The previous post ended by demonstrating that the time interval between successive ticks of a clock at the earth’s surface and a clock ticking infinitely far away from all masses are related by the formula

The gravitational potential is a number for all . This means that the time intervals measured by the clock at the earth’s surface is than the time interval measured far away from the earth. If you saw the movie “Interstellar“, you will hopefully remember that a year passed on Miller’s planet (the one with the huge tidal waves) while 23 years passed on the Earth, since Miller’s planet was close to the giant Black Hole Gargantua. So time appears to slow down on the surface of the Earth compared to a clock placed far away.

Time for some computations. The mass of the earth is , Earth’s radius is and the MKS units for . In addition, the speed of light . If , the clock on an orbiting satellite (assumed to be really far away from the earth) measures one second, the clock at the surface measures

this can be simplified to less than . In a day, which is , this is (microseconds are a millionth of a second).

In reality, as will be explained below, the GPS satellites are operating at roughly above the earth’s surface, so what’s relevant is the in the gravitational potential at and from the earth’s center. That modifies the difference in clock rates to per second, or in a day.

How does GPS work? The US (other countries too – Russia, the EU, China, India) launched several satellites into a distant orbit above the earth’s surface. Most of the orbits are designed to allow different satellites to cover the earth’s surface at various points of time. A few of the systems (in particular, India’s) have satellites placed in a Geo-Stationary orbit, so they rotate around the earth with the earth – they are always above a certain point on the earth’s surface. The key is that they possess rather accurate and synchronized atomic clocks and send the time signals, along with the satellite position and ID to GPS receivers.

If you think about how to locate someone on the earth, if I told you I was 10 miles from the Empire State Building in Manhattan, you wouldn’t know where I was. Then, if I told you that I was 5 miles from the Chrysler building (also in Manhattan), you would be better off, but you still wouldn’t know how high I was. If I receive a third coordinate (distance from yet another landmark), I’d be set. So we need distances from three well-known locations in order to locate ourselves on the Earth’s surface.

The GPS receiver on your dashboard receives signals from three GPS satellites. It knows how far they are, because it knows when the signals were emitted, as well as what the time at your location is. Since these signals travel at the speed of light (and this is sometimes a problem if you have atmospheric interference), the receiver can compute how far away the satellites are. Since it has distances to three “landmarks”, it can be programmed to compute its own location.

Of course, if its clock was constantly running slower than the satellite clocks, it would constantly overestimate the distance to these satellites, for it would think the signals were than they actually were. This would screw up the location calculation, to the distance travelled by light in , which is meters. Over a day, this would become kilometers. You could well be in a different city!

There’s another effect – that of time dilation. To explain this, there is no better than the below thought experiment, that I think I first heard of from George Gamow’s book. As with arguments in special and general relativity, the only things observers can agree on is the speed of light and (hence) the order of causally related events. That’s what we use in the below.

There’s an observer standing in a much–abused rail carriage. The rail carriage is travelling to the right, at a high speed . The observer has a rather cool contraption / clock. It is made with a laser, that emits photons and a mirror, that reflects them. The laser emits photons from the bottom of the carriage towards the ceiling, where the mirror is mounted. The mirror reflects the photon back to the floor of the car, where it is received by a photo-detector (yet another thing that Einstein first explained!).

The time taken for this up- and down- journey (the emitter and mirror are separated by a length ) is

That’s what the observer on the train measures the time interval to be. What does an observer on the track, outside the train, see?

She sees the light traverse the path down in blue above. However, she also sees the light traveling at the same (numerical) speed, so she decides that the time between emission and reception of the photon is found using Pythagoras’ theorem

So, the time interval between the same two events is computed to be larger on the stationary observer’s clock, than on the moving observer’s clock. The relationship is

How about that old chestnut – well, isn’t the observer on the track moving relative to the observer on the train? How come you can’t reverse this argument?

The answer is – who’s going to have to turn the train around and sheepishly come back after this silly experiment runs its course? Well! The point is that one of these observers has to actively come back in order to compare clocks. Relativity just observes that you cannot make statements about motion. You certainly have to accept relative motion and in particular, how observers have to compare clocks at the same point in space.

From the above, second on the moving clock would correspond to seconds on the clock by the tracks. A satellite at a distance from the center of the earth has an orbital speed of , which for an orbit km above the earth’s surface, which is from the earth’s center, would be roughly

which means that second on the moving clock would correspond to on the clock by the tracks. Over a day, this would correspond to a drift of , in the direction to the above calculation for gravitational slowing.

Net result – the satellite clocks run faster by microseconds in a day. They need to be continually adjusted to bring them in sync with earth-based clocks.

So, that’s three ways in which Mr. Einstein matters to you EVERY day!

]]>

The protagonist, sitting at the table tells you (and you are able to confirm this by a video taken by a nearby security camera run by a disinterested police officer), that he has managed to toss the same quarter (an American coin) thirty times and managed to get “Heads” of those times. What would you say about the fairness or unfairness of the coin in question?

Next, your good friend rushes to your side and whispers to you that this guy is actually one of a number of people (a little more than a billion) that were asked to successively toss freshly minted, scrupulously clean and fair quarters. People that tossed tails were “tossed” out at each successive toss and only those that tossed heads were allowed to toss again. This guy (and one more like him) were the only ones that remained. What can you say now about the fairness or unfairness of the coin in question?

