SCALAR FIELD DARK MATTER ENCASING A SUPER MASSIVE BLACK HOLE
BY
WILLIAM, A, DULANEY
A Thesis Submitted to the Graduate Faculty of
WAKE FOREST UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES
in Partial Fulfillment of the Requirements
for the Degree of
MASTER OF SCIENCE
Mathematics & Statistics
May 2021
Winston-Salem, North Carolina
Approved By:
Sarah G Raynor, Ph.D., Advisor
Jason Parsley, Ph.D., Chair
Abbey Bourdon, Ph.D. © 2021 William A. Dulaney ALL RIGHTS RESERVED
ii Acknowledgements
I am grateful for the support and guidance given by my thesis advisor Dr. Sarah Raynor. Her pa- tience has extended well beyond expectation and I will always keep in mind the “trivial” boundary terms in my future endeavors. Additionally I would like to acknowledge the magnitudes of support given in collaboration with my colleagues, Benjamin Hamm and Dr. Hubert Bray, at Duke Uni- versity. Above all, I would like to extend my thanks to my parents for the unlimited support they have given me during the COVID-19 pandemic. Again, thank you to my previous professors and mentors from Appalachian State University who have helped me achieve my goals. Thank you for all of your encouragement.
iii Contents
List of Figures v
List of Tables v
Abstract vii
1 Introduction viii
2 General Relativity 1 2.1 History and Motivation ...... 1 2.2 Basic Geometric Background ...... 2 2.2.1 Tangential Structures ...... 3 2.2.2 Tensors and Metrics ...... 5 2.2.3 Connections and Curvature ...... 6 2.3 Derivation of the Field Equations ...... 12
3 Scalar Field Dark Matter 18 3.1 Dark Matter ...... 18 3.2 Axioms and the Einstein Klein Gordon System ...... 20 3.3 Effective Field Limits ...... 22
4 Spherical Harmonics 27 4.1 Solution for Scalar Field Dark Matter ...... 28
5 Scalar Field Dark Matter Around a Super Massive Black Hole 32 5.1 Black Holes ...... 32 5.2 Singular Perturbations ...... 35 5.2.1 Zeroth Order and Energy ...... 38 5.2.2 First Order ...... 41 5.3 Solution Scheme and Plots ...... 45
6 Conclusion and Future Work 48
CV 51 List of Figures
1 Gravitational lensing from a concentration of mass can bend the light of a distant object as the light passes. For more details see [19]...... 2 2 Depiction of a tangent bundle of a manifold M [7] ...... 4 3 A graph relating possible dark matter candidates. Image: G. Bertone and T. Tait. [2] 19 4 “Exact solution to the Klein-Gordon equation in a fixed spherically symmetric po- tential well based on the Milky Way Galaxy at t = 0, t = 10 million years, and t = 20 million years. The pictures show the dark matter density (in white) in the xy plane. This solution, which one can see is rotationg, has angular momentum.” For more details see [3]...... 19 5 “Visual representations of the first few real spherical harmonics. Blue portions rep- resent regions where the function is positive, and yellow portions represent where it is negative. The distance of the surface from the origin indicates the absolute value m of Y` (θ, φ) in angular direction (θ, φ)(θ, φ) ” For more details see wiki- [20]. . . . 27 6 “The gravitational-wave event GW150914 observed by the LIGO Hanford (H1, left column panels) and Livingston (L1, right column panels) detectors. ” For more details see [1]...... 34 7 “EHT images of M87 on four different observing nights. In each panel, the white circle shows the resolution of the EHT. All four images are dominated by a bright ring with enhanced emission in the south.” For more details see [11]...... 34 2 8 Side view of |f0(r)| for the m = ` = N = 0 state of the dark matter field. . . . . 45 2 9 Top down view of |f0(r)| for the m = ` = N = 0 state of the dark matter field. . 46 2 10 A side view of |f0(r)| for the m = ` = 1,N = 0 state of the dark matter field. . . 47 2 11 Top down view of |f0(r)| for the m = ` = 1,N = 0 state of the dark matter field. 47
List of Tables
1 A Few Associated Legendre Polynomials ...... 31
v List of Symbols
Here the reader can find a list of symbols commonly used throughout this thesis. Each symbol is defined as follows unless otherwise stated or clear from context within the text.
Physical Constants
Υ Bosonic mass of dark matter c Speed of light in a vacuum inertial frame 299, 792, 458 m/s
−11 N·m2 G Newton’s Constant 6.67430 × 10 kg2 h Planck constant 6.62607 × 10−34 Js
Number Sets
C The set of complex numbers
Z The set of integers
R The set of real numbers
N The set of natural numbers, including zero Other Symbols
G The Einstein curvature tensor
T The stress energy tensor
∇ The connection or covariant derivative
g The Laplacian operator on the spacetime with respect to the metric g (D’Alembertian) ∆ The Laplacian operator in Euclidean space, will be used in spherical coordinates
Γ(T M) A section of the tangent bundle of a manifold M
i Γjk The connection coefficients
∂ ∂x The coordinate vector field ∂x corresponding to the coordinate function x a the patch g The spacetime metric
R The scalar curvature
Ricc The Ricci curvature tensor
Riem The Riemann curvature tensor
vi Abstract
This is a survey of various results in a model for dark matter called scalar field dark matter, which is expressed in terms of the Einstein-Klein-Gordon system. The low field limit of this system is derived in terms of a spherical symmetry and the result is found to be the Schrodinger-Poisson¨ system. An analysis of the Schrodinger-Poisson system is undertaken with a goal of finding a solution in terms of spherical harmonics. Further work is done to apply a 1/r like potential to the system in order to model a scalar field of dark matter around a black hole. After undergoing a perturbation expansion, a zeroth order solution has been completed and further analysis on the first order equation is underway.
vii 1 Introduction
Finding a complete description of the physical laws that govern our universe has proven to be a timeless endeavour of mankind. From the discovery of fire in prehistory, to the foundations of the standard model in particle physics, humans have collectively made massive strides toward achiev- ing this goal. Nonetheless, there is still much left to be discovered as both mathematics and physics continuously press further into the unknown.
As of the writing of this thesis, it is generally accepted that everything in the universe that we might consider as ordinary matter only accounts for about 5% of what is actually there. The other 95% is comprised of a mix of dark matter and dark energy, where the term “dark” is used simply because they do not absorb or emit photons detectable by any currently known means. The focus of this thesis will be to survey a current development in theoretical physics and extend the implications to a specific example around a black hole. This theory in question suggests dark matter can be mathe- matically expressed as a scalar field geometry.
We begin this journey by delving into some semi-Riemannian geometry to help readers become accustomed to the language used in both Einstein’s theory of general relativity and the subsequent extension to the Einstein-Klein-Gordon system. Sections 2-4 can be considered background mate- rial and a compilation of various literature sources on Riemannian geometry and scalar field dark matter (SFDM). The sections on the latter are mostly based on the works of Dr. Hubert Bray of
Duke University along with two of his previous Ph.D. students, Dr. Alan Parry and Dr. Andrew
Goetz [3], [4] et al. Sections 5-6 are new interests in a low field SFDM theory, which the author is currently exploring. Specifically new these works are an extension of known theory to the dynamics of such a dark matter scalar field encompassing a black hole, in what is known as the low field limit.
viii 2 General Relativity
In the 17th century, Sir Isaac Newton revolutionized mechanics when he formulated a theory of gravitation. This initial description of the force of gravity between two bodies of mass is given by
GMm F = (2.1) r2
where G is Newton’s gravitational constant, M, m are the masses of the two bodies in question, and r is the radial distance between them. This idea revolutionized physics over the next few centuries. In the mid 1800s, Urbain Le Verrier discovered a precession of Mercury’s orbit around the Sun that could not be described by Newton’s theory of gravity[21]. These findings, which were published in Determination´ nouvelle de l ’orbite de Mercure et de ses perturbations, once again opened discussions on how to correctly describe gravity.
2.1 History and Motivation
If a random person were asked what Albert Einstein was most famous for, more often that not the answer will be the equation E = mc2; an expression relating the energy and rest mass of a system.
This is a result derived in Einstein’s theory of special relativity, which was not even the research that earned him the Nobel Prize. Nonetheless, this work was originally published in 1905 in a paper titled On the Electrodynamics of Moving Bodies[8]. The special theory of relativity was merely a stepping stone into what would soon again revolutionize how humanity saw the world around them.
Throughout the early 1900s Einstein worked tirelessly with other prestigious physicists and math- ematicians, such as Grossman and Hilbert, to rigorously incorporate gravity into his 1905 theory.
This work was completed by 1915 and given the name general relativity[9]. Since published, re- searchers continue to conduct experiments and observations to confirm relativity. Fascinations such as black holes, gravitational lensing, and the procession of Mercury’s orbit have each been predicted and/or confirmed via this theory.
1 Figure 1: Gravitational lensing from a concentration of mass can bend the light of a distant object as the light passes. For more details see [19].
The following sections will focus on the mathematics needed to approach the derivation of what are called Einstein’s Field Equations, the central equations of general relativity.
2.2 Basic Geometric Background
The material in this section is based on well known developments in differential geometry, much like that found in [5],[7], [10] et al. Throughout this thesis a notation known as Einstein’s summa- tion notation will be incorporated. The convention requires that contravariant components will be
i denoted with an upper index as a where as covariant components will have a lower index as ej, for i, j ∈ N. These indices denote the i-th and j-th component of these objects. In this notation, summation over a repeated diagonal index is implied. This becomes necessary later on as we often have entities that are summed over many indexing sets at once and writing many different sigmas becomes tedious. To see how this works, consider a vector v whose basis elements are given as
{b1, b2, . . . , bn}. Then in terms of this basis
n X i 1 2 n v = a bi = a b1 + a b2 + ... + a bn (2.2) i=1
2 in the Einstein notation, (2.2) is expressed as
i v = a bi. (2.3)
2.2.1 Tangential Structures
Definition 2.1. Consider the smooth manifold M, with a tangent space TpM for each p ∈ M, we construct the tangent bundle T M which joins together all tangent vectors in M as the disjoint union of all tangent spaces,
· F S T M = TpM = {p} × TpM. p∈M p∈M
For this we construct the bundle projection map defined as, for v ∈ TpM,
π : T M → M , π(p, v) = p. (2.4)
We denote the corresponding dual space T ∗M, the cotangent bundle.
