Confirming Einstein’s Theory of General Relativity With Calculus, Part 1b: Precession of Mercury

In this series, I’m discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury’s orbit and thus confirm Einstein’s theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago.

The figure below shows the (greatly exaggerated) effect of precession on a planet’s otherwise elliptical orbit. In the figure, each perihelion is precessed by an angle of $40^circ$ . After nine orbits, the planet returns to its original position. Suppose, for the sake of argument, that each orbit of the planet depicted in the figure is four months, or one third of Earth’s year. Then the amount of precession would be $40^\circ$ per four months, or $120^\circ$ per year, or $12,000^\circ$ per century.

As I said, the figure above is greatly exaggerated. As we’ll see by the end of this series, Einstein’s general relativity predicts that, on top of the gravitational influences of the other planets, the orbit of Mercury should precess by 43″ of arc per century. That’s a really small angle, since 1 $^\circ$ is equal to 60′ (minutes) of arc and each 1′ is equal to 60″ (seconds) of arc, that means 1″ of arc is the same as $(1/3600)^\circ$ , so that 43″ of arc per century is about $0.012^\circ$ per century. That’s about a million times smaller than the precession of the fictitious planet in the above figure.

How small is $0.012^\circ$ , really?

Courtesy of Wikipedia, the pictures below are the Copernicus crater on the Moon as well as an indicator of its location on the Moon. It is visible with binoculars.

The diameter of the crater is 93 km. Since the Moon is 384,400 km from Earth, that means the angle subtended by the crater, as viewed from the Earth, is about

$\arctan \left( \frac{93}{384,400} \right) \approx 0.014^\circ$ .

So how much is 43″ of arc per century? That’s about the speed as, hypothetically, pointing at the left edge of this lunar crater (which cannot be seen by the naked eye) and then slowly moving your figure so that, about 115 years later, your finger is pointing at the right edge of the crater.

Said another way, the diameter of the Moon is about 3475 km, so that the angle subtended by the Moon, as viewed from the Earth, is about

$\arctan \left( \frac{3745}{384,400} \right) \approx 0.518^\circ$ .

So, at the rate of $0.012^\circ$ per century, it would take $0.518/0.012 \approx 43$ centuries, or about 43,000 years, to trace the angle subtended by the moon.

Needless to say, 43” of arc per century is really, really slow.

Nevertheless, and remarkably, this itty, bitty precession was observable by careful 19th century astronomers with the telescopes that were available then. At the time, this precession was the great unsolved mystery of Newtonian physics that was only answered after two generations later with the discovery of general relativity.

Snell’s Law and a mystery novel

Lately, for my own leisure reading, I’ve been enjoying the murder-mystery novels of Dorothy Sayers. Her books are an enjoyable trip back in time, as she paints a very vivid portrait of English life of during the interwar years of the 1920s and 1930s. (Of course, at the time she was writing, no one had any idea that the Great War would not actually be the war to end all wars, as was the popular sentiment of the time.) Indeed, her first novel was published literally a century ago in 1923. The lead character, Lord Peter Wimsey (back then, the aristocracy was still part of English culture), has a distinctive way of speaking that makes the novels so delightful. A hallmark of the Sayers novels is that she didn’t merely write whodunit stories; instead, she strove to write novels in which a detective story happens to happen.

As an aside, I learned in her novel Gaudy Night that the adjective Oxonian means “related to Oxford,” which led me to further learn that my hometown of Oxon Hill, Maryland was so named because somebody, centuries ago, thought that the landscape of that part of the state reminded him of Oxford, England. While that comparison might have been reasonable centuries ago, it certainly would raise eyebrows today.

Anyway, with all that as background, in her story Unnatural Death, the following figure depicts an aerial view of a witness’s testimony at a key point in the story. I think I can describe this much of the scene without giving away the plot: the witnesses stood just inside the door of elderly Miss Dawson’s bedroom. A screen blocked direct observation of Miss Dawson as she lay in bed, but the witnesses could see Miss Dawson in the mirror.

As I read the novel, I immediately noticed that the mirror in the figure was not a perfect reflector… at the mirror, the angles of reflection of the dashed path of light are quite different. Indeed, I pulled out my protractor: the angle where the word “Mirror” is located has a measure of about 52 degrees, while the opposite reflected angle has a measure of about 72 degrees.

