Title: An Introduction to General Relativity

Author: Xavier Ramos Oliv

Advisor: Narciso Romn-Roy
Department: atem!tica Aplicada I"
Academic year: #$%#&#$%'
De(ree in athematics
Universitat Polit`ecnica de Catalunya
Facultat de Matem`atiques i Estadstica
Bachelor Thesis - Treball de Fi de Grau
An Introduction to General Relativity
Xavier Ramos Olive
Advisor: Narciso Roman-Roy
Matem`atica Aplicada IV
Nature likes theories that are simple when
stated in coordinate-free, geometric language.
Nilsk Oeijord
Preface
This is a bachelors degree thesis of the Bachelor on Mathematics (Grau de
Matem`atiques), a degree included in the EHEA (European Higher Education
Area), oered by the Technical University of Catalonia (Universitat Polit`ecnica de
Catalunya). As I belong to the rst generation of EHEA students, it is the rst
year that this kind of thesis is done in this university, so I would like to thank the
eort that all the professors, employees and students of the university, and specif-
ically of the Faculty of Mathematics and Statistics (Facultat de Matem`atiques i
Estadstica), have made to help us during the elaboration of these thesis.
I would like to specially acknowledge and thank the eort of my tutor Professor
Narciso Roman-Roy, for his time and patience, and to Professor Xavier Gr`acia, for
allowing me to assist to his master lectures on Dierentiable Manifolds that helped
me a lot in this thesis. I would like to extend these acknowledgements to all the
people that have helped me to write this paper in one or another way. I hope you
will enjoy it!
Abstract
Key words: Pseudoriemannian Geometry, Dierential Geometry, General Relativity
MSC2010: 83C, 53Z05
This bachelors degree thesis is an introduction to the Theory of General Relativity (GR), a
relativistic theory of gravity, from the point of view of a recently graduated mathematitian.
The principles of GR are stated and some motivation on the formulation of the theory is
provided. It is shown that freely-falling particles move along geodesics of spacetime and
Einsteins equations are derived as a generalization of Newtons gravity. The uniqueness
of Einsteins equations and the presence of the cosmological constant are discussed.
No knowledge is assumed neither in Special or General Relativity, nor in Pseudoriemannian
Geometry, but knowledge of basic Dierential Geometry on Manifolds is highly recom-
mended, although there is a brief introduction on this topic at the beginning.
The thesis concludes nding Schwarzschild solution by assuming that there exists a spher-
ically symmetric metric that is a solution to Einsteins equations in vacuum and seeing
what properties should this metric have. The boundary conditions imposed are the ex-
istence of a punctual uncharged mass at the origin and atness of the metric at innity.
The result is a particular solution that can be applied in many contexts, such as in the
Solar System.
Contents
Introduction 1
Chapter 1. Pseudoriemannian Geometry 3
1. General notions on Manifolds 3
2. Covariant Derivative 4
3. Parallel Transport and Geodesics 6
4. Curvature and Torsion 6
5. Pseudoriemannian Manifolds 7
6. Example 10
Chapter 2. Foundations of General Relativity 11
1. Special Relativity 11
2. Postulates of General Relativity 14
3. The matter elds 16
Chapter 3. Einsteins Equations 19
1. An informal approach 19
2. Einsteins Tensor 21
Chapter 4. The Schwarzschild Solution 27
1. The time-dependent spherically symmetric metric 27
2. The Schwarzschild solution 30
Conclusions 33
References 35
xi
Introduction
This text is a bachelor thesis about the Theory of General Relativity. Many articles
have been written on this topic and there are plenty of related elds that can be
studied, but the goal of this paper is not so ambicious; its objective is to give
an introduction to the topic comprehensible from the point of view of a recently
graduated mathematician. More precisely, we will study the geometric background
needed to deal with the theory of General Relativity, we will introduce some results
of Special Relativity, we will deduce Einsteins Equations and we will nd a special
solution to them: Schwarzschilds solution. The only requirement to understand it
is to have some knowledge of dierential geometry on manifolds (although there is
a brief review on it in the thesis), and curiosity about the natural phenomena that
makes us keep our feet on the ground: gravity.
Einsteins papers from 1905 to 1915 were actually a revolution in the scientic
community; his new theory wasnt only important because of solving the incon-
sistencies between classical mechanics and electromagnetism, but also because his
new theory brought to physicists a new point of view of the mathematics involved
in physical fenomena: the noneuclidean geometries studied during the 19th century
by Gauss and Riemann, among others, were not just a mathematical game. Space
and time were described in Einsteins Theory of Special Relativity (SR) as a sin-
gle 4-dimensional manifold (called spacetime) provided with a non-positive-dened
constant and uniform metric.
However, Einstein realized that SR, although being successful in many experiments,
was not able to describe gravity; this brought him to develop the Theory of General
Relativity (GR), that is essentially a theory of gravity in the context of SR (GR
reduces to SR locally). This theory is, probably, one of the most beautiful theories
from a mathematical point of view. Contrasting with SR, where there existed a
special class of frames, called inertial frames (bodies moving at a constant velocity
from us), in GR any reference frame is valid for describing physics. Moreover, in
general, the variables involved in GR (the coordinate functions of spacetime) have
no longer a clear physical meaning, they are, as physicists would say, generalized
variables: there is no distinction between time and space variables. This is beautiful
from a mathematicians point of view: GR is a geometrical theory.
The outgrowth of SR and GR not only revolutionised physics, but also mathematics.
The geometry of the new theories needed the study of pseudoriemannian manifolds:
manifolds with non-positive-dened metrics, that were not so popular as they are
1
2 INTRODUCTION
today before Einsteins articles, maybe because of their unnatural properties (such
as that they have some vectors with negative norm, or the existence of non-zero
vectors with null norm). The rst chapter on this text is devoted entirely to the
study of pseudoriemannian geometries.
The second chapter formulates SR and introduces some relevant results for GR. It
also contains the principles of GR, some simple ideas that guided Einstein towards
his theory. One of them is specially remarkable from the point of view of geometry:
the laws of physics must be invariant under changes of coordinates. This points
out that they should be formulated in a coordinate-free environment: for example,
using equalities between tensor elds.
GR considers gravity as a result of spacetime curvature. As it was known since
Newtons times, mass is the source of gravity, so as gravity is now explained in
terms of the metric tensor, GR has to provide the relation between the metric of
spacetime and its mass distribution. Not only mass, but also, as Einstein saw in SR
that there exists an equivalence between mass and energy (with the famous formula
E = mc
2
), any sort of energy inuences the metric; all these sources of gravity are
expressed in a compact tensor called the Energy-Momentum Tensor. In the third
chapter of this thesis, I deduce Einsteins Equation, that gives the relation between
this tensor and the metric tensor, and that constitutes the heart of GR. I also show
why freely falling particles move along geodesics of spacetime (another beautiful
result of GR).
To conclude with, we should notice that Einsteins equations are not easy to solve
in general: they are a system of 10 second-order nonlinear PDEs (although just 6
of them are independent). Nevertheless, assuming spherical symmetry, we can nd
easily Schwarzschilds solution. This solution is really important, it was found by
the astronomer Karl Schwarzschild in 1916, and it describes the metric of spacetime
in vacuum, when there is just a spherical (or punctual) mass at the origin. It was
the rst nontrivial solution found and it describes the behavour of gravity near
spherical bodies, such as stars, planets or satellites. In the last chapter I will
discuss how to nd this solution.
Notation remarks: In the following, we will use Einstein summation convention,
i.e., if equal indices appear on dierent levels, summation over all the indices will
be assumed. For example: a
i
dx
i
= a
1
dx
1
+ a
2
dx
2
+ . . . + a
n
dx
n
. On the other
hand, all the objects that we will use (manifolds, maps, elds, etc.) are supposed
to be as regular as necessary; if it is not specied, we can assume that they are
always smooth objects.
Chapter 1
Pseudoriemannian Geometry
In this section we are going to see the main geometrical tools needed to properly
formulate the Theory of General Relativity (GR); it will be a rather short introduc-
tion to Pseudoriemannian (or Semiriemannian) Geometry, so the readers interested
in a deeper insight into this eld should see the references [15], [13] or [10].
1. General notions on Manifolds
This is a brief review from the basic notions learned in the courses of Dierential
Geometry and Dierentiable Manifolds; you can use the notes on [7] as a trustable
source. Lets remember some concepts:
Adierentiable manifold is a topological space M together with an m-dimensional
dierentiable structure, i.e., a maximal collection of pairs (U

