Numerical Relativity: Black Holes and Gravity - Part 3

 By: Unknown    6:47 PM  Christoffel symbols, deviation equation, Eddington-Finkelstein coordinates, general relativity, gravitational physics, metric, Minkowsky metric, Schwarzschild coordinates, spacetime, special relativity, tensors   No comments

Spacetime

The definition of spacetime comes from special relativity. It tells us that different inertial frames have different notions of time. Let's take three Cartesian coordinates (x, y, z). By moving these axes parallel to themselves from some origin, we create new values of x, y, and z. In addition, by moving these coordinates we are creating a notion of time, which is different in each of these inertial frames. This means that inertial frames are spanned by four coordinates (t, x, y, z).

Moving reference frames. Graphic from www.cv.nrao.edu.

Flat Spacetime and the Metric

Just like in geometry where we use the Pythagorean theorem, $  a^{2} + b^{2} =  c^{2} $ relativity uses something somewhat similar called a line element of flat spacetime $$  (\Delta s)^{2} = -(c\Delta t)^{2} + (\Delta x)^{2} + (\Delta y)^{2} + (\Delta z)^{2} $$ A line element, such as the one above, tells us the change in position vectors as time also changes. Combining these changes gives us the distance between points in spacetime - denoted by $ \Delta s $ . In fact, this is the one used in special relativity because it gives us flat spacetime (which is not true in general relativity). Since we have four coordinates that give us a quantity and a direction, they can be written as vectors and the  vectors can be written using index notation as shown below $$ \textbf{a} \cdotp \textbf{b} = (a^{\alpha} \textbf{e}_{\alpha}) \cdotp (b^{\beta} \textbf{e}_{\beta}) = (\textbf{e}_{\alpha} \cdotp \textbf{e}_{\beta}) a^{\alpha} b^{\beta} = \eta_{\alpha \beta} a^{\alpha} b^{\beta} $$ where $ \eta_{\alpha \beta} $ is the metric of flat space time and can be shown as a diagonal matrix

\(\eta_{\alpha \beta} =\begin{pmatrix}-c^2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix} =\begin{pmatrix}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix} = diag(-1,1,1,1)\)
Here the speed of light is made equal to one in order to make further calculations easier. This metric, called a Minkowsky metric, gives us information of the geometry and structure of spacetime. Although the matrix notation looks messy, it can be very useful for understanding calculations. Instead of writing $  (\Delta s)^{2} = \cdots $ we can just say $$  ds^{2} = \eta_{\alpha\beta} dx^{\alpha} dx^{\beta} $$ The reason why we changed $ \Delta $ into $ d $ is to show that the values are now changing by infinitesimal amounts. Our $ \alpha 's $ and $ \beta 's $ sum over our four coordinates. I recommend learning about Einstein notation so that the formula above becomes clearer. This notation will prove crucial in the posts of gr-qc publications.

Curved Spacetime and the Coordinate Problem

**NOTE: The gravitational constant, G, and the speed of light, c, will be equal to one from now on.** 

Inertial frames are useful because of the special symmetries of flat spacetime. However, in general relativity spacetime is curved and, generally, not symmetric. This means that we cannot take one coordinate system (and I am not talking only about Cartesian or spherical) to simplify the laws; instead, we have to chose a specific system for a specific problem. For example, if we use Schwarzschild coordinates (below)
$$ ds^2 = - \left( 1 - \frac{2M}{r} \right)dt^2 + \left(1 - \frac{2M}{r} \right)^{-1}dr^2 + r^2(d \theta ^2 + sin^2{\theta} d \phi^2) $$
we will find that it breaks down at r = 2M, 0. If instead we decide to use Eddington-Finkelstein coordinates (below)
$$ ds^2 = - \left( 1 - \frac{2M}{r} \right)dv^2 + 2dvdr + r^2(d \theta ^2 + sin^2{\theta} d \phi^2) $$ we will only get infinity at the physical singularity - when at r = 0. This shows us that sometimes, we just need to use a different coordinate system, and there is not one that can be used to solve all problems.

Just as we use Maxwell's equations for electromagnetic fields, we use Einstein's equation for gravitational fields. Einstein's equation is a set of ten second-order partial differential equations; they are also nonlinear. This set of equations tell us how a measure of local spacetime curvature is related to a measure of matter energy density. To find a measure of spacetime curvature, we must come up with a thought experiment on the motion of at least two test particles. Imagine two rings of test particles, one is in space and the other is falling towards the Earth. The particles in space feel no force, so they stay in the shape of a ring. However, the ones experiencing the Earth's gravitational force will start forming an ellipse. This occurs because the particles closer to the Earth feel a stronger pull than the ones further away, and the particles on the sides of the falling ring will begin to come closer together since they are pulled towards the center of the Earth. This change of shape in the rings is a measure of spacetime curvature. This is exactly what Newtonian gravity dictates and the derivation is short.

Unlike the Newtonian view, Einstein saw that the relative accelerations of these particles are not caused by changes in the gravitational force, but instead, they are caused by the fact that particles simply follow geodesics, or straight lines, in spacetime. And because we are seeing these particles accelerate, their geodesics must be curving relative to each other. This shows us that spacetime is actually being curved. To follow the derivation of the equation of geodesic deviation in general relativity and understand the implications in-depth, you must understand the use and meaning of Einstein notation, tensors, and Christoffel symbols. You will be able to follow the math without knowing about Christoffel symbols. In the derivations linked below, I will explain as much as I can without going off topic. While the first link is easier to follow, the second link will be mathematically rigorous.

Links from above:

1. Derivation of the Newtonian Deviation Equation

2. Derivation of the Geodesic Deviation Equation