Mathematics: God's gift to physicists!

It has taken me some time, but I have realized now what most physicists do at some point in their life: you can not possibly become a functioning physicist without having a strong hold on mathematics. Yes, experimental and theoretical alike! One might think that understanding the laws of physics must serve one well, but while that is the necessary condition it surely isn't a sufficient one (Pun #1). While understanding physics well itself requires a lot of mathematics, researching in physics is a whole another ball game. I say this because when you are involved in researching you attempt to expand the boundaries of our knowledge, and that, at least for a theorist, requires extreme mathematical rigor. Why? Because mathematics is the only tool known to mankind that one can use to justify one's hypothesis. You make assumptions based on the facts you already know (or believe) are true and based on those assumptions you derive the mathematical form of your idea. And it doesn't end there, your theory isn't proved until it can be proven experimentally, ask Peter Higgs! I had started this post to express how important mathematics is for a physicist, and have drifted to telling you about the plight of a theorist. What a subtle distraction indeed.

While physics answers the 'why' behind a physical phenomenon, mathematics answers the 'why' behind the formulation involved in answering the former 'why'. For example, while you might know about how quantum mechanics involves Hilbert spaces and Dirac notation and is able to precisely explain the quantum world, functional analysis tells you why use Hilbert spaces, among all others, in the first place. Mathematics is like a tool box a physicist must have, and knowing more doesn't hurt because you never know what tool may be required to solve what problem. When I was in secondary school, I always felt that matrices were a lame topic. You arrange numbers in a definite form to ease calculations. Only in college did I understand the vastness of the field. Think of it, matrix algebra, linear algebra to be general, is one of the most important topics for a physicist as all the basic concepts and subjects one learns involves these bad boys. 

And complex numbers, don't get me started on complex numbers. I can say this without even an iota of doubt that they are the most beautiful part of mathematics I have yet encountered (Pun #2). If you think about it, it is a very simple idea: what if we took the square root of -1? And there you have it, i, a letter that changed how people looked at the world. Moving from imaginary to some real stuff, no matter how much we get accused of being day dreamers, physicists work towards explaining things that are real. While we might get rigorous with complex numbers, at the end of the day, real numbers are what we return to. This itself makes it very important for a physicist to understand real numbers, and real analysis sure helps a lot in it.

I know what you are thinking, three paragraphs in and I haven't mentioned calculus! Well, do I need to? Calculus to me feels like a mathematicians' version of a theory of everything. Before you say it, I know calculus can not be used everywhere, but I say this because of the immense applicability of the subject. Well, I am sure Leibniz and Newton will be proud. 

Sometimes people claim that physics is just applied mathematics. While this may sound tempting to mathematicians, I will have to side with the physicists here. I know that a mathematician can very well derive the same laws that a physicist did and in probably a more elegant manner, but a mathematician might not be able to appreciate the meaning behind that law like a physicist can. The difference in what they appreciate is what differentiates between the two most amazing people in the word: the mathematician and the physicist.

Quantum Mechanics- First Impressions!

You know when you've heard about something being extremely mysterious, mind-boggling and makes people's world turn round, but you've never had the chance of getting introduced to it? That is what Quantum Mechanics was for me, until this January. I had read people say that if you do understand Quantum Mechanics, you don't know it well enough. Such a claim makes a curious mind anxious and excited at the same time. So, without anymore build up, let me tell you about my first impression of Quantum Mechanics after one month of study.

When one is introduced to QM, it does not take one much long to realize that QM is not a theory that illustrates human incompetency during experiments, but actually, one that sheds light on the 'spooky' way in which nature works at the smallest level, namely, the quantum level. What attracts some and repels others is the fact that initially, QM challenges intuition. But if you are exposed to it for long enough and in the right way, it redefines intuition and the way you see the world. Classically, we have been taught about discrete systems with definite states and properties. So, classical mechanics talks about certainty and some might stretch the laws to even talk about determinacy. Then comes QM which talks about probabilities and uncertainties. Not only that, the Heisenberg's uncertainty principle states that uncertainty is embedded in the working of the universe. In my opinion, this leap from classical to quantum in one's intuition is the difficult part when it comes to mastering QM. 

