final_year_project_rn
TRANSCRIPT
Quantum Tunneling near the horizons of
a Reissner-Nordstrom black hole
By Joseph O’Leary
2014
Project Report submitted in partial fulfillment of
examination requirements leading to the award of BSc in
Mathematical Sciences, Dublin Institute of Technology.
Supervised by Dr. Emil Prodanov.
Declaration
Project Title:
Student Name:
Student Number:
I certify that this Project which I now submit for examination in partial ful-
filment of the requirements for the award of BSc (Honours) Mathematical
Sciences/Industrial Mathematics is entirely my own work and has not been
taken from the work of others save and to the extent that such work has
beencited and acknowledged within the text of my work. This project was
prepared according to the regulations of the Dublin Institute of Technology.
Signed: Date:
Acknowledgements
I would like to thank Dr. Emil Prodanov for his guidance and understanding.
He has made a difficult area very understandable and his willingness to teach
has been invaluable. This project would not have been possible without such
a great mentor and supervisor.
Furthermore, I would like to personally thank Andrew Copeland for many
conversations in the area of Theoretical Physics. He has been a great friend
over the course of the research project.
Finally, I would like to thank my mother, Jackie O’Leary and my family,
their belief and their ability to empower has been a huge contributor to the
research project.
Contents
1 Introduction to tensors 1
1.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Transformation of coordinates . . . . . . . . . . . . . . . . . . 2
1.3 Contravariant and covariant tensors . . . . . . . . . . . . . . 5
1.4 Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 The metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Tensor calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Covariant differentiation . . . . . . . . . . . . . . . . . . . . . 11
1.8 Affine geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.9 The Riemann tensor . . . . . . . . . . . . . . . . . . . . . . . 15
2 Principles of General Relativity 18
2.1 Mach’s principle . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Mass in Newtonian Theory . . . . . . . . . . . . . . . . . . . 22
2.3 Principle of equivalence . . . . . . . . . . . . . . . . . . . . . 25
2.4 The field equations of general relativity . . . . . . . . . . . . 28
2.5 Schwarzschild geometry . . . . . . . . . . . . . . . . . . . . . 29
3 Quantum Mechanics 33
3.1 The Schrodinger equation . . . . . . . . . . . . . . . . . . . . 33
3.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
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3.4 Time-independent Schrodinger equation . . . . . . . . . . . . 38
3.5 The WKB approximation . . . . . . . . . . . . . . . . . . . . 40
3.6 Quantum tunneling . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Marginally bound particles in Reissner-Nordstrom
geometry 46
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 The Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Bound particles in Reissner-Nordstrom
geometry 57
5.1 The Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6 Neutral particles in Reissner-Nordstrom geometry 61
6.1 The Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
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Brief Introduction
The following research project is based in the area of Theoretical physics.
It can be divided into two parts. The first part being theory and the second
being applications.
We begin by introducing the language of general Relativity i.e. tensor anal-
ysis and calculus. We apply this language to briefly describe the geometry
of Schwarzschild and Reissner-Nordsom black holes.
It follows then that we incorporate a chapter on the theory behind quan-
tum mechanics in order to derive the Wentzel, Kramers, Brillouin (WKB)
approximation method in two separate ways.
The final chapters are dedicated to extending a research paper by Dr. Emil
Prodanov and also covering a new topic that is in relation to the motion of
a neutral particle.
1 Introduction to tensors
The mathematical language of tensors is fundamental when one is concerned
with relativity theory. It should be noted that there is a deep geometrical
significance behind tensors, however for the purpose of this research project
we will be concerned with what we can actually do with tensors rather than
taking the first approach. This chapter is intended to discuss tensor algebra
and calculus in preparation for the following chapter where we will derive a
key solution that plays an important role in the theory of general relativity.
1.1 Manifolds
We will begin by stating that a tensor is an object defined on a manifold [6].
To understand the idea of a manifold consider the following example: In
ancient times people believed the Earth to be flat, of course we know that
this is not the case, but if we considered the Earth on a much smaller scale
then it would seem that it was flat. Of course the theory of manifolds and
differential geometry is much deeper and complex, we will simply describe a
manifold as an object that appears flat on small scales i.e. we can compare
it to n-dimensional Euclidean space Rn. A coordinate patch is one where we
have a coordinate system that only covers a section of the manifold, however
we can use a set coordinate patches to cover the whole manifold in which the
coordinate systems overlap for e.g we need at least two coordinate systems
to cover the whole of a 2-sphere.
1
Figure 1: Representation of two overlapping coordinate patches in some
manifold M .
A key point in relation to tensors arises here,we know that a manifold can be
covered by multiple coordinate patches, then if we make a statement about
tensors we do not only wish for it to be true in a particular coordinate system
but rather for all coordinate systems. For this reason a clear understanding
about coordinate transformation is vital.
1.2 Transformation of coordinates
We will be concerned with points and subsets of points in a manifold which
can define curves and surfaces of different dimensions, for example we can
define a curve by the following parametric equation [6],
xα = xα(u) (1.1)
a curve only has one degree of freedom and hence depends only on one
parameter, u. Also α = 0, 1, . . . , n and this convention will always be used
2
unless specified. Let us consider the change of coordinates from xα to x′α
given by the following,
x′α = ξα(x1, x2, . . . , xn) (1.2)
where ξα are functions of the old xα coordinate system. This is what is
known as a passive transformation and we describe it as assigning to a point
new primed coordinates, (x′1, x′2, . . . , x′n) whose previous components where
(x1, x2, . . . , xn),
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Figure 2: Illustration of passively transforming from an old unprimed coor-
dinate system (x1, x2) to a new primed coordinate system (x′1, x′2). Notice
the point we are assigning the new primed coordinates does not change it is
merely the coordinates themselves that change.
We can also differentiate x′α with respect to xβ (again β = x1, x2, . . . , xn)
to produce an n× n transformation matrix [6],
∂x′α
∂xβ=
∂x′1
∂x1∂x′1
∂x2∂x′1
∂x3 . . . ∂x′1
∂xn
∂x′2
∂x1∂x′2
∂x2∂x′2
∂x3 . . . ∂x′2
∂xn
∂x′3
∂x1∂x′3
∂x2∂x′3
∂x3 . . . ∂x′3
∂xn
...
∂x′n
∂x1∂x′n
∂x2∂x′n
∂x3 . . . ∂x′n
∂xn
(1.3)
The Jacobian is defined as the determinant of equation (1.3) and is used
in calculating transformations from one coordinate system to another for
e.g. transforming from Cartesian coordinates (xα) = (x, y, z) to spherical
coordinates (x′α) = (r, θ, φ).
We can then define the total differential of an n-dimensional coordinates
system in three equivalent ways,
dx′α =∂x′α
∂x1dx1 +
∂x′α
∂x2dx2 + . . .+
∂x′α
∂xndxn
=n∑β=1
∂x′α
∂xβdxβ
=∂x′α
∂xβdxβ
(1.4)
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The final equation utilizes the Einstein summation convention where we sum
over repeated indices, such indices are called dummy indices and can take on
any other index for e.g. ∂x′α
∂xβdxβ = ∂x′α
∂xγ dxγ . Finally we define the Kronecker
delta as δαβ = ∂x′α
∂x′β= ∂xα
∂xβwhich takes the value 1 or 0. If α = β then δαβ = 1
and 0 if α 6= β [6].
1.3 Contravariant and covariant tensors
It is now possible to describe the transformation components of a given
vector in two different ways, namely, contravariant and covariant. We say a
vector has contravariant components if it transforms in the following way,
X ′α =∂x′α
∂xβXβ (1.5)
This can be defined as a contravariant tensor of rank 1.
We say that a vector has covariant components if it transforms as follows,
X ′α =∂xβ
∂x′αXβ (1.6)
This can be defined as as covariant tensor of rank 1. In equations (1.5) and
(1.6) the Einstein summation convention is utilized for repeated indices and
is assumed unless specified otherwise. To illustrate the convenience of the
notation we use, consider equation (1.6), this can be wrote equivalently in
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the following matrix equation (α, β = (0, 1, 2, 3))
X ′0
X ′1
X ′2
X ′3
=
∂x0
∂x′0∂x1
∂x′0∂x2
∂x′0∂x3
∂x′0
∂x0
∂x′1∂x1
∂x′1 . . . ∂x3
∂x′1
......
∂x0
∂x′3∂x3
∂x′3
X0
X1
X2
X3
(1.7)
Clearly equation (1.6) is more economical to write.
We can generalize equation (1.5) and (1.6) to obtain contravariant and co-
variant tensors of higher order, so for e.g. a second order contravariant
tensor is as follows,
X ′αβ =∂x′α
∂xγ∂x′β
∂xδXγδ (1.8)
higher order tensors follow in the same manner. Note that equation (1.8) is
a set of n2 quantities associated with some point.
