Science in Daily Life

New kinds of Cash & the connection to the Conservation of Energy And Momentum

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Its been difficult to find time to write articles on this blog – what with running a section teaching undergraduates (after 27 years of {\underline {not \: \: doing \: \:  so}}), as well as learning about topological quantum field theory – a topic I always fancied but knew little about.

However, a trip with my daughter brought up something that sparked an interesting answer to questions I got at my undergraduate section. I had taken my daughter to the grocery store – she ran out of the car to go shopping and left her wallet behind. I quickly honked at her and waved, displaying the wallet. She waved back, displaying her phone. And insight struck me – she had the usual gamut of applications on her phone that serve as ways to pay at retailers – who needs a credit card when you have Apple Pay or Google Pay. I clearly hadn’t adopted the Millennial ways of life enough to understand that money comes in yet another form, adapted to your cell phone and aren’t only the kinds of things you can see, smell or Visa!

And that’s the connection to the Law Of Conservation of Energy, in the following way. There were a set of phenomena that Wolfgang Pauli considered in the 1930s – beta decay. The nucleus was known and so were negatively charged electrons (these were called \beta-particles). People had a good idea of the composition and mass of the nucleus (as being composed of protons and neutrons), the structure of the atom (with electrons in orbit around the nucleus) and also understood Einstein’s revolutionary conceptions of the unity of mass and energy. Experimenters were studying the phenomenon of nuclear radioactive decay. Here, a nucleus abruptly emits an electron, then turns into a nucleus with one higher proton number and one less neutron number, so roughly the same atomic weight, but with an extra positive charge. This appears to happen spontaneously, but in concert with the “creation” of a proton, an electron is also produced (and emitted from the atom), so the change in the total electric charge is +1 -1 = 0 – it is “conserved”.  What seemed to be happening inside the nucleus was, that one of the neutrons was decaying into a proton and an electron. Now, scientists had constructed rather precise devices  to “stop” electrons, thereby measuring their momentum and energy. It was immediately clear that the total energy we started with – the mass-energy of the neutron (which starts out not moving very much in the experiment), in decaying into the proton and electron was more than the energy of the said proton (which also wasn’t moving very much at the end) and aforesaid electron.

People were quite confused about all this. What was happening? Where was the energy going? It wasn’t being lost to heating up the samples (that was possible to check). Maybe the underlying process going on wasn’t that simple? Some people, including some famous physicists, were convinced that the Law of Conservation of Energy and Momentum had to go.

As it turned out, much like I was confused in the car because I had neglected that money could be created and destroyed in an iPhone, people had neglected that energy could be carried away or brought in by invisible particles called neutrinos. It was just a proposal, till they were actually discovered in 1956 through careful experiments.

In fact, as has been rather clear since Emmy Noether discovered the connection between a symmetry and this principle years ago, getting rid of the Law of Conservation of Energy and Momentum is not that easy. It is connected to a belief that physics (and the result of Physics experiments) is the same whether done here, on Pluto or in empty space outside one of the galaxies on the Hubble deep field view! As long as you systematically get rid of all “known” differences at these locations – the gravity and magnetic field of the earth, your noisy cousin next door, the tectonic activity on Pluto, or small black holes in the Universe’s distant past, the fundamental nature of the universe is translationally \: \: invariant. So if you discover that you have found some violation of the Law of Conservation of Energy and Momentum, i.e., a perpetual motion machine, remember that you are announcing that there is some deep inequivalence between different points and time in the Universe.

The usual story is that if you notice some “violation” of this Law, you immediately start looking for particles or sources that ate up the missing energy and momentum rather than announce that you are creating or destroying energy. This principle gets carried into the introduction of new forms of “potential energy” too, in physics, as we discover new ways in which the Universe can bamboozle us and reserve energy for later use in so many different ways. Just like you have to add up so many ways you can store money up for later use!

That leads to a conundrum. If the Universe has a finite size and has a finite lifetime, what does it mean to say that all times and points are equivalent? We can deal with the spatial finiteness – after all, the Earth is finite, but all points on it are geographically equivalent, once you account for the rotation axis (which is currently where Antarctica and the Arctic are, but really could be anywhere). But how do you account for the fact that time seems to start from zero? More on this in a future post.