What if the number of coin tosses was rather than , with a larger number of initial subjects?

Just to make sure you think about this correctly, suppose you were the Director of a large State Pension Fund and you need to invest the life savings of your state’s teachers, firemen, policemen, highway maintenance workers and the like. You get told you have to decide to allocate some money to a bet based made by an investment manager based on his or her track record (he successively tossed “Heads” a hundred times in a row). Should you invest money on the possibility that he or she will toss “Heads” again? If so, how much should you invest? Should you stay away?

This question cuts to the heart of how we operate in real life. If you cut out the analytical skills you learnt in school and revert to how our “lizard” brain thinks, we would assume the coin was unfair (in the first instance) and express total surprise at the knowledge of the second fact. In fact, even though the second situation could well have happened to every similar situation of the first sort we encounter in the real world, we would still operate as if the coin was unfair, as our “lizard” brain would instruct us to behave.

What we are doing unconsciously is using Bayes’ theorem. Bayes’ theorem is the linchpin of inferential deduction and is often misused even by people who understand what they are doing with it. If you want to read couple of rather interesting books that use it in various ways, read Gerd Gigirenzer’s “*Reckoning with Risk: Learning to Live with Uncertainty” *or Hans Christian von Baeyer’s “*QBism*“. I will discuss a few classic examples. In particular Gigirenzer’s book discusses several such, as well as ways to overcome popular mistakes made in the interpretation of the results.

Here’s a very overused, but instructive example. Let’s say there is a rare disease (pick your poison) that afflicts of the population. Unfortunately, you are worried that you might have it. Fortunately for you, there is a test that can be performed, that is accurate – so if you do have the disease, the test will detect it of the time. Unfortunately for us, the test has a false positive rate, which means that if you don’t have the disease, of such tested people will mistakenly get a positive result. Despite this, the results look exceedingly good, so the test is much admired.

You nervously proceed to your doctor’s office and get tested. Alas, the result comes back “Positive”. Now, ask yourself, what the chances you actually have the disease? After all, you have heard of false positives!

A simple way to turn the percentages above into numbers, suppose you consider a population of people. Since the disease is rather rare, only have the disease. If they are tested, only of them will get an erroneous “negative” result. However, if the rest of the population were tested in the same way, people would get a “Positive” result, despite not having the disease. In other words, of the people who would get a “Positive” result, only actually have the disease, which is roughly – so such an accurate test can only give you a 7-in-10 chance of actually being diseased, despite its incredible accuracy. The reason is that the “false positive” rate is low, but not low enough to overcome the extreme rarity of the disease in question.

Notice, as Gigirenzer does, how simple the argument seems when phrased with numbers, rather than with percentages. To do this using standard probability theory, one writes, if we are speaking about Events and and write the probability that could occur once we know that has occurred as , then

Using this

and then we note

since I could test positive for two reasons – either I really among the positive people and additionally was among the that the test caught OR I really was among the negative people but was among the that unfortunately got a false positive.

Indeed,

which was the answer we got before.

The rather straightforward formula I used in the above is one formulation of Bayes’ theorem. Bayes’ theorem allows one to incorporate one’s knowledge of partial outcomes to deduce what the underlying probabilities of events were to start with.

There is no good answer to the question that I posed in the first paragraph. It is true that both a fair and an unfair coin could give results consistent with the first event (someone gets or even coin tosses). However, if one desires that probability has an objective meaning independent of our experience, based upon the results of an infinite number of repetitions of some experiment (the so-called “*frequentist*” interpretation of probability), then one is stuck. In fact, based upon that principle, if you haven’t heard something contrary to the facts about the coin, your *a priori* assumption about the probability of heads must be . On the other hand, that isn’t how you run your daily life. In fact, the most legally defensible (many people would argue the defensible) strategy for the Director of the Pension Fund would be to

- not assume that prior returns were based on pure chance and would be equally likely to be positive or negative
- bet on the manager with the best track record

At a minimum, I would advise people to stay away from a stable of managers that simply are the survivors of a talent test where the losers were rejected (oh wait, that sounds like a large number of investment managers in business these days!). Of course, the manager that knows they have a good thing going is likely to not allow investors at all for fear of reducing their returns due to crowding. Such managers also exist in the global market.

The Bayesian approach has a lot in common with our every-day approach to life. It is not surprising that it has been applied to the interpretation of Quantum Mechanics and that will be discussed in a future post.

]]>

Type 1a supernovas happen to white dwarf stars. A white dwarf is a kind of star that has reached the end of its starry career. It has burnt through its hydrogen fuel, producing all sorts of heavier elements, through to carbon and oxygen. It has also ceased being hot enough to burn Carbon and Oxygen in fusion reactions. Since these two elements burn rather less efficiently than Hydrogen or Helium in fusion reactions, the star is dense (there is less pressure from light being radiated by fusion reactions in the inside to counteract the gravitational pressure of matter, so it compresses itself) and the interior is composed of ionized carbon and oxygen (all the negatively charged electrons are pulled out of every atom, the remaining ions are positively charged and the electrons roam freely in the star). Just as in a crystalline lattice (as in a typical metal), the light electrons are good at keeping the positively charged ions screened from other ions. In addition, they also use them in turn to screen themselves from other electrons; the upshot is, the electrons behave like free particles.

