Ask Ethan: Does cosmic inflation violate energy conservation?

Here in our ever-changing Universe, there are a few things we can rely on remaining constant and unchanged. From place-to-place and moment-to-moment, the laws of physics don’t change, they remain constant and apply equally across the entire span of the Universe. The values of the fundamental constants — the parameters that determine the masses of particles and the strength of forces — don’t change with time or across space. And in every experiment we’ve ever performed, we’ve seen that certain quantities are always conserved: the total amount that you start with always equals the total amount that you end with.

Many conserved quantities abound: electric charge, linear momentum, angular momentum, and, of course, energy. The conservation of energy is maybe the best known of the conservation laws, and applies to objects on Earth just as well as it applies to the Solar System, the stars, and the entire Milky Way. But if you go way, way back in cosmic history, would you find that, just possibly, energy isn’t conserved after all? That’s what Michael Genovese wants to know, asking:

“If the energy field density remains the same during cosmic inflation, how does that also agree with the conservation of energy principle (since the energy would be increasing exponentially with the space increasing and density remaining steady)?”

The answer, shocking though it may be to hear, is that energy isn’t always conserved, and cosmic inflation actually violates energy conservation in a particularly extreme way. Here’s what’s going on.

Different frames of reference, including different positions and motions, would see different laws of physics (and would disagree on reality) if a theory is not relativistically invariant. The fact that we have a symmetry under ‘boosts,’ or velocity transformations, tells us we have a conserved quantity: linear momentum. The fact that a theory is invariant under any sort of coordinate or velocity transformation is known as Lorentz invariance, and any Lorentz invariant symmetry conserves CPT symmetry. This notion of invariance under constant motion dates all the way back to the time of Galileo.

It turns out that if you want to know “what’s conserved” in the Universe, you need something specific that underlies the rules that govern your system: a symmetry. There are profound connections between symmetries and conserved quantities, and they were first revealed more than a hundred years ago by one of the unsung heroes of physics and mathematics: Emmy Noether. She wrote two important proofs, published in 1915, whose content showed for the very first time that if you had the following three criteria in place:

• a physical system,
• where the forces at play are conservative (the force of one object on another leads to an equal-and-opposite force acting back on the first object),
• whose “action” has a differentiable symmetry,

then each and every independent differentiable symmetry that it possesses will also lead to an associated conservation law. If there’s a conservation law, then there’s a conserved quantity. That is the core of the deep connection between conserved quantities (like energy) and conservation laws and symmetries in general.

Emmy Noether, the person who proved Noether’s theorem, which connects symmetries and invariances of a theory with an associated conserved quantity. At right, the illustration of conservation of linear momentum, a consequence of spatial-translation invariance, is shown.

In other words, these proofs, now known as Noether’s theorem, tell us that every time you have a physical symmetry within your physical system, then there’s something physical associated with your system that will be conserved. There are many important examples we can look to that illustrate this powerful connection.

• If your system is symmetric under rotations, then that system’s angular momentum will be conserved.
• If your system is invariant under spatial translations (i.e., you can move to any location and still recover the same system), then that system’s linear momentum will be conserved.
• If your system is invariant under boosts (also known as velocity transformations), then it leads to the conservation of four-momentum: the center-of-mass theorem.
• And if your system is invariant under time translations (i.e., your system is identical to how it was in the past or will be in the future), then it leads to that system’s energy being conserved.

That last item is of particular relevance here: if your system doesn’t change with time — if its laws and properties are indistinguishable whether you examine it earlier, later, or right now — then energy must be conserved within it.

But the converse is also true: if the Universe evolves with time, and an example of evolution is the distance between two points within that Universe changing with time, then there isn’t anything mandating that energy be conserved at all.

A ball in mid-bounce has its past and future trajectories determined by the laws of physics, and only loses energy between bounces due to dissipative interactions of the ball with the ground. One cannot tell, based on this photo, whether the ball bounces to the left and gains energy, or bounces to the right and loses energy. In general, if all forms of energy are taken into account, energy is conserved for all four of the fundamental interactions on Earth. In the expanding Universe, however, gravitation experiences a different story.

This might puzzle and confuse you. After all, energy conservation isn’t something that we realized was a property of the Universe because of Noether’s theorem; it was something that was uncovered empirically: by quantifying and measuring the total amount of energy

• initially present in a system,
• that’s added to or removed from that system,
• and then that exists in that system in its final state.

If the initial energy, plus or minus the energy that’s added or removed to the system, is equal to the final amount of energy in the system, then energy conservation is indeed confirmed. If not, then energy conservation is violated.

For laboratory systems here on Earth, energy is indeed always conserved in exactly this fashion. When you collide two particles together, no matter what it is that goes into or comes out of your experiment, the sum total of all the different initial energies and the initial final energies will always match. When you release a particle in the gravitational field of the Earth, whether you shoot it up, throw it horizontally, or let it fall, the total sum of the particle-Earth system initially and at the end will always be equal. In fact, under all four fundamental interactions — gravitational, electromagnetic, or even the weak and strong nuclear forces — any physical system examined here on Earth has always been observed to conserve energy.

