This post was inspired by Don Lincoln’s YouTube video “Subatomic Stories: Is the Planck length really the smallest?”

In his Q&A (where he responds to questions from prior videos), he notes a caveat about the law of conservation of energy. Energy may not be conserved … because space-time can change.

He offers some links for more thorough and technical explanations (see below).

In what special cases is this important? Models of the early universe and the cosmological constant … black holes … quantum gravity … Anything less cosmic?

• YouTube > Fermilab > Don Lincoln > “20 Subatomic Stories: Is the Planck length really the smallest?” (Aug 19, 2020)

Conservation of energy isn’t always real. Now, me saying that conservation of energy isn’t really true is pretty staggering and requires some explanation. Bear with me, because this is somewhat technical.

According to the theory, a conserved quantity implies that the equations don’t care where you set your zero. For instance, for conservation of energy, the equations can’t care what moment you set as time zero. … If the laws of physics don’t care about that choice, energy will be conserved. But an additional requirement that is never mentioned is that space and time must be static and unchanging. But, of course, in general relativity, that last requirement is not satisfied. Space and time can change and warp and distort. Accordingly, the law of conservation of energy doesn’t necessarily apply. So that’s the reason that it appears that energy is lost. It’s because non-conservation is not only allowed, it’s expected. Now, the description I’ve given here is the gist, but, because of the limited time we have, it’s very brief. So I’ve put links to two more thorough and technical explanations in the video description below.

• Sean Carroll’s blog > “Energy conservation in general relativity” (2010)

In his 2010 post, Carroll discusses something that’s long been understood about General Relativity (GR), namely, that energy is not conserved because spacetime is not fixed (static).

The dynamic nature of spacetime is key to the theory of Big Bang Nucleosynthesis – the expansion and energy density of the universe.

He notes that some experts in cosmology and GR frame the physics in terms which preserve conservation by including gravitational field energy in the tally. He finds such an approach counterproductive due to:

1. Ambiguous mapping at points of space: “Unlike with ordinary matter fields, there is no such thing as the density of gravitational energy. The thing you would like to define as the energy associated with the curvature of spacetime is not uniquely defined at every point in space.“
2. Dubious pedagogical benefit – introducing negative energy rather than saying “spacetime can give energy to matter, or absorb it from matter.”

The point is pretty simple: back when you thought energy was conserved, there was a reason why you thought that, namely time-translation invariance. A fancy way of saying “the background on which particles and forces evolve, as well as the dynamical rules governing their motions, are fixed, not changing with time.” But in general relativity that’s simply no longer true. Einstein tells us that space and time are dynamical, and in particular that they can evolve with time. When the space through which particles move is changing, the total energy of those particles is not conserved.

It’s not that all hell has broken loose; it’s just that we’re considering a more general context than was necessary under Newtonian rules. There is still a single important equation, which is indeed often called “energy-momentum conservation.”

… energy and momentum evolve in a precisely specified way in response to the behavior of spacetime around them. If that spacetime is standing completely still, the total energy is constant; if it’s evolving, the energy changes in a completely unambiguous way.

In the case of dark energy, that evolution is pretty simple: the density of vacuum energy in empty space is absolute constant, even as the volume of a region of space (comoving along with galaxies and other particles) grows as the universe expands. So the total energy, density times volume, goes up.

This bothers some people, but it’s nothing newfangled that has been pushed in our face by the idea of dark energy. It’s just as true for “radiation” — particles like photons that move at or near the speed of light. The thing about photons is that they redshift, losing energy as space expands. If we keep track of a certain fixed number of photons, the number stays constant while the energy per photon decreases, so the total energy decreases. A decrease in energy is just as much a “violation of energy conservation” as an increase in energy, …

• University of California, Riverside > Math > “Is Energy Conserved in General Relativity?” (2017)

This article by Michael Weiss and John Baez addresses the question at a more technical level. In particular, the effort (indirectly referenced by Carroll) using “pseudo-tensors” to accomodate conservation.

In special cases, yes. In general, it depends on what you mean by “energy”, and what you mean by “conserved”.

In flat spacetime (the backdrop for special relativity), you can phrase energy conservation in two ways: as a differential equation, or as an equation involving integrals (gory details below). The two formulations are mathematically equivalent. But when you try to generalize this to curved spacetimes (the arena for general relativity), this equivalence breaks down. The differential form extends with nary a hiccup; not so the integral form.

The differential form says, loosely speaking, that no energy is created in any infinitesimal piece of spacetime. The integral form says the same for a non-infinitesimal piece. (This may remind you of the “divergence” and “flux” forms of Gauss’s law in electrostatics, or the equation of continuity in fluid dynamics. Hold on to that thought!)

As often in physics, Weiss’ discussion of energy flux across infinitesimal (spacetime) volumes involves artful modeling. [1] Particularly when you encounter non-linear equations with synergetic effects and decide what level of detail is good enough. [2] In this case, as noted, conservation requires wrangling something to have an invariant meaning.

The article concludes:

We will not delve into definitions of energy in general relativity such as the hamiltonian (amusingly, the energy of a closed universe always works out to be zero according to this definition), various kinds of energy one hopes to obtain by “deparametrizing” Einstein’s equations, or “quasilocal energy”. There’s quite a bit to say about this sort of thing! Indeed, the issue of energy in general relativity has a lot to do with the notorious “problem of time” in quantum gravity… but that’s another can of worms.

Notes

[1] As well as skillfully applying corresponding equations, as discussed by Chad Orzel. When is a cow like a sphere, when is a cow treated as a point?

• “The Hardest Thing To Grasp In Physics? Thinking Like A Physicist” by Chad Orzel (Aug 29, 2016)

This kind of simplified model-building isn’t completely unique to physics, but we seem to rely on it more heavily than other sciences.

You need to know not just how to do calculations that treat cows as spheres, but when it’s appropriate to do that. And that helps make thinking like a physicist the hardest part of the discipline to learn.

[2] For example, perhaps gravitational waves interact and contribute to the energy tally (and so an additional source of gravity)? As well as the scale at which effective theories apply (as in scale analysis).

Terms

Divergence

In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source at a given point. It is a local measure of its “outgoingness” – the extent to which there is more of the field vectors exiting an infinitesimal region of space than entering it. A point at which the flux is outgoing has positive divergence, and is often called a “source” of the field. A point at which the flux is directed inward has negative divergence, and is often called a “sink” of the field. The greater the flux of field through a small surface enclosing a given point, the greater the value of divergence at that point. A point at which there is zero flux through an enclosing surface has zero divergence.