# General Relativity Primer

General Relativity really takes an entire class to do it justice. Here, we just cover some important concepts and results without really going into detail. GR is notorious for getting bogged down in notation. We’ll give you a taste of the notation, but we’ll try to avoid the bog.

## Notation

When spacetime can curve, it becomes important to define how you measure distance. For this, we use the metric:

${\displaystyle (\Delta s)^{2}=-(c\Delta t)^{2}+(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}\,\!}$

As we know from special relativity that our metric, when defined this way, is invariant under Lorentz transformations, which means that all inertial observers can agree on it, and light travels a ${\displaystyle ds^{2}=0}$ path.

Let’s pick two inertial observers. Lorentz invariance dictates that two inertial frame ${\displaystyle (x,y,z,t)}$ and ${\displaystyle (x^{\prime },y^{\prime },z^{\prime },t^{\prime })}$, with one moving with respect to the other at velocity ${\displaystyle {\hat {v}}=v{\hat {x}}}$, are related by:

${\displaystyle {\begin{matrix}x^{\prime }=\gamma (x-vt),&y^{\prime }=y,&z^{\prime }=z,&t^{\prime }=\gamma \left(t-{v \over c^{2}}x\right)\end{matrix}}\,\!}$

where ${\displaystyle \gamma \equiv {1 \over {\sqrt {1-{v^{2} \over c^{2}}}}}}$. The Lorentz transformation relating two frames of reference is straightforward, but a bit cumbersome to write for an arbitrary velocity direction. We can use matrix notation to clean things up a bit.

We can write the Lorentz transformation as a matrix acting on a vector. The Lorentz transformation will be denoted by the 2-index object ${\displaystyle \Lambda _{\,\,\,\,\nu }^{\mu }}$. The transformed four-vector is given by

${\displaystyle (x^{\prime })^{\mu }=\Lambda _{\,\,\,\,\nu }^{\mu }x^{\nu }.\,\!}$

This is just matrix multiplication where

${\displaystyle x^{\mu }=\left({\begin{array}{c}t\\x\\y\\z\end{array}}\right)\,\!}$

and, for example,

${\displaystyle \Lambda _{\,\,\,\,\nu }^{\mu }=\left({\begin{array}{cccc}\gamma &-v\gamma &0&0\\-v\gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{array}}\right).\,\!}$

In the above example, ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are indices denoting reference frames (${\displaystyle x^{\mu }}$ and ${\displaystyle x^{\nu }}$, respectively. We don’t need the primed notation, because it is redundant: ${\displaystyle (x^{\prime })^{\mu }=x^{\mu }}$. Matrices map one coordinate system to another, with the top index (${\displaystyle \mu }$ in ${\displaystyle \Lambda _{\nu }^{\mu }}$) indicating the output coordinate system and the lower index (${\displaystyle \nu }$ in ${\displaystyle \Lambda _{\nu }^{\mu }}$) indicating the input coordinate system, and according to Einstein’s notation, if you see the same index in the top and bottom (e.g. ${\displaystyle \nu }$ in ${\displaystyle \Lambda _{\nu }^{\mu }x^{\nu }}$), you sum over it and it goes away. In this way, you can know, without knowing anything else about ${\displaystyle \Lambda _{\nu }^{\mu }}$, that it will operate on ${\displaystyle x^{\nu }}$ to put it into the ${\displaystyle \mu }$ frame.

We can also express our Lorentz invariant interval in this notation, where ${\displaystyle ds^{2}=-c^{2}dt^{2}+(dx^{2}+dy^{2}+dz^{2})}$ becomes:

${\displaystyle ds^{2}=dx^{\alpha }\eta _{\alpha \beta }dx^{\beta }\,\!}$

where ${\displaystyle \eta _{\alpha \beta }}$, the metric tensor, is given by:

${\displaystyle \eta _{\alpha \beta }\equiv {\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}\,\!}$

Note how you can see, just from the Einstein summing conventions, that the result ${\displaystyle ds}$ is independent of any reference frame because it has no indices.

## Key Concepts

Here is a list of key results in GR, along with some brief discussion.

### The Equivalence Principle

Gravity is pure acceleration. This means that the inertial mass (i.e. the ${\displaystyle m}$ in ${\displaystyle F=ma}$) is the same mass that shows up in ${\displaystyle F_{g}={\frac {Gm_{1}m_{2}}{r^{2}}}}$. According to GR, this comes about because gravity is, in fact, a pure acceleration associated with the curvature of spacetime. Spacetime is curved by the presence of a mass, ${\displaystyle m_{1}}$, causing a second mass, ${\displaystyle m_{2}}$, to experience an acceleration in its vicinity.

### Energy is mass and vice versa

According to Einstein’s famous theorem, ${\displaystyle E=mc^{2}}$, stating that rest mass has an associated energy (and vice versa). More generally, ${\displaystyle E={\sqrt {m^{2}c^{2}+p^{2}c^{2}}}}$.

The major consequence of this is that energy behaves the same as mass to warp spacetime.

### Clocks run slow in the vicinity of strong gravity

And, by the equivalence principle, they also run slow when being otherwise accelerated.

### Spacetime is bendy

There are lots of weird things that can happen when you are forced to abandon Euclidean geometry. Universes can have nontrivial topologies (they can close back in on themselves, for example). Space can stretch to make it look like everything is moving away from you. Space can stretch to make it look like things are moving away from each other at faster than the speed of light. Spacetime can be curved so steeply that not even light can escape (a black hole). Waves can be excited in the fabric of spacetime, carrying off energy as an oscillation in distance/time.

These are just a few of the effects.