Definition 2.2. A cotangent bundle to the n-manifold M is the space T ∗M of all covectors located at each point on M. Like with the tangent bundle, a point in T ∗M is a pair (p, α) where α is a
covector located at p ∈ M. That is
· ∗ F ∗ S ∗ T M = Tp M = {p} × Tp M p∈M p∈M ∗ As before we need a projection mapping, π, where for ω ∈ Tp M
π : T ∗M → M , π(p, ω) = p. (2.5)
We will think of a general vector bundle as the 4-tuple (E, M, Y, π) where E is the total space or
bundle space, M is the base space, Y is a vector space, and π is the projection mapping.
Recall that the projection mapping on a bundle gives a mapping from the bundle to the base mani-
fold. To move from the manifold up to the bundle, we define what is called a section.
Definition 2.3. Given the total space E of a vector bundle (E, M, Y, π), a smooth section of E is the smooth map
s : M → E
3 Figure 2: Depiction of a tangent bundle of a manifold M [7] such that for each p ∈ M we select some element of the fiber s(p) ∈ π−1(p). That is
π ◦ s = idM.
We denote the space of sections on a bundle E as Γ(E). It follows that since s is a vector valued
mapping, a vector space structure is induced on Γ(E).
Example 2.4. An element of the smooth section of a (co-)vector bundle is a smooth (co-)vector field. Let X ∈ Γ(T M) and ω ∈ Γ(T ∗M). Then in Einstein’s summation notation, the vector field
X and covector field ω are expressed respectively as
∂ X = Xi and (2.6) ∂xi j ω = ωjdx . (2.7)
Note the summation over the vector field’s components, Xi, is over diagonal repeated indices; likewise, the covector’s components are summed diagonally over j.
4 2.2.2 Tensors and Metrics
Definition 2.5. Let V be a vector space and V ∗ be its dual. Then we define an (r, s)−tensor as the multilinear map
∗ ∗ ∗ T : V × V × ... × V × V × V × ... × V → R | {z } | {z } r−times s−times
∗ where for v, u, w ∈ V or V as appropriate and a, b ∈ R.
T (v1, ..., au + bw, ..., vr+s) = aT (v1, ..., u, ..., vr+s) + bT (v1, ..., w, ..., vr+s).
There are many ways to define a tensor however, this definition will suffice for our purposes. As
quick examples, a (0, 0) tensor is a scalar, . We now will turn to a few fundamental objects of study
in general relativity – spacetime, events, and the structures therein.
Definition 2.6. A metric, g, on a smooth manifold M is a (0, 2)−tensor field satisfying, for smooth vector fields X,Y ∈ Γ(T M)
1. Symmetry: g(X,Y ) = g(Y,X) for all X,Y ∈ Γ(T M)
2. Non-degeneracy: For [ : Γ(T M) → Γ(T ∗M)
such that X 7→ [(X) , where [(X)(Y ) := g(X,Y ). It must be that [(X) = g(X, ·) is an
isomorphism and [ is C∞.
In the context of linear algebra it is generally required, for vectors v, w in an inner product space
(V, g), that g(v, w) ≥ 0 for all v, w and g(v, w) = 0 for all w if and only if v = 0. This is a positive
definite condition instead of non-degenerate. Here we allow the weaker non-degenerate condition
when building a semi-Riemannian manifold, which is needed for general relativity. The stronger
positive definite condition is used when building Riemannian manifolds. Note that positive definite
implies non-degeneracy.
Definition 2.7. A manifold M equipped with a symmetric and non-degenerate metric as defined above is called a semi-Riemannian manifold.
Furthermore, we can talk about a sort of “inverse” map that takes in covectors and spits out a scalar
function.
5 Definition 2.8. The symmetric (2,0) tensor field, g−1, with respect to a metric g is map g−1 : Γ(T ∗M) × Γ(T ∗M) → C∞(M) such that for ω, σ ∈ Γ(T ∗M),
(ω, σ) 7→ ω([−1(σ)).
Since the metric tensor connects a vector space with its dual in an isomorphic manner, it is also a
tool for “lowering” indices. Note that in this notation, we simply have
m ([(X))a = gamX (2.8)
−1 a −1 am am [ (ω) = (g ) ωm := g ωm and (2.9)
ab a g gab = δb . (2.10)
We generally think of the maps [, [−1 as operations to raise and lower indices. Such operations are a common practice when solving and manipulating the field equations.
Definition 2.9. An event is a point in time and space labeled by coordinates. A manifold of events endowed with a metric is called a spacetime.
It is important to note the event itself exists independent of a choice of coordinates and reference frame. Let us now move forward to defining more sophisticated structures on manifolds.
2.2.3 Connections and Curvature
Definition 2.10. Let T(M) be the set of (p, q)−tensor fields on M. A linear affine connection ∇ on a smooth manifold M is defined as a mapping
D : Γ(T M) × T(M) → T(M)
Such that
∞ 1. DV W is function-linear, or C (M)-linear, in V ,
2. DV W is R-linear in W ,
∞ 3. DV (fW ) = (V f)W + f(DV W ) for f ∈ C (M).
6 To make this more clear, let f ∈ C∞(M), X,T ∈ Γ(T M), ω ∈ Γ(T ∗M) and S, T be (1, 1) tensor
fields on M. Then
1. DX f = Xf
2. DX (T + S) = DX T + DX S
3. DX T (ω, Y ) = (DX T )(ω, Y ) + T (DX ω, Y ) + T (ω, DX Y )
4. DfX+Y T = fDX T + DY T .
Often times the connection is denoted ∇. The choice of this notation alludes to how a connection is most commonly used in the special case of the covariant derivative, which will be discussed shortly.
For now, consider two vector fields X,Y ∈ Γ(T M). Then
m ∂ i m ∂ i m ∂ ∇X Y = ∇Xi ∂ Y m = X Y ∇ ∂ m + X ∇ ∂ Y m (2.11) ∂xi ∂x ∂xi ∂x ∂xi ∂x ∂ ∂ ∂ = Xi Y m + XiY mΓq , (2.12) ∂xi ∂xm mi ∂xq
q where the Γmi are called the connection coefficients. More formally we make the following defini- tion.
Definition 2.11. Given a manifold M and a coordinate chart (U, x) for which U ⊂ M, the con- nection coefficients with respect to (U, x) are the (dim(M))3 many functions such that for p ∈ U
i Γjk : U → R, by i ∂ p 7→ dx ∇ ∂ ∂xj (p). ∂xk
In terms of components, it can be shown that the ith term of the connection is given by
∂ (∇ Y )i = Xm Y i + Γi Y nXm. (2.13) X ∂xm nm
As a side remark, on a chart domain U, the choice of connection coefficients suffices to fix the action of ∇ on a vector field. For more details on general connections, see [7] et al. From here we will develop a specific connection, known as the Levi-Civita connection. First we need to define the notions of torsion free and metric compatibility.
7 Definition 2.12. The torsion tensor T is defined as the mapping
T : Γ(T M) × Γ(T M) → Γ(T M) such that for a connection ∇ and vector fields X and Y
T (X,Y ) = ∇X Y − ∇Y X − [X,Y ].
Here [X,Y ] is the lie bracket of X and Y . We denote the connection being torsion free as T (X,Y ) =
∞ 0. Torsion free implies [X,Y ] = ∇X Y − ∇Y X. Then given f ∈ C (M), as a mild sanity check we can verify the connection is behaving as previously defined. Comparing the definition of the lie bracket on a function and the torsion free result,
[X,Y ]f = X(Y f) − Y (Xf) (from the definition of the lie bracket)
[X,Y ]f = (∇X Y − ∇Y X)(f) (from the torsion free condition)
= ∇X (Y f) − ∇Y (Xf)
= X(Y f) − Y (Xf).
i Definition 2.13. Consider a collection of connection coefficients Γba. We define the symmetric and antisymmectic connection coefficients respectively as:
i i i i • {Γ(ba)} = {Γba | Γba = Γab}
i i i i • {Γ[ba]} = {Γba | Γba = −Γab}.
a a Let Γ(bc) and Γ[bc] denote the symmetric and antisymmetric connection coefficients respectively.
In a chart, the torsion tensor is expressed as
i i ∂ ∂ i ∂ ∂ ∂ ∂ T = T dx , , = dx ∇ ∂ − ∇ ∂ − , (2.14) ab a b a b a a b ∂x ∂x ∂x ∂x ∂xb ∂x ∂x ∂x ∂ ∂ ∂ ∂ = dxi Γi − Γi − , (2.15) ab ∂xq ba ∂xq ∂xa ∂xb
i i i = Γab − Γba = 2Γ[ab]. (2.16)
8 Given a torsion free connection, it follows that the anti-symmetric connection coefficients vanish as
a a a Γ[bc] = Γbc − Γ(cb) = 0 and so
a a Γbc = Γ(bc). (2.17)
Definition 2.14. We define metric compatibility with the chosen connection, vector fields X,Y,Z, and the metric g as
X(g(Y,Z)) = g(∇X (Y ),Z) + g(Y, ∇X (Z))
Such a connection ∇ is called a metric connection.
Theorem 2.15. The Levi-Civita Connection: For a given semi-Riemannian manifold M, there exists a unique torsion free, metric connection ∇.
