# General Relativity

Quote

Spacetime tells matter how to move; matter tells spacetime how to curve.

— John Wheeler

## Equivalence Principle

Up to this point, we have thought of mass as a single thing - however, we can break it into two, as such:

#### Gravitational Mass

This is the magnitude of gravitational force between two objects.

#### Inertial Mass

This is an object's opposition to a change in motion. When we use equations for momentum and $F=ma$, inertial mass is the mass we are considering.

### What's the difference?

While gravitational mass and inertial mass relate to two different concepts, they are numerically equivalent and the physics with regard to each is identical - this is what the equivalence principle means. Gravitational mass and inertial mass can be thought of as one and the same, and are completely interchangeable.

### So, what's the point?

Why make the distinction if we immediately state that they're identical again? Well, it has some very interesting implications on the way we see and think about the world around us.

#### Frames of Reference

Imagine two people, Alice and Bob, standing inside of two identical rooms. Alice's is situated on Earth and is completely stationary - whereas Bob's room is inside a rocket in the middle of space. This means that currently, if both Alice and Bob tried to drop a ball in their rooms, Alice's ball would fall to the ground but Bob's ball would remain floating.

However, if Bob's rocket was accelerating forwards at exactly , where $g$ is the acceleration due to gravity on Earth (approximately ), then as per the equivalence principle, both Alice and Bob would observe the same results in their respective rooms. If they repeated the ball experiment, the two balls would take the exact same amount of time to hit the ground, appear to accelerate towards the ground at the same rate, and bounce off the ground in exactly the same way. They can be considered identical, even though Alice's ball is accelerating due to gravitational force and Bob's ball appears to be accelerating due to inertia (when the rocket accelerates forwards, the ball's inertia makes it resist that acceleration, causing it to hit the ground).

One more example would be if Bob's rocket stopped moving, and Alice's room was in freefall on Earth - like a falling lift (or "elevator" for you American oddballs). Again, Alice and Bob would observe the same physics relative to their respective frames of reference - both would experience the exact same sensation of weightlessness. If you were in a room in freefall, you would feel the exact same as an astronaut in the International Space Station (well, other than the rapidly growing terror of a crash landing)!

## Spacetime Diagrams

A spacetime diagram shows the movement of an object through space and time. It represents space, a temporal dimension, as a physical dimension by giving time its own axis in addition to the spatial axes.

For example, a spacetime diagram for 1D space (e.g. only moving horizontally through space) would look like this:

### Worldlines

If you traced out the path an object takes as it progresses through time, you would get its worldline - a line which represents how the object moves through both space and time. A vertical worldline represents a stationary object, whereas a slanted worldline would represent a moving object. A curved line would therefore represent an object with changing velocity, i.e. accelerating.

Since the speed of light would be one light-second per second, a diagonal line with gradient $1$ or $-1$ would be the worldline of the speed of light. Any gradient which falls outwith this would be faster than the speed of light, and therefore not possible - for a constant velocity from the origin, this results in an hourglass shape of accessible spacetime.