On Electrons

This article is sort of a continuation of the article on quarks. That one focused on quarks and the theory of its interactions, Quantum Chromodynamics. This article will focus on electrons and the theory of electromagnetism on a small scale and the theory of Quantum Electrodynamics (QED). Feynman, one of the founders of this theory, called it the ‘Jewel of Physics’ for its incredibly accurate predictions. Such accuracy has never been seen before. It predicted some observations accurately to thirteen decimal places!

The electron is a fundamental particle with a unit negative charge and an incredibly small mass. It is a lepton with a spin of magnitude half. The idea of spin is pretty hard to grasp, it is not like the usual classical idea of a spinning object, that cannot happen as calculations show that an electron has to spin at a speed faster than the speed of light. Spin, essentially means that a particle has a non-zero value of angular momentum just by the virtue of existence. Electrons are fermions, which implies that electrons are subject to the Pauli Exclusion Principle; no two electrons in the universe can have the same magnitude of energy. Like gluons in QCD, the carriers of electromagnetic force are virtual photons. Virtual particles are those that do not exist, that is, while we can’t physically show the existence of the particle, we can feel its effects.

First, a note on fermions and bosons. As I mentioned, the Pauli Exclusion Principle applies only to fermions, or particles with half-integral spin, and not to bosons, with integral spin. This idea is purely mathematical, so there’s really not much to explain. The effective wavefunction of fermions is anti-symmetric while, for bosons, it is symmetric. This basically means that if you have a wavefunction of two bosons, and you interchange the bosons, then the wavefunction still remains the same. However, in case of a wavefunction for two fermions, if you switch them, then the wavefunction picks up a negative sign.

Electrons (and other fermions) exhibit an interesting phenomenon, the formation of Cooper pairs. At very low temperatures, two electrons come together and behave as one bound pair. The energy of formation of a cooper pair is very low, so thermal excitations can easily break up the pairs. The electrons in the pair don’t have to be in contact, they behave as one pair even over long distances, like up to hundreds of nanometers. An electron in a metal normally behaves as a free particle. The electrons mutually repel each other and attract the positive ions that make up a metallic lattice. This attraction distorts an ionic lattice, moving some of the ions slightly toward the electron. This increases the positive charge density of the lattice in the vicinity of the electron. This positive charge can attract other electrons. Over long distances, the attraction between electrons due to the displaced ions can overcome the inter-electronic repulsion and can thus pair up. This is at best a semi-classical explanation of the effect. The actual quantum mechanical explanation is much more rigorous and requires a lot of mathematics so I won’t be going into that.

An interesting phenomenon in this theory is the idea of vacuum polarization. The measured charge of an electron is lesser than its actual charge. Because of the charge of the electron, it has its own electric field. Now, the existence of an electric field means that there is more energy in that region than there would be without the electron. This leads to the spontaneous formation of particle-antiparticle pairs around the electron for a short duration of time. However, since this process happens near the electron as long as it is present, it leads to the electron being perpetually surrounded by a covering of particles. The existence of this layer of particles damps out the charge on the electron, and thus the charge that we observe is lesser than the actual charge on the electron.

Before I discuss the positron, I want to mention one last very interesting phenomenon that is only shown by electrons. This is called spin-charge separation. Basically, under certain conditions, the spin and charge on an electron split up. That is, one electron exhibits only charge, and the other electron exhibits only the spin. No electron can exhibit both these properties. An electron, by itself, has charge and spin. Another way looking at this is that the electron itself is bound to the state of two particles, the spinon (carries the spin) and the chargon or holon (carries the charge). An electron in a bound system also has a third particle, the orbiton (carries the orbital angular momentum). In our case, we consider an electron as a bound state of a spinon and chargon. Under certain conditions these particles can become deconfined, that is, they break free and behave as individual particles. These particles, the chargon, orbiton and the spinon are called quasi-particles.

The antiparticle of an electron is the positron, with a unit positive charge. All other properties remain the same. The existence of antiparticles leads to an interesting effect termed as ‘Zitterbewegung’, which is German for ‘trembling motion’. When the relativistic effects are taken into consideration and the wavefunction of an electron is formulated, then, it is observed that the wave-packet solution of the electron interacts with the positron wave-packets, which occupy the negative energy states and this gives rise to rapid oscillations. These oscillations occur with the speed of light. On a side note, this effect is observed not only with electrons but with the relativistic hydrogen atom and has been observed in Bose-Einstein-Condensates (BECs).

QED is one of the only theories that we understand to a great extent, yet there is a lot to be done, in the larger scheme of things. This is just the starting point in the quest to find a Grand Unified Theory.

Stephen Hawking and the Big Bang

On the 14th of March, 2018, one of the greatest cosmologists of our time, Stephen Hawking, passed away. It was a great loss, not only for physics but for science as a whole. There is no better way to honour his memory than to follow in his footsteps, taking the complex concepts of science and making them accessible to the general public. In this article, I will be elaborating upon some of his most influential works concerning the universe. I will focus mostly on his contributions to understanding the universe, and not on his seminal work on black holes as I have covered most of that in my two articles on black holes.

The first of his revolutionary ideas was the notion of the Big Bang. He did this in collaboration with Roger Penrose. Penrose had put forth his singularity theorem, which proposed that at the center of every black hole is a point-sized singularity. Hawking thought that this might be true for the entire universe. He worked out what the universe should have looked like in the past, and concluded that considering the accelerated universe now, the universe must have, at some point of time in the past, been a point like a singularity. This was the idea of the beginning of the universe, the Big Bang. In 1970, the two of them jointly proved that if the universe were to be any of the proposed Friedmann models (the most widely accepted models that describe the universe), then the universe must have started off with the Big Bang.

After this, he shifted his focus to the study of black holes. He came with the concept of Black Hole Thermodynamics and Hawking Radiation, which laid the foundation for the famous Information Loss Paradox of Black Holes.

Following this, Andrei Linde and Alan Guth proposed the inflationary model of the Universe, as a consequence of the Big Bang. The Inflationary model was important as it explained the fluctuations necessary to create the complex matter that we see now. Inflation basically says the universe initially expanded rapidly, but then, the rate of expansion slowed down.

