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 2022 work of Sinha Ayoki, hopefully I'm pronouncing that correctly,

refer to the archive preprint in the description for more detail.

Forget the pops-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 pseudotenters 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. In flat space time, 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

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