# Relativity

## Relativistic mass

When Special Relativity is being introduced at school (if it is at all - the curriculum might depend on your country and it can change in time), one of the notions being discussed is so called "relativistic mass".

One of the consequences of relativity is that faster moving objects are harder to accelerate, which means that their inertia increases. And since it is being said from the beginning of the physics lessons that mass is the measure of inertia, it is tempting to try to explain this effect with an increase in mass. So, the notion of mass is being split into "rest mass" - the mass an object has at rest - and a "relativistic mass" - the mass of the object in motion, larger than the rest mass. The equations also become prettier right away, since if we denote the relativistic mass by $m$, we can always write $E = mc^2$, and momentum can be expressed using the formula known from classical physics $p = mv$ (versions with the rest mass also have an ugly square root in the denominator - we'll see it later). This is the life!

If you are following articles or discussions about relativity on the internet, you probably noticed relativistic mass being mentioned in multiple contexts. It is often used to explain the impossibility of reaching the speed of light ("because the mass would grow to infinity"), or sometimes someone will ask whether an object can become a black hole by going fast enough (it can't). The relativistic increase in mass is being treated as fact in such situations, as something certain.

Well, I'd like to disturb this state of affairs slightly with this article ;) Because, as it turns out, the notion of relativistic mass loses a lot of its appeal upon closer scrutiny. As a result, relativistic mass is rarely being used in academia and you can encounter it pretty much only at school, in discussions on the internet and in popular science publications. Let's take a closer look at the reasons behind that.

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## Part 3 - the metric

We already mentioned the notion of the magnitude of a vector, but we said nothing about what it actually is. On a plane it's easy - when we move by $v_x$ in the $x$ axis and by $v_y$ in the $y$ axis, the distance between the starting and the ending point is $\sqrt{v_x^2 + v_y^2}$ (which can be seen by drawing a right triangle and using the Pythagorean theorem - see the picture). It doesn't have to be always like that, though, and here is where the metric comes into play.

The metric is a way of generalizing the Pythagorean theorem. The coordinates don't always correspond to distances along perpendicular axes, and it is even not always possible to introduce such coordinates (but let's not get ahead of ourselves). We want then to have a way of calculating the distance between points $\Delta x^\mu$ apart, where $x^\mu$ are some unspecified coordinates.

## Part 2 - coordinates, vectors and the summation convention

The basic object in GR is the spacetime. As a mathematical object, formally it is a differential manifold, but for our purposes it is enough to consider it as a set of points called events, which can be described by coordinates. In GR, the spacetime is 4-dimensional, which means that we need 4 coordinates - one temporal and three spatial ones.

The coordinates can be denoted by pretty much anything (like $x$, $y$, $z$, $t$), but since we will refer to all four of them at multiple occasions, it will be convenient to denote them by numbers. It is pretty standard to denote time by 0, and the spatial coordinates by 1, 2 and 3. The coordinate number $\mu$ will be written like this: $x^\mu$ (attention: in this case it is not a power!). $\mu$ here is called an index (here: an upper one). By convention, if we mean one of the 4 coordinates, we use a greek letter as the index; if only the spatial ones are to be considered, we use a letter from the latin alphabet.

## Part 1 - partial derivatives

Let's remember the ordinary derivatives first. We denote a derivative of a function $f(x)$ as $f'(x)$ or $\frac{df}{dx}$. It means, basically, how fast the value of the function changes while we change the argument x. For example, when $f(x) = x^2$, then $\frac{df}{dx} = 2x$.
But what if the function depends on more than one variable? Like if we have a function $f(x,y) = x^2 + y^2$ that assigns to each point of the plane the square of its distance from the origin. How do we even define the derivative of such a function?