As this is was part of a murder-mystery novel, I thought: what could be the cause of this disparity? To be a good detective, any explanation, no matter how implausible, must be thought of and reasoned out.

One explanation of the different angles is that, somehow, the speed of light changed in the room. This is the same principle behind Snell’s Law, which explains the refraction of light as it travels between air and water. Since the speed of light in air ( $c_1$ ) is different than the speed of light in water ( $c_2$ ), the angle of incidence ( $\theta_1)$ is different from the angle of refraction ( $\theta_2$ ), but they are related through the formula

$\displaystyle \frac{\sin \theta_1}{c_1} = \frac{\sin \theta_2}{c_2}$ .

This relationship occurs because of Fermat’s principle, which says that light always travels in a path that requires the least amount of time. Ordinarily, this means that light travels in a straight line. However, if the speed of light should change (say, when traveling through both air and water), then the path of the light is refracted.

Fermat’s principle also explains why light reflects at equal angles if the speed of light is constant (as amusingly illustrated in this PBS video by Dianna Cowern, a.k.a. Physics Girl). However, if the speed of light should somehow change in the room at the point where the light reflects, then the light would bounce at a different angle for the same reason that Snell’s Law works.

In this case, the angles $\theta_1$ and $\theta_2$ are complementary to the 52-degree and 72-degree angles, respectively. By the cofunction trigonometric identities, this means that

$\sin \theta_1 = \cos 52^\circ \quad$ and $\quad \sin \theta_2 = \cos 72^\circ$ ,

so that Snell’s Law can be rewritten as

$\displaystyle \frac{c_1}{c_2} = \frac{\cos 52^\circ}{\cos 72^\circ} \approx 1.992$ .

In other words, one explanation for the unusual path of light is that the speed of light was almost exactly twice as fast in one part of room than in the other part… and the exact threshold of this change occurred at the point where the light hit the mirror. Perhaps there was some kind of fog, mist, or other contaminant in the air near poor Miss Dawson that was so thick that light slowed to half its usual speed. So that’s one explanation.

The other explanation, of course, is that the artist who drew the picture just did a lousy job depicting the reflected light.

As this was part of a murder-mystery, both options are still open to investigation. (Yes, that was tongue-in-cheek.)

For what it’s worth, the figure in my book was not exactly the same as Sayers’ original drawing — clearly, modern word processing was used that was unavailable in the 1930s. One of these days, I may visit the Wade Center in Wheaton, Illinois, which has an impressive collection of Sayers’ works, to peruse a first-run printing of Unnatural Death to see if the figure in my book is faithful to the one that appeared when the novel was first published.

Geometric Sequence from Rotating an Image

Source: https://xkcd.com/2671/

An algebra and trigonometry–based proof of Kepler’s First Law

The proofs of Kepler’s Three Laws are usually included in textbooks for multivariable calculus. So I was very intrigued when I saw, in the Media Reviews of College Mathematics Journal, that somebody had published a proof of Kepler’s First Law that only uses algebra and trigonometry. Let me quote from the review:

Kepler’s first law states that bounded planetary orbits are elliptical. This law is presented in introductory textbooks, but the proof typically requires intricate integrals or vector analysis involving an accidental degeneracy. Simha offers an elementary proof of Kepler’s first law using algebra and trigonometry at the high school level.
https://doi.org/10.1080/07468342.2022.2026089

Once upon a time, I taught Precalculus for precocious high school students. I wish I had known of this result back then, as it would have been a wonderful capstone to their studies of trigonometry and the conic sections.

The preprint of this result can be found on arXiv. (The proof only addresses Kepler’s First Law and not the Second and Third Laws.) The actual article, for those with institutional access, was published in American Journal of Physics Vol. 89 No. 11 (2021): 1009-1011.

Division Notation

Source: https://xkcd.com/2687/

Square roots and Logarithms Without a Calculator: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on computing square roots and logarithms without a calculator (with the latest post added).

Part 1: Method #1: Trial and error.

Part 2: Method #2: An algorithm comparable to long division.

Part 3: Method #3: Introduction to logarithmic tables.

Part 4: Finding antilogarithms with a table.

Part 5: Pedagogical and historical thoughts on log tables.

