After the memorial I posted earlier in the week, I have had a few readers ask me for more details about energy localization in the general theory of relativity. There are a few variations on this theory, and so I will try to focus on generic properties and as usual I will try to minimize formal equations in order to make this article accessible to a general audience.

The general theory of relativity was first published in 1915, and quickly confirmed by astrophysics experiments over the following decades. It successfully described the properties of gravitational forces and fields, and was more accurate in its predictions than the classical theory of Newtonian gravity. Even though there are still technical issues with the theory, such as how it behaves at high energies or small subatomic scales, it is generally accepted to be the correct model of gravity.

One of the key aspects of general relativity is that mass and energy cause space and time to curve. The more energy that is present at a point in spacetime, the greater the curvature will be. And it is the movement of objects in curved space that gives the appearance of a gravitational force, rather than an actual physical field as in electromagnetic or nuclear forces. However this leads to a very interesting and as yet unsolved problem in the theory of gravity.

It is expected that a gravitational field should contain energy as well. A gravitational wave can remove energy from one location in space and deliver it to another system, which would suggest that it carries energy through empty space. A cloud of dust particles will reduce its energy as gravity causes it to condense, and so this extra energy should be contained in its gravitational field (However contrary to popular belief, while these suggest the gravitational fields contain energy there are methods of cancelling out the energy in the gravitational fields so that no energy transfer occurs). And yet it is not clear how something that is caused by the curvature of spacetime rather than a physical field could contain energy. 

Perhaps even more important is the question of whether this gravitational energy can curve spacetime. The equations of general relativity only include the energy from non-gravitational sources, and yet there is no reason to expect that gravitational and non-gravitational energy are different from each other in nature. 

The issue of gravitational energy has been studied almost from the start. Over the decades many authors have proposed different methods of calculating the energy of a gravitational field and how they affect the curvature of spacetime. The issue has been debated extensively in the academic literature, and yet over a century later there is still no definitive answer. Each of the proposed definitions for gravitational energy contains their own technical difficulties, and unfortunately the effects of this energy on the forces of gravity are too small to be measured in any experiment currently in operation or planned to be in operation in the near future.

One interesting aspect of this debate is the localization of gravitational energy. It is possible that gravitational fields only contain energy when there are non-gravitational fields present. Through some as yet unknown mechanism, electromagnetic or nuclear forces, or the presence of matter, would cause gravitational fields to contain energy. This energy would be different, in that it would not contribute to the curvature of spacetime itself, but would instead be an effect of the curvature caused by non-gravitational energy. In regions of spacetime that do not contain other matter or forces, the gravitational field would have no energy.

There are other reasons to consider such an energy localization aside from those given above. In classical physics and quantum physics, energy density can be defined based on how a system changes over time. The details are too technical to include in this article, but the general idea is that a property of the system known as the action will in general change over time. The amount that it varies determines the energy of the system. (And once again I must apologize to those who quite rightly feel that this is an over-generalization of a very beautiful aspect of classical and quantum physics.).

When this same method is applied to systems in general relativity, the result is that the energy of a gravitational field vanish in vacuum. Gravitational energy will be localized in regions that contain non-gravitational energy. Unfortunately there are also technical difficulties with applying such methods to general relativity, and so this is not considered a proof of energy localization.

And so for the foreseeable future, the definition of gravitational energy in the general theory of relativity will remain unanswered. With no experimental data that can provide hints of its nature, it will remain for the theoretical physicists to discuss and debate for many years to come. 

For now energy localization will remain an unproven hypothesis.