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This thesis discusses the misconceptions in special relativity and argues for the augmentation of the modern physics course. It provides an introduction to Einstein's work on the electrodynamics of moving bodies and his theory of relativity. The thesis also discusses Einstein's field equations and the difficulty in solving them. It concludes with a proposal for a conversation on the topic and future prospects. a table of contents, acknowledgments, and references.
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Bachelor’s of Science in Chemistry
TABLE OF CONTENTS
iv A mention of faculty members would be incomplete without a mention of one who took me in as a freshman and has maintained herself as a friend and mentor, Dr. Crowe. She has always been my go-to for fervently honest and critical advice (even if I ignore her advice and take the harder path), and has encouraged me to improve in many aspects of my life. I would also like to thank my closest friends and peers on campus who helped me maintain my sanity outside of research, class, work, and athletics. Their feedback, scientific insight, and sense of humor have been invaluable throughout my undergraduate career. In particular, I would like to thank Bernie Tyson, Zachary Fralish, Ashley Norberg, Brett Walker, Megan Scranton, Christian Beauchemin, Brian Slivonik, Jacob Taminosian, and Daniel Bolding. Most importantly, I would like to thank my family for their never-ending love and support of me as I pursue my academic and career goals. Their willingness to listen, patience, and understanding have been crucial to my success in my undergraduate career. To always being continuous, but occasionally being non-differentiable.
On September 26, 1905, Albert Einstein published his work entitled “On the Electrodynamics of Moving Bodies”, where he reconciled the inconsistencies between Maxwell’s equations and Newtonian mechanics [1]. While this theory had implications that would take years to be accepted be the physics community, Einstein’s work did not cease there. Shortly after publishing his special theory of relativity, Einstein began work to incorporate gravity into his theory – generalizing his theory of relativity. Einstein’s work culminated 10 years later when he presented what are now known as Einstein’s field equations at the Prussian Academy of Science in 1915. At the end of that same year, Einstein would publish his paper “The Foundation of the General Theory of Relativity” [2]. Einstein’s field equations are a system of 16 partial, non-linear, coupled differential equations and as such are incredibly difficult to solve explicitly [3]. It is for this reason that General Relativity (GR) is never rigorously discussed in undergraduate and is not mandatory for graduate school programs. In fact, many physicists are never exposed to an honest discussion of GR. The mathematical machinery necessary for a rigorous discussion of GR to take place is no trivial obstacle to overcome. Instead,
fact, no Modern Physics textbooks to date include a discussion of acceleration in the context of special relativity [4,5]. As hinted at above, GR cannot be rigorously discussed at this level due to the prohibitive difficulty of the mathematics involved; however, many Modern Physics textbooks will include a qualitative discussion of GR and of spacetime. It is at this point that many Modern Physics textbooks state the Einsteinian equivalence principle, “The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime,” which is often phrased by educators as the “local equivalence of gravity and acceleration” [6]. It has been noted by relativists, such as Sean Carroll, that there is a misconception amongst physicists that special relativity only applies to inertial situations, and any physical situation that involves acceleration must be reconciled through the use of GR [7]. In his textbook, An Introduction to General Relativity: Spacetime and Geometry , Sean Carroll states, “The notion of acceleration in special relativity has a bad reputation, for no good reason”. The true statement is that the mass in question must be set up within an inertial coordinate system, but once the coordinate system is setup, any non-inertial trajectory of the mass is fully within the capabilities of special relativity. What makes special relativity special is that the
spacetime must have no curvature. In other words special relativity holds true in flat spacetime. Unfortunately, a literature search does not reveal any hard evidence that this misconception actually exists, or amongst whom it exists. One Modern Physics textbook even incorrectly states, “Special relativity is concerned only with inertial frames of reference, that is, frames that are not accelerated” [8]. One would expect that if this misconception exists, it will be most prevalent in those physicists who have the smallest amount of experience with relativistic mechanics. For many fields, physicists are not exposed to relativity whatsoever with the exception of a Modern Physics course. If this is true, it would be expected that physicists in fields such as statistical mechanics or atomic, molecular, and optical (AMO) physics would have a higher prevalence of this misconception than those of relativity or particle physics where relativistic mechanics is employed more frequently. 1.2 Method Constructing the survey In an effort to quantitatively investigate if and where this misconception exists, a survey was constructed carefully to probe at three misconception-rich areas in statistical mechanics, classical mechanics, and quantum mechanics. This was done in order to mask the goal of the survey from the participant in an effort to reduce bias.