) where U

is
an open set of M, M =

, and

: M R
m
are homeomorphisms such that,
in the corresponding domain,

and

are dierentiable functions (they

are dieomorphisms).
On every point p M we can dene its tangent space T
p
M as the classes of
equivalence of curves : I R M that have the same tangent vector, i.e., given
a chart (U, ) with p U, if D()(0) = D( )(0), assuming (0) = (0) =
p. In fact, there exists an isomorphism between T
p
M and the space of punctual
derivations of C

(M) in p, i.e., the R-linear maps : C

(M) R such that

(fg) = (f)g + f(g) (the isomorphism is the one that brings a tangent vector
u to the directional derivative along u, that we will denote /
u
). From this new
point of view, xed a chart (U, ), we can dene a basis of T
p
M as the directional
derivatives along the coordinate vectors

x
i
(p). The corresponding dual space,
T

p
M, is called cotangent space and has an associate dual basis dx
i
(p).
Then we can dene the tangent bundle as the set of all possible tangent vec-
tors on M, TM =

pM
T
p
M, and analogously the cotangent bundle, T

M =

pM
T

p
M. These sets are, indeed, manifolds of dimension 2m. Then we can
dene the projection
M
: TM M, and the set of vector elds X(M) as the
set of all sections of the tangent bundle; i.e., a vector eld is a map X : M TM
such that
m
X = Id
M
. Similarly, we dene the projection
M
: T

M M and
3
4 1. PSEUDORIEMANNIAN GEOMETRY
the set of one-forms
1
(M) is the set of all sections of the cotangent bundle, so a
one-form is a map : M T

M such that
M
= Id
M
.
In a more general context, we can dene at each point the tensor product space
Tens
k
l
(T
p
M) =

l
T

p
M

k
T
p
M, with k, l N, and the associate tensor bundle
Tens
k
l
(TM) =

pM

l
T

p
M

k
T
p
M, that brings us to dene a k-contravariant
l-covariant tensor eld as a section of Tens
k
l
(TM), and we will denote the set
of this kind of tensor elds as T
k
l
(M), and in general, the set of tensor elds of
any order as T(M). Notice that, p U M, we can write in local coordinates a
tensor eld R Tens
k
l
(TM) as:
R(p) = R
i1i2...i
k
j1j2...j
l
(p)

x
j1

x
j2

p
. . .

x
j
l

p
d
p
x
i1
d
p
x
i2
. . . d
p
x
i
k
Remembering that /
u
is the directional derivative along u, we can understand
vector elds as derivations:
Denition 1.1. Given X X(M), if W M is an open set and f : W R is
a function, the Lie derivative along X is the function /
X
f : W R dened as
(/
X
f)(p) = /
Xp
f(p).
With this new tool, we can dene a useful operator that measures up to what point
two vector elds commute:
Denition 1.2. If X, Y X(M) the Lie bracket of X and Y is the derivation
[X, Y ] := /
X
/
Y
/
Y
/
X
.
In coordinates, if (U, ) is a chart, X

U
= f
i
x
i
, Y

U
= g
i
x
i
, then:
[X, Y ]

U
=
_
f
i
g
j
x
i
g
i
f
j
x
i
_

x
j
In particular,
_

x
i
,

x
j

= 0
2. Covariant Derivative
In this thesis, we will be interested in writing down dierential equations that are
invariant under changes of coordinates to describe physical phenomena in the same
way from any reference frame. However, the natural concept of partial derivative
doesnt t this requirement. The concept of a covariant derivative is one of the
main geometrical tools that we are going to use and it has to be regarded as a
generalisation of the concept of partial derivative.
Denition 1.3. A covariant derivative or a connection dened on M is a map
: X(M) X(M) X(M), (X, Y ) =
X
Y , satisfying:
a)
X
Y is R-linear on Y
2. COVARIANT DERIVATIVE 5
b)
X
Y is C

(M)-linear on X
c)
X
(fY ) = (L
X
f)Y +f
X
Y , f C

(M)
Denition 1.4. If (U, ) is a chart of M and

x
i
are the corresponding coordinate
vector elds, that form a basis of X(U), we dene the Christoel symbols of
with respect to this basis,
k
ij
C

(M), as:

x
i

x
j
=
k
ij

x
k
This denition can be extended to an arbitrary basis (not necessarily a coordinate
basis) of X(U) E
i
.
Proposition 1.5. There exists a bijection between the set of connections on M and
the set of Christoel symbols (sets of n
3
functions
k
ij
C

(M)). If X = f
i
E
i
and Y = g
j
E
j
,
X
Y = (L
X
g
k
+
k
ij
f
i
g
j
)E
k
.
The notion of covariant derivative of vector elds can be extended to the covariant
derivative of a tensor eld.
Theorem 1.6. There exists a unique mapping
X
: T(M) T(M) such that:
f C

(M),
X
f = L
X
f
Y X(M),
X
Y is the connection dened in 1.1

X
is R-linear

X
applies T
k
l
(M) to T
k
l
(M)

X
(R S) = (
X
R) S +R (
X
S)

X
commutes with inner contractions
In particular,
x
i
dx
j
=
j
ik
dx
k
.
It can be seen that the covariant derivative
X
Y (p) depends only on the value of
X
p
and the value of Y in a neighbourhood of p. Moreover, it only depends on the
value of Y along a path through (t) such that (0) = p, when t (, ). We will
denote the set of vector elds along a path as X().
Proposition 1.7. If : I M is a path, there exists a unique map
t
: X()
X() such that:
(1) It is R-linear
(2) f C

(I), X(),
t
(f) =
df
dt
+f
t

(3) If Y X(M),
t
(Y )(t) =

(t)
Y
The operator
t
is called the covariant derivative along .
This operator can also be generalized to a covariant derivation of a tensor
eld along : the additional axioms we need to dene it are the same as in the
denition of the covariant derivation of a tensor eld.
6 1. PSEUDORIEMANNIAN GEOMETRY
3. Parallel Transport and Geodesics
When studying GR, geodesics (in a certain way, the straightest possible curves in
the manifold) are an imprescindible topic to cover, because in fact, as we will see
later, bodies inuenced by a gravitational eld travel along a geodesic of spacetime
(this is the simplest way they can behave, it is very similar to the idea of Newtons
rst law of motion). To dene a geodesic, rst we have to introduce the concept
of parallel transport: if we have the notion of parallel transporting a vector eld
along a path, then a path whose tangent vector is always parallel (the straightest
possible curve) will be dened as a geodesic.
Denition 1.8. Given a path : I M, a vector eld X X() is said to be
parallel transported along (t) if it satises:

t
X = 0, t I
In a chart of coordinates (U, = (x
i
)), where X = X
k
x
k
(assuming is entirely
dened in U), this condition can be expressed as a 1st order linear system of
dierential equations:

t
X =
_
dX
k
dt
+ (
k
ij
)
d(x
i
)
dt
X
j
_

x
k
= 0
As an immediate result of writing down the denition of parallel transport in co-
ordinates we obtain the next result, using Picards theorem:
Proposition 1.9. Given t
0
I and X
0
T
(t0)
M, there exists a unique X X()
such that it is parallel transported along and X(t
0
) = X
0
. We say that X is the
parallel transport of X
0
with respect to the connection.
Then we can dene the concept of geodesic. The denition comes as a generalisation
of the concept of a straight line in euclidean space; one of the properties of straight
lines is that their tangent vector is always parallel to themselves.
Denition 1.10. A path : I M is a geodesic of (M, ) if

is parallel along
, i.e.,
t

= 0. This can also be written in coordinates, and we obtain the 2nd

order non linear autonomous system:

=
_
d
2
dt
2
(x
k
) + (
k
ij
)
d(x
i
)
dt
d(x
j
)
dt
_

x
k
= 0
4. Curvature and Torsion
So far, we have introduced the necessary tools to construct some objects that de-
scribe the local geometry of M: the curvature and the torsion.
5. PSEUDORIEMANNIAN MANIFOLDS 7
Denition 1.11. Given a manifold M with a connection , the Riemann cur-
vature tensor or Riemann tensor of M is dened as the map R : X(M)
X(M) X(M) X(M), R(X, Y, Z) =
X

Y
Z
Y

X
Z
[X,Y ]
Z (notice that
R T
1
3
(M)). Sometimes it is denoted as R(X, Y )Z.
We can compute its expression in coordinates. If E
i
is a basis of X(U), E
i
being the
corresponding dual basis of X

