Showing posts from April, 2015

energy of photons in FLRW universe

Note that for photons , unlike particles for which
we know that , , where is an affine parameter. Also, . And calling

say implying that
and let’s call

differentiating this equation, we get

we need to solve this perturbatively i.e where Written with StackEdit

momentum of particles in FLRW universe

The expression for the line element in a FLRW universe is
The geodesic equation, used to describe the kinematics of particles or photons in the manifold is
Let’s look at specifically,
say and are zeroAnd if you compute all of the relevant christoffel symbols i.e s, you will see that is the only non-zero component.

and since we know that

If we substitute the value of , we get
Now, we also know that for particles.

Substituting the above expression in the expression below, we get


we wrote earlier that

where represents the spatial components.

where .Note that we are working in units of . If I’m missing factors anywhere, I apologize.We basically get that the spatial momentum of any free particle in an FLRW universe is inversely proportional to the scale factor and since is always positive and increasing, the momentum keeps decreasing with time.Written with StackEdit.

Seminars that go over my head and computational power

Well, I sat through a seminar during which a post-doc was talking about how many body systems can be understood using numerical simulations of their behavior, using density matrix formalism or the wavefunction formalism. He then went on to talk about how they studied various theoretical and (very) real systems and showed how their properties arise from ab-initio calculations. Although much of the talk went over my head, the one thing that was running through my mind was how I was taught the Hartree-Fock method in my second year, as part of a course on physical chemistry, and how it's seemingly invincible power went right over my head.

This is pretty much what I've been doing most of the summer. I am doing a course on numerical methods in programming and apart from the examples the prof mentions in classes and in our programming lab session, I've been trying to come up with or formulate computational problems related to physic topics that I've been learning over the la…

General relativity

The day was spent rewriting notes of the course on General Relativity. The professor has structured the course to take advantage of the assignments, teaching the necessary concepts and putting them to good use soon after by working out some problems. I wanted to keep a copy of the assignment solutions so I can refer to them during the exam but I thought I'd go one step further and fill all the gaps between the assignment sheets by rewriting the notes. The professor also leaves behind incomplete derivations for us to complete so it was good practice overall.

One other thing that came to my mind during the day was regarding Pocket, the offline reading app. I love it and I've been religiously using it for about 4 months now. Pocket tags article as 'Best Of' or 'Long Reads', and I'm guessing it creates the former label depending on how many people star the article in their library. Also, at the end of the year, Pocket tells you how many articles you've rea…

An eventful start to the weekend

Compared to the rest of the saturdays I've spent so far, I'd have to say that today was the most eventful. From 2 till ~4:30, I was at a panel discussion on research in basic science in India, with panellists mostly from the physics community, with the exception of one professor from the EE dept. Some really interesting points were raised on why Indian research isn't on par with the research output of western universities, how this can be tackled and overcome, how research in basic sciences can help the society and the country as a whole and the measures one needs to take to increase the research output of Indian institutes. With veterans and young researchers on the panel, it was a lively talk with a good amount of cross talk, a lot of anecdotes and lots of humour. I wanted to ask the panellists what role science popularization plays in increasing public and government interest in science as a whole and in drawing more students towards basic research in the sciences. I a…

Root finding, interpolation and python

I prototyped the codes to find the root of a function, using the bisection, newton-rhapson and secant methods. And then i tried writing a code to do polynomial interpolation. Both of these codes need refinement, the latter more than the former. But i finally wrote code, after two days of putting it off and doing other chores instead of this. And the reason i finally did so was, partially, python. Well, apart from my mood being better today than in the days before, I think trying to write the code in python is what kept me going, instead of say c or fortran.

The reason why I prefer python and ask beginners to use it is because of how easy it is to write a prototype in python. The number of steps from idea to a model are the least in python, in comparison to c or fortran. Relaxed syntax constraints, dynamic type allocation, variable initialization. I don't know which of these, and many other such differences, is the reason why i prefer python over C. Maybe I have an intrinsic bias …

Motion in electric and magnetic fields

I spent about an hour writing code that computes the trajectory of a particle in constant electric or constant magnetic fields, the first of which is available here and the second here. I apologize for the incomplete code that computes the motion of a particle in a magnetic field, I am still trying to figure out how to plot two different data sets with different x and y ranges on the same figure. There are convenience functions to duplicate the x axis or the y axis but not both, which is what I'm looking for.

Either way, the codes are still rough, in the sense that I used euler's method to do solve the differential equation that governs the motion, which is known to be errenous. I will have to implement one of the Runge-Kutta methods if I want better accuracy in position and velocity measurements. Moving further, it's a bit harder to solve the case of a particle in a magnetic field as there are cross terms i.e the force in the x direction depends on the velocity in the y …

How quantum physics democreatized music

Today was a wonder colloquium by Prof. Sir Michael Berry on the effect that science and research have on our day to day lives, on technology and on science as a whole.

He started talking about how CD players have democratized music and made it available to the millions. Sure, gramophone had been invented and the radio was also prevalent. But only with the advent of compact disc players did people get a portable device to listen to music. He then talked about lasers, without which there wouldn't be any CDs and digging deeper, quantum physics, without which we wouldn't have the theory of how lasers work. He talked about how Einstein couldn't have imagined that his theory explaining atomic transition using spontaneous absorption and spontaneous and stimulated emission could've led to the laser. And not just that, he gave many accounts of scientists who pursued research without worrying about applying them to *real world problems*, who would be surprised knowing how much …