FOLLOW-UP: Learning and Making Sense of Differential Geometry in General Relativity

This post is a follow-up to the previous post [LINK] in which I tried to make sense of the mathematics of differential geometry, especially in the context of general relativity, and proposed notation that may be less confusing, more consistent, and arguably more powerful than traditional notation given the existence of a metric in general relativity. With similar motivations, this post explores how some of the ideas of electromagnetic (EM) theory may change on curved manifolds. Follow the jump to see more.

Learning and Making Sense of Differential Geometry in General Relativity

I have been out of the field of physics for over 5 years, didn't get to make use of my training in quantitative analysis too much in my previous job as a postdoctoral researcher at UC Davis, and make even less use of my training in quantitative analysis in my current job as a transportation planner at Cambridge Systematics. When I was doing work in my PhD in physics and when I was a student in school and college before graduate school, I enjoyed solving problems in math & physics and growing & applying my toolbox of quantitative skills, so since leaving the field, and especially more recently, I have felt a slight itch to recover some of those skills purely for my own personal satisfaction. To that end, I've resolved to learn or relearn some parts of math that I did not learn at all or to the full extent that I should have in college or graduate school (because I could ultimately manage my work in my PhD without knowing those things to that extent). Currently, I'm interested in learning differential geometry as well as complex analysis. It remains to be seen whether there are other topics in physics-relevant math that I become interested in; if they are, then I will certainly make an effort to learn at least a little bit about them. I've enjoyed learning these topics in math to broaden my knowledge & skills, and I've particularly enjoyed pondering definitions & rules of these concepts in math almost like a lawyer (which also makes this useful for my current job in a very indirect way, because my current job in part involves analyzing laws & regulations and creating intuitive explanations of them for public sector clients).

I can't guarantee that I'll write a blog post about every topic in math that I learn about. However, I am writing this post specifically about differential geometry in parts because I feel that I have learned the basic ideas in it in the context of general relativity to my satisfaction (which was my original goal) and because I have some questions/concerns that I have not been able to satisfactorily resolve based on what I have read in lecture notes or textbooks. This post is a way to further flesh out those questions/concerns. Follow the jump to see more.

There are a few conventions & assumptions to note throughout this post.

• The dimension of the manifold will generically be denoted \( N \). For most nonrelativistic physics, \( N = 3 \), while for most relativistic physics (including general relativity), \( N = 4 \); exceptions include constrained low-dimensional systems or different physical models that have different dimensions analogous to differential geometry.
• I will consistently use Einstein summation unless otherwise specified. This means that indices that are repeated with one as an upper index and one as a lower index will be summed; indices should never be present more than twice at all and more than once in the same (upper versus lower) position.
• Although the convention in general relativity is to use lowercase Greek letters for indices, I will keep things easier to read by using lowercase English letters for indices.
• The motivation of general relativity means that I will only consider differentiable manifolds and specifically smooth (infinitely differentiable) scalar functions or tensor components on them.
• The motivation of general relativity also means that I will only consider torsion-free connection coefficients that can be expressed in terms of the metric.

Green's Functions and Violin Making

A few years ago, a friend of mine, who is a very accomplished player of the violin & viola and who is an amateur builder of violins & violas, sent me a video on YouTube [LINK] of the violin maker Peter Westerlund showing how to create violins (and violas, by the same logic and with only slightly different overall dimensions) that consistently produce sounds that people well-versed in European classical music judge to be of high quality. Many such people have praised the consistent warm & rich aural tones of violins & violas made in Italy by Andrea Guarneri around 1650 & Antonio Stradivari around 1700, yet it has been difficult for any violin maker to replicate such warm & rich tones in the violins that they have made, in part because there is little scientific consensus about the specific acoustic spectral qualities that make those violins so highly valued. Further complicating matters, as explained in a news article by Science magazine in 2017 [LINK], is the fact that many blind & double-blind studies have left musicians & keen listeners preferring contemporary violins over those made by Stradivari or Guarneri. Even if the instruments by made by Stradivari or Guarneri aren't consistently rated the best, they are rated very highly far more consistently than those made by other violin makers (though this may reflect selection bias in such tests), so there may be merit to considering how to replicate those instruments' aural qualities.

