#  Mimicking Black Holes in General Relativity

Thomas Berry

###### Abstract

The central theme of this thesis is the study and analysis of black hole
mimickers. The concept of a black hole mimicker is introduced, and various
mimicker spacetime models are examined within the framework of classical
general relativity. The mimickers examined fall into the classes of regular
black holes and traversable wormholes under spherical symmetry. The regular
black holes examined can be further categorised as static spacetimes, however
the traversable wormhole is allowed to have a dynamic (non-static) throat.
Astrophysical observables are calculated for a recently proposed regular black
hole model containing an exponential suppression of the Misner–Sharp quasi-
local mass. This same regular black hole model is then used to construct a
wormhole via the “cut-and-paste” technique. The resulting wormhole is then
analysed within the Darmois-Israel thin-shell formalism, and a linearised
stability analysis of the (dynamic) wormhole throat is undertaken. Yet another
regular black hole model spacetime is proposed, extending a previous work
which attempted to construct a regular black hole through a quantum
“deformation” of the Schwarzschild spacetime. The resulting spacetime is again
analysed within the framework of classical general relativity.

In addition to the study of black hole mimickers, I start with a brief
overview of the theory of special relativity where a new and novel result is
presented for the combination of relativistic velocities in general directions
using quaternions. This is succeed by an introduction to concepts in
differential geometry needed for the successive introduction to the theory of
general relativity. A thorough discussion of the concept of spacetime
singularities is then provided, before analysing the specific black hole
mimickers discussed above.

## Acknowledgments

I would like to thank everyone who has supported me throughout my degree and
in completing this thesis. During this time I have been lucky enough to be
supervised by Professor Matt Visser, whose tutelage, support, and insight has
been invaluable. The amount of time you have dedicated as a mentor to me has
not gone unnoticed, and no doubt the many priceless lessons and skills you
have taught me will stay with me for the rest of my life.

Along this line, I would also like to thank Alex Simpson. You’ve been a great
friend, and I look forward to the many future collaborations bound to come. I
am also thankful for the work you have put into our co-authored papers over
the past year, many of which have ended up in this thesis in some capacity.

I would also like to thank Francisco Lobo. Particularly for your insight and
input into chapter 7.

Special thanks goes to my partner Cassidy and to my family. Your endless
support and guidance throughout my time at university has made this thesis
possible.

This work was directly supported by a Victoria University of Wellington
Master’s by Thesis Scholarship, and also indirectly supported by the Marsden
fund, via a grant administered by the Royal Society of New Zealand.

###### Contents

  1. Acknowledgments
  2. 1 Introduction
  3. 2 Special relativity
    1. 2.1 Minkowski space and Einstein’s postulates
    2. 2.2 Lorentz transformations
  4. 3 Combination of velocities using quaternions
    1. 3.1 Quaternions
    2. 3.2 Combining two 3-velocities
      1. 3.2.1 Velocities in the ($x$,$y$)-plane
      2. 3.2.2 General 3-velocities
      3. 3.2.3 Calculating the Wigner angle
    3. 3.3 Combining three 3-velocities
      1. 3.3.1 Combining 3 half-velocities: $\mathbf{w}_{(1\oplus 2)\oplus 3}$
      2. 3.3.2 Combining 3 half-velocities: $\mathbf{w}_{1\oplus(2\oplus 3)}$
      3. 3.3.3 Combining 3 half-velocities: (Non)-associativity
      4. 3.3.4 Specific non-coplanar example
    4. 3.4 Summary
  5. 4 General relativity
    1. 4.1 Manifolds
    2. 4.2 The metric tensor
    3. 4.3 Covariant differentiation and connections
    4. 4.4 Curvature
    5. 4.5 Geodesics
    6. 4.6 Einstein’s equivalence principle and field equations
      1. 4.6.1 The equivalence principle
      2. 4.6.2 Einstein’s field equations and the stress-energy tensor
      3. 4.6.3 Energy conditions
  6. 5 Black hole mimickers: extensions to Einstein’s theory
    1. 5.1 Singularities
    2. 5.2 Regular black holes
    3. 5.3 Wormholes
  7. 6 Regular black hole with asymptotically Minkowski core
    1. 6.1 Introducing the spacetime
    2. 6.2 Geodesics and the effective potential
    3. 6.3 Photon spheres
      1. 6.3.1 Existence of photon spheres
      2. 6.3.2 Stability _versus_ instability for circular photon orbits
      3. 6.3.3 Turning points
    4. 6.4 Timelike circular orbits
      1. 6.4.1 Existence of circular timelike orbits
      2. 6.4.2 Stability _versus_ instability for circular timelike orbits
      3. 6.4.3 Intersection of ESCO and photon sphere
      4. 6.4.4 Explicit result for the angular momentum
    5. 6.5 Regge–Wheeler and quasi-normal modes
      1. 6.5.1 Regge–Wheeler potential
      2. 6.5.2 Quasi-normal modes
    6. 6.6 Summary
  8. 7 From regular black hole to thin-shell wormhole
    1. 7.1 Thin–shell wormhole framework
      1. 7.1.1 Background
      2. 7.1.2 Construction
      3. 7.1.3 Energy conditions in the bulk spacetime
      4. 7.1.4 Four-velocity, unit normal, and extrinsic curvature of the throat
      5. 7.1.5 Surface stress–energy
      6. 7.1.6 Stability analysis
    2. 7.2 Examples
      1. 7.2.1 Symmetrically vanishing $a$ parameter; asymmetric mass $m_{+}\neq m_{-}$.
      2. 7.2.2 Mirror symmetry: Both $m_{+}=m_{-}$ and $a_{+}=a_{-}$
      3. 7.2.3 Specific asymmetry: $m_{+}\neq m_{-}$ while $a_{+}=a_{-}=a$
      4. 7.2.4 Specific asymmetry: $a_{+}\neq a_{-}$ while $m_{+}=m_{-}$
    3. 7.3 Summary
  9. 8 General class of “quantum deformed” regular black holes
    1. 8.1 Introducing the spacetime
    2. 8.2 Geometric analysis
      1. 8.2.1 Metric components
      2. 8.2.2 Event horizons
      3. 8.2.3 Christoffel symbols of the second kind
      4. 8.2.4 Orthonormal components
      5. 8.2.5 Riemann tensor
      6. 8.2.6 Ricci tensor
      7. 8.2.7 Ricci scalar
      8. 8.2.8 Einstein tensor
      9. 8.2.9 Weyl tensor
      10. 8.2.10 Weyl scalar
      11. 8.2.11 Kretschmann scalar
    3. 8.3 Surface gravity and Hawking temperature
    4. 8.4 Stress-energy tensor
    5. 8.5 Energy conditions
      1. 8.5.1 Null energy condition
      2. 8.5.2 Weak energy condition
      3. 8.5.3 Strong energy condition
      4. 8.5.4 Dominant energy condition
    6. 8.6 ISCO and photon sphere analysis
      1. 8.6.1 Photon orbits
      2. 8.6.2 ISCOs
      3. 8.6.3 Summary
    7. 8.7 Regge–Wheeler analysis
    8. 8.8 Summary
  10. 9 Conclusions
    1. 9.1 Combination of relativistic velocities using quaternions
    2. 9.2 Regular black hole with asymptotically Minkowski core
    3. 9.3 Regular black hole to thin-shell wormhole
    4. 9.4 General class of “quantum deformed” regular black holes
    5. 9.5 Overall summary and outlook
  11. Appendix  List of publications

###### List of Figures

  1. 2.1 Light cone.
  2. 2.2 Co-moving observers.
  3. 5.1 Black hole.
  4. 5.2 Inter-universe wormhole.
  5. 6.1 Asymptotically Minkowski core regular black hole:Photon sphere.
  6. (a)
  7. (b)
  8. 6.2 Asymptotically Minkowski core regular black hole:Photon sphere zoomed in.
  9. (a)
  10. (b)
  11. 6.3 Asymptotically Minkowski core regular black hole:Existence of circular orbits.
  12. (a)
  13. (b)
  14. 6.4 Asymptotically Minkowski core regular black hole:Photon sphere and ESCO.
  15. (a)
  16. (b)
  17. 6.5 Asymptotically Minkowski core regular black hole:Photon sphere and ESCO stability regions.
  18. (a)
  19. (b)
  20. 6.6 Asymptotically Minkowski core regular black hole:ESCO zoomed in.
  21. (a)
  22. (b)
  23. 7.1 Thin-shell wormhole stability analysis: $a_{\pm}=0$.
  24. 7.2 Thin-shell wormhole stability analysis: Mirror symmetry.
  25. 7.3 Thin-shell wormhole stability analysis: Mirror symmetry contour plot.
  26. 7.4 Thin-shell wormhole stability analysis: $F_{2}(x,y)<0$.
  27. 7.5 Thin-shell wormhole stability analysis: $F_{2}(x,y)<-1$.
  28. 7.6 Thin-shell wormhole stability analysis: $a_{+}=a_{-}$ but $m_{+}\neq m_{-}$.
  29. (a) $\alpha=0.7$
  30. (b) $\alpha=0.9$
  31. 7.7 Thin-shell wormhole stability analysis: $a_{+}=a_{-}$ but$m_{+}\neq m_{-}$ contour plots.
  32. (a) $\alpha=0.7$
  33. (b) $\alpha=0.9$
  34. 7.8 Thin-shell wormhole stability analysis: $a_{+}\neq a_{-}$ but $m_{+}=m_{-}$.
  35. (a) $\beta=1.2$
  36. (b) $\beta=1.4$
  37. 7.9 Thin-shell wormhole stability analysis: $a_{+}\neq a_{-}$ but$m_{+}=m_{-}$ contour plots.
  38. (a) $\beta=1.2$
  39. (b) $\beta=1.4$
  40. 8.1 Quantum deformed regular black hole: Plots of the NEC

##  Chapter 1 Introduction

General relativity is currently our best theoretical model for gravity and has
made many predications that have been verified by astronomical observations.
Perhaps one of the best well-known predictions made by the theory is that of
black holes. However, within the framework of classical general relativity,
black holes contain a spacetime singularity at their core. As shown by the
Penrose singularity theorems, under suitable physically-reasonable assumptions
these singularities are unavoidable consequences of theory once a trapped
surface forms. These singularities present a host of issues from a physical
standpoint. Critically, the theory of general relativity ceases to be
predictive at the singularity. This is clearly an issue, as one of the most
important properties of any physical theory is that it can make accurate
predictions about the universe we live in.

In spite of this, in many cases singularities are not an issue for every-day
physics as they are hidden by an event horizon, and as such physics outside of
the black hole is often unaffected. However, there are regimes where the
presence of a singularity could be detected (at least, in theory). Examples of
such are the information loss paradox, or during the very last stages of a
black hole’s evaporation.

Although the issues surrounding singularities have not been resolved, there is
convincing astronomical data which suggests that general relativity is
extremely accurate in its description of black holes. Thus, we either have to
accept the reality of spacetime singularities at the centre of black holes,
thereby accepting that some of out most basic notions of physics no longer
hold, or we have to accept that physical black holes are different to their
mathematical counterparts. A proposed solution to this issue is the concept of
black hole ‘mimickers’. These are objects that are sufficiently similar to
black holes so that they agree with astronomical observations but, crucially,
do not contain singularities at their cores.

It is commonly believed that singularities will not be present in a consistent
theory of quantum gravity. However, it is unlikely that such a theory will be
achieved any time in the near future, and so black hole mimickers provide an
effective, classical approach to the resolution of singularities in general
relativity. In this thesis, I investigate a variety of black hole mimickers
within the scope of classical general relativity and discuss their validity as
real, physical alternatives to black holes as predicted by classical general
relativity.

In chapter 2, the reader is reminded about some of the main results of the
theory of special relativity, providing a starting point for discussing the
more general theory in later chapters. Familiarity with the theory of special
relativity is assumed herein.

Chapter 3 then studies the special-relativistic combination of velocities
using the quaternion number system. A new and novel result for combining
relativistic velocities is proposed and thoroughly investigated within the
framework of the theory of special relativity.

In chapter 4, we move on to discuss the theory of general relativity. The
mathematical framework needed to understand the results in later chapters is
provided and discussed alongside the key postulates which lead Einstein to the
development of the theory111The term “postulate” is often used to simply mean
a collection of very good experimental evidence. Einstein’s postulates have a
very solid experimental foundation.. This chapter is intended as a review of
classical general relativity, and as such no new results will be presented.

Chapter 5 discusses, in detail, singularities in general relativity. This
includes a rigorous definition of a singularity, as well as a general
discussion of the Penrose singularity theorems. This is then used to introduce
the idea of black hole mimickers, including regular black hole and traversable
wormhole spacetimes.

In chapter 6, a specific regular black hole model is analysed within the
context of classical general relativity. Specifically, the location of
timelike and null circular geodesics are investigated in detail, the spin-
dependent Regge-Wheeler potential is calculated, and a first-order WKB
approximation of the quasi normal modes is completed. The novel regular black
hole under investigation results in a far richer phenomenology than standard
(non-regular) black holes.

In chapter 7, the same regular black hole spacetime is used to construct a
thin-shell traversable wormhole via the “cut-and-paste” technique, thereby
constructing yet another black hole mimicker. A linearised stability analysis
is conducted for the wormhole throat and a series of specific examples are
investigated wherein the spacetime parameters are changed (and allowed to be
different) between the two manifolds used in the cut-and-paste thin-shell
construction. Again, the novelty of the spacetime results in a rich
phenomenology of potential interest to observational astronomers and
astrophysicists.

Chapter 8 introduces a family of regular black hole spacetimes, which is
analysed within the framework of classical general relativity. The family of
regular black hole spacetimes were inspired by a (non-regular) black hole
spacetime which arises as a quantum modification to the Schwarzschild black
hole.

Finally, in chapter 9 we provide a brief summary of the main results in this
thesis and provide an outlook on the future of the field and avenues of
potential future research.

##  Chapter 2 Special relativity

Before we introduce Einsteins theory of general relativity, we will provide a
brief overview of his theory of Special relativity, and provide some new
results on the combination of relativistic velocities. Note that this is not
intended to be a complete overview of the theory, and many results will be
assumed to be prior knowledge to the reader.

Special relativity is where one typically first encounters the notion of
spacetime: one time dimension $t$, and three space dimensions $(x,y,z)$
combined into one four-dimensional space representing the collective set of
points $\\{(t,x,y,z)\\}$. In Newtonian mechanics, there is no limit on how
fast an object may move through space, and notions such as ‘the length of an
object’, or ‘how fast a clock ticks’ are the same no matter who makes the
measurements. In special relativity, however, the situation is much different
– the ‘length of an object’ or ‘how fast a clock ticks’ is dependent on the
relative speed of the observer making the measurements.

The theory of special relativity is built from two ingredients:

  1. (1)

Minkowski space: the mathematical ‘space’ representing spacetime in which all
observers move along their ‘worldlines’.

  2. (2)

Einstein’s postulates: the ‘laws’ of physics, or mathematical ‘axioms’ (i.e.
summary of experimental evidence) which we use to derive equations of motion
governing how objects move throughout Minkowski space.

With these two ingredients, we can completely describe the kinematics of an
object, no matter how fast it is travelling, so long as we do not consider the
effects of gravity. Bringing gravity into the picture induces many additional
complications which will be captured by the full theory of general relativity
discussed in chapter 4. That is not to say that special relativity is not a
good, or useful theory. In many cases, one can ignore the effects of gravity
and work completely inside the framework of special relativity. In fact, this
is built into general relativity at a fundamental level: so long as you are
working in sufficiently small areas of spacetime, you need not worry about any
gravitational effects. Thus, for now, let us assume that we are working only
within the framework of special relativity and worry about gravitational
effects in later chapters.

###  2.1 Minkowski space and Einstein’s postulates

Special relativity is a theory of how objects move throughout spacetime. Thus,
first of all, we need to define a notion of what we mean by ‘spacetime’.

######  Definition 1.

Minkowski spacetime is the pair $(\mathbb{R}^{4},\eta)$, where $\eta$ is the
quadratic form given in matrix representation by

$\eta=\begin{pmatrix}-1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}.$
(2.1)

As one would expect, Minkowski spacetime is four-dimensional, and so we can
use it to construct a theory of spacetime. At this point, we need introduce
our laws of physics from which we can deduce equations of motion. These laws
are given by Einstein’s postulates:

######  Postulate 1 (Principle of relativity111It is worth noting that
postulate 1 holds for mechanical processes even in Galilean relativity. It is
postulate 2 which is unique to special relativity.).

The laws of physics are the same in all inertial frames of reference.

######  Postulate 2 (Invariance of $c$).

The speed of light has the same value in all inertial frames of reference.

Like all laws of physics, these postulates have been rigorously experimentally
tested, and so far have proven to be an accurate reflection of how the
universe works. Details of the experimental verification of special relativity
can be found in Ref. [96].

We first define a four-vector $X$ which has components
$X^{\mu}=(x^{0},\vec{x})=(ct,x,y,z)$ (note the components have dimension
length). The four-vector $X$ represents an event in spacetime (i.e. a point in
Minkowski space). Suppose now that we have two events $X_{1}$ and $X_{2}$ such
that a light ray passes from event 1 to event 2. From postulate 2, the
difference between these two events is $\Delta X=X_{2}-X_{1}=(c\Delta
t,\Delta\vec{x})^{T}$. Since a light ray connects the two events, we have that
$\absolutevalue{\Delta\vec{x}}/\absolutevalue{\Delta t}=c$. That is,
$\absolutevalue{\Delta\vec{x}}^{2}-(x^{0})^{2}=0$. We can write this as matrix
multiplication in terms of the quadratic form $\eta$ as $\eta(\Delta X,\Delta
X)\equiv(\Delta X)^{T}\eta\Delta X=0$.

This is our first notion of the idea of _causality_ : as nothing can travel
faster than a light signal, event 1 can only _cause_ event 2 if at least a
light signal can join the two points in spacetime (i.e. if $\eta(\Delta
X,\Delta X)=0$). Note also that if $\eta(\Delta X,\Delta X)=0$, then we also
have $\eta(\Delta(-X),\Delta(-X))=0$, and so (at least in this purely
mathematical framework) if event 1 can cause event 2, so too can event 2 cause
event 1. In most _physical_ situations, however, we will require that time
flows in the positive direction. Then we will have a well-defined time-
ordering of events 1 and 2, thereby removing any ambiguity about which event
‘caused’ the other. The set of points

$C(X_{1})=\\{X_{2}\mid\eta(\Delta X,\Delta X)=0\\},$ (2.2)

forms the surface of a double-cone with apex at $X_{1}$, called the “light-
cone” of $X_{1}$ (see figure 2.1). Any event $X_{2}$ that can be reached from
$X_{1}$ by a signal travelling slower than the speed of light will lie inside
the surface of the light-cone. That is $X_{2}$ is in the future of $X_{1}$.
(Note, if $X_{2}$ is in the past of $X_{1}$, it will lie in its past light-
cone). Conversely, any event that cannot be reached by $X_{1}$ without
travelling faster than the speed of light will lie outside of the light-cone
of event 1. That is, event 2 is _not_ in the future of event 1.