Part 6: Computation of square roots using a log table.

Part 7: Method #4: Slide rules

Part 8: Method #5: By hand, using a couple of known logarithms base 10, the change of base formula, and the Taylor approximation $\ln(1+x) \approx x$ .

Part 9: An in-class activity for getting students comfortable with logarithms when seen for the first time.

Part 10: Method #6: Mentally… anecdotes from Nobel Prize-winning physicist Richard P. Feynman and me.

Part 11: Method #7: Newton’s Method.

Part 12: Method #8: The formula $\sqrt{b} \approx \displaystyle \frac{a+b}{2\sqrt{a}}$

An elementary proof of the insolvability of the quintic

When I was in middle school, I remember my teacher telling me, after I learned the quadratic formula, that there was a general formula for solving cubic and quartic equations, but no such formula existed for solving the quintic. This was also when I first heard the infamous story of young Galois’s death from a duel.

Using my profound middle-school logic, I took this story as a challenge to devise my own formula for solving the quintic. Naturally, my efforts came up short.

When I was in high school, with this obsession still fully intact, I attempted to read through the wonderful monograph Field Theory and Its Classical Problems. Here’s the MAA review of this book:

Hadlock’s book sports one of the best prefaces I’ve ever read in a mathematics book. The rest of the book is even better: in 1984 it won the first MAA Edwin Beckenbach Book Prize for excellence in mathematical exposition.
Hadlock says in the preface that he wrote the book for himself, as a personal path through Galois theory as motivated by the three classical Greek geometric construction problems (doubling the cube, trisecting angles, and squaring the circle — all with just ruler and compass) and the classical problem of solving equations by radicals. Unlike what happens in most books on the subject, all three Greek problems are solved in the first chapter, with just the definition of field as a subfield of the real numbers, but without even defining degree of field extensions, much less proving its multiplicativity (this is done in chapter 2). Doubling the cube is proved to be impossible by proving that the cube root of 2 cannot be an element of a tower of quadratic extensions: if the cube root of 2 is in a quadratic extension, then it is actually in the base field. Repeating the argument, we conclude that it is not constructible because it is not rational. A similar argument works for proving that trisecting a 60 degree angle is impossible. Of course, proving that duplicating the cube is impossible needs a different argument: chapter 1 ends with Niven’s proof of the transcendence of π.
After this successful bare-hands attack at three important problems, Chapter 2 discusses in detail the construction of regular polygons and explains Gauss’s characterization of constructible regular polygons, including the construction of the regular 17-gon. Chapter 3 describes Galois theory and the solution of equations by radicals, including Abel’s theorem on the impossibility of solutions by radicals for equations of degree 5 or higher. Chapter 4, the last one, considers a special case of the inverse Galois problem and proves that there are polynomials with rational coefficients whose Galois group is the symmetric group, a result that is established via Hilbert’s irreducibility theorem.
Many examples, references, exercises, and complete solutions (taking up a third of the book!) are included and make this enjoyable book both an inspiration for teachers and a useful source for independent study or supplementary reading by students.

As I recall, I made it successfully through the first couple of chapters but started to get lost with the Galois theory somewhere in the middle of Chapter 3. Despite not completing the book, this was one of the most rewarding challenges of my young mathematical life. Perhaps one of these days I’ll undertake this challenge again.

Anyway, this year I came across the wonderful article The Abel–Ruffini Theorem: Complex but Not Complicated in the March issue of the American Mathematical Monthly. The article presents a completely different way of approaching the insolvability of the quintic that avoids Galois theory altogether.

The proof is elementary; I’m confident that I could have understood this proof had I seen it when I was in high school. That said, the word “elementary” in mathematics can be a bit loaded — this means that it is based on simple ideas that are perhaps used in a profound and surprising way. Perhaps my favorite quote along these lines was this understated gem from the book Three Pearls of Number Theory after the conclusion of a very complicated proof in Chapter 1:

You see how complicated an entirely elementary construction can sometimes be. And yet this is not an extreme case; in the next chapter you will encounter just as elementary a construction which is considerably more complicated.

I believe that a paid subscription to the Monthly is required to view the above link, but the main ideas of the proof can be found in the video below as well as this short PDF file by Leo Goldmakher.