(U), and writing

Ei
E
j
=

ij
E

, [E
i
, E
j
] = c

ij
E

and R = R
l
ijk
E
l
E
i
E
j
E
k
, we have:
R
l
ijk
=< E
l
, R(E
i
, E
j
)E
k
>
=< E
l
,
Ei

Ej
E
k

Ej

Ei
E
k

[Ei,Ej]
E
k
>
=< E
l
,
Ei
(

jk
E

)
Ej
(

ik
E

)
c

ij
E
E
k
>
=< E
l
, L
Ei
(

jk
)E

jk

i
E

L
Ej
(

ik
)E

ik

j
E

ij

k
E

>
= L
Ei
(
l
jk
) +

jk

l
i
L
Ej
(
l
ik
)

ik

l
j
c

ij

l
k
In a coordinate basis, c
k
ij
= 0, so the coecients R
l
ijk
are antisymmetric with
respect to the rst two indices, i.e., R
l
ijk
= R
l
jik
. Notice that the position of the
indices is important, and that the notation that I have decided to use may dier
from other notations used in the sources that appear in the bibliography.
Denition 1.12. The torsion tensor of is the map T : X(M)X(M) X(M),
T(X, Y ) =
X
Y
Y
X [X, Y ] (notice that T T
1
2
(M)).
In coordinates, similarly as what we have done before, if T = T
k
ij
E
k
E
i
E
j
, we
obtain:
T
k
ij
=
k
ij

k
ji
c
k
ij
Again, using a coordinate basis (c
k
ij
= 0) we obtain the symmetry T
k
ij
= T
k
ji
, and
also that T
k
ij
= 0
k
ij
=
k
ji
, i.e., the Christoel symbols are symmetric in
their lower indices. This is the reason to call symmetric or torsionless when
T = 0.
When dealing with GR, we need to dene some objects that derive from the cur-
vature; one of those objects is the Ricci tensor.
Denition 1.13. We dene the Ricci tensor as Ric(Y, Z) :=< E
i
, R(E
i
, Y )Z >.
Then, in a coordinate basis, we will usually use the notation for the components
R
jk
:= R
i
ijk
.
5. Pseudoriemannian Manifolds
As we mentioned above, in Einsteins theory of relativity space and time are hold
together in a 4-dimensional manifold that has associated a metric; this metric can
be though as a kind of inner product in the tangent bundle, but has some special
8 1. PSEUDORIEMANNIAN GEOMETRY
characteristics that doesnt allow us to say that it is an inner product in the usual
sense that we studied in Linear Algebra. A manifold together with this kind of
metric is called a Pseudoriemannian Manifold.
Denition 1.14. Given a manifold M, a pseudoriemannian or semirieman-
nian metric in M is a tensor eld g T
0
2
(M) symmetric and nondegenerate, i.e.,
it satises:
(1) g
p
(U, V ) = g
p
(V, U), U, V T
p
M, p M
(2) g
p
(U, V ) = 0, U T
p
M = V = 0, p M
The couple (M,g) is called a pseudoriemannian manifold. If g is also positive
dened (g
p
(U, U) 0, g
p
(U, U) = 0 U = 0, U TpM, p M), then the
metric and the manifold are called riemannian.
In coordinates g = g

. Notice that the metric allows us to stablish an

isomorphism g : TM T

M: < g
p
(u
p
), v
p
>= g
p
(u
p
, v
p
); this is what is known
colloquially as the operations of rising and lowering indices (because g

=
E

and using the inverse of the metric g

= E

, just by denition of the

metric coecients). Using this new operation, the metric allows us to compute a
new parameter related to the curvature of the manifold, the scalar curvature:
Denition 1.15. The scalar curvature is dened as R = R

= g

T
0
0
(M) = C

(M).
We see that the metric is a tool that allows us to dene the inner product of two
vectors. Now we may be interesed in studying the covariant derivatives on M
that make the inner product of X and Y remain constant when they are parallel
transported.
Denition 1.16. A connection is said to be metric compatible with (M, g) or
a metric connection if it satises g = 0, or equivalently
X
g = 0, X X(M).
Lemma 1.17. g = 0 L
Z
(g(X, Y )) = g(
Z
X, Y ) +g(X,
Z
Y ), X, Y, Z
X(M)
Proof. Using that g(X, Y ) =< g(X), Y > and the commutativity of with
respect to inner contractions:
L
Z
(g(X, Y )) =
Z
< g(X), Y >
=
Z
<< g, X >, Y >
=<<
Z
g, X >, Y > + << g,
Z
X >, Y > + << g, X >,
Z
Y >
= (
Z
g)(X, Y ) +g(
Z
X, Y ) +g(X,
Z
Y )
And then both implications are trivial. .
In general, if N M is a submanifold, N is not a semiriemannian manifold with
the semiriemannian metric induced by the metric on M (this is always true in a
riemannian manifold), so allowing the metric to be negative dened has deep con-
sequences. Nevertheless, there are some interesting properties that hold for any
5. PSEUDORIEMANNIAN MANIFOLDS 9
pseudoriemannian manifold, such as the fundamental theorem of pseudorie-
mannian geometry:
Theorem 1.18. On a pseudoriemannian manifold (M, g) there exists a unique
torsionless connection wich is compatible with the metric g. This connection is
called the Levi-Civita connection.
Proof. If exists, then by the condition of being metric compatible and using
the previous lemma, we can write the following expressions:
L
X
(g(Y, Z)) = g(
X
Y, Z) +g(Y,
X
Z)
L
Y
(g(Z, X)) = g(
Y
Z, X) +g(Z,
Y
X)
L
Z
(g(X, Y )) = g(
Z
X, Y ) +g(X,
Z
Y )
Adding up the rst two equations and subtracting the third one, we obtain:
L
X
(g(Y, Z)) +L
Y
(g(Z, X)) L
Z
(g(X, Y )) =
= g(X,
Y
Z
Z
Y ) +g(Y,
X
Z
Z
X) +g(Z,
X
Y +
Z
X)
Using now that is torsionless: T(X, Y ) =
X
Y
Y
X [X, Y ] = 0, we obtain:
L
X
(g(Y, Z)) +L
Y
(g(Z, X)) L
Z
(g(X, Y )) =
= g(X, [Y, Z]) +g(Y, [X, Z]) +g(Z, 2
X
Y [X, Y ])
Thus, now we can compute the Christoel symbols choosing X =

x
i
, Y =

x
j
and Z =

x
k
, where de Lie brackets are 0 because of Schwarzs Theorem:
2g(
l
ij

x
l
,

x
k
) =

x
i
g(

x
j
,

x
k
) +

x
j
g(

x
i
,

x
k
)

x
k
g(

x
i
,

x
j
)
So, as g(

x
i
,

x
j
) = g
ij
, we obtain:
2
l
ij
g
lk
=
g
jk
x
i
+
g
ik
x
j

g
ij
x
k
And thus we nd the corresponding Christoel symbols (so, by virtue of the bi-
jection between Christoel symbols and connections, we have eectively dened a
unique connection, and it satises de properties, as it can be checked):

l
ij
=
1
2
g
lk
_
g
jk
x
i
+
g
ik
x
j

g
ij
x
k
_
.
The curvature tensor has n
4
components, but it contains many symmetries. Some
of them appear just when considering the Levi-Civitta connection. For instance,
we have that with the Levi-Civitta connection:
Proposition 1.19. The Riemann curvature tensor has the properties:
10 1. PSEUDORIEMANNIAN GEOMETRY
(1) R

= R

(2) R

= R

(3) R

+R

+R

= 0
(4) The Bianchi identity:

= 0
We will not prove these identities here, but notice that to prove them we just need
to compute them in an adequate frame: since they are tensorial equations, they
will hold in any other frame. These symmetries induce symmetries on the Ricci
tensor too:
Lemma 1.20. The Ricci tensor is symmetric.
Proof. It is deduced from the Riemann tensors symmetries: in particular, from
the second property, that says that R

= R

, so: R

= R

= R

=
R

. Notice that we are assuming that is the Levi-Civitta connection. .