Figure 2.1: The light cone of an event in spacetime. Time runs vertically up
the page.

We can place the separation of two events into three distinct classes:

  1. (1)

Timelike separated: event 1 can reach event 2 by slower-than-light travel;
$\eta(\Delta X,\Delta X)<0$.

  2. (2)

Lightlike (null) separated: event 1 can only reach event 2 by speed of light
travel; $\eta(\Delta X,\Delta X)=0$.

  3. (3)

Spacelike separated: event 1 cannot reach event 2 without travelling faster
than the speed of light; $\eta(\Delta X,\Delta X)>0$.

###  2.2 Lorentz transformations

Postulate 2 leads to some interesting results that disagree with Newtonian
kinematics. Consider, for example, what happens if two observers $A$ and $B$
are travelling toward each other, each with a speed $v$ as measured by some
third observer (see figure 2.2). According to $A$, they will see $B$ moving
toward them at a speed $v+v=2v$; and vice versa for $B$. This is a perfectly
acceptable scenario within the framework of Newtonian kinematics. Now suppose
that the velocities $v$ approach a reasonable fraction of the speed of light,
say $v=0.75c$. We know from postulate 2 that nothing can travel faster than
$v=c$, and so A cannot possibly see B travelling toward them at a speed
$0.75c+0.75c=1.5c$. This suggests that the simple non-relativistic Galilean
transformations of Newtonian kinematics no longer work at relativistic speeds,
and that we need a new set of transformations which incorporate postulate 2 in
a fundamental way. These are known as the Lorentz transformations.

Figure 2.2: Two observers travelling towards each other in the non-
relativistic case. Each observer sees the other travelling towards them at
speed $2v$.

One of the biggest differences between Lorentz transformations and the
Galilean transformations of Newtonian kinematics is that the Lorentz
transformations also transform the time coordinate between the two reference
frames. For the case of co-linear velocities (along the $x$-axis), they take
the particularly simple form222Derivations can be found in any of the standard
introductory textbooks on special relativity.

$\displaystyle ct^{\prime}$ $\displaystyle=\gamma(ct-\beta x),$ $\displaystyle
x^{\prime}$ $\displaystyle=\gamma(x-\beta ct),$ $\displaystyle y^{\prime}$
$\displaystyle=y,$ $\displaystyle z^{\prime}$ $\displaystyle=z;$ (2.3)

where $\beta=v/c$, $\gamma=1/\sqrt{1-\beta^{2}}$, and the primed and un-primed
coordinates represent the coordinates in the different frames. Note that so
long as we are only working with two coordinate frames, we can always define
the coordinate axes in such a way as to ensure that they are co-linear, and so
equations (2.3) hold. This does become an issue, however, if one is dealing
with more than two coordinate systems. In such a case, a more complicated form
of the Lorentz transformations are needed:

$\displaystyle ct^{\prime}$
$\displaystyle=\gamma\left(ct-\vec{\beta}\cdot\vec{r_{\parallel}}\right),$
$\displaystyle\vec{r}^{\prime}_{\parallel}$
$\displaystyle=\gamma(\vec{r}_{\parallel}-\vec{\beta}ct),$
$\displaystyle\vec{r}^{\prime}_{\perp}$ $\displaystyle=\vec{r}_{\perp},$ (2.4)

where $\vec{r}=\vec{r}_{\parallel}+\vec{r}_{\perp}$ is the position vector
connecting the two frames, decomposed into its components parallel and
perpendicular to the relative velocity $\vec{\beta}=\vec{v}/c$. The Lorentz
transformations (2.4) are known as Lorentz boosts, or pure Lorentz
transformations, and form a subset of the group of all Lorentz
transformations, the Lorentz group.

Mathematically, the Lorentz group is isomorphic to $\text{O}(1,3)$, the
orthogonal group of one time and three space dimensions that preserves the
space-time interval

$\eta(\Delta X,\Delta X)=-t^{2}+x^{2}+y^{2}+z^{2}.$ (2.5)

Note that, here and hereafter, we adopt units where the speed of light is set
to unity. It is clear from this description that rotations of space-time are
included in the Lorentz group, as well as the more familiar Lorentz boosts
(2.4). In fact, the pure Lorentz transformations do not even form a subgroup
of the Lorentz group as, in general, the composition of two boosts $B_{1}$ and
$B_{2}$ is not another boost but in fact a boost and a rotation
$B_{12}R_{12}=B_{1}B_{2}$; whilst $B_{21}R_{21}=B_{2}B_{1}$. This rotation,
known as the Wigner rotation, was first discovered by Llewellyn Thomas in 1926
whilst trying to describe the Zeeman effect from a relativistic view-point
[139], and was more fully analysed by Eugene Wigner in 1939 [165]. (For more
recent discussions see [54, 55, 92, 109, 116].)

It is well–known that the composition of Lorentz transformations is non-
commutative. That is, applying two successive boosts $B_{1}$ and $B_{2}$ in
different orders results in the same final boost, $B_{12}=B_{21}$, but
different rotations, $R_{12}\neq R_{21}$. In the context of the combination of
two velocities $\vec{v}_{1}$ and $\vec{v}_{2}$, this means that the final
speed is the same no matter the order we combine the velocities,
$\norm{\vec{v}_{1}\oplus\vec{v}_{2}}=\norm{\vec{v}_{2}\oplus\vec{v}_{1}}$, but
the final directions they point in are different $\hat{v}_{1\oplus
2}\neq\hat{v}_{2\oplus 1}$. Although not immediately obvious, the angle
between $\vec{v}_{1}\oplus\vec{v}_{2}$ and $\vec{v}_{2}\oplus\vec{v}_{1}$ is
in fact the Wigner angle $\Omega$ [109].

In the following chapter, we will provide a way of using Lorentz
transformations to derive results for the combination of relativistic
velocities using a very special representation of the Lorentz group: the
quaternions.

##  Chapter 3 Combination of velocities using quaternions

Hamilton first described the quaternions in the mid-1800s, primarily with a
view to finding algebraically simple ways to handle 3-dimensional rotations.
With the advent of special relativity in 1905, and noting the manifestly
4-dimensional nature of quaternions once one adds a real part, multiple
authors have tried to interpret special relativity in an intrinsically
quaternionic fashion [49, 64, 100, 113, 125, 126, 140].

Despite technical success in applying quaternions to special relativity, the
use of quaternions in this subject has never really gained all that much
traction in the physics community. Perhaps one of the reasons for this is that
there are a number of sub-optimal notational choices in Silberstein’s original
work [125, 126], and the fact that there is no generally accepted way of using
quaternions to represent Lorentz transformations, with many different authors
employing their own quite distinct methods [48, 49, 64, 100, 113, 125, 126,
140]. Even in more recent, post-millennial, articles on “quaternionic special
relativity” there is considerable disagreement on notational choices [60, 61,
68, 168].

In this chapter, we shall introduce what we feel is a particularly simple and
straightforward method for combining relativistic 3-velocities using
quaternions. All of the interesting features due to non-commutativity
properties of non-collinear boosts are implicitly and rather efficiently dealt
with by the non-commutative algebra of quaternions. The method is based on an
extension of an analysis by Giust, Vigoureux, and Lages [65, 85], who (because
they were working with the usual complex numbers) were essentially limited to
motion in 2-space; their formalism is not really well-adapted to general
motions in 3-space. Related constructions can also be found in references [60,
61].

One could instead try to deal with the non-commutativity of the Lorentz
transformations by adapting the general formalism of the
Baker–Campbell–Hausdorff (BCH) theorem [2, 66, 145, 146, 147]. Unfortunately,
the general BCH formalism applied to this problem very quickly becomes
intractable, and we have found that the specifics of the quaternion formalism
yield much more useful and tractable results. Similarly, since the full
symmetry group of the Maxwell equations is the conformal extension of the
Poincare group, it is sometimes useful, (when looking at pure electromagnetic
effects), to work with this conformal extension. However physical observers,
(physical clocks and physical rulers), break the conformal invariance, and to
even meaningfully define 3-velocities one needs to restrict attention to the
Poincare group. We shall go even further and take translation invariance
(spatial and temporal homogeneity) for granted, and focus more specifically on
the Lorentz group.

###  3.1 Quaternions

The quaternions are numbers that can be written in the form
$a+b\,\mathbf{i}+c\,\mathbf{j}+d\,\mathbf{k}$, where $a,$ $b,$ $c,$ and $d$
are real numbers; and $\mathbf{i},$ $\mathbf{j},$ and $\mathbf{k}$ are the
quaternion units which satisfy the famous relation

$\mathbf{i}^{2}=\mathbf{j}^{2}=\mathbf{k}^{2}=\mathbf{i}\mathbf{j}\mathbf{k}=-1.$
(3.1)

They form a four–dimensional number system that is generally treated as an
extension of the complex numbers. We shall define the quaternion conjugate of
the quaternion $\mathbf{q}=a+b\,\mathbf{i}+c\,\mathbf{j}+d\,\mathbf{k}$ to be
$\mathbf{q}^{\star}=a-b\,\mathbf{i}-c\,\mathbf{j}-d\,\mathbf{k}$, and define
the norm of $\mathbf{q}$ to be
$\mathbf{qq}^{\star}=\absolutevalue{\mathbf{q}}^{2}=a^{2}+b^{2}+c^{2}+d^{2}\in\mathbb{R}$.
This allows us to evaluate the quaternion inverse as
$\mathbf{q}^{-1}=\mathbf{q}^{\star}/|\mathbf{q}|^{2}$.

Trying to define a “norm” as $\mathbf{q}^{2}=a^{2}-b^{2}-c^{2}-d^{2}$, while
superficially more “relativistic”, violates the usual mathematical definition
of “norm”, and furthermore is not useful when it comes to evaluating the
quaternion inverse $\mathbf{q}^{-1}$.

For current purposes we focus our attention on pure quaternions. That is,
quaternions of the form $a\,\mathbf{i}+b\,\mathbf{j}+c\,\mathbf{k}$. Many
quaternion operations become much simpler when we are dealing with pure
quaternions. For example, the product of two pure quaternions $\mathbf{p}$ and
$\mathbf{q}$ is given by
$\mathbf{p}\mathbf{q}=-\vec{p}\cdot\vec{q}+(\vec{p}\crossproduct\vec{q})\cdot(\mathbf{i},\mathbf{j},\mathbf{k})$,
where, in general, we shall set
$\mathbf{v}=\vec{v}\cdot(\mathbf{i},\mathbf{j},\mathbf{k})$. From this, we
obtain the useful relations

$\left[\mathbf{p},\mathbf{q}\right]=2(\vec{p}\crossproduct\vec{q})\cdot(\mathbf{i},\mathbf{j},\mathbf{k}),\mbox{\quad
and\quad}\\{\mathbf{p},\mathbf{q}\\}=-2\,\vec{p}\cdot\vec{q}.$ (3.2)

A notable consequence of (3.2) is
$\mathbf{q}^{2}=-\vec{q}\cdot\vec{q}=-q^{2}=-\absolutevalue{\mathbf{q}}^{2}$.
There is a natural isomorphism between the space of pure quaternions and
$\mathbb{R}^{3}$ given by

$\mathbf{i}\mapsto\hat{x},\quad\mathbf{j}\mapsto\hat{y},\quad\mathbf{k}\mapsto\hat{z};$
(3.3)

where $\hat{x},\,\hat{y},$ and $\hat{z}$ are the standard unit vectors in
$\mathbb{R}^{3}$.

One of the most common uses for quaternions today (2021) is in the computer
graphics community, where they are used to compactly and efficiently generate
rotations in 3-space. Indeed, if
$\mathbf{q}=\cos(\theta/2)+\hat{\mathbf{u}}\sin(\theta/2)$ is an arbitrary
unit quaternion and $\mathbf{v}$ is the image of a vector in $\mathbb{R}^{3}$
under the isomorphism (3.3), then the mapping
$\mathbf{v}\mapsto\mathbf{q}\mathbf{v}\mathbf{q}^{-1}$ rotates $\mathbf{v}$
through an angle $\theta$ about the axis defined by $\hat{\mathbf{u}}$. The
mapping $\mathbf{v}\mapsto\mathbf{q}\mathbf{v}\mathbf{q}^{-1}$ is called
quaternion conjugation by $\mathbf{q}$. Furthermore, an extension of the
quaternions, the dual quaternions, are used in the field of theoretical robot
kinematics, due to their ability to efficiently handle rotations and
translations of vectors [14].

###  3.2 Combining two 3-velocities

In the paper by Giust, Vigoureux, and Lages [65], see also [85], (and the
somewhat related discussion in reference [60]), a method is developed to
compactly combine relativistic velocities in two space dimensions, and by
extension, coplanar relativistic velocities in 3 space dimensions. In the
following subsection, we first provide a short summary of their approach, and
then in the next subsection extend their method to general non-coplanar
3-velocities.

####  3.2.1 Velocities in the ($x$,$y$)-plane

The success of this Giust, Vigoureux, and Lages approach relies on the angle
addition formula for the hyperbolic tangent function,

$\tanh(\xi_{1}+\xi_{2})=\frac{\tanh{\xi_{1}}+\tanh{\xi_{2}}}{1+\tanh{\xi_{1}}\tanh{\xi_{2}}}.$
(3.4)

The tanh function is a natural choice for combining relativistic velocities
since it is limited to the interval $\left[-1,1\right]$. Indeed, using the
rapidity $\xi$ defined by $v=\tanh(\xi)$, we can easily combine collinear
relativistic speeds using equation (3.4). In order to use this for the
combination of non-collinear relativistic 2-velocities, we replace each
2-velocity $\vec{v}$ by the complex number

$V=\tanh({\xi}/{2})\;\mathrm{e}^{i\varphi}.$ (3.5)

Here $\xi$ is the rapidity of the velocity $\vec{v}$, and $\varphi$ gives the
orientation of $\vec{v}$ according to some observer in the plane defined by
$\vec{v}_{1}$ and $\vec{v}_{2}$. Giust, Vigoureux, and Lages then define the
composition law $\oplus$ for coplanar velocities $\vec{v}_{1}$ and
$\vec{v}_{2}$ by

$W=\tanh\frac{\xi}{2}\;\mathrm{e}^{i\varphi_{1\oplus 2}}=V_{1}\oplus
V_{2}=\frac{V_{1}+V_{2}}{1+\overline{V_{2}}\,V_{1}}=\frac{\tanh\frac{\xi_{1}}{2}\,\mathrm{e}^{i\varphi_{1}}+\tanh\frac{\xi_{2}}{2}\,\mathrm{e}^{i\varphi_{2}}}{1+\tanh\frac{\xi_{2}}{2}\,\mathrm{e}^{-i\varphi_{2}}\tanh\frac{\xi_{1}}{2}\,\mathrm{e}^{i\varphi_{1}}},$
(3.6)

where $\overline{V}$ is the standard complex conjugate of $V$. By using
$\xi/2$ instead of $\xi$ in equations (3.5) and (3.6), we are actually dealing
with the “relativistic half–velocities”, $\tanh(\xi/2)$, (sometimes called the
“symmetric velocities”), where

$w=\tanh(\xi/2);\qquad v=\tanh(\xi)={2w\over 1+w^{2}}.$ (3.7)

That is:

$w=\tanh\left(\frac{1}{2}\tanh^{-1}(v)\right)=\frac{v}{1+\sqrt{1-v^{2}}}.$
(3.8)

Using equations (3.4) and (3.6) we can easily retrieve the real velocity from
the half-velocity by using $\oplus$ operator:
$v=\tanh\xi=\tanh{\xi/2}\oplus\tanh{\xi/2}=w\oplus w$.

As an aside, it is worth noting that these half-velocities are often first
encountered when working with Loedel diagrams [1]. Standard spacetime diagrams
(often called Minkowski diagrams) have orthogonal spacetime axes in a given
rest frame. As such, the axes of other reference frames (moving with a
relative velocity to the rest frame) form an acute angle. This asymmetry
between reference frames in a Minkowski diagram is often misleading, as
postulate 1 enforces the equivalence of any two frames of reference. Loedel
diagrams are constructed in a third reference frame travelling at the relative
half-velocity of the two initial frames, and so the symmetry between the two
frames of reference is manifest.

In terms of the half velocities, we can write the combined velocity as

$w_{1\oplus 2}\;\mathrm{e}^{i\varphi_{1\oplus
2}}=\frac{w_{1}\,\mathrm{e}^{i\varphi_{1}}+w_{2}\,\mathrm{e}^{i\varphi_{2}}}{1+w_{1}w_{2}\,\,\mathrm{e}^{i(\varphi_{1}-\varphi_{2})}}.$
(3.9)

The $\oplus$ addition law is non-commutative, which is most easily seen by
first setting $\theta=\varphi_{2}-\varphi_{1}$, then $\Omega=\varphi_{1\oplus
2}-\varphi_{2\oplus 1}$, and finally observing that the ratio

$\mathrm{e}^{i\Omega/2}=\frac{1+\tanh\frac{\xi_{1}}{2}\tanh\frac{\xi_{2}}{2}\mathrm{e}^{i\theta}}{1+\tanh\frac{\xi_{1}}{2}\tanh\frac{\xi_{2}}{2}\mathrm{e}^{-i\theta}}={1+w_{1}w_{2}\mathrm{e}^{i\theta}\over
1+w_{1}w_{2}\mathrm{e}^{-i\theta}}$ (3.10)

is not equal to unity for non–zero $\theta$, meaning that
$\Omega=\varphi_{1\oplus 2}-\varphi_{2\oplus 1}$ is non-zero.