All of this is pretty well known and there are a lot of resources one can find to read up on these ideas. Now, here is one of his ideas that really fascinated me, on which I did not find many articles. The “Hawking-Hartle No Boundary Proposal” of the universe. This idea basically says that the universe has no boundary. In the sense that if you go far enough and long enough in the same direction, you will end up where you started. This is sort of like an ant walking on the surface of an apple. Or, sort of like asking the question, How do I go north?, when you are at the North Pole. This is called a closed universe.

In the paper, they derived what they called the wavefunction of the universe. It seems absurd at first sight, a wavefunction of a particle is generally a function of either its position or momentum. How do you describe either of those two for the entire universe? You don’t. They calculated an integral of all the possible histories. This has an interesting consequence. The wavefunction now has the added information about all the possible different universes that could have come about after the Big Bang! It elegantly includes the idea of a multiverse. The wavefunction looks like any other quantum mechanical wavefunction, it can have places where the value of the wavefunction is zero. These correspond to places where there are breaks between universes. Basically, this corresponds to a boundary between two universes.

An idea like this has it’s fair share of opposition. One argument against this is that observation tells us that the universe is expanding, which means that the universe must be open. We can overcome this problem by choosing a suitable interpretation to the wavefunction. If we think of the wavefunction as a generator of space-time, then any bit of the wavefunction that corresponds to a universe can be thought of as an expanding universe. That is, we take it as the wavefunction generating spacetime in that region. This is, of course, glossing over a lot of technicalities and subtleties, but for an article of this type, that can be accepted.

Stephen Hawking, in a way, was the catalyst for my interest and love for the subject. The first scientific books that I read were the series of books about George, written by the man himself and his daughter Lucy. Soon after, I read the Grand Design and the Brief History of Time. While he may not be the greatest of his field, he is certainly one of the giants. And if we can see further than anyone before us, it is because we stand upon the shoulders of such giants. Very few have, and fewer still in the future will influence the field of arcana as he had. As he once said, “Science is not only a disciple of reason but, also, one of romance and passion. For not only does God play dice, but…… he sometimes throws them where they cannot be seen.”

On Quarks

The Standard Model of Particle Physics is the most elegant model of nature that we have currently. According to it, everything in the universe and every change in the universe is governed by 17 (confirmed) fundamental particles. Five of which are classified as bosons, and are of two types, there are 4 gauge bosons that are responsible for mediating the four fundamental forces and the one scalar boson, the Higgs Boson that is, in a way, responsible for matter having mass. According to the standard model, we feel a force when the particle mediating that force is exchanged between particles. If two electrons interact via the electromagnetic force, they exchange photons, as photons are the mediator of the electromagnetic force. The W and Z bosons are responsible for the Weak Nuclear Force, that underlies radioactivity and other phenomena, and the gluons, that mediate the Strong Nuclear Force. Gravitons have been predicted to exist (a particle that mediates the gravitational force) but not yet been confirmed.
The other 12 particles have mass and may or may not be charged. The interactions between the particles are very well represented by the famous Feynman diagrams. Feynman diagrams tell us in how many ways particles can interact by showing the incoming particles, the subsequent exchange of bosons and the outgoing particles. Based on some rules, we can tell how likely each of the interactions will be. Technically, there are infinite possible ways an interaction can happen, but some have incredibly low probabilities and can be ignored.
In this article, I will focus only on quarks, a set of 6 of the mass particles and their interaction via the gluons (responsible for the Strong Nuclear Force). The six kinds of quarks are the up, down, top, bottom, strange and charm quarks. They are termed as the flavors of quarks. Very, very creative, I know. The theory that explains this interaction is called Quantum Chromodynamics (QCD). The name is very misleading. There is nothing related to colours at this scale but due to the incredible creativity of physicists (Feynman called them as ‘idiot physicists’), a property that is called ‘colour charge,’ was introduced to explain the interactions between quarks.
Colour charge is a property that is exhibited only by quarks and gluons. It is named so because the interactions are quite analogous to how the three primary colors – red, green and blue interact. Each flavor of quark has three types and a different colour charge. Of course, their antiparticles also show colour. They are called anti-red, anti-blue, anti-green. A combination of all these three colours or the quark of a colour and the corresponding anti-color form a ‘colourless’ quark. This was needed to show how quarks could form hadrons (composite particles made up of three quarks, like protons and neutrons) and mesons (particles composed of quark-antiquark. But they do not annihilate each other as they are not a colour charge and its corresponding anti-colour charge), without violating the Pauli Exclusion Principle (which states that two or more particles with half-integral spins cannot occupy the same quantum state in a quantum system).
Like the idea of electric and magnetic field lines, there are field lines between quarks of different colours. Just like how in classical electrodynamics, one can draw field lines, we can also do so for QCD. The field lines exist between quarks of different colours. These field lines correspond to interactions that hold together quarks to form hadrons and mesons, via gluons. Quarks of different colours exchange gluons and experience the force due to colour charge. Colour charge is felt because of the exchange of ‘coloured’ gluons between the quarks.
Now, to the two most interesting and mind-blowing phenomena that occur in QCD – quark confinement and asymptotic freedom. To study the internal structure of composite particles, we bombard them with high energy particles to split them and study the resultant stream of released particles to understand what it consists of. When protons were bombarded with high energy quarks, we noticed something very weird. The binding energy of quarks inside protons was small. It makes sense to think that we could then observe free quarks right? Wrong. We have not yet isolated any sample of a free quark.
The quarks inside of a proton behaved like free particles. We know this directly from experiments. As we bombarded protons with high energy electron beams, we could clearly observe energy and momentum being imparted to the quarks. The way the quarks moved inside a proton was a lot like you would expect any free particle to move when they undergo similar collisions. The closer you push two quarks together, the weaker they are bound and the more like free particles they will behave. This is something very counter-intuitive. You would expect that when you push stuff closer, each individual particles will behave lesser and lesser like free particles. Instead, the closer you push quarks, the more like free particles they behave. This phenomenon is known as asymptotic freedom.
We can extend asymptotic freedom to get to quark confinement. In asymptotic freedom, we observe that the potential energy of a system of closely bound quarks is less, and as the quarks draw further apart, the potential energy of the system increases. The potential energy of the system is directly dependant on the distance of separation. The further two quarks are apart, the closer they appear to be bound. A very interesting thing happens at this point.
Before I go on, I need to state two things. One is Einstein’s mass-energy equivalence. Mass and energy are one and the same and they can be converted to each other. The other fact is that mass is a very stable expression of energy. Energy wants to exist in the form of mass rather than in the form of kinetic or potential energy. We observe kinetic and potential energy because their magnitudes are not enough to create any appreciable amount of mass at the classical scale. However, in the quantum limit, the energy that corresponds to a quark is not much.
Going back, as we pull two quarks further and further apart, the potential energy of the system increases. One can visualize this like a tube(called the flux tube) connecting two quarks. We will reach a point at which the potential energy between the quarks is sufficient enough to form a quark-antiquark pair. At this point, the tube snaps into two, resulting in the formation of two sets of tubes. One end of each of the flux tubes contains the original two quarks that were being pulled apart and in between the two, because of the high potential energy, a quark and an antiquark pair are formed. Now, each of the flux tube pairs contains one of the original quarks and one of the created particles.
This explains everything about why we cannot observe free quarks and why we observe only bound states of quarks. Every particle likes to be in its lowest possible energy state. So, quarks prefer to be bound, as they are more free that way. The only way to observe free quarks is to pull the quarks apart, but, we cannot do so as if we try to pull a pair of quarks apart the flux tubes split. Clearly, asymptotic freedom and quark confinement explain everything weird about quarks. If you have a more technical knowledge or are looking to understand these phenomena to a greater extent, without a great deal of math, I would highly suggest Franck Wilczek’s Nobel Prize lecture titled “Asymptotic Freedom: From Paradox to Paradigm”. It is a truly wonderful read.
On a slightly unrelated, yet cool note, I have stated that it is gluons that mediates the Strong Nuclear Force. That is not completely true. To be more precise, it is ‘glueballs’, that are responsible for the force. Glueballs are a result of gluons interacting with itself.