Traditionally, violin makers have carefully studied the exact dimensions, shapes, and materials used in instruments made by Stradivari or Guarneri and done their best to replicate them. However, the resulting aural qualities have typically been judged to be unsatisfactory. This is in various parts due to the effects of aging on the material components of violins (which generally cannot be artificially accelerated), one-off differences in instances of even the same materials used, slight deviations in the craftsmanship, shapes, or dimensions, and other factors that have yet to be explained. This motivated Westerlund (who is probably not the first person to come up with this method) to develop a method to reliably create instruments with highly consistent sounds even if they don't replicate sounds from instruments made by Stradivari or Guarneri per se. The method is as follows. Each violin or viola has a top plate and a back plate, and these are initially constructed separately, joined only later on corresponding sides of the ribs. For each plate, after getting the basic shape & thickness correct, the violin maker should further adjust the shape by tapping or rubbing the plate in various locations, ideally using the same tapping or rubbing motion with the same intensity every time. The violin maker should iteratively shape (by cutting or scraping) at & around those locations of tapping or rubbing and then continue to tap or rub at all locations, until eventually, the tapping or rubbing motion produces approximately the same tone at every location. Once that has been achieved, shaping of the plate is finished.

My background in theoretical & computational nanophotonics from graduate school, even though I was already a few years removed from that even by a few years ago (having changed fields to transportation research after graduate school [LINK]) when my friend shared that video with me, made me interested in trying to mathematically & physically understand why Westerlund's process is successful in creating instruments that have strong internal tonal consistency. However, I wasn't able to come up with a satisfactory answer until much more recently. Follow the jump to see more of my thinking about this, leading to the resolution.

Some Dangers of Confusing "Changing One's Mind" with "Bayesian Updating"

Recent conversations with friends & colleagues about probability theory reminded me of conversations with a friend of mine in graduate school about the supposed virtues of making one's own reasoning in one's daily life more systematic through Bayesian inference. The basic idea, in rough qualitative terms, is that one's belief in a hypothesis can be quantified through a prior probability, and when one observes some data related to that hypothesis, one can use the probabilities of observing that data when that hypothesis does or does not hold to update one's belief in (becoming the posterior probability of) that hypothesis based on the data. An example of quantitative & qualitative explanations can be found on the site LessWrong [LINK]. However, even in graduate school and again more recently, I realized that it is very easy for one to talk oneself into believing that one is using systematic Bayesian reasoning while actually just rationalizing one's own prior beliefs & changes in beliefs after the fact. This can be illustrated mathematically in a few ways that are not exhaustive. Follow the jump to see more.

Transitioning from microscopic to macroscopic and quantum to classical regimes

I recently read two things that were of interest to me having previously worked in physics. One was an article in The New Yorker magazine [LINK], in which the author does a good job of going over the successes of mathematical modeling in the physical sciences and contrasting this with the limitations of mathematical modeling in public health (showing, for example, how many models of the spread of contagions fail when governments & societies take fast & drastic collective actions to limit the spread), the failures of mathematical models in social sciences where the outputs of those models can create feedback loops with public sentiment (for example in political polling), and the way that many people who use machine learning models in different domains expect the fancy curve-fitting of those models to represent fundamental understanding when that might not really be so. The other was a journal article published in Physical Review Letters [LINK] about how it can be possible to test the extent to which a massive (as opposed to massless) object which exhibits the dynamics of a simple harmonic oscillator and prepared in a quantum coherent state can be tested for deviations from classical behavior using a protocol that does not depend on the mass of the object (although I question this given that the protocol depends on timed measurements that depend on the frequency of oscillation, and in many physics contexts the frequency does depend on the mass as \( \omega = \sqrt{k/m}\), but this is somewhat of a quibble). These two things got me to think about something that I realized I never got out of many years of formal undergraduate & graduate education in physics. This can be illustrated with the following example.

In introductory physics classes that focus on Newtonian mechanics, a prototypical problem involves a block, modeled as a point mass, sliding (with or without friction) down a fixed triangular incline in the constant gravitational field of the Earth. In the context of those classes, instructors will be careful to note that this is merely a model, and corrections could come from the inclusion of the variation of the Earth's gravitational field & surface curvature, the technical possibility of moving the triangular incline (which must be much more massive than the block in question), the shape of the block, variations in the touching surfaces, air resistance, et cetera. In later classes, instructors may point out corrections due to special relativity (i.e. the speed of light) and general relativity (as it relates to the Earth's gravitational field).

