Table Of Content´
Orbitas materiales en agujeros negros
Universidad de Murcia
Grado en Matem´aticas
Guillermo Fern´andez Melgarejo
29 de junio de 2015
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Indice general
Introduction 5
Introduccio´n 11
1. Nociones b´asicas 17
1.1. Tensores y variedades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2. Unidades Geom´etricas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
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1.3. Orbitas en Gravedad Cl´asica . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2. Ecuaciones de movimiento 33
2.1. Ecuaciones de campo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2. Accio´n de Einstein-Hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3. Part´ıcula a velocidad baja en un campo d´ebil . . . . . . . . . . . . . . . . . 39
3. La solucio´n de Schwarzschild 43
3.1. Obtencio´n de la m´etrica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2. Espacio-tiempo de Schwarzschild . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.1. Espacio-tiempo no conexo . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.2. Espacio-tiempo curvo . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
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3.3. Orbitas en Relatividad General . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4. La curvatura de la luz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3
Introduction
The aim of this document is the study of the orbits of particles in the Schwarzschild
metric. This metric is a black hole-type solution of the Einstein field equations. In order to
carry it out, we will study the orbits in the Newtonian classical theory and we will review
the main aspects of the theory of General Relativity.
In 1915, Einstein published the theory of General Relativity, which is a theory about
the gravitational field and the general reference frames. Einstein said that at a certain point
of our space-time, it is not possible to distinguish between an uniformly accelerated body
and the same body under an uniform gravitational field. Its name is due to the fact that it
generalises the Special Theory of Relativity. Its main ideas are: the curvature of space-time,
the principle of general covariance and the equivalence principle.
The curvature of the space-time is a consequence of the gravitational attraction. In other
words, the gravity makes that the geometry of the space-time gets curved and not plane.
The bodies in a gravitational field describe a spacial curved trajectory, although its world-
lines were as straight as possible. These lines are called geodesics and they have minimum
curvature. Rephrasing the American theoretical physicist John A. Wheeler, the space-time
tells the matter how to move, and the matter tells the space-time how to curve.
Based on the principle of special covariance used in the theory of Special Relativity,
Einstein wanted to extend this type of invariance to non-inertial reference frames. That is, he
was trying to construct such a theory whose equations had the same form for any observer,
either inertial or not. To do so, he introduced the principle of general covariance.
The principle of general covariance establishes that laws of Nature (equations) must be
the same for any reference frame. That is, they must remain invariant under any coordinate
transformation. We can interpret this principle as the fact that Nature has no privileged
reference frame. In addition, the concept of coordinate does not exist in Nature a priori and,
therefore, the chosen coordinate system must not play any important role in the formulation
of physical laws. Einstein thought that this principle ought to be one of the key ingredients
in the formulation of a new theory to accelerated reference frames.
Since the laws of Physics must be invariant under general coordinate transformations,
Einstein formulated his theory in terms of tensors. Due to its properties, tensors transform
in a certain way under any general coordinate transformation.
Once it has been introduced the principle of general covariance and the absence of privi-
leged reference frames, we can naturally understand the equivalence principle:
The outcome of any local non-gravitational experiment in a freely falling laboratory is
5
Introduction 6
independent of the velocity of the laboratory and its location in spacetime.
This physical principle says that if we fix an instant event p in a gravitational field, it
can be described by an accelerated observer who is at that point. In other words, there is
an accelerated observer which has no way to distinguish whether the particles are moving
within or without the gravitational field. Therefore, the laws of Physics are locally the same
for all observers. For example, if we fell at the same time as a stone from a cliff, we would
see that the stone is falling at constant velocity, as if there was no gravitational field that
causes the fall. Exactly the same happens to astronauts when they are in a spaceship: they
may think that they are floating and can imagine that they are not suffering the attraction
of the Earth or another planet.
Now, let us analyze the mathematical tools of the theory and its local and global im-
plications. The theory is formulated in terms of tensor fields defined on a space-time that
is represented by a Lorentz manifold. This is due to, as it has been mentioned above, the
condition of general covariance of the theory under general coordinate transformations. That
is, the tensors provide the general coordinate transformations required to the theory to be
invariant. The main element of the theory is the metric tensor g . The metric tensor,
mu nu
which is an order-2 tensor, is used to define metric concepts such as distance, angle and
volume in a locally Euclidean space.