The angle $\Omega=\varphi_{1\oplus 2}-\varphi_{2\oplus 1}$ is in fact the
Wigner angle $\Omega$, so an expression for this angle can be obtained by
taking the real and imaginary parts of equation (3.10):

$\tan\frac{\Omega}{2}=\frac{\tanh\frac{\xi_{1}}{2}\,\tanh\frac{\xi_{2}}{2}\,\sin\theta}{1+\tanh\frac{\xi_{1}}{2}\,\tanh\frac{\xi_{2}}{2}\,\cos\theta}={w_{1}w_{2}\sin\theta\over
1+w_{1}w_{2}\cos\theta}.$ (3.11)

This expression does not explicitly appear in reference [65] though something
functionally equivalent, in the form
$\Omega=2\;\hbox{arg}(1+w_{1}w_{2}\mathrm{e}^{i\theta})$, appears in reference
[85].

The $\oplus$ law can be applied to any number of coplanar velocities by
iteration:

$W=(((V_{1}\oplus V_{2})\oplus\dots\oplus V_{n-1})\oplus V_{n}).$ (3.12)

Thus, it would be desirable to cleanly extend this formalism to general three-
dimensional velocities. Note that the order of composition is important, as we
shall see in more detail below, the $\oplus$ operation is in general _not_
associative.

####  3.2.2 General 3-velocities

We now extend the result of Giust, Vigoureux, and Lages to arbitrary
3-velocities in three dimensions.

##### Algorithm

Suppose we have a velocity $\vec{v}_{i}$ in the $(x,y)$-plane, represented by
the pure quaternion
$\mathbf{w}_{i}=\tanh(\xi_{i}/2)\hat{\mathbf{n}}_{i}=\tanh(\xi_{i}/2)\;(\mathbf{i}\cos\theta_{i}+\mathbf{j}\sin\theta_{i})$.
Using the rules for quaternion multiplication, we can write this as

$\mathbf{w}_{i}=\tanh(\xi_{i}/2)\;(\cos\theta_{i}+\mathbf{k}\sin\theta_{i})\mathbf{i}.$
(3.13)

The term inside the brackets now looks very similar to what would be a natural
extension of the exponential function to the quaternions,
$\mathrm{e}^{\mathbf{k}\theta}=\cos\theta+\mathbf{k}\sin\theta$. To formalise
this, we define the exponential of a quaternion $\mathbf{q}$ by the power
series

$\mathrm{e}^{\mathbf{q}}=\sum_{k=0}^{\infty}\frac{\mathbf{q}^{k}}{k!}.$ (3.14)

To calculate an explicit formula for equation (3.14), we first consider the
case of a pure quaternion $\mathbf{u}$. We know from section 3.1 that for a
pure quaternion we have $\mathbf{u}^{2}=-\absolutevalue{\mathbf{u}}^{2}$, and
so we find
$\mathbf{u}^{3}=-\absolutevalue{\mathbf{u}}^{2}\mathbf{u},\,\mathbf{u}^{4}=\absolutevalue{\mathbf{u}}^{4},$
and so on. Thus, we can compute directly from the definition (3.14):

$\displaystyle\mathrm{e}^{\mathbf{u}}$ $\displaystyle=\left(1-{1\over
2!}\absolutevalue{\mathbf{u}}^{2}+{1\over
4!}\absolutevalue{\mathbf{u}}^{4}-\dots\right)+{\mathbf{u}\over\absolutevalue{\mathbf{u}}}\left(\absolutevalue{\mathbf{u}}-{1\over
3!}\absolutevalue{\mathbf{u}}^{3}+{1\over
5!}\absolutevalue{\mathbf{u}}^{5}-\dots\right)$
$\displaystyle=\cos\absolutevalue{\mathbf{u}}+\hat{\mathbf{u}}\sin\absolutevalue{\mathbf{u}}.$
(3.15)

Following the same procedure above, we find the exponential of a pure unit
quaternion $\hat{\mathbf{u}}$ and real number $\phi$ to be

$\mathrm{e}^{\hat{\mathbf{u}}\phi}=\cos\phi+\hat{\mathbf{u}}\sin\phi.$ (3.16)

This nice result reflects the expression for the exponential of a complex
number.

We can now extend this result to any arbitrary quaternion
$\mathbf{q}=a+\mathbf{u}$ by noting that the real number $a$ commutes with all
the terms in $\mathbf{u}$, thereby allowing us to write
$\mathrm{e}^{\mathbf{q}}=\mathrm{e}^{a}\mathrm{e}^{\mathbf{u}}$, where
$\mathrm{e}^{\mathbf{u}}$ has the same form as equation (3.15). Explicitly,

$\mathrm{e}^{\mathbf{q}}=\mathrm{e}^{a}(\cos\absolutevalue{\mathbf{u}}+\hat{\mathbf{u}}\;\sin\absolutevalue{\mathbf{u}}).$
(3.17)

The exponential of a quaternion possesses many of the same properties as the
exponential of a complex number. Two particularly useful ones we use below are

$\left(\mathrm{e}^{\hat{\mathbf{u}}\phi}\right)^{\star}=\mathrm{e}^{-\hat{\mathbf{u}}\phi}=\cos\phi-\hat{\mathbf{u}}\sin\phi,\mbox{\quad
and\quad}\absolutevalue{\mathrm{e}^{\hat{\mathbf{u}}\phi}}=1.$ (3.18)

Using these results, we are now justified in writing

$\mathbf{w}_{i}=\tanh(\xi_{i}/2)\,\mathrm{e}^{\mathbf{k}\theta_{i}}\,\mathbf{i}=w_{i}\,\mathrm{e}^{\mathbf{k}\theta_{i}}\,\mathbf{i}$
(3.19)

for our velocity in the $(x,y)$-plane.

Building on this result, we now find it appropriate to define the $\oplus$
operator for general 3-velocities, $\mathbf{w}_{1}=w_{1}\mathbf{\hat{n}_{1}}$
and $\mathbf{w}_{2}=w_{2}\mathbf{\hat{n}_{2}}$, by:

$\mathbf{w}_{1\oplus
2}=\mathbf{w}_{1}\oplus\mathbf{w}_{2}=(1-\mathbf{w}_{1}\mathbf{w}_{2})^{-1}(\mathbf{w}_{1}+\mathbf{w}_{2}).$
(3.20)

The usefulness of this definition is best understood by looking at a few
examples.

##### Example: Parallel velocities

We consider two parallel velocities $\vec{v}_{1}$ and $\vec{v}_{2}$
represented by the quaternions

$\mathbf{w}_{1}=\tanh\frac{\xi_{1}}{2}\,\mathbf{\hat{n}}\mbox{\quad
and\quad}\mathbf{w}_{2}=\tanh\frac{\xi_{2}}{2}\,\mathbf{\hat{n}},$ (3.21)

respectively. Our composition law (3.20) then gives

$\displaystyle\mathbf{w}_{1\oplus 2}$
$\displaystyle=\left(1+\tanh\frac{\xi_{1}}{2}\,\tanh\frac{\xi_{2}}{2}\right)^{-1}\left(\tanh\frac{\xi_{1}}{2}\,\mathbf{\hat{n}}\,+\,\tanh\frac{\xi_{2}}{2}\,\mathbf{\hat{n}}\right)$
$\displaystyle=\frac{\tanh\frac{\xi_{1}}{2}+\tanh\frac{\xi_{2}}{2}}{1+\tanh\frac{\xi_{1}}{2}\,\tanh\frac{\xi_{2}}{2}}\;\mathbf{\hat{n}}$
$\displaystyle=\tanh\left(\frac{\xi_{1}+\xi_{2}}{2}\right)\;\mathbf{\hat{n}},$
(3.22)

which is equivalent to

$\mathbf{w}_{1\oplus 2}={w_{1}+w_{2}\over 1+w_{1}w_{2}}\;\mathbf{\hat{n}},$
(3.23)

and hence, also equivalent to the well–known result for the relativistic
composition of two parallel velocities,

$\vec{v}_{1}\oplus\vec{v}_{2}=\frac{v_{1}+v_{2}}{1+v_{1}v_{2}}\;\hat{n}.$
(3.24)

##### Example: Perpendicular velocities in the $x$–$y$ plane

We now consider two perpendicular velocities in the $x$–$y$ plane. By rotating
around the $z$ axis, without loss of generality they can be taken to be given
by

$\mathbf{w}_{1}=w_{1}\mathbf{i},\quad\mathbf{w}_{2}=w_{2}\,\mathbf{j},$ (3.25)

where we have written $\tanh(\xi_{1}/2)=w_{1}$ and $\tanh(\xi_{2}/2)=w_{2}$
for brevity.

Our composition law then gives a combined velocity of

$\mathbf{w}_{1\oplus
2}=(1-w_{1}w_{2}\mathbf{i}\mathbf{j})^{-1}(w_{1}\mathbf{i}+w_{2}\mathbf{j})=\frac{w_{1}(1-w_{2}^{2})\mathbf{i}+w_{2}(1+w_{1}^{2})\mathbf{j}}{1+w_{1}^{2}w_{2}^{2}},$
(3.26)

which is definitely not commutative. In contrast the norm is symmetric:

$\absolutevalue{\mathbf{w}_{1\oplus
2}}^{2}=\frac{w_{1}^{2}(1-w_{2}^{2})^{2}+w_{2}^{2}(1+w_{1}^{2})^{2}}{(1+w_{1}^{2}w_{2}^{2})^{2}}=\frac{w_{1}^{2}+w_{2}^{2}}{1+w_{1}^{2}w_{2}^{2}}.$
(3.27)

Here the $\mathbf{w}_{i}$ are the “relativistic half–velocities”
$w_{i}=\tanh(\xi_{i}/2)$, so the full velocities are

$\absolutevalue{\mathbf{v}_{i}}^{2}=\absolutevalue{\mathbf{w}_{i}\oplus\mathbf{w}_{i}}^{2}=\frac{4w_{i}^{2}}{(1+w_{i}^{2})^{2}},$
(3.28)

and so give a final speed of

$\absolutevalue{\mathbf{v}_{1\oplus
2}}^{2}=\frac{4\left(w_{1}^{2}+w_{2}^{2}\right)}{\big{(}1+w_{1}^{2}w_{2}^{2}\big{)}\left[1+\frac{w_{1}^{2}+w_{2}^{2}}{1+w_{1}^{2}w_{2}^{2}}\right]^{2}}=\frac{4\big{(}w_{1}^{2}+w_{2}^{2}\big{)}\big{(}1+w_{1}^{2}w_{2}^{2}\big{)}}{\left[\big{(}1+w_{1}^{2}\big{)}\big{(}1+w_{2}^{2}\big{)}\right]^{2}}.$
(3.29)

The non-quaternionic result for the composition of two perpendicular
velocities is [109]

$\norm{\vec{v}_{1\oplus
2}}^{2}=v_{1}^{2}+v_{2}^{2}-v_{1}^{2}v_{2}^{2}=1-\big{(}1-v_{1}^{2}\big{)}\big{(}1-v_{2}^{2}\big{)}.$
(3.30)

Thus, we find

$\displaystyle\norm{\vec{v}_{1\oplus 2}}^{2}$
$\displaystyle=\frac{4w_{1}^{2}}{(1+w_{1}^{2})^{2}}+\frac{4w_{2}^{2}}{(1+w_{2}^{2})^{2}}-\frac{16w_{1}^{2}w_{2}^{2}}{(1+w_{1}^{2})^{2}(1+w_{2}^{2})^{2}}$
$\displaystyle=\frac{4(w_{1}^{2}+w_{2}^{2})(1+w_{1}^{2}w_{2}^{2})}{\left[(1+w_{1}^{2})(1+w_{2}^{2})\right]^{2}}.$
(3.31)

And so our composition law $\oplus$ gives the standard result for the
composition of two perpendicular velocities in the $x$–$y$ plane.

##### Example: Perpendicular velocities in general

For general perpendicular velocities $\vec{v}_{1}$ and $\vec{v}_{2}$ the
easiest way of proceeding is to simply rotate to point $\vec{v}_{1}$ along the
$x$-axis and $\vec{v}_{2}$ along the $y$-axis, and just copy the argument
above. If one wishes to be more direct then simply define

$\mathbf{w}_{1}=w_{1}\;\widehat{\mathbf{w}_{1}},\qquad\mathbf{w}_{2}=w_{2}\;\widehat{\mathbf{w}_{2}};\qquad\widehat{\mathbf{w}_{3}}=\widehat{\mathbf{w}_{1}}\;\widehat{\mathbf{w}_{2}}.$
(3.32)

In view of the mutual orthogonality of the vectors $\hat{w}_{1}$,
$\hat{w}_{2}$, and $\hat{w}_{3}$, the unit quaternions
$(\widehat{\mathbf{w}_{1}},\widehat{\mathbf{w}_{2}},\widehat{\mathbf{w}_{3}})$
obey exactly the same commutation relations as
$(\mathbf{i},\mathbf{j},\mathbf{k})$. Thence

$\mathbf{w}_{1\oplus
2}=(1-w_{1}w_{2}\widehat{\mathbf{w}_{1}}\widehat{\mathbf{w}_{2}})^{-1}(w_{1}\widehat{\mathbf{w}_{1}}+w_{2}\widehat{\mathbf{w}_{2}})=\frac{w_{1}(1-w_{2}^{2})\widehat{\mathbf{w}_{1}}+w_{2}(1+w_{1}^{2})\widehat{\mathbf{w}_{2}}}{1+w_{1}^{2}w_{2}^{2}}.$
(3.33)

This now leads to exactly the same results as above; there was no loss of
generality inherent in working in the $x$–$y$ plane.

##### Example: Reduction to Giust–Vigoureux–Lages result in the $x$–$y$ plane

It is important to note that our composition law $\oplus$ reduces to the
composition law of Giust, Vigoureux, and Lages when dealing with planar
velocities in the $x$–$y$ plane. As above, we define general velocities in the
($\mathbf{i},\mathbf{j}$)-plane by
$\mathbf{w}_{1}=\tanh(\xi_{1}/2)\mathrm{e}^{\mathbf{k}\phi_{1}}\mathbf{i}$,
and
$\mathbf{w}_{2}=\tanh(\xi_{2}/2)\mathrm{e}^{\mathbf{k}\phi_{2}}\mathbf{i}$,
then, using our composition law (3.20), we find

$\mathbf{w}_{1\oplus
2}=\left(1-\tanh\frac{\xi_{1}}{2}\mathrm{e}^{\mathbf{k}\phi_{1}}\mathbf{i}\;\tanh\frac{\xi_{2}}{2}\mathrm{e}^{\mathbf{k}\phi_{2}}\mathbf{i}\right)^{-1}\left(\tanh\frac{\xi_{1}}{2}\mathrm{e}^{\mathbf{k}\phi_{1}}\mathbf{i}+\tanh\frac{\xi_{2}}{2}\mathrm{e}^{\mathbf{k}\phi_{2}}\mathbf{i}\right).$
(3.34)

But, noting that
$\tanh(\xi_{2}/2)\mathrm{e}^{\mathbf{k}\phi_{2}}\mathbf{i}=\tanh(\xi_{2}/2)\mathbf{i}\,\mathrm{e}^{-\mathbf{k}\phi_{2}}$
and $\mathbf{i}^{2}=-1$, we can re-write this as

$\mathbf{w}_{1\oplus
2}=\left(1+\tanh\frac{\xi_{1}}{2}\mathrm{e}^{\mathbf{k}\phi_{1}}\tanh\frac{\xi_{2}}{2}\mathrm{e}^{-\mathbf{k}\phi_{2}}\
\right)^{-1}\left(\tanh\frac{\xi_{1}}{2}\mathrm{e}^{\mathbf{k}\phi_{1}}+\tanh\frac{\xi_{2}}{2}\mathrm{e}^{\mathbf{k}\phi_{2}}\right)\mathbf{i}.$
(3.35)

Now, writing

$\mathbf{w}_{1\oplus 2}=\tanh(\xi_{1\oplus
2}/2)\,\mathrm{e}^{\mathbf{k}\phi_{1\oplus 2}}\,\mathbf{i}$ (3.36)

we can cancel out the trailing $\mathbf{i}$, to obtain

$\tanh{\xi_{1\oplus 2}\over 2}\;\mathrm{e}^{\mathbf{k}\phi_{1\oplus
2}}=\left(1+\tanh\frac{\xi_{1}}{2}\mathrm{e}^{\mathbf{k}\phi_{1}}\tanh\frac{\xi_{2}}{2}\mathrm{e}^{-\mathbf{k}\phi_{2}}\
\right)^{-1}\times\\\
\left(\tanh\frac{\xi_{1}}{2}\mathrm{e}^{\mathbf{k}\phi_{1}}+\tanh\frac{\xi_{2}}{2}\mathrm{e}^{\mathbf{k}\phi_{2}}\right).$
(3.37)

This expression now only contains $\mathbf{k}$, so everything commutes, and we
can write

$w_{1\oplus 2}\,\mathrm{e}^{\mathbf{k}\phi_{1\oplus
2}}={w_{1}\,\mathrm{e}^{\mathbf{k}\phi_{1}}+w_{2}\,\mathrm{e}^{\mathbf{k}\phi_{2}}\over
1+w_{1}\,\mathrm{e}^{\mathbf{k}\phi_{1}}\,w_{2}\,\mathrm{e}^{-\mathbf{k}\phi_{2}}}$
(3.38)

which is equivalent to the result of Giust, Vigoureux, and Lages.

