A black hole is an object in space where the pull of its gravity is so strong
that no light can escape. This notion gives rise to the phenomenon called
gravitational lensing which is the effect where light is being bent by a massive
object, in our case a black hole. With these two concepts in mind we are
able to formulate the goal of this thesis: we aim to simulate and visualize the
distortion of a projected image caused by the gravitational field of a black
hole.
First of all we need to cover the relevant Physics to form some sort of under-
standing of the bigger picture and have an idea of all the factors involved in
reaching that goal. We are then able to create a concrete plan to reach our
goal in manageable consecutive steps.
We find that determining geodesics in a specified metric is one of the most
important factors of this plan. In order to do so we derive the geodesic equa-
tion which enables us to calculate these geodesics.
We continue by first applying the geodesic equation in two-dimensional Eu-
clidean space. This provides us with a system of differential equation which
we solve by means of numerical methods. These results are visualised and
proved to be correct.
We then move over to four-dimensional Minkowski space where we calcu-
lated and visualised the geodesics for this specific metric. In the Minkowski
space we make a start with actually visualizing the paths of light rays.
We continue to our final metric, the Schwarzschild metric. The Schwarzschild
geometry essentially describes the spacetime geometry of empty space sur-
rounding any spherical mass which in our case will be a black hole. We
calculate and visualize the geodesics thoroughly and created the image con-
structor for the Schwarzschild metric. This image constructor visualises how
an image will be altered by being projected in a Schwarzschild metric with
respect to that image in the Minkowski metric. Once the image constructor
is up and running a significant amount of time is specifically dedicated to
showcasing the constructed images.
We conclude that we have reached our defined goal since we are able to
simulate and construct the projected images. We look back at all the steps that played a key role in this process.
Besides the goal, we spent some time reflecting at all the unfamiliar Physics
and Mathematical theory that had to be understood and applied in order to
create the entire thesis

Modified Newtonian Dynamics (MOND) can account for a variety of phenomena on galactic scales without the need for dark matter, but it cannot fully explain the mass contained in galaxy clusters. We explore two possible solutions to this problem: relativistic extensions of MOND, and FEM simulations of the apparent matter distribution in clusters, utilising the non-linear AQUAL formulation to analyse non spherically symmetric systems. We consider Covariant Emergent Gravity, and we show that the theory is inconsistent with the original formulation of Emergent Gravity. Moreover, we show that either the theory is incompatible with observations, or it presents grave theoretical difficulties. We then suggest that a covariant formulation of EG can be obtained through a Generalised Einstein Aether (GEA) theory, which is capable of retrieving the MOND PDE, and is, at the same time, consistent with observational constraints. Regarding the FEM simulations of the apparent matter distribution in galaxy clusters, we construct a sample of 15 clusters from the catalogs of Reiprich and Abell for the baryonic mass distribution present in galaxy clusters. We choose FEM for its ability to treat the combination of continuous and discrete mass distributions without the need for smoothing. This is necessary, as in MOND we cannot apply the principle of superposition or the weak lensing formalism. We then utilise the FeniCS software package to study the properties of these clusters. We simulate each cluster with elements up to degree 3, thanks to a speedup by a factor of ∼ 100 obtained by the use of local mesh refinement for the serial case. In addition, we run the code in parallel on a single-threaded 8-core CPU, achieving near optimal weak scaling for the regime of interest. For the mass distribution in the galaxy clusters, we analyse the distribution of baryons, Phantom Dark Matter (PDM) and apparent mass both close to the core and around each galaxy. We find that the PDM tends to clump around the galaxies, regardless of the gas to galaxy mass ratio. Moreover, we show that both the apparent mass and the PDM can exhibit negative masses as predicted by Milgrom. Our observations on the density of PDM around the galaxies match recent observations of small scale weak lensing. In addition, our results for negative mass distributions provide an opportunity to test a prediction of MOND that can never be replicated in the dark-matter paradigm, and shed light on the properties of non-spherically symmetric mass distributions, that have, up to now, not been studied in the literature for the fully non-linear case.