Proof. Let M be a semi-Riemannian manifold equipped with a torsion free metric compatible con-
∂ ∂ nection ∇. Define ∂xa gbc := gbc,a and ∂xa := ∂a. We begin with the following three computations
n n 1. (∇∂a g)bc = gbc,a − Γbagnc − Γcagbn
n n 2. (∇∂b g)ca = gca,b − Γcbgna − Γabgcn
n n 3. (∇∂c g)ab = gab,c − Γacgnb − Γcbgna.
Note that each of the three computations above must be equivalently zero since ∇ is metric compat-
a a ible. Recall that the torsion free condition implies Γbc = Γ(bc). If we add the first two items above and subtract the second, we see
0 = (∇∂a g)bc + (∇∂b g)ca − (∇∂c g)ab
n n n n n n = gbc,a − Γbagnc − Γcagbn + gca,b − Γcbgna − Γabgcn − gab,c + Γacgnb + Γcbgna
n = gbc,a + gca,b − gab,c − 2Γbagnc
nc n = g (gbc,a + gca,b − gab,c) − 2Γba
n Solving for Γba yields
9 n 1 nc Γba = 2 g (gbc,a + gca,b − gab,c).
Since each the metric and its derivatives are unique, it follows that the connection coefficients are unique as well. Hence the connection is uniquely determined. A connection satisfying these prop- erties is known as both the Levi-Civita connection and covariant derivative.
Lemma 2.16. The connection coefficients themselves are not tensors, but their difference is.
Proof. Consider two coordinate charts (U, x) and (V, y) on a manifold M. It can be shown that the
connection coefficients transform as follows
i ∂yi ∂xn ∂xq m ∂yi ∂2xm Γ(y)jk = ∂xm ∂yj ∂yk Γ(x)nq + ∂xm ∂yj ∂yk .
Hence for a different connection on the same charts, we have
ˆi ∂yi ∂xn ∂xq ˆm ∂yi ∂2xm Γ(y)jk = ∂xm ∂yj ∂yk Γ(x)nq + ∂xm ∂yj ∂yk .
Thus the difference between the two, given as
i n q i ˆi ∂y ∂x ∂x m ˆm Γ(y)jk − Γ(y)jk = ∂xm ∂yj ∂yk Γ(x)nq − Γ(x)nq .
This is exactly the transformation law a tensor must satisfy.
Lemma 2.17. There exists a coordinate system (x1, . . . , xn) at a point p in a neighborhood U such
i that Γjk(p) = 0. This system is called normal coordinates.
Proof. See [5].
The rest of this chapter will assume a torsion free connection. No discussion of general relativity can happen without a solid notion of curvature. To this extent, we will consider four types of curvature; to begin, we define the Riemann curvature tensor as follows.
Definition 2.18. Let M be a semi-Riemannian manifold equipped with the connection ∇, X,Y,Z ∈ Γ(T M), and ω ∈ Γ(T ∗M). The Riemann curvature of the connection ∇ is the (1, 3) tensor field
Riem(ω, Z, X, Y ) = ω ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ]Z .
10 As with the previously discussed tensors, we can determine the coefficients of the Riemann curva- ture tensor with respect to a chart (U, x) in terms of the connection functions. To start, define the following vector fields
∂ ∂ ∂ ω = dxi,Z = := ∂ ,X = := ∂ ,Y = := ∂ . (2.18) ∂xj j ∂xk k ∂xm m
Then evaluating Riem(ω, Z, X, Y ) on these fields yields
i i i r i r i Riemjkm = Riem(dx, ∂j, ∂k, ∂m) = ∂kΓjm − ∂jΓkm + ΓjmΓrk − ΓjkΓrm. (2.19)
A fun application:
Theorem 2.19. Let M be a 1-manifold with the connection ∇. Then M has no curvature.
1 Proof. Note that on a 1 dimensional manifold, the only connection coefficient is Γ11. This implies
i 1 1 1 1 1 1 Riemjkm = ∂1Γ11 − ∂1Γ11 + Γ11Γ11 − Γ11Γ11 = 0.
i Therefore Riemjkm = 0, hence M has no curvature.
Definition 2.20. Again let M be a semi-Riemannian manifold equipped with the connection ∇ and let X,Y ∈ Γ(T M). Then the Ricci curvature tensor is a (0, 2) tensor defined as
m a b Rab(X,Y ) := RiemambX Y .
The reader should note that the Ricci curvature is essentially a contraction of the Riemann curvature,
m meaning we take a trace over a diagonal set of indices. Generally this is expressed as Riemamb =
Rab. Further contractions are done to define the scalar curvature R.
Definition 2.21. Given a semi-Riemannian manifold M equipped with the connection ∇, we define the scalar curvature R in the following equivalent ways
ab R = trgRab = g Rab .
Finally, we define the curvature most known to those who study general relativity, the Einstein
Curvature tensor.
11 Definition 2.22. Let M be a manifold equipped with a Lorentzian metric, a metric with signature (− + ++), and the connection ∇, then the Einstein Curvature tensor is defined to be
1 Gab = Rab − 2 gabR.
These constructions are central to the theory of relativity. Many of the developments that were cutting edge in the early 20th century just happened to be exactly what Einstein needed to formulate his theory of gravitation. This allowed much collaboration between him and some of the greatest mathematicians of the 20th century. In the next section we will derive the equations that changed physics as we know it.
2.3 Derivation of the Field Equations
As any physicist may know, variational principles and the Euler-Lagrange equations, as will be discussed below, are central to understanding the mechanics of a given system. The mathematics of variational calculus is now widely used to not only describe the classical kinematics of massive particles, but also to determine the complex nature of mass-less fields. The use of these principals will be key in understanding both the derivation of Einstein’s field equations as well as deriving the
Einstein-Klein-Gordon system for a scalar field dark matter theory in the next chapter.
To proceed, we need the following notions from variational calculus. A functional is map that takes in a function y(x) and assigns to it a real number in a well defined manner. So a functional is a a map much like a function, except its domain is a function space. This will be made clear in the following examples.
Example 2.23. Arclength, s, of a curve y(x) is a functional. One can see directly that
R b p 0 2 s = J(y) = a 1 + y (x) dx
maps the function y(x) to a real number s, its associated arclength between two points.
As with the original goal of generalizing the minimization problem in calculus, modern uses of
functionals often are associated with what is called the Lagrangian.
Example 2.24. Consider the following functional
12 R b 0 J(y) = a L(x, y(x), y (x))dx, y ∈ A.
The Lagrangian, L(x, y, y0), is some given function and A is a well defined class of functions. The
integrand of the arclength functional, p1 + y0(x)2, is a great example of a Lagrangian. Generally, these are functionals that we wish to minimize.
As stated above, these ideas were first developed in order to generalize the notions of minimization from calculus. For intuition, one can think of how a physicist might want to minimize energy in a given physical system. In calculus, one finds the local minimum by solving y0(x) = 0. For a functional, we will take a similar approach with the Gateaux derivative of a functional.
Definition 2.25. Let V be a normed linear space, A ⊂ V , and J : A → R be a functional on A.
For y0 ∈ A and h ∈ V such that, for a small > 0, y0 + h ∈ A; the Gaˆteaux derivative of a functional J at y0 in the direction of h is defined by
0 d δJ(y0, h) ≡ J (0) ≡ d J(y0 + h) =0 , if it exists. Here we define the function J() = J(y0 + h). The expression δJ(y0, h) is called the
first variation. Such a direction h for which δJ(y0, h) exists is called an admissible variation at y0.
One can show quite easily from the definition that variations act just like one would expect a direc- tional derivative. That is
J(y0 + h) − J(y0) δJ(y0, h) = lim . (2.20) →0
That is to say, the well known rules for differentiation hold for variations.
Theorem 2.26. Let J : A → R be a functional, A ⊂ V . If y0 ∈ A provides a local minimum for J relative to the norm || · ||, then
δJ(y0, h) = 0 for all admissible variations h.
Proof. See [12].
13 As a note, the action J(g) often may be denoted J[g], where g is the function to be varied. This
notation will be used extensively in the following discussion when we derive the field equations.
For more details on functionals and variational principles, see [12].
Einstein worked in close collaboration with Hilbert in formalizing his theory. As the story goes,
Einstein derived his field equations from physical intuition, whereas Hilbert used calculus of vari-
ations to derive the equations from varying a metric. The Hilbert action was postulated by Hilbert
himself in order to determine how matter should curve spacetime. Some speculate that the Hilbert
action was first written down as it is the simplest action one could write down involving curvature.
Definition 2.27. The Hilbert action is a first variation of the metric, gab given as
R √ ab 4 SH [g] = M −gRabg d x.
R √ ab 4 Theorem 2.28. The result of varying the action SH [g] = M −gRabg d x with respect to gab ab ab ab 1 ab will be the Euler-Lagrange equations G = 0 for the tensor G = R − 2 g R.
