The most basic mathematical concepts to start learning the Theory of Relativity and Spacetime
You have probably already heard about Albert Einstein and the Theory of Relativity. Quotes like “Nothing can go faster than light”, “Everything is relative” or even “Time is an illusion” are spread out all over the internet. It’s truly a marvelous theory but quite challenging to grasp what it means.
Well, where do we start? Of course, there’s a lot of necessary mathematics and complicated calculus you need to learn if you want to master the theory of relativity. To catch on the introduction to this gorgeous physics theory, we need to understand better a few mathematical concepts, and that’s what this article is supposed to do.
First of all, do you remember the cartesian plane?
There’s a lot of things we can do with these 3D and 2D cartesian planes, right? For example, we could represent the position of something as an Ordered Pair (x, y) or Ordered Triple (x, y, z). Imagining an arbitrary point in the cartesian space it’s rather easy to realize that it can move in all directions, up, down, sideways, diagonals and so on. But the thing is… What happens if we add a fourth axis? A 4D Cartesian Plane? You see, this is where things start to get weird. Primarily because there’s no way to visualize a 4D cartesian space here on the screen. Also, I don’t want to show you an image to represent a 4D space or objects because it’s a mandatory exercise to imagine it. At least, spend a few minutes trying to form a mental image to grasp the meaning. After that, search on google all about a french mathematician called Jean-Baptiste le Rond d’Alembert.
Now, let’s add a fourth variable and assign it with a letter t. It will be this weird (t, x, y, z).
You probably interpreted t as time, but a 4D space using (t, x, y, z) is pure mathematics. It can be a Euclidean space, for example.
What about using t as time? Well, this is where we draw the line to start understanding the theory of relativity.
Although Einstein came up in 1905 with his special theory of relativity, Henri Poincaré already introduced the so-called Spacetime using the fourth dimension t as time.
Ok, wait a second. I need you to truly understand that it isn’t space with time, space in time, or something else. It’s Spacetime! Just one word, unifying space and time. They cannot be defined separately from each other. Spacetime is a mathematical model to describe the Universe using a fourth-dimensional continuum.
However, it is imperative to know that in 1908, Hermann Minkowski presented a geometric interpretation for the special relativity theory. The so-called Minkowski space fuses time with those three spatial dimensions. It was crucial for Einstein and the general theory of relativity, which was published in 1915.
Now that we’re interpreting t as time in (t, x, y, z) it is possible to define an Event. One of the most important concepts to understand Spacetime. An Event is nothing more than a point in Spacetime, a coordinate of the type (t, x, y, z), therefore, some point located somewhere in three-dimensional space at some instant of time.
So, let me show you the Light Cone:
3D Spacetime diagram. The time is one of the axes, and the two others are spatial dimensions.
You don’t need to understand what is this Light Cone right now. This article is all about the most basic mathematical concepts required to learn the theory of relativity.
That red dot is an event occurring at an instant of time. It could be represented using (t, x, y, z) but remember this is a three-dimension spacetime diagram.
This is where everyone starts to be confusing because we’re used to interpreting dimensions as a spatial dimension. It’s almost like an instinct to see the image above just as a geometric object. Moreover, this is the point where you’ll start to feel smarter than before.
Dimension is defined quite informally as the minimum number of coordinates needed to specify any point within it. While a line has one dimension, a square has two, and a cube has three.
The dimension is an intrinsic property of an object. It’s independent of the dimension of the space in which the object is or can web embedded. For example, a circle has only one dimension, but you can’t draw it or describe it within a one dimension euclidean space.
But let me ask you a question. What about a system describing the velocity as a function over time?
It’s a curve with one dimension. We use a cartesian plane with two axes and variables to describe it. You might observe a triangle there and start using the Pythagorean theorem to calculate something, but the velocity is not a triangle. We are just using mathematical tools for physics. Therefore, when you see a Light Cone, it isn’t about the Universe dimensions but mathematical tools to describe physics.
Light Cone using two axes.
Light form an expanding or contracting sphere in 3D space rather than a circle in 2D but its more comfortable to visualize and grasp its meaning by reducing the spatial dimensions from three to two. The Light Cone would be a four-dimensional version of a cone whose cross-sections form 3D spheres. But a diagram with a Light Cone describes the path of a flash of light emanating from a single Event through Spacetime. It isn’t about a Universe with a cone shape!
Well, now that you understand a little better about dimensions I’ll leave you with this image for you to think and scratch your head.
