What is 'Energy,' Actually? [Graduate Level]

ELI5 / TLDR

Nobody actually knows how to define energy in general relativity, and this has been true for over a century. Einstein’s own fix — the pseudotensor — broke one of his theory’s core principles. Clean definitions only work in spacetimes with special symmetries, which most real spacetimes lack. The video walks through the increasingly uncomfortable ladder of partial solutions, ending at the suggestion that maybe the conserved quantity in cosmology is entropy, not energy at all.

Summary

The video, drawing heavily on a 2022 paper by Sinya Aoki, lays out why energy in general relativity is not merely hard to calculate but genuinely ill-defined. In flat spacetime, energy-momentum conservation falls out cleanly from the stress-energy tensor. In curved spacetime, the covariant derivative introduces extra terms — energy appears to leak into the gravitational field. Einstein patched this with the pseudotensor, a quantity that restores conservation but depends on your choice of coordinates, violating general covariance. The Killing vector approach gives a proper covariant energy, but only when the spacetime has a time-translation symmetry — which rules out most of the actual universe, including the FLRW expanding cosmology. A generalized conserved quantity S exists under weaker conditions and, applied to FLRW, produces something that looks suspiciously like the first law of thermodynamics, with S as entropy and an inverse-temperature parameter. Gravitational waves present the sharpest embarrassment: LIGO detects them, binary pulsars lose energy to them, yet fully covariant definitions assign them zero energy. The video concludes that after a century, no universal, coordinate-independent definition of energy exists in GR.

Key Takeaways

• Pop-science definitions of energy (“mass in motion,” “capacity to change,” E=mc^2) break down in curved spacetime. They are slogans, not definitions.
• The stress-energy tensor conservation law in GR (covariant divergence = 0) is not the same as ordinary conservation. The covariant derivative smuggles in connection terms that let energy “leak.”
• Einstein’s pseudotensor restores a conservation law but at the cost of coordinate dependence — it is not a tensor and therefore not a physical observable under GR’s own rules.
• Killing vector energy is genuinely covariant and conserved, but only exists in spacetimes with time-translation symmetry. Most realistic cosmological spacetimes have none.
• The Schwarzschild “vacuum solution” is a misnomer — it has a delta-function source at r=0. The Killing-vector energy correctly recovers the black hole mass M, but only because Schwarzschild has the requisite symmetry.
• For neutron stars, two reasonable energy definitions (covariant E and ADM mass E_A) give different answers. Which is “the” energy depends on what you’re asking.
• A generalized conserved quantity S, requiring weaker conditions than a Killing vector, obeys a relation resembling the first law of thermodynamics when applied to FLRW cosmology. The conserved thing in an expanding universe may be entropy, not energy.
• Gravitational waves carry energy by every practical measure (LIGO, pulsar spin-down), yet covariant stress-energy-based definitions assign them exactly zero energy. The Isaacson tensor resolves this but is an effective-field-theory approximation, not fundamental.
• The Bel-Robinson tensor is non-zero for gravitational waves but its status as a definitive energy-momentum measure is debated.

Detailed Notes

The Two Pillars and Their Tension

General relativity rests on general covariance (physics is independent of coordinate choice) and the equivalence principle (gravity equals local acceleration). Energy conservation, as understood in flat spacetime, collides with both. The covariant derivative in curved spacetime introduces connection terms that make the stress-energy tensor’s conservation equation messier than its flat-space cousin. Energy appears to flow into the gravitational field, but the gravitational field has no stress-energy tensor of its own in the standard formalism.

Einstein’s Pseudotensor: The Original Kludge

Einstein’s response was to define a pseudotensor t_uv representing gravitational field energy. Adding it to the matter stress-energy T_uv produces a quantity whose ordinary (not covariant) divergence vanishes — conservation restored. The cost: t_uv depends on coordinates, which is anathema in a theory built on coordinate independence. The justification sometimes offered — gravity vanishes locally via the equivalence principle, so of course its energy is coordinate-dependent — reads as a rationalization for a patch that violates the theory’s own philosophy.

Killing Vectors: Clean but Limited

When a spacetime admits a timelike Killing vector field (meaning the geometry is invariant under time translation), you can construct a genuinely covariant conserved energy. The math is clean: the conservation follows from the Killing equation and the symmetry of the stress-energy tensor. The catch is brutal: most physical spacetimes, including our expanding universe, have no exact Killing vectors. The definition works beautifully in exactly the cases where you least need it.