Proof. We begin with minimizing the action by
Z h √ ab √ ab √ ab i 4 0 = δSH [g] = δ −gg Rab + −gδg Rab + −gg δRab d x. (2.21) M
We consider each variational derivative separately. First calculate
√ det(g)gabδg 1√ δ −g = − √ ab = −ggabδg . (2.22) 2 −g 2 ab
Second, we consider
a ab δb = g gab
ab 0 = δ(g gab)
ab ab 0 = (δg )gab + g (δgab) and hence
ab am bn δg = −g g δgmn. (2.23)
14 Finally we consider the variation of the Ricci tensor, in normal coordinates. Recall that the terms involving the connection coefficients alone drop out by Lemma (2.17), therefore
m m δRab = δ (∂bΓab − ∂mΓab)
m m = ∂bδΓam − ∂mδΓab. (2.24)
i ˆi Note that by Lemma (2.16) the difference of two connection coefficients, Γjk − Γjk, is actually a m tensor. This is exactly what the δΓam means. Therefore we can rewrite our expression as
m m δRab = ∇∂b (δΓ)am − ∇∂m (δΓ)ab . (2.25)
ab m b ab m m Hence it follows that if we define the vectors g δΓ am = A and g δΓ ab = B , then
√ ab √ ab m m −gg δRab = −gg (∇∂b (δΓ)am − ∇∂m (δΓ)ab) m m √ ab ab = −g ∇∂b g δΓ − ∇∂m g δΓ am ab √ b m = −g ∇∂bA − ∇∂mB
√ b m = −g ∂bA − ∂mB . (2.26)
Collecting the terms, one obtains
0 = δSH [g] Z 1√ mn ab √ am bn √ b m 4 0 = −gg δgmng Rab − −gg g δgmnRab + −g ∂bA − ∂mB d x. M 2 (2.27)
√ b m Observe here that the −g ∂bA − ∂mB is a divergence, and hence has no effect on the system as a boundary term. So we may discard it. This leaves us with
15 Z √ 1 mn mn 4 0 = −gδgmn[ g R − R ]d x. (2.28) M 2
Since this action vanishes for an arbitrary variation, we must have the terms inside the bracket vanish. Thus we obtain
1 Gmn = Rmn − gmnR = 0. (2.29) 2
Thus from the variation on the Hilbert action, we have derived what is known as the Einstein vacuum equations as given in equation (2.29). These are the equations that are to govern a universe with no matter. However, these are not the full Einstein field equations. If we work in natural units where the speed of light, c, and Newton’s gravitational constant, G, are each equal to 1; then we can vary
the following Einstein-Hilbert action in combination with an action describing matter density
Z Z √ 4 √ ac bd 4 SE−H [g] + Smatter[A, g] = −g(R − 2Λ)d x + −gFabFcdg g d x. (2.30) M M
The result is the full system of Einstein’s field equations
Gab + Λgab = 8πTab. (2.31)
We omit the proof, however interested readers can see [6] for details. Here Λ is a scalar known
as the cosmological constant, a value which relates to what is known as dark energy. Dark energy
is simply a stress energy that exists everywhere in space, yet does not react with other matter even
though it accounts for about two thirds of the energy in the universe [14]. The G is the same Einstein
curvature tensor as derived from the Hilbert action. Finally the tensor T is the stress energy tensor,
which relates to the energy density of the system. It is interesting to note, Einstein had thought the
cosmological constant was his “biggest blunder.” However we now believe that the cosmological
constant, albeit small, is nonzero. The reader might take this time to think back to equation (2.1).
Though Newton’s description of gravity looks nothing like the field equations, he wasn’t entirely
16 wrong. With enough work the inverse square law of gravity can be derived from the field equations as an approximation in a special case. This however will not be shown here.
17 3 Scalar Field Dark Matter
The scope of this section is to motivate the significance of studying dark matter as a scalar field.
This view is developed axiomatically in [3], where it is conjectured that scalar field dark matter
(SFDM) can be studied as the deviation of the connection on spacetime from the standard
Levi-Civita connection on the tangent bundle. It is important to emphasize that no quantum me- chanical assumptions are made throughout this discussion even though the equations derived in the main discussion of this paper are reminiscent to those of the hydrogen atom. Further reading on
SFDM and the effects in spiral galaxies can be found in [3] , [16] , et al.
3.1 Dark Matter
Dark matter describes the abundance of mass within the universe that does not interact with the electromagnetic field, hence making it hard to detect. Even though dark matter is over five times more abundant than the types of matter we are used to, physicists have so far been unable to observe it directly. Despite this, there is an abundance of evidence for its existence. In particular this includes, but is not limited to, instances of gravitational lensing, orbiting speed of gas and dust in spiral galaxies and galactic clusters, and the overall distribution of galaxies throughout the universe
[3] et. al.
This evidence first appears to contradict the predictions of general relativity. However as we will explore in the next section, dark matter can actually be incorporated into the field equations if modeled as a scalar field. As the reader will see, altering the field equations in this way will only yield an extension to the field, analogous to the cosmological constant, and thus will not contradict general relativity.
Understanding dark matter has become one of the biggest problems in astrophysics. So far, many possible candidates of dark matter are being explored, see Figure (3). Quantum mechanics does well in describing baryonic matter, which dark matter is not, and although there do exist frameworks in which quantum mechanics attempts to describe dark matter as a particle; we will explore dark matter through the lens of general relativity. If this theory proves to be successful at describing dark matter, general relativity would then successfully account for 95% of the matter in the universe.
18 Figure 3: A graph relating possible dark matter candidates. Image: G. Bertone and T. Tait. [2]
Scalar field dark matter is also known as fuzzy dark matter and wave dark matter. A SFDM theory is a naturally cold dark matter model in a “homogeneous, isotropic universe sufficiently long after the big bang” [3]. Recent works have been successful in showing SFDM can describe observable features of the universe, such as spiral patterns in galactic structures [3]. However, our interests will now shift to the mathematical nature of the theory.
Figure 4: “Exact solution to the Klein-Gordon equation in a fixed spherically symmetric potential well based on the Milky Way Galaxy at t = 0, t = 10 million years, and t = 20 million years. The pictures show the dark matter density (in white) in the xy plane. This solution, which one can see is rotationg, has angular momentum.” For more details see [3].
19 3.2 Axioms and the Einstein Klein Gordon System
We first consider one of the three main axioms outlined in [3] that will lead to a more general form of Einstein’s field equations. A more detailed discussion of these can be found in [3] as well. We assume the spacetime has a general connection, ∇, instead of the standard Levi-Civita connection,
∇¯ . That is to say, for example, the connection need not be torsion free. We make this choice in order to view the general connection as a fundamental object of the spacetime. Let the manifold N be smooth, Hausdorff, and second countable; assume also both the metric g and the connection ∇ are smooth. Given a fixed chart let {∂i}, for i ∈ {0, 1, 2, 3}, be tangent vector fields to N . Denote:
0 0 gij = g(∂i, ∂j), Γijk = g(∇∂i ∂j, ∂k), M = {gij}, C = {Γijk}, M = {gij,k}, and C = {Γijk,`}.
P αβ αβ Definition 3.1. Define QuadY ({xα}) = α,β F (Y )xαyα, for some functions {F }, to be the quadratic expression of the {xα} with coefficients in Y .
Axiom 1: As found in [3].
4 1. For all coordinate charts Φ:Ω ⊂ N → R and open sets U whose closure is compact and contained within Int(Ω), (g, ∇) is a critical point of the functional:
Z 0 0 FΦ,U (g, ∇) = QuadM (M ∪ M ∪ C ∪ C)dVR4 (3.1) Φ(U)
with respect to smooth variations of the metric and connection, compactly supported in U for
some fixed quadratic functional QuadM with coefficients in M.
This is just an assumption which restricts the types of actions we want to allow. With a little bit of work one can show the action used to derive Einstein’s field equations satisfies this quadratic form of the action in the axiom above. Yet still there are many others for which the classification is an open question [3]. We now will formulate the scalar field that is to become our dark matter field.
Definition 3.2. Denote the Levi-Civita connection as ∇¯ and a general connection as ∇. Let X,Y,Z ∈ Γ(T M) and define the difference tensor as
D(X,Y,Z) = g(∇X Y,Z) − g(∇¯ X Y,Z). (3.2)
20 To start, fix a coordinate chart and let {∂i}, for 0 ≤ i ≤ 3 be the tangent vector fields on the manifold
M which correspond to the standard basis of the coordinate chart. By letting gij = g(∂i, ∂j) and
Γijk = g(∇∂i ∂j, ∂k) we find an expression for D in coordinates as
Dijk = Γijk − Γ¯ijk (3.3) where again the connection coefficients for the Levi-Civita connection are the standard ¯ 1 Γijk = 2 (gik,j + gjk,i − gij,k). As was previously discussed, the Levi-Civita connection is torsion free, therefore
Tijk = Dijk − Djik = Γijk − Γ¯ijk − Γjik − Γ¯jik
= Γijk − Γjik. (3.4)
We now will define the fully antisymmetric part of the difference tensor to be γijk, which has the form
1 γ = (T + T + T ) ijk 6 ijk jki kij 1 = (D − D + D − D + D − D ) 6 ijk jik jki kji kij ikj 1 = (Γ − Γ + Γ − Γ + Γ − Γ ) . (3.5) 6 ijk jik jki kji kij ikj
1 Note that γijk is also 2 the fully antisymmetric part of the torsion tensor T . The reader should also note here that since the difference in connection coefficients is a tensor, hence the γijk are components of a (3, 0) tensor field i.e. a 3−form. Taking the exterior derivative, we obtain
dγijk = γjk`,i − γk`i,j + γ`ij,k − γ`ijk,`. (3.6)
Let v ∈ Γ(T M) with the dual v∗ ∈ Γ(T ∗M) such that γ = ∗(v∗). Here ∗ denotes the Hodge star.
Recall that when the dimension of the manifold is 4, the Hodge dual is a map
21 ∗ : k − forms −→ (4 − k) − forms. (3.7)
Note that |γ|2 = −|v|2 and |dγ|2 = −(∇ · v)2. We now will define the scalar field
f = Υ∇ · v (3.8) for a real parameter Υ > 0. Using the above formulation for f and an action of the form (3.1), it can be shown [3] that the following action satisfies axiom 1
Z 2 |df| 2 F (g, f) = R − 2Λ − 16πµ0 2 + |f| dV. (3.9) Φ(U) Υ
As outlined in [3],[16] and much like the derivation of the field equations previously discussed; variations on the metric g yield (3.10) and varying the scalar field f yields (3.11) below. Thus the
Einstein Klein-Gordon system for a complex scalar field f is as follows.
df ⊗ df¯+ df¯⊗ df |df|2 G + Λg = 8πµ − + |f|2 g (3.10) 0 Υ2 Υ2
2 gf = Υ f (3.11) where G is the Einstein curvature tensor, f is a scalar field, Υ is a new constant of nature (currently an unknown value), and µ0 is the energy density of an oscillating scalar field of magnitude one and is solely a function of t. We will be working on a scale small enough for which it is fair to assume the cosmological constant Λ = 0. Recall f represents a measure of the deviation of ∇ from ∇¯ and so f = 0 implies ∇ = ∇¯ . This value of f recovers the vacuum field equations.