A Tribute To The Nobel Laureates

The Nobel Prize in Physics, 2017, was awarded to Kip Thorne, Barry Barish, and Rainer Weiss, for the discovery of Gravitational Waves. The detection of Gravitational Waves was a tremendous achievement. Gravitational Waves were theorized by Einstein, shortly after he published his papers on General Relativity. He noted that detecting these waves would be nearly impossible because of the great sensitivity that the instruments would have to possess. The instruments would have to detect shifts that were of similar length scales as the width of atomic nuclei. Personally, the fact that we can, quite literally, hear the universe, is mind-blowing.

The three Nobel Laureates were the masterminds behind doing so. They created LIGO, Laser Interferometer Gravitational-Wave Observatory. LIGO uses lasers to detect the shifts in spacetime. Gravitational Waves causes spacetime to expand in one direction and contract in the perpendicular direction, like any other transverse wave. These ripples happen in the very fabric of spacetime. So, earth, being embedded in this fabric would also contract in one direction and expand in the other.

Say you had a laser beam, that you split perpendicularly, and allowed each half-beam to reflect off mirrors and recombine, and let them be incident to a screen. Then you would observe an interference pattern. Now, say this happened while a Gravitational Wave hit you. Then, you would observe a change in the interference pattern. For a better understanding and a visual of how it works, I highly recommend the video titled ‘Gravitational Waves Hit The Late Show’ starring Brian Greene on the Late Night Show with Stephen Colbert.

The sheer importance of this discovery made itself apparent quite soon. It gave us the field of Gravitational Wave Astronomy. Astronomy is a field purely dependent on observation. The only thing that carries information to us about the universe is light. Photons, to be more precise. Initially, we could observe using only the Visible Spectrum. But as technology improved, new avenues opened up. We could “see”, and I use the term loosely, Gamma rays, Ultraviolet rays, and Microwave rays, each being immensely helpful in their own ways.

Now, the problem with light is that it can be affected by celestial bodies. The gravity due to stars and planets can change its path and increase the time taken for it to reach us. Gravitational Waves, however, are not affected by anything. They are, quite literally, the ripples in our very fabric of existence. They just cannot be stopped. This makes them the best way to detect and observe gravitationally dominated events, like the merging of Neutron Stars and Black Holes. We can watch as the fabric of spacetime comes alive and dances to the tune of a cataclysmic event.

LIGO has detected four sets of Gravitational Waves, all from the merging of Black Holes. But the interest in the fifth signal was for quite a different reason. A few seconds after the detection of these Gravitational Waves, some Gamma-Ray astronomers saw bright flashes in space, a Gamma-Ray Burst (GMB). They were not Black Holes, they were Neutron Stars. It was confirmed by later observations using Visible, Ultraviolet, and the Microwave ranges of the spectra of light. For the first time in the history of mankind, we observed the merging of Neutron Stars and studied it in every possible way.

Of course, there are thousands of people without whom LIGO would have never been made, and they deserve recognition, but, the aforementioned people deserve it the most. Gravitational Waves opens an entirely new field filled with unknown treasures and the fact that we are observing the universe using one of the fundamental forces makes it the clearest picture of the universe that we can get, with the current constraints of technology considered.

On Entanglement

Entanglement is by far one of the coolest things that can happen in the quantum realm. It defies all intuition. Before I go on to discuss what entanglement actually is, I need to clarify some ideas. One of them is locality, a cornerstone of Classical Relativity. Locality is the phenomenon that any event affects only the region near it, not everywhere instantaneously. That is, change at one region takes time to affect a region far away. This basically says that all changes, and thus information transfer take time. Einstein found out that information cannot travel faster than the speed of light. The other idea of importance here is Bohr’s view of reality. He suggested that only when you observe something, does it exist. If there is no one to observe the universe, it does not exist. It is like saying that if you look down while looking at the mirror, then your reflection and the mirror does not exist.

Einstein thought that Bohr’s view, based of Quantum Mechanics was absurd. He was of the view that the existence of an observer was irrelevant to the existence of the universe. In order to show that Quantum Mechanics was incomplete, he, along with Boris Podolsky and Nathan Rosen came up with a paper, now called EPR, in which they proved that particles could interact in ways which would allow you to measure observables more accurately than any of the uncertainty principles allow. It involved what Einstein called ‘spooky action at a distance’, which allowed information to be transmitted faster than the speed of light. This contradicted locality, and hence, Relativity too.