The fundamental core of this theory is the Einstein field equations,
1
R − g R = T , (1)
µν µν µν
2
where R is the Ricci tensor, R is the scalar curvature, g is the metric tensor and T is
µν µν µν
the energy-momentum tensor. Basically, these Einstein’s equations are a set of 10 non-linear
partial differentials equations in the variables g . They describe the gravitational interaction
µν
as a result that the spacetime is being deformed by the matter and the energy.
In Physics and Mathematics, it is important to distinguish between the local and global
structures. Since the measurements are made in a relatively small regions of the space-time,
it is reasonable to do a local study of the space-time. However, the determination of the
global structure of space-time is important in cosmological problems.
Knowing whether two space-times are locally the same or not is a complex problem in
General Relativity. Previously in the manifold theory, the problem of determining whether
two Riemannian manifolds of same dimensions were locally isometric was already addressed.
This problem has been solved and adapted to General Relativity by an algorithm1.
Going back to (3), we can see that the left-hand side of the equation is written in terms
which are purely geometric. On the other hand, in the right-hand side we have the energy-
momentum tensor, which parametrizes the matter and energy of our space-time. So, mathe-
matically, Einstein conjectured that the geometry of the Universe can be curved because of
the presence of mass and/or energy. Therefore, he thought that the universe was some kind
of curved space-time given by a pseudo-Riemannian manifold and whose field equations state
that the sectional curvature is related to the energy-momentum tensor at each point. This
tensor describes the energy and matter content in our spacetime. Hence, the particle trajec-
1That method is the Cartan-Karlhede algorithm (cf. [5]).
7 Introduction
tories are affected by the curvature and, reciprocally, the presence of mass/energy affects the
curvature.
Despite the difficulty to obtain solutions to these equations, there exist some of them,
although very few have direct physical applications. Some solutions describing black holes
with different properties are: the Schwarzschild solution, the Reissner-Nordstr¨om solution
and the Kerr metric. Finally, it is important to point out the Friedmann-Robertson-Walker-
Lemaˆıtre solution, which describes an homogeneous, isotropic and expansion (or contraction)
Universe. In Physics, this type of solution is used to describe our Universe.
Normally, it is common to rely on the numerical integration of the equations. In numerical
Relativity, there are powerful computers which simulate the geometry of the space-time and
solve Einstein’s equations in specific situations. For example, one case is the collision of two
black holes.
Let us interpret now the gravitational interaction in General Relativity and its differences
with the Newtonian theory. Let us assume a fixed particle of mass m in a certain space-time.
1
That particle bends space according to its mass m . Then, if we introduce a second particle
1
of mass m , space-time will be deformed and the perturbation (which is equivalent to a wave
2
on a lake) will propagate at the speed of light, c. Therefore, the first particle does not feel the
presence of the second instantaneously. These space-time perturbations, which propagate as
waves, are called gravitational waves. Furthermore, in the classical theories of gravitation, the
conceptofthegravityspeedisinterpretedasthespeedthatthegravitationalfieldpropagates.
In other words, it is the propagation speed of any change in the distribution of energy or
momentumofthemattercausedbythegravitationalfield.Inaphysicalsense,gravityvelocity
refers to the velocity of a gravitational wave. So, any change in the distribution of energy or
matter can produce gravitational waves.
This propagation is radically different from the classical theory. In the Newton’s law
Universal Gravitation, a particle interacts instantly with any other particle independently of
the distance that separates them. Gravitation is classically described by a scalar potential
which satisfies the Poisson equation and fits instantly in case of any change in the mass
distribution. This means that it is assumed that the propagation velocity is infinite. This
assumption helped to clarify many phenomena of the epoch, although it was not until the
nineteenth century, when an astronomical anomaly in the Mercury’s motion was observed
that could not be explained by the Newtonian model of instant action. Finally, in 1859, the
French astronomer Urbain Le Verrier determined that the elliptical orbit of Mercury has a
perihelion precession which differs from the prediction of Newtonian theory.
Another interesting aspect of the General Relativity occurs when we have very compact
and massive particles. In the case of supermassive stars (about 30 times the mass of the Sun),
they die and generate a supernova explosion or a gravitational collapse. Both processes lead
to the formation of black holes, although there is no evidence of any of them nowadays. A
black hole2 is a region of space-time which has an attractive gravitational force such that
no particle nor electromagnetic radiation (light) can escape from it. The theory of General
Relativity predicts that if there exists a certain body whose mass density is high enough,
2Notice that it is not necessary to define black holes in General Relativity, since this idea comes from the
conceptofescapevelocityinClassicalMechanics.Adarkstarisaverymassiveobject,whoseescapevelocity
is equal or greater than the speed of light.