At this point, the star is being pulled in by its own mass and is being held up by the pressure exerted by the gas of free electrons in its midst. The “lattice” of positive ions also exerts pressure, but the pressure is much less, as we will see. The temperature of the surface of the white dwarf is known from observations to be quite high, Kelvin. More important, the free electrons in a white dwarf of mass much greater than the Sun’s mass (written as ) are ultra-relativistic, with energy much higher than their individual mass. Remember, too, that electrons are a species of “fermion”, which obey Fermi-Dirac statistics.

The Fermi-Dirac formula is written as

What does this formula mean? The energy of an ultra-relativistic electron, that has energy far in excess of its mass, is

where is the “frequency” corresponding to the electron of momentum , while is the “reduced” Planck’s constant (here is the regular Planck’s constant) and is the speed of light. The quantity is called the Fermi wave-vector. The function is the (density of) probability of finding an electron in the momentum state specified by . In the study of particles where their wave nature is apparent, it is useful to use the concept of the de Broglie “frequency” , the de Broglie “wavelength” where is the particle velocity) and , the “wave-number” corresponding to the particle. It is customary for lazy people to forget to write and in formulas, hence, we speak of momentum for a hyper-relativistic particle travelling at speed close to the speed of light, when it should really be .

Why a factor of 2? It wasn’t there in the previous post!

From the previous post, you know that fermions don’t like to be in the same state together. We also know that electrons have a property called spin and they can be spin-up or spin-down. Spin is a property akin to angular momentum, which is a property that we understand classically, for instance, as describing the rotation of a bicycle wheel. You might remember that angular momentum is conserved unless someone applies a torque to the wheel. This is the reason why free-standing gyroscopes can be used for airplane navigation – they “remember” which direction they are pointing in. Similarly, spin is usually conserved, unless you apply a magnetic field to “twist” a spin-up electron into a spin-down configuration. So, you can actually have two kinds of electrons – spin-up and spin-down, in each momentum state . This is the reason for the factor of 2 in the formula above – there are two “spin” states per state.

Let’s understand the Fermi wave-vector . Since the fermions need to occupy momentum states two-at-a-time for each, and if they were forced into a cube of side , you can ask how many levels they occupy. They will, like all sensible particles, start occupying levels starting with the lowest energy level, going up till all the fermions available are exhausted. The fermions are described by waves and, in turn, waves are described by wavelength. You need to classify all the possible ways to fit waves into a cube. Let’s look at a one-dimensional case to start

The fermions need to bounce off the ends of the one-dimensional lattice of length so we need the waves to be pinned to 0 at the ends. If you look at the above pictures, the wavelengths of the waves are (for n=1), (for n=2), (for n=3), (for n=4). In that case, the wavenumbers, which are basically for the fermions need to be of the sort , where is an integer .

For a cube of side L, the corresponding wave-numbers are described by since a vector will have three components in three dimensions. These wave-numbers correspond to the momenta of the fermions (this is basically what’s referred to as wave-particle duality), so the momentum is . The energy of each level is . It is therefore convenient to think of the electrons as filling spots in the space of .

What do we have so far? These “free” electrons are going to occupy energy levels starting from the lowest and so on in a neat symmetric fashion. In space, which is “momentum space”, since we have many, many electrons, we could think of them filling up a sphere of radius in momentum space. This radius is called the Fermi wave-vector. It represents the most energetic of the electrons, when they are all arranged in as economically as possible – with the lowest possible energy for the gas of electrons. This would happen at zero temperature (which is the approximation we are going to work with at this time). This comes out from the probability distribution formula (ignore the 2 for this, consider the probability of occupation of levels for a one-dimensional fermion gas). Note that all the electrons are inside the Fermi sphere at low temperature and leak out as the temperature is raised (graphs towards the right).

It is remarkable and you should realize this, that a gas of fermions in its lowest energy configuration has a huge amount of energy. The Pauli principle requires it. If they were bosons, all of them would be sitting at the lowest possible energy level, which couldn’t be zero (because we live in a quantum world) but just above it.

What’s the energy of this gas? Its an exercise in arithmetic for zero temperature. Is that good enough? No, but it gets us pretty close to the correct answer and it is instructive.

The total energy in the gas of electrons is (in a spherical white dwarf of volume , with spins per state, is

The total number of electrons is obtained by just adding up all the available states in momentum space, up to

We need to estimate the number of electrons in the white dwarf to start this calculation off. That’s what sets the value of , the radius of the “sphere” in momentum space of filled energy states at zero temperature.

The mass of the star is . That corresponds to full atoms, where is the mass of the proton (the dominant part of mass) and is the ratio of atomic weight to atomic number for a typical constituent atom in the white dwarf. For a star composed of Carbon and Oxygen, this is . So, .

Using all the above

Next, the white dwarf has some gravitational potential energy just because of its existence. This is calculated in high school classes by integration over successive spherical shells from to the radius , as shown below

The gravitational potential energy is

A strange set of things happen if the energy of the electrons (which is called, by the way, the “degeneracy energy”) plus the gravitational energy goes negative. At that point, the total energy can become even more negative as the white dwarf’s radius gets smaller – this can continue – the star collapses. This starts to happen when you set the gravitational potential energy equal to the Fermi gas energy, this leads to

the (radius) of the star drops out and we are left with a unique mass where this happens – the calculation above gives an answer of . A more precise calculation at non-zero temperature gives .

