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.