Introduction 8
it can deform spacetime and become a black hole. The boundary of the region where it
is possible to escape is called the event horizon. The event horizon is a boundary of the
spacetime where the events can not affect to external observers. It is the point of no return,
that is, the point where the gravitational is strong enough that it is impossible to escape.
In the black hole, the light emitted can never escape or reach an external observer. Simi-
larly, any object which approaches to the event horizon near the outside observer, it seems
to go slow and never cross the horizon at a finite time. Nevertheless, this object does not
experience strange effects, so it crosses the horizon at a finite time and finally it will be
caught by the central singularity.
There is a theorem called the no-hair theorem which characterises the black holes in terms
of three classic parameters: mass, electric charge and angular momentum. The physicist John
Archibald Wheeler expressed this idea in the phrase: black holes have no-hair, who was the
creator of the name of this theorem. Since there is still no rigorous proof of the theorem,
mathematicians refer to it as the no-hair conjecture. In case that there is only gravity (wit-
hout electric fields), this conjecture has been partially proven by Stephen Hawking, Brandon
Carter, and David C. Robinson, under the hypothesis that event horizons are not degenerate
and the assumption of real analyticity of the continuous space-time.
In this work, we will focus on the study of the Schwarzschild black hole. According to
Birkhoff’s theorem, the Schwarzschild metric is the most general and spherically symmetric
solution of the Einstein field equations in vacuum (the electric charge of the mass, the an-
gular momentum of the mass and the cosmological constant are all zero). This solution is
useful to describe astronomical objects with a slow rotation and its name comes from Karl
Schwarzschild, the first one who published it in 1916. Thus, a Schwarzschild black hole is
a static black hole which has no charge or angular momentum and it is described by the
Schwarzschild metric,
(cid:18) 2M(cid:19) (cid:18) 2M(cid:19)−1
dτ2 = 1− dt2 − 1− dr2 −r2dφ2 −r2sin2φdθ2. (2)
r r
This kind of black hole can not be distinguished by another Schwarzschild one, except for its
mass.
The Schwarzschild solution is valid for any radial coordinate r > 0 and has some peculiar
properties. When we are in the black hole, r < 2M, the coordinate r is time-like and the
t coordinate becomes space-like. Thus, a constant curve r is not a world-line of particles or
observers. This is because the space-time is curved, so that the direction of cause and effect
(future light cone of the particle) points to the central singularity. The surface r = 2M is
what we have called the event horizon of the black hole. We will study more properties of
this solution in Chapter 3.
To summarize, we can say that General Relativity is a successful model for gravity and
the description of our Universe. Up to now, it has passed positively and predicted many
observational and experimental evidences. However, there are some hints pointing out that
thetheoryisstillincomplete.Theproblemofquantumgravityandquestionsaboutspacetime
singularities have not been answered yet. In addition, some experimental evidences about the
existenceofdarkenergyanddarkmattercouldindicateustheneedforanewphysicaltheory.
Many physicists and mathematicians are trying to understand the nature of gravitational
9 Introduction
interaction and go beyond the Einstein’s equations, having it as a essential ingredient of the
new ones. The struggle in the detection of gravitational waves still continues by means of
some experiments as LISA, LIGO, BICEP. Exactly 100 years after its publication, General
Relativity is and will be a very active area of research.
Finally, let us summarize this work, which is divided in 3 chapters. In the first chapter,
we review the basic theoretical concepts for the development of the theory. We start by
introducing the ideas of tensors and Lorentzian manifolds, where we stress the concepts of
metric tensor and proper time. Then, we indicate how it is possible to establish a geometric
system of units in General Relativity. Closing this chapter, the orbits of planets that rotate
around others will be calculated in the context of classical gravity.
In the second chapter, the Einstein field equations are introduced. Then, we will obtain
these equations from the minimization of an action. Finally, we will study how the metric
tensor is related to the gravitational potential of Newton’s theory in a limiting case.
In order to end this work, we will obtain the Schwarzschild metric under certain assum-
ptions and we will study some of its features of this solution. In particular, we will focus
in those that are a consequence of the curvature of this model. Once the metric has been
obtained, we will close the work with the study of the orbits of material and light particles.
Finally, some aspects like the perihelion precession and the bending of light will be explained.
Description:generalises the Special Theory of Relativity. Its main ideas are: the The main element of the theory is the metric tensor g mu nu. The metric tensor,.