However, in later classes about quantum mechanics & statistical mechanics, instructors explain how different the models are from models of Newtonian mechanics at human scales, but they often promise that appropriate treatments of aggregates of microscopic constituents can consistently recover results from Newtonian mechanics, yet this promise is almost never fulfilled. In particular, wavefunctions that describe pure states of single microscopic particles are quite far removed from the simple dynamical variables describing blocks on inclined planes, although statistical mechanics can probabilistically describe the solid states of the block & inclined plane as well as the gaseous state of the surrounding air, it is not usually extended to describe the dynamics of the block sliding down the inclined plane. For example, if a block sliding down a fixed inclined plane of horizontal angle \( \theta \) in a uniform gravitational field is described as having equations of motion \( m\ddot{x} = mg\sin(\theta) \) where the \( x \)-axis is defined as pointing downward parallel to the slope of the inclined plane for increasing \( x \) and the \( y \)-axis points outward in the normal direction from the inclined plane, then I wish to see corrections of the form \( m\ddot{\vec{x}} = \sum_{\mu = 0}^{\infty} \sum_{\nu = 0}^{\infty} \hbar^{\mu} k_{\mathrm{B}}^{\nu} \vec{f}^{(\mu, \nu)} \) where the lowest-order term is \( \vec{f}^{(0, 0)} = mg\sin(\theta)\vec{e}_{x} \). I have never seen these sorts of quantum or statistical corrections to Newtonian equations of motion in simple (in the context of Newtonian mechanics) systems. Similarly, it is rare to see how quantum or statistical mechanical systems can, in appropriate limits, reproduce classical systems; I can only think of the quantum coherent state of the simple harmonic oscillator as well as how the Moyal bracket in the phase space formulation of quantum mechanics reduces to lowest order in \( \hbar \) to the Poisson bracket, and in the latter case, intuitive construction of the quantum phase space quasiprobability function is made more difficult (compared to construction of a classical phase space probability density function, as I did in a post [LINK] from a few years ago) by the fact that unlike the classical phase space probability density function, the quantum phase space quasiprobability function cannot be arbitrarily localized in phase space, it can take on negative values for certain wavefunctions, it is compressible in phase space with respect to its own evolution over time, and it is not obvious how it should look for a system of many particles constituting a macroscopic object like a block (in contrast to a classical phase space probability density function, which for such a system could just be a product of Dirac delta functions localizing each microscopic constituent to a point in phase space).

These considerations reminded me of a discussion I had last year with friends from college, who also did course 8 (physics) with me. We came to a consensus that while people who do not become physics majors should, as usual, get exposure to Newtonian physics and the basics of electricity & magnetism, people who become physics majors should have a curriculum over 3-4 years that exhibits a sensible conceptual progression. In particular, after seeing Newtonian mechanics, such students should then be exposed to Lagrangian & Hamiltonian formulations of classical mechanics. The Lagrangian formulation of classical mechanics should then be used to develop intuitions about mechanical waves, which in turn can lead to introductions to classical field theory and development of classical electromagnetic theory as a rich example of a classical field theory. (I would also personally recommend using the introduction of mechanical waves to introduce the linear algebraic treatment of waves and then reintroduce the linear algebraic treatment of waves into the treatment of linear classical field theories in general & linear classical electromagnetic theory in particular.) The Hamiltonian formulation of classical mechanics should then be used to develop intuitions about probability distributions in classical mechanics, which in turn can be used to develop intuitions about statistical mechanics. Optionally, at this point, the Hamiltonian formulation of classical mechanics can also be used to develop intuitions about nonlinear dynamics & chaos theory, but while this is good for the broader education of physics students, it is less immediately relevant for the introduction of quantum theory to come soon after (because quantum mechanics is linear). Finally, only after these things happen should quantum theory be introduced, such that there are clear connections of the wavefunction formulation of quantum mechanics to mechanical waves, the phase space formulation of quantum mechanics to classical phase space probability distributions, and the linear algebraic framework of quantum mechanics to linear algebraic treatments of classical field theories (including linear classical electromagnetic theory); this will ensure that students understand how ideas like superposition, interference, rotation through a Hilbert space, statistical uncertainty, and related ideas are not unique to quantum mechanics (which is unfortunately too often a consequence of the way quantum mechanics is typically introduced in undergraduate curricula, at least in the US). We also came to a consensus that in each course, there should be clear explanations of what prototypical systems are analytically solvable, what prototypical systems are not analytically solvable, and why (in each case).

FOLLOW-UP: How to Tell Whether a Functional is Extremized

This post is a follow-up to an earlier post (link here) about how to tell whether a stationary point of a functional is a maximum, minimum, or saddle point. In particular, as I thought about it more, I realized that using the analogy to discrete vectors could help when formulating a more general expression for the second derivative of the nonrelativistic classical action for a single degree of freedom (i.e. the corresponding Hessian operator). Additionally, I thought of a few other examples of actions whose Hessian operators are positive-definite. Finally, I've thought more about how to express these equations for systems with multiple degrees of freedom (DOFs) as well as for fields and about how these ideas connect to the path integral formulation of quantum mechanics. Follow the jump to see more

How to Tell Whether a Functional is Extremized

I happened to be thinking recently about how to tell when a functional is extremized. Examples in physics include minimizing the ground state energy of an electronic system expressed as an approximate density functional \( E[\rho] \) with respect to the electron density \( \rho \) or maximizing the relativistic proper time \( \tau \) of a classical particle with respect to a path through spacetime. Additionally, finding the points of stationary action that lead to the Euler-Lagrange equations of motion is often called "minimization of the action", but I can't recall ever having seen a proof that the action is truly minimized (as opposed to reaching a saddle point). This got me to think more about the conditions under which a functional is truly maximized or minimized as opposed to reaching a saddle point. Follow the jump to see more. I will frequently refer to concepts presented in a recent post (link here), including the relationships between functionals of vectors & functionals of functions. Additionally, for simplicity, all variables and functions will be real-valued.