A pentagon is an example of a highly symmetric polygon in two-dimensional space. The three-and four-dimensional analogue of these polygons are the regular polyhedra and the regular polytopes. There exist five regular polyhedra in three-dimensional space and these are called the Platonic solids. These five Platonic solids are the tetrahedron, cube, octahedron, dodecahedron and the icosahedron. In four-dimensional space, the regular polytopes are the 5-cell, the 8-cell, also called the tesseract, the 16-cell, the 24-cell, the 120-cell and the 600-cell. The main focus in this thesis is to describe the symmetry group of the icosahedron, to introduce the Icosians, which are related to the rotation group of the icosahedron, and to study the action of the symmetries of the 600-cell on the twenty-five 24-cells it circumscribes. Firstly, the symmetry groups of the Platonic Solids, the regular polytopes in three dimensional space are established. Then it will be shown that there eixsts a two-to-one map from the the group H₁ of unit quaternions to the group SO(3), the group of 3 x 3 orientation-preserving matrices. This map will be used to describe the binary groups, which are double covers of the rotation groups of the Platonic solids. After that, the symmetry group of the tesseract will be studied both via an isomorphism between G := {± 1 }⁴ x S₄} and the symmetry group of the tesseract as well as geometrically via rotation planes. Then, the 24-cell and the 600-cell will be defined as the four-dimensional regular polytopes whose vertices are the quaternions from the binary tetrahedral group and the binary icosahedral group, the Icosians. It will be shown that twenty-five 24-cells inscribe a 600-cell and that there are 10 ways to decompose the vertices of a 600-cell into the vertices of 5 disjoint 24-cells. Next, it will be shown that these 10 decompositions are chiral, 5 being 'left-handed' and 5 being 'right-handed'. Finally, it is shown that the symmetry group of the 600-cell acts on these 5+5 decompositions by permutation, each permutation being described by an element from A₅ x A₅ ⋊ {±1}, where -1 acts on A₅x A₅ by interchanging the factors of A₅x A_5.

The concept of entanglement is one of the distinguishing features in quantum mechanics. Information about one particle can determine the state of another particle. This information and entanglement in subsystems is quantified by the entanglement entropy. The entanglement entropy has become an important research topic in recent years. Entanglement entropy is expected to grow with the boundary size for systems described by Hamiltonians with local interactions, described by the area law. For one dimension there exists a bound quantifying this area law S = O (2^1/ΔE) in terms of the energy gap ΔE between the ground-state energy and the first excited-state energy. It is an open problem whether the area law holds in higher spatial dimensions. Therefore it is of great interest to study the entanglement entropy in systems that violate the area law. In this thesis spin chain models with local interactions are studied for arbitrary spin. Hamiltonians with a unique ground state are constructed by mapping spin chains onto (coloured) Dyck and (coloured) Motzkin paths. We show that these models express logarithmic or power law violations of the area law for the bipartite entanglement entropy. These results are presented in the table below. The known bound of the area law in one dimensional systems is dependent on the energy gap of these models. The energy gap of the Motzkin-path model is investigated by constructing the an orthogonal excited state with small energy. In doing so we associate the Hamiltonian with the Laplacian of a graph. We present an alternative proof for the bound on the energy gap of the colourless Motzkin-path model than S. Bravyi et al and R. Movassagh et al. We show that the energy gap ΔE scales in terms of the chain length n as ΔE = O(n^-2) in the limit n →∞. Therefore ΔE → 0 in this limit, resulting in area-law violations of the entanglement entropy while maintaining the known bound on the entanglement entropy.

