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 chose 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 has interested me for ages. 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.

There is another situation of interest where Black Holes can form, from the fluctuations in the very early universe, around the Big Bang. These are called Primordial Black Holes, and are objects that are incredible hard to detect.  If such a Black Hole were to be found, it would be direct 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 (Check out On Black Holes – II for more details on this!). 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.