So, before you send me mail telling me you have built a perpetual motion machine, you really have to be Divine and if so, I am expecting some miracles too.

The Normal Distribution is AbNormal

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I gave a talk on this topic exactly two years ago at my undergraduate institution, the Indian Institute of Technology, in Chennai (India). The speech is here, with the powerpoint presentation accompanying it The Normal Distribution is Abnormal And Other Oddities. The general import of the speech was that the Normal Distribution, which is a statistical distribution that applies to random data of a variety of sorts, that’s often used to model the random data, is often not particularly appropriate at all. I presented cases where this is the case and the assumption (of a normal distribution) leads to costly errors.

Enjoy!

Mr. Olbers and his paradox

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Why is the night sky dark? Wilhelm Olbers asked this question, certainly not for the first time in history, in the 1800s.

That’s a silly question with an obvious answer. Isn’t that so?

Let’s see. There certainly is no sun visible, which is the definition of night, after all. The moon might be, but on a new moon night, the moon isn’t, so all we have are stars in the sky.

Now, let’s make some rather simple-minded assumptions. Suppose the stars are distributed equally throughout space, at all previous times too. Why do we have to think about previous times? You know that light travels at 300,000 \: km/s, so when you look out into space, you also look back in time. So, one has to make some assumptions about the distribution of stars at prior times.

Then, if you draw a shell around the earth, that has a radius R and thickness \delta R, the volume of this thin shell is 4 \pi R^2 \delta R.

Shell Radius R

Suppose there were a constant density of stars n stars per unit volume, this thin shell has n 4 \pi R^2 \delta R stars. Now the further away a star is, the dimmer it seems – the light spreads out in a sphere around the star. A star a distance R that emits I units of energy per second, will project an intensity of \frac{I}{4 \pi R^2} per unit area at a distance R away. So a shell (thickness \delta R) of stars at a radius R will bombard us (on the earth) with intensity \frac {I}{4 \pi R^2} n 4 \pi R^2 \delta R units of energy per unit area. This is = I n \ \delta R units of energy per unit area.

This is independent of R!. Since we can do this for successive shells of stars, the brightness of each shell adds! The night sky would be infinitely bright, IF the universe were infinitely big.

Some assumptions were made in the above description.

  1. We assumed that the stars are distributed uniformly in space, at past times too. We also assumed, in particular,
  2. isotropy, so there are no special directions along which stars lie.
  3. We also assumed that stars all shine with the same brightness.
  4. We didn’t mention it, but we assumed nothing obscures the light of far away stars, so we are able to see everything.
  5. In addition, we also assumed that the universe is infinite in size and that we can see all the objects in it, so it also must have had an infinite lifetime before this moment. Since light travels at a finite speed, light from an object would take some time to reach us; if the universe were infinitely old, we’d see every object in it at the moment we look up.
  6. We also assumed that the Universe wasn’t expanding rapidly – in fact with a recession speed for far away stars that increased (at least proportionately, but could be even faster than linear) with their distance. In such a universe, if we went far enough, we’d have stars whose recession speed from us exceeds the speed of light. If so, the light from those stars couldn’t possibly reach us – like a fellow on an escalator trying hard to progress upwards while the escalator goes down.

There are a tremendous number of population analyses of the distribution of light-emitting objects in the universe that make a convincing case (next post!) that the universe is isotropic and homogeneous on enormously large length scales (as in 100 Mega parsecs). We don’t see the kind of peculiar distributions that would lead us to assume a conspiracy of the sort implied in point 2.

We have a good idea of the life cycles of stars, but the argument would proceed on the same lines, unless we had a systematic diminution of intrinsic brightness as we looked at stars further and further away. Actually, the converse appears to be true. Stars and galaxies further away had tremendous amounts of hydrogen and little else and appear to be brighter, much brighter.

If there were actually dust obscuring far away stars, then the dust would have absorbed radiation from the stars, started to heat up, then would have emitted radiation of the same amount, once it reached thermodynamic equilibrium. This is not really a valid objection.