In 2019 werd er voor het eerst een foto van een zwart gat gepubliceerd. Het betreffende super zware zwarte gat ligt in het midden van het sterrenstelsel Messier 87. In dit onderzoek willen we simulaties maken van hoe een foto van een zwart gat eruit ziet. Zowel voor een achtergrond (zoals de sterrenhemel) als de accretieschijf die eromheen zit. De vraag is vervolgens welke informatie uit de foto gehaald kan worden. Om de foto te maken, werd een ray-tracer gemaakt. Hierbij kijk je waar een lichtstraal, die op het scherm komt (een oog of camera), vandaan komt. De bewegingsvergelijkingen zijn vereenvoudigd door constantes van de beweging te zoeken om de geodeetvergelijkingen te versimpelen. Vervolgens zijn de bewegingsvergelijkingen numeriek opgelost. Er is gevonden dat lichtstralen van veraf niet in een zwart gat vallen bij $\frac{\sqrt{27}}{2}R_s$. $R_s$ is de Schwarzschild straal. Dit is de straal waarvandaan licht niet meer kan ontsnappen van een zwart gat. De gesimuleerde plaatjes zijn 128x128 pixels. In de gesimuleerde plaatjes had het zwart gat ongeveer $2.5R_s$. Dit komt nauw overeen maar er wordt een aanzienlijke fout gemaakt door de resolutie van de plaatjes. Verder is kwalitatief de oriëntatie en rotatieparameter $a$ goed uit de plaatjes te halen.

In dit project worden vier vereenvoudigingen van de trebuchet onderzocht. Er wordt gekeken naar het rendement met betrekking tot de energie overdracht van het zware tegengewicht op het afgeschoten voorwerp. De invloed van het toevoegen van extra armen en wielen wordt onderzocht. Bij meer energie overdracht kan het voorwerp verder weg geschoten worden en zal het voorwerp meer impact hebben als het op de grond terecht komt.

This research contains the design, modeling, and analysis of the Terrascope. This apparatus intends to use the Earth’s atmosphere as a lens to bend light rays from celestial objects and focus them into a detector placed at a distance from the Earth, for example, the Moon. We develop an independent model from that of David Kipping, on whose Terrascope research this work is based. Our Terrascope model uses gradient-index optics to calculate the light bending through the Earth’s atmosphere. We also model three different atmospheric effects which modify the light passing through the atmosphere: turbulence, Rayleigh scattering, and ozone absorption. Our results show that turbulence has the largest impact on the light, and, consequently, on the functioning of our Terrascope. It causes the light to spread out, decreasing the image resolution and amplification. Without atmospheric effects, our model simulations predict a maximum amplification of about 55,000, the same results as Kipping. This occurs when using a 1m detector aperture, 1.5×10ዃm detector distance, and 1000nm wavelength light. Using the same parameters, when scattering and absorption are considered, the amplification decreases by 14%. When turbulence is considered, the amplification decreases by 99.98% to a total of 10. This is much lower than Kipping’s prediction of 22,500. We conclude that turbulence is the most important aspect of the Terrascope to consider in any future work. The Terrascope continues to be interesting concept for study and may have promise for observing celestial objects if a farther detector distance, longer light wavelength, and different atmosphere are considered.

context: The long-term evolution of a self-gravitating astrophysical disk can be modeled using secular perturbation theory. Recently, Batygin published a paper where he claims that such a disk with a special density can be described by a Schrödinger equation by using this method. aims: In this thesis, we will study the secular perturbation theory applied to an astrophysical disk with the same density as Batygin, using the Laplace-Lagrange equations. We will take the continuum limit of those equations, and try to find a wave equation like Batygin. methods: We first apply the Laplace-Lagrange equations to a disk with a large number of planets. Then we take the continuum limit of an infinite number of planets. We then compare the numeric results of the discrete disk to the analytic results of the continuum limit. results: The eigenmodes of the system are well approximated by damped sinusoids. Mode number n changes sign n times. The eigenvalues are linear in the mode number. conclusions: The eigenmodes do not satisfy a wave equation.

E8 is a famous root system in mathematics. Lisi claims in his paper “An Exceptionally Simple Theory of Everything” that there is a connection between the standard model of the elementary particles and this root system. He claims that standard model has the same structure as the root system E8. If this is true, there must be a mapping from the standard model to E8. This map must identify each elementary particle with an unique root of E8. Also the four charges (weak isospin, weak hypercharge, g3 and g8) of the standard model must have corresponding vectors in R8. These four charge vectors must have the property that taking the inner product of the charge vector and the root identified withan elementary particle, it would result in the charge of that particle. This is needed to guarantee that we have conservation of charge. In this paper we have shown that it is not possible to find a map with four charge vectors. This was done by studying root systems and how different root systems are present in E8 as sub root systems. We also proved that the intersection of E8 with the orthogonal complement of a charge, must be a root system. This limits the possibilities for the charge vectors. For these possible charges it has been checked whether there are enough roots for a given charge to identify the elementary particles. Since this was not the case, we concluded that no mapping with four charge vectors exists.