##### Example: Composition in general directions

For general velocities $\vec{v}_{1}$ and $\vec{v}_{2}$ the easiest way of
proceeding is to simply rotate to put $\vec{v}_{1}$ and $\vec{v}_{2}$ in the
the $x$–$y$ plane, and just copy the Giust–Vigoureux–Lages argument [65]
above. If one wishes to be more direct, then simply define

$\mathbf{w}_{1}=w_{1}\;\widehat{\mathbf{w}_{1}},\qquad\qquad\mathbf{w}_{2}=w_{2}\;\widehat{\mathbf{w}_{2}};\qquad\qquad\widehat{\mathbf{w}_{3}}={[\widehat{\mathbf{w}_{1}},\;\widehat{\mathbf{w}_{2}}]\over|\,[{\widehat{\mathbf{w}_{1}},\;\widehat{\mathbf{w}_{2}}]\,|}}.$
(3.39)

If $\widehat{\mathbf{w}_{1}}$ is not parallel to $\widehat{\mathbf{w}_{2}}$,
then $\widehat{\mathbf{w}_{3}}$ is well defined and perpendicular to both
$\widehat{\mathbf{w}_{1}}$ and $\widehat{\mathbf{w}_{2}}$. With these
definitions one can now write

$\widehat{\mathbf{w}_{2}}=\exp(\phi\widehat{\mathbf{w}_{3}})\;\widehat{\mathbf{w}_{1}}.$
(3.40)

Then, following the discussion above, we see

$\displaystyle\mathbf{w}_{1\oplus 2}$
$\displaystyle=(1+w_{1}\,w_{2}\,\mathrm{e}^{-\widehat{\mathbf{w}_{3}}\phi})^{-1}{(w_{1}\,\widehat{\mathbf{w}_{1}}+w_{2}\,\widehat{\mathbf{w}_{2}})}$
$\displaystyle=(1+w_{1}\,w_{2}\,\mathrm{e}^{-\widehat{\mathbf{w}_{3}}\phi})^{-1}{(w_{1}+w_{2}\,\mathrm{e}^{\widehat{\mathbf{w}_{3}}\phi})\widehat{\mathbf{w}_{1}}}.$
(3.41)

From this we can extract

$w_{1\oplus 2}\;\mathrm{e}^{\widehat{\mathbf{w}_{3}}\phi_{1\oplus
2}}=(1+w_{1}\,w_{2}\,\mathrm{e}^{-\widehat{\mathbf{w}_{3}}\phi})^{-1}{(w_{1}+w_{2}\,\mathrm{e}^{\widehat{\mathbf{w}_{3}}\phi})}={(w_{1}+w_{2}\,\mathrm{e}^{\widehat{\mathbf{w}_{3}}\phi})\over(1+w_{1}\,w_{2}\,\mathrm{e}^{-\widehat{\mathbf{w}_{3}}\phi})}.$
(3.42)

Finally,

$w_{1\oplus 2}\;\mathrm{e}^{\widehat{\mathbf{w}_{3}}\phi_{1\oplus
2}}={(w_{1}+w_{2}\,\mathrm{e}^{\widehat{\mathbf{w}_{3}}\phi})\over(1+w_{1}\,w_{2}\,\mathrm{e}^{-\widehat{\mathbf{w}_{3}}\phi})}.$
(3.43)

This finally is a fully explicit result for general velocities $\vec{v}_{1}$
and $\vec{v}_{2}$, which is manifestly in agreement with the
Giust–Vigoureux–Lages results [65].

##### Uniqueness of the composition law

Finally, we might note that the expression for the composition law (3.20) is
not unique. For example, by considering the power-series of
$(1-\mathbf{w}_{1}\mathbf{w}_{2})^{-1}$, we can re-write equation (3.20) as

$\mathbf{w}_{1\oplus
2}=(1-\mathbf{w}_{1}\mathbf{w}_{2})^{-1}(\mathbf{w}_{1}+\mathbf{w}_{2})=\sum_{n=0}^{\infty}(\mathbf{w}_{1}\mathbf{w}_{2})^{n}(\mathbf{w}_{1}+\mathbf{w}_{2}).$
(3.44)

But, as $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$ are pure quaternions, both
$\mathbf{w}_{1}^{2}$ and $\mathbf{w}_{2}^{2}$ are real numbers, and so commute
with $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$. Thus,

$\mathbf{w}_{1\oplus
2}=\sum_{n=0}^{\infty}(\mathbf{w}_{1}\mathbf{w}_{2})^{n}\mathbf{w}_{1}+\sum_{n=0}^{\infty}(\mathbf{w}_{1}\mathbf{w}_{2})^{n}\mathbf{w}_{2}=\mathbf{w}_{1}\sum_{n=0}^{\infty}(\mathbf{w}_{2}\mathbf{w}_{1})^{n}+\mathbf{w}_{2}\sum_{n=0}^{\infty}(\mathbf{w}_{2}\mathbf{w}_{1})^{n}.$
(3.45)

Consequently we find that our composition law can also be written as

$\mathbf{w}_{1\oplus
2}=(\mathbf{w}_{1}+\mathbf{w}_{2})\sum_{n=0}^{\infty}(\mathbf{w}_{2}\mathbf{w}_{1})^{n}=(\mathbf{w}_{1}+\mathbf{w}_{2})(1-\mathbf{w}_{2}\mathbf{w}_{1})^{-1}.$
(3.46)

Indeed, one could use either equation (3.20) or equation (3.46) as the
definition of the composition law $\oplus$. Nonetheless, we will stick with
the convention given in (3.20).

####  3.2.3 Calculating the Wigner angle

In this section we obtain an expression for the Wigner angle for general
3-velocities using our composition law (3.20). Our calculations are obtained
using the result that the Wigner angle is the angle between the velocities
$\mathbf{w}_{1\oplus 2}$ and $\mathbf{w}_{2\oplus 1}$.

We first note

$|\mathbf{w}_{1\oplus 2}|=|\mathbf{w}_{2\oplus
1}|=|1-\mathbf{w}_{1}\mathbf{w}_{2}|^{-1}|\mathbf{w}_{1}+\mathbf{w}_{2}|={|w_{1}\mathbf{\hat{n}}_{1}+w_{2}\mathbf{\hat{n}}_{2}|\over|1-w_{1}w_{2}\;\mathbf{\hat{n}}_{1}\mathbf{\hat{n}}_{2}|}.$
(3.47)

Thus, setting $\cos\theta=\vec{n}_{1}\cdot\vec{n}_{2}$ we explicitly verify

$|\mathbf{w}_{1\oplus 2}|=|\mathbf{w}_{2\oplus
1}|=\sqrt{w_{1}^{2}+w_{2}^{2}+2w_{1}w_{2}\cos\theta\over
1+w_{1}^{2}w_{2}^{2}+2w_{1}w_{2}\cos\theta}.$ (3.48)

Now note that because $|\mathbf{w}_{1\oplus 2}|=|\mathbf{w}_{2\oplus 1}|$ it
follows that $\left(\mathbf{w}_{1\oplus 2}\right)\left(\mathbf{w}_{2\oplus
1}\right)^{-1}$ is a unit norm quaternion. In fact, defining

$\mathrm{e}^{\mathbf{\Omega}}=\left(\mathbf{w}_{1\oplus
2}\right)\left(\mathbf{w}_{2\oplus 1}\right)^{-1},$ (3.49)

we will soon see that the norm of $\bf{\Omega}$ is the Wigner angle.
Explicitly,

$\mathrm{e}^{\mathbf{\Omega}}=\left((1-\mathbf{w}_{1}\mathbf{w}_{2})^{-1}(\mathbf{w}_{1}+\mathbf{w}_{2})\right)\left((1-\mathbf{w}_{2}\mathbf{w}_{1})^{-1}(\mathbf{w}_{2}+\mathbf{w}_{1})\right)^{-1}.$
(3.50)

But for a product of quaternions we have
$(\mathbf{q}_{1}\mathbf{q}_{2})^{-1}=\mathbf{q}_{2}^{-1}\mathbf{q}_{1}^{-1}$,
and so this reduces to

$\mathrm{e}^{\mathbf{\Omega}}=(1-\mathbf{w}_{1}\mathbf{w}_{2})^{-1}(1-\mathbf{w}_{2}\mathbf{w}_{1}).$
(3.51)

Now

$\mathbf{w}_{1}\mathbf{w}_{2}=-w_{1}w_{2}\cos\theta+(\vec{w}_{1}\times\vec{w}_{2})\cdot(\mathbf{i},\mathbf{j},\mathbf{k}).$
(3.52)

Let us define

${\hat{\Omega}}={(\vec{w}_{1}\times\vec{w}_{2})\over|\vec{w}_{1}\times\vec{w}_{2}|};\qquad\hbox{so}\qquad{\hat{w}_{1}}\times{\hat{w}_{2}}=\sin\theta\;\;{\hat{\Omega}}.$
(3.53)

Then setting
$\mathbf{\hat{\Omega}}={\hat{\Omega}}\cdot(\mathbf{i},\mathbf{j},\mathbf{k})$
so that $\mathbf{\Omega}=\Omega\,\mathbf{\hat{\Omega}}$, we have:

$\mathbf{w}_{1}\mathbf{w}_{2}=-w_{1}w_{2}(\cos\theta-\sin\theta\;\mathbf{\hat{\Omega}})=-w_{1}w_{2}\;\mathrm{e}^{-\theta\mathbf{\hat{\Omega}}}.$
(3.54)

Consequently the Wigner angle satisfies

$\mathrm{e}^{\mathbf{\Omega}}=\mathrm{e}^{\Omega\,\mathbf{\hat{\Omega}}}=\left(1+w_{1}w_{2}\,\mathrm{e}^{-\theta\mathbf{\hat{\Omega}}}\right)^{-1}\left(1+w_{1}w_{2}\,\mathrm{e}^{\theta\mathbf{\hat{\Omega}}}\right)={1+w_{1}w_{2}\,\mathrm{e}^{\theta\mathbf{\hat{\Omega}}}\over
1+w_{1}w_{2}\,\mathrm{e}^{-\theta\mathbf{\hat{\Omega}}}}.$ (3.55)

Equivalently,

$\mathrm{e}^{\Omega\mathbf{\hat{\Omega}}/2}=\frac{1+w_{1}w_{2}\,\mathrm{e}^{\theta\mathbf{\hat{\Omega}}}}{\absolutevalue{1+w_{1}w_{2}\,\mathrm{e}^{\theta\mathbf{\hat{\Omega}}}}}.$
(3.56)

Taking the scalar and vectorial parts of equation (3.56), we finally obtain

$\tan\frac{\Omega}{2}=\frac{w_{1}w_{2}\sin\theta}{1+w_{1}w_{2}\cos\theta}=\frac{\absolutevalue{\vec{w}_{1}\crossproduct\vec{w}_{2}}}{1+\vec{w}_{1}\cdot\vec{w}_{2}},$
(3.57)

as an explicit expression for the Wigner angle $\Omega$.

The simplicity of equation (3.57) compared to exisiting formulae for $\Omega$
in the literature, shows how the composition law (3.20) can lead to much
tidier and simpler formulae than other methods allowed for. This can be seen
as the extension of the result (3.11) to more general velocities.

We can write equation (3.57) in a perhaps more familiar (though possibly more
tedious) form by first noting that from equation (3.28) we have

$w_{i}=\frac{1-\sqrt{1-v_{i}^{2}}}{v_{i}}=\frac{\gamma_{i}-1}{\sqrt{\gamma_{i}^{2}-1}}=\sqrt{\frac{\gamma_{i}-1}{\gamma_{i}+1}}=\frac{\sqrt{\gamma_{i}^{2}-1}}{\gamma_{i}+1}=\frac{v_{i}\gamma_{i}}{\gamma_{i}+1},$
(3.58)

and so

$\tan\frac{\Omega}{2}=\frac{v_{1}v_{2}\gamma_{1}\gamma_{2}\sin\theta}{(1+\gamma_{1})(1+\gamma_{2})+v_{1}v_{2}\gamma_{1}\gamma_{2}\cos\theta}.$
(3.59)

We can check two interesting cases of equation (3.57) for when $\theta=0$
(parallel velocities) and when $\theta=\pi/2$ (perpendicular velocities). We
can see directly that, for parallel velocities, the associated Wigner angle is
given by $\tan(\Omega/2)=0$, so that $\Omega=n\pi$ for $n\in\mathbb{Z}$;
whilst for perpendicular velocities, the associated Wigner angle is simply
given by $\tan(\Omega/2)=w_{1}w_{2}$.

It is easiest to check our results against the literature using the somewhat
messier equation (3.59), in which case parallel velocities again give
$\tan(\Omega/2)=0$, whilst perpendicular velocities give

$\tan(\Omega/2)=\frac{v_{1}v_{2}\gamma_{1}\gamma_{2}}{(1+\gamma_{1})(1+\gamma_{2})},$
(3.60)

which agrees with the results given in [109].

###  3.3 Combining three 3-velocities

Let us now see what happens when we relativistically combine 3 half-
velocities.
We shall calculate, compare, and contrast $\mathbf{w}_{(1\oplus 2)\oplus 3}$
with $\mathbf{w}_{1\oplus(2\oplus 3)}$.

####  3.3.1 Combining 3 half-velocities: $\mathbf{w}_{(1\oplus 2)\oplus 3}$

Start from our key result

$\mathbf{w}_{1\oplus
2}=\mathbf{w}_{1}\oplus\mathbf{w}_{2}=(1-\mathbf{w}_{1}\mathbf{w}_{2})^{-1}(\mathbf{w}_{1}+\mathbf{w}_{2}),$
(3.61)

and iterate it to yield

$\mathbf{w}_{(1\oplus 2)\oplus
3}=\\{1-(1-\mathbf{w}_{1}\mathbf{w}_{2})^{-1}(\mathbf{w}_{1}+\mathbf{w}_{2})\mathbf{w}_{3}\\}^{-1}\\{(1-\mathbf{w}_{1}\mathbf{w}_{2})^{-1}(\mathbf{w}_{1}+\mathbf{w}_{2})+\mathbf{w}_{3}\\}.$
(3.62)

It is now a matter of straightforward quaternionic algebra to check that

$\displaystyle\mathbf{w}_{(1\oplus 2)\oplus 3}$
$\displaystyle=\\{(1-\mathbf{w}_{1}\mathbf{w}_{2})^{-1}(1-\mathbf{w}_{1}\mathbf{w}_{2}-(\mathbf{w}_{1}+\mathbf{w}_{2})\mathbf{w}_{3})\\}^{-1}\times$
$\displaystyle\qquad\quad\\{(1-\mathbf{w}_{1}\mathbf{w}_{2})^{-1}(\mathbf{w}_{1}+\mathbf{w}_{2})+\mathbf{w}_{3}\\}$
$\displaystyle=(1-\mathbf{w}_{1}\mathbf{w}_{2}-(\mathbf{w}_{1}+\mathbf{w}_{2})\mathbf{w}_{3})^{-1}(1-\mathbf{w}_{1}\mathbf{w}_{2})\times$
$\displaystyle\qquad\quad\\{(1-\mathbf{w}_{1}\mathbf{w}_{2})^{-1}(\mathbf{w}_{1}+\mathbf{w}_{2})+\mathbf{w}_{3}\\}$
$\displaystyle=(1-\mathbf{w}_{1}\mathbf{w}_{2}-(\mathbf{w}_{1}+\mathbf{w}_{2})\mathbf{w}_{3})^{-1}\\{(\mathbf{w}_{1}+\mathbf{w}_{2})+(1-\mathbf{w}_{1}\mathbf{w}_{2})\mathbf{w}_{3}\\}.$
(3.63)

Ultimately,

$\mathbf{w}_{(1\oplus 2)\oplus
3}=\\{1-\mathbf{w}_{1}\mathbf{w}_{2}-\mathbf{w}_{1}\mathbf{w}_{3}-\mathbf{w}_{2}\mathbf{w}_{3}\\}^{-1}\\{\mathbf{w}_{1}+\mathbf{w}_{2}+\mathbf{w}_{3}-\mathbf{w}_{1}\mathbf{w}_{2}\mathbf{w}_{3}\\}.$
(3.64)

An alternative formulation starts from

$\mathbf{w}_{1\oplus
2}=\mathbf{w}_{1}\oplus\mathbf{w}_{2}=(\mathbf{w}_{1}+\mathbf{w}_{2})(1-\mathbf{w}_{2}\mathbf{w}_{1})^{-1},$
(3.65)

which when iterated yields

$\mathbf{w}_{(1\oplus 2)\oplus
3}=\\{(\mathbf{w}_{1}+\mathbf{w}_{2})(1-\mathbf{w}_{2}\mathbf{w}_{1})^{-1}+\mathbf{w}_{3}\\}\\{1-\mathbf{w}_{3}(\mathbf{w}_{1}+\mathbf{w}_{2})(1-\mathbf{w}_{2}\mathbf{w}_{1})^{-1}\\}^{-1}.$
(3.66)

Thus, a little straightforward quaternionic algebra verifies that

$\displaystyle\mathbf{w}_{(1\oplus 2)\oplus 3}$
$\displaystyle=\\{(\mathbf{w}_{1}+\mathbf{w}_{2})+\mathbf{w}_{3}(1-\mathbf{w}_{2}\mathbf{w}_{1})\\}(1-\mathbf{w}_{2}\mathbf{w}_{1})^{-1}\times$
$\displaystyle\qquad\qquad\\{1-\mathbf{w}_{3}(\mathbf{w}_{1}+\mathbf{w}_{2})(1-\mathbf{w}_{2}\mathbf{w}_{1})^{-1}\\}^{-1}$
$\displaystyle=\\{(\mathbf{w}_{1}+\mathbf{w}_{2})+\mathbf{w}_{3}(1-\mathbf{w}_{2}\mathbf{w}_{1})\\}\\{(1-\mathbf{w}_{2}\mathbf{w}_{1})-\mathbf{w}_{3}(\mathbf{w}_{1}+\mathbf{w}_{2})\\}^{-1}.$
(3.67)

Ultimately,

$\mathbf{w}_{(1\oplus 2)\oplus
3}=\\{\mathbf{w}_{1}+\mathbf{w}_{2}+\mathbf{w}_{3}-\mathbf{w}_{3}\mathbf{w}_{2}\mathbf{w}_{1}\\}\\{1-\mathbf{w}_{2}\mathbf{w}_{1}-\mathbf{w}_{3}\mathbf{w}_{1}-\mathbf{w}_{3}\mathbf{w}_{2}\\}^{-1}.$
(3.68)

So we have found two equivalent formulae for $\mathbf{w}_{(1\oplus 2)\oplus
3}$, equations (3.64) and (3.68).

