The local energy of the gravitational field is holographic: the energy-momentum pseudotensor of the gravitational field

The energy density of the gravitational field is significantly different from that of matter. This is because the energy density of the gravitational field can be written in terms of the Christoffel connection Γ, which represents the gravity acting on particles, and the metric tensor g, and it can always be made zero in a local coordinate system around a certain spacetime point.

In other words, if one chooses a coordinate system well at any arbitrary spacetime point, the gravitational field vanishes there. This is the "equivalence principle," which was the guiding principle for Einstein when he created general relativity. It means that gravity does not act on a freely falling object. Then, in that local coordinate system, the energy density of the gravitational field also becomes zero. If the energy density of matter is not zero in a certain coordinate system, its energy density will not vanish in any other coordinate system, including local inertial frames, but the energy density of the gravitational field can be made to vanish.

In the first place, the energy and momentum density of the gravitational field are written not as components of a tensor, but as components of a "pseudotensor." A pseudotensor is a quantity that does not satisfy the transformation laws of a tensor defined under general coordinate transformations.

For example, Einstein initially considered an energy-momentum pseudotensor for the gravitational field that was asymmetric with respect to its indices.

He first focused on the curvature quantity:

$$
G^{\mu\nu}=R^{\mu\nu}-\frac{1}{2}g^{\mu\nu} R
$$

This quantity automatically satisfies the following by definition:

$$
\nabla_\mu G^{\mu\nu}=0
$$

Also, this G appears on the left side of the Einstein field equations.

$$
G^{\mu\nu}=\kappa T^{\mu\nu} (1)
$$

The κ on the right side is a constant obtained by multiplying the gravitational constant by 8π, and T is the energy-momentum tensor of matter. By taking the covariant derivative ∇ on both sides of this Einstein equation and contracting the indices, the following equation is obtained, which signifies the local conservation law of the energy-momentum of matter:

$$
\nabla_\mu T^{\mu\nu}=0 (2)
$$

As it stands, this does not look like an energy conservation law between matter and the gravitational field. Therefore, Einstein defined the following pseudotensor for the gravitational field:

$$
t^\nu_{E\mu}=\frac{1}{2\kappa\sqrt{-g}} \left( \partial_\mu ( \sqrt{-g} g^{\alpha\beta}) (\Gamma^\nu_{\alpha\beta} -\delta^\nu_\beta \Gamma^\lambda_{\alpha\lambda} ) -\delta^\nu_\mu \sqrt{-g} (\Gamma_{\alpha\beta}^\lambda \Gamma^\sigma_{\lambda\sigma}-\Gamma_{\alpha\sigma}^\lambda \Gamma^\sigma_{\beta \lambda} ) \right) (3)
$$

By performing the differentiation on the right side, it can be confirmed that this quantity is made only of Γ and g, and therefore it satisfies the equivalence principle. Also, from the Einstein equation in (1):

$$
\partial_\nu \left(\sqrt{-g}(T_\mu^\nu +t^\nu_{E\mu} ) \right)=0
$$

A local conservation law is obtained. Here, the pseudotensor in equation (3) derived by Einstein is not symmetric with respect to its indices. However, the following pseudotensor proposed by Landau and Lifshitz is symmetric with respect to its indices:

$$
t_{LL}^{\mu\nu}=-\frac{1}{\kappa}G^{\mu\nu}+\frac{1}{2\kappa(-g)}\partial_\alpha \partial_\beta \left((-g)(g^{\mu\nu} g^{\alpha\mu}-g^{\alpha\mu} g^{\beta\nu}) \right)
$$

This is called the "Landau-Lifshitz pseudotensor," and it also satisfies the following local conservation law:

$$
\partial_\mu \left( (-g)(T^{\mu\nu} +t_{LL}^{\mu\nu}) \right)=0
$$

Beyond this, many energy-momentum pseudotensors for the gravitational field have been proposed. Each one becomes zero in a local inertial frame and satisfies the equivalence principle, but the values of the quantities defined by each are different. Therefore, the energy density of the gravitational field is not uniquely determined even at the level of the defining equations.

However, if we limit ourselves to asymptotically flat spacetimes and decide to set up a coordinate system such that the metric tensor g asymptotically approaches the flat Minkowski metric η, every pseudotensor yields the same value for the total energy of the gravitational field. This is because the (00) component of the pseudotensor, which corresponds to the energy density, can be written as the spatial divergence of a certain quantity. When that energy density is integrated over a volume, it is replaced by a surface integral over a two-dimensional boundary at spatial infinity due to Stokes' theorem.

Although the energy distribution of the gravitational field at each point in spacetime is not determined due to the nature of gravity that satisfies the equivalence principle, the value of the total energy of the gravitational field in an asymptotically flat spacetime does have a proper physical meaning. In relativity, the region at infinity that becomes asymptotically flat can be regarded as a single boundary. And the energy of the gravitational field inside the space exists as "information" on this two-dimensional boundary. This is the "holographic principle" in classical general relativity.