# Falling through the earth part 2

In my previous post, I showed that an elevator falling from the surface through the center of the earth due to gravity alone would obey the dynamics of a simple harmonic oscillator. I did not know what would happen if the shaft went through some arbitrary chord through the earth. Rick Gerkin believed that it would take the same amount of time for all chords and it turns out that he is correct. The proof is very simple. Consider any chord (straight path) through the earth. Now take a plane and slice the earth through that chord and the center of the earth. This is always possible because it takes three points to specify a plane. Now looking perpendicular to the plane, you can always rotate the earth such that you see

Let the blue dot represent the elevator on this chord. It will fall towards the midpoint. The total force on the elevator is towards the center of the earth along the vector $r$. From the previous post, we know that the gravitational acceleration is $\omega^2 r$. The force driving the elevator is along the chord and will have a magnitude that is given by $r$ times the cosine of the angle between $x$ and $r$. But this has magnitude exactly equal to $x$! Thus, the acceleration of the elevator along the chord is $\omega^2 x$ and thus the equation of motion for the elevator is $\ddot x = \omega^2 x$, which will be true for all chords and is the same as what we derived before. Hence, it will take the same amount of time to transit the earth. This is a perfect example of how most problems are solved by conceptualizing them in the right way.

# Falling through the earth

The 2012 remake of the classic film Total Recall features a giant elevator that plunges through the earth from Australia to England. This trip is called the “fall”, which I presume to mean it is propelled by gravity alone in an evacuated tube. The film states that the trip takes 17 minutes (I don’t remember if this is to get to the center of the earth or the other side). It also made some goofy point that the seats flip around in the vehicle when you cross the center because gravity reverses. This makes no sense because when you fall you are weightless and if you are strapped in, what difference does it make what direction you are in. In any case, I was still curious to know if 17 minutes was remotely accurate and the privilege of a physics education is that one is given the tools to calculate the transit time through the earth due to gravity.

The first thing to do is to make an order of magnitude estimate to see if the time is even in the ballpark. For this you only need middle school physics. The gravitational acceleration for a mass at the surface of the earth is $g = 9.8 m/s^2$. The radius of the earth is 6.371 million metres. Using the formula that distance $r = 1/2 g t^2$ (which you get by integrating twice over time), you get $t = \sqrt{2 r / g}$. Plugging in the numbers gives 1140 seconds or 19 minutes. So it would take 19 minutes to get to the center of the earth if you constantly accelerated at $9.8 m/s^2$. It would take the same amount of time to get back to the surface. Given that the gravitational acceleration at the surface should be an upper bound, the real transit time should be slower. I don’t know who they consulted but 17 minutes is not too far off.

We can calculate a more accurate time by including the effect of the gravitational force changing as you transit through the earth but this will require calculus. It’s a beautiful calculation so I’ll show it here. Newton’s law for the gravitational force between a point mass m and a point mass M separated by a distance r is

$F = - \frac{G Mm}{r^2}$

where $G = 6.67\times 10^{-11} m^3 kg^{-1} s^{-2}$ is the gravitational constant. If we assume that mass M (i.e. earth) is fixed then Newton’s 2nd law of motion for the mass m is given by $m \ddot r = F$. The equivalence of inertial mass and gravitational mass means you can divide m from both sides. So, if a mass m were outside of the earth, then it would accelerate towards the earth as

$\ddot r = F/m = - \frac{G M}{r^2}$

and this number is g when r is the radius of the earth. This is the reason that all objects fall with the same acceleration, apocryphally validated by Galileo on the Tower of Pisa. (It is also the basis of the theory of general relativity).