A second order covariant tensor is as follows,
X ′αβ =∂xγ
∂x′α∂xδ
∂x′βXγδ (1.9)
How do we interpret tensors of mixed rank? i.e. tensors with both con-
travariant and covariant components? It follows in an analogous way to the
transformation equations given in equations (1.5) and (1.6),
X ′αβ =∂x′α
∂xγ∂xδ
∂x′βXγδ (1.10)
we say that this is a rank 2 tensor of type (1,1) i.e. it has one contravariant
index and also one covariant index. If we are given two tensors of type (a, b)
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and (c, d) their product would produce a tensor of type (a+ c, b+ d), so for
e.g. if we were concerned with the product Y αβ Z
γδ we would get the tensor
Xαγδβ .
Another method used in tensor analysis is one which is called contraction.
This is where we are given a tensor of type (a, b) we can produce a new
tensor of type (a− 1, b− 1) by setting a raised index equal to a lower index,
for e.g. given the tensor Xαβγδ we can contract on α and β (so that our
tensor is as follows Xααγδ) to produce Yγδ. Two tensors added together or
subtracted of type (a,b) produce another tensor of the same type, for e.g.
Y αβγ + Y αβ
γ = Xαβγ
1.4 Tensor fields
For the most part of this project we will be dealing with what are known
as tensor fields. They can be described in an almost analogous way to that
of a vector field in vector analysis i.e. a vector field is an association of a
vector to every point in a particular region.
7
Figure 3: Illustration to aid in visualizing general vector fields in 2 and 3
dimensions using Mathematica.
With this in mind we say a tensor field is an association of a tensor of the
same type to every point of a region in a manifold i.e. P → T ...α...β (P ), where
T ...α...β (P ) is the value associated with the tensor at some point P [6].
1.5 The metric tensor
A very important tensor we meet in relativity theory is called the metric
tensor. It can be used to describe infinitesimal distances between points in
a given coordinate system. As an example let us consider the infinitesimal
distance between two points, say dx1 and dx2 which we will denote ds, using
pythagoras theorem it is clear to see that,
(ds)2 = (dx1)2 + (dx2)2 (1.11)
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where we define (ds)2 as the line element,which can be equivalently expressed
as follows,
(ds)2 = δµνdxµdxν (1.12)
where δµν is the kronecker delta described in section (1.2). In matrix form
it is described as,
δµν =
1 0
0 1
(1.13)
However in equation (1.12) the kronecker delta function is only used to de-
scribe Euclidean (flat) space. For non-Euclidean space we express equation
(1.12) as,
(ds)2 = gµνdxµdxν (1.14)
The expression gµν is defined as the metric tensor. To illustrate equation
(1.14) let us consider spherical coordinates (r, θ, φ). The line element and
metric tensor read,
(ds)2 = dr2 + r2dθ2 + r2sin2θdφ2 (1.15)
gµν =
1 0 0
0 r2 0
0 0 r2sin2θ
(1.16)
Some properties of the metric tensor are as follows:
1. The metric tensor is symmetric i.e. gµν = gνµ and likewise for gµν =
gνµ.
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2. We can use the method of contraction with the metric tensor on a
vector to raise or lower indices i.e. gµνXν = Xµ and gµνXν = Xµ.
3. The determinant of the metric tensor is non-zero i.e. it is non-singular,
we define the inverse of gµν or gµν as gµνgνα = gναgµν = δαµ .
1.6 Tensor calculus
We can describe some quantity as tensorial if it converts a tensor into an-
other tensor [6].In this section we will describe new methods for determining
tensorial differentiation of contravariant and covariant tensors. First though
we encounter the partial derivative of a contravariant vector, Xα with re-
spect to xβ, denoted ∂βXα or ∂Xα
∂xβso for example,
∂′γX′α =
∂xδ
∂x′γ∂
∂xδ
(∂x′α∂xβ
Xβ)
=∂x′α
∂xβ∂xδ
∂x′γ∂δX
β +∂2x′α
∂xβ∂xδ∂xδ
∂x′γXβ
(1.17)
This is not a tensorial operation. To justify this consider the definition of
computing a derivative. We generally consider a quantity evaluated at two
neighboring points say A and B then divide by some parameter representing
the seperation of A and B and then take the limit as this parameter tends
to 0 [6]. However if we were to compute this for some contravariant vector
field Xα evaluated at these points then we would find that this involves the
transformation matrix being evaluated at different points which we can then
see that this would not be a tensor.
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1.7 Covariant differentiation
To illustrate the idea of covariant differentiation we must first consider what
is known as the affine connection. We will use the same method used in [6]
to arrive at this connection and in doing so introduce what is known as the
Christoffel symbol.
Consider some vector field Xα(x) that is being evaluated at a point B that
has coordinates xα + dxα, near a point A with coordinates xα. Applying
Taylor’s theorem we find that,
Xα(x+ δx) = Xα(x) + δxβ∂βXα (1.18)
By allowing the second term in equation (1.14) to equal δXα(x) then we
can write,
δXα(x) = Xα(x+ δx)−Xα(x) (1.19)
Again this is not tensorial as we are evaluating tensors at two different
points. To allow us to define a tensorial derivative we will introduce a
vector at B that is in a sense parallel to to Xα at A. It is plausible to
assume that our parallel vectors at A and B only differ by a small amount
denoted δXα(x)(not tensorial), Please see below,
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Figure 4: Illustration to aid in visualizing the parallel vector Xα + δXα at
B.
As already noted δXα(x) is not tensorial however we can contruct a new
vector or a difference vector that will be tensorial, namely,
δXα(x)− δXα(x) = Xα(x) + δXα(x)− [Xα(x) + δXα(x)] (1.20)
the easiest way to perceive this is to assume that δXα is linear in both Xα
and δxα which would mean that there exists some multiplicative factor, Γαβλ,
where
δXα(x) = −Γαβλ(x)Xβ(x)δxλ (1.21)
This multiplicative factor is what is known as the Christoffel symbol. It is
sometimes referred to as the connection coefficient as it allows us to connect
the value of a vector field Xα(x) at some point with the value of another [8].
Again note the methodology presented above can be seen in [6].
It is now possible to define the transformation equation of Γ′αβλ using the
covariant derivative of a contravariant vector, Xα denoted ∇λXα and also
define the covariant derivative of a covariant vector, again denoted ∇λXα.
12
To define ∇λXα we take the limit as δxλ tends to zero of the difference of
the vector field Xα at the point B and the vector at B parallel to the vector
at A and then then divide by the coordinate differences δxλ [6] then using
equations (1.14) and (1.17) we find that,
∇λXα = ∂λXα + ΓαβλX
β (1.22)
The covariant derivative of a covariant vector is defined as,
∇λXα = ∂λXα − ΓβαλXβ (1.23)
How do we determine the transformation equation for Γ′αβλ? This is obtained
by demanding that ∇λXα by of type (1,1) then we can deduce that,
Γ′αβλ =∂x′α
∂xδ∂xτ
∂x′β∂xρ
∂x′λΓδτρ −
∂xδ
∂x′β∂xτ
∂x′λ∂2x′α
∂xδ∂xτ(1.24)
This transformation equation is not tensorial. Any quantity transforming
in this way is known as the affine connection, the coefficients of the affine
connection can be given in terms of the metric tensor and its first derivative
by the following equation [4]
Γαβλ =1
2gαδ(gδλ,β + gδβ,λ − gβλ,δ) (1.25)
The notation gδλ,β = ∂β gδλ. We will generally be concerned with symmetric
connection coefficients namely Γαβλ = Γαλβ i.e. the torsion tensor equals
0 where we define the torsion tensor to be the anti-symmetric part of a
connection coefficient, namely, [6]
Tαβλ = Γαβλ − Γαλβ (1.26)
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1.8 Affine geodesics
Let us first define by what is meant by a congruence of curves in a manifold.
We say a congruence of curves is different sets of curves in which each set
has only one curve that passes through a particular point in the manifold.
In general we can write this as xα = xα(u) with u being the parameter in the
same way as we defined equation (1.1). We define the tangent vector field
to the congruence as being the derivative of each coordinate with respect to
the parameter u and is denoted dxα
du = Xα.
If we take a tensor T ...α...β then we can write,
D(T ...α...β )
Du= ∇XT ...α...β = Xδ∇δT ...α...β (1.27)
With ∇X being the covariant derivative contracted with X and DDu denoting
what is known as the total derivative [6]. Now we are at a position to describe
what we mean by an affine geodesic, it is a curve in which the tangent vector
is propagated or transported parallel to itself [6], namely,
14
Figure 5: A tangent vector portraying the idea of parallel transportation or
propagation.
We can write this mathematically in two equivalent ways,
∇XXα = λ(u)Xα (1.28)
d2xα
du+ Γαβδ
dxβ
du
dxδ
du= λ(u)
dxα
du(1.29)
λ represents the proportionality factor of the tangent vector to the parallely
propagated vector. Equation (1.28) is known as the geodesic equation. If
either equation equates to zero due to the parameterizing of λ then λ is
known as an affine parameter.