Twelfth Paper: "Near-field radiative heat transfer in many-body systems"

My twelfth paper has been published! It is in volume 93, issue 2 of Reviews of Modern Physics, and an older preprint of it is available too for those who don't have access to academic journals (it is identical in content and only differs in formatting). Unlike my previous blog posts about published papers that I have written, this one will not strictly use the thousand most common words in English. This is because unlike my previous papers, which put forth novel ideas advancing the field of nanophotonics, this is a review paper that gives a broad historical scientific overview of the subject, namely, the flow of heat through light (i.e. electromagnetic (EM) fields) between objects that are typically separated by less than 1 micron (approximately 1% of the width of a typical human hair). It goes over work that other scientists have done theoretically and experimentally in this subject, and this paper in particular is divided into two main sections.

The first section, to which my PhD advisor & I made most of our contributions, is about the flow of heat via EM fields between just 2 objects. Relevant issues include choices of materials (mainly metals/conductors versus insulators), choices of shapes for objects, advances in experimental measurement techniques, advances in computational simulation techniques, and derivations of upper limits to the flow of heat via EM fields between 2 objects (mostly referring to my previous 2 papers that were the subject of the following linked blog post).

The second section, which constitutes the bulk of the paper, is about the flow of heat via EM fields among more than 2 objects. Relevant issues include changes in temperature over time in objects that are very small compared to their separations, the fact that the heat flow among more than 2 objects involves very complicated interactions among them, the fact that material properties could depend on temperature so there could be many possible sets of object temperatures where heat flows but temperatures don't change, heat flow via EM fields over distances longer than 1 micron, applications of heat flow via EM fields to microscopy techniques, heat flow via EM fields in materials that can be attracted to permanent magnets, and applications of heat flow via EM fields to engineer new devices.

Eighth, Ninth, Tenth, and Eleventh Papers

My eighth, ninth, tenth, and eleventh papers have been published! These require subscriptions to read, so here are alternate links to older preprints for the eighth, ninth, tenth, and eleventh papers, respectively (which have most of the same content, with some minor changes to explanations, citations, and figures relative to the published versions). As with my previous papers, in the interest of explaining these ideas in a way that is easy to understand, I am using the ten hundred most used words in English (except for the two lines that came before this one), as put together from the XKCD Simple Writer. I will use numbers sometimes without completely writing them out, use words for certain names of things without explaining further, and explain less used words when they come up. Keep reading to see what comes next. While these papers aren't as closely related to each other as the previous three, there are enough relations that I'm putting them together in a single post. These papers need a lot more math (note: "math" isn't one of the ten hundred words) than the papers before, and because they need a lot of thinking to get, I actually won't say as much about them.

On another note, this is a milestone for me because these are the last papers from my PhD in which I was a leading author. I still have one more review paper left to be published, but as that has been submitted to the journal and as I'm not the leading author, I don't really need to worry about that at this point. (Of course, once that is published, I will write a blog post summarizing it, though as it is a review paper, that summary will probably be quite short.) Thus, I am truly done with the work from my PhD, and can fully shift my mindset away from physics toward thinking about problems in transportation policy, as I will do in my postdoctoral research at UC Davis.

Classical Phase Space Densities for One or a Few Particles

This is the first time in several years that I've done a post about physics that didn't have to do with my research. This came about from thinking about applying techniques in statistical physics to game theory; although I still have a lot more to learn about that and need to do more to flesh out those ideas, it occurred to me in the process that I never had such a good intuition for the phase space density in classical mechanics, and notes that I've found online focus almost exclusively on the phase space density of a large number of particles in an explicitly statistical treatment. I intend to use this post to shed light on why this may be the case, help build intuition for how things like the Liouville equation work for simple systems of one or a few particles, and reinforce the notion that there is no classical analogue to the phenomenon of a multi-particle entangled quantum state yielding a mixed single-particle state under a partial trace. Follow the jump to see more.

