## Kategorie: ‘physics’

## TTK Outreach: Special Relativity in a Nutshell

Einstein’s theory of relativity has seeped into popular culture like no other. But what is relativity? And why is it important to our day-to-day life? Today, we look at Special Relativity: the imagine-the-cow-to-be-a-sphere case of the complete or general theory of relativity.

The beauty of SR and probably one of the reasons for its ubiquity in popular science is its elegance and simplicity. An added benefit is that it’s possible to go quite a-ways with an intuitive understanding of SR and no complicated mathematics. At the heart of it, special relativity has two basic principles. Once we understand these two ideas, we basically understand all of special relativity and the ‘paradoxes’ that come with it. These two ideas are as follows:

**1. The laws of physics are invariant (identical) in all inertial reference frames.**

There is just one jargon-y term here which is ‘inertial reference frames’. A reference frame is a system of coordinates that you use when you perform an experiment. This system fixes the location and orientation of your experiment. An inertial reference frame is one that is not accelerating, i.e, it is either stationary or moving with a constant velocity. So, a car going in a straight line at 50 km/h is an inertial frame of reference. So is a physicist sitting at her desk. The Falcon Heavy during its trip to outer space is not: it accelerates. Neither is the Earth.

The first principle of SR states that physics should look the same in all inertial frames. In essence, if you perform your experiment on your way to work (provided you drive at a constant speed) you’ll get the same results as when you repeat it in your lab.

This also means that there is no ‘absolute’ frame of reference. Say you perform your experiment in a bleak, windowless container. Unbeknownst to you, the container is actually on a moving belt. This moving belt is on a ship on its way to the New World. Do you consider the ship to be your reference frame? Or the belt? Or just the container? It’s kind of an inverse Russian doll situation. But we don’t care. As long as the reference frames are inertial, the physics would remain the same and we get the same results either way.

**2. The speed of light in vacuum is the same for all inertial frames**

This one is slightly tricky because it’s counterintuitive. For any object, the speed you measure depends on the reference frame you’re in. For example, you’re in a car going at 50 km/h. On the seat next to you are six boxes of pizzas. For you, the pizzas are stationary. For a hungry person at a bus stop, the pizzas are going away from them at 50 km/h. For another car which is coming towards you at 25 km/h, the pizzas are coming towards it at a speed of 25 km/h. So, in general, speed is relative. But for light, we always measure the same speed irrespective of our motion with respect to the light source.

The constancy of the speed of light gives rise to a host of interesting results. The one most used in science-fiction is time dilation. And as it turns out, it’s pretty easy to understand time dilation if you understand these two principles of SR. So, let’s give this a shot!

**Time Dilation:**

Quick side-note before we begin: In special relativity, we assume that gravity plays no role (hence equating a cow to a sphere). Here, time dilation is a result of the velocity difference between two observers. If we consider the full picture, i.e., General Relativity, time dilation can also be caused by gravity. If you’ve seen Interstellar, this is why time runs slower closer to the black hole. And also the reason that clocks in outer space will tick slower than clocks on Earth (if their relative motion to Earth is zero).

A simple thought experiment to understand time dilation is as follows. Consider two scientists Alice and Bob inside their respective spaceships. Both have light-clocks. A light-clock consists of two mirrors at opposite ends of a cylinder. One end also has a light source. The way this clock measures time is by shooting a photon from one end of the cylinder and ‘ticking’ when the photon returns to the same end.

Now back to Alice and Bob. Alice gets tired of trying to convince Bob of the superiority of Firefly and flies away in her spaceship. For Alice, one tick on her light-clock corresponds to the process depicted in Fig 1 and with some middle-school math, we can calculate the time between ticks.

For Bob staring dejectedly at Alice’s ship realizing that he was wrong, the path that the photon takes is given in Fig 2. Again, employing some simple middle-school math, we can calculate the time between ticks from Bob’s perspective.

After a bit of algebra, we find that from Bob’s perspective/frame of reference, time appears to be running slower for Alice.

Δt_{B} = Δt_{A}γ

where,

γ = ( 1 – u^{2}/c^{2 })^{-1/2} so that γ > 1

So, when Alice sends a passive-aggressive email to Bob with the *one* season of Firefly –such injustice—her clock would be a little behind Bob’s. By extension, she would’ve aged slightly less than Bob (in Bob’s frame of reference)**.

And that, in principle, is how time dilation works. Keep in mind that this is not just an abstract thought experiment. We actually sent high-precision atomic clocks on plane rides around the earth and compared their time to the ones on the ground. The lag was exactly the one given by special relativity.

Of course, you can’t mention time dilation without talking about the Twin Paradox. But this post has already exceeded its word limit. So, I’ll leave that for the next one.

**For now, we’ve chosen to completely ignore Alice’s frame of reference. If we delve deeper, we’d find that for Alice, Bob would be the one aging more slowly. This is what eventually leads to the twin paradox. More on this in the next post!