1.9 The Riemann tensor
If we take two vector fields S and T the commutator of these vectors defines
a new vector and is expressed as [S, T ] = ST − TS. Then if we take the
commutator of two covariant derivatives of a tensor, Xα we will be at a
position to define the Riemann tensor. The commutator would read,
[∇δXα,∇τXα] = ∇δ∇τXα −∇τ∇δXα (1.30)
We can follow a simple rule for determining covariant derivatives for tensors
of any rank:
1. Take the ordinary derivative of the tensor.
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2. Add one Christoffel symbol having the same form and sign for a co-
variant tensor for every covariant index.
3. Add one Christoffel symbol having the same form and sign for a con-
travariant tensor for every contravariant index.
To solve equation (1.25) first consider ∇δ∇τXα, this is equivalent to writing
∇δ(∂τXα + ΓαλτXλ) which can be expressed as,
∂δ(∂τXα + ΓαλτX
λ) + Γανδ(∂τXν + ΓνλτX
λ)− Γντδ(∂νXα + ΓαλνX
λ) (1.31)
This is obtained using the fact that we demanded ∇λXα be of type (1,1)
(see section (1.7)) and the rules mentioned above. A similar expression can
be obtained for ∇τ∇δXα using the same method. Again remembering that
we are only interested in symmetric connection coefficients and subtracting
these expressions we can deduce that,
[∇δXα,∇τXα] = (∂δΓαλτ − ∂τΓαλδ + ΓνλτΓανδ − ΓνλδΓ
αντ )Xλ = RαλδτX
λ (1.32)
The right hand side of equation (1.27) is what is known as the Riemann
tensor and is of type (1,3). We can see that the commutator is in terms
of the Riemann tensor and the tensor itself, this holds true for any given
tensor. The vanishing of the Riemann tensor is a necessary condition for
the vanishing of the commutator of any given tensor [6].
Under contraction of the Riemann tensor we can derive the Ricci tensor,
Rααδτ = Rδτ (1.33)
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Under a contraction of the Ricci tensor we obtain the Ricci scalar,
Rδδ = R (1.34)
Both results will be utilized when describing the field equation of general
relativity.
17
2 Principles of General Relativity
For the purpose of this research project a chapter dedicated to the the theory
of general relativity is vital. Although entire textbooks are dedicated to such
a topic, we will try to give an understandable overview of the principles
which guided Albert Einstein to one of the greatest achievements of the
human mind. Topics to be included are Machs principle and the principle
of equivalence. A discussion on mass in Newtonian theory is also given.
2.1 Mach’s principle
To get an idea of Mach’s principle, we first must ask what happens to New-
ton’s laws in a non-inertial frame of reference. An inertial frame of reference
being one in which Newton’s first law holds. Inertial frames are in a state of
constant motion with respect to each other and are non-accelerating. When
we say non-accelerating we are concerned with acceleration that can be mea-
sured by an accelerometer.
We will consider the simple example given in [6] to mathematically define
what is meant when one is concerned with inertial forces.This is concerned
with Newton’s second law for a non-inertial frame, S′ being uniformly ac-
celerated relative to an inertial frame, S with acceleration given by a. As
is common when considering frames of references we’ll assume that a hypo-
thetical clock will be triggered when both observers meet which allows us to
18
describe the relationship between the frames as:
x = x′ + s
y = y′
z = z′
t = t′
(2.1)
By differentiating the first equation in equation (2.1) with respect to t = t′
twice and also considering some particle under some force ~F = (F, 0, 0) of
mass m moving in the x-direction, then in the view of the observer S′, we
can write Newton’s second law by
F −ma = mx′ (2.2)
This reduction by the amount ma is detected only by the observer S′ when
compared with S and such an additional force is called an inertial force. As
described in [6] some describe these particular forces as fictitious, however
we will consider them to be real forces judging them solely on their effects
in real life for e.g. such forces cause people on an aircraft to be pushed back
in their seats or a more extreme case such forces cause astronauts to pass
out due to the G-force that is associated with rapid acceleration.
Newtons attempted to answer the question of how do we detect inertial
frames by considering the thought experiment - the bucket experiment. He
considered the existence of absolute space in which he states [6]:
”Absolute space, in it’s own nature, without relation to anything
19
external, remains always similar and immovable” - I. Newton
In other words Newton considered absolute space as the backdrop for which
all motion is observed. We can then go on to say that an inertial observer
becomes an observer at rest or uniform motion relative to absolute space.
The bucket experiment helps us to understand the above motion and more
importantly is able to explain when a particular system is in absolute ro-
tation relative to absolute space [6]. The bucket experiment consists of a
bucket of water being suspended by a rope that is twisted and then released.
We can go on to describe the motion into four seperate phases:
1. Rotation of the bucket with the surface of the water remaining flat.
2. The friction between the bucket and the water communicating com-
mences rotation of the water. The water then becoming concave due
to the centrifugal force of the rotation.
3. In time the bucket will come to a stop with the water still rotating for
some time with the surface remaining concave.
4. The system will return to its original state with both the bucket at
rest and the water returning to being flat.
The concavity or curvature of the water clearly arises from the centrifugal
effects caused by the rotation of the water relative to absolute space [6]. We
can say that the curvature is not directly related to the rotation of the bucket
since in point 1 the bucket is rotating and the water remains flat and also
20
in point 3 the bucket has stopped rotating and the water is still curved. So
in conclusion the bucket experiment, although extremely simple in essence
can describe to us a method for detecting inertial observers. Simply an
observer is inertial when the water remains flat.However, Newton gives no
explanation for the curvature of the water when it is rotation relative to
absolute space.
The view taken by Mach was one where only relative motion for e.g. a body
in an empty universe cannot be described as being in motion as there is
nothing that such a body can be relative to. In our universe or in some other
populated hypothetical universe it is the interaction between the matter in
the universe that is the source of the inertial effects we have discussed. With
this in mind we can say that the bulk of the matter in our universe lies in
what is known as the ’fixed stars’ [6]. Thus it is these fixed stars which
determine a local inertial frame. This is what is known as Mach’s principle.
An example to illustrate the similarities in both the Newtonian postulate
of absolute space and the Machian principle is, again a very simple one but
very effective in portraying the ideas. Consider a pendulum rotating at the
North Pole. An observer on the Earth will see the pendulum rotating and
the time taken for for it to rotate 360◦is the same time it takes the Earth to
rotate 360◦with respect to absolute space. We can also measure the same
rotations relative to the fixed stars and correcting for experimental error the
times are the same. To summarize in Mach’s principle it is the fixed stars
21
which determine inertial frames [6].
2.2 Mass in Newtonian Theory
In Newtonian theory, one can talk about mass rather loosely. What we
mean by this is that we have different types of mass with different proper-
ties. However in most cases the mass of some body, m is not predefined or
described as having particular properties. We will outline three and show
that we can, in fact describe these as being equivalent. Their definitions
are: [6]
1. Inertial massmI : measure of the bodies resistance to change in motion.
2. Passive gravitational mass mP : measure of the bodies reaction to a
gravitational field.
3. Active gravitational mass mA: measure of the bodies source strength
for producing a gravitational field.
Inertial mass, mI is the quantity which we encounter in Newton’s second
law, often referred to as the bodies inertia which reads
~F =dmIv
dt= m~a (2.3)
If we let the gravitational potential at some point be denoted by φ, then we
can define the force associated with some passive gravitational mass, mP by
~F = −mP ~∇φ (2.4)
22
where ~∇φ denotes the gradient vector.
Finally, with mA measuring the strength of the gravitational field produced
by the body itself we can write the gravitational potential at any point a
distant r from the origin by
φ = −GmA
r(2.5)
where G represents the gravitational constant.
We can consider the observational result, that if two bodies, ignoring non-
fundamental forces are dropped from the same height they will reach the
ground simultaneously.Simply, both bodies experience the same acceleration
irrespective of their mass.With this in mind let us now show the relationship
between two inertial masses mI1 and mI
2 and two passive gravitational masses
mP1 and mP
2 . If we consider equation (2.3) and (2.4) we can write
mI1~a1 = ~F1 = −mP
1~∇φ
mI2~a2 = ~F2 = −mP
2~∇φ
(2.6)
We can say from our observational result above that ~a1 = ~a2 which leads to
mI1
mP1
=mI
2
mP2
(2.7)
then by taking the ratio mImP
for any given body and allowing it to equal
some universal constant say, α then by taking a suitable choice of units i.e.
