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Chaotic Curiosity or Curios Chaos

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  • Abhishek Tiwari

The word science comes from “scientia”, Latin word for knowledge. Webster’s dictionary defines science as “the knowledge covering the general truths of the operations of general laws, especially as obtained and tested through the scientific method and concerned with the physical world”. In a layman’s interpretation science is a systematic way to obtain the knowledge about the mystifying secrets of Mother Nature by few known facts, observations, and few approximate estimations. New theories and laws in science are proposed by using two points- 1) repetition or reproducibility of the data, and 2) computational simulation. But what happens if these fundamental rules are not followed in a system? As they say, exceptions to rules form new rules, these infringing systems fall into the category of “Chaotic systems”. Chaotic systems are those in which the distant results are practically unknowable. The Theory of Chaos rules this domain of science. Today chaos theory is a field of study in mathematics with several applications in the fields of biology, cosmology, economics, engineering, meteorology, and physics. It talks about deterministic dynamic systems which are not just highly but infinitely sensitive to the initial conditions. Under such circumstances even with a fraction of difference in the initial stages would yield into complete different outcomes every time the process is started hence defying the first basic rule in experimentation. Every technology is built with its own error, limitations and approximation, and therefore computer simulation of such a dynamic system to know about its fate at certain point in space-time would either require a huge amount of initially defined components or a long time calculations which would still differ with the actual outcome. In other words the deterministic nature of such dynamic systems may not help in determining their future. Meteorologist Edward Lorenz was the first person to talk about chaotic systems. He summarized this theory as, “when the present determines the future, but the approximate present does not approximately determines the future”. In 1972 Edward Lorenz wrote a paper titled as “Predictability: Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” on the theory of Chaos. Edward Lorenz accidentally came across this topic while doing his work on weather forecast. Using a simple computer of his time, he was running a weather simulation. However due to time constraints he started his work mid way and to his astonishment he got a completely diverted results from the previous output. He concluded that the difference was generated due to the rounding-off of the numbers in the input. The consensus of the difference should practically have no effect on the immediate prediction but in a long run produced a significant difference.

The word chaos is generally taken as complete disarray is a misnomer for this theory. The chaotic systems are not cluttered but are disciplined to an extent. Like we know for sure that the moon would not collide with earth in few weeks but the prediction remains uncertain for a longer time period. Similarly weather forecasts are known most accurately about a week before. Therefore chaotic systems are predictable for a while but later on with time it becomes random. The effectiveness of our prediction can be determined by following factors:-

  • The extent of randomness acceptable
  • The accuracy in defining the present state of the system
  • The relative time scale (called Lyapunov time).

In the hunt for defining the accurate state of the system the information increases tremendously and the uncertainty in the forecast increases exponentially over the time, thereby rendering distant forecasts meaningless. The state of the system is defined by a set number of points which tends to be accurate in describing it, but each one of it having a different fate over long duration. Thus, for a reason, chaotic systems are practically not possible in Euclidian Geometry. In 1982, Benoit Mandelbrot Published “The Fractal Geometry of Nature ” in which he argues that a twine ball when observed from a far distance looks like a point, from fair distance like a ball and from close like a curved strand. He argued that the appearance of an object is dependent upon the observer and they can be fractional. The book became one of the classics for the “chaologists” - as some of them called themselves. A double rod pendulum is the easiest example of a chaotic system. It follows the general rule and also oscillates almost in the same pattern as the pendulum but with time there arises difference of fractals which causes a large deviation. The slight initial conditions causes complete different trajectory.

The Theory of chaos very soon became a law governing many interdisciplinary subjects. The field of biology has great implications as the understanding of the nervous system, and the circulatory system, are proved to fit a fractal model. Biological evolution, eye tracking disorder in schizophrenics can also be explained in mathematical form using the chaos model. The natural phenomena like earthquakes, its intensities and its distribution became easier to explain with the model of fractals. More researches can be done on such knowledge to explore the darkest realms of reality.

Another such phenomena which is not completely explored by the present knowledge of science due to some limits is the Theory of Black Hole. Elbert Einstein proposed the existence of such phenomena years before astronomers actually spotted one. According to Einstein one of the fate of a giant size star is that when its nuclear fuel exhausts the force of gravity starts pulling in the mass around it. The star becomes as tiny as possible and the star becomes highly dense. When the size of the star reduces the gravitational pull increases square times and when the size decreases to such an extent that to escape from the field of gravitational influence one would need to attain the speed of light; a black hole is born. The radius at such stage is called as the event horizon. The point when black hole is formed, scientist believes that at the centre the space-time phenomenon is destroyed. It is also called the ideal black body, as it reflects no light.

