The concept of energy for the entire gravitational field is determined by the observer and the coordinate system.

In general relativity, gravity is described as a "field." It follows the same format as electromagnetic forces being described through electric and magnetic fields, but the nature of the "gravitational field" differs significantly from that of electromagnetic fields. One major difference is the energy-momentum density of the field. Mathematically, the energy-momentum density of an electromagnetic field is written as components of a "tensor" under coordinate transformations. If all components of the electromagnetic energy-momentum density tensor are zero in one coordinate system, the tensor remains zero in any other coordinate system. A tensor quantity is something that, if it does not "exist" for a certain coordinate system or observer, does not "exist" for any other coordinate system or observer either. Conversely, if there are non-zero tensor components in one coordinate system, it is impossible to make all tensor components zero in any other coordinate system. A tensor quantity is something that, if it "exists" for a certain coordinate system or observer, "exists" for all other coordinate systems and observers as well.

However, unlike electromagnetic fields, the energy-momentum density of a gravitational field is not a component of a tensor, but a component of a different quantity called a "pseudotensor." This pseudotensor can be zero for one coordinate system or observer, yet have a non-zero value for another. Furthermore, even if it is non-zero for one coordinate system or observer, there exists a specific coordinate system or observer for which the value is always zero; this is the nature of a pseudotensor quantity. In other words, it exists or does not exist depending on the coordinate system or observer. This property originates from the "equivalence principle," which Einstein assumed when formulating general relativity.

The equivalence principle states that in any gravitational field, the gravity can be eliminated by an observer in free fall or in a corresponding coordinate system known as a local inertial frame. The equivalence principle also asserts that even in a space without gravity, gravity will appear to other objects if the observer is undergoing accelerated motion. This property cannot be realized if the energy-momentum density of the gravitational field is defined as a strict tensor component. Therefore, Einstein considered it to be a component of a pseudotensor. I have also written about these topics in Chapter 2 of my book, "What is Nothing?" (SB Shinsho), without using mathematical formulas.

In this article, I will write about how the concept of energy in the gravitational field within general relativity is itself determined by the observer at spatial infinity and the choice of coordinate system.

First, for simplicity, let us fix the spacetime as a flat Minkowski spacetime and consider the energy-momentum tensor T of matter, such as an electromagnetic field, moving within it. Each of its components satisfies the local conservation law of equation (1).

Next, consider a four-dimensional vector field ξ that depends on the spacetime point,

and examine the quantity of this flow. To understand for what kind of ξ this J is locally conserved, let us perform the following calculation.

The first term of this final equation vanishes from equation (1). The remaining second term, due to the symmetry of the indices of T, can be written as

This term vanishes if ξ satisfies

Then, the original J becomes a locally conserved quantity.

Solving equation (5) as an equation for ξ, using a constant four-vector a and a constant antisymmetric tensor ω, we obtain

This ξ is actually a well-known vector field. First, recall the following infinitesimal coordinate transformation generated using ξ.

It can be seen that the a term in equation (6) generates time translation coordinate transformations and spatial translation transformations. From the perspective of Noether's theorem, the generators of these transformations are energy and momentum. Furthermore, the ω term in equation (6) represents spatial rotation transformations and Lorentz transformations in equation (7). These generators are the angular momentum and Lorentz boost generators. In other words, as a whole, the ξ in equation (6) generates infinitesimal Poincaré transformations. This is the transformation of spacetime symmetry that appeared in special relativity. The reason why the energy, momentum, and angular momentum of matter fields such as electromagnetic fields are locally conserved is that the ξ corresponding to the Poincaré group in flat spacetime satisfies equation (5). In other words, the attributes of spacetime give rise to these conservation laws.

The quantity on the left-hand side of equation (5) is mathematically the 'Lie derivative' of the metric tensor. In a general curved coordinate system, the difference in the form of the metric tensor under the infinitesimal coordinate transformation of equation (7) is expressed using the general Lie derivative as

In a general spacetime, a special vector field ξ that makes the Lie derivative of the metric tensor zero is called a 'Killing vector field.' It represents the property that the metric tensor does not change its form even after the coordinate transformation generated by ξ. Writing this as a mathematical formula, we get

or specifically

In particular, equation (10) is called the 'Killing equation' for a given spacetime. For the quantum J of the flow as a generator of ξ, the ξ that satisfies this equation derives a local conservation law originating from the symmetry of that spacetime.

