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.

Gravitational waves, what is it?

Gravitational waves, what is it?

Recently the research on cosmology has touched a level of difficulty that just a few people are able to understand. However, the cosmology is almost entirely based in the theory of general relativity. So the understanding of some points from that theory can allow us to understand a little of "classical cosmology". Current cosmology deals with multi-universes and other things like that, that is, effects that arises when are put together quantum mechanics and general relativity.

Gravitational waves are a immediate consequence of the general relativity. They are obtained even just in the classical cosmology, although its source are quantum effects. To understand what are gravitational waves we can make use of a basic analogy. Assume an electric charge and also that we are in a inertial system of coordinates in relation to the charge. Then if we are watching the charge moving, it's well known from the electromagnetism that we will to observe a electric field. Now, if we are in a accelerated motion in relation to the charge, also is well established that we will note a magnetic field as well. What we know as electromagnetic waves are just deformations on this field, called electromagnetic field.

In a complete analogy, the gravitational waves are just deformations on the gravitational field, which is our old friend since Galileo and Newton. But we need to use the general relativity to realize that. Nevertheless, we can not to measure signal of gravitational waves in a direct way yet. That happens due to the fact that the intensity of the gravitation interaction is much lower than other interactions, for instance, electromagnetic interaction. 

In the next post, I'll expose some devices which are candidates to measure directly signal of gravitational waves. Before, however, I must to show some sources from these waves.

Any question, it will be a pleasure to discuss about it.