What Is Energy Actually Graduate Level

TITLE: What is “Energy,” Actually? [Graduate Level] CHANNEL: Curt Jaimungal URL: https://youtu.be/hQk9GLZ0Fms

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 Sinya Aoki. Hopefully I’m
pronouncing that correctly. Refer to the archive preprint in the description for more detail.
Looking at the pop-sci soundbites 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 spacetime. 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. 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, cooked 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 pseudotensor,
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, Curt? 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 kludge. Is it? And is there a better way? Well, if your spacetime has symmetries,
then yes. If there’s a timelike Killing field, something called a Killing field,
meaning that spacetime looks the same along the flow of this vector field, then you can define a
genuinely conserved coordinate independent energy. 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 Gotterdammerung events. 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. The problem is
that most spacetimes, especially realistic cosmological ones, don’t have exact Killing
vectors, so this definition, while clean, is limited. Energy in GR is like my friend’s
veganism. It’s loudly declared, but suspiciously flexible. Now, let’s consider the Schwarzschild
black hole. Textbooks often call it a vacuum solution, because T equals zero everywhere,
but if that were true, then E would be zero. Thus, vacuum solution is somewhat of a misnomer,
because the Schwarzschild solution isn’t truly a vacuum. It has a delta function singularity,
exactly where R equals zero, representing that collapsed matter source. And yes,
mathematically, it should be noted that T equals zero whenever you have a positive radius, and R
equals zero is just a curvature singularity, but the parameter M comes from this source. Anyhow,
using the time-like Killing vector, which is Killing outside the horizon, and again, “Killing”
isn’t murderous, it’s just a name for a type of field, it correctly gives that mass, the M, the
black hole mass, after you handle a singularity. The vacuum story is a convenience that hides the
source, likely contributing to why this covariant definition wasn’t embraced sooner. Now, what about
something less singular, like a neutron star? For a static spherical star, E involves integrating
over a density, so I’ll place one here on screen. Now you compare this to the standard ADM energy.
This is often called the Misner-Sharp mass in this context, and is defined via integrals at spatial
infinity, assuming spacetime suitably flattens far from the source, which just integrates rho
without the volume factor. They are not the same. Quick aside, you should know there are resources
and citations in the description if you want to read more in-depth and want to understand what
these words actually mean. Now, in the Newtonian limit, so the weak gravity limit, this E sub A,
which is what I’m calling the ADM mass, looks like the total rest mass plus the gravitational
binding energy, a negative term. E, however, looks like the rest mass energy, evaluated in
the background potential. The issue is that they both seem reasonable, yet they differ. And E,
the regular E, is covariant, whereas E sub A is not. It relies on this asymptotic flatness. Okay,
so which one of these guys is THE energy? Well, it depends on what question you ask, perhaps. But
then does that mean that energy depends on what you ask? Also not cool. Further, if there’s no
Killing vector, what do you do? What if there’s no perfect symmetry? Now, we saw that energy,
or at least this form of energy, this form of E, was conserved if you satisfy a certain symmetry.
It turns out that there’s another quantity, let’s call it S, which is still conserved
if a more general condition holds. That condition is written here, but this condition just needs to
hold for some vector field and it doesn’t need to be Killing. But can we always find such a vector
field? And what does this S even mean? I talk about gravity, gravitons, and gravitational energy
here with Professor Claudia de Rham if you’d like more detail. For now, let’s look at the expanding
universe, so the FLRW metric. In it, there’s no time-like Killing vector, so the standard energy,
where A is the scale factor, is famously, or infamously, not conserved. As the universe
expands, energy gets diluted, oddly. However, you can find a vector field that does satisfy
that condition prior. And this condition forces this formula here and a conserved quantity. Which
turns out, by the way, to obey this lovely guy here, which should look familiar to you,
if you’ve studied physics before, because that’s awfully close to the first law of thermodynamics,
provided that you interpret S as total entropy and beta as inverse temperature. What does this mean?
Does it mean that in cosmology, the conserved thing isn’t energy? Maybe it’s entropy, and
this beta of T tells us that the universe cools as it expands. Well, I’ll be talking to Ted Jacobson
soon about the very topic of emergent gravity and what that has to do with entropy, so feel free to
subscribe to get notified. It may already be out, you can check the description. This brings us back
to gravitational waves. LIGO detects them, so something previously thought was impossible
because of how weak gravity waves are. And binary pulsars spin down exactly as predicted if they’re
losing energy to gravitational waves. So surely gravitational waves carry energy? Well, yes,
in effective field theoretical approximations, or using pseudotensors, such as the Isaacson
effective stress energy tensor, which is derived from averaging metric perturbations. But if you
try to use a fully covariant stress energy based definition, like E or S, then pure gravitational
waves are vacuum solutions, so these definitions give zero energy. Interesting. Now, as an aside,
I should say there are other covariant quantities like the Bell-Robinson tensor, which are non-zero
for gravitational waves, but their physical interpretation as a definitive energy momentum
measure is debated, and the point about there not being a unique characterization of energy
still stands. 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? Perhaps we do need a way to account for gravitational energy, but the pseudotensor
isn’t it, unfortunately. So, where does that leave us? Defining energy in GR is not trivial. Well,
technically, defining even what light is isn’t trivial either. You can check out the Substack
for that. Pseudotensors give conservation, but they break covariance. Covariant definitions,
which are linked to the stress energy, big T, work cleanly when there are symmetries,
but they fail for general spacetimes or pure gravity. Generalizations, such as
the entropy-like S that we had before, they hint at some deeper structure, but it still lacks a
universal interpretation. Perhaps the seemingly indubitable structure of Einstein’s equation,
so that is that the matter sources curvature, but curvature doesn’t directly source itself in
the same manner, maybe that implies that only matter energy is truly well-defined. Or maybe
after 100 plus years, we just still haven’t figured it out and the search continues.