####  3.3.2 Combining 3 half-velocities: $\mathbf{w}_{1\oplus(2\oplus 3)}$

In contrast, the situation for $\mathbf{w}_{1\oplus(2\oplus 3)}$ is
considerably more subtle. Start from the key result that

$\mathbf{w}_{2\oplus
3}=\mathbf{w}_{2}\oplus\mathbf{w}_{3}=(1-\mathbf{w}_{2}\mathbf{w}_{3})^{-1}(\mathbf{w}_{2}+\mathbf{w}_{3}),$
(3.69)

and iterate it to yield

$\mathbf{w}_{1\oplus(2\oplus
3)}=\\{1-\mathbf{w}_{1}(1-\mathbf{w}_{2}\mathbf{w}_{3})^{-1}(\mathbf{w}_{2}+\mathbf{w}_{3})\\}^{-1}\\{\mathbf{w}_{1}+(1-\mathbf{w}_{2}\mathbf{w}_{3})^{-1}(\mathbf{w}_{2}+\mathbf{w}_{3})\\}.$
(3.70)

The relevant quaternionic algebra is now a little trickier

$\displaystyle\mathbf{w}_{1\oplus(2\oplus 3)}$ $\displaystyle=$
$\displaystyle\\{1-\mathbf{w}_{1}(1-\mathbf{w}_{2}\mathbf{w}_{3})^{-1}(\mathbf{w}_{2}+\mathbf{w}_{3})\\}^{-1}(1-\mathbf{w}_{2}\mathbf{w}_{3})^{-1}\times$
(3.71)
$\displaystyle\qquad\quad\\{(1-\mathbf{w}_{2}\mathbf{w}_{3})\mathbf{w}_{1}+(\mathbf{w}_{2}+\mathbf{w}_{3})\\}$
$\displaystyle=$
$\displaystyle\\{(1-\mathbf{w}_{2}\mathbf{w}_{3})(1-\mathbf{w}_{1}(1-\mathbf{w}_{2}\mathbf{w}_{3})^{-1}(\mathbf{w}_{2}+\mathbf{w}_{3})\\}^{-1}\times$
$\displaystyle\qquad\quad\\{(1-\mathbf{w}_{2}\mathbf{w}_{3})\mathbf{w}_{1}+(\mathbf{w}_{2}+\mathbf{w}_{3})\\}$
$\displaystyle=$
$\displaystyle\\{1-\mathbf{w}_{2}\mathbf{w}_{3}-(1-\mathbf{w}_{2}\mathbf{w}_{3})\mathbf{w}_{1}(1-\mathbf{w}_{2}\mathbf{w}_{3})^{-1}(\mathbf{w}_{2}+\mathbf{w}_{3})\\}^{-1}\times$
$\displaystyle\qquad\quad\\{\mathbf{w}_{1}+\mathbf{w}_{2}+\mathbf{w}_{3}-\mathbf{w}_{2}\mathbf{w}_{3}\mathbf{w}_{1}\\}.$

To proceed we note that

$\displaystyle(1-\mathbf{w}_{2}\mathbf{w}_{3})\mathbf{w}_{1}(1-\mathbf{w}_{2}\mathbf{w}_{3})^{-1}$
$\displaystyle=\left(1-\mathbf{w}_{2}\mathbf{w}_{3}\over|1-\mathbf{w}_{2}\mathbf{w}_{3}|\right)\mathbf{w}_{1}\left(1-\mathbf{w}_{2}\mathbf{w}_{3}\over|1-\mathbf{w}_{2}\mathbf{w}_{3}|\right)^{-1}$
$\displaystyle\vphantom{\Big{|}}=e^{-\mathbf{\Omega}_{2\oplus
3}/2}\;\mathbf{w}_{1}\;e^{+\mathbf{\Omega}_{2\oplus 3}/2}.$ (3.72)

Thus,

$\mathbf{w}_{1\oplus(2\oplus
3)}=\\{1-\mathbf{w}_{2}\mathbf{w}_{3}-(e^{-\mathbf{\Omega}_{2\oplus
3}/2}\mathbf{w}_{1}e^{+\mathbf{\Omega}_{2\oplus
3}/2})(\mathbf{w}_{2}+\mathbf{w}_{3})\\}^{-1}\times\\\
\\{\mathbf{w}_{1}+\mathbf{w}_{2}+\mathbf{w}_{3}-\mathbf{w}_{2}\mathbf{w}_{3}\mathbf{w}_{1}\\}.$
(3.73)

While structurally similar to the formulae (3.64) and (3.68) for
$\mathbf{w}_{(1\oplus 2)\oplus 3}$ the present result (3.73) for
$\mathbf{w}_{1\oplus(2\oplus 3)}$ is certainly different — the Wigner angle
$\mathbf{\Omega}_{2\oplus 3}$ now makes an explicit appearance, also the form
of the triple-product $\mathbf{w}_{2}\mathbf{w}_{3}\mathbf{w}_{1}$ is
different.

####  3.3.3 Combining 3 half-velocities: (Non)-associativity

From (3.64) and (3.68) for $\mathbf{w}_{(1\oplus 2)\oplus 3}$, and (3.73) for
$\mathbf{w}_{1\oplus(2\oplus 3)}$, it is clear that relativistic composition
of velocities is in general not associative. (See for instance the discussion
in references [131, 141], commenting on reference [130].)

A sufficient condition for associativity, $\mathbf{w}_{(1\oplus 2)\oplus
3}=\mathbf{w}_{1\oplus(2\oplus 3)}$, is to enforce

$e^{-\mathbf{\Omega}_{2\oplus 3}/2}\mathbf{w}_{1}e^{+\mathbf{\Omega}_{2\oplus
3}/2}=\mathbf{w}_{1},\qquad\hbox{and}\qquad\mathbf{w}_{1}\mathbf{w}_{2}\mathbf{w}_{3}=\mathbf{w}_{2}\mathbf{w}_{3}\mathbf{w}_{1}.$
(3.74)

That is, a sufficient condition for associativity is

$[\mathbf{\Omega}_{2\oplus
3},\mathbf{w}_{1}]=0,\qquad\hbox{and}\qquad[\mathbf{w}_{1},\mathbf{w}_{2}\mathbf{w}_{3}]=0.$
(3.75)

But note $\mathbf{\Omega}_{2\oplus 3}\propto[\mathbf{w}_{2},\mathbf{w}_{3}]$
and $\mathbf{w}_{2}\mathbf{w}_{3}={1\over
2}\\{\mathbf{w}_{2},\mathbf{w}_{3}\\}+{1\over
2}[\mathbf{w}_{2},\mathbf{w}_{3}]$. Since
$\\{\mathbf{w}_{2},\mathbf{w}_{3}\\}\in\mathbb{R}$, we then have
$[\mathbf{w}_{1},\mathbf{w}_{2}\mathbf{w}_{3}]={1\over
2}[\mathbf{w}_{1},[\mathbf{w}_{2},\mathbf{w}_{3}]]$. This now implies that
these two sufficiency conditions are in fact identical; so a sufficient
condition for associativity is

$[\mathbf{w}_{1},[\mathbf{w}_{2},\mathbf{w}_{3}]]=0.$ (3.76)

This sufficient condition for associativity can also be written as the
vanishing of the vector triple product

$\vec{w}_{1}\times(\vec{w}_{2}\times\vec{w}_{3})=0.$ (3.77)

Equivalently,

$\vec{v}_{1}\times(\vec{v}_{2}\times\vec{v}_{3})=0.$ (3.78)

####  3.3.4 Specific non-coplanar example

As a final example of the power of the quaternion formalism, let us consider a
specific intrinsically non-coplanar example. Let
$\mathbf{w}_{1}=w_{1}\mathbf{i}$, $\mathbf{w}_{2}=w_{2}\,\mathbf{j}$, and
$\mathbf{w}_{3}=w_{3}\mathbf{k}$ be three mutually perpendicular half-
velocities. (So this configuration does automatically satisfy the
associativity condition discussed above.)

Then we have already seen that

$\mathbf{w}_{1}\oplus\mathbf{w}_{2}=\frac{w_{1}(1-w_{2}^{2})\mathbf{i}+w_{2}(1+w_{1}^{2})\mathbf{j}}{1+w_{1}^{2}w_{2}^{2}},\mbox{\quad
and\quad}w_{1\oplus 2}^{2}={w_{1}^{2}+w_{2}^{2}\over 1+w_{1}^{2}w_{2}^{2}}.$
(3.79)

Furthermore, since $\mathbf{w}_{1}\oplus\mathbf{w}_{2}$ is perpendicular to
$\mathbf{w}_{3}$, we have

$(\mathbf{w}_{1}\oplus\mathbf{w}_{2})\oplus\mathbf{w}_{3}=\frac{w_{1\oplus
2}(1-w_{3}^{2})\mathbf{\hat{n}}_{1\oplus 2}+w_{3}(1+w_{1\oplus
2}^{2})\mathbf{k}}{1+w_{1\oplus 2}^{2}w_{3}^{2}},$ (3.80)

and

$w_{(1\oplus 2)\oplus 3}^{2}={w_{(1\oplus 2)}^{2}+w_{3}^{2}\over 1+w_{(1\oplus
2)}^{2}w_{3}^{2}}={w_{1}^{2}+w_{2}^{2}+w_{3}^{2}+w_{1}^{2}w_{2}^{2}w_{3}^{2}\over
1+w_{1}^{2}w_{2}^{2}+w_{2}^{2}w_{3}^{2}+w_{3}^{2}w_{1}^{2}}.$ (3.81)

A little algebra now yields the manifestly non-commutative result

$(\mathbf{w}_{1}\oplus\mathbf{w}_{2})\oplus\mathbf{w}_{3}=\frac{1}{1+w_{1}^{2}w_{2}^{2}+w_{2}^{2}w_{3}^{2}+w_{3}^{2}w_{1}^{2}}\bigg{\\{}(1-w_{2}^{2})(1-w_{3}^{2})\mathbf{w}_{1}+\\\
+(1+w_{1}^{2})(1-w_{3}^{2})\mathbf{w}_{2}+(1+w_{1}^{2})(1+w_{2}^{2})\mathbf{w}_{3}\bigg{\\}}.$
(3.82)

In this particular case we can also explicitly show that

$(\mathbf{w}_{1}\oplus\mathbf{w}_{2})\oplus\mathbf{w}_{3}=\mathbf{w}_{1}\oplus(\mathbf{w}_{2}\oplus\mathbf{w}_{3}),$
(3.83)

though (as discussed above) associativity fails in general.

###  3.4 Summary

In this chapter, we analysed a simple, elegant, and novel algebraic method for
combining special relativistic 3-velocities using quaternions:

$\mathbf{w}_{1\oplus
2}=\mathbf{w}_{1}\oplus\mathbf{w}_{2}=(1-\mathbf{w}_{1}\mathbf{w}_{2})^{-1}(\mathbf{w}_{1}+\mathbf{w}_{2})=(\mathbf{w}_{1}+\mathbf{w}_{2})(1-\mathbf{w}_{2}\mathbf{w}_{1})^{-1}.$
(3.84)

The construction also leads to a simple, elegant, and novel formula for the
Wigner angle:

$e^{\mathbf{\Omega}}=e^{\Omega\;\mathbf{\hat{\Omega}}}=(1-\mathbf{w}_{1}\mathbf{w}_{2})^{-1}(1-\mathbf{w}_{2}\mathbf{w}_{1}),$
(3.85)

in terms of which

${\mathbf{\hat{w}}}_{1\oplus
2}=\mathrm{e}^{\mathbf{\Omega}/2}\;\;{\mathbf{w}_{1}+\mathbf{w}_{2}\over|\mathbf{w}_{1}+\mathbf{w}_{2}|};\qquad\qquad{\mathbf{\hat{w}}}_{2\oplus
1}=\mathrm{e}^{-\mathbf{\Omega}/2}\;\;{\mathbf{w}_{1}+\mathbf{w}_{2}\over|\mathbf{w}_{1}+\mathbf{w}_{2}|}.$

We saw how all of the non-commutativity associated with non-collinearity of
3-velocities is automatically and rather efficiently dealt with by the
quaternion algebra.

This concludes our discussion of the theory of special relativity, and we will
now move onto the theory of general relativity in the following chapters.

##  Chapter 4 General relativity

We now move to the setting of general relativity. As the name suggests,
general relativity is a generalisation of the theory of special relativity.
Indeed, it is a generalisation in the sense that it moves from the “flat
space” of special relativity (Minkowski space) to more general “curved
spaces”. These curved spaces are represented by manifolds, and the
mathematical language used to describe them is differential geometry. In this
chapter, we will present some of the main mathematical tools from differential
geometry used in general relativity and introduce the idea of a spacetime
rigorously. From there, we will discuss Einstein’s equivalence principle and
how it leads to the theory of general relativity. This will then allow us to
analyse specific spacetime models within the framework of general relativity
in the following chapters.

The concept of a spacetime in general relativity is developed by adding
additional mathematical structure to a four-dimensional manifold. Rigorously,
we may define a spacetime as follows [70, 156].

######  Definition 2.

A spacetime is a pair ($\mathcal{M},\mathbf{g}$), where $\mathcal{M}$ is a
connected, four-dimensional $C^{\infty}$ manifold and $\mathbf{g}$ is a metric
with Lorentzian signature.

Although most of the terms in this definition have not yet been defined (they
will be in the following sections), we can see a familiar theme with special
relativity: spacetime is four-dimensional. As in special relativity, three of
these are spacial dimensions, whilst the fourth is a temporal dimension. Thus,
intuitively, one may think of spacetime in general relativity as a four-
dimensional object that encodes both spatial and temporal information.
However, in order to understand all of the terms in definition 2, we will need
the mathematical framework presented in the following sections.

###  4.1 Manifolds

Intuitively, a manifold is a space which may be curved on a global scale
(characterised by curvature tensors defined below), yet locally it must
resemble Euclidean (flat) space. We formalise the notion of ‘resembling flat
space’ with the following definition.

######  Definition 3.

A $n$-dimensional locally Euclidean space is a set $\mathcal{E}$ together with
a $C^{r}$ atlas $\\{U_{\alpha},\phi_{\alpha}\\}$, i.e. a collection of charts
$(U_{\alpha},\phi_{\alpha})$ where the $U_{\alpha}$ are subsets of
$\mathcal{E}$ and the $\phi_{\alpha}$ are injective maps from the $U_{\alpha}$
into open subsets of $\mathbb{R}^{n}$ (endowed with the standard topology)
such that:

  1. (1)

the $U_{\alpha}$ cover $\mathcal{E}$, i.e.
$\mathcal{E}=\bigcup_{\alpha}U_{\alpha}$;

  2. (2)

if $U_{\alpha}\cap U_{\beta}$ is non-empty, then the map

$\phi_{\alpha}\circ\phi_{\beta}^{-1}:\phi_{\beta}(U_{\alpha}\cap
U_{\beta})\rightarrow\phi_{\alpha}(U_{\alpha}\cap U_{\beta})$

is a $C^{r}$ map from an open subset of $\mathbb{R}^{n}$ to an open subset of
$\mathbb{R}^{n}$.

Here we define a $C^{r}$ map to be one that is $r$-times continuously
differentiable. We can now define a manifold as follows [148].

######  Definition 4.

A manifold $\mathcal{M}$ is a locally Euclidean space which:

  1. (1)

has the same dimension everywhere,

  2. (2)

is Hausdorff,

  3. (3)

has at least one countable atlas.

A space is said to be Hausdorff if whenever $p$ and $q$ are distinct points in
$\mathcal{M}$, there exists disjoint open subsets $U$ and $V$ of $\mathcal{M}$
such that $p\in U$ and $q\in V$. A topological space is said to be connected
if it is _not_ the union of two disjoint non-empty open sets. Both properties
of connectedness and Hausdorffness are simply “physically reasonable”
constraints that we place on the manifold in order to ensure that the
mathematical notion of a spacetime resembles as closely as possible the
universe we observe. The canonical example of a two-dimensional manifold is
the 2-sphere, $S^{2}$. Globally it is a curved surface, yet a small patch on
the surface of the sphere will look like a flat piece of $\mathbb{R}^{2}$ if
we ‘zoom in’ close enough.

###  4.2 The metric tensor

Perhaps the most important mathematical object in classical general relativity
is the metric tensor which one endows on their manifold. In general, we have
the following definition [70].

######  Definition 5.

A metric tensor $\mathbf{g}$ (usually just called “the metric”) at a point
$p\in\mathcal{M}$ is a symmetric tensor of type $(0,2)$ at $p$.

If a coordinate basis $\\{\partial/\partial x^{a}\\}$ is used, then one can
express the metric in terms of its components as

$\mathbf{g}=g_{ab}\differential{x^{a}}\otimes\differential{x^{b}}.$ (4.1)

The metric is extremely useful as it allows us to define the notion of a path
length between two points along a curve in a manifold. Indeed, suppose that
for two points $p,q\in\mathcal{M}$ there is a smooth curve $\gamma$
parametrised by some parameter $\lambda\in\mathbb{R}$ such that $p=\gamma(a)$
and $q=\gamma(b)$. Then the path length between the points $p$ and $q$ is
given by

$L=\int_{a}^{b}\sqrt{g_{ab}\derivative{x^{a}}{\lambda}\derivative{x^{b}}{\lambda}}\differential{\lambda}.$
(4.2)

We may express equations (4.1) and (4.2) via one relation as

$\differential s^{2}=g_{ij}\differential x^{i}\differential x^{j},$ (4.3)

which represents the infinitesimal arc determined by the coordinate
displacement $x^{i}\rightarrow x^{i}+\differential x^{i}$. In the context of
general relativity, the quantity $\differential s^{2}$ as defined by equation
(4.3) is commonly called the “line element” of the spacetime, and is in direct
one-to-one correspondence with the metric endowed on the spacetime manifold.
As such, one may see the terms ‘metric’ and ‘line-element’ used
interchangeably within the context of a spacetime.

One of the powerful properties of the metric is that it allows us to
interchange between contravariant and covariant tensor quantities. We call a
metric non-degenerate at a point $p\in\mathcal{M}$ if the matrix $(g_{ab})$ of
components of $\mathbf{g}$ is non-singular at $p$ (i.e. the matrix is
invertible at $p$). For a non-degenerate metric $\mathbf{g}$, we can always
define a unique non-degenerate symmetric tensor of type $(2,0)$ (sometimes
called the “contravariant-metric”) by the relation

$g^{ab}g_{bc}=\delta^{a}_{c}.$ (4.4)

That is, $(g^{ab})$ is the matrix-inverse of $(g_{ab})$. Thus, if $X^{a}$ are
the components of a contravariant vector, then we can uniquely define a
covariant vector with components given by $X_{a}=g_{ab}X^{b}$. Hence, one may
also write $X^{a}=g^{ab}X_{b}$. Similarly, for a type $(0,2)$ tensor $X_{ab}$
we may write $X\indices{{}^{a}_{\,b}}=g^{ac}X_{cb}$,
$X\indices{{}_{a}^{b}}=g^{bc}X_{ac}$, and $X^{ab}=g^{ac}g^{bd}X_{cd}$.

Using the metric tensor, we can classify non-zero vectors into three distinct
classes:

  1. (i)

Timelike: $g_{ab}X^{a}X^{b}<0$,

  2. (ii)

Null: $g_{ab}X^{a}X^{b}=0$,

  3. (iii)

Spacelike: $g_{ab}X^{a}X^{b}>0$.

Analogous to special relativity, this corresponds to (i) massive particles,
(ii) massless particles, and (iii) tachyonic particles.

