Coming to grips with Special Relativity

Contrary to popular opinion, Special Relativity is not a more specialized, more involved part of General Relativity. It is the easier of the two Relativity theories, involving only thought experiments and Linear Algebra. However, despite having been exposed to ideas from this theory right from school, and also taking an advanced course (and doing well) in it, I have always felt that I don’t really understand this theory. And many people I know in grad school, those who have taken this and more advanced courses, feel the same way.

Reason for not really knowing what’s going on: Time dilation is explained by light clocks. But that’s just one kind of clock!! What if we had a different kind of clock? Would it still show that time is slowing down in a moving frame? These and other misunderstood thought experiments give one the impression that only our perception of time and length are changing. Time and length aren’t really changing. And this is despite accepting easily the two postulates of Special Relativity: that the speed of light is same in all frames, and that the laws of Physics are valid in all inertial frames.

The motivation of this article is that we need better thought experiments to understand Special Relativity. And the author, recently fueled by the brilliantly written autobiography of Einstein by Walter Isaacson, hopes to do just the same.

Length contraction

Maxwell’s laws specify the speed of light, and their formulation suggests that it should be the same in all inertial frames. Now imagine that you’re traveling in a train, and you have a 1 ft wide window. A window that is 1 ft while stationary, will appear to be 1 ft long when it is moving, if you’re moving with it. Hence, lengths don’t change while you’re in the same frame. Now if you’re observing from the platform, the light will travel across the window in time t. Similarly, if you’re inside the train, light will travel across the window in time t. All good. So what has changed? If I stand on the platform and observe the moving train, I can see that the relative velocity of the train and the light beam is low. Hence, if the window becomes shorter in the direction of motion of the light beam, all will be well. Hence, the window remains 1 ft in the frame of the moving train. But it shortens in the frame of reference of the platform.

Does this mean that there is no absolute length? There is! It is the length measured in the frame of reference of the window.

Time dilation

Now we’ll have to move perpendicular to the motion of the train. We all know the famous time clock, in which a light beam bounces off mirrors that are placed parallel to the motion of the train.

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Here’s what you need to remember: when the light beam hits the mirror, you see it, the person sitting inside the train sees it, everyone sees it at the same moment. Alright. Here we go.

The light travels a longer distance, if you’re observing from the platform. Hence, as the speed of light is the same in both frames, the person standing on the platform should see the light beam reflecting from the top mirror and arriving at the bottom mirror in t seconds, while the person sitting inside the train should see the light beam arriving at the bottom mirror in, say, t' seconds. Clearly, t>t'. However, they both see the light arrive at the bottom mirror at the same moment. There can be no discrepancy about this. It is almost like t' expanded, although remaining of the same magnitude, and became equal to t. This is what is called time dilation. For the person standing on the platform, if they were to look inside the train, they would imagine the world moving at a slower rate. Just imagine a slow motion movie running inside the train.

But has time really expanded? Have lengths really contracted? No! For the observer sitting in the train, the same length contractions and time time dilations will happen for phenomena on the platform. Basically there are two kinds of length- the length observed from the frame of the object, and the length observed from a moving frame. And the length observed from the moving frame is always shorter. The same can be said about time dilation.

Published by ayushkhaitan3437

Hello! My name is Ayush Khaitan, and I'm a graduate student in Mathematics.

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