Fifth, Sixth, and Seventh Papers

My fifth, sixth, and seventh papers have been published! These require subscriptions to read, so here are alternate links to older preprints for the fifth, sixth, and seventh papers, respectively (which have most of the same content, with some minor changes to explanations, citations, and figures relative to the published versions). As with my previous papers, in the interest of explaining these ideas in a way that is easy to understand, I am using the ten hundred most used words in English (except for the two lines that came before this one), as put together from the XKCD Simple Writer. I will use numbers sometimes without completely writing them out, use words for certain names of things without explaining further, and explain less used words when they come up. Keep reading to see what comes next. I'm putting these three papers together in a single post because they form a trilogy of sorts, all having to do with finding the biggest number for how much heat, through light, can go from one body to another when they are really close together, or can go from one body into outer space. These papers need a lot more math (note: "math" isn't one of the ten hundred words) than the papers before, and because they need a lot of thinking to get, I actually won't say as much about them.

The fifth paper is called "T Operator Bounds on Angle-Integrated Absorption and Thermal Radiation for Arbitrary Objects", and is in volume 123, issue 5 of Physical Review Letters. This is the one that has to do with how much heat, through light, can go from one body to outer space. People knew before that the number for how much heat really big bodies can put through light into outer space grows like the surface area of the body, but for really small bodies it grows like the space of the whole body (volume), and they were not sure how these two things join in between. This paper lets people figure out what the most heat is that can go from a body through light into outer space no matter what the largest shape the body can sit in, and shows how to join the things that people knew before for middle-size bodies of different shapes. (Another press release from my department can be found here.)

The sixth paper is called "Fundamental limits to radiative heat transfer: Theory", and is in volume 101, issue 3 of Physical Review B, while the seventh paper is called "Fundamental Limits to Radiative Heat Transfer: The Limited Role of Nanostructuring in the Near-Field", and is in volume 124, issue 1 of Physical Review Letters. Those two papers go together, so I'll write about them together. The sixth paper is about the math behind figuring out the biggest number for heat, through light, to go between two bodies. The seventh paper shows that heat, through light, going between two big flat bodies that are close together can be pretty close to the biggest number possible, so making the shapes of the bodies less simple than just flat surfaces is of no use.

Fourth Paper: "Impact of nuclear vibrations on van der Waals and Casimir interactions at zero and finite temperature"

My fourth paper has been published! It is in volume 5, issue 11 of Science Advances, which is an open-access journal, so anyone can read it. As with my previous papers, in the interest of explaining these ideas in a way that is easy to understand, I am using the ten hundred most used words in English (except for the two lines that came before this one), as put together from the XKCD Simple Writer. I will use numbers sometimes without completely writing them out, use words for certain names of things without explaining further, and explain less used words when they come up. Keep reading to see what comes next.

In papers that came before this one, I looked at how to do a better job of figuring out the van der Waals (vdW) forces, which are the forces that let geckos (small animals with hard skin over which your finger can slip easily) stick to anything no matter what it is made of, between molecules, which are the little things that make up most of the stuff we see and are in turn made of smaller things called atoms; I also looked at how to do a better job of figuring out how heat (through light) goes between different molecules, especially when they are near larger bodies, and that needed me to do a better job of considering how molecules can make changes on each other through light, and that means that I need to better consider how the full atoms within molecules move toward and away from each other in a way that repeats itself. In this paper, I used the new way from the paper that came before of considering how molecules can make changes on each other through light to show what happens with vdW forces between molecules and larger bodies, especially when they aren't as cold as possible but are as hot as a room you might go into each day. It turns out that for small or long thin molecules made of carbon atoms, our work can do a pretty good job of showing how being as hot as a room can make vdW forces look very different. For some kinds of large thin sheets of atoms, like boron nitride in which every other atom is boron or nitrogen, our work still does a pretty good job because the electrons, which are the parts of the atoms that are the smallest, lightest, and move around the most, are still pretty close to the centers of the atoms. On the other hand, for other kinds of large thin sheets of atoms, like graphene in which every atom is carbon, our work has some more problems, because electrons in graphene can move around a lot more than our work might make you think, and the ways in which those electrons change how the rest of the atoms move around when they are as hot as a room (instead of as cold as possible) makes vdW forces harder to figure out than our work can say. This means we still need to do more work to better figure out how vdW forces look for those kinds of sheets.

My Time in Luxembourg and at the META19 Conference in Lisbon

This is an update from my recent travel to Luxembourg and Lisbon. I had the privilege of traveling to Luxembourg to deliver an invited talk to my collaborator's group; we have published 2 papers together, are about to have a third published (for which I'll write a post once the publication is official), and are working on a fourth. It was productive to meet with my collaborators to hash out the details of our fourth paper together, and I think I did reasonably well delivering my talk, for which the first part was about the work with my collaborators modeling fluctuational electromagnetic phenomena at the interface between small objects that have to be treated at the atomic scale and large objects that can be treated as continuous, and the second part was about my more recent work purely within my group putting upper bounds on thermal radiation and heat transfer. Additionally, I appreciated the interesting conversations I had after my talk with various members of that group.

