Why doesn't cosmic inflation violate conservation of energy?

[RogerWaters]

TL;DR Summary

Can you help me understand Lawrence Krauss on how cosmic inflation does not violate conservation of energy - due to: (a) the net-zero gravitational energy of objects; and (b) their "negative" relativistic pressure?

Hi all, I'm not a physics student (although I have a PhD in a different field) and so don't have the math, but I'm trying to interpret a key passage from Krauss' book 'A Universe from Nothing' where he is (trying?) to explain, in 'layman's terms', what Alan Guth termed 'the ultimate free lunch': namely, 'How can a microscopically small region end up as a universe-sized region today with enough matter and radiation within it to account for everything we can see'?

I'm trying to follow Krauss' reasoning and I'm not if he is just not including enough in the reasoning for it to make sense because it is too complicated without the math. I understand part (a) of his explanation for why cosmic inflation does not violate conservation of energy, in the (no doubt crude) terms used by Krauss for laypeople, but not part (b).

...(a) The net-zero gravitational energy of objects...

Krauss starts off stating: 'including the effects of gravity in thinking about the universe allows objects to have "negative" as well as "positive" energy'. As a result, 'gravity can start out with an empty universe and end up with a filled one'. He explains this by asking the reader to think about the net zero effect of kinetic energy and gravitational potential energy according to classical mechanics, using the example of throwing a ball up in the air on earth. I understand this. He then asks: 'What has all this got to do with the universe in general, and inflation in particular, you may ask? Well, the exact same calculation I just described for a ball that I throw up from my hand at the Earth's surface applies to every object in our expanding universe'. He then asks the reader to consider a galaxy at the edge of a spherical region of our universe large enough to encompass a lot of galaxies but small enough so that it is well within the largest distances we can observe. He states: 'just as for the ball from the Earth, we can ask whether the galaxy will be able to escape from the gravitational pull of all the other galaxies within the sphere. And the calculation we would perform to determine the answer is precisely the same as the calculation we performed for the ball' (in the example things are moving well under the speed of light etc.). Further 'we simply calculate the total gravitational energy of the galaxy, based on its motion outwards (giving it positive energy), and the gravitational pull of its neighbours (providing a negative energy piece). If its total energy is grater than zero, it will escape, and if it less than zero, it will stop and fall inward'. Fine.

He now moves between classical mechanics and relativity: 'Now, remarkably, it is possible to show that we can rewrite the simple Newtonian equation for the total gravitational energy of this galaxy in a way that reproduces *exactly* Einstein's equation from general relativity for an expanding universe. And the term that corresponds to the total gravitational energy of the galaxy becomes, in general relativity, the term that describes the curvature of the universe'. Ok, I'm still with him. Then: 'So what do we find? In a flat universe, and *only* in a flat universe, the total average Newtonian gravitational energy of each object moving with the expansion is *precisely zero*! This is what makes a flat universe so special. In such a universe the positive energy of motion is exactly canceled by the negative energy of gravitational attraction'. Further: 'In a flat universe, even one with a small cosmological constant, as long as the scale is small enough that velocities are much less than the speed of light, the Newtonian gravitational energy associated with every object in the universe is zero'. The end point of this line of reasoning is this: 'As each region of the universe expands to ever larger size, it becomes closer and closer to being flat, so that the total Newtonian gravitational energy of everything that results after the vacuum energy during inflation gets converted to matter and radiation becomes precisely zero'.

To my understanding, this means that cosmic inflation made the observable universe flat, and by doing so created the 'arena' wherein the net energy in the universe is allowed to be zero, meaning there is nothing to explain, energy wise (although of course one has to explain how the inflaton field originally collapsed to produce matter, but let's disregard this today).

What we can't disregard, based on the above, is how the universe 'expands to ever larger size', creating the (flat) arena wherein the net energy in the universe is zero. Indeed, Krauss states 'But you can still ask, Where does all the energy come from to keep the density of energy constant during inflation, when the universe is expanding exponentially'?

My first question is: Why does the idea of constant energy density suddenly appear here?? Is it related to the net-zero energy of objects which, as we have seen, Krauss explains via the idea of gravitational potential energy?

..(b) The "negative" relativistic pressure of objects...

What comes to rescue [to explain where the energy comes from the keep the density of energy constant during inflation] is 'another remarkable aspect of general relativity': 'Not only can the gravitational energy of objects be negative, but their relativistic "pressure" can be negative'. He attempts to explain negative "pressure" as follows: 'Gas, say, in a balloon, exerts pressure on the walls of the balloon. In so doing, if it expands the walls of balloon, it does work on the balloon. The work it does causes the gas to lose energy and cool'. Fine. But then: 'However, it turns out that the energy of empty space is gravitationally repulsive precisely because it causes empty space to have a "negative" pressure. As a result of this negative pressure, the universe actually does work *on* empty space as it expands. This work goes into maintaining the constant energy density of space as the universe expands'.