α = 1, we get the result that inertial mass is equal to passive gravitational
mass [6]. To find a relationship between passive and active gravitational
masses we must use the observation that nothing can be shielded from a
23
gravitational field and that all matter is both acted upon by a gravitational
field and is itself a source of a gravitational field [6]. Consider two bodies at
seperate points, say Q and R which are moving under their mutual gravita-
tional interaction, then their gravitational potential due to each body can
be written [6]
φ1 = −GmA1
r
φ2 = −GmA2
r
(2.8)
Again the force they both experience is given by equation (2.4)
~F1 = −mP1~∇Qφ2
~F2 = −mP1~∇Rφ1
(2.9)
taking Q to be at the origin, the gradient vectors read,
~∇R = r∂
∂r= −~∇Q (2.10)
We are now at a position to get an expression for each force separately,
namely
~F1 =GmP
1 mA1
r2r
~F2 =GmP
2 mA2
r2r
(2.11)
By using using Newton’s third law, ~F1 = − ~F2 we arrive at the result
mP1
mA1
=mP
2
mA2
(2.12)
By using the same argument from equation (2.7) we see that passive grav-
itational mass is equal to active gravitational mass and hence we can see
24
that in Newtonian theory we can in fact simply refer to the massm m of a
body where [6]
m = mI = mP = mA (2.13)
2.3 Principle of equivalence
A gravitational test particle is defined to be a particle which experiences a
gravitational field but in no way alters or contributes to the field. We are
going to build our understanding of general relativity on what is known as
the strong form of the principle of equivalence which says that the motion of
a gravitational test particle in a gravitational field is independent of its mass
and composition [6]. In the theory of general relativity this is an essential
hypothesis where if it fails then so too does the theory. However, in New-
tonian theory this is regarded as merely a coincidence or an observational
result. If it failed in Newtonian theory it would not alter or upset the the-
ory which could accommodate such a failure. We will make the assumption
that both matter and energy, respond to, and is a source of a gravitational
field [6]. The inclusion of energy in the above statement stems from the fact
that in the theory of special relativity matter and energy are deemed equiv-
alent [6]. If we have a strong form of the principle of equivalence one could
safely assume that we also have a weak form of the principle of equivalence
which states that the gravitational field is coupled to everything. This is
merely a different way of stating what we have noted in the previous section,
25
namely nothing can be shielded from a gravitational field.
If we consider some frame of reference that is in free fall or in other words,
co moving with a gravitational test particle, it is possible to remove gravi-
tational effects locally and return to the theory of special relativity. Here,
locally is defined to mean that observations are confined over a region in
which variations of the gravitational fields are observably small [6]. If our
free falling frame is non rotating then we return to the concept of a locally
inertial frame. Gravitational test particles in such inertial frames remain
either at rest or move in a straight line. This leads us to the following
statement about the principle of equivalence,
There are no local experiments which can distinguish non rotat-
ing free fall in a gravitational field from uniform motion in space
in the absence of a gravitional field [6].
We can compare the force of gravitation with inertial frames in the sense
that both are proportional to the mass of some body. This is seen simply
by dropping a given body in the Earth’s gravitational field, hence it will
experience the force mg. This led Einstein to believe that both effects stem
from the same origin. Hence, Einstein then preceded to suggest that we
should treat gravity as an inertial effect aswell. By comparing the inertial
force ma from equation (2.2) with the force mg led Einstein to the following
and our last statement with regards to the principle of equivalence, namely,
A frame linearly accelerated relative to an inertial frame in spe-
26
cial relativity is locally identical to a frame at rest in a gravita-
tional field [6].
We will now attempt to clarify the previous two versions of the principle of
equivalence by detailing the famous Albert Einstein thought experiment -
the lift experiments.
Imagine an observer confined in a lift with no rooms or the ability to com-
municate at all. There are four states of motion that will be discussed,
namely,
1. Imagine the lift was inserted into some rocket that accelerated the lift
through space with a constant acceleration g relative to some inertial
observer, if the observer let some body drop from rest, the observer in
the lift would see such a body fall to the floor with acceleration g.
2. Consider the case where the rocket motor was turned off then the
rocket will travel through space with uniform motion relative to that
of an inertial observer. The observer in the lift would see the released
body remain at rest relative to the observer.
3. The rocket has now been abandoned and the lift is on the surface of
the earth. A released body will fall to the floor with acceleration g.
4. For the last case the lift is inserted in an evacuated lift shaft and is
falling freely towards the Earth, the released body is found to remain
at rest relative to the observer.
27
We can say that from the point of view of the observer, cases 1 and 3 can
be considered the same and so can cases 2 and 4 [6]. These cases reinforce
the statements made in this section.
2.4 The field equations of general relativity
The field equations are given as,
Gµν + gµνΛ = 8πTµν (2.14)
Note from this point we are using geometrized units in which c = G = 1.
The first term on the left is known as the Einstein tensor, the components
of which are as follows,
Rµν −1
2gµνR (2.15)
We can recognize these terms from section (1.9) as the Ricci tensor and
the Ricci scalar. We can also recognize the gµν as the metric tensor. The
second term on the left is the so called cosmological constant. Originally
when Einstein was developing his general theory of relativity equation (2.14)
read
Gµν = 8πTµν (2.16)
which are known as the standard field equations. Einstein was trying to
construct static models of the universe in which he found that the universe
was either expanding or contracting. It was believed at the time that the
Milky Way represented the entire universe, it was later realized that this was
28
not the case and in 1929 Edward Hubble discovered that the universe was
in fact expanding and the cosmological constant was no longer needed [5].
Einstein is reported to have regarded his introduction of the cosmological
constant as his ”biggest blunder”. On the right hand side of the field equa-
tions is what is known as stress-energy-momentum tensor or equivalently
known simply as the energy-momentum tensor and has components of the
form, T00 represents the energy density of the particles, T0ν = Tµ0 represents
the momentum density and T11 = T22 = T33 represents the pressure. We say
in general that Tµν contains all forms of energy and momentum, a region of
spacetime where the energy-momentum tensor equals zero is called empty
it then follows that the field equations in empty space, referred to as the
empty space field equations read [5],
Rµν = 0 (2.17)
2.5 Schwarzschild geometry
The approach taken in deriving the Schwarzschild solution is the same as
that of [5]. We are going to consider how to solve the field equations of
general relativity. We will look for solutions whose spacetimes possess sym-
metries. The metric of the Schwarzschild solution will represent a static
spherically symmetric gravitational field in the empty space surrounding
some massive spherical object such as a black hole or star [5]. We are trying
to find a set of coordinates, xµ where the metric tensor components do not
29
depend on a timelike coordinate, namely x0 and the line element is invariant
under the transformation from x0 to −x0 if these conditions are met we say
the metric is static, we say the metric is isotropic (i.e. space is the same in
any direction) if the line element depends only on rotational invariants. [5].
We can define a general line element in spherical coordinates that satisfy
these conditions,
ds2 = A(r)dt2 −B(r)dr2 − r2dθ2 − r2sin2θdφ2 (2.18)
To determine A(r) and B(r) we must solve the empty space field equations.
By using equation (1.27) we can write the Ricci tensor as [5],
Rµν = Γδµδ,ν − Γδµν,δ + ΓρµδΓδρν − ΓρµνΓδρδ (2.19)
The coefficients of the affine connection in terms of the metric are given by
equation (1.25) (of course making the appropriate index substitution). We
calculate the appropriate Γδµν by first calculating the corresponding metric
components defined in equation (1.25) and then substitute the Γδµν into the
expression for the Ricci tensor given by equation (2.21). The corresponding
non zero components of the Ricci tensor are given as follows [5],
R00 = −A′′
2B+A′
4B
(A′A
+B′
B
)− A′
rB(2.20)
R11 =A′′
2A− A′
4A
(A′A
+B′
B
)− B′
rB(2.21)
R22 =1
B− 1 +
r
2B
(A′A
+B′
B
)(2.22)
R33 = R22sin2θ (2.23)
30
where the primed expression are first and second derivatives with respect
to r. By equating these components equal to zero we are able to solve
equation (2.17) from which we can deduce two expressions for A(r) and
B(r) respectfully, namely [5],
A(r) = α(
1 +k
r
)(2.24)
B(r) =(
1 +k
r
)−1(2.25)
With k representing a constant of integration and α being defined as a con-
stant determined by the product AB. We can determine both of which by
considering what is known as the weak field limit [5] in which we determine
that k represents the mass, M of the object producing the gravitational field.
The weak field limit states that we require A(r)c2
tend to 1 + 2φc2
where φ rep-
resents the gravitational potential presented at the beginning of this chapter
in which case we have φ = −GMr and hence we can write the Schwarzschild
metric in geometrized units (c = G = 1) as [5],
ds2 =(
1− 2M
r
)dt2 −
(1− 2M
r
)−1dr2 − r2dθ2 − r2sin2θdφ2 (2.26)
Reissner-Nordstrom geometry is expressed in an analogous manner. The
solution allows for a charged mass point. To obtain the solution one must
search for static, spherically symmetric solution to the Einstein-Maxwell
equations, which read,
Gµν = 8πTµν (2.27)
31
where Tµν is the Einstein-Maxwell energy momentum tensor and the solution
is given by [1],
ds2 = −∆
r2dt2 +
r2
∆dr2 + r2dθ2 + r2sin2dφ2 (2.28)
Expressions in equation (2.28) are explained rigorously in chapter 4.
32
3 Quantum Mechanics
The need for a chapter on quantum mechanics is again, vital for the un-
derstanding of the purpose of this research project. The approach taken is
the same as that of [4], in the sense that rigorous details and derivations
are not presented. The idea is to outline and discuss some very fundamen-
tal concepts and mathematics that will allow us to approach the problem
outlined in [1] in an understandable and coherent way. We will begin by in-
troducing the famous Schrodinger equation and then progress to solving the
time-independent Schrodinger equation and finally discuss quantum tunnel-
ing and the Wentzel, Kramers, Brillouin (WKB) approximation method.