6. Example
An easy but useful example of a pseudoriemannian manifold is the Minkowski
spacetime; it is the spacetime of Special Relativity (SR), a 4-dimensional manifold
with the metric

dened as
0
=
0

and
ij
=
i
j
, i.e.:
=
_

_
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
_

_
Sometimes it is alternatively dened as , but we wont care about that here.
Notice that this metric allows a vector to have positive, null or negative norm; we
will call it timelike, lightlike or null or spacelike respectively.
Using the formula of the section above, we can nd the Christoel symbols corre-
sponding to its associated Levi-Civita connection; since the metric is constant, it
is clear that they are the constant functions:

l
ij
= 0
Now, using the coordinate expression of Riemanns curvature tensor:
R

= 0
And, thus, R

= 0 and R = 0. This explains why the Minkowski spacetime is said

to be a at spacetime.
Chapter 2
Foundations of General Relativity
Now that we have dened the main tools we need to formulate the Theory of
General Relativity, we are going to see what ideas and postulates are used to build
it up. There are many sources covering this topic; I have used mainly [3], [14], [2]
and [4], but also, specially for a rigorous discussion, [8] and [12]. To start with,
we are going to see a brief review of the Theory of Special Relativity (SR), that
mainly can be found in [9].
1. Special Relativity
SR is the solution to the incongruencies between classical mechanics and electromag-
netism that appeared in the 19
th
century. It was a problem that had been studied
by many physicists and mathematicians, such as Lorentz, Poincare or Hilbert. But
it was Albert Einstein who in his famous articles from 1905 proposed the solution
(Minkowski gave it a geometrical interpretation later on).
One of the problems that 19
th
century physicists dealt with is the noninvariance
of Maxwells equations under galilean transformations. SR tries to reformulate
newtonian mechanics accepting as true the Maxwell equations, and looking for a
new set of coordinate transformations (the Lorentz transformations) that leave
all the physical laws invariant in any inertial frame transformation.
Its startpoint is the fact that the velocity of light is a constant of the Universe
(something intrinsic in Maxwell equations and conrmed by several experiments),
together with the natural idea that the laws of physics have to be the same in any
inertial frame (a specic equivalence class of frames). This discards the idea of
an absolute time and space, it has to be replaced by a 4-dimensional space, called
spacetime.
Moreover, we can observe that, if the velocity of light has to be the same in any
frame, there exists a quantity relating the points of spacetime that are connected
by a light ray that is invariant under changes of frames: the relativistic interval
s
2
= c
2
t
2
x
2
y
2
z
2
. In fact, in SR this is always an invariant quantity,
and can be thought as the pseudoriemannian metric of Minkowski spacetime that
we saw in Chapter 1. It is a Lorentzian metric, a metric with signature (1, n) (it
11
12 2. FOUNDATIONS OF GENERAL RELATIVITY
has one positive and n negative eigenvalues) in a manifold of dimension n+1. This
shows a hyperbolic symmetry on spacetime: the set of points of constant distance
is given by the equation of the hyperbola (x
0
)
2
(x
1
)
2
(x
2
)
2
(x
3
)
2
= C.
Summarizing, we can formulate the basic postulates of SR as:
(1) Time and space are not independent. They are held together in the Minkowski
spacetime M, a 4-dimensional semiriemannian manifold with signature (1,3),
whos metric is constant and is denoted as (
00
= 1,
i
=
i
).
(2) There exists an equivalence class of frames (the inertial frames) where the
laws of physics (mechanics and electromagnetism) are valid. The velocity
between them is always constant.
(3) The velocity of light is a constant of the Universe (it takes the same value in
any inertial frame).
From here, the Lorentz transformations are deduced as the linear isometries of
(M, ), as they have to keep constant s (the metric); the ones that do not reverse
neither the time nor any spacial variable are of special interest in physics. If v =
(v
x
, v
y
, v
z
) is the velocity between two inertial frames, and we denote
i
=
vi
c
(c is
the velocity of light in vacuum), =
_

2
x
+
2
y
+
2
z
and =
1
1
2
, then a general
Lorentz transformation that does not reverse any variable would be:
_
_
_
_
ct

_
_
_
_
=
_
_
_
_
_
_

x

y

z

x
1 + ( 1)

2
x

2
( 1)
xy

2
( 1)
xz

y
( 1)
yx

2
1 + ( 1)

2
y

2
( 1)
yz

z
( 1)
zx

2
( 1)
yz

2
1 + ( 1)

2
z

2
_
_
_
_
_
_

_
_
_
_
ct
x
y
z
_
_
_
_
+
_
_
_
_
ct
0
x
0
y
0
z
0
_
_
_
_
1.1. Relativistic Kinematics
Denition 2.1. The proper time interval between two events is the elapsed time
measured by an observer for which the two events happen in the same point of space.
The proper time for a worldline : I R M between two events s
1
= (t
1
)
and s
2
= (t
2
) is:
=
1
c
_
t2
t1
ds
where ds =

dx

dx

.
Denition 2.2. The proper length between two events is the spacial distance
between them measured in an inertial frame S where both events happen in points
that are at rest.
Some easy results that can be derived from the Lorentz transformations (when
written in a friendlier way, choosing adequately the two inertial frames) are the
Proper Time Dilation and the Proper Length Contraction:
1. SPECIAL RELATIVITY 13
Proposition 2.3. If relates the inertial frame in which a time T is measured
between two events and the frame where proper time T
0
is measured, then:
T = T
0

Proposition 2.4. If relates the inertial frame in which a length L is measured

between two events and the frame where proper length L
0
is measured, then:
L =
L
0

Notice that, since Galilean transformations are no longer valid, if a rst inertial
frame S sees a second one S

moving with velocity v, and the second one observes

a particle moving with velocity v

on the same direction (it can be considered as a

third inertial frame), it is not true that the rst one sees the third one moving at
velocity v +v

, as one would expect. The correct version of the Law of addition

of velocities is:
Theorem 2.5. Consider three inertial reference frames S
0
, S
1
and S
2
such that
S
1
moves with velocity v
1
with respect to S
0
, S
2
moves with velocity v
2
with respect
to S
1
, their axes are parallel and they are moving along one of them. Then, the
velocity between S
2
and S
0
is:
v =
v
1
+v
2
1 +
v1v2
c
2
1.2. Relativistic Dynamics
Consider a worldline x

() in M (a curve in M with timelike tangent vector)

parameterised by the proper time; it can be though to be the trajectory of some
massive particle.
Denition 2.6. The four-velocity of a particle with worldline x

() is U

() =
x

.
Using the chain rule, and that
t

= (because of time dilation), it is clear that in an

inertial frame where a particle moves with velocity v = (v
1
, v
2
, v
3
) (not necessarily
constant) its four-velocity is:
U = (c, v
1
, v
2
, v
3
)
Denition 2.7. The four-acceleration of a particle with worldline x

() is A

=
U

=

2
x

2
.
Denition 2.8. The rest mass or proper mass of a particle is the mass m
0
measured in a reference frame where the particle is at rest.
Denition 2.9. The four-momentum or four-vector enregy-momentum is
P

= m
0
U

, so P = m
0
(c, v
1
, v
2
, v
3
). The magnitude m = m
0
is called inertial
relativistic mass.
14 2. FOUNDATIONS OF GENERAL RELATIVITY
Notice that in many of the denitions we are giving is implicit the fact that c
is a limit velocity for massive particles; for instance, the inertial relativistic mass
represents the mass measured by an inertial frame in which the particle is moving
at velocity v, where =
1

1
v
2
c
2
. So we have that m when v c.
Denition 2.10. The Total Relativistic Energy is E = P
0
c = mc
2
.
Then P = (E/c, p), where p represents the linear momenum of the particle, and it
can be seen that E =
_
(m
0
c
2
)
2
+c
2
[[p[[
2
. As E = m
0
c
2
, E
0
= m
0
c
2
is called the
energy at rest.
Denition 2.11. The four-force is dened as F