When light is emitted from a source, it has a particular wavelength. The longer it must travel through the expanding Universe before being absorbed by an observer, the greater the amount that the wavelength of that light will be redshifted, or stretched to longer values, compared to the wavelength it has when it was emitted.

But if you leave our cosmic backyard behind, things are no longer so simple. If you were to shine a light outward into the broader Universe from planet Earth, it would start to appear as though energy was indeed still conserved. As the photon — the quantum of light — that you created rises up through Earth’s atmosphere and into space, it loses energy, causing its wavelength to get longer. This isn’t a violation of energy conservation, however; energy is conserved because the photon’s energy loss is exactly balanced by the change in gravitational energy as it moves farther away from the center of the Earth.

This same process continues as the photon leaves the Solar System: it loses energy (and lengthens in wavelength) as it moves farther and farther away from the Sun and the planets, trading a gain in gravitational potential energy (climbing out of the gravitational potential well) for a loss in the photon’s kinetic energy. As the photon escapes from the gravitational pull of the Milky Way, the same thing happens: energy loss in the photon’s kinetic energy is balanced by the gravitational potential energy it gains by leaving the gravity field of the object it originated from. This continues all the way up to its escape from the Local Group, which is the last time that total energy is conserved.

Once the photon enters true intergalactic space, the story changes dramatically, however. As it travels through the Universe, it now loses energy, with no other form of energy experiencing a corresponding gain in energy.

As the fabric of the Universe expands, the wavelengths of any radiation present will get stretched as well. Within a bound structure, like a star system, galaxy, or galaxy group, space doesn’t expand, but between bound structures, in deep intergalactic space, space does expand. This stretching of wavelength (and loss of energy) applies just as well to gravitational waves and matter (de Broglie) waves as it does to electromagnetic waves; any form of radiation has its wavelength stretched as the Universe expands.

What’s happening here?

Once a photon — or any form of energy, for that matter — leaves the Local Group of galaxies, it’s no longer part of a (gravitationally) bound structure. Instead, it’s just traveling through space: through the expanding space of the broader cosmos. After all, we’ve known for right around 100 years that the Universe is expanding, as it was both predicted theoretically and measured observationally back in the 1920s. When the Universe expands, the distance between any two points gets larger over time, including the two points that define the size of a “wavelength” for any individual photon, or quantum of light.

In other words, as the Universe expands and a photon travels through it, its wavelength stretches to longer and longer wavelengths, which in turn takes it to lower and lower energies. (A photon’s energy is defined by its wavelength!) This time, the energy doesn’t “go” anywhere; the final energy is simply different from the initial energy because energy is not conserved within the expanding Universe. After all, without time-translation symmetry, there is no corresponding conserved quantity of energy, and a Universe that expands (or contracts, for that matter) is fundamentally different — particularly in terms of the distance separating any two points — from one moment to the next.

For radiation traveling through an expanding Universe, energy isn’t conserved; it’s lost.

This simplified animation shows how light redshifts and how distances between unbound objects change over time in the expanding Universe. Note that the objects start off closer than the amount of time it takes light to travel between them, the light redshifts due to the expansion of space, and the two galaxies wind up much farther apart than the light-travel path taken by the photon exchanged between them.

But radiation isn’t the only type of energy allowed to be present within our Universe. There’s also matter (both normal and dark), neutrinos (which behave like radiation when they have lots of kinetic energy and behave like matter when they have very little), topological defects (such as cosmic strings and domain walls, which may be theoretical only), and energy inherent to the fabric of space itself. That last form of energy shows up in many different fashions.

• A cosmological constant, known as Λ in Einstein’s general relativity, behaves as a form of energy inherent to space.
• The vacuum energy of quantum field theory, also known as the zero-point energy of space, behaves as energy inherent to space.
• Dark energy, which drives today’s modern accelerated expansion of the Universe, is a form of energy inherent to space.
• And early on in cosmic history, during a period of cosmic inflation that preceded and set up the hot Big Bang, the energy inherent to space was enormous, and drove the Universe to expand at an exponential rate: doubling in size in all three dimensions with each tiny fraction-of-a-second that elapses, all while its energy density remains constant.

That’s the key feature to a Universe that has energy inherent to space: as it expands, new space gets created, but the energy density remains constant. Same density but more volume means more total energy, overall. Whereas a radiation-filled Universe (i.e., a Universe filled with photons) loses energy as it expands, a Universe filled with energy inherent to space (i.e., the inflating Universe) gains energy as it expands.

How matter (top), radiation (middle), and dark energy/inflationary energy (bottom) all evolve with time in an expanding Universe. As the Universe expands, the matter density dilutes, but the radiation also becomes cooler as its wavelengths get stretched to longer, less energetic states. Dark energy’s (or inflationary energy’s) density, on the other hand, will truly remain constant if it behaves as is currently thought: as a form of energy intrinsic to space itself. The overall energy in an expanding patch of space is not conserved.

