Technische Universiteit Delft Faculteit Elektrotechniek, Wiskunde en Informatica Delft Institute of Applied Mathematics Omtrent Covariante Emergente Zwaartekracht (Engelse titel: On Covariant Emergent Gravity) Verslag ten behoeve van het Delft Institute of Applied Mathematics als onderdeel ter verkrijging van de graad van BACHELOR OF SCIENCE in TECHNISCHE WISKUNDE & TECHNISCHE NATUURKUNDE door Arthur Carel Platschorre Delft, Nederland 12 juli 2019 Copyright © 2019 door Arthur Carel Platschorre. Alle rechten voorbehouden. BSc verslag TECHNISCHE WISKUNDE & TECHNISCHE NATUURKUNDE “Omtrent Covariante Emergente Zwaartekracht ” (Engelse titel: “On Covariant Emergent Gravity ”) Arthur Carel Platschorre Technische Universiteit Delft Begeleiders Dr. Paul M. Visser Dr. Jos M. Thijssen Overige commissieleden Prof. Dr. ir. C. Vuik Dr. Stephan W. H. Eijt Dr. Jeroen G. Spandaw 12 juli, 2019 Delft Bachelor Thesis On Covariant Emergent Gravity Arthur Platschorre Delft University of Technology Delft University of Technology Bachelor Thesis On Covariant Emergent Gravity Author: Supervisors: Arthur C. Platschorre Dr. Paul M. Visser Dr. Jos M. Thijssen Committee members: Dr. Paul M. Visser Dr. Jos M. Thijssen Prof. Dr. ir. C. Vuik Dr. Stephan W. H. Eijt Dr. Jeroen G. Spandaw A thesis submitted in fulfillment of the requirements for the degree of Bachelor of Science in Applied Physics & Applied Mathematics at Mathematical Physics Faculty of Electrical Engineering, Mathematics & Computer Science July 12, 2019 Cover image copyright © ESA/Hubble iii Declaration of Authorship I, Arthur C. Platschorre, declare that this thesis titled, “On Covariant Emergent Gravity ” and the work presented in it are my own. I confirm that: • This work was done wholly or mainly while in candidature for a research degree at this University. • Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. • Where I have consulted the published work of others, this is always clearly attributed. • Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. • I have acknowledged all main sources of help. • Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. Signed: Date: v “Oh leave the Wise our measures to collate One thing at least is certain, light has weight One thing is certain, and the rest debate— Light rays, when near the Sun, do not go straight.” Sir Arthur Eddington vii DELFT UNIVERSITY OF TECHNOLOGY Abstract On Covariant Emergent Gravity by Arthur C. Platschorre This report focuses on covariant emergent gravity (CEG), a coordinate-free formulation by Sabine Hossenfelder of Erik Verlinde’s emergent gravity (EG). In (C)EG, gravity is considered an emergent, entropic force arising from shifts in the entropy content of the universe due to matter. The main goal of this report is to identify experimentally verifiable results for CEG through a theoretical analysis of the covariant field equations. The initial goal of this report was to understand both EG and CEG. We have expanded upon this goal both theoretically and numerically. Our theoretical expansion consists of two parts: a gravitational lensing formalism for CEG and an attempt at a cosmological model. The lensing formalism includes a lensing potential that can be applied to general lensing systems. We show that current assumptions in CEG on the imposter field result in a lensing equation that predicts that the amount of baryonic matter in a galaxy as measured by strong lensing should differ from the amount of baryonic matter as measured from the rotation curves. We have also included an attempt at a cosmological model in which we solve for a vacuum dominated and a matter dominated universe in CEG. Our numerical expansion also consists of two parts. In our first numerical expansion, we propose and test an iterative algorithm to solve for CEG and MOND in cases of cylindrical symmetric baryonic densities. This is done using an iterative algorithm based on Fourier- Bessel transforms. The algorithm can also be applied to arbitrary baryonic densities using just Fourier methods. In our second numerical expansion, we fitted both CEG and MOND to 131 galaxies from the SPARC database. The fittings were done by employing a Markov Chain Monte Carlo (MCMC) fitting algorithm using three fit parameters. Both MOND and CEG provide good fits to the rotation curves. Out of the 131 fits, 9 fits (7%) were poor fits (R2 < 0) and 62 galaxies (47%) were excellent fits (R2 > 0:9) and 94 gaxlies (72%) good fits (R2 > 0:7). We also provide fits to galaxies that are traditionally considered hard to fit with dark matter maximum-disk models. Such goods fit for CEG are in contrast to EG, which was found to be a bad