The best explanation is that either the universe hasn’t been around infinitely long, or the distant parts are receding from us so rapidly that they are exiting our visibility sphere. Or both.

And that is the start of the study of modern cosmology.

The Great American Eclipse of 2017

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I really had to see this eclipse – met up with my nephew at KSU, then eclipse chasing (versus the clouds) all the way from Kansas to central and south-east Missouri. The pictures I got were interesting, but I think the videos (and audio) reflect the experience of totality much better. The initial crescent shaped shadows through the “pinholes” in the leafy branches,

With the slowly creeping moon swallowing the sun

Followed by totality

and the sudden disappearance of sunlight, followed by crickets chirping (listen to the sounds as the sky darkens)

I must confess, it became dark and my camera exposure settings got screwed up – no pictures of the diamond ring. Ah, well, better luck next time!

This is definitely an ethereal experience and one worth the effort to see it. Everybody and his uncle did!

Coincidences and the stealthiness of the Calculus of Probabilities

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You know this story (or something similar) from your own life. I was walking from my parked car to the convenience store to purchase a couple of bottles of sparkling water. As I walked there, I noticed a car with the number 1966 – that’s the year I was born! This must be a coincidence – today must be a lucky day!

There are other coincidences, numerical or otherwise. Carl Sagan, in one of his books mentions a person that thought of his mother the very day she passed away in a different city. He (this person) was convinced this was proof of life after/before/during death.

There are others in the Natural World around us (I will be writing about the “Naturalness” idea in the future) – for eclipse aficionados, there is going to be a total solar eclipse over a third of the United States on the 21st of August 2017. It is a coincidence that the moon is exactly the right size to completely cover the sun (precisely – see eclipse photos from NASA below)

170316134335-total-solar-eclipse-process-nasa-exlarge-169.jpgwhite_light_corona

Isn’t is peculiar that the moon is exactly the right size? For instance, the moon has other properties – for instance, the face of the moon that we see is always the same face. Mercury does the same thing with the Sun – it also exhibits the same face to the Sun. This is well understood as a tidal effect of a small object in the gravitational field of a large neighbor. There’s an excellent Wikipedia article about this effect and I well explain it further in the future. But there is no simple explanation for why the moon is the right size for total eclipses. It is not believed to be anything but an astonishing coincidence. After all, we have 6000 odd other visible objects in the sky that aren’t exactly eclipsed by any other satellite, so why should this particular pair matter, except that they provide us much-needed heat and light?

The famous physicist Paul Dirac discovered an interesting numerical coincidence based on some other numerology that another scientist called Eddington was obsessed with. It turns out that a number of (somewhat carefully constructed) ratios are of the same order of magnitude – basically remember the number 10^{40}!

  • The ratio of the electrical and gravitational forces between the electron and the proton (\frac{1}{4 \pi \epsilon_0} e^2 vs. G m_p m_e) is approximately 10^{40}
  • The ratio of the size of the universe to the electron’s Compton wavelength, which is the de Broglie wavelength of a photon of the same energy as the electron – 10^{27}m \: vs \: 10^{-12} m \: \approx 10^{39}

On the basis of this astonishing coincidence, Dirac made the startling observation that this could indicate (since the size of the universe is related to it’s age) the value of G would fall with time (why not e going up with time, or something else?). Whereas precision experiments in measuring the value of G are beginning now, there would have been cosmological consequences if G had indeed behaved as 1/t in the past! For this reason, people discount this “theory” these days.

I heard of another coincidence recently – the value of the quantity \frac {c^2}{g}, where c is the speed of light and g is the acceleration due to gravity at the surface of the earth (9.81 \frac{m}{s^2}), is very close to 1 light year. This implies (if you put together the formulas for g, and 1 light year), a relationship between the masses of the earth, the sun, the earth’s radius and the earth-moon distance!

The question with coincidences of any sort is that it is imperative to separate signal from noise. And this is why some simple examples of probability are useful to consider. Let’s understand this.