3.1 The Schrodinger equation
In classical mechanics we apply Newton’s laws to determine different dy-
namical properties of a given system. In particular, we apply Newton’s
second law to determine the position of a particle at any given time, x(t).
With this, we are then at a position to determine said dynamical properties.
Quantum mechanics deals with phenomena on the atomic and sub-atomic
level that does not adhere to the trusted laws of Newton. Due to this, in
the late nineteenth and early twentieth century there was need to formulate
a new theory: quantum mechanics.
Quantum mechanics approaches the problem outlined above in quite a dif-
ferent way. We are concerned with a particles wave function, denoted Ψ(x, t)
33
and it is obtained by solving the famous Schrodinger equation,
ih∂Ψ
∂t= − h2
2m
∂2Ψ
∂x2+ VΨ (3.1)
here we have introduced, h, this is what is known as Planck’s constant. In
the same way that the velocity c of light plays a central role in special rela-
tivity, so too, does Planck’s constant in quantum mechanics [7].
A question arises here, how can a wave function (spread out in space) rep-
resent a particle when we know a particle is localized at a particular point?
To answer this, we consult Born’s statistical interpretation of the wave func-
tion, which says that | Ψ(x, t) |2 is the probability of finding the particle at
point x at a time t, or more precisely [4]
∫ a
b| Ψ(x, t) |2 dx (3.2)
equation (3.2) is the probability of finding the particle between a and b at a
time t. The wave function is defined as complex, and so cannot represent a
probability, as a probability must be real, however | Ψ |2= Ψ∗Ψ, where Ψ∗
is the complex conjugate of Ψ and hence Ψ∗Ψ is real and nonnegative [4].
We will digress at this point and talk further about the probabilistic nature
of the wave function and also discuss what is known as normalization of the
wave function and Schrodinger’s equation.
34
3.2 Normalization
If we consider equation (3.2) and take the integral to be over the entire real
line, then we know from statistics that
∫ ∞−∞| Ψ(x, t) |2 dx = 1 (3.3)
In other words, equation (3.3) is stating that the particle must be some-
where. If Ψ(x, t) is a solution of equation (3.1), then it is not hard to see
that if Ψ(x, t) is in fact a solution then so too is AΨ(x, t), where A is any
complex constant. However, we must pick A in such a way that equation
(3.3) is satisfied and this is what is known as normalization [4]. If we were
to normalize the wave function at time t = 0, how can we be sure that
the wave function will stay normalized as time goes on? The Schrodinger
equation possesses the property that it preserves the normalization of the
wave function. This is important property displaying that the Schrodinger
equation is in fact compatible with the statistical interpretation of the wave
function and the proof is outlined below:
d
dt
∫ ∞−∞| Ψ(x, t) |2 dx =
∫ ∞−∞
∂
∂t| Ψ(x, t) |2 dx =
∫ ∞−∞
∂
∂t(Ψ∗Ψ)dx (3.4)
By using the product rule in equation (3.4) we will have an integral con-
taining ∂Ψ∂t and ∂Ψ∗
∂t , these expressions can be obtained from equation (3.1)
simply by multiplying across by ih and taking the complex conjugate of
equation (3.1) where necessary. Upon substituting these expressions into
equation (3.4) and doing some simple arithmetic and tidying up we obtain
35
the result
d
dt
∫ ∞−∞| Ψ(x, t) |2 dx =
ih
2m
(Ψ∗
∂Ψ
∂x− ∂Ψ∗
∂x
)∣∣∣∞−∞
(3.5)
As x goes to ±∞ the wave function must go to 0 otherwise the wave function
would not be normalizable [4] and hence,
d
dt
∫ ∞−∞| Ψ(x, t) |2 dx = 0 (3.6)
equation (3.6) tells us that the integral is independent of time and that if
the wave function is normalized at t = 0 then it will stay normalized.
3.3 Momentum
We can define the expectation value of x in state Ψ to be the average of
repeated measurements on an ensemble of identically prepared systems [4]
and is denoted mathematically by,
〈x〉 =
∫ ∞−∞
x | Ψ(x, t) |2 dx (3.7)
It’s important to note that this is not the same as saying that the integral
will be equal to the average of repeated measurements of the position of a
particle.
We know that as time goes on then 〈x〉 will change and using simple calculus
we can determine how fast it moves, or in other words the expectation value
of the velocity, that is,
〈v〉 =d〈x〉dt
=
∫ ∞−∞
x∂
∂t(Ψ∗Ψ) dx (3.8)
36
Then by taking the derivative of the product in equation (3.8) and using
integration by parts twice we obtain that,
〈v〉 = − ihm
∫ ∞−∞
Ψ∗∂Ψ
∂xdx (3.9)
Then taking the classical definition for momentum, i.e. p = mv, we can get
an expression for the expectation value for momentum in quantum mechan-
ics,namely,
〈p〉 = md〈x〉dt
= −ih∫ ∞−∞
Ψ∗∂Ψ
∂xdx (3.10)
We can re-write equations (3.7) and (3.10) in two equivalent ways which will
allow us too see more clearly what is meant when discussing what is known
as operators,
〈x〉 =
∫ ∞−∞
Ψ∗ (x) Ψ dx (3.11)
〈p〉 =
∫ ∞−∞
Ψ∗( hi
∂
∂x
)Ψ dx (3.12)
We can describe an operator as an instruction. So in the case of equation
(3.12) the instruction is to differentiate with respect to x and multiply by
hi . The operator (x) can be descirbed as representing position, likewise for
the operator(hi
∂∂x
), this represents the momentum.
There is an important problem presented in [4], which is to prove Ehrenfest’s
theorem,
d〈p〉dt
=⟨− ∂V
∂x
⟩(3.13)
The right hand side of equation (3.13) could also be denoted as the nega-
tive gradient of the potential, namely −∇V for higher dimensions of course.
37
Equation (3.13) states that the laws of classical mechanics hold for expec-
tation values. So in one sense it is a connection between both classical and
quantum mechanics. For the sake of completeness a brief proof is outlined
below,
d〈p〉dt
=h
i
∫ ∞−∞
(∂Ψ∗
∂t
∂Ψ
∂x+ Ψ∗
∂
∂x
∂Ψ
∂t
)dx (3.14)
Equation (3.14) is obtained simply by using equation (3.12) and applying
the product rule and taking the derivative with respect to time. In the same
fashion from the previous section using equation (3.1) we can obtain ex-
pressions for both ∂Ψ∗
∂t and ∂Ψ∂t . Substituting these expressions into equation
(3.14) and applying some simple algebra,arithmetic and integration by parts
we are able to deduce the following,∫ ∞−∞
h2
2m
(∂2Ψ
∂x2
∂Ψ∗
∂x− ∂2Ψ
∂x2
∂Ψ∗
∂x
)dx −
∫ ∞−∞
(Ψ∗
∂V
∂xΨ)dx =
⟨− ∂V
∂x
⟩(3.15)
3.4 Time-independent Schrodinger equation
We are now at a position to solve equation (3.1) for some specified potential
energy function V and determine Ψ(x, t).It is assumed that the potential
does not depend on time i.e. V = V (x). We will use the method of separa-
tion of variables and look for solutions of the form,
Ψ(x, t) = ψ(x)ϕ(t) (3.16)
We can take first and second derivatives of equation (3.16) with respect to
time and position seperately i.e. ∂Ψ∂t = ψ dϕ
dt and ∂2Ψ∂x2 = d2ψ
dx2 ϕ. Substituting
38
these expressions into equation (3.1) we find,
ih1
ϕ
dϕ
dt= − h2
2m
1
ψ
d2ψ
dx2+ V (3.17)
however the left hand side of equation (3.17) depends on t where the right
hand side depends on x. The only way we could have such an equality is if
both sides were constant, we allow either side to be equal to the seperation
constant E [4], and hence we can obtain two equations from equation (3.17)
namely,
ϕ(t) = e−iEth (3.18)
h2
2m
d2ψ
dx2+ V ψ = Eψ (3.19)
Equation (3.19) is what is known as the time-independent Schrodinger equa-
tion. There is of course a constant of intergration in equation (3.18), how-
ever we say that is absorbed into ψ since if we consider equation (3.16) we
are in fact concerned with the product ψ(x)ϕ(t). We can also write the
time-independent Schrodinger equation as an eigenvalue problem. We know
from classical mechanics that the total energy is equal to kinetic energy plus
potential energy, this is also called the Hamiltonian and takes the form,
H =p2
2m+ V (x) (3.20)
We can use our definition for the momentum operator in equation (3.12) to
define the Hamiltonian operator, H and hence write equation (3.19) as,
Hψ = Eψ (3.21)
39
3.5 The WKB approximation
We apply what is known as the WKB approximation to determine approx-
imate solutions to the one dimensional time-independent Schrodinger equa-
tion. We consider two seperate derivations of the WKB approximation one
presented by [7] which utilizes expanding h in a power series and a seperate
method presented by [4]. Please note that the derivation presented by [7]
will be non-rigorous. We are concerned with the situation when our poten-
tial energy function is a slowly varying function of position, x.