Furthermore, in general relativity, there is a peculiar property where the volume integral of a density quantity that satisfies this local conservation (that is, the 0th component of J) can be expressed as a surface integral on a spherical surface at spatial infinity. This is because the flow quantity J can be written as the total derivative of a quantity Q, as shown in

below.

Then, by the Gauss-Stokes theorem, the volume integral becomes

which is a surface integral. Although it was originally a volume integral of the 0th component of J, detailed information about how that 0th component is distributed as a density within the volume is not necessary to know its value. This property is also called 'classical holography,' because a volume quantity becomes a lower-dimensional surface quantity.

From this property of classical holography, the peculiarity of the concept of energy of the gravitational field in general relativity becomes visible. As can be seen from equation (13), the behavior of the gravitational field in the boundary region at spatial infinity is most important for the definition of the energy of the gravitational field. This is because energy is defined from ξ that satisfies an equation like (10) only in the vicinity of the boundary.

First, let us consider a metric tensor that asymptotically approaches a flat Minkowski metric in the boundary region at infinity. That is, for

we consider the radial coordinate r defined by

and the metric tensor in the vicinity of the boundary is

can be written as follows, and we assume that each component decreases with the behavior of

If we express this using the Lie derivative of the metric tensor, we consider an equation of the form

in the vicinity of the boundary at infinity. The O on the right-hand side of this equation specifies only the power of r, and its coefficient can be arbitrary. This means that as r increases, it approaches zero at the speed of that power. Equation (18) is called the "asymptotic Killing equation" for ξ. Furthermore, the symmetry of the coordinate transformation group generated by the solution ξ through Equation (7) is called the "asymptotic symmetry" of that spacetime.

To define the asymptotic symmetry of this spacetime, it is also required to specify how each component on the right-hand side of Equation (18) falls off, as it approaches the background metric tensor. This is done by the observer who sets up the spacetime measuring instrument. This fall-off behavior of each component (the speed at which it approaches zero) is also called the "fall-off condition." Each time this fall-off condition is specified, the asymptotic Killing equation of Equation (18) is also defined.

The solution ξ to this asymptotic Killing equation does not exist for many background spacetimes. A solution appears only when the background spacetime is chosen well. This is the same as the fact that the exact Killing equation of Equation (10) has solutions only for special spacetimes. However, in the asymptotic Killing equation, even after choosing a background spacetime, if an arbitrary fall-off condition is assumed, ξ often does not exist as a solution in the first place. It is important to choose the fall-off condition well. The fact that this choice of fall-off condition strongly influences the definition and concept of the total gravitational energy conserved in general relativity is one of the very interesting points. This property is also caused by the fact that the original energy density of the gravitational field is one component of a pseudotensor. Moreover, it originates from the equivalence principle, which is unavoidable as the origin of general relativity.

For example, let us choose flat spacetime as the background spacetime. In the standard Minkowski coordinate system, the metric tensor is given by

However, here, letting a be a positive constant,

obtained by applying the coordinate transformation

to Equation (19), let us consider the energy density of the gravitational field with this metric tensor.

Since we have only re-chosen the coordinates in flat spacetime, all components of the spacetime curvature tensor calculated from this metric tensor remain zero. However, the Christoffel connection Γ itself, obtained from the first derivative of the metric tensor, has not vanished. Although this Γ was zero in the original Minkowski coordinate system, the fact that Γ itself is not a tensor quantity causes the following non-zero components to survive.

Here, as an example of the energy-momentum pseudotensor of the gravitational field, we consider the following Landau-Lifshitz pseudotensor.

Assuming the equations of motion in a general spacetime, this quantity satisfies the following local conservation law with the energy-momentum tensor T of the matter field.

Using this quantity, we will evaluate the energy density of the gravitational field for the metric tensor of equation (21) in the case of a vacuum without matter. Then,

It becomes non-zero as shown in, and furthermore, when this is integrated over the volume, it is found to diverge due to the exponential behavior of the spatial component x. In the original coordinate system that had the metric tensor of equation (19), Γ vanished, so all components of the Landau-Lifshitz pseudotensor were zero, as is appropriate for a correct gravitational field in a vacuum; however, in the current coordinate system, it turns out that the space is filled with an infinite amount of gravitational field energy. Thus, even after the spacetime is determined, the value of the gravitational field energy can become zero or infinite depending on the coordinate system within it and the observer who chooses that system. The equivalence principle gives rise to such peculiar behavior of the gravitational field.