Engaging students: Finding the equation of a circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Emma White. Her topic, from Precalculus: finding the equation of a circle.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Ironically, this morning on the way to class I received a notification saying Coldplay dropped a new album called “ Music of the Spheres” and I couldn’t help but look into it more! Although we are talking about circles, as mathematicians (or other people who came across this blog), we realize that circles and spheres are related in some ways. Although that is a discussion for another time, I want to focus on this album and how it relates to our world. Circles are used in various ways when it comes to the “circle of life” or “time on a ticking clock”. One song talks about “Humankind” and how we’re designed. This is a continuous cycle as humans pass away and are born and the cycle continues. While this may be a more serious thing to think about, life happens and cycles (we also see this in history and cycles of conflicts, wars, and much more). Furthermore (and maybe on a more lighthearted feel), we see the concept of circle in “The Circle of Life” as seen in “The Lion King”. I encourage you to look at the lyrics below:

“From the day we arrive on the planet

And, blinking, step into the sun

There’s more to see than can ever be seen

More to do than can ever be done

There’s far too much to take in here

More to find than can ever be found

But the sun rolling high

Through the sapphire sky

Keeps great and small on the endless round

It’s the circle of life

And it moves us all

Through despair and hope

Through faith and love

‘Til we find our place

On the path unwinding

In the circle

The circle of life.”

Source: LyricFind

Songwriters: Elton John / Tim Rice

Whatever your background may be, we can agree that much in life happens in cycles (think of cells as well!) and that is done in a metaphorical circular motion. The moon rotates around the sun, the planets rotate around the sun, and so forth. Many songs capture the concept of “circling” or time (think of the Sundial), and I bet if we took the time to really dig deep, we could find more songs with this concept more than we think.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

According to many articles, the discovery of the circle goes way back before recorded history. It started with the Egyptians (the inventors of Geometry) who invented the wheel. I find this intriguing that the people following the Egyptians “investigated” a simple man made tool, the wheel, to go about finding the equation of a circle. I want to emphasize this point because there is so much in life relating to math if only we stop to look and/or think about it more in depth! Furthermore, Euclid (naturally), contributed to the finding of the properties of the circle and “problems of inscribing polygons” (“Circle”, n.d.). Around 650 BC, Thales, a mathematical philosopher who contributed to various elementary geometry theorems, contributed to the theorems regarding circles. Nearly 400 years later, Apollonius, “a Greek mathematician known as ‘The Great Geometer’”, also contributed to the finding of the equation for a circle, specifically the equation itself (J J O’Connor and E F Roberts). He founded the bipolar equation “ $mr^2 + nr'^2=c^2$ represent[ing] a circle whose centre divides the line segment between the two fixed points of the system in the ratio n to m” (“Circle”, n.d.). Needless to say, the people who helped create this equation were years apart and it’s pretty cool to see how their work built off of each other over time.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

When it comes to the equation of a circle, using technology would be a great way to visually show students what is happening and understand where the equation comes from. KhanAcademy is a great resource for students to work through problems and furthermore, Desmos could be a resource for students to use at home for homework to check their work and understand how different values for ‘x’ and ‘y’ change the circle. A beneficial video to share/watch with your students would be “Lesson Video: Equation of a Circle”, for it provides a visual representation of how to derive the equation (I think exposing students to how to derive the equation will make the equation easier to understand and how the equation formulated). Giving your students technological resources is beneficial and I bet the students appreciate having multiple resources to help them become more understanding of the subject matter.

Resources: http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Kim/emat6690/instructional%20unit/circle/Circle/Circle.htm

http://britanica.com

http://mathworld.wolfram.com/Circle.html

https://mathshistory.st-andrews.ac.uk/Curves/Circle/

https://mathshistory.st-andrews.ac.uk/Biographies/Apollonius/

https://www.nagwa.com/en/videos/370167476508/

Engaging students: Vectors in two dimensions

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Fidel Gonzales. His topic, from Precalculus: vectors in two dimensions.

How can this topic be used in your students’ future courses in mathematics?