The famous physicist S. Chandrasekhar, after whom the Chandra X-ray space observatory is named, discovered this (Chandrasekhar limit) while ruminating about the effect of hyper-relativistic fermions in a white dwarf.

He was on a cruise from India to Great Britain at the time and had the time for unrestricted rumination of these sorts!

Therefore, as is often the case, if a white dwarf is surrounded by a swirling cloud of gas of various sorts or has a companion star of some sort that it accretes matter from, it will collapse into a denser neutron star in a cataclysmic collapse precisely when this limit is reached. If so, once one understands the type of light emitted from such a supernova from some nearer location, one has a Standard Candle – it is like having a hand grenade that is of the same quality at various distances. By looking at how bright the explosion is, you can tell how far away it is.

After this longish post, I will describe the wonderful results from this analysis in the next post – it has changed our views of the Universe and our place in it, in the last several years.

]]>

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)

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 !

- The ratio of the electrical and gravitational forces between the electron and the proton ( vs. ) is approximately
- 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 –

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 would fall with time (why not going up with time, or something else?). Whereas precision experiments in measuring the value of are beginning now, there would have been cosmological consequences if had indeed behaved as in the past! For this reason, people discount this “theory” these days.

I heard of another coincidence recently – the value of the quantity , where is the speed of light and is the acceleration due to gravity at the surface of the earth (), is very close to light year. This implies (if you put together the formulas for , and 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 – 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 people and you ask for match of birthdays, there are 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 matches – that is , which is for the first non-match, then 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.

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.

]]>

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 two paragraphs above). Additionally, consider if you, Dear Reader, own a Call Option, with strike price , 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 Let’s say , while the stock price was initially . At the end of the year, suppose the stock price became , then the difference , so you, Dear Reader and Fortunate-Call-Option-Owner, would make . On the other hand, if the stock price unfortunately sank to , then the difference 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 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 and not ? 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 which goes to as . 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 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 . Can we find a portfolio that would decline by exactly the same amount if the stock price was up by . 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 increase in the stock price. But once I do this, I have a portfolio (Option plus this “short” position) that is 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 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).

One photon rushes from the bottom of the apparatus to the top. Let’s assume it has a frequency at the bottom of the apparatus and a frequency 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 ), 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 ) 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.

In the above, , 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 (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

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

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

so , for a weak gravitational field,

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

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,

which means

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 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 seconds 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.

]]>

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 , rather than Earth’s radius ). This means (which is the earth’s equatorial circumference) is roughly 26,000 miles, which is close to the correct figure to within . The extreme height at Denver and Cheyenne has something to do with it! The sun have risen later in Denver and Cheyenne if they had been at lower elevations, so 1700 miles would have corresponded to a few minutes more than 2 hours, which would have meant a lower estimate for the earth’s equatorial circumference.

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.

]]>

Babu is sitting in a railway carriage, manufactured by the famous C-Kansen company, that travels at speeds close to that of light. Babu’s sitting exactly in the middle of the carriage and for reasons best known to himself (I guess the pantry car was closed and he was bored), decides to shoot a laser beam simultaneously at either end of the carriage from his position. There are detectors/mirrors that detect the light at the two ends of the carriage. As far as he is concerned, light travels at and he will challenge anyone who says otherwise to a gunfight – note that he is wearing a cowboy hat and probably practices open carry.

Since the detectors at the end of the carriage are equidistant from him, he is going to be sure to find the laser beams hit the detectors simultaneously, from his point of view.

Now, consider the situation from the point of view of Alisha, standing outside the train, near the tracks, but safely away from Babu and his openly carried munitions. She sees that the train is speeding away to the left, so clearly since thinks light also travels at , she would say that the light hit the detector first before the detector. She doesn’t that the light hit the two detectors simultaneously. If you asked her to explain, she’d say that the right detector is speeding towards the light, while the left detector is speeding away from the light, which is why the light strikes them at different times.

Wait – it is worse. If you had a third observer, Emmy, who is skiing to the at an even higher speed than the C-Kansen (some of these skiers are crazy), she thinks the C-Kansen train is going off to the right (think about it), not able to keep up with her. As far as is concerned, the laser beam hit the detector before the other beam hit the detector.

What are we finding? The Events in question are – “Light hits Left Detector” and “Light hits Right Detector”. Babu claims the two events are simultaneous. Alisha claims the second happened earlier. Emmy is insistent that the first happened earlier. Who is right?

They are ALL correct, in their own reference frames. Events that appear simultaneous in one reference frame can appear to occur the one before the other in a different frame, or indeed the one after the other, in another frame. This is called the Relativity of Simultaneity. Basically, this means that you cannot attribute one of these events to have the other, since their order can be changed. Events that are separated in this fashion are called “space-like separated”.

Now, on to the topic of this post. In the physics of quantum field theory, particles interact with each other by exchanging other particles, called gauge bosons. This interaction is depicted, in very simplified fashion so we can calculate things like the effective force between the particles, in a sequence of diagrams called Feynman diagrams. Here’s a diagram that depicts the simplest possible interaction between two electrons

Time goes from the bottom to the top, the electrons approach each other, exchange a photon, then scoot off in different directions.

This is the simplest diagram, though and to get the exact numerical results for such scattering, you have to add higher orders of this diagram, as shown below

When you study such processes, you have to perform mathematical integrals – all you know is that you sent in some particles from far away into your experimental set-up, something happened and some particles emerged from inside. Since you don’t know where and when the interaction occurred (where a particle was emitted or picked up, as at the vertexes in the above diagrams), you have to sum over all possible places and times that the interaction have occurred.