= m
0
A

.
Theorem 2.12. The relation between the four-momentum and the four-force is:
P

= F
This is the general equation of relativistic dynamics.
1.3. Gravity
Up to this point, we may ask ourselves: if we have constructed a dynamical theory
of relativity, why doesnt SR describe gravity?
One of the main experimental facts that currently supports GR is that light is
aected by gravity; a gravitational eld can bend a light ray, so one of the postulates
of SR is not true! This was rst observed by Eddington in 1919, and has been
observed more recently using the Hubble telescope. Of course, as this was not
observed until 1919, it wasnt the reason for constructing GR.
Another reason is Bondis thought experiment, whih provided a perpetual motion
machine. He imagined a vertical circular belt submitted to gravity, with some
atoms attached. The atoms in the right-hand side are excited, so by the formula
E = mc
2
they are heavier than the ones in the left-hand side; this makes the belt
move. When an excited atom reaches the lowest point in the belt, it emmits a
photon that, using a mirror, is absorved by the non-excited atom in the top, and
this keeps the belt in a perpetual motion. The way out in GR is that photons lose
energy as they climb through the gravitational eld: they are red-shifted. Such
a shift was measured directly by Pound and Rebka in 1959. But this red-shift is
incompatible with SR, as it considers that gravity can not aect lights velocity.
But the denitive reason was the one concerning the equivalence principle, that we
will explain in the next section.
2. Postulates of General Relativity
Classical mechanics and the special theory of relativity distinguish between two
kinds of bodies: reference-bodies relative to which the recognised laws of nature
2. POSTULATES OF GENERAL RELATIVITY 15
can be said to hold, and reference-bodies relative to which these laws do not hold.
We would like to generalize the theory of SR to accelerating frames. In particular,
Einsteins principle of relativity has to be modied, it has to be extended to these
frames.
Let me reproduce an example that appears in [4]. Consider a Galileian reference-
body, for example a room with an observer inside in a large portion of empty space,
far from stars and other appreciable masses. A rope attached in the middle of the
ceiling of the room is pulled with a constant force by an immaterial being. The
room together with the observer, then, begin to move upwards with uniformly
accelerated motion, from the point of view of an external observer.
But the observer inside the room notices the acceleration by the reaction on the
oor. He is then standing in the room as anyone stands in a room on earth. If he
releases a body which he previously had in his hand, the acceleration of the room
will no longer be transmitted to this body, and will then approach the oor of the
room with an accelerated relative motion. After several experiments, the observer
will convince himself that the acceleration of the body towards the oor of the
room is always of the same magnitude, whatever kind of body he may happen to
use for the experiment. Then, because of the experimental fact that inertial and
gravitational masses are equal (this is called the Weak Equivalence Principle),
the man can not distinguish this situation from being at rest in a gravitational
eld, because two bodies in the same gravitational eld behave in the same way
independently of their inertial masses, just as the observer in the room is measuring.
Notice that the observer can not measure the absence of tidal forces, as he is inside
a room (i.e., mathematically, these measurements are local, not global).
This mental experiment justies adopting the General Postulate of Relativity
(also called the Equivalence Principle), the main idea of which is that all bodies
of reference are equivalent for the description of natural phenomena, whatever may
be their state of motion. An accelerated observer with respect to us will just
consider the accelerations of the bodies that are at rest from our point of view as
the result of some gravitational eld (not necessarily produced by Newtons law of
universal gravitation).
This postulate has deep implications: some basic postulates of SR, such as the
invariance of the velocity of light, are no longer valid. For instance, if an observer
K sees a light ray at constant velocity c travelling in a straight line path, an observer
K

in rotation with respect to K will see it travelling along a curved path, and he
will explain this phenomena because of the presence of a gravitational eld. So the
idea of straight line has to be reconsidered.
Moreover, the way of measuring time and distances has to be changed too: consider
an observer K in a nearly Galileian frame as before, and an observer K

located
on a rotating disc (from the point of view of K). By the time dilation, from the
point of view of K, a clock located in the center of the rotating disc and a clock
located on the disc at any other point will not measure the same time (the latter
goes at a rate permanently slower than the rst one); as this would be oberved too
by any other observer, in general, we can say that in every gravitational eld, a
clock will go more quickly or less quickly according to the position in which the
16 2. FOUNDATIONS OF GENERAL RELATIVITY
clock (at rest) is situated. A similar result is obtained when measuring distances:
the size of a rod to measure distances depends on the position where the rod is
situated.
This fact introduces the idea of allowing non-euclidean geometries as possible ge-
ometries for spacetime, and here appears the necessity of considering spacetime as
a manifold with a given metric that contains all the gravitational eld information.
This point of view implies that gravity is omnipresent: there is no such thing as
a gravitationally neutral object, thus it makes no sense to try to measure an
objects acceleration due to gravity; instead, it is better to dene an unaccelerated
body as a freely falling object.
Not only that, but also, notice that it is not necessary that the coordinates of
spacetime have any physical signicance as they had in Minkowski spacetime, where
x
i
represented space coordinates and x
0
the time coordinate. Why is that so? Any
event is a point in spacetime, and a particle (a being with some duration) can be
represented by a curve in spacetime, called its worldline; the only actual evidence
of a time-space nature with which we meet in physical statements appears when
two worldlines intersect, and the intersection of worldlines is well described even if
the coordinates have no physical interpretation.
Putting all this information together, we can formulate some postulates or main
ideas over which the whole theory is constructed:
(1) Spacetime is a 4-dimensional semiriemannian manifold whose metric g con-
tains all the information about the gravitational eld and is locally lorentzian.
Notice that any frame is, locally, a Lorentz frame, using Riemann Normal Co-
ordinates (see for example [1] p.335-341 or [5] p.158-160 to see the properties
and denition of Riemann Normal Coordinates).
(2) Einsteins Equivalence Principle: In any and every local Lorentz frame,
anywhere and anytime in the universe, all the (nongravitational) laws of
physics must take on their familiar special-relativistic forms.
The relation between the metric and the mass and energy distributions will be
discussed in chapter 3, and constitutes the main piece of the theory: Einsteins
eld equation.
3. The matter elds
If we want to formulate a physical law in a manifold satisfying Einsteins Equiva-
lence Principle, we should announce it using an equality between tensors, because
a law formulated in terms of tensors is independent of the coordinates chosen in the
manifold. As we want to relate the geometry of the manifold with the mass and
energy distribution, we have to single out adequate tensors for both concepts. We
have already studied some tensor candidates containing geometrical information in
the preceding chapter, such as Riemanns curvature tensor and its inner contrac-
tions. Now we should study how to dene a tensor containing information about
the mass and energy distribution of the Universe: a matter eld. This tensor we
3. THE MATTER FIELDS 17
are looking for is not a new concept: it appears in the Special Theory of Relativity,
and it can be regarded as a generalization of the four-vector energy-momentum P

.
The denition of this tensor eld will depend on the problem we are studying.
There will be various elds on M which describe the matter content of space-time:
a part from the mass distribution, we have to take into account the electromagnetic
eld, the neutrino eld, etc. The theory of gravity one obtains depends on what
matter elds one incorporates in it. These matter elds should obey a couple of
postulates:
a) Local causality: the equations governing the matter elds must be such that
a signal can be send between two points if and only if these points can be joined
by a non-spacelike curve (a curve whose tangent vector is not spacelike).
b) Local conservation of energy and momentum: The equations governing
the matter elds are such that there exists a symmetric tensor eld T T
2
0
(M),
called the Energy-Momentum Tensor or the Stress-Energy Tensor, which
depends on the elds, their covariant derivatives, and the metric, and which has
the properties:
T vanishes on an open set U if and only if all the matter elds vanish on U
T

obey the equation

= 0 (this expresses the conservation of en-

ergy in the presence of a gravitational eld; in fact, the operator

is the
generalization of the divergence operator

x

to a curved space, as it will

be discussed in the next chapter).
From the second postulate, we get that in vacuum conditions T

= 0, as anyone
would expect. On the other hand, if we are considering a uid with particle density
k C

(M) at rest:
Denition 2.13. The Particle Density 4-vector N is dened as P = kU, where
U is the 4-velocity and k C

(M) is the density of particles of the medium at rest.

In coordinates we can write it N

= k U

.
Denition 2.14. The Energy-Momentum Tensor or Stress-Energy Tensor,
is the symmetric tensor eld T T
2
0
dened as T = P N, where P and N are
the 4-momentum and the particle density 4-vector, respectively.
In coordinates, T

= P

.
Chapter 3
Einsteins Equations
Up to this point, it only remains to characterize the way the space-time is curved
because of the matter elds presence; we have to state Einsteins equation. Good
discussions on this topic can be found on the same references that I used in chapter
2; particularly, I used [3] for the informal approach.
1. An informal approach
Lets rst consider a freely falling particle. Freely falling particles in at space (SR)
move in straight lines , thus the second derivative of the parameterized path x

()
vanishes:
d
2
x

d
2
= 0
According to Einsteins Equivalence Principle, this equation should hold in curved
space if we are using Riemann Normal Coordinates. But this equation is not an
equation between tensors, so it doesnt hold in an arbitrary coordinate system.
However, there is a unique tensorial equation which reduces to this one when the
Christoel symbols vanish; it is:

= 0
d
2
x
k
d
2
+
k
ij
dx
i
d
dx
j
d
= 0
This is, in fact, the usual way of generalizing the operator

to a covariant ex-
pression. We have obtained the geodesic equation; therefore, in GR free particles
move along geodesics. Here it is clear why sometimes the covariant derivative is
thought as a possible way of dening the concept of acceleration on a manifold.
So far, we have seen that curvature is necessary to describe the motion of freely
falling particles. Lets see that it is sucient: we will deduce the usual results of
Newtonian gravity from it.
Lets dene the Newtonian limit by the requirements:
19
20 3. EINSTEINS EQUATIONS
(1) The particles are moving slowly with respect to the speed of light. This means
that
dx
i
d
<<
dt
d
, so the terms
dx
i
d
dx
j
d
can be neglected, and then the geodesic
equation becomes:
d
2
x

d
2
+

00
_
dt
d
_
2
= 0
(2) The eld is static, it doesnt depend on time. Then, the Christoel symbols
of the Levi-Civitta connection can be simplied:

00
=
1
2
g

_
g
0
t
+
g
0
t

g
00
x

_
=
1
2
g

g
00
x

(3) The gravitational eld is weak, it can be considered as a perturbation of

at space. Then, we can write g

+ h

, with [h

[ << 1, and its

inverse becomes g

, where h

. Substituting this
expression into the previous one, and neglecting the terms where h

appears
multiplying (because of its small norm):

00
=
1
2

h
00
x

The geodesic equation is therefore:

d
2
x

d
2
=
1
2

h
00
x

_
dt
d
_
2
Using again that the eld is static,
h00
t
= 0, and that
0i
= 0, the equation above,
when = 0, becomes:
d
2
t
d
2
= 0
So
dt
d
is constant. Then, the spacelike components of the geodesic equation ( = i)
are:
d
2
x
i
d
2
=
1
2
h
00
x
i
_
dt
d
_
2

d
2
x
i
dt
2
=
1
2
h
00
x
i
And dening =
1
2
h
00
we recover Newtons eld equation:

2
x
t
2
=
Now, we know that the curvature of the manifold is sucient to describe the gravity
in the Newtonian limit. It remains, of course, to nd eld equations for the metric
that allow it to take the form g

+ h

, and to see that in the case of a

single gravitating punctual body of mass M, h
00
=
2GM
r
; but this will come soon.
2. EINSTEINS TENSOR 21
We know how to generalize the laws of physics from a at spacetime to a curved
one: the only thing we have to do is to change partial derivatives by covariant
derivatives. This is a good idea that comes from Einsteins Equivalence Principle,
but it is not a well stablished rule: obviously, there are many ways the same physical
law can be generalized to curved space. For instance, consider a law of the form:
Y

= 0
The partial derivatives are commutative operators, but the covariant ones are not!
So there are two possible ways of generalizing this result, and they dier in:
Y

= R

We will not care about that now; in order to decide which of them is the correct
one we should consider their behaviour and their compatibility with other physical
laws. Now, lets try to generalize, in an informal way, Newtons eld equations of
the gravitational potential to curved spacetime. Consider Poissons equation for
the gravitational potential:
= 4G
Where represents the mass density. The equations we want to nd has to reduce
to this one in the Newtonian limit: on the left-hand side we have a second-order
dierential operator acting on the gravitational potential, whereas the right-hand
side contains a measure of the mass distribution. As we have already discussed, a
relativistic generalization should take the form of an equation between tensors. We
know how to generalize the mass distribution, using the energy-momentum tensor
T

, or analogously, lowering its indices, T

. On the other hand, the gravitational

potential should get replaced by the metric tensor. Now, we have to choose a proper
second-order dierential operator to replace the laplacian.
2. Einsteins Tensor
This operator should be such that the resulting tensor, that we will call G

, is
symmetric (as T

is), and is compatible with energy-momentum conservation in

curved spacetime, i.e.,

= 0. It should be derived using only geometric

information: the metric tensor g

and Riemanns curvature tensor R

(that in
fact is derived from g); notice that I am not considering torsion here because we
use the Levi-Civitta connection, that is torsionless. As it has to be a generalization
of the laplacian, we will consider just 2nd order tensors, and as it has to equal T

,
we are just going to consider second-rank tensors.
Proposition 3.1. The most general second-rank symmetric tensor constructable
from g

and R

and linear in R

is:
22 3. EINSTEINS EQUATIONS
G

= aR

+bRg

+ g

And

= 0 b =
1
2
a. Moreover, it vanishes in at spacetime if and only
if = 0.
Proof. Lets consider all the possible elements that we can construct from g

and R

that have rank 2:

First of all, we have g

, a second-order tensor that is symmetric and does

not depend on the curvature. G

will contain a multiple of it: g

.
The curvature tensor hasnt got second order. In order to obtain a second-
order tensor, we have to contract two of its indices: this can be done in several
ways.
(1) The Ricci tensor R

= R
i
i
is one of these contractions. It is sym-
metric, as it was seen in the rst chapter, and depends linearly on the
Riemann curvature tensor, so it can contribute to G

in the form aR

.
(2) Using the same idea, we can construct two other second-order tensors:
R
i
i
and R
i
i
. On the other hand, we can consider other contractions
using the metric to rising and lowering indices. But Ricci tensor is essen-
tially the unique nontrivial way of contracting Riemann curvature tensor.
I will not discuss this here.
The scalar curvature R = g

is another contraction that we can consider:

as it is a scalar eld it is symmetric, but it hasnt got second order. Neverthe-
less, we can multiply a second-order tensor by R and we will obtain another
possible element of G

. Since we want the tensor G to be linear with R

,
R can only be multiplied by g

, and will contribute to G

in the form bRg

.
So we have seen that the most general second-rank symmetric tensor constructable
from g

and R

and linear in R

is G

= aR

+ bRg

+ g

. Now we
have to see that

= 0 b =
1
2
a.
First of all, lets nd a useful relation. Consider the Biancchi identity rising its
indices:
0 = g

)
Using that R
abcd
= R
cdab
and that R

= R

:
0 =

= 2

Notice that:
R = g

= g

And using that R

abcd
= R
dbca
:
R = g

= g

R


= R

2. EINSTEINS TENSOR 23
So introducing this in the previous form of the Biancchi identity (using that

, that can be checked easily) we obtain that:

=
1
2

R
Now we can compute G =

:
G

= aR

+bRg

+ g

= g

= aR

+bR

= a

+b

R =
_
1
2
a +b
_

R
So, at last, we see that G = 0 b =
1
2
a.
Finally, we should proof that in at space G = 0 = 0. But this is trivial,
as we have seen in the rst chapter that in at space R

= 0 and R = 0, so
G

= 0

= 0 = 0. .
It can be proved that there exists no tensor with components constructable from
the ten metric coecients and their rst derivatives. In fact, the only tensors
constructable from the ten g

, the fourty

and the one hundred

that are linear in the latter are:

(1) R

(2) g

(3) Tensors constructed from the Riemann curvature tensor that are linear in
R

(such as the Ricci tensor).

The tensor G

= R

1
2
Rg

+g

is the only second-rank, symmetric tensor

such that:
a) Has components constructable solely from g

and

.
b) Has components linear in

.
c)

= 0
If we add the condition that G vanishes in at space, then G

= R

1
2
Rg

.
The uniqueness of this tensor, that is called Einsteins tensor, was proven, ac-
cording to [6], by Poincare. From these propositions, we deduce the unicity (under
the restrictions above) of Einsteins tensor G

= R

1
2
Rg

, and thus we are

able to formulate Einsteins eld equation:
G

= T

Where is some real constant.