Let us first consider equation (3.19) with our potential being constant,
namely V (x) = V0 and we are considering a particle that is traveling with
energy E > V0. Then our solution to equation (3.19) would be written as
linear combinations of the basic plane waves [7],
ψ(x) = Ae±ihp0x (3.22)
where the positive sign represents a wave traveling to the right and the
negative representing a wave traveling to the left and p0 =√
2m(E − V0).
If we consider equation (3.22) for the constant potential then this would lead
us to believe that in the case for a slowly varying potential (in which case
we can write equation (3.19) as d2ψdx2 = p2
h2ψ where p(x) ≡√
2m[E − V (x)] is
defined as the classical momentum of a particle with energy E. ) we would
seek solutions to equation (3.19) to be of the form,
ψ(x) = AeihQ(x) (3.23)
40
substituting equation (3.23) into equation (3.19) we obtain a non-linear or-
dinary differential equation,namely, [7],
− ih
2m
d2Q(x)
dx2+
1
2m
[dQ(x)
dx
]2+ V (x)− E = 0 (3.24)
If we consider equation (3.23) we can see that Q(x) is proportional to h and
hence will disappear as h tends to 0. Using a power series with h as the
parameter we can expand Q(x) in the following power series, [7]
Q(x) = Q0(x) + hQ1(x) +h2
2Q2(x) +
h3
6Q3(x) + . . . (3.25)
Substituting equation (3.25) into equation (3.24) and using the normal
method for solving power series we are able to determine our coefficients.By
solving these equation and again assuming that E > V (x) we find,
Q0(x) = ±∫p(x)dx (3.26)
and,
Q1(x) =i
2ln p(x) (3.27)
It is important to note that when considering the WKB approximation
method we are only concerned with the first two terms of the power se-
ries [7]. It is now possible to formulate the WKB approximation using
equations (3.26) and (3.27), namely,
ψ(x) ≈ A√p(x)
e±ih
∫p(x)dx (3.28)
41
Note that when we are considering the case E > V (x) this is what is known
as a classical region where motion of a particle is confined to a certain range.
Also if E ≈ V this situation defines a classical turning point,
Figure 6: Illustration of classical turning points when E ≈ V (x)
We now consider the approach taken by [4]. Again we are concerned with a
classical region and we will express our wave function as,
ψ(x) = A(x)eiφ(x) (3.29)
differentiating equation (3.29) with respect to x twice and substituting into
d2ψdx2 = p2
h2ψ yeilds,
d2A
dx2+ 2i
dA
dx
dφ
dx+ iA
d2φ
dx2−A
(dφdx
)2= −p
2
h2A (3.30)
42
This can be expressed as two real equations by equating one equation for
the real part and one equation for the imaginary part, namely,
d2A
dx2= A
[(dφdx
)2− p2
h2
](3.31)
d
dx
(A2dφ
dx
)= 0 (3.32)
Clearly the latter has the solution,
A =C√dφdx
(3.33)
Equation (3.31) is where we introduce an assumption which gives rise to the
idea of an approximation. We assume that d2Adx2 ( 1
A) is negligible in relation
to the other terms i.e. (dφdx )2 and p2
h2 [4] and hence we can solve equation
(3.31),
φ(x) = ± ih
∫p(x)dx (3.34)
By substituting equations (3.33) and (3.34) into equation (3.29) it follows
that we have again derived an approximate solution for ψ, namely the WKB
approximation. The general solution (for arbitrary constants A and B) will
be a linear combination of this, namely,
ψ(x) ≈ A√p(x)
eih
∫p(x)dx +
B√p(x)
e−ih
∫p(x)dx (3.35)
3.6 Quantum tunneling
Up to this point we have dictated that E > V (x) which makes our momen-
tum real, however if we considered the case of a where E < V (x) then the
43
WKB approximation reads,
ψ(x) ≈ C√| p(x) |
eih
∫|p(x)|dx (3.36)
We can define the Gamow factor by considering the example of the rectangle
barrier with bumpy top,
To the left of the barrier (x < 0) the wave function is defined by,
ψ(x) = Aeikx +Be−ikx (3.37)
We define A to be the incident amplitude and B to be the reflected amplitude
with k ≡√
2mEh [4]. To the right of the barrier the wave function reads,
ψ(x) = Feikx (3.38)
where F is defined to be the transmission amplitude and the transmission
probability is given by [4],
T =| F |2
| A |2(3.39)
44
Finally within the region (0 ≤ x ≤ a) the wave function is a linear combi-
nation of equation (3.36),
ψ(x) ≈ C√| p(x) |
eih
∫ x0 |p(x)|dx + ψ(x) ≈ D√
| p(x) |eih
∫ x0 |p(x)|dx (3.40)
The amplitudes of the transmitted wave relative to the incident wave are
determined by the total decrease of the exponential over the nonclassical
region [4],
T ≈ e−2γ where γ =1
h
∫ a
0| p(x) | dx (3.41)
This is what is known as the gamow factor.
45
4 Marginally bound particles in Reissner-
Nordstrom
geometry
4.1 Introduction
With regards to applications of the previous chapters, namely General Rel-
ativity and Quantum Mechanics, it is proposed to study two situations re-
garding motion of a charged particle and one situation regarding the motion
of a neutral particle. The idea for the latter is mentioned in [1] as an inter-
esting case to consider and is fully analyzed in the following chapters. We
will study cases for (marginally)bound and unbound particles. To get an
idea of the difference we can imagine a bound particle approaching the black
hole and remaining in orbit similar to that of the planets in our solar system
orbiting the sun. For unbound particles it approaches the black hole from
infinity but can not enter a stable enough orbit and exits the gravitational
field of the black hole to infinity. We will denote the specific energy of the 3-
dimensional motion with ε, where ε = Em . We can define a marginally bound
particle for ε = 1. Physically this means that its total energy equals its rest
energy and the particle falls towards the black hole from rest and escapes
the black hole to infinity. For the case of a bound particle we have 0 < ε < 1.
Cases with ε < 0 are not studied as a part of this research project, however
such particles are known as antiparticles and for our situation it would be a
46
particle with the same mass as a bound or unbound particle however having
an opposite charge.
Consider the following equation which governs radial motion: [1]
r2 dr
dλ= ±
√Er2 − qQr)2 −∆(m2r2 + J2) (4.1)
Where λ = τm denotes proper time per unit mass,E and J are the conserved
energy and the conserved angular momentum, ∆ = r2 − 2Mr + Q2 and q
and m are the charge and the mass of the particle, respectively. Equation
(4.1) can be written in two equivalent ways for the specific kinetic energy of
the particle,namely:
r2 =(ε− qQ
r
)2− ∆
r2
(1 +
j2
r2
)(4.2)
1
2r4r2 = Ar4− (qQε−M)r3− 1
2[j2 +Q2(1− q2)]r2 +Mj2r− 1
2Q2j2 (4.3)
We denote M and Q as the mass and charge of the center of the black
hole. We have A = ε2−12 denoting the specific energy of the one dimensional
motion, also q = qm is the specific charge of the particle [1].We say that
for a Reissner-Norstrom black hole to exist Q2 < M2. This can be seen
by considering when ∆ = 0. By solving this we find that we encounter
coordinate singularities at r = r±, where r± = M ±√M2 −Q2. We must
consider the three situations that arise for our parameters, M and Q. The
first being if Q2 > M2, then r± are both imaginary and this situation will
not be considered. The second being where Q2 < M2, and r± are both real.
As noted in [5] the situation when r = r+ is very similar to a Schwarzschild
47
black hole at r = 2M , that is when a particle enters the surface at r = r+
it will move in the direction of decreasing r and is defined to be an Event
Horizon. For r = r− this is what is known as a Cauchy surface. The
third and final case is when we have M2 = Q2, this is very similar to the
the second situation, however, with the event horizon being at r = M and
∆ > 0 everywhere except at r = M = 0. This is what is known as an
extreme Reissner-Nordstrom black hole [5].
With equation (4.1) representing the radial motion it can be considered as
a one-dimensional non-relativistic effective motion [1]:
1
2r2 + Veff(r) = A, (4.4)
Where the effective potential is represented by,
Veff(r) =qQε−M
r+j2 +Q2(1− q2)
2r2− Mj2
r3+Q2j2
2r4(4.5)
As mentioned above motion is possible for r2 > 0, considering this and
equation (4.2), we can see there are possibilities for the specific kinetic
energy to become negative, which leads to classically forbidden regions, it
helps to study both equations (4.2) and (4.3) simultaneously to analyze the
forbidden regions.
Consider equation (4.2) when ∆ < 0 i.e. between r− and r+ the expression
r2 > 0, if this were not the case then we would have a situation where the
turning points are on the horizons,although not considered it interesting to
mention that in kerr spacetime the horizons are in a forbidden region. If
48
for example ∆ = 0 then r2 =(ε − qQ
r
)2> 0, unless, of course if ε =
r,
then we have the case where ∆ = r2 = 0.
