then using the fact that

γ′ab = γ′ab −12ηabγ

′

where we can write, using the same gauge freedom equation, and the fact thatγ = γaa is the trace of γab, we have

γ′ = γ − ∂aVa − ∂aVa,

so that combining these two facts, we observe that

γ′ab = γab − ∂aVb − ∂bVa −12ηabγ +

12ηab∂

cVc +12ηab∂

cVc

= γab − ∂aVb − ∂bVa + ηab∂cVc,

and then differentiating both sides of the equation, we obtain

∂bγ′ab = ∂bγab − ∂b∂aVb − ∂b∂bVa + ηab∂b∂cVc

= ∂b∂bVa − ∂b∂aVb − ∂b∂bVa + ∂c∂bVc

= ∂b∂aVb − ∂b∂aVb= 0

where the second to last line comes from the fact that a, b, c are dummy indices,so ∂c∂bVc = ∂b∂aVb. Thus, we can see that by solving the differential equation

34

∂b∂bVa = ∂bγab, we can conclude that under our gauge change of coordinates, thatthe equation

∂bγab = 0must be satisfied. Thus, it is clear that under this change of coordinates, the

Einstein equation in the linearized gravity approximation becomes

∂c∂cγab = −16πTab

Thus, we can see that if we are in a region of space-time where the stress-energytensor Tab is relatively simple, then solving the Einstein’s equation in the linearizedgravity approximation reduces to solving an equation involving the Laplacian op-erator essentially, which is a lot less hard then the original problem. We will shownext how this can be used to convince the non-believers that under this linearizedgravity approximation, and under the right assumptions about the stress-energytensor Tab, that the laws of general relativity reduce to those of good old fashionedNewtonian Mechanics.

12. The Newtonian Limit

The culmination of this paper will be to show that all of the work that has goneinto developing this theory of gravitation has not been in vain. For it would be agreat atrocity if we had devoloped a beautiful mathematical theory of gravitationwhich did not accurately predict phenomena that occurs in nature. Since we knowthat Newton’s laws of gravitation are a very good model, it will be very pleasing,and even a huge relief if we can show that under the same set of circumstances(assumptions) the theory of gravitation introduced here reduces to the theory in-troduced by Newton. This goal will be what occupies the rest of the paper (thoughmost of the work towards this has already been done in just deriving the Einsteinequation in the linearized gravity approximation that we have given above).

Now, the first thing we need to assume is that in the Newtonian Limit, grav-ity will be relatively week. This means that the linearized gravity approximationshould hold, and that velocities will be very slow when compared to the speed oflight. Thus, in this approximation the four-velocity of any world line should beapproximately

V a ∼ (1, 0, 0, 0).Here we are using units where the speed of light c = 1, and we are assuming

that the proper time parameter τ ≈ t, since velocities are slow. Then, we can seefrom the definition of the stress energy tensor, that in this global inertial coordinatesystem, the energy density is given by

TabVaV b = T00V

0V 0 + T0jV0V j + Ti0V

iV 0 + TijViV j

where i, j 6= 0. But since in our approximation we have V i ≈ 0 for i 6= 0, itfollows that we have

Tabvavb ≈ T00v

0v0.

From this we can writeTab ≈ ρtatb,

where ρ is the energy density of the space-time in question, and ta = ∂/∂x0 isthe time basis vector in any local coordinate system. One can see from the definitionof the stress energy tensor, that the lack of space-space components represents the

12. THE NEWTONIAN LIMIT 35

fact that stresses are small in the Newtonian limit, and the lack of the time-spacecomponents represent the fact that velocities (momenta) are small in this limit.

Using the fact that we can approximate Tab ≈ ρtatb, and Einstein’s equationthat we have just derived for linearized gravity, ∂c∂cγab = 16πTab, we see thatsolving this equation in any local coordinate system in the Newtonian limit amountsto solving the system of differential equations

∆γij = 0

for i, j 6= 0 and∆γ00 = −16πρ,

where we are using ∆ to denote the standard Laplacian operator in space.Since we are looking for solutions to these equations that are “well-behaved” atinfinity, meaning that the solutions do not blow up at infinity, since that wouldseem to make very little physical sense in an environment that is very far awayfrom gravitation causing matter, it follows that the unique solution to ∆γij = 0 isjust γij = 0. One might observe that we could also take γij = constant, but thena further gauge transformation could take us back to the zero solution, so we cansee that we lose no information by taking our solution to be 0 for i, j = 1, 2, 3.

Thus, if we define a scalar function φ ≡ − 14γ00, we can see that φ must satisfy

Poisson’s equation∆φ = 4πρ

and furthermore, since all the other terms of γab other than the time-time com-ponent drop out of the equation, we can see that the pertubation γab is completelydefined by the equation

γab = γab −12ηabγ = − (4tatb + 2ηab)φ.