The last term to unpack in our definition of a spacetime (definition 2) is the
concept of the signature of a metric. We have the following definition [70].

######  Definition 6.

The signature of a metric $\mathbf{g}$ at a point $p\in\mathcal{M}$ is defined
as the number of positive eigenvalues of the matrix $(g_{ab})$ at $p$, minus
the number of negative ones.

Furthermore, if $\mathbf{g}$ is non-degenerate, the signature of $\mathbf{g}$
will be constant on $\mathcal{M}$ [70]. Commonly, one may see the signature of
a metric presented as a $n$-tuple of $`+^{\prime}$ and $`-^{\prime}$ signs,
representing the positive ($+$) and negative ($-$) eigenvalues of the metric.
For example, $(-,+,+)$ or $(+,+,+,+,+)$. This encodes the signature of the
metric, as in definition 6, but is not so easily adapted to the description of
metrics on high-dimensional manifolds. However, as a spacetime is restricted
to four dimensions, this representation of the metric signature is commonly
seen in classical general relativity.

######  Definition 7.

A positive-definite metric (sometimes called a Riemannian metric) on a
$n$-dimensional manifold is a metric with signature $n$ (i.e. all positive
eigenvalues). Contrastingly, a Lorentzian metric on a $n$-dimensional manifold
is a metric with signature $n-2$ (i.e. one negative eigenvalue).

Now that we have all of the framework necessary to understand the definition
of a spacetime (c.f. definition 2), we can begin to investigate some of the
more geometrical properties of manifolds and the associated spacetimes.

###  4.3 Covariant differentiation and connections

In general, the partial derivative of a tensor is not a tensor in of itself.
There are three standard, distinct ways to define a notion of the derivative
of a general tensor quantity which invoke additional complications in order to
bypass this problem. The first of these notions (the exterior derivative)
restricts the class of tensors it operates on so as to ensure that it produces
tensor quantities, whilst the second (the Lie derivative) restricts the
‘directions’ that you can differentiate tensors in to again ensure that it
produces tensor quantities. Both of these notions of a derivative have a
multitude of uses in classical general relativity, but will not be needed for
any of the analysis presented in this thesis. As such, details of exterior-
and Lie derivatives will not be discussed here for purposes of keeping this
chapter compendious. Thus, we will focus our analysis on the third way of
defining a derivative of a tensor quantity, the covariant derivative. The
covariant derivative solves the issue of differentiating a tensor by adding
additional mathematical structure by way of an “affine connection”111Sometimes
simply called a “connection”, or the “Christoffel symbols”. However, as we
will soon see, the Christoffel symbols are in fact a very specific, special
type of connection.. Various ways of defining or constructing the covariant
derivative can be found in the literature [70, 99, 148, 161]. We define the
covariant derivative for vectors axiomatically below, as in [148]. The
definition adopted here greatly simplifies the discussion with regards to
covariant differentiation, but does not provide much insight into the
construction of the affine connection. As such, many authors prefer to start
by constructing the affine connection and then go on to use this in their
definition of the covariant derivative. However, as covariant differentiation
will be used explicitly in the following sections, I have used the following
construction to simplify the discussion around it. If the reader would like
additional insight into the affine connection and its relationship to the
parallel transport of vectors, these details may be found in any of the
standard textbooks on general relativity [35, 69, 70, 99, 161].

######  Definition 8.

Suppose we have two contravariant vector fields $u^{a}$ and $v^{a}$, and
define the vectors $u$ and $v$ to be the directional derivatives
$u=u^{a}\partial_{a}$ and $v=v^{a}\partial_{a}$. We now define our covariant
derivative operator to be the linear operator $\nabla_{u}=u^{a}\nabla_{a}$
which satisfies

  1. (1)

$\nabla_{u}f=u^{a}\partial_{a}f$,

  2. (2)

$\nabla_{fu}v=f\nabla_{u}v$,

  3. (3)

$\nabla_{u}(fv)=(\nabla_{u}f)v+f\nabla_{u}v$,

where $f$ is a scalar function.

Now, note that $\partial_{a}$ is a vector field, and so
$\nabla_{\partial_{a}}\partial_{b}$ is also a vector field and can hence be
expanded as a linear combination of $\partial_{c}$’s. That is, in terms of the
expansion coefficients $C\indices{{}^{a}_{bc}}$, we have
$\nabla_{\partial_{a}}\partial_{b}=C\indices{{}^{a}_{bc}}\partial_{c}$. We can
now use this result to derive expressions for the covariant derivative of
tensor quantities of general rank. Full details of this process van be found
in any of [70, 161, 148], but here I will just present the main results:

  * •

$(1,0)$ tensor:

$\nabla_{a}X^{b}=\partial_{a}X^{b}+C\indices{{}^{b}_{ca}}X^{c},$ (4.5a)

  * •

$(0,1)$ tensor:

$\nabla_{a}X_{b}=\partial_{a}X_{b}-C\indices{{}^{c}_{ba}}g_{c},$ (4.5b)

  * •

general $(r,s)$ tensor:

$\nabla_{a}X\indices{{}^{b_{1}b_{2}\dots b_{r}}_{c_{1}c_{2}\dots
c_{s}}}=\partial_{a}X\indices{{}^{b_{1}b_{2}\dots b_{r}}_{c_{1}c_{2}\dots
c_{s}}}+\sum_{i=1}^{r}C\indices{{}^{b_{i}}_{ma}}X\indices{{}^{b_{1}b_{2}\dots
b_{i-1}mb_{i+1}\dots b_{r}}_{c_{1}c_{2}\dots c_{s}}}\\\
-\sum_{j=1}^{s}C\indices{{}^{m}_{c_{j}a}}X\indices{{}^{b_{1}b_{2}\dots
b_{r}}_{c_{1}c_{2}\dots c_{j-1}mc_{j+1}\dots c_{s}}}.$ (4.5c)

The $C\indices{{}^{a}_{bc}}$ form the components of the affine connection,
which plays an extremely important role in general relativity. In fact,
classical general relativity imposes a geometrical constraint on the
connection. This extra condition is that it is torsion-free. That is,
$C\indices{{}^{a}_{[bc]}}\equiv\frac{1}{2}[C\indices{{}^{a}_{bc}}-C\indices{{}^{a}_{cb}}]=0$.
This extra condition has been experimentally checked, and so far all evidence
seems to suggest that we do indeed live in a torsion-free universe.
Furthermore, it has the added benefit that it allows one to use an extremely
natural choice of connection for their spacetime, the Christoffel connection.
This is typically denoted by $\Gamma$, and has components defined in terms of
partial derivatives of the metric:

$\Gamma\indices{{}^{a}_{bc}}=\frac{1}{2}g^{ad}\big{(}\partial_{b}g_{cd}+\partial_{c}g_{bd}-\partial_{d}g_{bc}\big{)}.$
(4.6)

In fact, the Christoffel connection is the unique metric-compatible222Metric
compatible simply means $\nabla_{a}g_{bc}=0$., torsion-free affine connection
one can construct. It should be noted that there are alternative theories of
gravity that do not impose zero torsion on their connection (for example,
certain string-inspired models, teleparallel gravity, etcetera). However, this
thesis is primarily concerned with classical general relativity, and so we
will we use the torsion-free Christoffel connection throughout the rest of the
thesis.

We can now start with a metric for a given manifold, use this to construct the
Christoffel connection via equation (4.6), and hence define a notion of
covariant differentiation for general tensors in our spacetime via equations
(4.5). In the following sections, we will use our covariant derivative to
obtain various notions for the ‘curvature’ of a spacetime, and then use this
to define the concept of the ‘straightest possible path’ in a curved space by
way of the geodesic equation. Finally, we will use these concepts to construct
the theory of general relativity.

###  4.4 Curvature

In general, covariant derivatives do not commute. As a result of this, if one
starts at a point $p\in\mathcal{M}$ and parallel transports a vector $v_{1}$
along a curve $\gamma_{1}$ that also ends at $p$, you will obtain a vector
$v_{2}$ which is in general different to $v_{1}$. Furthermore, if one
parallelly transports along a different curve $\gamma_{2}$ that also ends at
$p$, you will in general obtain yet another vector $v_{3}\neq v_{2}\neq
v_{1}$. This ‘non-integrability’ of parallel transport directly corresponds to
the non-commutativity of the covariant derivative. Thus, we can define a
tensor $R$ which measures the extent of this non-commutativity when acting on
vector fields $X,Y,$ and $Z$:

$R(X,Y)Z=\nabla_{X}(\nabla_{Y}Z)-\nabla_{Y}(\nabla_{X}Z)-\nabla_{[X,Y]}Z.$
(4.7)

This tensor is called the Riemann (curvature) tensor, and has components [99,
161]

$R\indices{{}^{a}_{bcd}}=\partial_{c}\Gamma\indices{{}^{a}_{bd}}-\partial_{d}\Gamma\indices{{}^{a}_{bc}}+\Gamma\indices{{}^{a}_{mc}}\Gamma\indices{{}^{m}_{bd}}-\Gamma\indices{{}^{a}_{md}}\Gamma\indices{{}^{m}_{bc}}.$
(4.8)

As we will soon see, the Riemann tensor plays a central role in the
mathematical formulation of general relativity.

Using the Riemann tensor, we can construct a whole host of other tensor
quantities. Below, I list some of the more commonly used tensors relevant to
general relativity.

Contracting on the first and second indices of the Riemann tensor we yield
another tensor, the Ricci tensor:

$R_{ab}\equiv
R\indices{{}^{c}_{acb}}=\partial_{c}\Gamma\indices{{}^{c}_{ab}}-\partial_{b}\Gamma\indices{{}^{c}_{ac}}+\Gamma\indices{{}^{c}_{dc}}\Gamma\indices{{}^{d}_{ab}}-\Gamma\indices{{}^{c}_{db}}\Gamma\indices{{}^{d}_{ac}}.$
(4.9)

We may contract once more to obtain a scalar, the Ricci scalar333Sometimes
called the “scalar curvature”.:

$R\equiv
g^{ab}R_{ab}=g^{ab}\big{(}\partial_{c}\Gamma\indices{{}^{c}_{ab}}-\partial_{b}\Gamma\indices{{}^{c}_{ac}}+\Gamma\indices{{}^{c}_{dc}}\Gamma\indices{{}^{d}_{ab}}-\Gamma\indices{{}^{c}_{db}}\big{)}.$
(4.10)

Note that as this is a scalar quantity it is invariant under changes of
coordinates. As the name suggests, another tensor quantity of central
importance in general relativity is the Einstein tensor:

$G_{ab}\equiv R_{ab}-\frac{1}{2}Rg_{ab}.$ (4.11)

Contacting all indices on the Riemann tensor we obtain the Kretschmann scalar:

$K\equiv R^{abcd}R_{abcd}.$ (4.12)

By defining the Weyl (conformal) tensor (for a manifold of dimension $n\geq
4$) as

$C_{abcd}\equiv
R_{abcd}+\frac{1}{n-2}\big{(}g_{ad}R_{bc}+g_{bc}R_{ad}-g_{ac}R_{bd}-g_{bd}R_{ac}\big{)}\\\
+\frac{R}{(n-1)(n-2)}\big{(}g_{ac}g_{bd}-g_{ad}g_{bc}\big{)},$ (4.13)

we can re-write the Kretschmann scalar as

$K=C^{abcd}C_{abcd}+\frac{4}{n-2}R^{ab}R_{ab}-\frac{2}{(n-1)(n-2)}R^{2}.$
(4.14)

All of the tensors presented above give slightly different notions of the
geometrical curvature of a spacetime. As a spacetime is intrinsically four-
dimensional, it is often hard to obtain a mental picture of what this
physically means for a spacetime (how can time be curved?). In section 4.6, we
will provide a way of connecting this mathematical framework back to physical
reality. Before we get to this, however, we still need one more piece of
mathematical machinery: geodesics.

###  4.5 Geodesics

Intuitively, we want to define a geodesic to be a curve that is “as straight
as possible” in a curved manifold. Building on this intuition, we wish to
require that the tangent vector $T^{a}$ to the curve points in the same
direction as itself when parallel propagated. That is, we require

$T^{a}\nabla_{a}T^{b}\propto T^{b}.$ (4.15)

With a suitable parameterisation of our curve, we can turn this equation in to
a much simpler condition:

$T^{a}\nabla_{a}T^{b}=0.$ (4.16)

Such a parameterisation is called an “affine” parameterisation, and we will
assume throughout the rest of this thesis that we are using such a
parameterisation unless otherwise stated. Choosing some affine parameter
$\lambda$, our tangent vector is simply $T^{a}=\differential
x^{a}/\differential\lambda$, and so using equation (4.5a) we can re-write
equation (4.16) as

$\derivative[2]{x^{a}}{\lambda}+\Gamma\indices{{}^{a}_{bc}}\derivative{x^{b}}{\lambda}\derivative{x^{c}}{\lambda}=0.$
(4.17)

This equation is commonly referred to as the geodesic equation, and any curves
$x^{a}(\lambda)$ which satisfy it are called geodesics.

###  4.6 Einstein’s equivalence principle and field
equations

We now have constructed the mathematical framework necessary to understand
Einstein’s general theory of relativity. However, as with all physical
theories, we have to start with a set of fundamental axioms, or “laws” of
physics, on top of which we can apply our mathematical framework to obtain the
fundamental equations of motion. It is worth noting again that, as with all
“laws” of physics, these axioms are just a summary of very good experimental
evidence. Using the equations of motion, we can then make predictions about
how the universe behaves. In Newtonian mechanics, we have Newton’s three laws;
in special relativity, we have Einstein’s postulates; in general relativity,
we have the equivalence principle. In all cases, laws of physics are just
statements which encode and summarise a lot of rigorous experimental evidence.
The situation is no different in general relativity.

####  4.6.1 The equivalence principle

The equivalence principle asserts that all freely falling particles follow the
same trajectories independent of their internal composition [156]. In the
language of Newtonian mechanics, this is equivalent to asserting that an
objects gravitational mass (how the mass responds to-/generates a
gravitational field) is identical to its inertial mass (how the mass ‘resists’
changes to its velocity). The assertion that gravitational mass and inertial
mass are the same quantity is now an extremely accurately tested principle: In
1999 Baessler et al. experimentally verified this to around one part in
$10^{13}$ [8]. Although this was not so well experimentally verified at the
time Einstein developed his theory, it was still commonly believed to be true
by the physicists of the time. Thus, Einstein decided to fundamentally encode
the equivalence principle into his theory with the following postulates,
commonly called the Einstein equivalence principle.

######  Postulate 3 (Equivalence principle).

Given a spacetime $(\mathcal{M},\mathbf{g})$, the gravitational field is
represented by the Christoffel connection and free fall corresponds to
geodesic motion. Furthermore, in the flat spacetime limit, the metric must
reduce to the Minkowski metric $\eta$ of special relativity.

The last of these two statements implies, that in suitably small local
coordinate patches, we must be able to reproduce the laws of special
relativity from the more general theory. Hence, we have the following
hierarchy

$\text{General relativity}\underset{\text{flat
space}}{\longrightarrow}\text{Special relativity}\underset{\text{low
energy}}{\longrightarrow}\text{Newtonian mechanics}.$

####  4.6.2 Einstein’s field equations and the stress-energy
tensor

In order for general relativity to be a complete physical theory, we need a
set of equations of motion. Like all field theories, it is desirable to derive
the equations of motion via an action principle. Therefore, consider the
action

$S=-\frac{1}{16\pi}\int_{\mathcal{M}}\sqrt{-g}\,(R-2\Lambda)\,\differential[4]x+\int_{\mathcal{M}}\sqrt{-g}\,\mathcal{L}_{M}\,\differential[4]x,$
(4.18)

where $\Lambda$ is a constant (usually called the “cosmological” constant),
$g=\det(g_{ab})$, and $\mathcal{L}_{M}$ is a Lagrangian which is related to
the matter-content (or stress-energy) of the spacetime. Generally, one may
write

$S=S_{gr}+S_{M},$ (4.19)

highlighting the gravitational and matter-content contributions of a given
spacetime to the general action. The action (4.18) has been accredited to
David Hilbert, who published it less than a week after Einstein published his
own formulation of the field equations [74]. As such, it is typically referred
to as the Einstein-Hilbert action444Traditionally, the Einstein-Hilbert action
is only defined to be the $(\sqrt{-g}\,R)$ term in the action presented above.
However, including the other terms allows for the derivation of the most
general form of the Einstein equations.. Varying the first integral in the
action yields

$\delta
S=-\frac{1}{16\pi}\int_{\mathcal{M}}\bigg{[}\delta\big{(}\sqrt{-g}\,\big{)}\,(R-2\Lambda)+\sqrt{-g}\,\delta
R\bigg{]}\differential[4]x.$ (4.20)

The variation of $\sqrt{-g}$ is a standard result [70, 161]:

$\int_{\mathcal{M}}\delta(\sqrt{-g})\differential[4]x=\int_{M}-\frac{1}{2}\sqrt{-g}\,g_{ab}\,\delta
g^{ab}\,\differential[4]x.$ (4.21)

However, the variation of the Ricci scalar is a little more subtle:

$\displaystyle\int_{\mathcal{M}}\delta R\differential[4]x$
$\displaystyle=\int_{\mathcal{M}}\delta(g^{ab}R_{ab})\differential[4]x=\int_{\mathcal{M}}\bigg{[}\delta(g^{ab})R_{ab}+g^{ab}\delta(R_{ab})\bigg{]}\differential[4]x$
$\displaystyle=\int_{\mathcal{M}}\bigg{[}\delta(g^{ab})R_{ab}+\nabla^{a}v_{a}\bigg{]}\differential[4]x,$
(4.22)

where $v_{a}=\nabla^{b}\delta(g_{ab})-g^{cd}\nabla_{a}\delta(g_{cd})$ [161].
Thus, equation (4.20) can be written as

$\delta
S=-\frac{1}{16\pi}\int_{\mathcal{M}}\sqrt{-g}\,\bigg{[}R_{ab}-\frac{1}{2}R\,g_{ab}+g_{ab}\,\Lambda\bigg{]}\delta(g^{ab})\,\differential[4]x\\\
-\frac{1}{16\pi}\int_{\mathcal{M}}\nabla^{a}v_{a}\sqrt{-g}\,\differential[4]x.$
(4.23)

But the second integral is just the integral of the divergence
$\nabla^{a}v_{a}$ with respect to the natural volume element
$(\sqrt{-g}\,\differential[4]x)$, and so by Stokes’ theorem contributes only a
boundary term – the Gibbons–Hawking surface term [156]:

$-\frac{1}{16\pi}\int_{\mathcal{M}}\nabla^{a}v_{a}\sqrt{-g}\,\differential[4]x=\delta\left(\frac{1}{8\pi}\int_{\partial\mathcal{M}}K\sqrt{{}^{3}g}\,\differential[3]x\right),$
(4.24)

where $K=K\indices{{}^{a}_{a}}$ is the trace of the extrinsic-curvature of the
spacetime, and ${}^{3}g$ is the determinant of the induced three-metric. It is
standard to ignore this term in the derivation of the equations of motion, as
one can just simply subtract it from the original action presented in equation
(4.18). Furthermore, it will be identically zero for variations where $g^{ab}$
and its derivatives are held fixed on the boundary [161]. Hence, moving
forward, we will ignore this term in the derivation of the equations of
motion.