What EPR predicted was what we now call entanglement. Entanglement, as the name suggests, is when the properties of a pair of particles are correlated. There is a property of fundamental particles called spin (take the name with a grain of salt, as it is very misleading) which can be measured. We can force the particles to interact in a way which will cause the spins of the two particles to get correlated. So, knowing the spin of one particle will allow us to predict, with a hundred percent certainty, the spin of the associated pair. An example of this would be pair creation. All particle pairs that are created, will have their properties mutually entangled. This can be justified based on conservation laws. Since spin must be conserved, if one has a certain spin up, then the other must have a spin down with the same value. Also, charge must be conserved, so if one is positively charged, the other must be negatively charged. The knowledge of the properties of one of the particles created allows us to know all the properties of the entangled particle, without any measurement of that particle.

This seems to make sense at first glance. The problems arise at large scales. Assume an entangled pair of particles that are separated by millions of kilometers. Consider only spin of the particles. If we measure the spin of the particle close to us, then we will instantaneously know the spin of the particle that is millions of kilometers away. This clearly violated locality. Information must travel faster than the speed of light for this to be true. Hence, it also violates Relativity.

This sparked off an intense debate between the two giants of physics, Neils Bohr, and Albert Einstein. Einstein’s argument hinged on the fact that Quantum Mechanics was incomplete, there was some variable that was hidden from physicists. To resolve this debate, a physicist, Bell, came with an inequality. A mathematical statement that gave correlations between observable properties of particles. This inequality was based on ideas of Einstein. If the inequality was true, then Einstein was right. If the inequality was violated, Bohr was. The experiment was carried out by the French physicist, Alain Aspect. He used photons with entangled polarizations. His observations violated Bell’s inequality. Einstein was wrong. Quantum Mechanics was complete.

To any doubters, the experiment carried out by Alain Aspect had the accurate set-up. The readings were not very accurate. Similar experiments were set up later and carried out to a higher degree of accuracy, and in all cases, Bell’s inequality was violated. There is nothing wrong with any of the instruments used, or with the setup of the experiment, or any fault of the experimentalist. Bell’s inequality is always violated.

However, we can still save our classical ideas. I have talked about the Einstien Field Equations in previous articles. Einstein and Rosen solved those equations to get a structure that was like a bridge that could connect two distant parts of space-time. This was called an Einstein-Rosen bridge, or as it is more popularly known, a wormhole. If the entanglement between particles can be thought of as a wormhole, then the problem of information being transferred faster than the speed of light is solved, as information can travel through a short-cut. An interesting note here would be that in movies and sci-fi books, wormholes are often showed to be handle like structures. However, in this case, that cannot happen. If it is like a handle then, information would take up some time to be transmitted. However, information is received instantaneously. So, the two points at which the entangled pair of particles are, have to be equivalent. It is not a handle connecting two points, but two points glued together so that there is no handle needed to connect them. They are the same point.

All that I have stated here is work that has been done on it, but if you analyze the Quantum Theory and General Relativity at its roots, this problem is bound to arise. In the formulation of General Relativity, time and space are taken as relative, with no absolute measure of either, united together in the manifold of spacetime. Quantum Theory, however, is formulated on the assumption of absolute space and absolute time, the same principles as those in Newtonian Mechanics. In the Quantum Theory, there is no universal speed limit, but relativity has a universal speed limit, the speed of light. There is no way in which this can be avoided unless we try to formulate the Quantum Theory after including the assumption of the universal speed limit.

On Antimatter

After the second article on Black Holes, to make the idea of Quantum Field Fluctuations clearer, a discussion on antimatter is needed.

A very common misconception about antimatter is that it is made up of negative energy. It is not. It is made up of what normal particles are made up of. Normal ‘positive’ energy. What sets it apart from normal matter is that it can occupy negative energy states while normal matter cannot. To elaborate, we observe quantized energy levels because of the nature of the potential that the system in consideration is subjected to. This can be easily justified. Consider a particle in free space, subjected to no potential. A solution of the Schrodinger Wave Equation for this case gives plane waves as a solution. If you act on the wavefunction with the energy operator. it will give you the possible energy states that the particle can take. Doing this on the aforementioned solution, you get the possible energy values of that particle. You will observe that the particle in free space can take any energy, there is no restriction to the energy values. In this case, energy is not quantized.

Now, consider the particle subjected to a potential. On solving the wave equation and acting on it with the energy operator, we see that only certain possible discrete values of energy are applicable. The separation between the ladder of energy states changes based on the potential. For the infinite square well potential, we get discrete equidistant energy level. But, for a central potential (like the hydrogen atom), the harmonic oscillator potential and others, we observe discrete non-equidistant energy levels.

A historical note is needed to completely grasp why this came about. With Relativity and Quantum Mechanics being created at around the same time, it was natural to try and unite the two. This attempt at unifying the two opened an entire can of worms. We got more problems than solutions. The Schrodinger Wave Equation (SWE) is an energy equation. Einstein gave the Relativistic mass-energy equivalence. Klein and Gordon came managed to incorporate relativity into the SWE, many problems existed. Dirac proposed another equation the rectified most of these problems. Dirac’s equation was successful only electrons, particles with spin equal to one-half. His equation was very successful, he managed to predict the fine structure of hydrogen. However, the pathology of negative energy states, which was there in the Klein-Gordon equation, remained in Dirac’s equation.

Negative states are considered a pathology because they cause apparent paradoxes. In a normal ladder of energy states, to go to a lower state, energy has to be given out, and the particle will slow down at the ground state, the particle has minimum energy and slowest velocity. However, the existence of negative energy states means that the particle can go to lower states than the so-called ground state. So, now, the particle will give out energy but will gain velocity. So, a particle with lesser energy moves faster and needs to gain energy to be brought to rest. Another thing that can easily be seen is that for an electron, its antiparticle beaves a lot like a positively charged particle. So, if an electron undergoes a transition from positive to negative energy state, it will change its sign. This will violate the law of conservation of charge. For a more comprehensive understanding of these paradoxes, I would highly recommend the paper by Dirac himself, titled ‘A Theory of Protons and Electrons’. The paper is not very mathematical and is a good read.