Now comes the strange bit. Look at what might happen when you sum over all possible paths for a collision between an electron and a photon.

In the above diagram, the exchange was simultaneous.

In the next one, the electron emitted a photon, then went on to absorb a photon.

and then comes the strange bit –

Here the electron emitted a photon, then went backwards in time, absorbed a photon, then went its way.

When we sum over all possible event times and locations, this is really what the integrals in quantum field theory instruct us to do!

Really, should we allow ourselves to count processes where two events occur simultaneously, which means we would then have to allow for them to happen in reverse order, as in the third diagram? What’s going on? This has to be wrong! And what’s an electron going backwards in time anyway? Have we ever seen such a thing?

Could we simply ban such processes? So, we would only sum over positions and times where the intermediate particles had enough time to go from one place to another,

There’s a problem with this. Notice the individual vertexes where an electron comes in, emits (or absorbs) a photon, then moves on. If this were a “real” process, it wouldn’t be allowed. It violates the principle of energy-momentum conservation. A simple way to understand this is to ask, could a stationary electron suddenly emit a photon and shoot off in a direction opposite to the photon, It looks deceptively possible! The photon would have, say, energy and momentum . This means that the electron would also have momentum , in the opposite direction (for conservation) but then its energy would have to be from the relativistic formula. This is higher than the energy of the initial electron : is bigger than ! Not allowed !

We are stuck. We have to assume that energy-momentum conservation is violated in the intermediate state – in all possible ways. But then, all hell breaks loose – in relativity, the speed of a particle is related to its momentum and its energy by – since and can be , the intermediate electron could, for instance, travel faster than light. If so, in the appropriate reference frame, it would be absorbed before it was created. If you can travel faster than light, you can travel backwards in time (read this post in Matt Buckley’s blog for a neat explanation).

If the electron were uncharged, we would probably be hard-pressed to notice. But the electron is charged. This means if we had the following sequence of events,

– the world has -1 net charge

– electron emits a photon and travels forward in time

– electron absorbs a photon and goes on.

This sequence doesn’t appear to change the net charge in the universe.

But consider the following sequence of events

– the world has -1 net charge

– the electron emits a photon and travels backwards in time

– the electron absorbs a photon in the past and then starts going forwards in time

Now, at some intermediate time, the universe seems to have developed two extra negative charges.

This can’t happen – we’d notice! Extra charges tend to cause trouble, as you’d realize if you ever received an electric shock.

The only way to solve this is to postulate that an electron moving backward in time has a positive charge. Then the net charge added for all time slices is always -1.

ergo, we have antiparticles. We need to introduce the concept to marry relativity with quantum field theory.

There is a way out of this morass if we insist that all interactions occur at the same point in space and that we never have to deal with “virtual” particles that violate momentum-energy conservation at intermediate times. This doesn’t work because of something that Ken Wilson discovered in the late ’70s, called the renormalization group – the results of our insistence would be we would disagree with experiment – the effects would be too weak.

For quantum-field-theory students, this is basically saying that the expansion of the electron’s field operator into its components can’t simply be

but has to be

including particles being destroyed on par with anti-particles being created and vice versa.

The next post in this sequence will discuss another interesting principle that governs particle interactions – C-P-T.

Quantum field theory is an over-arching theory of fundamental interactions. One bedrock of the theory is something called C-P-T invariance. This means that if you take any physical situation involving any bunch of particles, then do the following

- make time go backwards
- parity-reverse space (so in three-dimensions, go into a mirror world, where you and everything else is opposite-handed)
- change all particles into anti-particles (with the opposite charge)

then you will get a process which could (and should) happen in your own world. As far as we know this is always right, in general and it has been proven under a variety of assumptions. A violation of the C-P-T theorem in the universe would create quite a stir. I’ll discuss that in a future post.

Addendum: After this article was published, I got a message from someone I respect a huge amount that there is an interesting issue here. When we take a non-relativistic limit of the relativistic field theory, where do the anti-particles vanish off to? This is a question that is I am going to try and write about in a bit!

]]>

This post is about an experiment that directly points to the fundamental weirdness of small (and these days, not so small) particles. While quantum effects matter at the sub-atomic particle level, these effects can coalesce into macroscopic phenomena like superconductivity, the quantum Hall effect and so on, so they can't be ignored. This experiment, usually referred to as the "Double-Slit" experiment, is described and explained in detail in Vol. 3 of the Feynman Lectures in Physics. While it would be silly of me to try and outdo Feynman's explanation, which by the way was one of the reasons why I was enthused to study physics in the first place, I want to go beyond the Double-Slit experiment to discuss the Delayed-Choice experiment – this extra wrinkle on the Double-Slit experiment was invented by the famous scientist John Wheeler (who was Feynman's Ph.D advisor) and displays for all to see, even more aspects of quantum weirdness.

Let's get started.

The Double-Slit experiment is carried out by shooting electrons at a pair of closely placed slits – the electron flux is sufficiently small that one is able to count the number of times electrons hit various points on a television screen placed past the slits. If no measures are taken to identify which of the two paths the electrons actually took to reach the screen, then the probability density of arrival at various points on the television screen displays an “interference'' pattern. If, however, the experiment is set up so as to identify which slit the electron went through, for example by shining an intense beam of photons at the slits that scatter off the electrons, then the interference pattern for those “which-path'' identified electrons switches to a “clump'' pattern, centered around the open slit. The standard experiment is displayed, schematically, below, where since both slits are open and we don't bother to check which slit the electron goes through, we see an "interference" pattern. If we used photons (light) instead of electrons, we'd see alternate light and dark fringes.