fit to the rotation curves by the SPARC team. However, scatter plots show a clear correlation between the MOND and CEG fit parameters, making it hard to distinguish between the two experimentally on the basis of rotation curves or other Newtonian features. Another experimentally verifiable result is the proposed lensing formalism, which could be tested by comparing rotation curves of galaxies to strong lensing. We recommend further research into covariant features of both theories in order to arrive at experimentally verifiable differences. Other interesting topics of follow-up research include using the numerical method to solve for the evolution of the galactic disk or in cases that the magnetic term in CEG or MOND becomes non-negligible. The connection to both dark matter and dark energy also make CEG an interesting theory to further research in cosmology. ix Acknowledgements Even-though the rest of this report contains general relativity, curved space-times and tensors, this might be the hardest part of all. I would like to express my sincere and utmost gratitude to my supervisor Dr. Paul Visser for the countless meetings we had, the countless hours he has put into checking and correcting this report, but most of all for the work he has put in me and my development as a physicist. During the span of this project, I have been free to explore general relativity, classic field theory, cosmology etc. and experiment and discuss my own ideas with mister Visser. Such a mentor and guide is priceless and has allowed me to grow into the physicist I am today. Thank you for the time you have put into this report! Thank you for the time you have put in me! And most of all, thank you for your faith in me! We would also like to thank Dr. Jos Thijssen for his opinions and guidance on the numer- ical implementation of the presented algorithm and the other committee members for their willingness to review this report. xi Contents Declaration of Authorship iii Abstract vii Acknowledgements ix 1 Introduction1 1.1 Introduction...................................1 1.2 Structure of the report..............................1 1.3 Reading routes.................................2 1.3.1 Bachelor route.............................2 1.3.2 Astrophysical route...........................2 1.3.3 Theoretical route............................3 2 Astrophysics and galaxies5 2.1 Rotation curves.................................5 2.2 Observed velocities and dark matter......................7 2.3 Modified gravity................................7 2.4 The structure of galaxies............................9 2.5 Galactic velocities................................ 11 2.6 The SPARC database.............................. 11 2.6.1 Gas profile............................... 12 2.6.2 Stellar disk............................... 13 2.6.3 Mass-to-light ratio........................... 13 2.7 Dark matter in galaxies............................. 14 2.8 Covariant Emergent Gravity.......................... 14 2.8.1 Quantum gravity............................ 14 2.8.2 Covariant Emergent Gravity...................... 14 3 From Newtonian to Einsteinian gravity 17 3.1 Classical Gravity................................ 17 3.2 Particles and Field, What’s the difference?................... 18 3.3 Particles and Fields: The field equations.................... 19 3.4 Curved space-time............................... 21 3.5 Covariant Derivatives.............................. 22 3.6 Other uses of the metric............................. 24 3.7 General relativity................................ 24 3.8 Cosmology................................... 25 4 Emergent Gravity 27 4.1 Entropic Gravity................................ 27 4.1.1 Emergent properties.......................... 27 4.1.2 Entropy and emergent forces...................... 28 xii 4.1.3 Bekenstein’s entropy.......................... 29 4.1.4 Entropic gravity: Verlinde’s first paper................. 30 4.2 Emergent Gravity; Verlinde’s first paper.................... 31 4.2.1 Newton’s law of gravity........................ 32 4.2.2 General matter distributions...................... 33 4.2.3 Critique................................. 33 4.3 Apparant dark matter; Erik Verlinde’s second paper.............. 34 4.3.1 Entropy and the cosmological constant................ 34 4.3.2 Introducing mass............................ 36 4.3.3 Mass and entropy............................ 36 4.3.4 Entropy difference........................... 37 4.3.5 Interpretation of results......................... 38 4.4 Stresses and Strains............................... 38 4.4.1 Theory of Strain and Stress......................