The Symmetry of Space and Time

You are in 9th grade, your mind completely relaxed and oblivious to the harrowing journey it is about to be taken on. Your first physics class starts, and the teacher spews some jargon. Two words catch your attention: ‘position’ and ‘time’. It turns out that for the next two years of physics, these two words become extremely common. Most physics turns out to be described in terms of position and time, and these become the two variables you find most commonly in your numerical calculations.

You are now in 11th grade. Position and time take on a whole different meaning. Position is now an abstract mathematical quantity referred to as a vector, and it is usually expressed as a function of time. For the rest of your high school career as a student interested in physics, things you had learned before take on a more rigorous form, integrating a deep mathematical structure with a physical understanding of the world. Earlier, you only thought of space as the place outside the atmosphere. Now you realise space is where your position vectors live and evolve as time goes on independently. You learn that physics is written in the language of differential equations, and you learn how to solve these equations. All of this is extremely exciting, and you can’t wait to learn more when you go off to university. If only you knew…

What you just read is not my life story. This is how millions of students have been introduced to physics, and it’s not a bad place to start. But our understanding of physics has evolved so far beyond just some vectors and functions.
The first great leap in our understanding of physics happened over a century ago, in 1905, also known to some people as Einstein’s annus mirabilis – his year of miracles. That year, he published four papers that changed the entire face of physics, but for this discussion, I am interested in only one – “On the Electrodynamics of Moving Bodies.”

(The original paper was in German, but let’s stick to English. Mein Deutsch ist nicht so gut.) This paper addressed the discrepancies in Maxwell’s calculations of the speed of light, which seemed to be frame independent and questioned our understanding of Galilean relativity and came to a revolutionary conclusion. Speed of light is invariant with the change of frame of reference. Time is not. This changed how we viewed our universe as a three-dimensional vector space parameterised by time as an independent variable; time also changed with speed, same as position! This gave rise to the almighty “Covariant Derivative”, which treated space and time on equal footing with only minor differences in constants. Our space was no longer Euclidean space; it was Minkowskian. Vector calculus was no longer enough to study physics; tensor calculus was needed.

Ten years after Einstein shook the very foundation of our understanding, he came back again with General Relativity which further solidified the views of unified space-time. Bodies of high mass and energy warping the space and time coordinates in their vicinity and mathematics so complicated that it took nearly ten years to formulate.

All this progress presented a problem. Since contemporary ways of calculating conserved quantities were no longer valid, could we consider them as conserved anymore? Enter the saviour, Emmy Noether, who proved that conserved quantities were a product of the symmetries of nature, such as momentum was conserved because of continuous space symmetry, and energy was conserved because of conserved time symmetry. As if we needed more proof of the underlying deeper meanings to space-time….

Today, these subjects are widely studied and even more widely known among physicists and other scientists alike. In fact, when the next big thing came along, aka Quantum Mechanics, physicists were not happy till they found a covariant way of doing quantum mechanics which held space and time of equivalent footing and was invariant under Lorentz transformations. Even modern theories such as String theory and Quantum Loop Gravity are often held to the test of covariance. Tensor calculus (the way Einstein re-invented it) has become the norm of writing physics expressions.

You might be thinking, what if what we’re studying right now isn’t deep enough? Maybe there is a more elegant formulation of physics? You wouldn’t be wrong since there is already some research going on in probing the base postulates of quantum mechanics and the nature of our reality. But it might be a while before we get the next big idea in physics. After all, it took nearly 250 years since Newton to get Special Relativity.

– Shobuj Paul

Header image from here.

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