I can safely say that, as a piece of popular science writing (albeit one produced by a credentialed physicist), this utterly fails to bestow understanding. Is this because the matter is impenetrable without the math? I'm pretty sure I've encountered some writing by Sean Carroll who highlights the challenge someone gave of explaining "false vacuum energy" in intelligible terms, and attempts to meet the challenge but also fails.

Two further questions:

- What *is* negative pressure and what is this 'work' negative pressure does on empty space?
- Why does this 'work' not need its own 'energy balance' to explain how it was able to be carried out (thus violating conservation of energy')?

 

[PeroK]

First, the Krauss book is popular science, so it's never going to have the whole story. It's really just telling you about the physics, rather than doing the physics.

I'm not sure how important the issue of conservation of energy really is. Ultimately, it's a consequence of Noether's theorem and time symmetry. Time symmetry is lost in an expanding universe, so global conservation of energy is lost too. Here is Sean Carroll's perspective on things:

https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

If you have an hour to spare, you might be interested in this lecture by Alan Guth on inflationary cosmology. He presents, among other good stuff, the conservation of energy argument.

Two further questions:

- What *is* negative pressure and what is this 'work' negative pressure does on empty space?
- Why does this 'work' not need its own 'energy balance' to explain how it was able to be carried out (thus violating conservation of energy')?

Negative pressure is what it says: work must be input to expand a volume of vacuum. Ultimately, it's somewhat illogical to insist on conservation of energy, without re-examining the fundamental reasons that you believe energy should be conserved in the first place. The laws of physics are there to be tested and sometimes a domain of applicability is identified for a law that used to be universal. E.g. conservation of (rest) mass in Newtonian physics is replaced in Special Relativity by conservation of energy-momentum.

 

[Bandersnatch]

My first question is: Why does the idea of constant energy density suddenly appear here?? Is it related to the net-zero energy of objects which, as we have seen, Krauss explains via the idea of gravitational potential energy?

The reason it appears there is that constant energy density is what is needed to drive exponential expansion - such as during inflation, or in a dark energy dominated universe. I don't know if Krauss expounded on this before or maybe it simply went under the radar during your reading. In any case, that's what this is.
So, given how inflation flattening any prior curvature was provided as a justification for being able to treat today's universe as having zero nett energy, one needs to follow up with asking how the energy driving the inflation fits into this claim. Otherwise it'd be just moving the problem further back in time, rather than explaining it.
I.e. it says something along the lines: now that we've used inflation to justify zero energy in today's universe, let's move on to inflation itself.

Do note here that constant energy density in expanding space means that total energy 'amount' is increasing, as you have ever increasing volume times a constant. So the question being scrutinised here is: where does this ever increasing stock of energy come from?

- What *is* negative pressure and what is this 'work' negative pressure does on empty space?
- Why does this 'work' not need its own 'energy balance' to explain how it was able to be carried out (thus violating conservation of energy')?

The answer Krauss provides through the analogy with gas doing work on a balloon is precisely that of energy balance, but this time it's not adding up to zero.
In a balloon the gas has some energy associated with the motion of its molecules, but loses it as it does work to expand. I.e. energy of gas - work = less energy in expanded gas. Here we have an entity that does the opposite - it adds energy as it causes expansion. I.e. energy of empty space + work = more energy in expanded space (= constant energy density).
The point of this argument is as follows: If we imagined a universe (we're talking during inflation here) that were behaving like expanding gas, we could measure it at some moment in its history, and say 'There is now some energy in this universe, and it is expanding because the gas is doing work. Therefore there was more energy in the past.' And we'd end up asking ourselves where did this energy that's been driving the expansion come from in the beginning.
With negative pressure we measure the universe, find it expanding and having some energy, but it is not mysterious where the energy came from - the negative pressure component keeps adding it. As you look back towards earlier times during inflation, there is always less and less total energy, because the energy density is constant and the volume shrinks.
So in the end the argument goes: today, the universe doesn't need energy input to expand because of inflation. And the inflating universe did not need any energy input to expand because it's 'naturally' produced.

As to what it *is*, I'm not sure there's a satisfactory answer to be had. The negative pressure for dark energy (of which inflation is a cousin) shows up in the equations, and in relativity pressure with negative sign causes gravitational repulsion just like pressure with positive sign causes gravitational attraction (something that's not present in Newtonian gravity).It should be perhaps noted that this entire zero energy argument is not necessarily beloved by all, as PeroK's post above might have hinted at. I think you're more likely to find opinions that the question of the energy of the universe is rather ill defined. That is not to say that Krauss is massaging the facts, but is perhaps a bit too keen to justify a preconceived notion.

 

[RogerWaters]

Thanks PeroK and Bandersnatch.

I should have done a search for similar posts, as I see there are many, but it was informative for me to even formulate the question in writing and reading your answers.

I can't hope to understand the issue of net energy without knowing general relativity, and that ain't going to happen any time soon. But it's at least informative to know when one's understanding stops and why (i.e. it's because of a limitation in one's knowledge and not simply because confused about stuff one*does* have a basic handle on (i.e. classical mechanics)).