4.2 The Horizons
To begin the analysis when ε = 1, consider first what happens to equation
(4.3) and the different possibilities for the sign changes. Studying the roots
of the above equation will help to determine where classically forbidden
regions occur and also will help in determining where there is a possibility
for quantum tunneling, that is to study when:
1
2r4r2 = (qQ−M)r3 +
1
2[j2 +Q2(1− q2)]r2 −Mj2r +
1
2Q2j2 = 0 (4.6)
Notice the quartic term has now become zero. Using Descartes’ Rule of
signs,a method that is used in determining roots of some polynomial,P (x)
[2]. Depending on how many sign changes there is, Descartes’ Rule indi-
cates the maximum number of real positive roots for some polynomial, if
for example P (x) had n real positive roots the rule states that there is no
possibility for n−1 real positive roots only (n−2,n−4 . . .) i.e it can only go
down in twos from the maximum. With this in mind we can now consider
the different possibilities for the signs and also the maximum number of real
positive roots.
For our first term, that is qQ −M , depending on whether M > qQ or if
M < qQ we can have either a positive or negative sign, likewise with the
49
second term, if (j2+Q2) > Q2q2 or if (j2+Q2) < Q2q2 will be either positive
or negative. This leaves 4 different possibilities for the sign changes which
are, −+−+, −−−+, +−−+ and finally + +−+.
Consider first the case where there is three different sign changes, that is
− + −+. From Descartes’ Rule we know that there is a possibility for a
maximum of 3 real positive roots. The other situations(involving positive
roots) are that there could be 1 positive and two negative roots or 1 positive
with 2 complex.
Figure 7: Illustration (not to scale) of the case with 2 negative roots and 1
positive root
50
Figure 7 represents a situation where we have 1 positive root and two
negative. The case with 1 positive and 2 complex will be considered similar.
The three roots r1, r2, r3 range from left to right with r1 < r2 < r3 and are
at the points where the kinetic energy crosses the r-axis.
For the case above (along with the case for 1 positive root and 2 complex
roots) the event horizon is between 0 < r± < r3. This is because in this
region ∆ < 0 and therefore r2 is positive it cannot be to the right of r3
because r2 is in a forbidden region.
Consider now the case of three positive roots.
51
Figure 8: Illustration (not to scale) of the case with 3 positive roots
For the above case we need to decide whether the horizons are between 0
and r1 or between r2 and r3. Begin by assuming that the horizons r± are
located so that r2 < r± < r3 and we will show that this is not the case. By
making M and Q approach zero it will shift the ∆ to the left, also up as for
a black hole to exist Q2 < M2.
With Q and M tending to zero the roots r1, r2 will be squeezed to zero aswell
as long as r3 does not tend to ∞ [1]. However using Vieta’s Formula [3]
which states that given some cubic polynomial, say, ax3 + bx2 + cx+ d = 0
where a 6= 0 then the roots x1, x2, x3 satisfy:
x1 + x2 + x3 = − ba
x1x2 + x2x3 + x1x3 =c
a
x1x2x3 = −da
By letting a = (qQ−M), b = 12 [j2 +Q2(1− q2)], c = −Mj2, d = 1
2Q2j2,
remembering that for the case in question a < 0. Applying this to Vieta’s
formula we can see that in no situation does r3 tend to ∞. Which leads
us to the conclusion that the horizons can only be between 0 < r± < r1.
This particular case is of interest as there is the possibility for particles from
the black hole to tunnel across the forbidden region from r1 to r2. There
will be gravitationally trapped particles radially oscillating between r2 and
r3 which will give rise to the possibility for quantum tunneling back to the
52
allowed region where the horizons are. [1]
For the cases where there are two sign changes that is + +−+ and +−−+,
Figure 9: Illustration (not to scale) of the case with 2 positive roots and 1
negative
By the same method as above we can determine the position of the event
horizon by first allowing the horizons to be to the right of r3, where again
r1 < r2 < r3. Then using Vieta’s Formula and allowing M and Q to tend
to zero, ∆ will squeeze r2 and r3 to zero as long as r1 does not tend to −∞
this time. Again, applying Vieta’s formula along with the above signs for
our coefficients we see that r3 will not tend to −∞ and again the horizons
53
can only be located between 0 and r2.
The only case left to consider is when the signs are − − −+, that is, with
1 sign change and hence only a possibility for 1 positive root. This case
is covered (or contained within) Figure 7. The only possibility is that the
horizons are located between 0 < r± < r3.
Considering now the case of interest, that is, where we had three positive
roots and the horizons are located between 0 < r± < r1. As seen in [1] we
will be able to let r1 → r+,the Event horizon.
Consider again equation (4.2), for any fixed finite energy ε and for any region
r between 0 and r1 the term(ε− qQ
r
)2will be fixed and finite. Also if we
are given M and Q and at the same region r the term∆
r2will also be fixed
and finite, remembering that ∆ = r2−2Mr+Q2. Consider now the specific
angular momentum of the particle, j. If we allow j2 to increase so that
the term∆
r2
(1 +
j2
r2
)→(ε− qQ
r
)2then our kinetic energy r2 will start to
approach zero. If we take r just to the right of r+ then r2 will approach zero
just to the right of r+ i.e. r1 → r+.
We can approximate r near the event horizon, r+ with a Taylor series.
From the Taylor Series we will be able to get an expression for j2 that will
guarantee, for any energy ε that r1 → r+ as long as the equality is satisfied.
By writing equation (4.2) as
r2 =(
1− qQ
r
)2−(
1− 2M
r+Q2
r2
)(1 +
j2
r2
)(4.7)
54
The Taylor series is expressed as
r2(r) = r2(r+) + r2 ′(r+)(r − r+) + . . .
=(
1− qQ
r+
)2+
2
r2+
[(1− qQ
r+
)qQ−
(1 +
j2
r2+
)(M − Q2
r+
)](r − r+)
(4.8)
If r is indeed a root then r2(r) = 0, which yields, [1]
r1 ' r+
1 +r+
(ε− qQ
r+
)2
2(
1 + j2
r+
)√M2 −Q2 − 2qQ
(ε− qQ
r+
) (4.9)
The Taylor Series was truncated after the linear term, however for this to
be a legitimate truncation we need the fraction in the brackets to be r+
(ε− qQ
r+
)2
2(
1 + j2
r+
)√M2 −Q2 − 2qQ
(ε− qQ
r+
)� 1 (4.10)
The reason for this is as follows, if you consider the opposite of what we
have above i.e. allowing the fraction term be � 1, then when expanded the
expression for r1 in equation (4.7), not only will end up with r+ but will
also have some significant mulitple of r+ to be added, thus letting r1 → r+
would not be valid. However with the fraction being � 1 then adding this
term would still make r1 → r+ a valid argument.
An expression for j2 is still required, by rearranging equation (4.8), we see
that the angular momentum must satisfy [1]
j2 � r+
[(εr+)2 − (qQ)2
2√M2 −Q2
− r+
](4.11)
Now we have to determine the other two positive roots, r2 and r3. These are
found using Vieta’s Formulas. For simplicity allow the fraction in equation
55
(4.8) to be denoted ξ+ i.e. r1 = r+ + ξ+.
We will use the first and second of Vieta’s formulas to find the roots,namely,
r1 + r2 + r3 =j2 +Q2(1− q2)
2(qQ−M)≡ α (4.12)
r1r2 + r2r3 + r1r3 =Mj2
(qQ−M)≡ β (4.13)
Again, α and β are introduced to simplify the equations. Firstly, eliminate
r2 in terms of r1 and r3,that is,
r2 = α− r+ − ξ+ − r3
From here we can get a quadratic in r3 and solve for our root. That is,
r3 'α− ξ+ − r+
2+
1
2
√α2 − 4β + 2αξ + 5ξ2 + (2α− 6ξ − 3r+)r+ (4.14)
Finally, upon substituting r3 back into our expression for r2 we find
r2 'α− ξ+ − r+
2− 1
2
√α2 − 4β + 2αξ + 5ξ2 + (2α− 6ξ − 3r+)r+ (4.15)
56
5 Bound particles in Reissner-Nordstrom
geometry
5.1 The Horizons
Next we consider the case when the specific energy is between 0 < ε <
1. Again we know that there is possibilities for equation (4.2) to become
negative and hence for forbidden regions to exist. This time however we
have the quartic term of equation (4.3) where the coefficient of r4 is strictly
negative. We will have two different situations to discuss, that is for two
positive roots (along with two negative or complex) and with four positive
roots as the different sign combinations are −+++−, −+−+−, −−−+−
and finally − − + + −. As before we will determine the positions of the
horizons. For the case of a maximum of 2 positive roots we have:
57
Figure 10: Illustration (not to scale) of the case with 2 Negative roots and
2 positive.
By labeling the roots from left to right as r1 < r2 < r3 < r4, we can see that
there is a forbidden region to the left of r3 and the only possible situation
is that the horizons are between r3 < r± < r4.
For the remaining case we have the following:
Figure 11: Illustration (not to scale) of the case with 4 positive roots.
