Curt Jaimungal: What Is Energy, Actually?
episodePreviously titled “What is “Energy,” Actually?” — renamed by the publisher on Aug 3, 2026
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What common misconceptions exist about the definition of energy?
Think you know what energy is? You probably don't, and that's okay. Einstein likely didn't know either, at least not in the context of his own masterpiece, General Relativity. By the way, this whole analysis is heavily inspired by the twenty twenty two work of Sinya Aoki, hopefully I'm pronouncing that correctly, refer to the archive preprint in the description for more detail. Forget the pop size sound bites that you hear from people like Neil deGrasse Tyson. Energy is not simply mass in motion, or mass because E equals MC squared, or the capacity to change, or even the neatly conserved currency of our universe, whatever that means. These definitions, to the degree they're even definitions, don't hold up in dynamically curved space-time.
Most likely your GR instructor glossed over energy, perhaps mumbled something about pseudotensors under their breath, then quickly changed the subject. So why the rush? Why the evasion on such a supposedly fundamental concept? Physics professors skip the energy talk like dad skipped the sex talk, awkward mumbling and then hoping you never ask again. The full, honest treatment is extremely messy, it's deeply controversial and fundamentally unresolved, even after a century. Einstein himself wrestled with it, and the compromises he made are still being debated today, so let's talk about that mess. The heart of the problem is that general relativity has two foundational pillars. There's general covariance, which is another way of saying that physical laws don't depend on coordinate choices, and then there's the principle of equivalence, which is that gravity is the same as local acceleration.
How do general covariance and the equivalence principle create problems for energy conservation in GR?
In flat spacetime, energy momentum conservation is actually quite neat. It's written here, where this T is just the stress energy. tensor of matter. Now in GR it looks similar. However, that little upside triangle is what's called the covariant derivative, and that requires some extra machinery, something called a connection to employ. In coordinates, expanding this formula out gets you extra terms, like as follows here. Energy seems to leak into or out of the gravitational field itself. Einstein wanting something conserved, of course, could be Up a fix. Now he cooked up a fix before with the cosmological constant, calling that his biggest blunder, so it's not like this was new. Physics is largely a game of whack-a-mole, whereby fixing one problem creates another.
Anyhow, Einstein added a term here with a little t this time. This is the infamous pseudo-tenser, meant to represent the energy of the gravitational field itself. This combination here actually does satisfy a simple conservation law. Seems fine, so what's the problem, Kurt?
Why does Einstein’s pseudo‑tensor fail as a true gravitational energy definition?
Well, if you examine it, you realize the price was relatively steep. Yes, that's a pun. It was deceptively steep. TUV is not a tensor. That means it depends entirely on your chosen coordinates. So, not cool, bro. In GR, non-tensorial quantities are usually considered mathematical artifacts, so they're not physical realities. This is made blatant when you study. the bundle differential geometric view. Anyhow, this breaks the whole spirit of one of those foundational pillars, namely general covariance. Now saying TUV is gravity's energy and gravity vanishes locally via the equivalence principle, so its energy should be coordinate dependence, that sounds suspiciously like a post hoc justification for a Kluge, is it?
And is there a better way? Well, if your space-time has symmetries, then yes. If there's a time like killing field, something called a killing field, meaning that space-time looks The same along the flow of this vector field, then you can define a genuinely conserved coordinate independent energy.
When can a coordinate‑independent conserved energy be defined using Killing vectors?
Just as an aside, this isn't a murderous field. It's named after Wilhelm. There is a concept of Thanos-like annihilation in possibility space, though, called Guter Damon. Yeah. Now, why is this expression here conserved? It's because of this other expression. Now, notice that the first term here is zero, and the second term vanishes, because the capital T this time is symmetric, and the killing equation becomes this.
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Chapters
8 chapters
1
What common misconceptions exist about the definition of energy?
0:00–1:38
2
How do general covariance and the equivalence principle create problems for energy conservation in GR?
1:38–2:47
3
Why does Einstein’s pseudo‑tensor fail as a true gravitational energy definition?
2:47–3:57
4
When can a coordinate‑independent conserved energy be defined using Killing vectors?
3:57–4:57
5
How do black‑hole and neutron‑star examples illustrate the limits of GR energy definitions?
4:57–6:43
6
What does the expanding FLRW universe teach us about energy non‑conservation and entropy?
6:43–7:50
7
Do gravitational waves really carry energy, or are our definitions incomplete?
7:50–9:25
8
What are the open questions and future directions for defining energy in general relativity?
9:25–11:26
Speakers
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