If you have two specific people in mind, the probability of both having the same birthday is \frac{1}{365} – there are 365 possibilities for the second person’s birthday and only 1 way for it to match the first person’s birthday.

If, however, you have N people and you ask for any match of birthdays, there are \frac {N(N-1)}{2} pairs of people to consider and you have a substantially higher probability of a match. In fact, the easy way to calculate this is to ask for the probability of NO matches – that is \frac {364}{365} \times \frac {363}{365} \times ... \frac {365 - (N-1)}{365}, which is \frac {364}{365} for the first non-match, then \frac {363}{365} from the second non-match for the third person and so on. Then subtracting from 1 gives the probability of at least one match. Among other things, this implies that the chance of at least one match is over 50% (1 in 2) for a bunch of 23 un-connected people (no twins etc). And if you have 60 people, the probability is extremely close to 1.

Probability Graph for Birthdays

The key takeaway from this is that a less probable event has a chance to become much more probable when you have the luxury of adding more possibilities for the event to occur. As an example, if you went around your life declaring in advance that you will worry about coincidences ONLY if you find a number matching the specific birth year for your second cousin – the chances are low that you will observe such a number – unless you happen to hang about near your second cousin’s home! On the other hand, if you are willing to accept most numbers close to your heart – the possibilities stealthily abound and the probability of a match increases! Your birthday, your age, room number or address while in college, your current or previous addresses, the license plates for your cars, your current and previous passport numbers – the possibilities are literally, endless. And this means that the probability of a “coincidence” is that much higher.

I have a suggestion if you notice an unexplained coincidence in your life. Figure out if that same coincidence repeats itself in a bit – a week say. You have much stronger grounds for an argument with someone like me if you do! And then you still have to have a coherent theory why it was a real coincidence in the first place!

 

Addendum: Just to clarify, I write above “It is a coincidence that the moon is exactly the right size to completely cover the sun …” – this is from our point of view, of course. These objects would have radically different sizes when viewed from Jupiter, for instance.

Arbitrage arguments in Finance and Physics

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Arbitrage refers to a somewhat peculiar and rare situation in the financial world. It is succinctly described as follows. Suppose you start with an initial situation – let’s say you have some money in an ultra-safe bank that earns interest at a certain basic rate r. Assume, also, that there is a infinitely liquid market in the world, where you can choose to invest the money in any way you choose. If you can end up with {\bf {definite}} financial outcomes that are quite different, then you have an arbitrage between the two strategies. If so, the way to profit from the situation is to “short” one strategy (the one that makes less) and go “long” the other strategy (the one that makes more). An example of such a method would be to buy a cheaper class of shares and sell “short” an equivalent amount of an expensive class of shares for the same Company that has definitely committed to merge the two classes in a year.

An argument using arbitrage is hard to challenge except when basic assumptions about the market or initial conditions are violated. Hence, in the above example, suppose there was uncertainty about whether the merger of the two classes of shares in a year, the “arbitrage” wouldn’t really be one.

One of the best known arbitrage arguments was invented by Fischer Black, Myron Scholes and Robert Merton to deduce a price for Call and Put Options. Their argument is explained as follows. Suppose you have one interest rate for risk-free investments (the rate r two paragraphs above). Additionally, consider if you, Dear Reader, own a Call Option, with strike price \$X, on a stock price.  This is an instrument where at the end of (say) one year, you look at the market price of the stock and compute \$S - \$X. Let’s say X = \$100, while the stock price was initially \$76. At the end of the year, suppose the stock price became \$110, then the difference \$110 - \$100 = \$10, so you, Dear Reader and Fortunate-Call-Option-Owner, would make \$10. On the other hand, if the stock price unfortunately sank to \$55, then the difference \$55 - \$ 100 = - \$45 is negative. In this case, you, unfortunate Reader, would make nothing. A Call Option, therefore, is a way to speculate on the ascent of a stock price above the strike price.

