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## 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!