I also got to go to the META19 Conference in Lisbon, which is centered around research in nanophotonics and metamaterials; there, I presented an invited talk about the aforementioned work on upper bounds to heat transfer, which my advisor graciously let me do in his place. It was great to meet people that I knew associated with my advisor but hadn't seen in a few years, and we got to discuss each other's work in great depth; I got to also meet a few other people for the first time, and had some good conversations with them about our respective projects too. That said, while I think the delivery of my presentation there was good, I did feel like I could have done a better job of preparing my presentation in a way that better fit the context: my presentation may have been too theoretical, so for an audience of engineers who care more about using metamaterials and nanophotonic architectures to make new devices, I could have been clearer about broad physical insights and done a better job connecting the findings in my work to ideas of interest to such researchers. Additionally, I could have probably done a better job networking, though my reticence was in part due to uncertainty on my part regarding my future plans.

I also had a day in each city to explore, but there were issues in each case. For one, neither Luxembourg nor Lisbon are especially tourist-oriented cities compared to places like London, Paris, Madrid, Barcelona, or Berlin, as far as I've heard: both cities have a lot of nice restaurants and cafes as well as many nice viewpoints and other areas to simply hang out and relax, but there aren't so many specific attractions, as Luxembourg is known more for housing EU governmental offices as well as international finance companies, while Lisbon seems to generally not have invested in tourism as much. I did hear in each place that the surrounding countrysides are beautiful, so perhaps that is where more tourism happens and where more interesting things can be seen. (Also, it didn't help that I was in Lisbon on a Monday, when many things are closed.) For another, both cities are not especially good for people like me in power wheelchairs who want to travel independently; people who are able-bodied, in manual wheelchairs, or have disabilities that require them to be assisted by partners or personal care attendants full-time may be better served. In Luxembourg, I didn't feel like I missed too much because it's a fairly small city anyway, and the public transit buses were accessible (though there were some issues even there). In Lisbon, I felt like I missed more because it is a bigger city, yet its Metro rapid transit train system, which is supposedly accessible, actually has large gaps between the train floors and platforms, and most of the buildings have large steps to enter. I do hope to visit Europe again, but when I do, I hope to make sure it will be a city/area that is more friendly to people in power wheelchairs.

Book Review: "The Structure of Scientific Revolutions" by Thomas Kuhn

I've recently read the book The Structure of Scientific Revolutions by Thomas Kuhn. This is a classic treatise from 1962 expounding Kuhn's view of scientific progress not as cumulative and incremental but instead as comprising paradigms in each field and discipline which drive most scientific research while being subjected to drastic changes from time to time; this is the book that popularized the notions of scientific paradigms and shifts therein. It starts with a description of what "normal science" (in the sense of science comprising and being driven by existing paradigms) is, defining the notion of a "paradigm" in the context of science, and how people do science in that framework. It then moves onto the notion of a scientific crisis, and shows how that may or may not develop into a fully-fledged scientific revolution. Finally, it shows how new paradigms may take root and how scientific revolution may ultimately be resolved in one way or another.

While Kuhn did not perform serious sociological research for this treatise (though the book seemed to me like an informal sociological review of the scientific community at large), and while he later in life turned his attention more to fundamental questions of scientific philosophy, he was a historian of science and identified most strongly as that; I feel this may have helped shape this book into something far more clear and engaging for a layperson like myself than what I may have expected from a book about the philosophy of science, as the book is chock-full of relevant and easy to understand references to the history of science (though it may also have helped that Kuhn, having been a theoretical physicist before becoming a historian of science, focused almost exclusively on the historical development of theoretical frameworks in the physical sciences). Moreover, because this was meant as an extended essay, this book is not particularly long, though it is reasonably well-referenced with illuminating footnotes too; in fact, the chapters are called sections, as would befit an essay/treatise. One question to which I kept returning through the book was about how to distinguish between discoveries that answer open questions within a paradigm versus those which more fundamentally threaten the existence of such an established paradigm; Kuhn masterfully addresses the various aspects of this question in a clear progression over the course of the book, to the extent that I almost felt like he was speaking directly to me in order to answer my questions as I read the book. I do have a few criticisms of the book, though these should themselves be taken with a grain of salt and subjected to criticism too, as I am a layperson in the context of the philosophy of science; follow the jump to read those. Beyond that, though, I think this is a really interesting and valuable perspective on the practice of science at the level of groups/communities, and would be useful for anyone interested in how the sausage of science is made, discarded, and remade.