## TTK Outreach: A Universe of ~~Possibilities~~ Probabilities

~~The universe may not be full of possibilities –most of it is dark and fatal– but what it does have in abundance are probabilities. Most of us know about Newton’s three laws of motions. Especially the third which, taken out of context, apparently makes for a good argument justifying revenge. For centuries, Newton’s laws made perfect sense: an object’s position and velocity specified at a certain time gives us complete knowledge of its future position and velocity aka its trajectory. Everything was neat and simple and well-defined. So imagine our surprise when we found out that Newton’s laws, valid as they are on large scales, completely break down, on smaller ones. We cannot predict with 100% certainty the motion of an atom in the same way that we can predict the motion of a car or a rocket or a planet. And the heart of this disagreement is quantum mechanics. So today let’s talk about two of the main principles of quantum mechanics: duality and uncertainty.~~

**Duality:**

We begin with light. For a long time, no one seemed to be quite sure what light is. More specifically, we didn’t know if Light was a bunch of particles or a wave. Experiments verified both notions. We could see light interfering and diffracting much like two water waves would. At the same time, we had phenomena such as the photoelectric effect which could only be explained if Light was assumed to be made of particles. It is important to dwell on this dichotomy for a bit. Waves and particles lie on the opposite ends of a spectrum. At any given instant of time, a wave is spread out. It has a momentum, proportional to the speed with which it is traveling, but it makes no sense to talk of a definite, single position of a wave by its very definition. A particle, on the other hand, is localized. So the statement, ‘Light behaves as a wave and a particle’, is inherently non-trivial. It is equivalent to saying, ‘I love and hate pineapple on my pizza’, or ‘I love science fiction and hate Doctor Who.’

But nature is weird. And Light is both a particle and a wave, no matter how counter-intuitive this idea is to our tiny human brains. This is duality. And it doesn’t stop just at Light. In 1924, de Broglie proposed that everything exhibits a wave-like behavior. Only, as things grow bigger and bigger, their wavelengths get smaller and smaller and hence, unobservable. For instance, the wavelength of a cricket ball traveling at a speed of 50km/h is approximately 10^{-34 }m.

And it is duality which leads us directly to the second principle of quantum mechanics.

**Uncertainty:**

The idea of uncertainty, or Heisenberg’s Uncertainty principle, is simple: you can’t know the exact position and momentum of an object simultaneously. In popular science, this is often confused with something called the observer’s effect: the idea that you can’t make a measurement without disturbing the system in some unknowable way. But uncertainty is not a product of measurement, neither a limitation imposed by experimental inadequacy. It is a property of nature, derived directly from duality.

From our very small discussion about waves and particles above, we know that a wave has a definite momentum and a particle has a definite position. Let’s try to create a ‘particle’ out of a wave, or in other words, let’s try to localize a wave. It’s not that difficult actually. We take two waves of differing wavelengths (and hence differing momenta) and superimpose them. At certain places, the amplitudes of the waves would add up, and in others, they would cancel out. If we keep on adding more and more waves with slightly differing momenta, we would end up with a ‘wave-packet’, which is the closest we can get to a localized particle.

Even now, there is a small, non-zero ‘spread’ in the amplitude of the wave-packet. We can say that the ‘particle’ exists somewhere in this ‘spread’, but we can’t say exactly where. Secondly, we’ve already lost information on the exact momenta of the wave and so there is an uncertainty there as well. If we want to minimize the position uncertainty, we’d have to add more waves, implying a larger momentum uncertainty. If we want a smaller momentum uncertainty, we would need a larger wave-packet and hence automatically increase the position uncertainty. This is what Heisenberg quantified in his famous equation:

### Δx Δp ≥ h/4π

And so we come to probabilities. At micro-scales statements such as, ‘the particle is in the box’, are meaningless. What we can say is, ‘the particle has a 99% probability of being in the box’. From Newton’s deterministic universe (which is still valid at large scales) we transition to quantum mechanics’ probabilistic one where impossible sounding ideas become reality.

The Doctor once said, “The universe is big, it’s vast and complicated, and ridiculous. And sometimes, very rarely, impossible things just happen and we call them miracles.” Or you know, at small enough scales, a manifestation of quantum mechanics. And that is *fantastic.*

## TTK Outreach: A Beginner’s Guide to Dark Matter

In the post-truth society that we live in it is easy to fall down the rabbit hole of doubting every scientifically held belief. To wonder if NASA is hiding proof of intelligent extra-terrestrial life (*they’re not*), or if people at CERN are rubbing their hands plotting something nefarious (*nope*) or whether the Big Bang theory is a Big Bad Lie (*it really…isn’t*). But don’t worry, we at TTK have got you covered. Every Wednesday we answer your questions live on Twitter and every whenever-this-author-stops-procrastinating-day we give you a more elaborate explanation of some of the most frequently asked questions.

Today on the agenda: Dark Matter — what it is and why you should be reasonably sure of its existence.

