Cracking the Code

Well, if the goal is to simulate the merger of two black holes, then indeed there is one very big simplification that can be made. Black holes, though they have mass, are actually considered to be vacuum solutions. That is, the stress-energy-momentum tensor (the right-hand side of the Einstein-Hilbert field equations) is zero.

Another simplification can be made by noting that the cosmological constant, Λ_μν is quite small. In SI units, it is measured to be on the order of 10^-52m^-2.Though it might contain the majority of the energy in the observable universe, this is only when taken on vast, cosmological scales of the Mega-parsec range. In the few hundred billion-km or so that would be used to simulate even the largest black hole mergers, the effect of dark energy is thought to be minuscule by comparison to the other terms. Thus, for our purposes, setting Λ_μν to zero is a safe bet.

Thus, the version of the Einstein-Hilbert field equations that we would to wrestle with is:

Okay, this is beginning to look easy!

Relatively speaking, yes. Still, let's take a look at what the vacuum version is made up of before jumping to any such conclusion. First, note that the tensor equations we are left with have two indices. And, for four-dimensional space-time, that means there are 4 x 4 = 16 interdependent equations to solve.

Well, we can do a little better than that, since the Einstein tensor, E_μν , is symmetric (E_μν = E_νμ ) there are then just 10 non-redundant nonlinear partial differential equations to deal with. This is also true of the Ricci curvature tensor, Rμν .

Then, the system of equations we would like to solve looks like:

This looks like progress. But, just what is the Ricci curvature tensor? Ricci curvature is a measure of how much the local curvature deviates from Euclidean (i.e., non-curved) space in a Riemannian manifold. It can be calculated as a twice contraction of the Riemannian curvature, Rμbνc, with the metric g^bc

In this last equation, the Einstein summation convention is used, which is to say it can also be written in perhaps a more standard mathematical form as:

So then, what is the Riemannian curvature? The Riemannian curvature is a measure of curvature for manifolds with dimensions greater than 2 and is calculated as (note that α is used to indicate summation in the last two terms on the right hand side of the equation below):

where Γ ^λ _μσ  is the Christoffel symbol, which is calculated as:

So, when you unravel the Einstein-Hilbert field equations into their raw form, you find that they represent the relation between the space-time metric (g_αβ or g^αβ) and the slope of the space-time metric (∂_βg_μν.) So, given the value of the metric at all points in our 4-dimensional grid at some initial time, we should be able to figure how, based on the slope of the metric, it evolves.

(Actually, if you had the metric (and its slope) for all points in a 4-dimensional grid, you'd have the answer for all time. Which, I have to admit, sounds handy.)

It turns out that is a little tricky. For example, what does "initial time" in a 4-dimensional space-time even mean? 

Maybe it would be nice to formulate the general relativity field equations into a more standard initial value problem. 

This is the goal of a method called the 3+1 numerical relativity technique, which separates out time from space time (hence "3" spatial dimensions "+1" time dimension.)

Next: Moving Puncture BSSN Evolution Equations -->

<-- Back: Background on General Relativity
<--Back: Overview