If only one slit were open, we'd get a "clump" pattern as below

Note – no interference "bumps" at places far away from the peak.

This behavior is also what we'd get for light – photons.

Quantum mechanics is the theory that was constructed to "explain" this behavior. We construct a quantity called the "amplitude". The "amplitude" is a complex number that has a (complex) value at every point in space and time. Complex numbers have two properties – a magnitude and a phase. The magnitude squared of the amplitude at some time times a small volume in space is the probability of finding the particle (if its an amplitude for the electron, then the probability of finding the electron etc.) in that volume at time . Since you need to multiply the magnitude squared by a little volume element, the squared magnitude of the amplitude is referred to as the "Probability Density".

Schrodinger's equation writes down how this amplitude evolves in time – from the electron gun to the screen. To this equation, you need to add the "Born" prescription – that you have to square the magnitude of the amplitude to get the probability density.

Feynman found a neat, equivalent interpretation of Schrodinger's equation – his method basically said – if you want to find the amplitude for the electron (say) at some point in the screen, just write down path-amplitudes for all the different ways the electron could get from the electron gun to the screen. Add these path-amplitudes and then call the net sum the "total" amplitude for the electron to be found at the particular spot on the screen. Square the magnitude of this "total" amplitude and you will get the probability density for the electron to be found at that spot (times the little volume around that spot will give you the probability to find the electron at that spot).

All this discussion of path-amplitudes would be academic if the amplitudes were real numbers. The phase is a critical piece of the amplitude. Though the magnitude (squared) is the physical quantity (related to a probability), the magnitude of a sum of complex numbers depends delicately on the phase difference between the two. As an example, have the same magnitude . However, all of which have magnitudes very different from just the sum of the individual magnitudes. That's the reason we get alternate light and dark fringes on the television screen – the phases of the amplitudes for the electron to get to those spots from either of the two slits sometimes causes the sum amplitude to be 0 (which is called destructive interference) and sometimes causes the sum amplitude to add to the maximum (which is called constructive interference) and all magnitudes between these two extremes. While this behavior is extremely counter-intuitive for particles, it resembles behavior we are used to with waves, as this YouTube video shows, so does this one. This is usually referred to as wave-particle duality.

The thing you need to take away from this experiment is that if you don't force the electron to get to the screen through only one slit, by say, closing off the other slit, it appears to behave like it goes through both slits.

Wait, it gets even more interesting.

The Delayed-Choice experiment was proposed by Marlan Scully (who is an inspiring scientist in his own right) and is portrayed in the accompanying figure.

An SPDC (spontaneous parametric down conversion) setup is used – basically, each of the red regions in the picture above produces two photons, when one photon hits it. Since the laser's photon can go through either of the two slits, this is just like the Double-Slit experiment, except that things are arranged so that after going through a slit, the photon would produce two other photons. The photons that travel towards the interferometer/detector are referred to as “signal'' photons. The photons that travel towards the prism are the “idler'' photons. After passing through the prism, the “idler'' photons pass through a beam-splitter, that has a probability of deflecting the incoming photon to detectors and respectively and probability of letting the photon pass on to the fully silvered (reflecting mirrors) at the bottom of the picture. Another beam-splitter is placed between the detectors and , so photons that are detected at and have their “which-path'' information obliterated – for instance, an “idler'' photon arriving at could have come along either of two paths. The actual experiment was performed by Kim et. al.

The detector accumulates “signal'' photons – a coincidence counter correlates it to “idler'' photons detected at the detectors and (the “idler'' photons arrive at those detectors a few nanoseconds after the “signal'' photons are received. From the accumulated “signal'' photons received at , if we separate the ones received in coincidence with the detectors or , since the “which-path'' information is clear in those cases, the pattern of interference observed (in the spatial density of the “signal'' photons) is the “clump'' pattern. However, the accumulated “signal'' photons received at that are coincident with the ones received at display an interference pattern, since one cannot infer the path taken by the “idler'' photons that show up at detector , which means one cannot tell which slit the “signal'' photon came through before arriving at detector . Similarly, the accumulated “signal'' photons received at that are coincident with the ones received at display an interference pattern in their spatial distribution, however this pattern is spatially offset half a wavelength off the one due to . This is just enough to put peaks where the other pattern has dips and vice versa. So, if you aren't careful to note which detector or detected the photon coincident with the "signal" photon detector , you get a clump pattern. The reason for this offset is little tricky to explain – its called "unitarity" or conservation of probability at the beam-splitter, which forces some delicate phase assignments for the amplitudes we spoke about earlier.

Note, however, that someone could hide the specific arrival times of the photons at and from you for months and then tell you, say, a year later. All this time, you wouldn't have known there was an interference pattern "hiding" under the massive "clump" pattern you see. When you selectively look at the coincident detections separately for and , it is then and only then, that you see the interference pattern.

Curious! This experiment, as I said, has been done and the results are as described above.

With a friend, I put together another set up in an interesting locale, a black hole, that we tried to make work. Quantum mechanics defeated us in our attempts too, but its an interesting problem to work through.

