Fish do not swim at the speed of light, but if they did, it would be beneficial to study special relativity. They could start with consulting Einstein and Minkowski’s work!

Hermann Minkowski was a mathematician and his paper Space and Time (1908) was written a few years after Albert Einstein’s On the Electrodynamics of Moving Bodies (1905). Einstein wrote about special relativity, the science that ties space, time, mass, momentum, and energy together. He used thought experiments, which literally just means he thought about it, to suggest that time and distance were relative to the observer; the two are linked, but not combined. Imagine. A whole genre of physics started by this guy’s thoughts. There’s a funny quote from physicist Philipp Von Jolly who, in 1875, said “in this field, almost everything is already discovered, and all that remains is to fill a few unimportant holes.” Poor guy. He was so wrong. 

A classic thought experiment involves a train–one observer inside and another looking from afar. There are two lighting strikes on either side of the train, equidistant to the observer (Sasuke) outside. The speed of light, c= ~3*10^8 m/s, is constant no matter where you are and how you are moving in the Universe, and reaches Sasuke at the exact same time. Naruto is inside the train, which is moving at a speed close to c right towards one of the strikes. This means Naruto is closer to one of the strikes than the other, whose light will reach him first and cause him to perceive the lightning strikes to hit the Earth at different times. Again, observations are relative to the observer. 

While Einstein was a physicist, Minkowski was a mathematician. Minkowski combined space and time into spacetime, and used math like geometry, linear algebra, and tensor calculus to support it. His work wasn’t popular with many physicists since they saw it as a way of over-complicating special relativity–Einstein himself said that he stopped understanding special relativity once mathematicians got into it.

Minkowski didn’t care what they had to say and did it anyway. He started his paper by mentioning Newtonian mechanics (classical physics) and how the equations in Newtonian mechanics show two-fold invariance. Invariance is used to describe things (systems, objects, etc) that do not change under different operations, transformations, or perspectives. Two-fold refers to space and time. Basically, Newton’s Laws don’t change if you rotate/shift a coordinate system, begin measuring time at a different point, or observe from a moving reference frame (point in space where the observer measures) with a constant velocity. 

Hopefully it didn’t get too confusing…Main point: laws of classical physics don’t change throughout the Universe. 

Minkowski then introduced the idea of a world-point, a point in space with the coordinates (x,y,z,t). x, y, and z represent 3D space, and t is time. The collection of the world-points is the world, also known as spacetime, where we can never perceive space without time, nor time without space. 

Minkowski said that tracking world-points over a period of time creates a world-line, a path through spacetime. The world is a collection of world-lines. Calculating distance in Euclidean (“normal” 3D) space only requires the Pythagorean Theorem. Since spacetime involves time, it needs the space time interval.

I’m going to be honest with you. Things are going to get more math-y from here…it’s Minkowski we’re talking about.

We need to understand Galilean transformations to continue through his paper. They are equations used to compare time and space between two inertial frames. Inertial frames are coordinate systems with objects experiencing no forces, either not moving or at a constant velocity. Check the picture to see the Galilean transformations: 

However, when velocities travel close/at c, things start to change. Here we use Lorentz transformations, which treat space and time differently. THIS is where we can really start merging space and time. 

In these Lorentz transformations, you can see gamma, the Lorentz factor. Let’s derive it–we really don’t need high level math, just algebra and a little geometry. We can begin the derivation with a visual, drawn below. Valeria is in a train (trains are popular amongst physicists) that has a laser shooting beams towards the ceiling, which is reflected by a mirror attached to the ceiling. According to Valeria, it bounces the beam back to the floor. Caleb is watching the train zoom by, and can see the beam. From his POV though, the beam seems to be moving in a triangular shape. Observations are different once again! 

Try to use the image and see how we got our result. Use the Pythagorean Theorem to find distance for Caleb's observed time. The factor of 2 in the distance comes from the light travelling to the mirror on the ceiling and back down to the floor.

H = height of train

v = velocity

t = t sub v = Valeria’s observed time

t’ = t sub c = Caleb’s observed time

c = speed of light, approx. 2.99 * 10^8 m/s

This is about as far as I’m going to get into it. Hope you reached the end! Check out Minkowski’s paper here.

Kinda interesting, right? This has been the talk of the reef for a while glub glub

-Val