Simply stated, dark matter is a kind of matter that doesn’t interact with light. This means we can’t “see” it in the conventional sense. As you would expect, this makes studying dark matter a bit difficult. But if there is one redeeming quality in humankind, it is that we don’t shy away from the seemingly impossible. Of course, the question remains, if we can’t see dark matter and if it doesn’t interact all that much with other things, how do we know that it exists in the first place? The answer comes to you in four parts.

**1. Galaxy Rotation Curves**

Some of the earliest indirect evidence of dark matter comes from galaxy rotation curves. A rotation curve is a plot of the orbital speed of stars or visible gas present in a galaxy as a function of their distance from the galactic center. If we assume that the total mass of a galaxy is only composed of normal or ‘visible’ matter, the farther we move away from the center (where most of this mass is concentrated), the lower the orbital speeds should get. This is what happens in the Solar system. Since the Sun accounts for most of the mass percentage, the planets farthest from it revolve slowly as compared to the ones close by.

However, measurements of galactic rotation curves don’t agree with this prediction at all. Instead of decreasing with distance, the orbital speeds of outlying stars appear to either stagnate to a constant value or increase. This points towards the possibility of an additional contribution to the mass of a galaxy from something we can’t see. Maybe something dark?

## 2. The Bullet Cluster

Another smoking gun for dark matter is the Bullet Cluster. It is composed of two colliding galaxy clusters, the smaller of which looks like a bullet. Galaxy clusters are a busy place and when they collide, chaos ensues. The stars, far apart as they are, mostly survive the collision without a story to tell (aka pass through). The particles present in the galactic plasma, however, smash and ricochet and radiate a lot of energy.

Galactic plasma makes up most of the baryonic (visible) mass of a cluster so we can derive a mass-profile for the cluster from this radiated energy. We can also model the mass-profile by studying the lensing effects of clusters. Because massive objects bend light, we can figure out their mass distribution by studying how they distort light from surrounding clusters. If the entire mass of a cluster is just the baryonic mass, these two mass-profiles should coincide. What we find instead, is that they are in exact opposition. In the image above, the pink regions are where the baryonic mass is present. The blue regions show where the *total* mass of each cluster is concentrated. The zero-overlap between the two implies the presence of a non-baryonic, invisible source of mass. Moreover, it purports that *most* of the mass of a cluster is non-baryonic or dark. (In fact, roughly 80% of the universe’s matter content is dark!)

Quick Side Note: Keep in mind that the colors are for purely representative purposes! The radiation emitted by the galactic plasma doesn’t fall within the visible spectrum. Similarly, the blue is where the experiments tell us dark matter is concentrated.

## 3. Large Scale Structure Formation

An interesting question to ask cosmologists is why does the universe have a structure? How do we go from a more or less homogeneous particle soup to well-defined clusters of galaxies and then even to clusters of clusters of galaxies? The simple answer to this question is fluctuations. Tiny fluctuations right after the Big Bang lead to overdensities and underdensities of matter. As the universe expands, these fluctuations also grow on account of gravity and we end up with clumps of matter which would eventually form stars, galaxies, galaxy clusters, etc. There is one small problem with this line of reasoning though. We know that the early universe was dominated by radiation (or light). And light exerts pressure. So even as the fluctuations would cause matter to clump, radiation would cause it to homogenize. In the end, the fluctuations would be nearly wiped out and we wouldn’t have the kind of structure that we see today.

Dark matter solves this problem. It is massive and it doesn’t interact with light. Formation of dark matter lumps would aid the ‘clumping’ of normal, baryonic matter and give rise to structure despite the homogenizing effect of radiation.

## 4. Cosmic Microwave Background

The CMB can be regarded as a picture of the baby universe. And though at first glance it might look like random splotches of paint, it provides deep insights into what the universe looked like billions of years ago. Any cosmological model that we create has to be in agreement with this map. By specifying initial conditions — for instance, percentage of matter, dark matter and radiation — we should end up with density fluctuations as observed here. The best model we currently have is the ΛCDM. As you might have guessed, the DM here stands for dark matter. It is only when we include dark matter in the model that our predictions line up with the data.

These are just a few of the reasons we believe that dark matter exists. And even though we haven’t detected anything like a dark matter particle (yet), everywhere we look the universe seems to suggest that it *must* be there. If you still don’t understand why you should believe in it, (and as a reward for reading these 1000+ words), here’s a (dark) analogy:

## Bouncing robots and deformed planets

On a long hiking trip we were bored on our way back. So what do you do if you are a physicist and if you are outside and have no data and nothing to look up? You bring up some Fermi-Problem to solve as a pastime. Our up-to-date problem chosen was concerning the marvelous landing of Philae on the comet 67P/Churyumov–Gerasimenko. After the first “landing” the little robot bounced back and it took it around 2 hours to touch the comet again (resulting in at least one more bounce). The question at hand is: How far up did the robot bounce? We tried to answer this question, but this lead us to some more questions with quite unintuitive results.