So, if the equation works so well, making predictions, can this be ignored? Dirac thought no, and came up with an interpretation. He theorized that almost all the negative energy states were filled except for a few that are close to the ground state were filled. Any excitation would result in one of the electrons becoming observable, and a hole would exist in the sea. To ensure conservation of charge and other physical properties, the hole would behave contrary to the particle. If the negatively charged electron moved to the right, the positively charged hole would move to the left. This theory does explain some things, but the fact these electrons are ‘unobservable’ makes this idea doubtful. So, now, instead of saying the there is an invisible sea and holes, we say the for a negatively charged electron that can occupy the positive energy states, there exists a positron, a positively charged particle that occupies the negative energy states.

Perhaps, on a parting note, I should mention another fact. I will not go deep into explaining it, but will just tell it to you. If you use Feynmann diagrams, then, an antiparticle is basically a normal particle that travels back in time. It is mind-blowing, to think about antiparticles in this way, but it works. Feynman diagrams are basically squiggles that trace out the interaction between various particles. And it has been seen, mathematically, that antiparticles behave in the same way as the particle would if it were to travel back in time.

On Black Holes – II

In the previous article, I talked about the formation of Black Holes. In this one, I will talk about the Black Hole Information Loss Paradox. This is the problem that I had been working on. I will explain what most likely causes this paradox and about the most famous attempt to resolve it. I may go off on relevant tangents in the middle. You can expect a lesser number of rubber sheet analogies here.

The Black Hole Information Loss Paradox arises out of the fact that Black Holes shrink and die. Stephen W. Hawking, in a paper in the 1970’s, proved that Quantum Effects at the Event Horizon of a Black Holes cause a Black Hole to shrink. While the physical interpretation of this remains ambiguous, the mathematics he used was beautiful and the entire scientific community took it as law.

A possible physical interpretation is as follows. There exists an energy-time uncertainty relation in Quantum Mechanics. From this, it is possible to derive a result that is fundamental to Quantum Field Theory. The result we derive is that particle-antiparticle pairs can randomly pop into existence, and annihilate each other. This is really surprising, but, not a single law of physics is violated. For every amount of positive stuff created, there is an equivalent amount of negative stuff created. So, the effective amount of stuff in the universe still remains the same. This is called Quantum Field Fluctuation.

In fact, we have actually observed this effect. We have observed what we call the Casimir Effect, which is caused due to Quantum Field Fluctuations. If we place two metallic plates very close to each other, like a separation of the order of microns or so, effects of Quantum Field Fluctuations will cause plates to come together. We can explain this as follows. Keeping the plates so close together, we are effectively restricting the wavelengths of particles that can form. The smaller the wavelength of particle that forms, the greater, the energy of the created particle-antiparticle pairs. The greater the energy of the pairs, the smaller duration of time they can exist for. Outside the plate, however, there is no restriction on the possible wavelengths. So, all kinds of particle-antiparticle pairs can form and they can exist for larger time periods. So, effectively, outside the plates, there is more stuff than what can in between the plates. We can say that there is a pressure difference developed with ‘low’ pressure between the plates and ‘high’ pressure outside the plates. This pressure difference causes the plates to collide. Though the plates seem to attract each other, they are actually pushed together.

Now, the Quantum Field Fluctuations can happen anywhere and everywhere. The fluctuations happening outside and inside the Black Hole are of no consequence to us, the pairs form and subsequently annihilate each other. It all gets really interesting when it happens at the boundary of a Black Hole, at the Event Horizon. Say an arbitrary pair forms at the Event Horizon. Normally, they would just collide and annihilate themselves. However, the one the form inside the Black Hole (behind the Event Horizon), will fall into the Black Hole, and the other that forms outside the Black Hole (outside the Event Horizon), is free to escape. An observer outside the Black Hole will see only one particle that appears to have been created by the Black Hole itself (actually, the observer would see an entire stream of particles being “emitted” from the Black Hole, this is Hawking Radiation). The observer will feel that an extra amount of energy is added to the universe because the external observer has no way of knowing about the twin particle that fell into the Black Hole. The external observer feels that there is an addition of energy into the universe. However, energy just cannot be formed. So, this is accompanied by a loss in energy of the Black Hole itself. The loss in energy of the Black Hole causes it to shrink. The Black Hole continues to shrink until it disappears completely.

But, the shrinking of Black Holes presents us with a major problem. Nothing can escape a Black Hole once it has fallen inside it. Every particle inside the Black Hole is trapped. As a Black Hole shrinks due to Hawking Radiation, the particles remain trapped inside it. Once a Black Hole has evaporated completely, none of that information is available to us anymore, all that has been lost. Information appears to have been lost due to this. Information Loss violates the principle of conservation of information. This is the Black Hole Information Loss Paradox.

Information just cannot be lost. Physicists will declare you an outcast if you dare suggest that as even a remote possibility. Information must be conserved. Many attempts have been made at resolving this paradox. The most famous of which is Gerard T’Hooft’s Holographic Theory, for which the String Theoretic interpretation was provided by Leonard Susskind.

The Holographic Theory suggests that the Universe is really a 2-dimensional reality of which we are the 3-dimensional projection. For resolving the Paradox, Susskind suggests that there a Holographic Plate surrounding Black Holes. The plate is a 2-dimensional sheet, of which there exists so called ‘pixels’. Each ‘pixel’ has an area equal to the Plack Area. Each ‘pixel’ can contain only one unit (bit) of information. A ‘pixel’ is said to be saturated if light from a particle or the particle itself passes through the pixel. So, when a particle falls into the Black Hole, it has to pass through this sheet. Once it does, a copy of that information remains in that sheet but the particle itself is lost. It is like having a photocopy and losing the original document.

The other postulated resolution to the paradox is that the information is stored in a Planck-sized remnant, another suggests that the information is stored in a relatively large remnant, one suggests that information leaks out during the life of a Black Hole or just bursts out of a Black Hole in its final stages. There are many other postulates, these are just the ones that make the most sense to me. I mentioned the Holographic Theory just because it is really cool, however, I do have some reservations about it. Personally, I feel that of the possible solutions suggested, the information leak proposal and the information burst proposal make the most sense to me. They seem more intuitive than the Holographic Proposal.

On Black Holes – I

I choose to write an article on Black Holes as it was one of the first things that really interested me in the world of physics and astronomy. Moreover, the Black Hole Information Loss Paradox is something that I have been recently working on. Also, the fact that Black Holes are one of the coolest and most counter-intuitive things out there helps.