To distribute the survey, emails were sent from the Principal Investigator’s email address with the following text: To whom it may concern, I am a physics professor at Florida Southern College. As part of my undergraduate student’s honors thesis, we have formulated a survey with the goal of identifying misconceptions in physics at all levels. This survey is meant to be administered to whoever is willing to participate at the undergraduate, graduate, and post-doctoral levels as well as faculty who are willing to participate at your institution. This is a very quick survey that should take no longer than five minutes. The survey consists of four questions that probe misconceptions in classical, quantum and statistical mechanics, as well as relativity theory. As outlined in the attached consent form, we would like each anonymous participant to disclose their level of education and possibly their field of research. This will help us determine which misconceptions exist at which levels of education and/or specialty. The link below will take you directly to a consent form, which must be signed to gain access to the survey, as well as the survey itself. Thank you for your time and consideration. Sincerely, Asst. Prof. Ron Pepino The consent form is attached to this email if you have interest in reviewing it before administering the survey within your department. Prior to taking the survey, participants were required to read the informed consent form below from within a google form, and sign their understanding and agreement to the conditions of the survey prior to receiving a link to the survey. Florida Southern College Informed Consent Information Project Title: Persistent Misconceptions in Physics
Principal Investigator: Dr. Ron Pepino Phone: (863)680- 3776 Email: rpepino@flsouthern.edu Department: Chemistry, Biochemistry, Physics Hello, I am researcher at Florida Southern College. You are being invited to participate in a research study regarding common misconceptions in physics. The goal of this study is to identify which misconceptions persist at different levels of education including: undergraduate, graduate and postdoctoral. In this particular study, we are concerned with topics in classical mechanics, relativistic dynamics, quantum mechanics and thermal physics. The anonymous data collected in this survey will be used to develop “guided learning” or “case-study” teaching approaches that can be administered in the appropriate courses. This data may also become part of physics education publication. As part of this study, we ask you to complete a brief four question survey that should take roughly 5-10 minutes. The survey will be multiple choice, with a section to briefly provide a written response to justify your answer. Since the goal of this survey is to obtain information about misconceptions, please refrain from using outside sources to answer these questions; simply answer each question based on your current understanding of each topic. The data collected will be stored in a secure file, and your privacy and research records will be kept confidential to the fullest extent of the law. Any personal information that accompanies the data will be removed immediately upon reception. Your signing of this consent form is in no way linked to the data obtained in the survey. Furthermore, your institution affiliation is in no way traceable to your responses to the survey questions. Thus, there are no risks associated with taking this survey. Only authorized research personnel, employees of the Department of Health and Human Services, and the FSC Institutional Review Board may inspect the records from this project. Your decision to participate is completely voluntary and there is no monetary compensation for taking this survey. If you have any questions about this study, contact Dr. Ron Pepino at the phone number or email at the top of this form. If you have questions about your rights as an individual taking part in a research study, you may contact the Chair of the Florida
Table 1.2: Table showing the demographics of each subpopulation of participants. **Field # of Faculty
PhD/Postdocs
Master’s program students
undergraduates Astrophysics** 0 4 2 0 AMO 2 0 0 3 Biophysics^2 0 0 Condensed Matter
GR/Cosmology 5 3 1 0 Particle/Nuclear 13 5 0 4 PER^4 0 0 Other 4 3 0 2 Table 1. 3 : Table showing the percentage of each subpopulation which answered the relativity question correctly. Specialty Percent correct Astrophysics 80 % AMO 40 % Biophysics^0 % Condensed Matter 57.1% GR/Cosmology 66.7% Particle/Nuclear 45.5% PER^50 % Other 55.6% Total Population 52.1% Table 1.4: Table showing the percentage of correct answers by professional level. Education level Percent correct Faculty 56.41% PhD/Postdoc 36.84% Masters 66.67% Undergraduate 50%