We have not yet considered the contribution of the matter-content action
$S_{M}$. To do this, we simply define a $4\times 4$ type $(0,2)$ tensor
$T_{ab}$ by

$T_{ab}\equiv-\frac{1}{8\pi}\frac{1}{\sqrt{-g}}\frac{\delta S_{M}}{\delta
g^{ab}},$ (4.25)

where, as above, the action $S_{M}$ represents the matter-content of the
spacetime:

$S_{M}=\int_{\mathcal{M}}\sqrt{-g}\,\mathcal{L}_{M}\,\differential[4]x.$
(4.26)

As such, the tensor $T_{ab}$ is referred to as the stress-energy tensor of the
spacetime (sometimes called the ‘stress-energy-momentum’ tensor). Its
components are related to the matter in the spacetime by

$T_{ab}=\left(\begin{array}[]{c | c}\rho&S_{j}\\\ \hline\cr S_{i}&\pi_{ij}\end{array}\right),$ (4.27)

where $\rho$ is the energy density, $S_{i}$ is the energy-flux, and $\pi_{ij}$
is the stress ($i,j\in\\{1,2,3\\}$). Typically, $S_{i}$ is considered a
generalisation of the Poynting vector and $\pi_{ij}$ is considered a
generalisation of the notion of pressure [99, 156]. In an orthonormal frame,
all of the components have dimension [energy/volume].

Finally, requiring that the action (4.18) be stationary under variations with
respect to the metric, $\delta S/\delta g^{ab}=0$, we obtain the equations of
motion for a general spacetime in general relativity (the Einstein equations):

$R_{ab}-\frac{1}{2}Rg_{ab}+g_{ab}\Lambda=8\pi T_{ab}.$ (4.28)

In most instances, the cosmological constant is set to zero as the latest
(2021) experimental evidence seems to suggest that it is a very small number
(roughly $10^{-52}\,\text{m}^{-2}$ [110, 132]). In such instances, the
Einstein equations are commonly written in terms of the Einstein tensor
$G_{ab}$ as555In SI units: $G_{ab}=(8\pi G_{N}/c^{2})T_{ab}$.

$G_{ab}=8\pi T_{ab}.$ (4.29)

Even in cases where the cosmological constant is not set to zero, one can re-
define the stress energy tensor as
$(T_{ab})_{\text{new}}=(T_{ab})_{\text{old}}-\Lambda g_{ab}/8\pi$, thereby
writing the full Einstein equations (4.28) in the form of equation (4.29).

Although it may look simple, equation (4.29) actually constitutes a system of
ten, coupled, non-linear, second order, partial differential equations. As
such, finding exact solutions to the Einstein equations is, in general, a very
hard task. Historically, the Einstein equations have only been solved in
situations involving high degrees of mathematical symmetry [84]. For example,
the Schwarzschild solution is the unique, non-rotating (static), time-
independent (stationary), spherically symmetric solution; whilst the Kerr
solution is a rotating (non-static), time-independent (stationary), solution
with azimuthal symmetry. The Schwarzschild solution was discovered very
shortly after Einstein published his theory666Although, it took another 40 odd
years before it was realised that the solution represented a black hole
spacetime (to be defined in chapter 5). [122], whilst the Kerr solution was
not discovered until around 50 years later [81].

####  4.6.3 Energy conditions

In the case of a spherically symmetric spacetime, the stress-energy tensor of
equation (4.27) takes the form [156]

$T_{\hat{a}\hat{b}}=\begin{pmatrix}\rho&0&0&0\\\ 0&p_{r}&0&0\\\ 0&0&p_{t}&0\\\
0&0&0&p_{t}\end{pmatrix}_{\hat{a}\hat{b}},$ (4.30)

where $\rho$ is the energy density, $p_{r}$ is radial pressure, $p_{t}$
denotes the transverse pressure, and hats on the indices denote components in
an orthonormal basis.

It is commonly believed all physically reasonable classical matter will
satisfy a set of seven equations, known as the energy conditions. These
conditions are: the null, weak, strong, and dominant energy conditions (NEC,
WEC, SEC, DEC); as well as the averaged null, weak, and strong energy
conditions. In this thesis, we will primarily be concerned with the four non-
averaged energy conditions.

##### Null energy condition

The null energy condition is the statement that

$T_{ab}k^{a}k^{b}\geq 0,$ (4.31)

for any _null_ vector $k^{a}$. In terms of the components of the stress-energy
tensor (4.30), this is the statement that

$\text{NEC}\iff\rho+p_{r}\geq 0\mbox{\quad and\quad}\rho+p_{t}\geq 0.$ (4.32)

##### Weak energy condition

The weak energy condition asserts that

$T_{ab}V^{a}V^{b}\geq 0,$ (4.33)

for any _timelike_ vector $V^{a}$. In terms of the components of the stress-
energy tensor (4.30):

$\text{WEC}\iff\rho\geq 0\mbox{\quad and\quad}\text{NEC}.$ (4.34)

Thus, satisfaction of the WEC will imply that the NEC holds also.
Equivalently, if the NEC fails to hold, then so will the WEC. Physically, the
WEC is the statement that the local energy density as measured by a timelike
observer must be non-negative.

##### Strong energy condition

The strong energy condition is the statement that

$\left(T_{ab}-\frac{T}{2}g_{ab}\right)V^{a}V^{b}\geq 0,$ (4.35)

for any _timelike_ vector $V^{a}$, where
$T=T\indices{{}^{a}_{a}}=-\rho+p_{r}+2p_{t}$ is the trace of the stress-energy
tensor. In terms of the components of the stress-energy tensor (4.30):

$\text{SEC}\iff\rho+p_{r}+2p_{t}\geq 0\mbox{\quad and\quad}\text{NEC}.$ (4.36)

Thus, satisfaction of the SEC will automatically imply satisfaction of the
NEC, but not necessarily the WEC. Equivalently, if the NEC fails to hold, then
so will the SEC.

##### Dominant energy condition

The dominant energy condition asserts that for any _timelike_ vector $V^{a}$,

$T_{ab}V^{a}V^{b}\geq 0,\mbox{\quad and $T_{ab}V^{a}$ is a non-spacelike
vector.\quad}$ (4.37)

In terms of the components of the stress-energy tensor (4.30):

$\text{DEC}\iff\rho\geq 0\mbox{\quad and\quad}p_{r},p_{t}\in[-\rho,+\rho].$
(4.38)

Physically, this is the statement that the local energy density appears non-
negative _and_ that the energy flux is timelike or null. Note that
satisfaction of the DEC automatically implies satisfaction of the WEC and the
NEC, but not necessarily the SEC. Equivalently, if the NEC _or_ the WEC fails
to hold, then so will the DEC.

The interested reader can find a discussion of how the various energy
conditions are related to the singularity theorems in Ref. [156, pp. 118–119].
A somewhat non-technical discussion regarding the validity of the energy
conditions outside of general relativity is given in Ref. [11].

##  Chapter 5 Black hole mimickers: extensions to Einstein’s theory

In this chapter, we discuss what is considered to be one of the main issues
with classical general relativity – the prediction of spacetime singularities.
We then discuss how black hole mimickers may provide a classical resolution in
the specific case of black hole singularities, and then give some examples of
the different types of black hole mimickers commonly studied in the literature
and in this thesis.

###  5.1 Singularities

As we saw in chapter 4, if we have a spacetime $(\mathcal{M},\mathbf{g})$, we
can calculate various curvature tensors which provide insight into the
geometrical nature of the manifold. For example, consider the line element for
the Schwarzschild spacetime in curvature coordinates

$\differential s^{2}=-\left(1-\frac{2m}{r}\right)\differential
t^{2}+\frac{\differential
r^{2}}{1-\frac{2m}{r}}+r^{2}\big{(}\differential\theta^{2}+\sin^{2}\theta\differential\phi^{2}\big{)}.$
(5.1)

The Kretschmann scalar for this spacetime is

$K\equiv R^{abcd}R_{abcd}=\frac{48m^{2}}{r^{6}},$ (5.2)

and so $K\rightarrow\infty$ as $r\rightarrow 0$. That is, the Kretschmann
scalar is singular (in the mathematical sense) at $r=0$. In this instance, one
would say that there is a curvature singularity at $r=0$ in the Schwarzschild
spacetime.

Note, however, that the although the metric presented in equation (5.1) is
singular at $r=2m$, this is purely a result of the coordinate system we are
using. That is, a suitable change of coordinate system would remove the
singular nature of the metric components at $r=2m$111For example, the
Painlevé–Gullstrand coordinates (details may be found in references [9, 99].
See also reference [161]).. Thus, this type of singular nature of the metric
is typically referred to as a “coordinate artefact”, and is not considered to
be a true spacetime singularity. That being said, defining precisely what one
means by a ‘true’ spacetime singularity is in fact a very difficult task.

Naively, one may wish to define a singularity as a point in the spacetime
where at least one curvature invariant diverges to infinity. However, one
could simply remove such a point from the spacetime manifold and then claim
that the resulting spacetime is non-singular. Thus, the issue defining whether
or not a spacetime has a singularity has now become a problem of determining
whether or not any singular points have been removed. The notion of whether or
not a space has any ‘holes’ in it is easily formalised for Riemannian
manifolds.

Consider a manifold $\mathcal{M}$ and take a curve
$\gamma:[a,b]\subset\mathbb{R}\rightarrow\mathcal{M}$. Intuitively, one can
see how the point $p=\gamma(b)$ could be considered an endpoint of the curve
$\gamma$, and how in a ‘space without holes’ we would want any endpoints of a
curve to be contained in the space itself. We can formalise the notion of the
endpoint of a curve $\gamma:I\rightarrow\mathcal{M}$ for metric spaces (i.e.
Riemannian manifolds) with the following definition [70].

######  Definition 9.

A point $p$ in a manifold $\mathcal{M}$ is said to be an endpoint of a curve
$\gamma:I\rightarrow\mathcal{M}$ if for every neighbourhood $U$ of $p$, there
is a $\lambda\in I$ such that $\gamma(\lambda_{1})\in U$ for every
$\lambda_{1}\in I$ with $\lambda_{1}\geq\lambda$.

This allows us to formalise what we mean by a ‘space without holes’.

######  Definition 10.

A Riemannian manifold $\mathcal{M}$ is said to be metrically complete if every
$C^{1}$ curve of finite length has an endpoint. Furthermore, it is said to be
geodesically complete if every geodesic can be extended to arbitrary values in
its affine parameter.

For the case of Lorentzian metrics, one cannot define a metric space, and so
neither a notion of metric completeness. However, as Lorentzian metrics admit
the construction of geodesics, geodesic completeness can still be defined.
Lorentzian metrics further allow the classification of timelike, null, or
spacelike geodesic completeness depending on the sign of the norm of its
tangent vector. If a timelike geodesic is incomplete, this would imply the
possibility of a physical observer whose history does not exist after, or
before, a finite amount of proper time, which is certainly a physically
objectionable property of a (classical) spacetime. Thus, we will say that if a
spacetime is either timelike or null geodesically incomplete, it contains a
singularity. Note that we do not require the converse statement to be true, as
there are examples of spacetimes, such as the one constructed by Geroch [63],
which exhibit singularity-like properties but are still geodesically complete.
As such, some authors now require a more general condition for a spacetime to
be considered singularity-free [70].

######  Definition 11.

A spacetime $(\mathcal{M},\mathbf{g})$ is said to be bundle complete if every
affinely-parameterised $C^{1}$ curve of finite length has an endpoint.

######  Definition 12.

A spacetime is said to be singularity free, or contain no singularities, if it
is bundle complete.

We will adopt definition 12 as our formal definition of a singularity.
However, most of the time geodesic completeness is enough to classify
singularities, and bundle completeness is only needed for very technical
reasons. The interested reader may find more details in any of [63, 70, 161].

Defining singular spacetimes by the presence of incomplete curves of certain
classes is necessary for proving the singularity theorems (discussed below),
but does not provide us with any information about the location or type of
singularity present in the spacetime. However, all spacetimes which contain
curvature singularities are definitely bundle (and hence geodesic) incomplete.
Thus, in practice, one would usually look for singular points in the curvature
invariants of a particular spacetime of interest.

In the following chapters, we consider a type of singular spacetime known as a
black hole spacetime. In these spacetimes it will be clear where the
singularity is located by analysing the curvature invariants. In fact, if we
know we have a black hole spacetime, or the weaker condition of a trapped
surface, then in classical general relativity we can prove that such a
spacetime must necessarily contain a singularity. This is known as the Penrose
singularity theorem. We only present a simplified version of the theorem
below, as the full details are not directly relevant to the main work in this
thesis.

######  Theorem 1.

A spacetime $(\mathcal{M},\mathbf{g})$ cannot be null geodesically complete if

  1. (1)

$R_{ab}k^{a}k^{b}\geq 0$ for any null vector $k^{a}$;

  2. (2)

there is a Cauchy surface $\mathscr{H}$ in $\mathcal{M}$;

  3. (3)

$\mathcal{M}$ contains a closed trapped surface $\mathscr{T}$.

Condition (1) is known as the null convergence condition, and is implied by
the weak energy condition. Thus, a sufficient condition for it to hold is that
the energy density of the spacetime is positive for any observer (a physically
reasonable assumption for macroscopic spacetimes).

A Cauchy surface $\mathscr{H}$ is a subset of the manifold $\mathcal{M}$ which
is intersected _exactly once_ by every differentiable timelike curve in
$\mathcal{M}$. There are spacetimes which satisfy conditions (1) and (3) which
do not contain singularities, and as such condition (2) must be included. The
technical reasons for as to why the spacetime must contain a Cauchy surface
are not germane to the theme of this thesis, and are best understood by
completing the proof of the theorem. Details of the proof may be found in [70,
pp. 263–265].

Formally, a closed trapped surface is a closed, compact, spacelike two-surface
without boundary, such that the two families of null geodesics orthogonal to
$\mathscr{T}$ are converging at $\mathscr{T}$ [70, 99, 161]. Intuitively, one
may view $\mathscr{T}$ as being the surface at which even outgoing null
geodesics are dragged back inwards and forced to converge. As nothing physical
can travel faster than light, any matter inside $\mathscr{T}$ will be trapped
inside a succession of two-surfaces of ever decreasing area. Eventually, we
will get to a situation where we a forcing a massive object into a region so
small that it can no longer be considered physically reasonable. This apparent
breakdown of the theory is neatly captured by Penrose’s singularity theorem.
Note that theorem 1 implicitly assumes that the spacetime metric $\mathbf{g}$
is an exact solution to the Einstein equations (4.28), and so is strictly a
theorem within the framework of classical general relativity.

Perhaps the most important class of spacetimes containing trapped surfaces is
the black hole spacetimes. The formal definition of a black hole spacetime is
very technical, and will not be necessary for the work in this thesis. As
such, we will use a simplified definition of a black hole spacetime which
contains all of the ingredients necessary for the work in the following
chapters. The definition used here follows that of [156].

######  Definition 13.

For each asymptotically flat region, the associated future/past event horizon
is defined as the boundary of the region from which causal curves can reach
asymptotic future/past null infinity. Furthermore, we define a black hole to
be the region contained inside an event horizon, and say that a spacetime
containing a black hole is a black hole spacetime.

Note the important fact: _an event horizon is a trapped surface_. As such, any
black hole spacetime satisfying conditions (1) and (2) in theorem 1 will
necessarily contain a singularity.

Figure 5.1: A black hole spacetime embedded as a two-surface in four-
dimensional spacetime. The black hole contains a curvature singularity at its
core.

From a purely mathematical point of view, the fact the black holes contain
singularities causes no issues. However, as black holes are valid solutions to
the Einstein equations, and furthermore they can be shown to form by real,
physical stellar collapse models, this presents a real issue from a physical
standpoint. In spite of this, most of the time singularities are not an issue
for every-day physics as they are ‘hidden’ behind an event horizon, and so
physics outside of the black hole is unaffected. However, there are cases
where singularities do become an issue, such as during the very last stages of
a black hole’s evaporation – a time that plays an important role in
determining the compatibility of quantum mechanics and general relativity.
Many researchers now believe that a consistent theory of quantum gravity will
invoke sufficient changes to black holes at the Planck scale, which will stop
singularities forming [30, 31, 32, 33]. However, it is unlikely that such a
theory will be achieved any time in the near future, and as such it is useful
to consider classical modifications to black hole spacetimes which remove any
singularities which may be present. If such a modified spacetime mimics the
observable qualities of the singular black holes of classical general
relativity, we will call it a black hole mimicker. Note that as black hole
mimickers are non-singular geometries, by Penrose’s theorem they can not be
exact solutions to the vacuum Einstein equations. However, we can always
assume that they satisfy the non-vacuum equations, and then calculate what the
required stress-energy tensor will be for the non-singular spacetime.

We will now move on to discuss various types of black hole mimickers, specific
models of which will be analysed in subsequent chapters. Note that this is not
intended to be an exhaustive list; there are types of black hole mimickers
that we will not discuss in this thesis (for example, the Mazur–Mottola
gravastars [98, 97] and subsequent refinements [37, 87, 89, 93, 166]). We
will, however, discuss in some detail regular black holes and traversable
wormholes.

###  5.2 Regular black holes

The first class of black hole mimickers we will discuss is the so-called
regular black holes. A regular black hole is a black hole (in that it has a
well-defined event horizon) that does not contain a singularity at its core.
In this sense, one may say that the singularity has been “regularised”.