]]>

Consider the following scenario. You’ve organized a birthday party for a lot of different looking kids (no twins, triplets, quadruplets, quintuplets …). Each kid has equal access to a large pot of M&Ms in the center of the room. Each kid can grab M&Ms from the pot and additionally when they bounce off each other while playing, can exchange a few M&Ms with each other. After a long while, you notice that all the M&Ms in the central pot are gone. Let’s suppose there are truly a number of kids and a truly number of M&Ms .

Interesting question – how many M&Ms is each kid likely to have? A simpler question might be – how many kids likely have 1 M&M? How many likely have 2?..How many likely have 55?… How many likely have 5,656,005?…

How do we answer this question?

If you use the notation for the number of kids that have M&Ms?, then, we can easily write down

is the total number of kids.

is the total number of M&Ms.

But that isn’t enough to tell! We need some additional method to find the most likely distribution of M&Ms (clearly this wouldn’t work if I were there; I would have all of them and the kids would be looking at the mean dad that took the pot home, but that’s for a different post). The result, that Ludwig Boltzmann discovered, at the end of the 19th century, was simply the one where everybody has an equal number of M&Ms. The most likely distribution is the one with the most number of possible ways to exchange the roles of the kids and still have the same distribution. In other words, maximize the combinatoric number of ways

which is the way of distributing these kids so that have M&M, have M&Ms, have M&Ms…, have M&Ms and so on.

Boltzmann had a nervous breakdown a little after he invented the statistical mechanics, which is this method and its consequences, so don’t worry if you feel a little ringing in your ears. It will shortly grow in loudness!

How do we maximize this ?

The simplest thing to do is to maximize the logarithm of , which means we maximize

but we have to satisfy the constraints

The solution (a little algebra is required here) is that where is some constant for this ‘ere party. For historical reasons and since these techniques were initially used to describe the behavior of gases, it is called the inverse temperature. I much prefer “inverse gluttony” – the lower is, the larger the number of kids with a lot of M&Ms.

Instead of the quantity , which is the number of M&Ms the children have, if we considered , which is (say) the dollar value of the M&Ms, then the corresponding number of kids with is

Few kids have a lot of M&Ms, many have very few – so there you go, Socialists, doesn’t look like Nature prefers the equal distribution of M&Ms either.

If you thought of these kids as particles in a gas and as one of the possible energy levels (“number of M&Ms”) the particles could have, then the fraction of particles that have energy would be

This distribution of particles into energy levels is called the Boltzmann distribution (or the Boltzmann rule). The essential insight is that for several particles the probability that a particular particle is in a state of energy is proportional to .

After Boltzmann discovered this, the situation was static till the early when people started discovering particles in nature that were . It is a fascinating fact of nature that every photon or electron or muon or tau particle is identical to every other photon or electron or muon or tau particle (respectively and for all other sub-atomic particles too). While this fact isn’t “explained” by quantum field theory, it is used in the construction of our theories of nature.

Back to our party analogy.

Suppose, instead of a wide variety of kids, you invited the largest -tuplet the world has ever seen. kids that look identical. They all have the same parents (pity them , but hopefully were born in some physically possible way, like test-tubes. You cannot tell the kids apart, so if one of them has 10 M&Ms, its indistinguishable from of the kids having 10 M&Ms.

Now what’s the distribution of the number of kids with value in M&Ms? The argument I am going to present is one I personally have heard from Lubos Motl’s blog (I wouldn’t be surprised if its more widely available, though given the age of the field) and it is a really cute one.

There are a couple of possibilities.

Suppose there was a funny rule (made up by Ms. Fermi-Dirac, a well known and strict party host) that said that there could be at most kid that had, say value in M&Ms (for every ). Suppose were the probability that kid had of value in M&Ms. Then the probability that that 1 kid has of value in M&Ms is – remember the Boltzmann rule! Now if no other possibility is allowed (and if one kid has M&Ms, it is indistinguishable from any of the other kids, so you can’t ask which one has that many M&Ms)

since there are only two possibilities, the sum of the probabilities has to be 1.

This implies

And we can find the probability of there being kid with value in M&Ms. It would be

The number of kids with value in M&Ms would be

But we could also invite the fun-loving Mr. Bose to run the party. He has no rules! Take as much as you want!

Now, with the same notation as before, again keeping in mind that we cannot distinguish between the particles,

which is an infinite (geometric) series. The sum is

which is solved by

The expected number of kids with value in M&Ms is

which is

Now, here’s a logical question. If you followed the argument above, you could ask this -could we perhaps have a slightly less strict host, say Ms. Fermi-Dirac-Bose-2 that allows up to 2 kids to possess a number of M&Ms whose value is ? How about a general number kids that are allowed to possess M&Ms of value (the host being the even more generous Ms. Fermi-Dirac-Bose-L). More about this on a different thread. But the above kinds of statistics are the only ones Nature seems to allow in our 4 – dimensional world (three space and one time). Far more are allowed in 3 – dimensional worlds (two space and one time) and that will also be in a different post (the sheer number of connections one can come up with is fantastic!).

