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RWTH Aachen Particle Physics Theory

Schlagwort: ‘scicomm’

TTK Outreach: Special Relativity in a Nutshell

February 27th, 2018 | by

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).

                  Fig. 1

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.

                                        Fig. 2

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

ΔtB = ΔtAγ

where,

γ = ( 1 – u2/c)-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

January 30th, 2018 | by

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:

new doc 2018-01-28 16.38.08_1We 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.

Screen Shot 2018-01-28 at 4.45.06 PM

                                                                                      Image taken from these lecture notes.

 

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.