Black-Scholes-Merton wanted to find a formula for the price that you should logically expect to pay for the option. The simplest assumption for the uncertainty in the stock price is to state that \log S follows a random walk. A random walk is the walk of a drunkard that walks on a one-dimensional street and can take each successive step to the front or the back with equal probability. Why \log S and not S? That’s because a random walker could end up walking backwards for a long time. If her walk was akin to a stock price, clearly the stock price couldn’t go below 0 – a more natural choice is \log S which goes to - \infty as S \rightarrow 0. A random walker is characterized by her step size. The larger the step size, the further she would be expected to be found relative to her starting point after N steps. The step size is called the “volatility” of the stock price.

In addition to an assumption about volatility, B-S-M needed to figure out the “drift” of the stock price. The “drift”, in our example, is akin to a drunkard starting on a slope. In that case, there is an unconscious tendency to drift down-slope. One can model drift by assuming that there isn’t the same probability to move to the right, as to the left.

The problem is, while it is possible to deduce, from uncertainty measures in the market, the “volatility” of the stock, there is no natural reason to prefer one “drift” over the other. Roughly speaking, if you ask people in the market whether IBM will achieve a higher stock price after one year, half will say “Yes”, the other half will say “No”. In addition, the ones that say “Yes” will not agree on exactly by how much it will be up. The same for the “No”-sayers! What to do?

B-S-M came up with a phenomenal argument. It goes as follows. We know, intuitively, that a Call Option (for a stock in one year) should be worth more today if the stock price were higher today (for the same Strike Price) by, say \$1. Can we find a portfolio that would decline by exactly the same amount if the stock price was up by \$1. Yes, we can. We could simply “short” that amount of shares in the market. A “short” position is like a position in a negative number of shares. Such a position loses money if the market were to go up. And I could do the same thing every day till the Option expires. I will need to know, every day, from the Option Formula that I have yet to find, a “first-derivative” – how much the Option Value would change for a \$1 increase in the stock price. But once I do this, I have a portfolio (Option plus this “short” position) that is {\bf {insensitive}} to stock price changes (for small changes).

Now, B-S-M had the ingredients for an arbitrage argument. They said, if such a portfolio definitely could make more than the rate offered by a risk-less bank account, there would be an arbitrage. If the portfolio definitely made more, borrow (from this risk-free bank) the money to buy the option, run the strategy, wait to maturity, return the loan and clear a risk-free profit. If it definitely made less, sell this option, invest the money received in the bank, run the hedging strategy with the opposite sign, wait to maturity, pay off  the Option by withdrawing your bank funds, then pocket your risk-free difference.

This meant that they could assume that the portfolio described by the Option and the Hedge, run in that way, were forced to appreciate at the “risk-free” rate. This was hence a natural choice of the “drift” parameter to use. The price of the Option would actually not depend on it.

If you are a hard-headed options trader, though, the arguments just start here. After all, the running of the above strategy needs markets that are infinitely liquid with infinitesimal “friction” – ability to sell infinite amounts of stock at the same price as at which to buy them. All of these are violated to varying degrees in the real stock market, which is what makes the B-S-M formula of doubtful accuracy. In addition, there are other possible processes (not a simple random-walk) that the quantity \log S might follow. All this contributes to a robust Options market.

An arbitrage argument is akin to an argument by contradiction.

Arguments of the above sort, abound in Physics. Here’s a cute one, due to Hermann Bondi. He was able to use it to deduce that clocks should run slower in a gravitational field. Here goes (this paraphrases a description by the incomparable T. Padmanabhan from his book on General Relativity).

Bondi considered the following sort of apparatus (I have really constructed my own example, but the concept is his).

Bondi Apparatus.JPG

One photon rushes from the bottom of the apparatus to the top. Let’s assume it has a frequency \nu_{bottom} at the bottom of the apparatus and a frequency \nu_{top} at the top. In our current unenlightened state of mind, we think these will be the same frequency. Once the photon reaches the top, it strikes a target and undergoes pair production (photon swerves close to a nucleus and spontaneously produces an electron-positron pair – the nucleus recoils, not in horror, but in order to conserve energy and momentum). Let’s assume the photon is rather close to the mass of the electron-positron pair, so the pair are rather slow moving afterwards.