However, the earth is not a point but an extended ball where each point of the ball exerts a gravitational force on any mass inside or outside of the ball. (Nothing can shield gravity). Thus to compute the force acting on a particle we need to integrate over the contributions of each point inside the earth. Assume that the density of the earth $\rho$ is constant so the mass $M = \frac{4}{3} \pi R^3\rho$ where $R$ is the radius of the earth. The force between two point particles acts in a straight line between the two points. Thus for an extended object like the earth, each point in it will exert a force on a given particle in a different direction. So the calculation involves integrating over vectors. This is greatly simplified because a ball is highly symmetric. Consider the figure below, which is a cross section of the earth sliced down the middle.

The particle/elevator is the red dot, which is located a distance r from the center. (It can be inside or outside of the earth). We will assume that it travels on an axis through the center of the earth. We want to compute the net gravitational force on it from each point in the earth along this central axis. All distances (positive and negative) are measured with respect to the center of the earth. The blue dot is a point inside the earth with coordinates (x,y). There is also the third coordinate coming out of the page but we will not need it. For each blue point on one side of the earth there is another point diametrically opposed to it. The forces exerted by the two blue points on the red point are symmetrical. Their contributions in the y direction are exactly opposite and cancel leaving the net force only along the x axis. In fact there is an entire circle of points with radius y (orthogonal to the page) around the central axis where each point on the circle combines with a partner point on the opposite side to yield a force only along the x axis. Thus to compute the net force on the elevator we just need to integrate the contribution from concentric rings over the volume earth. This reduces an integral over three dimensions to just two.

The magnitude of the force (density) between the blue and red dot is given by

$\frac{G m \rho}{(r-x)^2+y^2}$

To get the component of the force along the x direction we need to multiple by the cosine of the angle between the central axis and the blue dot, which is

$\frac{r-x}{((r-x)^2+y^2)^{1/2}}$

(i.e. ratio of the adjacent to the hypotenuse of the relevant triangle). Now, to capture the contributions for all the pairs on the circle we multiple by the circumference which is $2\pi y$. Putting this together gives

$F/m = -G\rho \int_{-R}^{R}\int_{0}^{\sqrt{R^2-x^2}}\frac{r-x}{((r-x)^2+y^2)^{3/2}} 2\pi y dy dx$

The y integral extends from zero to the edge of the earth, which is $\sqrt{R^2-x^2}$. (This is R at $x=0$ (the center) and zero at $x=\pm R$ (the poles) as expected). The x integral extends from one pole to the other, hence -R to R. Completing the y integral gives

$2\pi G\rho \int_{-R}^{R}\left. \frac{r-x}{((r-x)^2+y^2)^{1/2}} \right|_{0}^{\sqrt{R^2-x^2}}dx$

$= 2\pi G\rho \int_{-R}^{R}\left[ \frac{r-x}{((r-x)^2+R^2-x^2)^{1/2}} - \frac{r-x}{|r-x|} \right]dx$ (*)

The second term comes from the 0 limit of the integral, which is $\frac{r-x}{((r-x)^2)^{1/2}}$. The square root of a number has positive and negative roots but the denominator here is a distance and thus is always a positive quantity and thus must include the absolute value. The first term of the above integral can be completed straightforwardly (I’ll leave it as an exercise) but the second term must be handled with care because r-x can change sign depending on whether r is greater or less than x. For a particle outside of the earth r-x is always positive and we get

$\int_{-R}^{R} \frac{r-x}{|r-x|} dx = \int_{-R}^{R} dx = 2R, r > R$

Inside the earth, we must break the integral up into two parts

$\int_{-R}^{R} \frac{r-x}{|r-x|} dx = \int_{-R}^{r} dx - \int_{r}^{R} dx = r+R - R + r = 2r, -R \le r\le R$

The first term of (*) integrates to

$\left[ \frac{(r^2-2rx+R^2)^{1/2}(-2r^2+rx+R^2)}{3r^2} \right]_{-R}^{R}$

$= \frac{(r^2-2rR+R^2)^{1/2}(-2r^2+rR+R^2)}{3r^2} - \frac{(r^2+2rR+R^2)^{1/2}(-2r^2-rR+R^2)}{3r^2}$