3.3 Effective Field Limits
We now begin an analysis of the Einstein-Klein-Gordon system in a low field and low energy setting.
As the reader will soon see this simplification is chosen to reduce the Einstein-Klein-Gordon system to the Schrodinger-Poisson¨ system. Not only does it greatly simplify the equations in question, but there also exists interesting applications to cosmological objects such as galaxies and black holes.
22 Both of these applications allow for possible verification through astronomical observations. To start choose the low field metric, which has a line element of the form
ds2 = −(1 + 2V )dt2 + (1 − 2V )dr2 + (1 − 2V )r2 dθ2 + sin2(θ)dφ2 . (3.12)
This is simply a perturbation of the metric as gij = ηij + hij, where ηij = δij is the Kronecker delta. In other words, we are considering small perturbations of the Minkowski spacetime. For our
purposes, we are interested in an equation involving the potential and energy density of the scalar
field. This means we must solve the following for our scalar field f and some vector field v0:
G(v0, v0) = 8πT(v0, v0) 1 = Ricc(v , v ) − Rg(v , v ) 0 0 2 0 0 1 = Ricc(v , v ) + R. (3.13) 0 0 2
Luckily the resulting Einstein curvature tensor over the low field metric is well known [6]. For our conditions, this is found to be
G00 = 2∆V = 8πT00. (3.14)
Here we are using the convention t, r, θ, φ are the 0, 1, 2, 3 coordinates respectively. Therefore we arrive at the equation
∆V = 4πT00. (3.15)
For this setup, it is known [6] that T00 encodes the energy density ρ of the system. Thus we can think of equation (3.15) as
∆V = 4πρ. (3.16)
This is known as the Poisson equation. We still need to evaluate T00 on our scalar field f to obtain
23 the coupled equation we desire. To begin, define the following vector field:
v0 = −(1 + 2V )∂0. (3.17)
It should be stated here the low field conditions assert the following:
∂ψ E = i << 1 (3.18) ∂t f = ψeiΥt. (3.19)
Calculate the exterior derivative of the scalar field f to find
0 1 2 3 df = f0dx + f1dx + f2dx + f3dx (3.20)
and
0 1 2 3 df¯ = f¯0dx + f¯1dx + f¯2dx + f¯3dx . (3.21)
We then calculate
|df|2 = g(df, df¯)
0 0 1 1 2 2 3 3 = f0f¯0g(dx , dx ) + f1f¯1g(dx , dx ) + f2f¯2g(dx , dx ) + f3f¯3g(dx , dx )
2 2 2 = −f0f¯0(1 + 2V ) + f1f¯1(1 − 2V ) + f2f¯2(1 − 2V )r + f3f¯3(1 − 2V )r sin (θ)
2 2 2 2 2 2 2 = −|f0| (1 + 2V ) + (1 − 2V ) |f1| + r |f2| + r sin (θ)|f3| . (3.22)
Then we can calculate the component T00 by evaluating on the vector field defined in (3.17). Ob- serve then,
24 2 |df|2 T(v , v ) = µ df(v )df(¯v ) − + |f|2 g(v , v ) 0 0 0 Υ2 0 0 Υ2 0 0 2 |df|2 = µ f (1 + 2V )f¯ (1 + 2V ) + + |f|2 (1 + 2V ) 0 Υ2 0 0 Υ2 2 |df|2 = µ |f |2(1 + 2V )2 + + |f|2 (1 + 2V ) 0 Υ2 0 Υ2 (1 + 2V )(1 − 2V ) = µ (1 + 2V )|f|2 + |f |2 + |f |2 + r2|f |2 + r2 sin2(θ)|f |2 0 Υ2 0 1 2 3 (3.23)
2 2 2 2 2 2 2 Observe here |f0| + |f1| + r |f2| + r sin (θ)|f3| is really ∇f · ∇f for a flat metric. So by [4], we apply the result that
∇f · ∇f << 1. (3.24) Υ2
Likewise, we apply the low energy condition for V << 1, then we have the result
2 T(v0, v0) = µ0|f| . (3.25)
As shown in [3] it will suffice for our purposes to consider units where µ0 = 1. Therefore we have
2 finally T(v0, v0) = |f| and hence as desired
∆V = 4π|f|2. (3.26)
For the following results, let f be a general complex scalar field. Expanding the d’Alembert operator in terms of the metric g we find:
1 p ij ij ¯k gf = ∇ · ∇f = ∂i |g|g ∂jf = g ∂i∂jf − Γij∂kf (3.27) p|g|
p We make a note here that with the low energy condition, |g| ∼ 1 − 4V and ∂i(1 − 4V ) ∼ 0. So
2 when we expand gf = Υ f we find for µ ∈ {1, 2, 3}
25 2 µµ 2 2 − ∂0 f + g ∂µf = −2V Υ f. (3.28)
Note that with f = ψ(x, t)e−iΥt, we have
−iΥt ∂0f ∼ −iΥψ(x, t)e (3.29)
2 −iΥt 2 −iΥt ∂0 f ∼ −2iΥ∂0(ψ(x, t))e + (iΥ) ψ(x, t)e . (3.30)
Placing this substitution into equation (3.28), we observe
2 −iΥt 2 2 −iΥt 2 −iΥt − 2iΥ∂0(ψ(x, t)) + Υ ψ(x, t) e − ∇ − Υ ψ(x, t)e = −2V Υ ψ(x, t)e (3.31)
2 2 − (2iΥ∂0(ψ(x, t))) − ∇ ψ(x, t) = −2V Υ ψ(x, t) (3.32)
Thus as desired, the Einstein-Klein-Gordon system reduces to
1 i∂ ψ = − ∇2ψ + ΥV ψ (3.33) t 2Υ ∆V = Υ2ψ. (3.34)
26 4 Spherical Harmonics
In this section we will use spherical harmonics in an attempt to find a full solution to the Schrodinger–¨
Poisson system. Spherical harmonics are defined as the eigenfuctions of the angular component of
3 3 the Laplacian operator in R . They are often used when solving partial differential equations in R , specifically when there is a known spherical structure. Such a structure is just what we will soon utilize in the following sections. These spherical harmonic solutions are known to work nicely with some radially dependent potentials and spherical separation of variables, as was done historically with the hydrogen atom in quantum mechanics. As we will soon show, an analogous solution can be found for the Schrodinger–Poisson¨ system with a gravitational potential for a point mass.
Figure 5: “Visual representations of the first few real spherical harmonics. Blue portions represent regions where the function is positive, m and yellow portions represent where it is negative. The distance of the surface from the origin indicates the absolute value of Y` (θ, φ) in angular direction (θ, φ)(θ, φ) ” For more details see wiki- [20].
27 4.1 Solution for Scalar Field Dark Matter
As previously discussed, the low field limit of the Einstein-Klein-Gordon system is the Schrodinger-¨
Poisson system. This is true both in cases where the scalar field f is complex and where it is a real scalar field. The low field limit of complex and real scalar fields are equivalent. We now aim to analyze the system in a special case where the potential is only radially dependent. Recall the system we are considering is:
1 i∂ f = − ∇2f + ΥV f (4.1) t 2Υ ∆V = Υ2f. (4.2)
Using the ansatz f = F (r)Y (θ, φ)eiωt = F Y eiωt, V (r) = V , where Υ is the “bosonic” mass, we
find solutions reminiscent to those of the hydrogen atom. Since the potential depends only on the radius, we need only consider the Schrodinger¨ part of the system for now. Observe,
1 i∂ f = − ∇2f + ΥV f (4.3) t 2Υ 1 iF Y ∂ eiωt = − ∇2(F Y eiωt) + ΥV F Y eiωt. (4.4) t 2Υ
So by expanding the Laplace operator in spherical coordinates and evaluating, we find
2 1 2 1 1 −2Υ VFY − 2ΥF Y ω = − ∂r(r FrY ) − ∂ (sin(θ)∂ (FY )) − ∂ (FY ) r2 r2 sin(θ) θ θ r2 sin2(θ) φφ
2 2 1 1 −2r ΥFY (ω + ΥV ) = −∂r(r FrY ) − ∂ (sin(θ)∂ (FY )) − ∂ (FY ) sin(θ) θ θ sin2(θ) φφ ∂ (r2F ) ∂ (sin(θ)Y ) Y 2r2Υ(ω + ΥV ) − r r = θ θ + φφ . (4.5) F sin(θ)Y sin2(θ)Y
It then follows that since each side of the equation is equal for all values of r, θ, φ; they each must
28 be equal to the same constant B. Observe then
∂ (r2F ) r2Υ(ω + ΥV ) − r r = B (4.6) 2F ∂ (sin(θ)Y ) Y θ θ + φφ = B. (4.7) sin(θ)Y sin2(θ)Y
Suppose now the angular components are also separable, that is Y (θ, φ) = Θ(θ)A(φ) = ΘA. Then
(4.7) separates from
∂ (sin(θ)(ΘA) ) (ΘA) θ θ + φφ = B (4.8) sin(θ)ΘA sin2(θ)ΘA to
sin(θ)∂ (sin(θ)Θ ) θ θ − sin2(θ)B = m2 (4.9) Θ (A) φφ = −m2. (4.10) A
We choose m ∈ N such that (4.10) has the solution
m imφ −imφ A (φ) = c1e + c`2 e . (4.11)
Since we wish to have a symmetry such that Am(φ) = Am(φ + 2π), (4.11) reduces to
Am(φ) = ceimφ (4.12)
for a to be determined constant c and m = ±m.