It all started with Einstein’s field equations. After all, a lot of developments in those times started with that one equation. The field equations were important because it described gravity and gravitational effects beautifully. It tells us (as so aptly put by John Wheeler) that – “matter tells space-time how to curve and the curvature of space-time tells matter how to move.”

Schwarzschild, a great physicist of that era, took the field equations and solved them considering the matter to be confined to a point. With the solution, he developed what we now call the Schwarzschild metric in his honor. A metric is basically a function that allows you to measure distances on any surface. Metrics vary from surface to surface, but, why do we need metrics? Why can we just not use a ruler and measure the length for us? This is because the very thing on which we measure distances is curved, with arbitrary bumps and valleys. It is easy to visualize. Take a rubber sheet and draw a straight line on it. Then, stretch the rubber sheet in any way. Depending on how you stretch the sheet, the length of the line will vary. The variation in the length of the line from a surface to another is encoded in the surface specific metric that you use. The metric contains the structure of the surface. So, just by analyzing the metric, one can derive a lot of crucial information about the surface which one wants to study.

Before I tell you about the Schwarzschild metric, I need to also clarify what a singularity means. In physics, a singularity is basically our math fails. The math that we have developed gives us infinities that do not correspond to any possible physical scenario. This metric gave us two singularities. Deeper analysis into this told us that one of the singularities from the metric corresponds to what we call the Event Horizon. The Event Horizon is literally “the point of no return”. We describe this as the region at which the gravitational effects are so strong that not even light escapes it.

To understand how light is affected by gravity; we need to understand how gravity affects its surroundings. Relativity tells us that we can describe these effects of gravity using a mathematical structure called space-time. Space-time is the very fabric of the universe. All kinds of motion of bodies in the universe are defined on space-time. So does light. Gravity causes space-time itself to curve. So, light traveling on space-time will end up getting deflected by gravity, as the very fabric on which we define its motion itself is curved. You can use the rubber sheet example again. If you curve the rubber sheet then the straight line becomes curved.

We realized that one of the singularities in the Schwarzschild metric corresponds to Event Horizon when we translated the Schwarzschild solution into alternate coordinates, we saw only one singularity. On analysis, we realized that it was just the Event Horizon. Now, this is all interesting, but it very un-intuitive how such objects could actually exist. The mechanism of their formation is pretty intuitive though.

Stars have a life-cycle. They are born and they can die. We need to understand this mechanism to talk about how certain types of Black Holes form. Stars are formed when a bunch of interstellar gas, mostly hydrogen (hydrogen is the simplest atom, therefore it is the most abundant matter in the universe, but how that happened needs an article all to itself). So, as the cosmic dust of hydrogen collected together, more and more matter started to coagulate. The coagulation caused a gravitational field to be established that in turn caused more cosmic dust to collect. As the dust collected two things happen, the gravitational field becomes stronger and stronger and pressure builds up at the core. Once a critical amount of pressure is reached, the nuclei of hydrogen are forced to gather to form helium. This process is called nuclear fusion. Once this happens, the star is live and is called a main-sequence star. It is now like any other star that we see.

Clearly, in the duration of the life of a star, two forces dominate. The fusion of elements at the core of the star causes energy to be liberated. This is what we feel as heat and also causes the star to be pushed outwards. This force is called Radiation Pressure. This is countered by gravity, which pulls stuff towards the core. Stars have the radii they do because only at that radius are the two forces balanced.

As the time progresses, hydrogen fuses to form helium, helium becomes Lithium and so on. This gets on until elements as heavy as iron is formed at the core. Iron cannot be fused further; it needs more energy than what the star can provide. This means that the radiation pressure will fall off. Now, clearly, gravity will start to dominate. Now an explosion will happen. The result of the explosion will be determined by the mass of the star. If it is less than the Chandrasekhar Limit (less than approximately 3.3 times the mass of our sun), then it will be either a dwarf or a neutron star (pulsars too can be formed, but they are basically a special case of neutron stars). However, these are not relevant for this article; I will probably write about them something in the future.

It all becomes interesting when a star is above the Chandrasekhar Limit. Now, something interesting happens. The radiation pressure falls once iron has been formed. The gravitational force dominates. The gravitational field is so intense that the iron core itself starts to shrink, collapsing on itself. In fact, there is nothing that can prevent the collapse, so it goes on unhindered. As it collapses, the density increase, which in turn causes the gravitational field strength to further increase. The collapse goes on until the entire mass is concentrated at the point. Now, the gravitational field at the center (what is now the singularity) is infinite. This is a Black Hole. Now, from that as the center, you can calculate a radius within which the escape velocity is greater than the speed of light. That boundary is called the Event Horizon. Once anything crosses this, it cannot escape or be observed by an external observer.

This is how a certain Black Hole is formed. Thinking of the matter being concentrated at a point is counterintuitive; it implies that the density is infinite. That implies the strength of the gravitational field too must be infinite. However, the fact that the singularity must be a point is clearly established by a piece of brilliance by Roger Penrose and Stephen Hawking, called the Penrose-Hawking singularity theorems. It states that the singularity must be a point. Another interesting thing pointed out was that singularities can never be observed, i.e., no naked singularities are observed. This is more intuitive, as the singularity has an immense effect on the neighboring fields, clearly causing the fields themselves to shield the singularity from direct observation.

Black holes can form in other ways too; these Black Holes are mainly of two types, Primordial Black Holes, and Kugelblitz. A Kugelblitz is only hypothetical, not a physical Black Hole. It is a Black Hole formed out of light. Theoretically, if one compresses a lot of light into a point, one can form a Black Hole. That is a Kugelblitz. A Primordial Black Hole, on the other hand, is slightly more complicated. It would be evidence of the inflationary model of the universe. The inflationary model of the universe talks about how the universe expanded form being a point to something of the order of light years in a time scale of the order of milliseconds. Such an expansion would have caused an uneven mass distribution of “stuff” in the universe. Some of this stuff may have been so close together that they may have collapsed onto themselves to form black Holes. But that would mean that we should be observing lots of Black Holes in the universe right?

But we do not. This is because Black Holes can shrink. The shrinking of Black Holes if due to what we call Hawking Radiation. Those Black Holes have shrunk so much that they are on the scale of nanometers as of now, therefore rendering them more or less unobservable. However, as our technology improves, so does our ability to probe the universe and hopefully detect these objects.