Using the fact that $(r \pm R)^2 = r^2 \pm 2rR + R^2$, we get

$= \frac{|R-r|(-2r^2+rR+R^2)}{3r^2} - \frac{|R+r|(-2r^2-rR+R^2)}{3r^2}$

(We again need the absolute value sign). For $r > R$, the particle is outside of the earth) and $|R-r| = r-R, |R+r| = r + R$. Putting everything together gives

$F/m = 2\pi G\rho \left[ \frac{6r^2R-2R^3}{3r^2} - 2 R\right] = -\frac{4}{3}\pi R^3\rho G\frac{1}{r^2} = - \frac{MG}{r^2}$

Thus, we have explicitly shown that the gravitational force exerted by a uniform ball is equivalent to concentrating all the mass in the center. This formula is true for $r < - R$ too.

For $-R \le r \le R$ we have

$F/m = 2\pi G\rho \left[ \frac{4}{3} r - 2r\right] =-\frac{4}{3}\pi\rho G r = -\frac{G M}{R^3}r$

Remarkably, the gravitational force on a particle inside the earth is just the force on the surface scaled by the ratio r/R. The equation of motion of the elevator is thus

$\ddot r = - \omega^2 r$ with $\omega^2 = GM/R^3 = g/R$

(Recall that the gravitational acceleration at the surface is $g = GM/R^2 = 9.8 m/s^2$). This is the classic equation for a harmonic oscillator with solutions of the form $\sin \omega t$. Thus, a period (i.e. round trip) is given by $2\pi/\omega$. Plugging in the numbers gives 5062 seconds or 84 minutes. A transit through the earth once would be half that at 42 minutes and the time to fall to the center of the earth would be 21 minutes, which I find surprisingly close to the back of the envelope estimate.

Now Australia is not exactly antipodal to England so the tube in the movie did not go directly through the center, which would make the calculation much harder. This would be a shorter distance but the gravitational force would be at an angle to the tube so there would be less acceleration and something would need to keep the elevator from rubbing against the walls (wheels or magnetic levitation). I actually don’t know if it would take a shorter or longer time than going through the center. If you calculate it, please let me know.

# Duality and computation in the MCU

I  took my kindergartener to see Avengers: Endgame recently. My son was a little disappointed, complaining that the film had too much talking and not enough fighting. To me, the immense popularity of the Marvel Cinematic Universe series and so-called science fiction/fantasy in general is an indicator of how people think they like science but really want magic. Popular science-fictiony franchises like MCU and Star Wars are couched in scientism but are often at odds with actual science as practiced today. Arthur C Clarke famously stated in his third law that “Any sufficiently advanced technology is indistinguishable from magic.” A sentiment captured in these films.

Science fiction should extrapolate from current scientific knowledge to the possible. Otherwise, it should just be called fiction. There have been a handful of films that try to do this like 2001: A Space Odyssey or more recently Interstellar and The Martian. I think there is a market for these types of films but they are certainly not as popular as the fantasy films. To be fair, neither Marvel nor Star Wars (both now owned by Disney) market themselves as science fiction as I defined it. They are intended to be mythologies a la Joseph Campbell’s Hero’s Journey. However, they do have a scientific aesthetic with worlds dominated by advanced technology.

Although I find the MCU films not overly compelling, they do bring up two interesting propositions. The first is dualism. The superhero character Ant-Man has a suit that allows him to change size and even shrink to sub-atomic scales, called the quantum realm in the films. (I won’t bother to discuss whether energy is conserved in these near instantaneous size changes, an issue that affects the Hulk as well). The film was advised by physicist Spiros Michalakis and is rife with physics terminology and concepts like quantum entanglement. One crucial concept it completely glosses over is how Ant-man maintains his identity as a person, much less his shape, when he is smaller than an atom. Even if one were to argue that one’s consciousness could be transferred to some set of quantum states at the sub-atomic scale, it would be overwhelmed by quantum fluctuations. The only self-consistent premise of Ant-Man is that the essence or soul if you wish of a person is not material. The MCU takes a definite stand for dualism on the mind-body problem, a sentiment with which I presume the public mostly agrees.