The thing to understand is that particles that obey Fermi-Dirac statistics (a maximum of one particle in every energy state) have a “repulsion” for each other – they don’t want to be in the same state as another Fermi-Dirac particle, because Nature forces them to obey Fermi-Dirac statistics. If the states were characterized by position in a box, they would want to stay apart. This leads to a kind of outwards pressure. This pressure (described in the next post) is called Fermi-degeneracy pressure – its what keeps a peculiar kind of dense star called a white dwarf from collapsing onto itself. However, beyond a certain limit of mass (called the Chandrasekhar limit after the scientist that discovered it), the pressure isn’t enough and the star collapses on itself – leading to a colossal explosion.

These explosions are the next kind of “standard candle”.

:

I feel the need to address the question I asked above, since I have been asked informally. Can one get statistics with choosing different values of in the above party? The answer is “No”. The reason is this – suppose you have kids at the party, with a maximum of kids that can carry M&Ms of value . Then we should be able to divide all our numbers by (making a scale model of our party that is times smaller) that has kids, with a maximum of kid that is allowed to hold M&Ms of value . You’d expect the expected number of kids with M&Ms of value to be, correspondingly, times smaller! Then, the expected number of particles in a state (with a limit of particles in each state) is just times the expected number with a limit of particle in each state.

So all we have are the basic Fermi-Dirac and Bose statistics (1 or many), in our three-space-dimensional party!

]]>

If I told you to estimate how far away a bulb was, you could probably make an estimate based on how bright the bulb seemed. For this you need two things. You need to know how bright the bulb is – this is the absolute luminosity and its measured in which is . Remember, however, that a 100 watt bulb right next to you appears brighter (and hotter) than the same 100 watt bulb ten miles away! To account for that, you could use the fact that the bulb distributes its light almost uniformly into a sphere around itself, to compute what fraction of the light energy you are actually able to intercept – we might have a patch of CCD (like the little sensor inside your video camera), of area capturing the light emitted by the bulb. Putting these together, as in the figure below, the amount of light captured is watts while the bulb puts out watts.

where if you dig into your memory, you should recall that the area of a sphere of radius is !

you can compute

You know how big your video camera’s sensor area is (it is in that manual that you almost threw away!) You know how much energy you are picking up every second (the apparent luminosity) – you’d need to buy a multimeter from Radio Shack for that (if you can find one now). But to actually compute the distance, you need to know the or luminosity of the light source!

That’s the problem! To do this, we need a set of “standard candles” (a light source of known actual luminosity in watts!) distributed around the universe. In fact the story of cosmology really revolves around the story of standard candles.

The first “standard candles” could well be the stars. If you assume you know how far away the Sun is, and if you assume other stars are just like our Sun, then you could make the first estimates of the size of the Universe.

We already know that the method of parallax could be used with the naked eye to calculate the distance to the moon. Hipparchus calculated that distance to be 59 earth radii. Aristarchus measured the distance to the sun (the method is a tour de force of elementary trigonometry and I will point to a picture here as an exercise!)

His calculation of the Earth-Sun distance was only 5 million miles, a fine example of a large experimental error – the one angle he had to measure was , he got wrong by a factor of 20. Of course, he was wise – he would have been blinded if he had tried to be very accurate and look at the sun’s geometric center!

Then, if you blindly used this estimate and ventured bravely on to calculate distances to other stars based on their apparent brightness relative to the sun, the results were startlingly large (and of course, still too small!) and people knew this as early as 200 B.C. The history of the world might have well been different if people had taken these observers seriously. It was quite a while and not till the Renaissance in Europe that quantitative techniques were re-discovered for distance measurements to the stars.

The problem with the technique of using the Sun as a “standard candle” is that stars differ quite a bit in their luminosity based on their composition, their size, their age and so on. The classification of stars and the description of their life-cycle was completed with the Hertzsprung-Russell diagram in 1910. In addition, the newly discovered nebulae had been resolved into millions of stars, so it wasn’t clear there was a simple way to think of stellar “standard candles” unless someone had a better idea of the size of these stellar clusters. However, some of the nearby galaxy companions of the Milky Way could have their distances estimated approximately (the Magellanic Cloud, for instance).

Enter Henrietta Leavitt. Her story is moving and representative of her time, from her Radcliffe college education to her $0.30 / hour salary for her work studying variable stars (she was a human computer for her academic boss), as well as the parsimonious recognition for her work while she was alive. She independently discovered that a class of variable stars called Cepheids in the Magellanic clouds appeared to have a universal connection between their intrinsic luminosity and the time period of their brightness oscillation. Here’s a typical graph (Cepheids are much brighter than the Sun and can be observed separately in many galaxies)

If you inverted the graph, you simply had to observe a Cepheid variable’s period to determine the absolute luminosity. Voila! You had a standard candle.

A little blip occurred in 1940, when Walter Baade discovered that Cepheids in the wings of the Andromeda galaxy were older stars (called Population II, compared to the earlier ones that are now referred to as Population I) and were in general dimmer than Population I Cepheids. When the Luminosity vs. Period graph was drawn for those, it implied the galaxy they were in was actually even further away! The size of the universe quadrupled (as it turned out) overnight!

Henrietta Leavitt invented the first reliable light-based distance measurement method for galaxies. Edwin Hubble and Milton Humason used data collected mainly from an analysis of Cepheids to derive the equation now known as Hubble’s law.

Next post will be about something called Olbers’ paradox before we start studying the expansion of the Universe, the Cosmic Microwave background and the current belief that we constitute just 4% of the universe – the rest being invisible to us and not (as far as we can tell) interacting with us.

]]>