We now rewrite (4.9) as follows
sin(θ)∂ (sin(θ)Θ ) θ θ − sin2(θ)B = m2 Θ ∂ (sin(θ)Θ ) m2 θ θ + −B − Θ = 0. (4.13) sin(θ) sin2(θ)
Define x(θ) = cos(θ) and P (cos(θ)) = Θ. Note that
29 d dx d d = = − sin(θ) (4.14) dθ dθ dx dx
− sin2(θ) = cos2(θ) − 1 = x2 − 1. (4.15)
Then equation (4.13) becomes
d d m2 − − sin2(θ) P (x) + −B − P (x) = 0 dx dx sin2(θ) d d m2 (1 − x2) P (x) + −B − P (x) = 0 dx dx (1 − x2) m2 (1 − x2)P 00(x) − 2xP (x)0 + −B − P (x) = 0. (4.16) (1 − x2)
This is an associated Legendre type differential equation, for which we have the solutions
∞ ∞ ! m m X a2n 2n X a2n+1 2n+1 P (x) = (1 − x) 2 a x + a x (4.17) ` 0 a 1 a n=0 0 n=0 1 where the ` dependence is in B(`) = −B = `(` + 1) and
(n + m)(n + m + 1) − B(`) a = . (4.18) n+2 (n + 1)(n + 2)
After substituting x = cos(θ) back into the solution, the associated Legendre polynomials can also be obtained[18] explicitly by
dm P m(cos(θ)) = (−1)m(sin(θ))m/2 (P (cos(θ)), where (4.19) ` d(cos(θ))m ` 1 d` P (cos(θ)) = [(cos2(θ) − 1)`]. (4.20) ` 2``! d(cos(θ))`
A chart of a handful of these polynomials will be supplied below in Table (1). This leaves the radial equation from (4.6) to become
30 ∂ (r2F ) r r − 2r2Υ(ω + ΥV ) − B(`) = 0 F 1 `(` + 1) ∂ (r2F ) − 2Υ(ω + ΥV )F − F = 0 r2 r r r2 2F `(` + 1) F + r − 2Υ(ω + ΥV )F − F = 0. (4.21) rr r r2
In conclusion, the equation (4.21) yields the second part of the system of equations, coupled still with ∆V = 4π|f|2. Additionally, we have full solutions to the angular and azimuthal equations for
m m m Y` (θ, φ) = Θ` (θ)A (φ) as follows,
s (2` + 1)(` − m)! Y m(θ, φ) = Θm(θ)Am(φ) = (−1)m P m(cos(θ))eimφ. (4.22) ` ` 4π(` + m)! `
m The first handful of associated Legendre polynomials P` (cos(θ)) are listed in the following table [18]:
Table 1: A Few Associated Legendre Polynomials 0 0 0 1 2 P0 = 1 P1 = cos(θ) P2 = 2 3cos (θ) − 1 1 1 1 3 2 P1 = − sin(θ) P2 = −3 sin(θ) cos(θ) P3 = − 2 5 cos (θ) − 1 sin(θ) 2 2 2 2 P2 = 3 sin (θ) P3 = 15 cos(θ) sin (θ)
mq (2`+1)(`−m)! ` m imφ iωt ` The full solution is f(r, θ, φ) = (−1) 4π(`+m)! F (r)P` (cos(θ))e e , where F (r) solves equation (4.21) and the coupled Poisson equation. We now consider a special case for f, namely for a field of dark matter around a super massive black hole.
31 5 Scalar Field Dark Matter Around a Super Massive Black Hole
5.1 Black Holes
In 1783 an amateur astronomer named John Michell used Newton’s equations to postulate the exis- tence of cosmic objects whose escape velocity was the speed of light[14]. John Michell calculated that a star 500 times larger than the sun, with a comparable density, would have such an escape velocity. One can naively substitute the speed of light, c, into the escape velocity equation to obtain an expression known as the Schwarzchild radius,
2GM R = (5.1) s c2 where G is Newton’s gravitational constant. Since the Schwarzchild radius seemed to be unreal- istically small at the time, especially too small to be within the realm of observation, the thought experiment held little interest. For example, the Schwarzchild radius for a 1 solar mass object is just under 3 kilometers.
This of course changed with the development of General Relativity. The Schwarzchild solution to
Einstein’s field equations falls seemingly into our laps with minimal assumptions in fact, this is one of the simplest solutions to the field equations. A Schwarzchild black hole is known to be a static, non-rotating, uncharged, black hole described by the metric
!2 2 p dr 2 2 (ds)2 = cdt 1 − 2GM/rc2 − − (rdθ) − (rsinθdφ) . (5.2) p1 − 2GM/rc2
As one can quickly see, the Schwarzchild radius is the point at which the radicals in the metric go
to zero. Loosely, this yields the implications that motion in the dt direction freezes and we gain divergence in the dr direction. Additionally, by setting ds = dθ = dφ = 0, we find the coordinate speed of light in the radial direction to be
dr 2GM 2 R 2 = c 1 − = c 1 − s . (5.3) dt rc2 r
32 This tells us for r >> Rs, the radial velocity dr/dt ∼ c as in an asymptotically flat spacetime. Here asymptotically flat means the perturbation from Eurclidean space drops off by at least 1/r.
Furthermore, we find at r = Rs, that dr/dt = 0 which tells us light is frozen at the Schwarzchild radius. The spherical surface at r = Rs is called the event horizon. At the center of the event horizon is the black hole’s singularity, a point of zero volume and infinite density. This singularity is where the entirety of the black hole’s mass lives and is a region where spacetime becomes infinitely curved.
It is conjectured that the singularity can never be observed, as naked singularities are prevented by the “Law of Cosmic Censorship” [14]. For more basic information on black holes, see section 17.3 of [14].
Although these black holes were theorized in the early 1900s as solutions to Einstein’s field equa- tions, it was not until recently that significant observational evidence for black holes was published.
First in 2015 it is evident that the first detected gravitational waves [1] occurred due to a pair of black hole mergers. Even more recently, a large collaboration has successfully imaged [11] a black hole for the first time. Each of these instances are illustrated in the figures below.
33 Figure 6: “The gravitational-wave event GW150914 observed by the LIGO Hanford (H1, left column panels) and Livingston (L1, right column panels) detectors. ” For more details see [1].
Figure 7: “EHT images of M87 on four different observing nights. In each panel, the white circle shows the resolution of the EHT. All four images are dominated by a bright ring with enhanced emission in the south.” For more details see [11].
We now wish to examine the behavior of scalar field dark matter around a black hole in the same spherical symmetry as described previously. To do so, we will consider a perturbation method using the low field limit radial equation (4.21).
34 5.2 Singular Perturbations
For a super massive black hole, we will approximate the potential energy in the low field limit as a Newtonian gravitational potential. Namely we will use the potential Veff = Vself + VBH where
Vself is unknown and, sufficiently far from the singularity,
−GM V = BH . (5.4) BH r
For simplicity here let F denote the radial solution F `(r). We also will define
d2 2 d `(` + 1) F `(` + 1) L F := + − F = F + 2 r − F. (5.5) ` dr2 r dr r2 rr r r2
Thus the reduced Schrodinger–Poisson¨ system is
L`F = 2Υ(ω + Υ(Vself + VBH ))F
2 ∆(Vself + VBH ) = 4π|f| . (5.6)
We first will view the SFDM-Black Hole system as perturbations on the dark matter field due to a large mass. Since the gravitational contributions near the event horizon will be mostly due to the black hole itself, we will initially solve the Schrodinger–Poisson¨ system with a negligible dark matter self potential using a singular perturbation method. This implies the radial component of the scalar field will be small compared to the potential of the black hole, that is for a small parameter > 0
F = O(VBH ). (5.7)
1 It is well known that the Laplacian of function proportional to r , in spherical coordinates, is as follows
1 ∆ = −4πδ(r − r0) (5.8) ||r − r0|| where δ in this case is the Dirac delta measure. However, sufficiently far from the singularity
35 ∆VBH = 0, hence it follows that ∆V = ∆(Vself + VBH ) = ∆Vself . As previously derived, we
` m −imφ iωt know f = F (r)P` (cos(θ))e e . Therefore the wave part of the system becomes
2 ∆Vself = 4π|f| . (5.9)
It then follows for a fixed `, m
2 ∆Vself = 4π|f|
` m −imφ iωt 2 = 4π|F (r)P` (cos(θ))e e |
` m 2 −imφ iωt 2 = 4π|F (r)P` (cos(θ))| |e e |
` 2 m 2 = 4πF (r) P` (cos(θ)) . (5.10)
From equations (5.9) and (5.10), we use Green’s Functions to express the self potential as
1 V = ∗ (|f|2) self r 1 = ∗ (F 2P m(cos(θ))2). (5.11) r `
Here ∗ denotes the convolution. It follows directly from equations (5.7) and (5.11) that
1 V ∼ ∗ F 2 = O(2). (5.12) self r
Now suppose F can be expressed as the perturbation expansion
2 F = F0 + δF1 + δ F2 + ... (5.13)
σ where δ ∼ for some σ ∈ N. This leads equation (5.11) to become
36 1 V = ∗ (F 2P m(cos(θ))2) self r ` 1 = ∗ ((F + δF + δ2F + ...)2P m(cos(θ))2) r 0 1 2 ` 1 = ∗ ((F 2 + 2δF F + δ2F 2 + ...)P m(cos(θ))2). (5.14) r 0 0 1 1 `
For the lowest term in the potential expansion above, define the re-scaling
1 1 ∆Vˆ = (F P m cos(θ))2 = ∆V = O(1) and so, (5.15) F0 2 0 ` 2 F0 1 Vˆ = ∗ (F P m cos(θ))2 . (5.16) F0 2r 0 `
Then by substituting these expansions into the radial equation in (5.6) and separating each equation in powers of δ, we find
L`F0 = 2Υ(ω + ΥVBH )F0 (5.17)
2 2 ˆ δL`F1 = 2δΥ(ω + ΥVBH )F1 + 2 Υ VF0 F0 (5.18) . .