The second is that magic has immense computational power. In the penultimate Avengers movie, the villain Thanos snaps his fingers while in possession of the complete set of infinity stones and eliminates half of all living things. (Setting aside the issue that Thanos clearly does not understand the the concept of exponential growth. If you are concerned about overpopulation, it is pointless to shrink the population and do nothing else because it will just return to its original size in short time.) What I’d like to know is who or what does the computation to carry out the command. There are at least two hard computational problems that must be solved. The first is to identify all lifeforms.  This is clearly no easy task as we to this day have no precise definition of life. Do viruses get culled by the snap? Do the population of silicon-based lifeforms of Star Trek get halved or is it only biochemical life? What algorithm does the snap use to find all the life forms? Living things on earth range in size from single cells (or viruses if you count them) all the way to 35 metre behemoths, which are comprised of over $10^{23}$ numbers of atoms. How do the stones know what scales they span in the MCU? Do photosynthetic lifeforms get spared since they don’t use many resources? What about fungi? Is the MCU actually a simulated universe where there is a continually updated census of all life? How accurate is the algorithm? Was it perfect? Did it aim for high specificity (i.e. reduce false positives so you only kill lifeforms and not non lifeforms) or high sensitivity (i.e. reduce false negatives and thus don’t miss any lifeforms). I think it probably favours sensitivity over specificity – who cares if a bunch of ammonia molecules accidentally get killed. The find-all-life problem is made much easier by proposition 1 because if all life were material then the only way to detect them would be to look for multiscale correlations between atoms (or find organic molecules if you only care about biochemical life). If each lifeform has a soul then you can simply search for “soulfulness”. The lifeforms were not erased instantly but only after a brief delay. What was happening over this delay. Is magic propagation limited by the speed of light or some other constraint? Or did the computation take time? In Endgame, the Hulk restores all the Thanos erased lifeforms and Tony Stark then snaps away Thanos and all of his allies. Where were the lifeforms after they were erased? In Heaven? In a soul repository somewhere? Is this one of the Nine Realms of the MCU? How do the stones know who is a Thanos ally? The second computation is to then decide which half to extinguish. The movie seems to imply that the choice was random so where did the randomness come from? Do the infinity stones generate random numbers? Do they rely on quantum fluctuations? Finally, in a world with magic, why is there also science? Why does the universe follow the laws of physics sometimes and magic other times. Is magic a finite resource as in Larry Niven’s The Magic Goes Away. So many questions, so few answers.

# Science and the vampire/zombie apocalypse

It seems like every time I turn on the TV, which only occurs when I’m in the exercise room, there is a show that involves either zombies or vampires. From my small sampling, it seems that the more recent incarnations try to invoke scientific explanations for these conditions that involve a viral or parasitic etiology. Popular entertainment reflects societal anxieties; disease and pandemics is to the twenty first century what nuclear war was to the late twentieth. Unfortunately, the addition of science to the zombie or vampire mythology makes for a much less compelling story.

A necessary requirement of good fiction is that it be self-consistent. The rules that govern the world the characters inhabit need to apply uniformly. Bram Stoker’s Dracula was a great story because there were simple rules that governed vampires – they die from exposure to sunlight, stakes to the heart, and silver bullets. They are repelled by garlic and Christian symbols. Most importantly, their thirst for blood was a lifestyle choice, like consuming fine wine, rather than a nutritional requirement. Vampires lived in a world of magic and so their world did not need to obey the laws of physics.