Black Holes, Neutron Stars, and Competing Definitions

The Schwarzschild black hole is routinely called a “vacuum solution” (T=0 everywhere), which would imply zero energy — obviously wrong for an object with mass M. The resolution is that Schwarzschild has a delta-function singularity at r=0 acting as the source; calling it a vacuum hides this. The Killing vector method correctly recovers M.

For a neutron star, the covariant energy E and the ADM mass E_A (defined via integrals at spatial infinity, assuming asymptotic flatness) give different values. In the Newtonian limit, E_A looks like rest mass plus gravitational binding energy; E looks like rest mass in the background potential. Both are defensible. Neither is uniquely “the” energy. This is not a computational subtlety — it is a conceptual one.

The Entropy Detour: FLRW and the First Law

In FLRW cosmology (our expanding universe), there is no timelike Killing vector. The standard energy is not conserved; it dilutes as space expands. However, a generalized conserved quantity S can be constructed under weaker symmetry conditions. Applied to FLRW, the resulting conservation law takes the form of the first law of thermodynamics: dS = beta * dE, where S plays the role of entropy and beta is inverse temperature. The implication: in cosmology, energy is not the conserved currency. Entropy might be. The universe cools as it expands, and this cooling is baked into the structure of the conserved quantity.

The Gravitational Wave Embarrassment

LIGO detects gravitational waves. Binary pulsars lose angular momentum at exactly the rate predicted by gravitational wave emission. By every empirical standard, gravitational waves carry energy. But under fully covariant definitions tied to the stress-energy tensor, pure gravitational waves are vacuum solutions — T=0 — so their energy is identically zero. The Isaacson effective stress-energy tensor resolves this by averaging over metric perturbations, but it is an approximation, not a fundamental definition. The Bel-Robinson tensor is covariant and non-zero for gravitational waves, but whether it constitutes a proper energy-momentum measure remains contested.

This is the sharpest version of the problem: a quantity that is empirically real and measurable has no clean fundamental definition in the theory that predicts it.

Quotes / Notable Moments

“Physics professors skip the energy talk like dad skipped the sex talk. Awkward mumbling and then hoping you never ask again.”

“Energy in GR is like my friend’s veganism. It’s loudly declared, but suspiciously flexible.”

“Physics is largely a game of whack-a-mole, whereby fixing one problem creates another.”

“Does it mean that in cosmology, the conserved thing isn’t energy? Maybe it’s entropy.”

“Does this mean that gravitational waves don’t carry energy fundamentally in general relativity? Or does it mean that our covariant definitions based solely on the stress energy are incomplete?”

Claude’s Take

This is a genuinely good explainer of a genuinely hard problem. Jaimungal is not overstating the difficulty — the non-existence of a universal energy definition in GR is a real and well-known issue in mathematical physics, not a YouTube clickbait mystery. The framing around Aoki’s 2022 paper is appropriate; that work does provide a useful organizing framework for the various partial definitions.

What’s solid:

• The explanation of why the pseudotensor violates general covariance is accurate and important. This is the central tension and he presents it clearly.
• The Killing vector treatment is correct — it is the textbook-clean case, and the limitation to symmetric spacetimes is real.
• The Schwarzschild “vacuum” point is a good pedagogical catch. The delta-function source is often swept under the rug in introductory treatments.
• The gravitational wave problem is stated accurately. This is one of the genuinely uncomfortable corners of GR.

Where it gets thin:

• The entropy/thermodynamics connection in FLRW is presented suggestively but without much rigor. The leap from “this conserved S obeys something like the first law” to “maybe entropy is the real conserved quantity” is a large interpretive jump. The formal resemblance to thermodynamics does not automatically mean it is thermodynamics. This is an active area of research (Jacobson’s work on emergent gravity, Verlinde’s entropic gravity, etc.) and the video gestures at it without fully acknowledging how speculative it remains.
• The Bel-Robinson tensor gets a single mention as an aside. Given that it is one of the more serious attempts at a covariant gravitational energy measure, this deserves more than a footnote.
• The video does not mention the Bondi mass or the Komar mass, which are other important partial definitions that would complete the picture of how many competing notions exist.

Overall: The video delivers what it promises — a graduate-level tour of why energy in GR is a mess. It is well-sourced, mostly careful with claims, and does not pretend to have an answer where none exists. The production style (equations on screen, references to specific papers) is aimed at people with at least an undergraduate physics background, which is appropriate for the content. The humor lands without undermining the seriousness of the topic. Worth the 15 minutes, especially if your GR instructor did in fact mumble about pseudotensors and change the subject.