FNERGY

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Here’s a bullet-point summary of the key ideas from the article:

What is Energy, Actually?

• Energy is a deeply controversial and unresolved concept in general relativity (GR).

• Popular definitions (e.g., "mass in motion" or "the conserved currency of the universe") do not hold up in dynamically curved spacetime.

• Even Einstein struggled with defining energy in GR.

Energy Conservation in Curved Spacetime

• GR is built on general covariance (laws of physics independent of coordinates) and the equivalence principle (gravity = local acceleration).

• In flat spacetime, energy-momentum conservation is straightforward: ∂ᵤ Tᵘᵛ = 0.

• In curved spacetime, it becomes ∇ᵤ Tᵘᵛ = 0, but extra terms (from the covariant derivative) suggest that energy "leaks" into the gravitational field.

Einstein’s "Fix": The Pseudo-Tensor

• To maintain energy conservation, Einstein introduced a pseudo-tensor (tᵘᵛ) to represent gravitational energy.

• Problem: tᵘᵛ is not a true tensor, meaning its value depends on coordinates, violating the principle of general covariance.

• This makes gravity's energy more of a mathematical artifact than a physical reality.

A Better Way? Symmetries & Killing Fields

• If a spacetime has a timelike Killing field (ξᵥ), a well-defined conserved energy can be derived.

• Schwarzschild black holes have such a field, allowing us to define mass (M) properly.

• However, most real-world spacetimes lack exact Killing vectors, making energy definition tricky.

Misconceptions About Vacuum Solutions

• Schwarzschild black holes are often called "vacuum solutions" (Tᵘᵥ = 0 everywhere).

• But at r = 0, there's a singularity in the stress-energy tensor, meaning it's not truly a vacuum.

• The mass (M) actually comes from this singularity—the "vacuum" explanation is a simplification.

Energy in Neutron Stars & Other Systems

• Different energy definitions arise in neutron stars:

  • ADM Energy (Misner-Sharp mass): Defined at spatial infinity (requires asymptotic flatness).

  • Covariant Energy (E): Integrates density in curved spacetime.

• These two definitions do not always agree, leading to ambiguity in what "energy" really means.

Final Takeaway

• The definition of energy depends on context, making it far more complex than commonly thought.

• There is no universal agreement on how to define energy in general relativity.

• Energy might not be an absolute concept, but rather one tied to the specific question being asked.

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