Table 1. 5 : Comments made by participants as justification for their answer to the relativity question. Professional Level Comment Faculty Special relativity applies only to constant velocity motion, in the absence of mass. Acceleration necessarily involves general relativity, but Mach's principle. Faculty Special relativity has no problem with acceleration; one just needs to stick to inertial frames. Undergraduate Special Relativity explains constant motion at speeds close to c, while general relativity explains non-uniform motion near c. Master’s program The answer, "Both special and general relativity" doesn't make sense, as special relativity is a subset of general relativity. Special relativity alone cannot explain what happens during acceleration, only what comes before and after. PhD program Special relativity is a subset of general relativity, where spacetime is flat, i.e., where there is no acceleration PhD program Special relativity only deals with inertial reference frames, which are non- accelerating PhD program Special relativity is just for fast things
Figure 1.3: Column diagram showing the number of correct and incorrect answers as a function of presentation level. The undergraduates were correct 50% of the time, while the PhD/Postdocs performed the worst out of any education level. The faculty were correct 1.4 Discussion While this survey was sent to 62 institutions across the United States, a total of 72 responses were received (including responses from Harvard, MIT, and Caltech). The demographics (Figure 1.1) reveal that the largest professional level represented was faculty members at their respective institutions. The fields’ of physics and their respective representation can be seen in Table 1.1. Table 1.2 shows the distribution of professional levels represented within each field of physics. It is worth mentioning that there was an error in the google form which caused one of the astrophysicists’ responses 0 5 10 15 20 25 Faculty PhD/Postdocs Masters Program Undergraduate
to be lost. The demographics were performed for n=72, but all subsequent analysis was performed with n=71. The largest field represented was nuclear/particle physics which consisted of 30.5% of the total population. The smallest field represented by this data is biophysics at 4.1% of the total population. While this is not the ideal distribution of responses by field that we would have hoped to obtain, this is the nature of conducting surveys. While each field may not have as many participants as would be ideal, there is good diversity of representation in our sample and as such the information is still useful. Because of the nature of the information obtained from the participants, there are a few interesting ways to look at the data. The first interesting way to analyze the data is by looking at the correct versus incorrect responses of the participants as a function of their field (Figure 1.2, Table
its own. On the other hand, when this question was incorrectly answered, the justification was that special relativity is only for constant velocity motion at velocities near the speed of light. It can also be seen in Table 1.4 that when the participant knew what they were talking about, the non-sensical nature of two of the four answers was obvious to them (Table 1. 4 , line 5). The professional levels seen in Table 1.4 reveal that this misconception is persistent to the level of PhD candidates and faculty members. It would seem obvious that this misconception is likely formed in a Modern Physics course where special relativity is only discussed in the inertial case, and acceleration is only discussed in the context of GR. This must be the case because the majority of physicists may never see relativity again in their career unless they go into a field where it is relevant. This would imply that there is a problem with the way that relativity is being discussed in Modern Physics. 1.5 Conclusions and future prospectus While the data obtained in this study was not as ideal as one would prefer, the data seems to imply that the physics community is not as well aware of the capabilities of special relativity as they should be. This data also hints that the issue is a systemic one, as faculty may not truly understand what they are teaching at the level necessary to teach the subject. The rectifying of this misconception is not difficult, especially with the mathematical background every physicist must obtain over the course of their
undergraduate career. The only mathematical tools necessary to eradicate this misconception from the students’ (or faculty members’) mind is a background through Calculus 2, which is a course taken by most physicists their freshman year in college. Since Modern Physics is sophomore level course, there appears to be no good reason not to address this misconception head-on by solving non-inertial problems in a Modern Physics course. It would be prudent moving forward to identify and employ an expedient teaching method that can effectively rectify this misconception directly, and this will be a topic of future work. 1.6 References [1] A. Einstein, Annalen Phys. , 17, 891- 921 ( 1905 ). [2] A. Einstein, Sitzungsber. 44, 778 - 799 (19 15 ). [3] H. Heintzmann, Zeitschrift Für Phys. 228, 489- 493 (1969). [4] J. Morrison, Modern Physics for Scientists and Engineers, Chap. 1, 1 st^ ed., (Academic press, 2010). [5] S. T. Thornton and A. Rex, Modern Physics for Scientists and Engineers , Chap. 1, 4th ed. (Brooks/Cole, 2013). [6] A. L. Harvey, Ann. Phys. , 29, 383-390, (1964). [7] S. Carroll, Spacetime and Geometry: An Introduction to General Relativity , 1st ed., Chap. 1, Sec. 2 (Pearson, 2003). [8] R. Harris, Modern Physics , Second Edition, Chap. 1, (Pearson: Addison-Wesley, 2008).
