# General Relativity

General Relativity is perhaps one of the most enlightening theories in all of physics that reconciles many fundamental ideas about gravity, mass, and spacetime. The understanding of such a theory requires, in some sense, a certain conceptual jump which in the past has been explained through a “stretchy fabric” where mass acts as wells and other masses fall into these wells acting as a conceptual picture of general relativity. If you don’t know what I’m talking about, watch this video. This is immensely flawed in many ways. Not only does it use gravity to explain gravity but it also fails to give a strong mathematical basis to the views to really see what happens fundamentally. I am not saying this post will provide the actual mathematics of how general relativity works because that is way too complicated but it definitely give better intuition into what is actually going on. (Note: there is a decent amount of mathematical rigor in the middle but it can easily be skipped over as I do provide a conceptual note at the end that anyone can really delve into. You do not need to understand the details of the math to understand the concept).

In order to really understand the origin of many concepts of relativity, one must digest two ideas: the Copernican principle and that of metrics. The first can be demonstrated with the following image constructed by the SDSS.

The Copernican principle acknowledges the idea that, in our universe, there is no “special” observer. Discounting local observations, this principle enforces the idea that from any point in the universe, there is, in general, the same density of “stuff.” Up until modern science, this has been a matter of philosophy where first countries, then the planet, and then the solar system were seen as the center of the universe. The survey above along with many others have shown that galaxies are actually evenly distributed across our universe. Many other astronomical objects like quasars and clusters are also similarly distributed implying that our universe is, in some sense, homogeneous. Although such an observation may seem messy and sort of ill-defined, it is actual a very powerful principle that motivates the development of the next idea: metrics.

A metric is a concept that is usually overlooked by many but it is actually the center of many large fields of both mathematics like differential geometry and mathematical analysis and physics such as quantum mechanics and obviously relativity. The most common metric known to all is the Euclidean metric as shown below (simply the Pythagorean theorem).

$\textup{d}s^2=\textup{d}x^2+\textup{d}y^2$

In three dimensions, this is just slightly modified

$\textup{d}s^2=\textup{d}x^2+\textup{d}y^2+\textup{d}z^2$

Many times, people will see this written in other coordinate systems like cylindrical coordinates …

$\textup{d}s^2=\textup{d}r^2+r^2\textup{d}\theta^2+\textup{d}z^2$

or spherical coordinates …

$\textup{d}s^2=\textup{d}\rho^2+\rho^2\textup{d}\theta^2+\rho^2\textup{sin}^2\theta\textup{d}\phi^2$

Cylindrical coordinates is simply polar coordinates with one extra term for the third dimension. In order to understand conceptually spherical coordinates, imagine your position on the earth. $\inline&space;r$ represents your distance from the center, $\inline&space;\theta$ represents which side or time zone of the earth you are in like longitude, and $\inline&space;\phi$ represents your latitude. Below is a representation of different distance metrics in order to visualize of space is contorted (they are 2 dimensional examples for the sake of simplicity).

As seen, different distance metrics contort space in different manners and expanding our notion of distance to various types of metrics brings us closer to the true nature of the geometry of space and time in our universe.

So how will these two concepts be reconciled to form the fundamental ideas of general relativity? Well, the Copernican principle states that space is homogeneous so the metric must be isotropic. Notice the last one has an obvious center so it is not isotropic. In order to create an isotropic metric, the curvature of space must be the same everywhere which can be conceptually realized if one imagines our space as the surface of a four dimensional sphere. All spheres have the same curvature everywhere on the surface. Let’s see if an appropriate notion of distance for our universe can be derived from this idea.

Consider the four spatial coordinates of this four dimensional sphere $\inline&space;(x,y,z,w)$ where $\inline&space;\textup{d}s^2=\textup{d}x^2+\textup{d}y^2+\textup{d}z^2+\textup{d}w^2$. Putting this into spherical coordinates (or I guess a 4D cylinder) $\inline&space;(\rho,\theta,\phi,w)$ yields

$\textup{d}s^2=\textup{d}\rho^2+\rho^2\textup{d}\theta^2+\rho^2\textup{sin}^2\theta\textup{d}\phi^2+\textup{d}w^2$

Because the space is the surface of a sphere, we may create the constraint $\inline&space;x^2+y^2+z^2+w^2=\rho^2+w^2=R^2$ where $\inline&space;R$ is the radius of the sphere. We differentiate and rearrange as follows.