On Gravitational Waves

With LIGO (pioneered by the greats like Rainer Weiss, Kip Thorne, and Ronald W. P. Drever) going nuts in September 2015, and twice later, gravitational waves seem to be physical reality, not just predictions. They are significant not only because they are another confirmation of Einstein’s Theory of Gravity, but also because they are key to a new and promising method of observing the universe. They were predicted out of the fundamental field equations of gravity that were established by Einstein. Given that all my articles seem to be centered around Einstein, it seems that I am a fanboy. I do not deny that I am one, but the fact remains that a lot of developments in physics in the 20th century happened due to that man. He laid the foundations of relativity and formulated a theory of gravity that explained everything that Newton could not and more, as well as was responsible for the development of Quantum Mechanics. So really, being a fanboy is not abnormal. But I digress.

Gravitational waves are the best example (after Quantum Mechanics) of how nature is fundamentally “nutty” and “fuzzy”. The fundamental counterintuitiveness of these aspects of physics is what brings abstractness into a concrete attempt at understanding nature. The Field Equations of Gravity, as given by Einstein, tell us how spacetime and matter interact. Spacetime is the mathematical structure that underlies the relativistic description of nature. The motion of all bodies in nature is encapsulated in this fabric.

Thinking of spacetime as having three dimensions of space and one of time is not completely accurate. It is just the most accurate way that we can represent it. In reality, it is just a big mashed-up mess. We cannot describe directions without space and we use these directions themselves to describe space. We are stuck in this kind of a loop using two interdependent quantities to describe each other. However, thinking of this fabric as a rubber sheet and creating analogs is just fine. It helps make visualizations easier and develop intuition.

In the previous article, I mentioned about metrics and other aspects of the Field Equations. Metrics describe the structure of the surface of spacetime. It is basically a function that helps us define distances on a given surface. The Field Equations tell us (as so aptly stated by John Wheeler) that “spacetime tells matter how to move and matter tells spacetime how to curve”. The existence of matter causes spacetime to curve. This curvature tells matter how to move. It is this curvature that causes planets to be in the orbits that they are in. It is this curvature that tells light how to bend around heavy objects.

An interesting side note, Newton’s equations tell us that gravity can only be felt between particles with masses. So, particles that do not have any rest mass have should not ideally be affected by gravity. But we observe that light is actually curved by Stars and other bodies with high gravitational effects. Einstein’s theory tells us that light curves, not because it is affected by gravity, but because the path of light is affected by gravity. Since the path itself is curved, light too must follow a curved path. This is analogous to taking a rubber sheet and drawing a straight line on it. Consider this line to be the path of light. Now, if there is an object with significant gravity in its path, we know from the field equations that it will curve spacetime. So, in our analogy, we press down on the rubber sheet (to simulate gravity) and we can see that the straight line itself is curved. Basically, light always travels in a straight line. It always takes the shortest path between two objects, once we consider all the constraints.

Now, going back to the topic, Gravitational Waves. Gravitational Waves are, as the name suggests, are waves in spacetime itself. For such waves to be formed, an intense amount of energy has to be injected into spacetime. When enough energy is injected into this fabric, the very fabric itself starts to oscillate. Waves are generated. These waves are called gravitational waves.

The first gravitational wave was suggested by Joseph Taylor Jr. and Russell Hulse. They observed a binary system of a pulsar in an orbit around a neutron star. They observed that the orbital radius of the orbiting pulsar was reducing. This would mean the loss of energy. No energy burst was observed from the binary system, so where was this energy going? Turns out, this energy was being injected into the fabric of spacetime. It was causing space-time itself to oscillate. It can also be caused due to a binary system of Black Holes. As the Black Holes rotate around each other, their speeds go close to the speed of light. They affect the neighboring spacetime to a great extent.

Think of it like dropping a stone into a pond. You can observe ripples. Now, instead of the stone, imagine two of them rotating around each other at unimaginably fast speeds. This would generate incredibly big ripples in the pond. That is exactly what happens. The pond is like spacetime, and the stones are like the two Black Holes.

For Black Holes, it goes one step further. Gravitational Waves are generated due to their mutual rotation, but when Black Holes merge, their final mass is less than the mass of the sum of the masses of the merging Black Holes. Where does this mass go? Turns out, according to mass energy equivalence, this mass gets converted into energy and injected into spacetime. This would be seen as a final burst of gravitational waves.

Interestingly, I had a few friends who had asked me that if we consider wave-particle duality, then why do gravitational waves not imply the existence of a particle for gravity, the graviton? The answer is that gravitational waves do not correspond to any particles. This is because gravitational waves do not exist by themselves, they exist as an effect of certain phenomena. All the fundamental particles can exist as waves by themselves, they do not require any other phenomena to create it. Also, another reason why this fails is that wave particle duality is seen quantum mechanically, Gravitational Waves are classical. General Relativity, is the most accurate description of the classical world that we have till date. So, this association will not work.

The intense interest of observing gravitational waves is not because it is another confirmation of General Relativity, but because of its immense applications in astronomy and cosmology. This opens a new field of observational cosmology or astronomy called gravitational wave astronomy. It will improve our ability to observe warped space-time and definitely improve our chances of observing and understanding Black Holes.

On Time

What is time?

This is perhaps the most clichéd way of writing an article on this topic. Unfortunately, it is the best start that I could think of. Regardless, it does seem to have a simple answer at first glance. Seemingly, the intuitive answer is – time is what allows changes to happen. Or at least something along those lines. In fact, this was what everyone thought about time until one brilliant man Albert Einstein came along and introduced General Relativity (Einstein is not the only one deserving credit, there are many other people without whom relativity would have never worked out – Minkowski, Lorentz, Schwarzschild and many more). General Relativity just knocked over all understanding of space and time that we had. Einstein used beautiful mathematics to show that gravity can be described geometrically by the curvature of a surface. The surface was a union of space and time. This was the first instance that our understanding of time was questioned.

It is not surprising that our notion of time is flawed. Our intuition was developed by men throwing spears and rocks. Our idea of reality was limited to and shaped by our ability to observe. Our perception of nature was developed by classical definiteness, not quantum fuzziness. Newton, when he single-handedly developed what we now call Classical Mechanics, used the notion of absolute space and absolute time. There were these ever constant entities called space and time. His formulation of physics also did not establish an upper bound on the speed limit. Infinite speeds were allowed.

