The relationship between geometry and energy for gravitational waves

By continuing the last post, we will talk a little about the sources of gravitational waves. Of course that the subject can be found in many other places, for example the wikipedia site. However, my hope is to add something besides the content at Wikipedia.

Well, we know that gravitational waves are predicted by the Einstein's general relativity. The main equation of the theory is called the "Einstein's equation" which is a tensor equation. We can see it below.

R_μν -g_μνR/2 = 8πGT_μν/3

Here, the R_μν and R terms are elements of geometry and the T_μν term represents elements of density of mass and energy; g_μν is the metric of space and the other terms are constants values.

As we can note we have geometry elements on left-side and matter elements on right-side. This equation shows a way of interacting the geometry of the universe (of a piece of it) and the distribution of mass (or matter, or energy) on the space-time. Thus, we will have just a space-time completely smooth if there is no matter in all space-time. From the moment that we have the presence of a small density of matter, the geometry of space-time is not smooth any more. What we need know more is that all space time geometry is defined by the metric.

Now if we want gravitational waves, which are deformations on the gravitational field (represented by the metric) we need to insert something else in the metric. We need just insert a small perturbation on it. Thus mathematically speaking, gravitational waves are a kind of perturbations on the space-time metric. By inserting  a perturbative term in the metric and by solving the Einstein equation it's possible to visualize gravitational waves.


So in order to visualize the solution for gravitational waves from the a certain massive body, we must insert in the Einstein equation the elements of geometry of the region where the body is localized and its distribution of matter (or energy). Some examples of possible sources of gravitational waves are black hole, extremely massive stars, and the main one, an inflationary period which took place in the first stages of our universe. I say main because gravitational waves that were generated in this special period can carry very important information about the first minutes or even seconds of the universe.