$\rho^2+w^2=R^2$

$2\rho\textup{d}\rho+2w\textup{d}w=0$

$w\textup{d}w=-\rho\textup{d}\rho$

$\textup{d}w^2=\frac{\rho^2\textup{d}\rho^2}{w^2}$

$\textup{d}w^2=\frac{\rho^2\textup{d}\rho^2}{R^2-\rho^2}$

This is substituted into the original distance equation.

$\textup{d}s^2=\textup{d}\rho^2+\rho^2\textup{d}\theta^2+\rho^2\textup{sin}^2\theta\textup{d}\phi^2+\frac{\rho^2\textup{d}\rho^2}{R^2-\rho^2}$

Grouping the first and last term then leads to some simplification of the equation.

$\textup{d}s^2=\textup{d}\rho^2\left(1+\frac{\rho^2}{R^2-\rho^2}&space;\right&space;)+\rho^2\textup{d}\theta^2+\rho^2\textup{sin}^2\theta\textup{d}\phi^2$

$\textup{d}s^2=\frac{\textup{d}\rho^2}{1-\frac{\rho^2}{R^2}}+\rho^2\textup{d}\theta^2+\rho^2\textup{sin}^2\theta\textup{d}\phi^2$

Realizing that $\inline&space;1/R^2$ is simply the Gaussian curvature $\inline&space;k$ of the four dimensional sphere and that it can be multiplied by a scalar value $\inline&space;A^2$ without removing the property of isotropy, we arrive at our final equation for the general form of an isotropic distance metric.

$\textup{d}s^2=A^2\left(\frac{\textup{d}\rho^2}{1-k\rho^2}&space;+\rho^2\textup{d}\theta^2+\rho^2\textup{sin}^2\theta\textup{d}\phi^2\right&space;)$

This is called the Robertson-Walker metric and it can tell us a lot about the cosmology of our universe. Let’s look at how varying different variables in the metric can change the space that it creates.

Imagine $\inline&space;A$ starts to grow. What does this geometrically mean? It means the space is expanding (the fact that it is squared is a matter of convention which will become more clear when we rigorously define it in another post). Everything is getting farther from everything else. In fact, in our universe, it actually is! This part of the equation represents the expansion of our universe and, as a result, should actually be written $\inline&space;A(t)$. The discovery that this part actually exists is one of the biggest breakthrough in cosmology.

Assume $\inline&space;k=0$. The four dimensional sphere would have infinite radius and would then just be a flat plane. The distance metric would simplify down to the regular Euclidean space that we know so well and there would really be nothing interesting.

Now picture a universe where $\inline&space;k>0$. The four dimensional sphere would have some finite radius and as a result, our space would start to look contorted and curved. In fact, in a universe that is essentially the surface of a sphere, two parallel lines would actually converge. It would be like an imaginary force in the fabric itself pushing everything together.

If $\inline&space;k<0$, we would actually be on the surface of a hyperbola and two parallel lines will diverge. Now the imaginary force does the opposite. Look at the following images I created to illustrate my point.

From this view, it is easy to see how changing the value of $\inline&space;k$ will change the rate at which the lines converge or diverge.

You might be thinking: we still don’t know how this has anything to do with general relativity! Well, think about what gravity does. It pushes things together similar to how the sphere with positive curvature pushes the convergent lines together. In fact, gravity is just the result of these contortions in spacetime pushing things together and apart. Curvature should really be written $\inline&space;k(m)$ because mass is what causes these contortions. One may argue that curvature should be constant from the Copernican principle so it can’t be a function of mass but this is only a global fact. Locally, homogeneity and isotropy disappear as a result of the variations in mass and curvature. The exact calculations of this curvature will be left for another post.

If you are still not convinced that the stretching and contracting of space affects the motion of objects, consider this example. If you are walking and both sides of your body experience the same space, in some sense, you walk straight forward assuming both your legs go at the same speed. However, if the space on just your right side of your body is squeezed, your right leg will move through that space a lot slower because two meters on your right side might be the equivalent of one on your left. Now you will start to turn towards the right as your right leg acts as a sort of pivot because it is moving too slow. In this way, the contorted space altered your path of motion.

If you want to know more or see where I learned this from, you can take the Cosmology course from the MIT-Harvard edX or read these lecture notes. The former should be on the “Sites” page of this site.