This isn’t a bug; this is something that’s mandatory in our expanding Universe. Because there’s no time-translation symmetry — because the Universe at any moment is different from the Universe at either the previous or the subsequent moment — energy is not conserved at all. It’s tempting to state that, depending on the type or species of energy present within the Universe, the total energy is something that can either decrease (like for radiation) or increase (like for inflationary energy), but a more truthful statement would be that energy is not even well-defined in a Universe without time-translation symmetry. In an expanding Universe, there is no consistent definition of energy at all.

Why is it impossible to define energy in a Universe without time-translation symmetry, such as in an expanding Universe?

Consider a comparison with electromagnetism. When things with electric charges move apart, for example, it’s because either some conservative force (like the electric force) is pushing them apart from one another, or some external force is being applied over a distance, performing work on those particles in order to move them. Since work is just another form of energy, and because the equal-and-opposite forces that act on these charges conserve the total (kinetic + electric potential) energy, it’s easy to see that energy conservation still holds. But for gravitation, things (including things with mass and/or energy) can move apart because there’s a conservative force (like a periodic comet departing from the Sun), or they can move apart because the fabric of space itself is actually expanding.

Both inside and outside the event horizon of a Schwarzschild black hole, space flows like either a moving walkway or a waterfall, depending on how you want to visualize it. Unlike the case of an expanding Universe, the case of a non-rotating point mass in spacetime, making a Schwarzschild black hole, is a static spacetime solution with time-translation symmetry. In this spacetime, all forces are conservative and energy, too, is always conserved.

In the first case, of a comet orbiting the Sun, the changes in distance and speed of the comet-Sun system result in energy being conserved, as the total initial energy and the total final energy will be equal. But in the second case, of a Universe that’s expanding, there isn’t a good way to define what the total energy of the system is, because there’s no objective, unchanging reference frame to define it with respect to.

Because we can observe and measure that the Universe is expanding, that means that relative positions and distances change over time. Consequently, so do any energy-dependent quantities that depend on the amount distance or space that objects occupy, including quantities such as wavelength, gravitational potential energy, and the volume of space (with a positive, constant energy density during inflation) enclosed within the observable Universe.

People often ask whether it costs energy to expand the Universe, or whether we would gain energy if the Universe stopped expanding and began contracting instead. But this assumes that “separating objects with energy” is a thing that costs energy in the first place; that’s only true in a non-expanding, non-contracting Universe! The expansion of space is “free” in the sense that it doesn’t cost any energy at all; it occurs over time, for free, as an inevitable consequence of the expanding Universe.

The ‘raisin bread’ model of the expanding Universe, where relative distances increase as the space (dough) expands. The farther away any two raisins are from one another, the greater the observed redshift will be by the time the light is received. This expansion causes the radiation density to drop and the energy-per-photon to drop, but allows the dark energy density to remain constant. This combination can raise or lower the total energy one would calculate for the observable Universe, apparently violating the conservation of energy.

It’s kind of funny, when you think about it, that this is one very, very important case where the old saying that “there’s no such thing as a free lunch” doesn’t apply. When the Universe expands at an exponential rate — both during inflation at early times and during dark energy domination at late times — there is something that you do get for free: extra amounts of space. Empty space, under these conditions, doesn’t cost a thing; as it gets created, it comes along with the same energy density that the space that created it also had.

Does that mean we get “free energy” from inflating, or any exponentially expanding, space?

I wouldn’t look at it that way; global energy is not something that’s well-defined for an expanding Universe. In fact, if you don’t have a time-translation symmetry to your Universe — including for your spacetime under Einstein’s relativity — that ensures that energy is not conserved.

It is possible to define pressure, work, and energy in such a way that each patch of space that expands does negative work on its surroundings, with dark energy doing the work with a pressure of P = -ρc². However, this is an ad hoc definition, not one imposed by the equations. In the reality of the expanding Universe, global energy isn’t only not conserved, it’s not defined.

It’s tempting to attempt to redefine energy in such a way to include the work done by a patch of space on its surroundings, both in positive (e.g., from radiation) and negative (e.g., from dark energy) forms. But the only thing you gain from that is your own personal satisfaction from having concocted a definition that allows this new “thing” to still be conserved in the expanding Universe; there is nothing useful, extractable, or applicable about it, however. It doesn’t behave like energy in the conventional sense; it’s just a redefinition driven by our own biased sense of aesthetics.

Cosmic inflation does indeed violate energy conservation, but not because there’s anything special about dark energy or a spacetime that’s exponentially expanding: with a constant energy density. As long as your spacetime lacks a time-translation symmetry, energy conservation is violated. Moreover, an energy definition is not rigorously possible under such conditions. Symmetries play a key role in physics, but there are many symmetries our Universe simply doesn’t display. As long as your Universe expands, contracts, or changes with time, energy won’t be conserved there, either.

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