Is there an energy density associated with a Gravitational Field? We've just been taught electrostatics in school and one of the things we were taught was that there is an energy density associated with the presence of an electric field given by $\frac{\epsilon}{2}{E}^2$. I was wondering if same is the case with a Gravitational Field since Newton's law of gravitation and Coulomb's law are analogous. If not so, then why?
 A: Surely there is! The classical gravitational field has an associated energy that can be computed exactly as the energy of an electric field, and it is proportional to the square of the modulus of the field. You just have to repeat every step that brings you to the expression of the energy substituting the gravitational field at every step.
EDIT: as suggested by the comments, it is worth noting that the energy density will always be negative, in this case. This is due to the fact that the gravitational potential is always attracting. Any configuration of (at least two) masses has a negative energy because, once you put the first pieces of the configuration in place, the others are attracted by the configuration. You can also see that by expressing the energy density as $\frac{1}{2}\rho\phi$, where $\rho$ is the mass distribution and $\phi$ is the gravitational potential: $\phi$ is always negative (when taking as reference $r\to\infty$, so the interaction energy density is always negative.
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P.s.: in fact, people have written to write Maxwell's equations for the gravitational field, search things like "gravitomagnetism". But then Einstein came and changed the point of view, and his theory was more predictive and powerful. EDIT: as suggested in the comments, this postscriptum does not address the question, but is just an hint of what happens when you take the gravitational field <-> electrostatic field analogy further, and perform all steps that are done to write Maxwell's equations. But this description of gravity is incomplete, and must be replaced by General Relativity.
A: A straightforward calculation of the energy density of a gravitational field, done the same way we were taught to calculate the energy density of the electromagnetic field, indicates that the energy density of the gravitational field is negative: increase the distance between two infinite sheets of matter, and the energy change in the field as the two sheets are separated is negative and proportional to the square of the field strength.
However, this calculation is at odds with the observed fact that gravitational waves have positive energy!  As stated by Steve Carlip: "To make gravity attractive in such a [vector-like] theory, you must require that the gravitational field has negative energy, which (apart from the obvious instabilities) would drastically disagree with binary pulsar observations".  It is not easy to reconcile this disagreement.  However, it can possibly be reconciled by assuming that gravitational waves amount to propagating rarefactions in the gravitational energy density rather than "ordinary" alternating rarefactions and compressions.
