Theories of Everything with Curt Jaimungal - What is “Energy,” Actually?

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Starting point is 00:00:55 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. Forget the pop-size sound bites that you hear from people like Neil deGrasse Tyson, energy is not simply

Starting point is 00:01:21 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?

Starting point is 00:01:54 Physics professors skip the energy talk like dads skip 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

Starting point is 00:02:36 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,

Starting point is 00:03:14 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 the little t this time. This is the infamous pseudo tensor 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? Well, if you examine it, you realize the price was relatively steep. Yes, that's a pun.

Starting point is 00:03:47 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 bundled differential geometric view. Anyhow, this breaks the whole spirit of one of those foundational pillars, namely general covariance.

Starting point is 00:04:19 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.

Starting point is 00:04:52 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 called Guderdamerung events. Now, why is this expression here conserved? It's because of this other expression. Now notice that the first term here is 0, and the second term vanishes because the capital T this time is symmetric and the killing equation becomes this.

Starting point is 00:05:21 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.

Starting point is 00:05:51 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 timelike killing vector, which is killing outside the horizon,

Starting point is 00:06:19 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 the 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's sharp mass in this

Starting point is 00:06:56 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

Starting point is 00:07:25 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

Starting point is 00:08:04 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?

Starting point is 00:08:33 I talk about gravity, gravitons, and gravitational energy here with Professor Claudia de Romm 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

Starting point is 00:09:16 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

Starting point is 00:09:48 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-theoretic approximations, or using pseudotensors, such as the Isaacson

Starting point is 00:10:23 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,

Starting point is 00:10:58 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.

Starting point is 00:11:37 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 acts 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

Starting point is 00:12:11 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.

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Theories of Everything with Curt Jaimungal - What is “Energy,” Actually?

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