Third Paper: "Phonon-Polariton Mediated Thermal Radiation and Heat Transfer among Molecules and Macroscopic Bodies: Nonlocal Electromagnetic Response at Mesoscopic Scales"

My third paper has been published! It is in volume 121, issue 4 of Physical Review Letters, and an older preprint of it is available too for those who don't have access to academic journals (it has all of the same figures and ideas, though it is missing a few sentences of further explanation as well as a couple of new citations that were inserted for the final publication). As with my previous papers, in the interest of explaining these ideas in a way that is easy to understand, I am using the ten hundred most used words in English (except for the two lines that came before this one), as put together from the XKCD Simple Writer. I will use numbers sometimes without completely writing them out, use words for certain names of things without explaining further, and explain less used words when they come up. Keep reading to see what comes next.

In the paper that came before this one, I looked at how to do a better job of figuring out the van der Waals (vdW) forces, which are the forces that let geckos (small animals with hard skin over which your finger can slip easily) stick to anything no matter what it is made of, between molecules, which are the little things that make up most of the stuff we see and are in turn made of smaller things called atoms. I tried to figure out how these forces look at distances where the fact that molecules are made of atoms is important, but if those molecules are near much larger bodies, the fact that the larger bodies are made of atoms and molecules should be less important; it turns out that at those distances, how fast light goes matters a lot, and using ways to figure out these forces exactly instead of using easier ways to figure out those forces makes a big change in what those forces are. That paper was able to show how to bring together all of these different ideas from considering large and small bodies in a single way where none of those ideas can be ignored. That would be important when considering new kinds of molecules like graphene, which is made of a lot of carbon atoms in a thin sheet, or really long molecules like DNA or those found in foods, when those molecules are near larger bodies that we make.

This paper looks at the same sorts of molecules, but not at vdW forces anymore. Instead, in this paper, I look at how heat (through light) goes between different molecules, especially when they are near larger bodies. For that, I need to do a better job of considering how molecules can make changes on each other through light, and that means that I need to better consider how the full atoms within molecules move toward and away from each other in a way that repeats itself. By doing that, I can now show how heat through light goes between different molecules, whether they are close together or far away from each other. When people considered heat going through light between larger bodies, they found that the heat would keep growing as the bodies came closer together, and that growing wouldn't stop; from knowing how things work every day, we know that once bodies come close enough, they touch each other, and the growing stops at some point. In this paper, I've shown that when molecules come close together, the heat grows for a while, but if they come close enough, that growing does stop, so I've been able to show what distance we can say two molecules touch each other, so that heat going between molecules happens through them touching instead of through light. This is really important for things like graphene, which are used as part of things used for making power from the sun by getting its light and heat, and also for making new things that can become part of computers made of really small things like molecules that work because of heat going between different parts.

On the Nobel Prize for Gravitational Waves

This is a short post about the news from last week, that the Nobel Prize in Physics went to 3 of the founders of the LIGO project that first observed gravitational waves. I was quite happy to see the news, for several reasons. One of the PIs, Rainer Weiss, is an emeritus professor from MIT, so there was the alma mater pride factor there; plus, that professor, along with the two other co-laureates, are all quite old, and it would have been a real shame if they hadn't survived to be recognized by the Nobel Prize committee for this incredible work, so it really was time for them to be recognized as such. From an applied science perspective, the amount of high-precision science and engineering they had to do, building laser systems, mirrors, and large-scale interferometers to measure atomic-scale and smaller displacements where quantum effects are important, has been amazing and will undoubtedly influence future technological development in those areas, even as gravitational waves themselves are at energy scales that are far too small to be useful in the near future (though people used to say that about quantum mechanics, and then transistors came about, so it's always better to be open to the possibility, while being mindful that this particular possibility is more remote).

Mostly, though, I'm thrilled to see this discovery being recognized from the perspective of fundamental science. My research is in fluctuational electrodynamics, which means that I deal with the propagation and scattering of electromagnetic fields at small length scales as they originate from quantum or thermal fluctuating atomic or electronic sources. Thus, all I think about on a daily basis is really electromagnetic interactions (and occasionally quantum exchange effects). But then, most physicists, with the exception of high-energy physicists (who study very short-ranged weak & strong interactions anyway), tend to make use of electromagnetism almost exclusively in their research, especially on the experimental side, as any spectroscopy, interferometry, and so on, would necessarily involve the propagation and scattering of light. In fact, if I think about how humans sense the world around them, vision is of course electromagnetic in nature, but so are hearing (detection of pressure waves caused by scattering of atoms and molecules in the air via short-range electromagnetic repulsion from atoms and molecules in the ear) and touch (short-range electromagnetic forces between atoms and molecules on the skin and a given surface), while electrical signals travel through neurons in the body by virtue of electromagnetic forces. Thus, in the context of sensing and understanding the world around us through truly long-range interactions, gravitational waves are fundamentally different, and it's weird to think that the interaction that keeps us on this planet can also be used in such a different way (i.e. propagating waves) to get information about faraway astronomical objects that wouldn't be available through purely electromagnetic means.