24 3. EINSTEINS EQUATIONS
These equations can be derived from a variational principle. In fact, Hilbert
published the correct equations deriving them from a variational principle (us-
ing Hilberts action) even before Einstein published the correct ones (althought
Hilbert knew the work of Einstein, and always deended Einsteins authorship of the
equations). You can nd a modern proof of the unicity of these equations in [11],
where Lovelock shows using the variational formulation, that in a 4-dimensional
space the Einstein eld equations (with the term g

) are the only permissible

second order Euler-Lagrange equations.
Lemma 3.2. Einsteins eld equation is equivalent to the equation:
R

=
_
T

1
2
Tg

_
Proof. From Einsteins eld equation: R

1
2
g

R = T

. Rising its indices

with the inverse of the metric:
g

_
R

1
2
g

R
_
= g

(T

)
R

1
2

R = T

Now, contracting the indices and :

R

1
2

R = T

R
1
2
4R = T
Then, we deduce that R = T, where T is called Laues scalar. So we can
rewrite Einstein eld equation, just substituting R by this expression:
R

=
_
T

1
2
Tg

_
.
This alternative way of writing Einsteins equations is very useful, specially when
we are looking for the solutions in vacuum space, because then Einsteins equation
reads:
R

= 0
Although apparently simple, this equation is still very dicult to solve unless ad-
ditional conditions are supposed. For instance, one of the rst solutions that was
found, Schwarschilds metric, has the additional assumption of spherical symmetry;
this solution is very useful, as it describes spacetime near a celestial body such as
a star, or a planet. We will study it later; now, lets nd the constant .
2. EINSTEINS TENSOR 25
To nd it, we will impose that Einsteins equation predicts Newtonian gravity in
the weak-eld, time-independent, slowly-moving-particles limit. Notice that in a
mass distribution with slowly-moving particles, the terms T
i
can be neglected
in front of T
00
, so we are going to study the case where = = 0 of Einsteins
equation. As we are in the weak-eld limit, g
00
= 1 +h
00
and g
00
= 1 h
00
. Then,
up to rst order, T = g
00
T
00
T
00
. So:
R
00
=
1
2
T
00
To obtain the explicit relation with the metric we need to compute R
00
= R

00
;
as R
0
000
= 0, we just need to compute R
i
j00
:
R
i
j00
=

x
j

i
00

x
0

i
j0
+
i
j

00

i
0

j0
Using that we are in the static approximation, up to rst order in :
R
i
j00


x
j

i
00
=

x
j
_

1
2
g
ik
g
00
x
k
_
Now contracting its indices, substitutin the expression of g
00
above and neglecting
the terms with order h:
R
00
= R
i
i00

1
2

ik

2
x
i
x
k
h
00
=
1
2
h
00
So comparing this with the expression R
00
=
1
2
T
00
, we nd that:
h
00
= T
00
And this is precisely Newtons equation if we take = 8G, because Newtons
equation tells us = 4 = 4T
00
and we already knew that h
00
= 2. Thus,
we have computed the adequate constant that makes Einsteins equation predict
Newtonian gravity in the specied limit conditions.
Summarizing, the equations that govern the curvature of spacetime are:
R

1
2
Rg

= 8GT

Here, as we have seen before, an additional term g

could be added; this term

was considered at rst by Einstein, but then he discarted it. is the famous cos-
mological constant that Einstein considered one of the worst mistakes in his life.
He introduced it to obtain a static (non-spanding) universe; but experimental data
showed that the universe is actually expanding, so by introducing this term Ein-
stein lost the opportunity of predicting the expansion of the universe. Nowadays,
it is known that the universe is not only expanding, but its expansion is being
26 3. EINSTEINS EQUATIONS
accelerated (the source of this acceleration is not already understood, and is called
the dark energy). Thus, some physicists have argued that the cosmological con-
stant should be reintroduced to the equations to predict the accelerated expansion.
Whether it has to be considered or not, I dont know, but it has been shown that,
currently, if is not zero, it has such a small value that if we are not working in a
cosmological scale, the equation above is the only one need.
Chapter 4
The Schwarzschild Solution
In this chapter we will nd a particular solution to Einsteins equation. This chapter
is mainly based on Olesens lecture notes [14], but similar discussions can be found
in many other sources. Schwarzschild solution is a spherically symmetric solution
that has many applications: for instance, it describes the spacetime metric near a
massive spherically symmetric body such as a planet or a star, so it is useful to
describe the Solar System. Our discussion will not be strictly rigorous, specially
when dealing with the concept of symmetry, but it can be formalized using some
additional dierential geometry tools, such as Lie groups and Lie algbras.
1. The time-dependent spherically symmetric met-
ric
A compact way of expressing the metric tensor is using the proper time interval:
d
2
= g

dx

dx

. If we consider, rst, the cartesian variables of space x

1
, x
2
and x
3
, together with the time t, a spherically symmetric metric intuitively is a
metric for which its proper time only depends on t, r, dt, xdx = rdr and (dx)
2
=
dr
2
+r
2
(d
2
+sin
2
d
2
), where r =
_
x
2
+y
2
+z
2
. Thus:
d
2
= A(r, t)dt
2
B(r, t)dr
2
C(r, t)drdt D(r, t)r
2
(d
2
+sin
2
d
2
)
As we have said in chapter 3, the coordinates in GR are arbitrary, so we are free to
make transformations x

to simpify the expression of d

2
. For example, we
can do the transformation r

= r
_
D(r, t) (here we assume that r

is not constant)
and d becomes dependent on new functions A

, B

and C

, which are functions of

t and r

. Dropping the primes for simplicity, we get:

d
2
= A(r, t)dt
2
B(r, t)dr
2
C(r, t)drdt r
2
(d
2
+sin
2
d
2
)
The term drdt can also be removed by using a new time coordinate t

dened by:
27
28 4. THE SCHWARZSCHILD SOLUTION
dt

(r, t) = (r, t)
_
A(r, t)dt
1
2
C(r, t)dr
_
Where (r, t) is an integrating factor that makes dt

be a perfect dierential with

A =
t

t
,
1
2
C =
t

r
. Then, from here, we deduce that:
1
A
2
dt
2
= Adt
2
Cdtdr +
C
2
4A
dr
2
and the expression of the proper time becomes:
d
2
=
1

2
A
dt
2

_
B +
C
2
4A
_
dr
2
r
2
(d
2
+sin
2
d
2
)
or, renaming the corresponding functions:
d
2
= E(r, t)dt
2
F(r, t)dr
2
r
2
(d
2
+sin
2
d
2
)
This is the standard form of the metric, derived for the rst time by Weyl. Then
we have:
g
rr
= F, g

= r
2
, g

= r
2
sin
2
, g
tt
= E
g
rr
=
1
F
, g

=
1
r
2
, g

=
1
r
2
sin
2
, g
tt
=
1
E
1.1. The Christoel Symbols
Now that we have a general expression for the metric, we can compute the Christof-
fel symbols. We will do this computation using the formula that we saw in chapter
1:

=
1
2
g

_
g

+
g

_
There are some other methods to compute

that may take less time, such as

the variational method proposed in [14], but we will compute them in the classical
way, using the good properties of this particular metric.
First of all, lets compute
r

. There are 16 such functions, but as

only
10 of them are independent. Notice that, since g
r
=

r
1
F
, the only terms in
the sumation that do not vanish are those corresponding to = r. So the formula
above becomes:

=
1
2
g
rr
_
g
r
x

+
g
r
x

r
_
1. THE TIME-DEPENDENT SPHERICALLY SYMMETRIC METRIC 29
Clearly, if , and r are all dierent,
r

=
r

= 0 because g
r
= g
r
= g

= 0,
so we get that:

r
t
=
r
t
=
r
t
=
r
t
=
r

=
r

= 0
If = ,= r, g
r
= g
r
= 0, thus
r

=
1
2
g
rr
g
r
, and we obtain:

r
tt
=
1
2F
E
r
,
r

=
r
F
,
r

=
rsin
2

F
Finally, if r = and is arbitrary, as
gr
r
=
gr
r
, we have that
r
r
=
r
r
=
1
2
g
rr grr
x

, and remembering that F = F(r, t) we obtain:

r
rr
=
1
2F
F
r
,
r
rt
=
r
tr
=
1
2F
F
t
,
r
r
=
r
r
=
r
r
=
r
r
= 0
Using a similar analysis, we can compute
t

and

. The nonvanishing
terms are:

t
tt
=
1
2E
E
t

r
=

r
=
1
r

r
=

r
=
1
r

t
tr
=
t
rt
=
1
2E
E
r

= sincos

=
cos
sin

t
rr
=
1
2E
F
t
1.2. The Ricci tensor
Once we have the connection coecients, we need to compute the Riemann curva-
ture tensor, and then contract its indices to get the Ricci tensor, that we need to
study Einstein equation. This computation, even in the most simple cases, is really
tedious. For example, lets compute R
tr
; we only need the terms R
i
itr
. Remember
the formula:
R
l
ijk
=

l
jk
x
i
+

jk

l
i


l
ik
x
j

ik

l
j
Then, we compute:
R
r
rtr
=
1
2F
2
F
r
F
t
+
1
2F
F
tr
+
_
1
2F
_
2
F
t
F
r
+
1
2F
F
t
1
2E
E
r
+