When a student learns about vectors in two dimensions, they worry about the magnitude of the vector and the direction that it goes. The direction is kept within its limitations which are up, down, left, and right. A student might be curious as to how this topic can be extended further. The way it extends further is by extending vectors into higher dimensions. It is even possible to extend vectors to the sixth dimension! However, for the sake of showing how vectors in two dimensions extend to future courses in math, we will stick to three-dimensions. Learning about vectors in the second dimension creates groundwork to learn about vectors in the third dimension. With the third dimension, vectors could be seen from our point of view compared to seeing it in the two dimensions on paper. The new perspective of the third dimension in vectors includes up, down, left, right, forward, and backwards. Having the new dimension to account for will give students a bigger tie into how mathematics applies into the real world.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Vectors in the two dimension is used all around our everyday life and we as people rarely notice it. The most common use of vectors in our culture is a quantity displaying a magnitude and direction. This is normally done on a x and y graph. Now you might be asking yourself, I do not play any types of games that sound like this. I am here to tell you that you do. One game that iPhone users play without noticing this would be a game on gamepigeon called knockout. The game appears to be an innocent game of knocking out your friends’ penguins while keeping yours in the designated box. However, math is involved, and you probably didn’t notice. First you must anticipate where the enemy is going. Then you must decide how strong you want to launch your penguin troopers without making them fall out of the ring. Does that sound familiar? Having to apply a force (magnitude) and direction to a quantity. Congratulations, you have now had fun doing math. Next time you are playing a game, try to see if there is any involvement of vectors in two dimensions involved.

How could you as a teacher create an activity or project that involves your topic?

Vectors in two dimensions has many ways to be incorporated in the classroom. A way to do so while connecting to the real world would be having an activity where the students tell a robot where to go using vectors. The students will have a robot that can walk around and in need of directions. The students will be given maps and asked to create a path for the robot to end up in its destination. Essentially, programming the robot to navigate though a course solely using vectors. If the robot falls or walks too far, then the student will realize that either the magnitude was wrong or the direction. Some students might seem to think this would be impractical to the real world, however, there is always a way to show relevance to students. Towards the end of the activity, the students will be asked to guide me to around the class using vectors. Then to sweeten the deal, they will also be asked to show me on a map being projected to them how to get to McDonald’s. Students will realize that vectors in the second dimension could be used to give directions to somewhere and can be applied to everyday life. They will walk outside of the classroom seeing math in the real world from a different perspective.

References:

https://www.khanacademy.org/science/physics/two-dimensional-motion/two-dimensional-projectile-mot/v/visualizing-vectors-in-2-dimensions

Engaging students: Compound interest

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Lydia Rios. Her topic, from Precalculus: compound interest.

How did people’s conception of this topic change over time?

While this concept is tied with business which is something that started rapidly changing in the early nineteen hundreds, we have understand that there is an accrued interest on loans long before then. People would loan out seeds or cattle and the interest would be paid after a harvest or with the young of the cattle. Of course now we use this concept mathematically but the concept still holds. We understand that there is a base fee and you must return that fee along with a little more. We then started using this with loose change and then as our currency changed from the gold standard we adapted to a new understanding of compound interest. Today we use the equation $A = P \left(1 + \frac{r}{n} \right)^{nt}$ , where $A$ is the amount accumulated, $P$ is the principal, $r$ is interest rate, $n$ is the compound period and $t$ is the number of periods.

Compound Interest Is Responsible for Modern Civilization (businessinsider.com)

What are the contributions of various cultures to this topic?

We have all experienced trade over the years. Native Americans would trade corn for other goods and offered payment plus interest with their corn harvest. The Silks Roads was a network of trading routes where China and other countries would trade textiles and other materials. They established the concept of payment and interest for purchases. Banks in America and other countries also have a set principal and a interest, whether this be in reference to your savings account or the billed interest on your credit card purchases. Even the invention of cars played a part on this and how our interest can decrease with the deterioration of the car. Over the years your interest payment can go down as the worth of the car goes down.

How have different cultures throughout time used this topic in their society?

Native Americans used compound interest to create trade deals a maintain some status of peace. China had their silk roads where they turned a profit and tried to maintain a sense of livelihood. For some cultures this was their only source of income, if they didn’t’ make some sort of trade then they had nothing to bring home. For others, such as the Native Americans, the trade itself was to protect their lives and the interest was something they owed. We can even see reference to this with the trade markets in Disney’s Aladdin, Aladdin cannot make a trade as he is a peasant but we see other village people making trades and we grasp the concept of worth from each object.