In General Relativity (GR), the concept of energy is far more subtle and complex than in any other physical theory.

Here’s a breakdown, from the simple analogy to the core of the problem.

1. The Short, Simple Answer

In General Relativity, energy (and mass and momentum) is the source of spacetime curvature. It tells spacetime how to bend. This is encapsulated in the famous Einstein Field Equations:

$G_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$

• The left side, $G_{\mu \nu }$, describes the geometry of spacetime—its curvature.

• The right side, $T_{\mu \nu }$, is the Stress-Energy Tensor. This is the GR definition of energy, mass, and momentum.

The Stress-Energy Tensor is a multi-component object that includes not just energy density, but also pressure, momentum density, and shear stress. All of these contribute to gravity.

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2. The Core Problem: Why Energy is Tricky in GR

In Special Relativity (SR), we have a perfectly well-defined concept of energy. It's a conserved quantity, thanks to Noether's theorem, which links conservation laws to symmetries. Specifically, conservation of energy is linked to time-translation symmetry—the laws of physics are the same today as they were yesterday.

Here is the central issue in GR: In a general, curved spacetime, this universal time-translation symmetry does not exist.

• Spacetime is dynamic and can be warped in complex ways.

• There is no universal "time" to translate. Your time flow depends on your location and motion in the gravitational field.

Because the fundamental symmetry is lost, there is no general, coordinate-invariant definition of a localized energy density for the gravitational field itself.

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3. The Different "Flavors" of Energy in GR

Because of this fundamental problem, physicists have defined different concepts of energy for different situations:

A. Local Energy: The Stress-Energy Tensor ($T_{\mu \nu }$)

This is straightforward. The Stress-Energy Tensor describes the energy, mass, and momentum of everything except gravity itself—that is, all matter and non-gravitational fields (electromagnetic, nuclear, etc.). At any point in spacetime, you can calculate the energy density measured by a local observer. This energy is locally conserved.

B. Gravitational Field Energy: The "Problem Child"

We know the gravitational field carries energy—you have to do work to lift an object, and that energy is stored in the field. But there is no tensor (a mathematically well-defined object) that can describe the energy density of the gravitational field at a single point.

Why? Because in GR, gravity is geometry. You can always choose a local inertial frame (a freely-falling frame) at any point where gravity "disappears" momentarily. In that frame, the gravitational field energy density would be zero. If it can be zero in one coordinate system and non-zero in another, it can't be a proper tensor.

Instead, we have pseudo-tensors (e.g., the Einstein pseudo-tensor). These are mathematical objects that are not tensors and are not coordinate-invariant, but they can be crafted to give a pseudo-local definition of gravitational energy. Most physicists find them ugly and unintuitive, and they are rarely used in practical calculations.

C. Global Energy: Mass-Energy for Isolated Systems

This is where we find the most useful and robust definitions of energy in GR. For a system that is isolated and asymptotically flat—meaning it's self-contained and spacetime becomes flat (Minkowskian) far away from it—we can define a total, conserved mass-energy.

The two most important concepts are:

1. ADM Mass (Arnowitt-Deser-Misner): This is the total mass-energy of an isolated system as measured by an observer at spatial infinity. Think of it as the total energy you would need to assemble the system from infinity. It's a measure of the total curvature caused by the system.

2. Bondi Mass: This is the mass-energy as measured by an observer at null infinity (far away, but along light rays). The Bondi mass is crucial for systems that lose energy, for example, through gravitational waves. The rate at which the Bondi mass decreases tells us exactly how much energy is being carried away by the waves.

For a black hole, we have the beautiful result of the Irreducible Mass, which is related to its event horizon's area, and the fact that a black hole's mass can be calculated entirely from its external properties (its charge, angular momentum, and mass), as stated by the No-Hair Theorem.

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Summary and Analogy

Think of it like this:

• The Stress-Energy Tensor ($T_{\mu \nu }$): Is like the fuel and cargo you put into a truck. It's the "non-gravity" stuff that creates gravity.

• The Gravitational Field Energy: Is like the "shape" of the truck's suspension and frame. It takes energy to compress it, and it releases energy when it bounces back, but you can't point to a single part and say "that's the energy." It's a property of the whole system.

• The ADM Mass: Is the total weight of the entire truck (including its cargo and the energy in its suspension) as measured on a giant, perfect scale far away.

In conclusion: Energy in GR is not a single, simple concept. It is:

• Well-defined locally for everything except gravity ($T_{\mu \nu }$).

• Not locally definable for the gravitational field itself.

• Well-defined globally for complete, isolated systems (ADM Mass, Bondi Mass).

This complexity is not a flaw of the theory but a profound reflection of the intimate connection between energy and the dynamic fabric of spacetime itself.