Solutions beyond first order will not be considered in this thesis. Nonetheless by analyzing equation
(5.18) we see the first interesting solution, beyond zeroth order, will occur at second order in δ.
Equivalently, let δ = 2. Then the system of interest becomes
L`F0 = 2Υ(ω + ΥVBH )F0 (5.19)
2 ˆ L`F1 = 2Υ(ω + ΥVBH )F1 + 2Υ VF0 F0 (5.20) 1 Vˆ = ∗ (F P m(cos(θ))2 . (5.21) F0 2r 0 `
Solution methods for this system will be considered in the following sections. Let us begin with the
37 leading order solution, where the potential is dominated by a super massive black hole.
5.2.1 Zeroth Order and Energy
−GMBH α In the unperturbed problem, we consider Veff ∼ VBH = r = − r . This causes (5.19), again away from the singularity, to become
2(F ) `(` + 1) (F ) + 0 r − 2Υ(ω + ΥV )F − F = 0 0 rr r n BH 0 r2 0
∆VBH = 0. (5.22)
The choice of ωn to denote the zeroth order frequency will be apparent by the end of this section. The Laplacian of the black hole potential is included to emphasize it is consistent with the Laplacian of the sum of the potentials. Note the radial equation in this system is equivalent to
1 `(` + 1) ∂2(rF ) − 2Υ ω + ΥV + F = 0. (5.23) r r 0 n BH 2Υr2 0
Substitute VBH = −α/r and a change of variables u = rF0. This causes (5.23) to become
1 d2u `(` + 1) Υα − + − u = −ω u. (5.24) 2Υ dr2 2Υr2 r n
Make another change of variables r = y/(2Υ2α). This transforms (5.24) into
d2 `(` + 1) 1 −ω − + − u = n u = −k2u, (5.25) dy2 y2 y 2Υ3α2
p ωn where k = 2Υ3α2 . Finally let x = ky to ease calculations. Then we land on the following equation,
d2 `(` + 1) 1 − + − u = −u. (5.26) dx2 x2 kx
Consider the following analysis. For large values of x, we see (5.26) has the following behavior
d2u x → ∞ ≈ u, which implies u ≈ e±x. (5.27) dx2
38 `+1 d2u `(` + 1) x x → 0 ≈ u, which implies u ≈ . (5.28) dx2 x2 x−`
Thus to yield the appropriate behavior, the ansatz will include the negative exponential from (5.27) and the positive exponential in (5.28). That is, we suppose the radial function has the form
u(x) = x`+1γ(x)e−x (5.29) for a to be determined function γ(x).
Substituting this ansatz into equation (5.26), we arrive at
d2γ(x) dγ(x) 1 x + 2(` + 1 − x) + − 2(` + 1) γ(x) = 0. (5.30) dx2 dx k
Suppose now that γ(x) can be realized as the power series
∞ X j γ(x) = ajx . (5.31) j=0
By substituting the series (5.31) into equation (5.30), we find the following recursion relation for
the coefficients of γ
j(j + 1)aj+1 + 2(` + 1)(j + 1)aj+1 − 2jaj − (2(` + 1) − 1/k)aj = 0. (5.32)
From which we can deduce the recursion formula
a 2(j + ` + 1) + 1/k j+1 = . (5.33) aj (j + 1)(j + 2` + 2)
The following analysis will be done to show the series must terminate in order to be a solution.
Observe that as j → ∞,
aj+1 2j 2 2 ≈ 2 = ≥ . (5.34) aj j j j + 1
This approximation is chosen to give a lower bound to this ratio in terms we can solve explicitly. It is known that
39 2 if a = a (5.35) j+1 j + 1 j 2ja then a = 0 (5.36) j j!
Therefore,
∞ ∞ X X 2j γ(x) = a xj ≥ a xj = a e2x (5.37) j 0 j! 0 j=0 j=0 which diverges for large x. Thus the series must terminate. We want a polynomial of degree N, that is aN 6= 0 but aN+1 = 0. It then follows that
aN+1 1 aN+1 = 0 ⇒ = 0 = 2(N + ` + 1) − , (5.38) aN k and therefore
1 = 2(N + ` + 1) ≡ 2n (5.39) k
≥0 ≥1 where N + ` + 1 is defined to be n by convention. We observe that N, ` ∈ Z and hence n ∈ Z . Note that there exists degeneracy for each value of n. Substituting back for k, we arrive at an expression for the frequency
Υ3G2M 2 ω = BH . (5.40) n 2n2
Therefore, the full zeroth order radial solution is
40 ` `+1 `+1 −Anr j F0 (r) = An r e Σajr , for (5.41) p An = 2ωnΥ a 2(j + ` + 1) + 1/k j+1 = aj (j + 1)(j + 2` + 2) r ω k = n 2Υ3α2 Υ3G2M 2 ω = BH . n 2n2
The reader may note here that F0(r) can have different states. These states are dependent on the specific value of the frequency number n. For example, if n = 1 then both N = ` = 0 and we have the following solution
2 0 2 −Υ GMBH r F0 (r) = a0Υ GMBH re . (5.42)
5.2.2 First Order
We now will begin our first order analysis. To reiterate, we assume F can be expressed as a sum of
a component dominated by the black hole interaction and another from its self interactions. This is
expressed as the approximation F = F0 + F1, where F0 is the solution from the previous section
where the black hole’s potential dominated and F1 ∼ F0, for a small parameter > 0, is the self interacting component. We now will consider the remaining equations in the system
2 ˆ L`F1 = 2Υ(ωn + ΥVBH )F1 + 2Υ VF0 F0 (5.43) 1 Vˆ = ∗ (F P m cos(θ))2 . (5.44) F0 2r 0 `
Note that ω has been replaced with ωn to reflect the findings of the previous section. To begin our analysis of this system, first define A such that equation (5.43) is expressed as
41 2 (L` − 2Υ VBH )F1 = 2ΥωnF1 + A (5.45) 2Υ2 A = 2Υ2Vˆ F = ∗ (F P m cos(θ))2 F . (5.46) F0 0 2r 0 ` 0
In other words, A is an the in-homogeneity of the system. The Fredholm Alternative says if λ is an eigenvalue of M, for Mvˆ = λvˆ +a ˆ, then the system has solutions if and only if ha,ˆ vˆλi = 0. Thus
F1 must solve the following orthogonality condition.
ZZZ 3 F1A dx = 0. (5.47) V
This implies it must be the case that
ZZZ Z π Z 2π Z ∞ 3 2 2 1 2 m 2 F1A dx = 2Υ r sin(φ)F1F0 ∗ (F0 P` (cos(θ)) ) drdθdφ (5.48) R3 0 0 0 r Z 2π Z ∞ 2 2 1 2 m 2 = 4Υ r F1F0 ∗ (F0 P` (cos(θ)) ) drdθ 0 0 r = 0
Since the Green’s function is dependent only on r, and given that the convolution of two scalar
functions is defined to be [13]
ZZZ V (r, θ) = G(r) ∗ ρ(r, θ) = ρ(ξ, θ)G(r − ξ)dξdηdζ (5.49)
it follows that, in spherical coordinates, the convolution in the expression of the potential can be
given as
Z ∞ Z 2π Z π 2 1 2 m 2 m 2 ξ 2 ∗ (F0(r) P` (cos(θ)) ) = P` (cos(θ)) sin(φ) F0(ξ) dφdθdξ r 0 0 0 |r − ξ| Z 2π Z ∞ 2 m 2 ξ 2 = 2 P` (cos(θ)) dθ F0(ξ) dξ 0 0 |r − ξ| Z 2π m 2 1 2 2 = 2 P` (cos(θ)) dθ ∗ (r F0(r) ) . (5.50) 0 r
42 Making this substitution into the above inner product yields
Z 2π Z ∞ Z 2π 2 m 2 1 2 2 0 = r F1F0 P` (cos(θ)) dθ ∗ (r F0 ) drdθ 0 0 0 r Z 2π Z ∞ m 2 2 1 2 2 = P` (cos(θ)) dθ r F1F0 ∗ (r F0 ) dr. (5.51) 0 0 r
Since there exists choices of ` and m such that the integral over θ is nonzero, it must be that
Z ∞ 2 1 2 2 r F1F0 ∗ (r F0 ) dr = 0. (5.52) 0 r
This yields a sort of orthogonality condition for the solution. We now will consider a bit of quali- tative analysis of the first order system. Observe that as r → ∞ and r → 0 respectively, equation
(5.43) is asymptotic to
√ ± 2Υωnr r → ∞ (F1)rr ≈ 2ΥωnF1, which implies F1 ≈ e (5.53) `+1 `(` + 1) r r → 0 (F ) ≈ F , which implies F ≈ . (5.54) 1 rr r2 1 1 r−`
As we did with the zeroth order solution, we choose the following ansatz to yield appropriate be- havior in the limits
√ `+1 − 2Υωnr F1(r) = r γ1(r)e (5.55)
where γ1(r) is yet to be determined. Substituting this result into (5.43) will yield the slight simpli- fication
√ √ Υα F `+1 − 2Υωnr 2Υωnr `+1 ˆ 0 L` r γ1(r)e e = 2Υ ωn − r γ1(r) + 2ΥVF0 √ . (5.56) r e− 2Υωnr
Though this is a step towards solving the first order equation, the solution itself depends heavily on
43 2 0 2 −Υ GMBH r which states are present in F0. For example, we know for n = 1, F0 (r) = a0Υ GMBH re , and so (5.56) becomes
√ √ Υα `+1 − 2Υωnr 2Υωnr `+1 3 ˆ L0 r γ1(r)e e = 2Υ ω0 − r γ1(r) + 2a0Υ GMBH VF 0 r (5.57) r 0
where
1 2 ˆ 2 −Υ GMBH r m 2 VF 0 = ∗ a0Υ GMBH re (P0 (cos(θ))) 0 2r 2 R 2π m 2 ! 2a0Υ GMBH P (cos(θ)) dθ 1 2 = 0 0 ∗ r3e−Υ GMBH r . (5.58) 2 r
For fun we will consider the m = 0 state. Observe then
2 R 2π 0 2 ! 2a0Υ GMBH P (cos(θ)) dθ 1 2 ˆ 0 0 3 −Υ GMBH r VF 0 = ∗ r e 0 2 r
2 2 Z ∞ 3 −Υ GMBH ξ 4πa0Υ GMBH ξ e = 2 dξ 0 r − ξ 2 ∞ 3 4πa Υ GM Z r 2 0 BH 2 2 −Υ GMBH ξ = 2 r + rξ + ξ + e dξ 0 ξ − r 2 2 2 2 2 2 3 Z ∞ −Υ GMBH ξ (Υ GMBH ) r + Υ GMBH r + 2 2a0Υ GMBH r e = 4πa0 2 2 + 2 dξ. (Υ GMBH ) 0 ξ − r (5.59)
As it turns out, the remaining integral can only be expressed in terms of the exponential integral, meaning there is no closed form solution. With that said, there does exist methods for approximat- ing this integral, which is termed Ei in the literature, however we will not press farther into this direction. This does imply that there does not exist a general exact closed form solution to γ1 and
hence the same is true for F1. It is unclear at this time whether or not there does exist a state that can be solved for with a closed form using this method.