In this thesis we consider the reconstruction of albedo maps of exoplanets. This is done with a new variant of spin-orbit tomography that has been described in [Cowan and Agol, 2008] and more in depth in [Fujii and Kawahara, 2012]. This method reconstructs the albedo map from the reflected-light curve, the total intensity of the light that originates from the host star and is reflected by the planet. In the mentioned papers, the surface map of the planet is modeled as a sum of finite sized surface elements with constant albedo, and the relation between this approximation of the map and the light-curve in the time domain is determined. In this report, we use that the signal is quasi periodic due to diurnal and annual motion, and work with the Fourier peaks of the light-curve. We also approximate the map in a different way, writing it as the sum of spherical harmonics, and neglecting spherical harmonics with high spatial frequencies. This has the advantage that the relation can be worked out analytically (for edge-on and face-on observations) without the use of complex mathematics, and that both the surface map and the light-curve contain a daily frequency. We derive an equation for the reflective light-curve under the assumption that the surface map is not a function of time (no clouds), and that the reflection is Lambertian (equal in magnitude in all directions). This transformation is found to be a linear function of the surface map. This equation is worked out for edge-on and face-on observations with arbitrary axial tilt, which describes the orientation of the spin axis with respect to the observer and the orbital plane. Furthermore, we describe how to invert this relation if the axial tilt is known to the observer. We also aimed at recovering the map if the axial tilt is unknown to the observer, since this would make sure that the reconstruction does not rely on other observations. In contrast to what was found in papers like [Fujii and Kawahara, 2010] and [Fujii and Kawahara, 2012], we did not succeed in this. A number of methods were used for this. The first two looked at the problem from a mathematical perspective: the minimization of the distance between the measured light-curve and the light-curve from the reconstructed map, and Tikhonov regularization. The two failed because both the column space and the singular values respectively are not a function of the axial tilt. The third method that has been treated and tested involved the maximization of the ‘amount’ of positive albedo on the reconstructed map, but a test showed that the distinction that this method makes is in the same order of magnitude as the numerical error, thus proving that this method was not useful as well. Further study might show what causes the results of the two methods to differ in this respect

This thesis will focus on verifying the area-law for the entanglement entropy SN for spin-1 2 lattice systems in 2 dimensions, with ferromagnetic interactions defined by the nearest neighbour XX-model. The area-law implies that the SN of a subsystem is expected to scale proportionally to the size of the boundary through which that subsystem interacts with the rest of the system. By writing a numerical model in MATLAB, which uses computational devices to circumvent the huge memory requirement that quantum simulations usually demand, it is possible to analyse 2 dimensional systems up to a lattice size of 5 × 5 for the first time. These finite systems have been assigned periodic boundary conditions, as if it were a torus. It doesn’t seem possible to evaluate such systems analytically, unlike 1 dimensional systems, as we will show. The area-law has been shown in a direct way by looking at entanglement entropy, as well as the indirect way by analysing the connected spin correlation.

