We have already explored what happens if you travel at high velocities, or more than 10% of the speed of light. (We call this relativistic speed). This is called special relativity, and can cause a whole lotta problems if you were to, say, turn around. (Read: twin paradox) This is why we have an even more mind-blowing type of relativity: the general relativity.
As mentioned in our post on the twin paradox (see link above), the gravitational force isn't a actually a force. The geometry of spacetime shapes itself according to the masses present there. Something very massive, like a black hole, will have a huge impact on how the geometry behaves, and we will therefore experience a huge gravitational force near one. (Or not, it's generally a good idea to stay away from them)
To describe this geometry, we are going to introduce something called the Schwarzschild geometry. We describe a displacement in spacetime as \(\Delta s^2\), which we call a Schwarzschild line element. Let's dive into it, shall we?
\(\Delta s^2 = \left (1 - \frac{2M}{r} \right) \Delta t^2 - \frac{\Delta r^2}{\left (1 - \frac{2M}{r} \right)} - r^2 \Delta \phi^2\)
As we can judge by the presence of an angle \(\phi\), this is written in polar coordinates. This is the displacement in the tangential direction. \(\Delta r\) is the displacement in the radial direction. \(\Delta t\) is the time measured by an observer in their own frame of reference.
This is a good time to mention local inertial frames. If you are falling freely, which means that no other force than gravity affects you, you are in a local inertial frame. In this type of frame of reference you can use Lorentz geometry, where the line element is given as
\(\Delta s^2 = \Delta t^2 - \Delta x^2 - \Delta y^2 = \Delta t^2 - \Delta r^2 - r^2 \Delta \phi^2\)
The most noticeable difference between the two types of geometries is \(\left ( 1 - \frac{2M}{r} \right)\). This is an important part of the geometry of spacetime, as the mass is shaping spacetime. Including this factor also showcases why the rules of physics sort of breaks down when you get close enough to a black hole.
So what is "close enough"? We call this the event horizon. The black hole will have a mass \(M\), and as you get to a distance \(r=2M\), the factor \(\left ( 1 - \frac{2M}{r} \right)\) will approach 0 as \(\frac{2M}{r}\) approaches 1. If we look at the Schwarzschild line element again, we notice that this is the denominator in one of the components. And what happens if we divide by 0?
...Nothing. Because you can't. It's strictly illegal, even for physicists. (Who generally have a flexible approach to math) We simply don't know how the geometry of spacetime behaves inside the event horizon!
The observant reader noticed that we mentioned something about a distance being equal to a mass. (\(r=2M\)) The less observant reader didn't, but is now equally confused. General relativity is about to become even weirder: we now measure everything in meters! The mass of the Sun? 1475 meters. The mass of the Earth? 0.044 meters. Your mass? That will be left as an exercise to the reader, but let's give you the general method first.
Lets look at some units. We know that speed is measured in \(\frac{m}{s}\). Acceleration is \(\frac{m}{s^2}\). What if we divided a speed by an acceleration?
\(\cfrac{\cfrac{m}{s}}{\cfrac{m}{s^2}} = \cfrac{s^2m}{sm} = s\)
We now have something measured in seconds! If we divided speed by speed the units would cancel each other out, and we would get a dimensionless number. Now, how about that weight in meters?
\(M_{m} = \frac{GM_{kg}}{c^2}\)
You take your mass in kilograms times the gravitational constant divided by the speed of light squared. As the whole Earth weighs a measly 0.044 meters, it's safe to say that this is an easy way to lose a lot of weight.
We also have natural units, where constants like \(c\) and \(G\) equals 1, as we are more interested in the relation between something, rather than the actual numbers. Say, if something is proportional with the square of the distance, it's not very interesting if it's the two times or three times the distance squared, as the difference will be negligible.
The next thing we need to talk about is observers. We have three types of observers: the far away observer, the shell observer and the free falling observer. The first one is theoretical, as there is nowhere you aren't completely unaffected by any forces, but we imagine this observer for theoretical reasons. The shell observer lives on "shells" around a mass with a gravitational pull. (Let's call it that for the sake of simplicity) What we mean when we say shell is actually just a fixed distance from a mass, not... an actual shell. The free falling observer is an observer who is only affected by gravity.
And this will be our tool box when we dive deeper into the general relativity!