The idea of such a massive body from which no matter can escape was put forth by John Michell in 1783. Earlier in times, such massive objects were not seen of any importance or rather a fancy sci-fi stuffs because scientists failed to understand how gravity would affect a mass-less wave such as light. All the black holes have mass, some have angular momentum and also electric charge. Black holes are classified on the basis of their masses and independent of the other two properties. They are, namely, supermassive black holes (~400A.U), intermediate mass black hole (~1000km), stellar black hole (~30km), and micro black hole (~0.1mm). The size of the black hole is determined by the radius of the event horizon. A particle can move randomly in space when it is away from the black hole. As the particle nears the black hole more paths lead it towards event horizon and less paths away from it. Once inside event horizon the particle has only one path and that is towards the centre where the mass is concentrated. It is then no longer possible for that particle to escape. Event horizon gets its name because any event occurring inside that boundary cannot reach outside observer by any means thus making one impaired to notice if such event took place. Another phenomena that takes place to an object nearing the black hole is that the time slows down. To an observer falling into the black hole the clock ticks normally but to a distant observer the time difference is significant. This phenomena is called Gravitational time dilation and was first proposed by Elbert Einstein in his theory of Relativity. Consequently experiments were carried out and it was found that atomic clocks placed at different altitudes that caused them to experience different gravitational pull differed in nanoseconds. The difference in the time between a clock on the surface of the earth and the satellite revolving around it is significantly different making it compulsion to correct the global positioning system’s clock after equal intervals. The Gravitational Time dilation was wonderfully showed in the 2014 Hollywood blockbuster Interstellar.

With the clock slowing down from a free falling object into the black hole the light emitted or reflected by this object changes to red. This is known as the gravitational red shift or Einstein’s shift. This primarily happens because the frequency of the light wave is slowed down by the gravitational pull of the black hole changing the color to red. Red shift is denoted by dimensionless variable (z).

,

Where, λo stands for wavelength of the electromagnetic radiation recorded by the observer, λe stands for the wavelength of the radiation at the source of emission.

Thus red shift can be defined as the fractional change in the wavelength of the emitted radiation.

There is a corresponding phenomenon to red shift known as blue shift and it occurs when some object is pulled towards a stronger gravitational pull from an area of weak gravity. However the observer free falling into the black hole does not realize these changes. Basically this means that the observer does not realize when he entered the event horizon. As per one theory when the person reaches the point of singularity i.e. the centre of the black hole where the space-time graph extends to infinity, they are crushed to infinite density and its mass is added to the mass of the black hole. Just before reaching the point of singularity the object is torn apart by the tidal forces which is often called as “spaghettification”. However this can be avoided in a charged black hole or reissner-nordstrom black hole. It is hypothetically said that the black hole takes the object to another space-time by acting as a wormhole. Since no information can be transmitted out of the event horizon, knowing exactly of what takes place inside is impossible and hence the mystery is still unsolved

The gas falling into a gravitational well will form a disc like structure due to the conservation of angular momentum. Friction caused within this formation should heat up the gases and they would start radiating x-rays. This is known to happen around neutron stars and white dwarfs. It is called accretion of mass. It is the most efficient way of producing large amounts of energy still known about 40% as compared to nuclear fusion generating 0.7% of mass as energy. It is now accepted that the centre of every galaxy contains a supermassive black hole. The evidence of this can be the motion of stars around the centre of any galaxy. Astronomers are studying a region call Sagittarius A from the year 1995. They found that there is about 4.3 million solar mass contained in a radius of less than 0.002 light years. This mass-distance ratio is 3000 times more than the mass-distance ratio of event horizon. It is therefore conclusive that the centre homes a supermassive black hole and nothing else like a star cluster. Stephen Hawking, in 1974, proposed that the black holes might release some radiation. These radiations are hypothetical and are called Hawking radiations. It is suggested that if black holes emit radiations they may shrink. To this, Hawking suggested that the microwave radiation entering the black hole has far more energy in comparison to the energy emitted. In 1970’s he proposed that the total area of a black hole can never decrease even if two of them collide and merge. This became the second law of black hole mechanics and has striking similarity to second law of thermodynamics which states that the entropy of universe continues to increase. As per the old thinking it was considered that the black holes having absolute zero temperature had zero entropy. If this was the case then the second law of thermodynamics would be violated and the entropy of the universe would decrease. Therefore, Bekenstein along with Hawking formulated the entropy of a black hole as follows:-

Where,

Entropy = S; speed of light = c; Area of black hole = A ; Newton’s constant = G; Boltzmann constant = k; and reduced Plank constant = H.

One striking thing to be noticed in the equation is that the entropy is related to the area of the black hole and not its volume which is rather the case in normal entropy calculation.

With many such theories put forth by the great minds the human knowledge has greatly increased. Not just that the knowledge but also the interpretation of the universe around us has greatly changed. From discovering the way to measure time to discussing about its deviation with gravity and from the time when we placed earth in the centre of universe till proving the big bang theory not just the technology has evolved but also the way of living life on the planet and knowing how greatly we affect the surroundings has been brought to daily prospects.

Can this evolution or the progress of human mind be infinite? Will someday religion be taken over by pure and fundamental science? Will human evolve as the creator and destroyer of the universe? What will happen then? Among these few questions are unknown and few are unknowable which will be revealed in the course of time. But one thing is assured that the curiosity in the human brain will keep driving it to discover the perplexity of the world around him. This will eventually help science to progress in every possible dimension.

The counter side to the above promises is far bigger, which justifies the fact that humans have to face limitations, some created by their own laws and some by the nature. Everything that humans make has its own errors, and it is well known that nature does not work on approximations. The point where humans cannot visualize and the power to understand gives up is the region where religion rules. May be one day science will enable mankind to have super powers or may be one day humans will succumb to the super powers of the Mother Nature.


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