This Bachelor thesis is on Covariant Emergent Gravity (CEG): a covariant formulation by Sabine Hossenfelder of the ideas of Erik Verlinde on gravity as an emergent force. In this work, the ideas of Erik Verlinde are presented as well as the field equations of CEG. The aim of this thesis is to find experimentally verifiable results for CEG. From the field equations we derive a general gravitational lensing formalism for CEG that is applicable to general lensing systems. This thesis also includes an attempt at an expanding universe model for a vacuum or matter dominated universe. Subsequently, a numerical algorithm is presented to solve for the non-linear differential equations in CEG and MOND in Newtonian regimes for general matter distributions using Fourier transformations. A faster version of this algorithm for cylindrical symmetric matter distributions using Fourier-Bessel transformations is also presented. This algorithm is tested on both the Sun and galaxy NGC6503. Next, the rotation curves as predicted by CEG and MOND of 132 galaxies are compared to the observed velocities using the SPARC data set. The velocities are fitted using three fit parameters and a Markov Chain Monte Carlo algorithm. Both theories show good fits to the observed velocities. We conclude that no differences between the predictions of CEG and MOND can be made on the basis of rotation curves, but that covariant features (such as gravitational lensing) might be used to distinguish between the two theories. An important test for both theories can be provided by a comparison between strong lensing measurements and rotation curves for a single galaxies.

In this text we used a quantum system of two quantum particles, specificallytwo qubits. One qubit acts as a detector in an equal weight superposition ofspin up and spin down. The two qubits in the system are entangled. To domultiple measurements on the system, we want to rotate the system back to theequal weight superposition, which is destroyed after a measurement.We have calculated the time evolution of the density matrix of this quantumsystem using a standard differential equation. We altered this differential equa-tion using the counting fields method. After this we used the density matrix tocalculate the optimal angle of rotation to return the qubit to the equal weightsuperposition. We did this for an ideal system without outside influences andfor a system with outside influences. We derived a relationship between theoutside influences on the system,γ, whereγ= 2 represents an ideal system,and a variable inside the system,z. This relation is2γ=z.We also described an algorithm to simulate a random quantum trajectory.We found that the value of the spin in the z-direction does go to the expectedvalue of 0 when the amount of single trajectories in the average trajectory islarge. This is true for all values ofγ. For the value of the spin in the x-directionwe found that the theoretical curve correctly predicts the behavior of an idealsystem. We found that for a non ideal system, whenγ >2, the theoreticalcurve does not correctly predict the behavior of a trajectory. The theory fails topredict the speed with which the spin in the x-direction goes to 0 in a non idealsystem, where the theory is significantly slower at going to 0. Also the theorypredicts the starting value to be lower than 1 in a non ideal system, which isnot the case in practice

Quantum thermodynamics takes over from classical thermodynamics when systems are of the scale of single particles and quantum fluctuations have a noticeable effect. An interesting topic of research of this relatively new field is the quantum battery, which in this thesis consists of an array of N identical electron spin qubits. In an article by Binder et al. [4], it is proven that in theory, an N-times decrease in charging time of the battery is achieved when global operations on qubits are permitted. This thesis investigates if a similar advantage can be achieved by using a local qubitqubit interaction operator on a one-dimensional chain of exchange-coupled electron spin qubits that are driven by microwave radiation in the presence of decoherence. This system is described
by a density matrix in order to include the presence of external influences. The time-evolution of the state of the system is calculated by solving the Von-Neumann equation both analytically and numerically, which is then used to calculate the extractable work. It is shown that exchange interaction does not have a direct effect on the extractable work, since it creates entanglement between two states of the same energy level and the operator commutes with the Hamiltonian of the system. The effect of the CNOT gate on the state of the system is then investigated. While it does have an effect on the extractable work, it did not achieve a decrease in charging time. These results are only relevant for the specific system used in this thesis. For other methods and systems, exchange interaction could lead to faster charging.

In dit verslag wordt besproken hoe je deelwortelsystemen in het wortelsysteem E8 kan vinden. In het begin wordt er algemene theorie over wortelsystemen gegeven. Vervolgens worden er twee manieren besproken hoe je deelwortelsystemen in wortelsystemen kan vinden. Uiteindelijk worden de twee manieren toegepast op E8 om deelwortelsystemen in E8 te vinden.