Similar to the situation we had for marginally bound particles, we need to
decide whether the horizons are between r1 < r± < r2 or between r3 <
r± < r4. Ideally we would like for the horizons to be between r1 < r± < r2
58
as this will lead to the possibility for quantum tunneling and we will show
that this is the case. Begin by assuming the horizons are located between
r3 < r± < r4. We can show this isn’t the case using Vieta’s Formula and
the fact that we can allow the parameters of ∆, M and Q to vary. Similar
to the situation above with 3 positive roots if we let M and Q tend to 0,
then r1, r2 and r3 will be squeezed to 0. However, r4 is shown to be finite
and by the argument that was used in the previous chapter this tells us that
the horizons cannot be between r3 < r± < r4 and the only other situation
is that it they are located between r1 < r± < r2 .
To determine the 4 positive roots for the above situation we apply the same
method as we did in section (4.3), i.e we allow r1 → r− and r2 → r+. This is
obtained by expanding in a Taylor series about the horizons, r±. From this
we will get two expressions that our angular momentum needs to satisfy.
From our Taylor series we see that
r2,1 ' r±
1 +r±
(ε− qQ
r±
)2
2(
1 + j2
r±
)√M2 −Q2 ∓ 2qQ
(ε− qQ
r±
) (5.1)
Once again the Taylor series was truncated after the linear term and we need
the fraction in equation (5.1) to be� 1. Taking each case individually, that
is for r+ and for r− we get our expressions for j2:
j2 � r±
[(εr±)2 − (qQ)2
2√M2 −Q2
− r±
](5.2)
Thus, guaranteeing that for any energy ε, r1 → r− and r2 → r+ [1]. Note
that in this situation ξ± will denote the fraction in equation (4.1).
59
Once we know the two smaller roots we can use Vieta’s formula to obtain
the remaining two. In this case we will use the first and the third equation.
They are set up as follows:
r1 + r2 + r3 + r4 =2(qQ−M)
ε2 − 1≡ ν (5.3)
r4 = ν − 2M − ξ+ + ξ− − r3 (5.4)
r1r2r3r4 =Q2j2
(ε2 − 1)(5.5)
r3r4 =Q2j2
(ε2 − 1)(Q2 − ξ+ξ− − r+ξ− + r−ξ+)≡ µ (5.6)
Again by substituting our expression for r4 into equation (5.3) we get a
quadratic in r3 and by solving it we find that:
r3 ' −M +ν + ξ− − ξ+
2+
1
2
√−4µ+ (2M − ν − ξ− + ξ+)2 (5.7)
Again by substituting back into our expression for r4 we find that:
r4 = −M +ν + ξ− − ξ+
2− 1
2
√−4µ+ (2M − ν − ξ− + ξ+)2 (5.8)
60
6 Neutral particles in Reissner-Nordstrom geom-
etry
As mentioned in [1] it is interesting to study the situation where we are
given a neutral particle, that is where q = 0. In this situation, equations
(4.2) and (4.3) now read
r2 = ε2 − ∆
r2
(1 +
j2
r2
)(6.1)
1
2r4r2 = Ar4 +Mr3 − 1
2[j2 +Q2]r2 +Mj2r − 1
2Q2j2 (6.2)
Consider the same situations as above where the energy, ε = 1 and when
0 < ε < 1. Also consider the case where ε > 1.
6.1 The Horizons
For marginally bound particles, there is a possibility for a maximum of 3
real positive roots. There is a situation where we could have 1 positive root
and 2 negative/complex roots. In this case, the horizons are to the right of
the biggest root, r3 (using our description we had earlier for the roots i.e.
from left to right with r1 < r2 . . . etc), and therefore there is no forbidden
region to be studied here. However for the situation of 3 positive roots there
does exist a forbidden region between r2 and r3. As in the previous cases we
need to determine the position of the horizons. We can determine this by
allowing the horizons to be to the right of r3 and allowing M,Q → 0. We
can see that this situation cannot be realized as the denominator in each of
61
the cases for Vieta’s formula will be M which is tending to 0 and therefore
the only situation that is realizable is that the horizons are r1 < r± < r2.
Consider now the case of unbound particles (ε > 1). There is a possibility
for a maximum of 3 (or 1) positive roots again. This time however we are
dealing with a quartic, therefore there will be either 3 positive roots with 1
negative or 1 positive root with 3 negative. Dealing with the latter first, we
know that the horizons have to be to the right of the biggest root r4 and
there is no forbidden region here. For the other case we are faced with the
same situation as in [1] and we can deduce that once again our horizons are
between r2 < r± < r3.
For the case of a bound particle, there is a possibility for a maximum of
4 positive roots or a possibility of 2 positive and 2 negative. We have the
same situation as we encountered previously and can deduce from the results
above that for the case of two positive roots our horizons are positioned
between r3 and r4. Finally for the first case with 4 positive roots our horizons
are located between r1 < r± < r2.
Using the same argument as before, we will be able to expand r2 in a Taylor
series about the horizons r±. From this we can deduce that we will able to
let r1 → r− and r2 → r+. For the case of an unbound particle where our
horizons are located between r2 and r3 we can ignore the negative root r1
as it is of no interest. Hence we can still say that for ε > 1 we can allow
r1 → r− and r2 → r+.
62
The Taylor series yields the result
r2,1 ' r±
1 +
ε2
r±
2r2±
(1 + j2
r2±
)√M2 −Q2 ∓ 2ε
(6.3)
Again the Taylor series was truncated after the linear term and we need the
fraction in the brackets in equation (6.3) to be � 1. Again using equation
(6.3) we can get an expression for the angular momentum that must be
satisfied to guarantee for any energy ε, r1 → r− and r2 → r+ that is
j2 �r2±2
[( ε
2
r±+ ε)r2
±√M2 −Q2
− 1
](6.4)
From here we can determine the roots using Vieta’s formula in each case.
For simplicity reasons denote the fraction in equation (6.3) as ϕ± so that
r2,1 ' r± ± ϕ±.
For ε = 1 we get that
r3 '12 [j2 +Q2]
M− 2M + ϕ− − ϕ+ (6.5)
For 0 < ε < 1 using Vieta’s Formula we get that r4 = 2Mε2−1−2M−ϕ++ϕ−−r3
Substituting this expression into r1 r2 r3 r4 = −Q2j2
ε2−1, then dividing by r1r2
and allowing Q2j2
(ε2−1)r1r2≡ ζ, we get a quadratic in r3, when solved we get
r3 =1
2
[Υ
(1 +
√4ζ
Υ2
)](6.6)
where Υ = 2Mε2−1
− 2M − ϕ+ + ϕ− from this we can see that
r4 =1
2
[Υ
(1−
√4ζ
Υ2
)](6.7)
63
Finally for the case ε > 1, we are disregarding the negative root and again
using Vieta’s Formula tells us that
r3 = − 2M
ε2 − 1− 2M + ϕ− − ϕ+ (6.8)
Determining the Gamow factor for the computed roots in chapter 4,5 and 6
follows an analogous manner presented in [1]. It is obtained by allowing the
limits of the integral to be the roots that were found and solved respectfully.
Conclusion
The project has been thoroughly enjoyed from start to finish and due to
it a keen interest in the area of Theoretical physics has been developed.
Understanding the language of tensors has been one of the most rewarding
mathematical accomplishments to date.
As can be seen material from both quantum mechanics and general relativ-
ity has been introduced and extended as much as possible and some famous
derivations have been presented, namely the Schwarzschild solution. We
have completely extended the research paper of Dr. Emil Prodanov by de-
termining the roots of the kinetic energy for marginally bound and bound
particles, not to mention incorporating a new section on neutral particles
that could be added to Dr. Emil’s Prodanov’s research paper.
There is a possibility for further research, namely the understanding of ro-
tating black holes i.e. Kerr-Newman black holes. It has also been proposed
64
by Dr. Emil Prodanov that there is possibilities for further studies on the re-
search papers of Demetrios Christodoulo and Remo Ruffini which is entitled
”Reverse transformations of a charged black hole”.
References
[1] E.M. Prodanov,Evaporation Screening, Scattering and Quantum Tunneling near
the event horizons of Schwarzschild and Reissner-Nordstrom Black Holes (2013).
[2] E.W. Weisstein, ”Descartes’ Sign Rule.” From MathWorld–A Wolfram Web Re-
source. http://mathworld.wolfram.com/DescartesSignRule.html
[3] E.W. Weisstein, ”Vieta’s Formulas.” MathWorld-A Wolfram Web Resource.
http://mathworld.wolfram.com/VietasFormulas.html
[4] D.J Griffiths, Introduction to Quantum Mechanics, Pearson Prentice Hall (2005).
[5] M.P Hobson, G.P Efstathiou and A.N Lasenby, General Relativity: An Introduc-
tion for Physicists, Cambridge University Press (2006).
[6] R.A d’Inverno , Introducing Einstein’s Relativity , Oxford University Press (1992).
[7] B.H Bransden and C.J Joachain, Introduction to Quantum Mechanics, Addison
Wesley Longman Limited (1989).
[8] L. Ryder, Introduction to General Relativity, Cambridge University Press (2009).
65