Regular black holes were first suggested as alternatives to black holes by
Bardeen in 1968, where he advocated the metric (now known as the Bardeen
regular black hole) [13]

$\differential
s^{2}=-f(r)\differential{t^{2}}+\frac{\differential{r^{2}}}{f(r)}+r^{2}\differential{\Omega}^{2},\quad
f(r)=1-\frac{2mr^{2}}{\big{(}r^{2}+q^{2}\big{)}^{3/2}}.$ (5.3)

Here
$\differential\Omega^{2}=\differential\theta^{2}+\sin^{2}\theta\,\differential\phi^{2}$
is the infinitesimal solid angle in spherical symmetry, and
$q=(2ml^{2})^{1/3}$ where $l$ is a length scale typically associated with the
Planck length. As $l$ is such a small number, $l\approx 1.616\times 10^{-35}m$
[132], the Bardeen spacetime was devised to be a regular spacetime which
differs only minimally from the Schwarzschild spacetime (c.f. equation (5.1)).
A simple (but tedious) calculation verifies that this is indeed a black hole
spacetime in the sense of definition 13, with a horizon characterised by the
location at which $g^{ab}\nabla_{a}r\nabla_{b}r=g^{rr}=f(r)$ changes sign.

One can easily check that all curvature invariants remain finite everywhere in
the spacetime222This is best done with computer algebra software packages.
Commonly used packages include Maple, Mathematica, and Sage Math., but we will
only present the calculation of the Ricci scalar for purposes of brevity:

$R=\frac{6mq^{2}(4q^{2}-r^{2})}{(r^{2}+q^{2})^{7/2}}.$ (5.4)

Note that as $r\rightarrow 0$ we have $R\rightarrow 24m/q^{3}=12/l^{2}$, and
so the spacetime is indeed regular. Also worthy of note is that in the
$r\rightarrow 0$ limit we have $R\rightarrow\mathcal{O}(l^{-2})$, that is, the
regular corrections are on order of magnitude of the Planck scale.

Since this is a black hole spacetime without a singularity, it must have a
non-zero stress-energy tensor (this can easily be seen by that fact that
$R\neq 0$). Indeed, the stress-energy tensor is found by the non-vacuum
Einstein equations (4.29) to have components

$T\indices{{}^{t}_{t}}=T\indices{{}^{r}_{r}}=-\frac{6mq^{2}}{8\pi(r^{2}+q^{2})^{5/2}},$
(5.5a)
$T\indices{{}^{\theta}_{\theta}}=T\indices{{}^{\phi}_{\phi}}=\frac{3mq^{2}(3r^{2}-2q^{2})}{8\pi(r^{2}+q^{2})^{7/2}}.$
(5.5b)

As we are working in spherical symmetry, the stress-energy tensor has the form
of equation (4.30), and we can write
$T\indices{{}^{a}_{b}}=\text{diag}(-\rho,\,p_{r},\,p_{t},\,p_{t})\indices{{}^{a}_{b}}$.
Note that _inside_ the horizon, $\rho$ and $p_{r}$ switch places, and so the
stress-energy tensor has the form
$T\indices{{}^{a}_{b}}=\text{diag}(p_{r},\,-\rho,\,p_{t},\,p_{t})\indices{{}^{a}_{b}}$.
Thus, outside the horizon, we have

$\rho=-p_{r}=\frac{6mq^{2}}{8\pi(r^{2}+q^{2})^{5/2}},$ (5.6a) whilst
$p_{t}=\frac{3mq^{2}(3r^{2}-2q^{2})}{8\pi(r^{2}+q^{2})^{7/2}}.$ (5.6b)

This then implies that $\rho+p_{r}=0$, whilst

$\rho+p_{t}=\frac{6mq^{2}}{8\pi(r^{2}+q^{2})^{5/2}}+\frac{3mq^{2}(3r^{2}-2q^{2})}{8\pi(r^{2}+q^{2})^{7/2}}=\frac{15mq^{2}r^{2}}{8\pi(r^{2}+q^{2})^{7/2}}\geq
0.$ (5.7)

Thus, we can conclude from equation (4.32) that the NEC is satisfied for the
Bardeen regular black hole. Similarly, since $\rho\geq 0$, we can conclude
that the WEC is globally satisfied (i.e. the local energy density as measured
by a timelike observer will be non-negative). The SEC, however, is a little
more subtle. We have that

$\rho+p_{r}+2p_{t}=\frac{3mq^{2}(3r^{2}-2q^{2})}{4\pi(r^{2}+q^{2})^{7/2}},$
(5.8)

and so from equation (4.36), we can conclude that the SEC will only be
satisfied in the region

$r\geq\frac{2}{3}q=\frac{2}{3}(2ml^{2})^{1/3}.$ (5.9)

This is a common theme that we will see with black hole mimickers: we can
force a regular black hole metric to satisfy the Einstein equations by
constructing the relevant stress-energy tensor, however said stress-energy
tensor does not globally satisfy all of the classical energy conditions. This
has lead to much discussion regarding the physical viability of black hole
mimickers [15, 16, 30, 31, 32, 33]. All black hole mimickers studied in this
thesis will fail to globally satisfy at least one of the classical energy
conditions. This is not necessarily an issue, however, as there are a number
of physical systems from a wide range of areas of physics which violate one or
more of the energy conditions. Perhaps the best well-known instance of this is
the Casimir effect, in which a physical system is shown to have negative
energy density, thereby violating the WEC and DEC. That being said, the scales
at which the Casimir effect is detectable induces a negative energy density
orders of magnitude smaller than what would be necessary to form, or stop the
collapse of, most black hole mimickers. The interested reader can find a
discussion of the Casimir effect and its relation to the energy conditions in
Ref. [156, pp. 121–126] (see also the original paper by Casimir [36], and the
technical references [17, 18]).

###  5.3 Wormholes

The next class of black hole mimicker we will introduce is the (traversable)
wormhole. Wormholes have a long history in general relativity: within only a
year of Einstein finalising his formulation of the field equations, Flamm
published an article hinting of possible spacetime “shortcuts” [58]. From
here, wormholes were then investigated as solutions to the Einstein equations
by Weyl in the 1920s [163], Einstein and Rosen in the 1930s [51], and by
Wheeler in the 1950s [164]. After this, though, the field lay relatively
dormant for close to 30 yers until the wormhole ‘renaissance’ in the late
1980s with the seminal papers by Morris, Thorne, and Yurtsever [103, 104].
This lead to rapid investigation into wormhole physics with many metrics of
interest being investigated [151, 152], and new ways of ‘constructing’
wormholes being discovered [111, 150]. Since then, the amount of research into
wormholes has slowed, but there is still a consistent and significant flow of
wormhole physics papers being published [15, 20, 47, 91, 128, 160].

A particularly lucid definition of a traversable wormhole spacetime is
provided by Morris and Thorne333Morris and Thorne actually provide many more
conditions than just the ones presented in definition 14. However, the extra
conditions they provide are primarily concerned with making the wormhole
physically traversable for a real, living human being – something we will not
worry about in this thesis. [103].

######  Definition 14.

A spacetime is said to contain a traversable wormhole if it satisfies the
following four conditions:

  1. (1)

The metric must satisfy the Einstein field equations (4.28) for some non-zero
stress-energy tensor.

  2. (2)

The spacetime must contain a region known as the ‘throat’. The throat region
must satisfy two conditions:

    1. (i)

it must connect two asymptotically flat regions of spacetime;

    2. (ii)

it must satisfy the ‘flare-out’ condition: the areas of the induced spatial
hypersurfaces on either side of the throat must be strictly increasing
functions of the distance from the throat.

  3. (3)

The spacetime cannot contain any horizons.

  4. (4)

The spacetime cannot contain any singularities.

Figure 5.2: An inter-universe wormhole embedded as a two-surface in four-
dimensional spacetime.

Condition (1) simply ensures that we stick to the confines of classical
general relativity. As with regular black holes, in practice one will
typically specify a metric that satisfies conditions (2)–(4) then use the
Einstein equations (4.28) or (4.29) to calculate the necessary stress-energy
tensor required to support such a geometry. Condition (2) specifies the
geometry of the wormhole spacetime. This agrees with our intuitive notion of a
wormhole as a “tunnel” between two regions of spacetime (c.f. figure 5.2).
Conditions (3) and (4) enforce the ‘traversability’ of the spacetime. If the
spacetime contained a horizon, only one-way propagation of timelike observer
would be allowed (anytime timelike or null geodesic would be dragged back
inwards towards the wormhole throat). If the spacetime contained a
singularity, the immense tidal forces it would produce would be sure to
destroy anything physical. Furthermore, the singularity would cause the throat
to be highly unstable and collapse in on itself in a very short amount of time
[103]. Note that in spherical symmetry, conditions (2) and (3) imply condition
(4).

Although traversable wormholes do not contain horizons, we will still consider
them to be black hole mimickers. The reason for this is two-fold:

  1. (1)

By definition, wormhole geometries do not contain singularities. As a result,
they rectify the physical concerns that result from singular black hole
spacetimes.

  2. (2)

We can construct the spacetime so that the wormhole throat lies arbitrarily
close to any would-be horizons in a corresponding black hole geometry (see
chapter 7). Thus, it would be hard for any astronomical observations to
distinguish between the two objects (the wormhole is “mimicking” the black
hole in an observational sense).

The canonical example of a wormhole spacetime is the Morris-Thorne traversable
wormhole [103, 104]. In curvature coordinates, this spacetime has line element

$\differential s^{2}=-\differential t^{2}+\differential
r^{2}+(r^{2}+b^{2})(\differential\theta^{2}+\sin^{2}\theta\differential\phi^{2}),$
(5.10)

where $b$ is a constant with dimensions of length. One can easily check that
this spacetime satisfies conditions (2)–(4) in definition 14, and represents a
static, spherically symmetric spacetime. Requiring that it be a traversable
wormhole solution to the Einstein field equations with $\Lambda=0$ (i.e.
condition (1)), forces the stress-energy tensor to have components

$\rho=p_{r}=-p_{t}=-\frac{b^{2}}{8\pi(r^{2}+b^{2})^{2}}.$ (5.11)

Thus,

$\rho+p_{r}=-\frac{b^{2}}{4\pi(r^{2}+b^{2})^{2}}<0,$ (5.12)

and so the NEC is violated globally in the Morris-Thorne wormhole spacetime.
This guarantees that the WEC, SEC, and the DEC will also be similarly violated
globally. Violation of the energy conditions is a common theme amongst
wormhole geometries, and is one of the reasons why physicists question their
physical validity [78, 156, 160].

Wormholes also bring to light many questions regarding causality and time
travel [72, 104, 152, 154]. If one were to construct (or discover) a
traversable intra-universe wormhole (i.e. connecting two regions of the same
universe), they could travel through the throat, along a specially chosen
timelike geodesic, and arrive back before they ever left. Many other paradoxes
arise when one considers using wormholes as time travel machines [72, 88, 86,
103, 156], which is yet another reason why some physicists believe that they
cannot possibly exist in nature. This is perhaps best summarised by Hawking’s
chronology protection conjecture, which asserts that the laws of physics will
conspire in such a way as to prevent using wormholes for time travel, except
for perhaps on microscopic or quantum scales. This is elegantly encapsulated
by Hawking’s famous quote [72]: “It seems there is a chronology protection
agency, which prevents the appearance of closed timelike curves and so makes
the universe safe for historians.”

Of course, there are other avenues of thought on the physical reality of
wormholes/time travel. One such example is Novikov’s self-consistency
conjecture. This states that if time travel via wormholes is possible, then
the laws of physics describing it will be such that the overall result is
always consistent. In their 1990 paper, Novikov et al. state [59]: “…the only
solutions to the laws of physics that can occur locally in the real Universe
are those which are globally self-consistent.” This is particularly nice, as
it does not conjecture any new law of physics that we have no experimental
evidence for. Instead, it simply claims that the universe will continue to
behave in the same way that we have observed it to so far; we have no evidence
that the universe is in any way internally inconsistent. Novikov’s conjecture
also agrees with Hawking’s (pseudo)-experiment [72]: “we have not been invaded
by hordes of tourists from the future.”

##  Chapter 6 Regular black hole with asymptotically Minkowski core

In this chapter, we analyse a specific example of a regular black hole
spacetime with an asymptotically Minkowski core. We calculate the radius of
the photon sphere and the extremal stable timelike circular orbit (ESCO),
which are (at least, in theory) physically observable quantities. The manner
in which the photon sphere and ESCO relate to the presence (or absence) of
horizons is much more complex than for the Schwarzschild black hole. We find
situations in which photon spheres can approach arbitrarily close to (near
extremal) horizons, situations in which some photon spheres become stable, and
situations in which the locations of both photon spheres and ESCOs become
multi-valued, with both ISCOs (innermost stable circular orbits) and OSCOs
(outermost stable circular orbits). This provides an extremely rich
phenomenology of potential astrophysical interest.

###  6.1 Introducing the spacetime

Without any loss of generality, any static spherically symmetric spacetime can
be described by a metric of the form

$\differential{s}^{2}=-\mathrm{e}^{-2\Phi(r)}\left(1-\frac{2m(r)}{r}\right)\differential{t^{2}}+\frac{\differential{r^{2}}}{1-\frac{2m(r)}{r}}+r^{2}\left(\differential{\theta^{2}}+\sin^{2}\theta\differential{\phi^{2}}\right).$
(6.1)

For the standard Schwarzschild metric (equation (5.1)) one sets $\Phi(r)=0$
and $m(r)=m_{0}$. Over the past century, a vast host of black hole spacetimes,
qualitatively distinct from that of Schwarzschild, have been investigated by
multiple researchers [9, 10, 80, 82, 105, 106, 115, 142, 143, 144, 149, 162,
167].

Furthermore, the field has now grown to not only include classical black
holes, but also quantum-modified black holes [4, 28, 29, 79], regular black
holes [6, 13, 30, 62, 73], and various other exotic spherically symmetric
spacetimes that are fundamentally different from black holes but mimic many of
their observable phenomena (_e.g._ traversable wormholes [20, 39, 47, 78, 91,
103, 104, 111, 127, 128, 150, 151, 152, 154, 156, 160], gravastars [37, 87,
89, 93, 97, 98, 166], ultracompact objects [45, 46], etcetera [33, 157, 159];
see [31] for an in-depth discussion).

The model spacetime investigated in this work is a regular black hole with an
asymptotically Minkowski core, as discussed in [15, 129]. This is an example
of a metric with an exponential mass suppression, and is described by the line
element

$\differential
s^{2}=-\left(1-\frac{2m\,\mathrm{e}^{-a/r}}{r}\right)\differential{t}^{2}+\frac{\differential{r}^{2}}{1-\frac{2m\,\mathrm{e}^{-a/r}}{r}}+r^{2}\left(\differential{\theta}^{2}+\sin^{2}\theta\differential{\phi}^{2}\right).$
(6.2)

A rather different (extremal) version of this model spacetime, based on
nonlinear electrodynamics, has been previously discussed by Culetu [41], with
follow-up on some aspects of the non-extremal case in references [42, 43, 44].
See also [77, 117].

Most regular black holes have a core that is asymptotically de Sitter (with
constant positive curvature) [6, 13, 62, 73]. However, the regular black hole
described by the metric (6.2) has an asymptotically Minkowski core (in the
sense that the stress-energy tensor asymptotes to zero). This model has some
attractive features compared to the more common de Sitter core regular black
holes: the stress-energy tensor vanishes at the core, greatly simplifying the
physics in this region; and many messy algebraic expressions are replaced by
simpler expressions involving the exponential and Lambert $W$ functions,
whilst still allowing for explicit closed form expressions for quantities of
physical interest [129]. Additionally, the results obtained in this work
reproduce the standard results for the Schwarzschild metric by letting the
parameter $a\rightarrow 0$. Thus, the value of the parameter $a$ determines
the extent of the “deviation” from the Schwarzschild spacetime.

If $0<a<2m/\mathrm{e}$ then the spacetime described by the metric (6.2) has
two horizons located at

$r_{H^{-}}=2m\;\mathrm{e}^{W_{-1}\left(-\frac{a}{2m}\right)},\qquad\hbox{and}\qquad
r_{H^{+}}=2m\;\mathrm{e}^{W_{0}\left(-\frac{a}{2m}\right)}.$ (6.3)

Here $W_{-1}(x)$ and $W_{0}(x)$ are the real-valued branches of Lambert $W$
function. We could also write

$r_{H^{-}}={a\over|W_{-1}\left(-\frac{a}{2m}\right)|},\qquad\hbox{and}\qquad
r_{H^{+}}={a\over|W_{0}\left(-\frac{a}{2m}\right)|}.$ (6.4)

Perturbatively, for small $a$ we have

$r_{H^{+}}=2m-a+\mathcal{O}(a^{2}),$ (6.5)

nicely reproducing Schwarzschild in the $a\to 0$ limit. For the inner horizon,
since $r_{H_{-}}<2m$ then

$r_{H^{-}}={a\over\ln(2m/r_{H_{-}})}$ (6.6)

implies $r_{H^{-}}<a$, whence we have a strict upper bound given by the simple
analytic expression:

$r_{H^{-}}<{a\over\ln(2m/a)}.$ (6.7)

Certainly $\lim_{a\to 0}r_{H^{-}}(m,a)=0$ as we would expect to recover
Schwarzschild; but the form of $r_{H^{-}}(m,a)$ is not analytic. This bound
can also be viewed as the first term in an asymptotic expansion [38] based on
(as $x\to 0^{+}$)

$W_{-1}(-x)=\ln(x)+\mathcal{O}(\ln(-\ln(x)))=-\ln(1/x)+\mathcal{O}(\ln(\ln(1/x))).$
(6.8)

This leads to

$r_{H^{-}}={a\over\ln(2m/a)+\mathcal{O}(\ln(\ln(2m/a)))}={a\over\ln(2m/a)}+\mathcal{O}\left(a\ln(\ln(2m/a))\over(\ln(2m/a))^{2}\right).$
(6.9)

More specifically (as $a/m\to 0$ or $m/a\to\infty$)

${r_{H^{-}}\over
a}={1\over\ln(2m/a)}+\mathcal{O}\left(\ln(\ln(2m/a))\over(\ln(2m/a))^{2}\right).$
(6.10)

If $a=2m/\mathrm{e}$ then the two horizons merge at $r_{H}=2m/\mathrm{e}=a$
and one has an extremal black hole. If $a>2m/\mathrm{e}$ then there are no
horizons, and one is dealing with a regular horizonless, extended but highly
localised object, (the energy density peaks at $r=a/4$).

This object could either be extended all the way down to $r=0$, or