Starting as a TA

I figured I might post a short update on progress through my PhD. Through the summer and the start of the fall, I've been learning more about how to consistently incorporate molecular vibrations into descriptions of fluctuational phenomena, including van der Waals interactions and radiative heat transfer. It's been a very fun and interesting process, learning more about such fluctuational phenomena, revisiting and expanding on what I've learned in linear algebra and numerical analysis, and gaining a better understanding of the broader impacts that my work could have.
Also, two days ago marked my first day as a TA: I'm a TA for the graduate-level class ELE 511 — Quantum Mechanics with Applications. The discussions I've had with people during TA training and with the lecturer for this class, along with the preparation that I've had to do on my own and the questions that I've encountered from students so far, have really made me appreciate how much work it takes to prepare for teaching such classes and to be ready for students' questions, whatever they may be; plus, I've come to realize how annoying I must have been for TAs when I was a student, expecting them to be ready for literally any question, no matter how tangential to the material, and expecting them to be completely perfect in grading assignments, et cetera. To those TAs whose classes I took, I now have a better understanding of what your jobs were really like, and I thank you for putting up with me. I've enjoyed what I've been able to do so far as a TA, and I intend, with more time and practice, to become better at these variegated aspects of teaching as the semester progresses.

Second Paper: "Unifying Microscopic and Continuum Treatments of van der Waals and Casimir Interactions"

My second paper has been published! It is in volume 118, issue 26 of Physical Review Letters, and an older preprint of it is available too for those who don't have access to academic journals (it has all of the same figures and ideas, though it is missing a few sentences of further explanation as well as a couple of new citations that were inserted for the final publication). As with my first paper, in the interest of explaining these ideas in a way that is easy to understand, I am using the ten hundred most used words in English (except for the two lines that came before this one), as put together from the XKCD Simple Writer. I will use numbers sometimes without completely writing them out, use words for certain names of things without explaining further, and explain less used words when they come up. Keep reading to see what comes next.

Classical Damping of Gases and Oscillators

I was on vacation last week, and during some quiet time, I randomly happened to be thinking about explanations for damping in physical systems. I remember learning in ELE 456 — Quantum Optics, from last spring, that the phenomenological linear damping of a classical oscillator could be derived by coupling a quantum oscillator to a thermal bath of quantum oscillators; each linear oscillator is microscopically undamped, but by treating the bath through statistical thermodynamics, the coupling of the oscillator in question to a bath essentially produces a linear damping coefficient dependent on the spectrum of the bath (and the coupling too). Microscopically, the quantization of energy levels in a linear oscillator makes it easy to interpret how discrete excitations can move from one oscillator to another coupled oscillator, but I was wondering if quantum mechanics is really necessary to explain damping. Follow the jump to see an extremely rough sketch of ideas that may (or may not) justify the use of classical mechanics by itself. (Added after finishing: this turns out to be a rambling and possibly ultimately pointless post with a much clearer and more self-consistent explanation linked at the end, so for the time being, humor me.)

Green's Functions and Correlations

I had the idea of writing this post a couple of weeks ago, but I didn't feel like I had enough stuff to write here at that time. Now I do, so here goes. (Also, here's hoping that inputting LaTeX into this post works once more.)

When I took 18.03 — Differential Equations in 2010 fall, one of the topics covered was linear time-invariant systems. The general system of interest was $Lu(t) = f(t)$ where $L$ is a linear time-invariant operator. The technique of course is to find a weight function $w(t)$ where $Lw(t) = \delta(t)$, and once that is done, the solution is $u(t) = \int_{-\infty}^{\infty} f(t') w(t - t') dt'$ which is a convolution of the input $f$ with the weight $w$. The professor mentioned that it is essentially akin to inverting the operator $L$, but while I could see the general utility in this method, I never quite understood why it might be considered inversion on any deeper level.

Last semester, I took 8.07 — Electromagnetism II, and there we discussed Green's functions a little more in the context of electromagnetism & electrodynamics. In a static situation, the Green's function comes up in solving the Poisson equation $\nabla^2 \phi = -\rho$. In this case, $\nabla^2 G(\vec{x}, \vec{x}') = -\delta(\vec{x} - \vec{x}')$ is solved by the familiar potential of a unit point charge $G(\vec{x}, \vec{x}') = \frac{1}{4\pi |\vec{x} - \vec{x}'|}$. I started to see a little more clearly why this worked, because if a general charge distribution was some superposition of point charges, then a general potential distribution should be the same superposition of point charge potentials. However, it still wasn't entirely clear to me how this was "inversion" per se. Follow the jump to see what changed.