The aim of this research is to study a method to find exomoons and to find specific characteristics that point to the existence of exomoons. Exomoons are natural satellites orbiting an extrasolar (exo)planet in an extrasolar system. By looking at eclipses on exoplanets, we can find these characteristics.
We consider the reflected light signals of exoplanets with an exomoon. The reflected light signal is the intensity of light the planet reflects from the host star towards the observer.
We derive the reflected light signal in a planetary system with one exoplanet and in a system with an exoplanet and an exomoon. We made the assumptions that the exoplanets and exomoons have
a homogeneous surface (albedo is 1) and move in circular orbits around a star with an inclination angle between the orbital planes. Along the orbit the planet and the moon have changing phases. The moon is always close to its planet, so the phases of the exomoon and the exoplanet are the
same. For exoplanets, it is not possible to spatially separate a moon from its planet. One only sees the total signal of both bodies, the light originating from the star, reflected by the two bodies towards the observer.
Maybe we can find an exomoon by examining eclipses. Eclipses occur when the exoplanet and exomoon are aligned with the star, so that the body closest to the star blocks the light towards the
body farthest from the star. Finding eclipses is one of the few methods to discover exomoons. The systems are modeled with the assumption that any total eclipse happens every time r = R or −r = R, we see short dips in the reflected light signal. These dips are periodic and make the complete signal quasi-periodic. That is why we also calculate the Fourier transform of the reflected light signal. This quasi-periodicity causes the Fourier spectrum to have side bands that are repeated and are copies of themselves.
In this research, the orbit of the exomoon is tilted to see the effect of inclination in the reflected light signal. The result is fewer eclipses. At most twice a year for a short amount of time an eclipse
can occur. In the Fourier domain, this results in more peaks, but the pattern is still repeated.
From this research, we cannot conclude whether or not an exomoon is present from measured data.
What we learned, is that if an exomoon is present, short dips in the received light signal occur and this results in periodic side bands in the Fourier domain with respect to the system of one planet.
The duration of an eclipse is very short, so the detection of the eclipse can easily be missed. The Fourier transform signal becomes stronger when a longer time has been measured.

In this thesis we look at the most cost-
effective trajectory for power limited rockets, i.e. the trajectory which costs the least amount of propellant. First some background information as well as the differences between thrust limited and power limited rockets will be discussed. Then the optimal trajectory for thrust limited rockets, the Hohmann Transfer Orbit, will be explained. Using Optimal Control Theory, the optimal trajectory for power limited rockets can be found. Three trajectories will be discussed: Low Earth Orbit to Geostationary Earth Orbit, Earth to
Mars and Earth to Saturn. After this we compare the propellant use of the thrust limited rockets for these trajectories with the power limited rockets. Here we made this comparison between a conventional thrust limited rocket with a specic impulse of 455 seconds and the VASIMR rocket for the power limited rocket. Also the initial mass of both rocket types was taken as
5 * 10^5 kg. Lastly, we take a look at a gravity assist. Gravity assists can help reduce propellant use even further. Therefore we will once more look at the trajectory for a power limited rocket
from Earth to Saturn. Only this time we use a gravity assist from Jupiter. Then we can see if the propellant use is indeed even lower when using the gravity assist. We find that the the power limited rocket becomes more fuel-effcient as travel time increases. Also using a gravity assist further reduces propellant consumption.

In dit onderzoek wordt gekeken of het model voor elementaire deeltjes voorgesteld door de Amerikaanse fysicus Antony G. Lisi bevestigd of weerlegd kan worden. In dit model worden elementaire deeltjes gekoppeld aan wortels uit het wortelsysteem van E8 en worden ladingen gekoppeld aan gewichten van dit wortelsysteem. Interacties tussen deeltjes moeten in het model voldoen aan optellingsregels van het wortelsysteem zodanig dat aan ladingsbehoud is voldaan voor elk type lading dat in het standaarmodel voorkomt.

Door het probleem te programmeren en het programma uit te voeren, wordt geconcludeerd dat alle ladingen afzonderlijk en combinaties van ladingen wel in het wortelsysteem van E8 passen, maar het niet mogelijk is alle ladingen te laten werken. Hiermee wordt de theorie van Lisi weerlegd. We hebben echter verondersteld dat de gewichten die horen bij de ladingen orthogonaal zijn en er kan in een vervolgonderzoek gekeken worden of dit probleem ook valt op te lossen zonder deze aanname.