Once the electron and positron are produced (each with momentum of magnitude p_{top}), they experience a strong magnetic field (in the picture, it points out of the paper). The law that describes the interaction between a charge and a magnetic field is called the Lorentz Force Law. It causes the (positively charged) positron to curve to the right, the (negatively charged) electron to curve to the left. The two then separately propagate down the apparatus (acquiring a momentum p_{bottom}) where they are forced to recombine, into a photon, of exactly the right frequency, which continues the cycle. In particular, writing the energy of the photons in each case.

h \nu_{top} = 2 \sqrt{(m_e c^2)^2+p_{top}^2 c^2} \approx 2 m_e c^2

h \nu_{bottom} = 2 \sqrt{(m_e c^2)^2+p_{bottom}^2 c^2} \approx 2 m_e c^2 + 2 m_e g L

In the above, p_{bottom} > p_{top}, the electrons have slightly higher speed at the bottom than at the top.

We know from the usual descriptions of potential energy and kinetic energy (from high school, hopefully), that the electron and positron pick up energy  m_e g L (each) on their path down to the bottom of the apparatus. Now, if the photon doesn’t experience a corresponding loss of energy as it travels from the bottom to the top of the apparatus, we have an arbitrage. We could use this apparatus to generate free energy (read “risk-less profit”) forever. This can’t be – this is nature, not a man-made market! So the change of energy of the photon will be

h \nu_{bottom} - h \nu_{top} =2 m_e g L \approx h \nu_{top} \frac{g L}{c^2}

indeed, the frequency of the photon is higher at the bottom of the apparatus than at the top. As photons “climb” out of the depths of the gravitational field, they get red-shifted – their wavelength lengthens/frequency reduces. This formula implies

\nu_{bottom} \approx \nu_{top} (1 + \frac{g L}{c^2})

writing this in terms of the gravitational potential due to the earth (mass M) at a distance R from its center

\Phi(R) = - \frac {G M}{R}

\nu_{bottom} \approx \nu_{top} (1 + \frac{\Phi(top) - \Phi(bottom)}{c^2})

so , for a weak gravitational field,

\nu_{bottom} (1 + \frac{ \Phi(bottom)}{c^2}) \approx \nu_{top} (1 + \frac{\Phi(top)}{c^2})

On the other time intervals are related to inverse frequencies (we consider the time between successive wave fronts)

\frac {1}{\Delta t_{bottom} } (1 + \frac{ \Phi(bottom)}{c^2}) \approx \frac {1}{\Delta t_{top}} (1 + \frac{\Phi(top)}{c^2})

so comparing the time intervals between successive ticks of a clock at the surface of the earth, versus at a point infinitely far away, where the gravitational potential is zero,

\frac {1}{\Delta t_{R} } (1 + \frac{ \Phi(R)}{c^2}) \approx \frac {1}{\Delta t_{\infty}}

which means

\Delta t_{R} =  \Delta t_{\infty} (1 + \frac{ \Phi(R)}{c^2})  

The conclusion is that the time between successive ticks of the clock is measured to be much smaller on the surface of the earth vs. far away. Note that \Phi(R) is negative, and the gravitational potential is usually assumed to be zero at infinity. This is the phenomenon of time dilation due to gravity. As an example, the GPS systems are run off clocks on satellites orbiting the earth at a distance of $20,700$ km. The clocks on the earth run slower than clocks on the satellites. In addition, as a smaller effect, the satellites are travelling at a high speed, so special relativity causes their clocks to run a little slower compared to those on the earth. The two effects act in opposite directions. This is the subject of a future post, but the effect, which has been precisely checked, is about 38 \museconds per day. If we didn’t correct for relativity, our planes would land at incorrect airports etc and we would experience total chaos in transportation.

The earth is flat – in Cleveland

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I stopped following basketball after Michael Jordan stopped playing for the Bulls – believe it or not, the sport appears to have become the place to believe and practice outlandish theories that might be described (in comparison to the Bulls) as bull****.