We are almost done showing how this case will boil down to the known equationin the Newtonian limit. The next step is simply to recall that the motion of testbodies in a gravitational field in general relativity are determined by the geodesicequation;

d2xi

dτ2+ Γijk

(dxj

dτ

)(dxk

dτ

)= 0

As we said in the beginning of the section, we are assuming that speeds are verylow when compared to the speed of light, it follows that in any local cooridnates,the four velocity of any test particle is given by

V a ∼ (1, 0, 0, 0)

so that the termsdxi

dτ= 0

for i = 1, 2, 3. The only terms that stay in the geodesic equation are the space-space terms,

dx0

dτ=dt

dτ≈ 1,

and since, in this approximation, the coordinate time t is approximately equal tothe proper time τ, t ≈ τ, the geodesic equation that governs motin in the Newtonianlimit becomes

d2xi

dt2= −Γµ00

36

in any local coordinate system. Now, to calculate the value of the time-timeChristoffel symbols, Γi00, we use the equation that we derived much earlier for theChristoffel symbols;

Γcab =12ηcd (∂aγbd + ∂bγad − ∂dγab)

which can be written in coordinates as

Γkij =12ηkl(∂γjl∂xi

+∂γil∂xj− ∂γij∂xl

).

From this it is easy to calculate the time-time components, Γi00,

Γi00 =12ηij(∂γ0j

∂x0+∂γ0j

∂x0− ∂γ00

∂xj

)= −1

2ηij

∂γ00

∂xj

= −12ηi0

∂γ00

∂t− 1

2ηi1

∂γ00

∂x1− 1

2ηi2

∂γ00

∂x2− 1

2ηi3

∂γ00

∂x3,

and thus we can read off the individual values for each of the symbols Γi00 byobserving that ηab is given by the matrix

ηab = ηab =

−1 0 0 00 1 0 00 0 1 00 0 0 1

.

Thus

Γ000 =

12∂γ00

∂t

Γ100 = − 1

2∂γ00

∂x1

Γ200 = − 1

2∂γ00

∂x2

Γ300 = − 1

2∂γ00

∂x3

Then, if we restrict our attention to the components i = 1, 2, 3, we have theequations

Γi00 = −12∂γ00

∂xi.

This equation can be further simplified by recalling the fact that in our New-tonian limit, we have the expression for the pertubation tensor γab

γab = − (4tatb + 2ηab)φ,

so that calculating directly the component γ00, we have

γ00 = − (4 (1) + 2 (−1))φ= −2φ,

and thus we have the following equation for the second time derivative of thespacial components

d2xi

dt2=

∂φ

∂xi.

12. THE NEWTONIAN LIMIT 37

From this we finally have have the following equation:

a = −∇φ

where ∇ denotes the ordinary spacial gradient operator, and a is the ordinaryspacial acceleration vector.

The equation that we have just derived, a = −∇φ, is precisely the equationused to describe the motion of test particles in Newtonian gravitation theory. Thus,we can see that in the Newtonian limit, the theory of gravitation described by Ein-stein matches up precisely with Newton’s own theory of gravitation. This is whatone would hope for, since as we said before, we know that Newtonian mechanicsaccurately describes nature for a very wide range of physical phenomena. Oneshould notice that something very amazing here has happened, however. Newton’stheory of gravitation describes the motion of test particles under a force of gravity.The theory states that particles will fail to travel in a straight line due to the forceacting on them. Einstein’s theory, however, asserts that although there may beno forces acting on these particles, that the geodesics that describe their motionunder the curvature of the space-time geometry are found by solving the same ex-act differential equation that is used to describe the motion of the particles underNewton’s theory of gravitation.

Throughout this paper, we have attempted to show how the abstract mathemat-ical theory of Riemannian geometry can be used in practice to describe real physicalphenomena, via Einstein’s theory of General Relativity. We also showed that in theright approximation (very low velocities) this theory agrees with Newton’s theoryof gravitation, which was formulated almost two centuries before Riemann was evenalive, and certainly long before Einstein developed the theory of General Relativity.It is our hope and goal that this paper has not only motivated some of the veryabstract definitions and theorems in Riemannian geometry, but has also exhibitedsome of the beauty behind the mathematics. It is our belief that most of the mostprofound questions known to man, the questions that have driven the pursuit ofknowledge for centuries, are deeply correlated with this beautiful theory of geom-etry and its applications to physics. It is for this reason that we have decided towrite this paper, and it is for this reason that we hope any reader might get a slightsense of how incredible the math is upon reading this paper.

Bibliography

Huggett, N. (1999). Space from Zeno to Einstein. The MIT Press.Lee, J. M. (1997). Riemannian Manifolds: An Introduction to Curvature. Springer.Lee, J. M. (2002). Introduction to Smooth Manifolds. Springer.Spivak, M. (1979). A Comprehensive Introduction to Differential Geometry. Publishor Perish.

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