44 5.3 Solution Scheme and Plots
The following plots are generated via a MATLAB program using a basic mesh grid. The plots depict the square of the radial component of the scalar field f using the analytical approximations derived in the previous chapter. The program uses units such that a0 = G = c = Υ = MBH = 1. Each of these plots can be thought of as the radial density of the field.
2 Figure 8: Side view of |f0(r)| for the m = ` = N = 0 state of the dark matter field.
45 2 Figure 9: Top down view of |f0(r)| for the m = ` = N = 0 state of the dark matter field.
These plots give a clear depiction that at zeroth order, the dark matter is focused in a halo around the black hole. Although the first plot looks interesting, the second is more intuitive. Note that these are plots of the magnitude of scalar field for an increasing radius. The top down view can be thought of as a slice through a spherical annulus, corresponding to the top spherical harmonic in Figure 5.
The following two plots depict the zeroth order solution again, but at a higher energy state. Namely the m = ` = 1 state. Again the first image (side view) is included for aesthetics while the second
(top down view) is the more intuitive depiction. In this case the reader can visualize Figure 11 as a slice through one of the spherical harmonics in the second row of Figure 5.
46 2 Figure 10: A side view of |f0(r)| for the m = ` = 1,N = 0 state of the dark matter field.
2 Figure 11: Top down view of |f0(r)| for the m = ` = 1,N = 0 state of the dark matter field.
47 6 Conclusion and Future Work
To recap, we have explored a derivation of field equations along with an extension to the Einstein-
Klein-Gordon system. The Einstein-Klein-Gordon system was reduced to the low field/energy limit.
We then solved the Schrodinger-Poisson¨ system for a general separable ansatz and solved the angu- lar component with spherical harmonics. Using the radial equation from the separation of variables we derived an approximate description of a scalar field encasing a black hole for which we have zeroth order solution. This solution was then plotted and we note it confirms general intuition on how the dark matter should behave in this scenario.
There is still much work to be done. Though the first order equations will yield small perturbations on the zeroth order solution, further approximation methods with or without numerical methods could be utilized to progress these solutions. Future work in this direction could involve the follow- ing:
• A general algorithm for computing plots/solutions of different dark matter states at zeroth and
first order (and beyond)
• Investigation into stable excited states and time evolution
• Investigation into whether energy in the initial dark matter system dissipates into the black
hole
• Can angular momentum emerge from this system?
The Einstein-Klein-Gordon system presents a promising description of dark matter that is so far compatible with observation. A successful theory that describes 95% of the energy in our universe would be a monumental step toward a complete picture of the world around us. The interested reader is welcome to investigate further.
48 References
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[3] Bray, H. L. On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity. AMS Contempt Math 599, 1-64(2013) arXiv:1004.4016.
[4] Bray, H. L. & Goetz, A. Wave Dark Matter and the Tully Fischer Relation. (2014) arXiv:1409.7347.
[5] Carmo, M. P. (1992). Riemannian Geometry (1st ed.). Springer.
[6] Carroll, S. M. (2020). Spacetime and geometry: An introduction to general relativity. Cambridge: Cambridge University Press.
[7] Dulaney W. A. (2019, May). A Brief Survey Of Manifolds And Vector Bundles. http://libres.uncg.edu/ir/uncg/ listing.aspx?styp=ti&id=25897
[8] Einstein, A. Zur Elektrodynamik bewegter Korper¨ . (German) [On the electrodynamics of moving bodies]. An- nalen der Physik, 322(10):891–921, 1905.
[9] Einstein, A. (n.d.). Uber¨ die spezielle und die allgemeine Relativitatstheorie¨ : (Gemeinverstandlich).¨ ECHO. Retrieved April 21, 2021, from https://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/ RB68AZVS/pageimg&pn=6&mode=imagepath
[10] Frankel, T. The Geometry of Physics, An Introduction. Cambridge University Press, Cambridge, 1997.
[11] Langevelde H. J. v. Focus on the First Event Horizon Telescope Results. https://iopscience.iop.org/journal/ 2041-8205/page/Focus on EHT
[12] Logan, D. J. (2013). Applied Mathematics (4th ed.). Wiley.
[13] Olver, P. J. (2016). Introduction to partial differential equations. New York, NY: Springer International.
[14] Ostlie, D. A., & Carroll, B. W. (2007). An Introduction to Modern Stellar Astrophysics (2nd ed.). Pearson Addison-Wesley.
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[16] Parry, A. R. Wave Dark Matter and Dwarf Spheroidal Galaxies. (2013) arXiv:1311.6087v1.
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49 [19] Wikipedia. Gravitational Lens. https://en.wikipedia.org/wiki/Gravitational lens.
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50 William Arthur Dulaney linkedin.com/in/williamadulaney [email protected]
Education Wake Forest University August 2020 – Present M.A. Mathematics Winston-Salem, NC Appalachian State University August 2016 – 2020 B.Sc Mathematics Boone, NC Experience Research Assistant May 2019 – May 2020 Department of Mathematical Sciences, Appalachian State University Boone, NC • Research assistant to Dr. Quinn Morris • Numerical PDEs • Mathematical Ecology Research Assistant August 2016 – May 2019 Department of Physics and Astronomy, Appalachian State University Boone, NC • Research assistant to Dr. Anthony Calamai • Primary and secondary source research in atomic phosphorescence and spontaneous quantum emissions • Optimization of EMF potential in octo-pole RF ion trap (NC Space Grant funded) Classic/Informal Research March 2016 – May 2020 Dark Sky Observatory Boone, NC • Research in observational astronomy with Dr. Daniel B. Caton • Spectroscopic observations on multiple eclipsing binary star systems • Manual use of 32-inch optical telescope at the Dark Sky Observatory
Teaching Lecturer Spring 2020 Department of Mathematics, Appalachian State University Boone, NC • MAT1035: Independant lecturer for Business Math with Calculus Assistant Lecturer Fall 2020 Department of Mathematics, Appalachian State University Boone, NC • MAT1035: Assisted Mrs. Genie Griin with instruction Business Math with Calculus
Technical Skills, Language Skills, and Interests OS: Linux, Windows Programming: Excel, IRAF, Maple, MathCad, Matlab Writing:LATEX, Oice Languages: English (native), German (Intermediate) Interests: Cosmology, Geometry, Mathematical Physics, Topology
Publications “Analytic solution to a three-level optical pumping system with constant coeicients.” Proceedings of the National Conference of Undergraduate Research 2017, 30 November 2017. William Arthur Dulaney
Thesis "Scalar Field Dark Matter Encasing A Super Massive Black Hole .”Masters Thesis, May 2021
"A Brief Survey of Manifolds and Vector Bundles.”Senior Honors Thesis, May 2019
51 Honors, Awards, and Conferences Honors and Awards - Undergraduate - • Summa cum laude, Pi Mu Epsilon(Mathematics), Phi Theta Kappa, Chancellor’s List, • Jeremy and Rebecca Ehret Scholarship in Physics and Astronomy, NC Space Grant Undergraduate Research, NC Space Grant Community College STEM Scholarship Conferences - Undergraduate - • “Analytic solution to a three-level optical pumping system with constant coeicients.”State of North Carolina Undergraduate Research and Creativity Symposium, Durham, NC • “Analytic solution to a three-level optical pumping system with constant coeicients.” National Conference of Undergraduate Research, Memphis, TN • "A new senior-level laboratory experience in atomic phosphorescence." Southeastern Section of the Ameri- can Physical Society, Georgia College, Department of Physics, Milledgeville, GA
Outreach and Service Dark Sky Observatory: Participated in numerous public night events at the Dark Sky Observatory and Rankin Sci- ence Observatory. Duties include operation of various optical telescopes ranging from the large 32-inch telescope to smaller Celestrons, engagement in informal discussion and Q&A with the general public during visitations, and giving outdoor “star talks” to guide visitors through the night sky.
References Available upon request
52