Energy Density of the gravitational field

[Geometrick]

Why is there no satisfactory way to define the energy density of a gravitational field? Why must we use global energy concepts? I'm currently reading proofs of the Positive Energy Theorem, but none of the references explains why this is so. Can anyone explain it or point me to a good reference?

 

[haushofer]

Because such an expression would depend on your coordinates due to the equivalence principle. If I measure a graviational field and say it has energy density in point x, then you can Lorentz-transform yourself to an accelerating observer and say that there is no gravitational field, so the energy density is 0.

This is completely different from, let's say, the electromagnetic field.

 

[Entropia]

The energy density of a gravitational field refers to the amount of energy contained within a certain volume of space due to the presence of a gravitational field. However, unlike other types of energy, such as electromagnetic or kinetic energy, there is no satisfactory way to define the energy density of a gravitational field. This is because the concept of energy density is based on the idea of a local energy density, or the amount of energy contained within a small region of space.

In the case of a gravitational field, the energy is not localized to a specific region, but rather is distributed throughout space. This is because gravity is a long-range force, meaning that it can act over large distances. Therefore, it is not possible to assign a specific energy density to a particular point in space.

Instead, in order to understand the energy associated with a gravitational field, we must use global energy concepts. This means looking at the total energy of the system as a whole, rather than trying to define the energy density at a specific point. This is particularly important when considering the Positive Energy Theorem, which states that the total energy of a closed system, including the energy of the gravitational field, must be positive.

There are several reasons why there is no satisfactory way to define the energy density of a gravitational field. One reason is that the concept of energy density assumes a background spacetime, which is not the case in general relativity. In general relativity, the curvature of spacetime is determined by the distribution of matter and energy, including the energy of the gravitational field itself.

Another reason is that the concept of energy density relies on the idea of a local reference frame, which is not well-defined in the presence of a gravitational field. This is due to the fact that gravity affects the motion of objects, making it difficult to define a fixed reference frame.

In summary, the energy density of a gravitational field cannot be defined in a satisfactory way due to the fundamental nature of gravity and the limitations of energy density as a concept. Instead, we must use global energy concepts to understand the energy associated with a gravitational field.