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.