The fact that the speed of light is a constant was hinted at by many physicists. There is an urban myth that there was a physicist who liked to play about with constants. Once, as he was shuffling about with the values of the permittivity of free space (a constant used in calculations related to electrical phenomena) and the permeability of free space (the constant used in calculations related to magnetic phenomena), he got the speed of light as the result of the computation. This should not be surprising as light is an electromagnetic wave. The fact that the speed of light is a constant was firmly established by the Michelson-Morley experiment. Ironically, that experiment was developed to show that light needed a medium, hypothesized as aether, to move it. Everyone expected to observe that the speed of light would vary based on the direction in which we make the measurement. However, the speed of light was measured to be the same in all directions. This gave Einstein the very basic ingredient that he needed – confirmation the speed of light is a constant. He assumed the speed of light in vacuum to be an absolute constant, something that cannot be exceeded even relatively. He made another assumption, the laws of physics apply in the same fashion in all directions (to put it slightly technically, the invariance of the laws of physics). Just with these assumptions, he shook the world.

The brilliance of Einstein lay in his unique ability to conduct gedankenexperiments (a physicist’s fancy way of saying thought experiment). And the best way to understand why space and time are not absolute, due to these two assumptions is to conduct one gedankenexperiment. Imagine a single photon propagating unhindered through the cosmos. Now consider yourself accelerating towards the photon. As you are accelerating you start noticing weird effects. Any other photon that is traveling towards you will take longer to reach you, even though they are all from the same starting point. An interesting note on photons, they are responsible for visual imagery. So, if a photon takes longer to reach you, that means that information itself takes longer to reach you. So, the faster you travel, the slower information will reach you and this will give the appearance of time slowing down. At the speed of light, time will appear to stop. And photon emitted after you have reached the speed of light will never reach you. The only photons that you will receive are the ones that reach you the moment you start traveling at the speed of light are the ones that hit you at that instant. So, you get no new information. It appears as though time has frozen for you. (This is to put to it intuitively, however, relativity tells us that time actually does not exist when one travels at the speed of light.) interestingly, if you were allowed to break the speed of light barrier, it would appear as though you were traveling back in time as you could now reach the photons that had already passed you by.

The conclusion that we draw from this is that we experience time due to information processing. Information is bounded by the speed of light. Therefore, our speeds will determine the passage of time. This was the first time the notion of absolute time was, not only questioned, but also annihilated. There is another gedankenexperiment that we can do that will cement this idea.

Consider a single photon between two perfectly reflecting mirrors, in vacuum. Perfectly reflecting means that the photon will not be absorbed into any of them. The speed of light in vacuum is a constant. So, the time interval between two reflections can be considered to be constant. So, this allows to accurately measure time. Now, set them in motion with some arbitrary velocity (lesser than the speed of light). For an observer in the mirror, he observes no difference. For another external observer who observes the moving mirrors in his frame of reference will see that the actual path between two reflections is not superimposing straight lines, but a series of continuous triangles. The distance that the external observer observes the light to travel is much more. Therefore, the observer will conclude that time will slow down for the moving observer.

This firmly established the fact that time is not absolute. It must be relative.

There are other questions related to the time that we must consider. Ones that will have a deeper impact. Let me put forth a question so basic that at first glance it seems to be a trivial one. Why is there only one direction of time? Physics has no answer to this. If you think about it, not a single law in physics talks about time changing. They only talk about changes that occur in intervals of time. In fact, there is only one law in physics that hints at a direction of time. The much famed second law of thermodynamics. In simple words, it says that a thermodynamical quantity that we call entropy (in layman terms, entropy is a measure of disorder) always increases. It can never decrease with time. This does not tell us why entropy or disorder in nature must increase, all that we know that is just does. There is another flaw in using this to talk about only one fixed direction of time. Entropy can remain constant. When entropy remains constant, we cannot say if time is constant or if time is actually increasing.

Classically, the notion of time having one direction cannot be concluded. Does Quantum Mechanics say anything about this? Turns out, no. Quantum Mechanics itself is built on the notion of absolute time. In fact, there is another thing in Quantum Mechanics that makes us question time. Every observable quantity in Quantum Mechanics is represented by a mathematical beast called an operator. An operator is actually a simple thing. Think about the wave function describing a quantum system as a dictionary. Then, the operator corresponds to a shortcut to find the word. Applying an operator on the wave function gives us the possible values of the chosen property that the quantum system can take. For example, on applying the momentum operator on the wave function will give us information about all the possible momentum states that the quantum system can exist in. Time is very much a measurable quantity. However, there is no corresponding operator that someone has managed to develop that can give us information about time.

This is not the only problem with our idea of time. Quantum Field Theory too has no answer for us. The final answer comes from our efforts to develop a Quantum Theory of Gravity. In the fundamental Equation of Quantum Gravity, the Wheeler De-Witt equation, time does not appear! Seemingly, time does not seem to exist in the most fundamental theory of nature. Is it possible to describe a theory of nature independent of time?

A brilliant physicist, Julian Barbour did exactly this. He asked if it was possible to develop a theory of the world that did not require time as a fundamental property. And he succeeded, to a large extent. In a time independent theory of the world, with time as an emergent phenomenon, he described changes and such as configurational changes. He developed the theory of Shape Dynamics. Now, as studies have gone deeper into Quantum Gravity and Shape Dynamics, we have come to the conclusion that time is not needed to describe nature at a fundamental level. Time does not necessarily have to exist.

We have come a long way from our ideas of absolute time. We first realized that time was relative and then the time is not a necessary condition for describing nature. This has definitely failed to answer the clichéd question that I mentioned at the beginning of this article.

Clearly, we experience time. We remember the past, we form memories in the present and we mostly cannot predict the future. The answer to the question why we experience time boils down to memories. It is our ability to form and retain memories that help us distinguish the past, present and the future. Memories are what forms our notion of time. This can be made more intuitive through a gedankenexperiment.

Consider an individual who is unable to form memories, i.e., he or she has absolutely no form of a short-term memory. The only thing that an individual is able to experience is the present. It is like a video-camera that is switched on but is not recording. The person cannot make any memories, therefore, cannot be aware of the existence of a past. Clearly, that individual will have no notion of time. The individual’s entire existence will be independent of the variable – time. What we call and experience as time is clearly a psychological phenomenon.