Once you try to make vampirism or zombism a disease and scientifically plausible in our world, you run into a host of troubles. Vampires and zombies need to obey the laws of thermodynamics, which means they need energy to function. This implies that the easiest way to kill one of these creatures is to starve them to death. Given how energetically active vampires are and how little caloric content blood has by volume, since it is mostly water, vampires would need to drink a lot of blood to sustain themselves. All you need to do is to quarantine all humans into secure locations for a few days and all vampires should either starve to death or fall into a dormant state. Vampirism is self-limiting because there would not be enough human hosts to sustain a large population. This is why only small animals can subsist entirely on blood (e.g. vampire bats weight about 40 grams and can drink half their weight in blood). Once, you make vampires biological, it makes no sense why they can only drink blood. What exactly is in blood that they can’t get from eating flesh? Even if they don’t have a digestive system that can handle solid food, they could always put meat into a Vitamix and make a smoothie. Zombies eat all parts of humans so they would need to feed less often than vampires and thus be harder to starve. However, zombies are usually non-intelligent and thus easier to avoid and sequester. It seems like any zombie epidemic could be controlled at very early stages. Additionally, why is it that zombies don’t eat each other? Why do they only like to eat humans?  Why aren’t they hanging around farms and eating livestock and poultry?

Vampires and sometimes zombies also have super-strength without having to bulk up. This means that their muscles are much more efficient. How is this possible? Muscles are pretty similar at the cellular level. Chimpanzees are stronger than humans by weight because they have more fast twitch than slow twitch muscles. There is thus always a trade-off between strength and endurance. In a physically plausible world, humans should always find an edge in combating zombies or vampires. The only way to make a vampire or zombie story viable is to endow them with nonphysical properties. My guess is that we have hit peak vampire/zombie; the next wave of horror shows will feature a more plausible threat – evil AI.

# Musical selection

Here is a 1975 performance of Chopin’s Piano Concerto No. 2 by Artur Rubenstein and the London Symphony Orchestra with Andre Previn conducting.

# Technology and inference

In my previous post, I gave an example of how fake news could lead to a scenario of no update of posterior probabilities. However, this situation could occur just from the knowledge of technology. When I was a child, fantasy and science fiction movies always had a campy feel because the special effects were unrealistic looking. When Godzilla came out of Tokyo Harbour it looked like little models in a bathtub. The Creature from the Black Lagoon looked like a man in a rubber suit. I think the first science fiction movie that looked astonishing real was Stanley Kubrick’s 1968 masterpiece 2001: A Space Odyssey, which adhered to physics like no others before and only a handful since. The simulation of weightlessness in space was marvelous and to me the ultimate attention to detail was the scene in the rotating space station where a mild curvature in the floor could be perceived. The next groundbreaking moment was the 1993 film Jurassic Park, which truly brought dinosaurs to life. The first scene of a giant sauropod eating from a tree top was astonishing. The distinction between fantasy and reality was forever gone.

The effect of this essentially perfect rendering of anything into a realistic image is that we now have a plausible reason to reject any evidence. Photographic evidence can be completely discounted because the technology exists to create completely fabricated versions. This is equally true of audio tapes and anything your read on the Internet. In Bayesian terms, we now have an internal model or likelihood function that any data could be false. The more cynical you are the closer this constant is to one. Once the likelihood becomes insensitive to data then we are in the same situation as before. Technology alone, in the absence of fake news, could lead to a world where no one ever changes their mind. The irony could be that this will force people to evaluate truth the way they did before such technology existed, which is that you believe people (or machines) that you trust through building relationships over long periods of time.

# More ways to waste time

In case you missed it.  Here is the famous OKGo Rube Goldberg machine video.

# Fun with zero gravity

Here is the video of the band OK Go filmed on a plane doing parabolic arcs. OK Go is famous for having the most creative videos, which combine Rube Goldberg contraptions with extreme synchronized choreography. The video of Upside Down and Inside Out is a single shot. Each zero gravity arc is about 30 seconds long. The intervening hyper gravity arcs are compressed in the video although it is very hard to detect in the first viewing.