There’s a basketball star, that plays for the Cleveland Cavaliers. His name is Kyrie Irving. He believes that the earth is flat. He wishes to leave the Cleveland Cavaliers – but not go away too far, since he might fall off the side of the earth. However, he has inspired a large number of middle-schoolers (none of whom I have had the pleasure of meeting, but apparently they exist) that the earth is flat and that the “round-earthers” are government-conspiracy-inspired, pointy-headed, Russian spies – read this article if you want background. In fact, there is a club called the Flat Earth Society, that has members around the globe, that all believe the earth is flat as a pancake.

It would be really interesting, I thought, if, like my favorite detective – Sherlock Holmes – I decided to write the “Intelligent Person’s Guide to Why the Earth is Round”. I would ask you, dear Skeptical Reader, to use no more than tools readily available, some believable friends who possess phones with cameras and the ability to send and receive pictures by mail or text, as well as not being in the pay of the FSB  (or the North Koreans, who decidedly are trying very hard to check the flat earth theory by sending out ICBMs at increasing distances).

I live  in south New Jersey. At my location, the sun rose today at 5:57 am (you could figure this out by typing it out on Google search or just wake up in time to look for the sun). I have two friends, that live in Denver (Colorado) and Cheyenne (Wyoming). Their sunrises occur at 6:00 am and 5:53 am (their time) – averages to 5:56:30 am roughly. I realize that Denver is a mile high, which is also roughly Cheyenne’s height, but hey, you don’t pick your friends. I also live at an elevation of roughly 98′, which isn’t much and I ignore it. They sent me pictures of when the sun rose and I was able to prove they weren’t lying to me or part of a government conspiracy.

The distance from my town to these places is 1766 miles (to Denver) and 1613 miles (to Cheyenne). I used Google to calculate these, but you could schlep yourself there too. Based on just these facts, I should conclude that the earth curves between New Jersey and those places. To my mind, this should clinch the question of whether the earth is round. Since the  roughly 1700 mile separation equals 2 hours of time difference (in sunrises), a 24-hour time difference corresponds to 20,400 miles. This is roughly equal to 24,000  miles times the cosine of 40 degrees, which is the latitude of both New York City, Denver and Cheyenne (which is the circumference for radius r, rather than Earth’s radius R). This means 2 \pi R (which is the earth’s equatorial circumference) is roughly 26,000 miles, which is close to the correct figure to within 4%. The extreme height at Denver and Cheyenne has something to do with it! The sun {\bf should} have risen later in Denver and Cheyenne if they had been at lower elevations, so 1700 miles would {\bf really} have corresponded to a few minutes more than 2 hours, which would have meant a lower estimate for the earth’s equatorial circumference.

Earth Is Round

By the way, I picked Cheyenne because of its auditory resemblance to the town that Eratosthenes picked for his diameter-of-Earth measurement, Syrene in present-day Libya. Yes, the first person to measure the Earth’s diameter was Libyan!

Some objections to these entirely reasonable calculations include – if the earth is actually rotating, why doesn’t it move under you when you go up in a balloon. Sorry, this has been thought of already! When I was young, I was consumed by Yakov Perelman’s “Astronomy for Entertainment” – a book written by the tragically short-lived Soviet popularizer of science who died during the siege of Leningrad (St. Petersburg) in 1942. Perelman wrote about a young, enterprising, French advertising executive/scammer at the turn of the 19th century that dreamed up a new scheme to separate people from their money. He advertised balloon flights that would take you to different parts of the world without moving – just go up in a balloon and stay aloft till your favorite country comes up beneath you. It doesn’t happen because all the stuff around you is moving with you. Why? Its the same reason why the rain drops don’t fly off your side-windows even when you are driving on the road at high speed in the rain – forgetting for a second about the gravitational force that pulls things towards the earth’s center. There is a boundary layer of material that rotates or moves as fast as a moving object – its a consequence of the mechanics of fluids and we live with it in various places. For instance, it is one reason why icing occurs on airplane wings – if there was a terrible force of wind all the time, ice wouldn’t form.

So, if you are willing to listen to reason, no reason to restrict yourself to Cleveland. The world is invitingly round.

Addendum : a rather insightful friend of mine just told me that Kyrie Irving was actually born in Australia on the other side of the Flat Earth. If so, I doubt that even my robust arguments would convince him to globalize his views.