1
2F
2
F
t
F
r
+
1
2F
F
rt
_

_
1
2F
_
2
F
t
F
r

1
2E
F
t
1
2F
E
r
= 0
R
t
ttr
=

t
tr
t
+

tr

t
t


t
tr
t

tr

t
t
= 0
R

tr
=

tr

r
t

t
=
r
tr

r
=
1
2Fr
F
t
30 4. THE SCHWARZSCHILD SOLUTION
R

tr
=
r
tr

r
=
1
2Fr
F
t
So we nally obtain:
R
tr
= R

tr
=
1
rF
F
t
In the lecture notes [14] Olesen derives a method for computing the Ricci tensor
components in a more straighforward way. Be careful, because he uses a dierent
sign convention for the metric coecients and for the Ricci tensor components! If
we compute the other components, the nonvanishing ones are:
R
rr
=
1
2E

2
E
r
2
+
1
4E
2
_
E
r
_
2
+
1
4EF
E
r
F
r
+
1
rF
F
r
+
1
2E

2
F
t
2

1
4E
2
E
t
F
t

1
4EF
_
F
t
_
2
R
tt
=
1
2F

2
E
r
2

1
4F
2
E
r
F
r
+
1
rF
E
r

1
4EF
_
E
r
_
2

1
2F

2
F
t
2
+
1
4F
2
_
F
t
_
2
+
1
4EF
E
t
F
t
R

= 1
1
F
+
r
2F
2
F
r

r
2EF
E
r
R

= sin
2
R

R
tr
=
1
rF
F
t
2. The Schwarzschild solution
Lets consider now the case where space is empty except for a mass M situated
at r = 0. Thus, except for r = 0, we must satisfy the vacuum Einstein equations
R

= 0. Using the expression of R

tr
above, we see that R
tr
= 0 =
F
t
= 0.
Notice that then, in the expressions above, we can remove all the time derivatives of
R

. As R

= sin
2
R

, the equations will be satised if and only if F = F(r)

and R
rr
= R
tt
= R

= 0. Observing that R
rr
and R
tt
contain similar terms, we
impose:
0 =
R
rr
F
+
R
tt
E
=
1
rF
_
F
r
+
E
r
_
This is equivalent to
(ln F)
r
=
(ln E)
r
. Integrating this rst order ODE we obtain:
E(r, t)F(r) = f(t)
2. THE SCHWARZSCHILD SOLUTION 31
Where f(t) is a certain function. As we want the metric to be at when r ,
then, E(r, t) = F(r) = 1 at innity, so f(t) = 1 everywhere (as f(t) does not
depend on r). So we have seen that E(r, t) = E(r) =
1
F(r)
.
Now, we just need to impose R

and R
rr
to vanish. Using the expression we have
just derived, we get:
R

= 1 E r
dE
dr
R
rr
=
1
2E(r)
d
2
E
dr
2

1
rE
dE
dr
=
1
2rE
dR

dr
Thus, if R

= 0, then automatically R
rr
= 0. We can write R

= 0 as:
d
dr
(rE(r)) = 1
So if C is a constant:
rE(r) = r +C
So we have found the functions E(r) and F(r) in terms of the constant C, which
can be xed by demanding compatibility with the Newton limit: g
00
12GM/r.
So we obtain that C = 2GM, and then:
E(r) = 1
2GM
r
; F(r) =
1
1
2GM
r
So we obtain the metric:
d
2
=
_
1
2GM
r
_
dt
2

dr
2
1
2GM
r
r
2
(d
2
+sin
2
d
2
)
This is the Schwarzschild solution. Notice that this expression contains two singu-
larities: one of them, r = 2MG, is ctitious, it appears because of a bad choice of
coordinates (why have we chosen spherical coordinates if at spacetime has hyper-
bolic symmetry?), the other one r = 0 is, in fact, a physical singularity (it appears
in any reference frame), and it can related to black holes event horizon. Of course,
as this solution was found from Einsteins Equations in vacuum, it makes no sense
to consider what happens at r = 0, because there is matter there, so we can not
derive any conclusion from it.
Conclusions
To conclude with, we have seen that GR is geometrical theory that describes gravity
not as a force, but as a consequence of the curvature of the spacetime manifold. In
this context, objects with no external forces acting on them, such as the Earth, are
no longer considered as massive bodies submitted to a gravitational force made by
the sun on them, but as freely falling bodies moving along the most similar thing
to a straight line in spacetime: geodesics.
We have studied the geometrical tools needed to formulate GR, in particular we
have seen what is a pseudoriemannian manifold, what is a connection, how can we
construct a connection using the metric of the manifold, how this connection allows
us to measure curvature using Riemann and Ricci tensors, etc. All these tools where
used to formulate Einsteins equations: a tensorial (thus coordinate-free) equation
that relates the geometry of the manifold with the mass and energy distribution of
spacetime (the matter elds).
In the thesis it has been discussed the physical principles that brings us to formulate
the theory in this way: it reduces to SR locally, it avoids the problem of distinguish-
ing between inertial and non-inertial frames (how could we discover wheather we
are in an inertial frame?), and the most important thing, it explains what SR can
not explain. Moreover, we have studied the uniqueness, up to a certain point, of
Einstein tensor and, thus, of Einsteins equation. Of course, many other equations
can be proposed, as we have seen whean dealing with the cosmological constant.
Moreover, why do we only consider secon-order tensors? Why dont we consider
third-order tensors? Well, Einstein knew this possibility and it is still being stud-
ied nowadays, in fact [11] shows the unicity of the third-order tensor that can be
constructed from the metric. I think that here is where Einstein would apply the
criterion of choosing the most beautiful explanation available (thus, if the result is
a good description of nature, it is always better to keep things simple and choose
second-order tensors).
Finally, we have given a particular solution to the equations: Schwarzschild solu-
tion. This has been derived from the assumption of having a spherically symmetric
solution in vacuum, with a massive particle in some point of space (that is just
a boundary condition, the solution we get is not valid there). The importance of
this solution should be clear, because as I said above, it describes, for example, the
situation of any planet in the Solar System. Of course, this was already predicted
by Newtons gravity, but in this new theory, we see that not only massive bodies
33
34 CONCLUSIONS
are aected by the sun, but also the light we see from distant stars (and this is
something that has been observed several times, and that supports GR).
Further work: once Einsteins equations are formulated, we can study many other
topics that I dont even had time to comment. For example, one can study the
three classical predictions that support GR: the perihelion precession of Mercury,
the deection of light by the Sun and the gravitational redshift of light. Not only
this, but also many topics on cosmology can be studied such as the existence of
the Big Bang, black holes or the cosmic microwave radiation background (although
that, to do this, some other metrics should be studied, such as the Robertson-Walker
metric). In addition, it could be studied a new system of coordinates that made
nonsingular the Schwarzschild metric in r = 2GM, we could study the physical
singularity that it has in r = 0, or we could look for new similar metrics for
dierent boundary conditions, such as having a charged punctual mass. As you
can see, many elds can be studied from this point! I hope to be able to study
them in the future.
References
[1] William M. Boothby. An Introduction to Dierentiable Manifolds and Riemannian Geometry.
Academic Press, Inc., 2nd edition, 1986.
[2] James J. Callahan. The Geometry of Spacetime - An Introduction to Special and General
Relativity. Springer, 2000.
[3] Sean M. Carroll. Lecture notes on general relativity, December 1997.
[4] Albert Einstein and Roger Penrose. Relativity: The Special and General Theory. Penguin
Group US, 2006.
[5] Bernard F.Schutz. A rst course in general relativity. Cambridge University Press, 7th edi-
tion, 1993.
[6] Joan Girbau. Geometria diferencial i relativitat. Universitat Aut`onoma de Barcelona, 1993.
[7] Xavier Gr`acia. Geometria diferencial 2 - denicions i resultats, 2011.
[8] Stephen W. Hawking and G.F.R. Ellis. The large scale structure of space-time. Cambridge
University Press, March 1975.
[9] M.C. Mu noz Lecanda and N. Roman-Roy. Notas sobre teora de la relatividad especial y
general, 2013.
[10] John M. Lee. Riemannian manifolds: an introduction to curvature. Springer, 1997.
[11] David Lovelock. The uniqueness of the einstein eld equations in a four-dimensional space.
Archive for Rational Mechanics and Analysis, 33(1):5470, 1969.
[12] Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. Gravitation. W. H. Freeman,
1st edition, September 1973.
[13] Mikio Nakahara. Geometry, Topology and Physics. Taylor & Francis, 2nd edition, 2003.
[14] Poul Olesen. General relativity and cosmology, lecture notes, 2008.
[15] Barrett ONeill. Semi-Riemannian Geometry With Applications to Relativity. Academic
Press, 1st edition, July 1983.
35