What is “Energy,” Actually? [Graduate Level]

This episode argues a provocative thesis about general relativity (GR): energy, one of physics’ supposedly bedrock concepts, is not just hard to conserve in curved spacetime — it may not even be uniquely definable. Drawing heavily on a 2022 analysis by Sinya Aoki, the host (Curt Jaimungal) walks through why the popular soundbite definitions of energy break down, why Einstein’s own fix was a flawed compromise, and how competing definitions give different answers depending on the spacetime and the question asked.

The core problem: GR’s two pillars clash with energy conservation

• GR rests on two foundational principles: general covariance (physical laws don’t depend on coordinate choices) and the equivalence principle (gravity is locally indistinguishable from acceleration).
• In flat spacetime, conservation of energy-momentum is clean, expressed via the stress-energy tensor of matter (T).
• In GR the formula looks similar but uses the covariant derivative, which requires extra machinery (a connection). Expanding it produces extra terms — energy appears to “leak” into or out of the gravitational field itself.

Einstein’s fix and its hidden cost: the pseudotensor

• Wanting something conserved, Einstein added a term (lowercase t), the infamous pseudotensor, meant to represent the energy of the gravitational field. The combined quantity does satisfy a conservation law.
• The catch: the pseudotensor is not a tensor, so it depends entirely on chosen coordinates. In GR, non-tensorial quantities are typically dismissed as mathematical artifacts rather than physical realities.
• This breaks general covariance. Justifying it by appeal to the equivalence principle (gravity vanishes locally, so its energy is coordinate-dependent) sounds like a post-hoc rationalization for a kludge.

The clean alternative: Killing vectors (only works with symmetry)

• If a spacetime has a timelike Killing field (a vector field along which spacetime looks unchanged), you can define a genuinely conserved, coordinate-independent energy.
• This works because the relevant terms vanish: one is zero, and the other vanishes because T is symmetric and satisfies the Killing equation. (The name comes from Wilhelm Killing — nothing murderous.)
• The limitation: most realistic spacetimes, especially cosmological ones, have no exact Killing vectors, so this clean definition is sharply restricted.

Case study — the Schwarzschild black hole

• Textbooks call it a “vacuum solution” because T = 0 everywhere, but if that were literally true, energy would be zero.
• In fact it hides a delta-function singularity at r = 0 representing the collapsed matter source; the mass parameter M comes from there. So “vacuum” is a misnomer.
• Using the timelike Killing vector (valid outside the horizon) and properly handling the singularity correctly recovers the black hole mass M. The “vacuum” framing conceals the source — likely part of why the covariant definition was overlooked.

Case study — neutron star: two definitions that disagree

• For a static spherical star, the covariant energy E integrates density with a volume factor.
• The standard ADM energy (here also called Misner-Sharp mass), defined via integrals at spatial infinity and assuming asymptotic flatness, integrates density without that volume factor.
• In the Newtonian/weak-gravity limit: the ADM energy looks like total rest mass plus (negative) gravitational binding energy, while the covariant E looks like rest mass energy evaluated in the background potential.
• Both seem reasonable, yet they differ. E is covariant; ADM relies on asymptotic flatness. Which one is “the” energy seems to depend on the question asked — itself an uncomfortable conclusion.

A more general conserved quantity (S) and cosmology

• Beyond Killing vectors, there’s a quantity S that stays conserved under a weaker condition that needs only some vector field (not necessarily Killing) — but whether such a field always exists, and what S means, is unclear.
• In the expanding universe (FLRW metric) there is no timelike Killing vector, so the standard energy is famously not conserved — energy dilutes as the universe expands.
• However, one can find a vector field satisfying the weaker condition, yielding a conserved quantity that obeys a relation closely resembling the first law of thermodynamics, if S is interpreted as total entropy and beta as inverse temperature.
• This suggests that in cosmology the truly conserved thing may be entropy, not energy, with beta(T) encoding that the universe cools as it expands.

Gravitational waves: do they carry energy?

• LIGO detects them, and binary pulsars spin down exactly as predicted if they lose energy to gravitational waves — strong evidence they carry energy.
• This works in effective field theory approximations or via pseudotensors (e.g., the Isaacson effective stress-energy tensor, from averaging metric perturbations).
• But fully covariant, stress-energy-based definitions (E or S) treat pure gravitational waves as vacuum solutions, giving zero energy.
• Other covariant quantities like the Bell-Robinson tensor are nonzero for gravitational waves, but their interpretation as a definitive energy measure is disputed — the lack of a unique characterization of energy stands.
• This forces a dilemma: either gravitational waves don’t fundamentally carry energy in GR, or our covariant stress-energy definitions are incomplete (and the pseudotensor isn’t the right remedy).

Where this leaves us

• Pseudotensors give conservation but break covariance.
• Covariant, stress-energy-linked definitions work cleanly only with symmetries and fail for general spacetimes or pure gravity.
• The entropy-like generalization S hints at deeper structure but lacks a universal interpretation.
• One possibility: the asymmetry of Einstein’s equation — matter sources curvature, but curvature doesn’t source itself the same way — may mean only matter energy is truly well-defined.
• The alternative: after 100+ years, the question of what energy is in GR simply remains unresolved.