# A musical selection

Since I gave up posting a musical selection every week, I changed the title. I still intend to post selections occasionally. Here is Joseph Haydn’s Quartet in E-flat major, Op. 33, No. 2, nicknamed “The Joke”, played by the Ariel Quartet. Make sure you listen to the end. I could think of nothing more appropriate for our times.

# Selection of the week

Sorry for the long radio silence. However, I was listening to the radio yesterday and this version of the Double Violin Concerto in D minor, BWV 1043 by JS Bach came on and I sat in my car in a hot parking lot listening to it. It’s from a forty year old EMI recording with violinists Itzhak Perlman and Pinchas Zukerman with Daniel Barenboim conducting the English Chamber Orchestra. I’ve been limiting my posts to videos of live performances but sometimes classic recordings should be given their due and this is certainly a classic. Even though I posted a version with Oistrakh and Menuhin before, I just had to share this.

# Selection of the week

It has been a rather tragic week around the world.  Here is the incomparable Herbert von Karajan conducting the Vienna Philharmonic Orchestra and Vienna Singverein in a 1986 rendition of Mozart’s Requiem.

# Selection of the week

The third movement of Felix Mendelssohn’s violin concerto played by Swedish prodigy Daniel Lozakovitj at age 10 with the Tchaikovsky Symphony Orchestra at the Tchaikovsky Concert Hall in 2011.

Here is the version by international superstar and former violin prodigy Sarah Chang with Kurt Masur and the New York Philharmonic in 1995 when she was about 15.

# Selection of the week

English Composer Frederick Delius’s Piano Concerto in C minor, played by Justin Bird with the Indiana University (ad hoc) Orchestra.

# Selection of the week

Bohemian (as from Bohemia, not lifestyle) composer Bedrich Smetana’s Symphonic Poem The Moldau, transcribed for harp, and played by Valerie Milot. This is Smetana’s evocation of the sounds of the Vltava or Moldau river.

# Selection of the week

The first movement of Beethoven’s Violin and Piano Sonata No.5, Op. 24, dubbed the Spring Sonata, played by Gidon Kremer and Martha Argerich. I was fortunate enough to attend a concert by Kremer in the 1980’s. I don’t think I really understood what great musicianship was, as opposed to virtuosity, until that concert.  For Kremer, every note is part of a bigger whole. In this video, it is not clear that Kremer and Argerich are on the same page though.

Below is the whole thing with Anne Sophie Mutter and Lambert Orkis, which has better balance.

# Selection of the week

The great Russian pianist Sviatoslav Richter playing Beethoven’s last piano sonata, No. 32 in C minor, Op 111, which really pushed the boundaries of music at that time. Beethoven was completely deaf when he composed it. Richter was considered to be a musical genius; he was admired by Glenn Gould. Richter also insisted that American pianist Van Cliburn should be the winner of the first International Tchaikovsky Piano Competition in Moscow in 1958. It was a controversial decision to say the least but Richter prevailed and that moment still resonates both musically and and geopolitically. It certainly launched Cliburn’s career and one could argue that it laid a path to the end of the cold war. Music can matter.

# Selection of the week

The second movement of Franz Schubert’s Trio No. 2 in E flat, op. 100 played by le Trio Wanderer.  This piece (or rather a small fraction of it) was popularized by the film Barry Lyndon.

Here is the whole thing played by Trio Cleonice

# Selection of the week

The Kyrie from Mozart’s Great Mass in C minor, K427, played  by the Chor und Symphonieorchester des Bayerischen Rundfunks conducted by Leonard Bernstein. The solo is sung by the American soprano Arleen Auger, who died in 1993 of brain cancer at the age 53.  Bernstein died in 1990 so this must have been performed sometime in the 1980’s.

Addendum: Actually it is from 1990 so it must have been right before Bernstein died.

Here is the whole mass if you have an hour.