QM owes its success to its mathematical formulation. We can mathematically calculate the outcome of an experiment without even conducting it. And when we do end up conducting the experiment, the theoretical and the experimental outcomes are in complete agreement. Whenever this happens, it lightens up the day of a physicist. Knowing this, in my opinion, one needs to realize that the best way to understand QM is to approach it mathematically and master the mathematics that makes up the theory. Once you have accomplished that mighty task, you can begin with developing the intuition for QM such that the much talked about 'spooky' action now becomes 'expected'. 

At present, I am only in the beginning stages of exploring this vast field. I started by learning about vector spaces, Dirac notation including the ket and the bra spaces, operators and their eigenvectors and eigenvalues, and their action on kets resulting in quantum states. So, you see, I really haven't done much. But the much I have done makes me excited about what is going to come and what more can be done using QM. I am curious to know how the basic principles will give rise to mysterious concepts of quantum entanglement, etc. and how and where I will be able to use my knowledge in this field.

I read about Einstein's trouble with QM and justifiably so, given the fact that QM and General Relativity don't go hand-in-hand. This is a major problem in Physics, a problem I am already interested in. It does create a sense of zeal when trying to connect a theory that works at the largest scale to one that works at the smallest. This reminds me, I am also taking a course in Theory of Relativity this semester which aims to make me proficient in Special Relativity and its consequences, and also introduce me to the formulation and aspects of General Relativity. But, talking about that is a whole other post!

Hi there!

Hi. My name is Ish Mohan Gupta. I am 19 years old and currently pursue my Masters in Physics at BITS Pilani KK Birla Goa Campus, India. My current interests lead to me desire a career as a Theoretical Physicist, in service to the field which in my opinion is the most awe-inspiring and rewarding.

As a 10 year-old, I was fascinated with planets and stars, who isn't, but to become a physicist was never an ambition. My aspirations varied from being an astronaut to a cricketer. But, later in life, by the time I was 15 years old I began questioning things I didn't understand. Well, to be honest, I even questioned the validity of things I did understand. In short, I didn't take things to be for granted. I wanted to confirm each phenomenon that interested me by myself.

In the way, I got introduced to the genius of Albert Einstein who somehow made me believe, even if that wasn't his intention, that if understanding the world is not simple enough, then it is just not understood well enough. So, whenever I read about a theory and understood all that I could given my knowledge as a mid-teen, I would question its complexity while also admiring the inherent beauty it had.  But, some of the unexplained phenomena and conjectures around them made me ask myself that why does something have to be this difficult to explain. which lead me to formulate my own theories.

If you view all the posts in this blog, from the earliest ones to the latest ones, you might appreciate how I have grown as a Physicist (if I am at all allowed to call myself that). While the earliest ones seem extremely childish and far from being scientific, the latest ones I am proud of. It took me time to accept the role of mathematical rigor in Physics and as you will observe that somewhere down the line, I have indulged in that too. This journey, though of merely 3-4 years now has made me embrace the bond Physics and Mathematics share which has largely contributed to my frequent adventures in Mathematics.

This blog does not only contain my theories, but also my views on everyday Physics, which includes current advancements, opinions about different theories, my experience studying different subjects in Physics as I complete the remaining years of my education, and anything else worth sharing. I might also talk about Mathematics, though seldom, as I attempt to explore the seemingly abstract field.

Finally, I often ponder over what I want in life? Is it to win a Nobel Prize? Is it to become a genius? I think genius is nature-given, one is born with it, it is not really in our hands. What is in our hands is mastery. Hence, I aim to gain mastery over my field, to know anything and everything that the field has to offer. While that might seem vague, abstract and unattainable, you must remind yourself that I am a Physicist, what else did you expect? ;-)

Dynamic Viscosity of Spacetime

Okay, so, time for a new idea! :-) A few days ago, while pondering over a biological experiment, a friend of mine mentioned fluid dynamics, particularly viscosity. For those of you are are not familiar with the term, viscosity is the property of a fluid to resist motion between its layers. It can be understood bluntly to be like the friction between two surfaces. Interesting property indeed!

Now flashback to about four months ago, where I was trying to figure out how much force will a body need to be applied to, in order to reach a certain acceleration, in a relativistic world. The formula I could derive was very different of what we know of through the Newtonian laws of Physics. The interesting thing, though, was the fact that force required in order to reach a certain acceleration in a relativistic world is always greater than that required in a classical world.

Thus, this difference in magnitudes of forces in relativistic world and classical world can be seen as a resistance that a body has to overcome in the relativistic world in order to reach a certain acceleration, while no such resistance is faced if we go by the Newtonian laws. Moreover, this resistance increases as our speed increases.

Here, γ = 1/sqrt(1-v²/c²) and is called the Lorentz factor, m is the absolute mass or rest mass of the body, a is the constant acceleration it possesses, v is its instantaneous speed, subscript c refers to classical and subscript R refers to relativistic, P refers to momentum and F refers to force.

This gave me an idea. I had read that research is going on in the area where they consider spacetime to be a superfluid, i.e. a fluid without viscosity. What if spacetime is a viscous fluid, because if it is, then I can easily relate Stoke's Law of Fluid Mechanics to my drag force! But, wait a minute, what is Stoke's Law?

Stoke's Law in fluid mechanics gives the formula for the force that will be experienced by a spherical object of radius R when it tries to move forward in a viscous liquid with a speed v. They call this resistive force as drag force. So, my idea here is to equate this drag force with the difference between relativistic force and classical force in order to find µ , i.e. the dynamic viscosity of spacetime, or in order to make you understand, the measure of stiffness of spacetime.

Here, E=mc^2. We all know where that came from! ;-) So, if my calculations are right (believe me they are, I rechecked like 20 times), then that is the formula for the dynamic viscosity of spacetime. But what is the significance of this dynamic viscosity? 

Well, it is not difficult to imagine that if the density of the fluid increases, its viscosity or stiffness will increase. Let us go the other way round, if viscosity if increasing, and no ther factor is affected, then density of the fluid must be increasing. Hence, as a body moves with faster velocity, viscosity of spacetime around it increases and so does the density. Spacetime becoming denser, if you visualize, is similar to warping of spacetime around the body, which is rightfully predicted by General Relativity. So, the theory explains why the spacetime is getting warped around a moving body! 

You would say that its relativistic mass is increasing as its speed increases, so the warping. But tell me this then, why is the mass increasing in the first place? I think that due to the movement of the body, viscosity of spacetime around it increases, resulting in denser spacetime, which indirectly means warping of spacetime around the body, which results in its increased gravitational effect! Well, something to think about! :-)

Report on Mathematical Explanations of the Spring Theory

The theory proposes that the spacetime fabric has spring-like characteristics and examines the outcomes of the assumption. The spacetime fabric’s spring-like nature results in it executing Simple Harmonic Motion (SHM) just like a compressed spring. Mathematically,

Symbol	For SHM by spacetime fabric	For SHM by particle attached to the spring
x	Present radius of observable universe	Position of particle attached to spring
A	Maximum radius that the observable universe can attain	Amplitude or Maximum displacement of the particle about mean point
ω	2*π*f (where f=frequency)	2*π*f (where f=frequency)
t	Age of the observable universe	The time that has passed since start of the SHM
φ	Phase difference	Phase difference

              

                             

·        Outcomes-

• Big Bang and the fate of the Universe- The hypothesis explains how initially the entire universe was compressed into a small particle (singularity) just like a compressed spring. After the bang, the space time fabric expanded with increasing acceleration in all directions and it still is expanding. Once the natural length of the fabric is reached, it will still expand but with decreasing acceleration, eventually coming to a stop with radius as A (defined above) same as the maximum displacement of a particle at the end of a spring. From there, the fabric will begin deceleration finally converging back to a singularity creating the same conditions as were before the Big Bang.
• Dark Energy- Just like the potential energy stored in a spring, dark energy might be nothing else but just the potential energy stored in the spacetime fabric due to its spring-like nature.
• On solving the given equations,

1. ω = 2.16 x 10^(-78) rad/s
2. A = 5.36 x 10^(86) m = 5.65 x 10^(70) light years
3. T = 9.29 x 10^(70) years

Results-

If the theory holds true,

• The maximum radius of the observable universe is 5.65 x 10^(70) light years.
• Total life of the observable universe is 9.29 x 10^(70) years.