Of Particular Significance

# The Next Webpage: The Zero-Point Energy of a Cosmic Field

#### POSTED BY Matt Strassler

ON 02/16/2024

My two new webpages from earlier this week addressed the zero-point energy for the simple case of a ball on a spring and for the much richer case of a guitar string; the latter served as a warmup to today’s webpage, the third in this series, which explains the zero-point energy of a field of the universe. This subject will lead us head-first into the cosmological constant problem. As before, the article starts with a non-mathematical overview, and then obtains the results stated in the overview using pre-university math (except for one aside.) [As always, please comment if you spot typos or find some of the presentation especially confusing!]

(See the first post announcing this series for a brief summary of the hierarchy puzzle, which motivates this whole series, and for links to longer related discussions of it.)

The next webpage after this one will be an extension to today’s, covering other types of fields. That will lead us deeper into the cosmological constant problem, and begin to touch on the hierarchy puzzle.

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### 3 Responses

1. While I can follow what happens if you put a cosmological constant into the Friedmann equation, I don’t get this intuitively:

How can we understand intuitively why gravity might cause this positive energy and negative pressure to expand, opposite to what we expect gravity to do?

In a star in hydrostatic equilibrium, the positive pressure of the gas counteracts gravity and prevents contraction. Thus — intuitively but naively — one might think that reversing the sign of the pressure would make both the gravity and the pressure act in unison such that both produce contraction. Why is that thought wrong?

1. Right — this is something that’s counter-intuitive, and subtle. I tried to phrase it carefully, but it’s tricky.

You’re right that positive pressure itself causes any gas to expand. But this has nothing to do with gravity at all; it’s all due to the collisions among the objects in the gas, due to various forms of electric forces. Also, you’re imagining the gas is finite in extent. In an infinite gas, the pressure would still be there, but because there’s equal pressure on all sides, it wouldn’t do anything, just as the positive pressure of the atmosphere doesn’t locally cause the air to expand. You’d only know the pressure was there if you put an empty box, with nothing in it, inside the gas; then the positive pressure of the gas outside would crush the box.

You’re also used to the idea that mass creates gravity and causes a gas to contract. You’re probably even okay with the idea that mass is a form of energy, and that it’s energy, in Einstein’s theory, that causes the gas to contract.

What you’re not used to (because it occurs neither in Newton’s gravity nor in special relativity) is that *positive pressure also creates gravity and causes a spacetime full of such a gas to contract faster.* In Einstein’s theory of gravity, high pressure gases feel more gravity than lower pressure gases, even though the non-gravitational forces reflected in the higher pressure may well be able to counteract that gravity.

That’s, of course, with the positive pressure always present in ordinary gases.

But now we come to spacetime and its negative pressure. This negative pressure is everywhere, so just like the positive pressure in an infinite gas, which has no effect in the absence of gravity, negative pressure throughout all of space has no effect in the absence of gravity. You could only tell it was there if you could place inside it a box with zero pressure inside it, in which case the negative pressure outside would blow the box apart. Unfortunately, no such box exists — this is space itself that we’re talking about, and there’s no box without space inside it. So this experiment is even hard to carry out as a thought experiment.

Now, to this essentially infinite space full of negative pressure that doesn’t have any effects — we now add Einstein’s gravity.

If we had a substance whose pressure was small and negative, then the gravitational attraction caused by the energy density would still win, and the object would contract like any gas. This is also true of a spacetime that is full of such a gas. But the larger the negative pressure, the weaker the gravitational self-attraction of hte gas. And once the pressure equals minus the energy density/3 (which makes the energy-momentum tensor traceless, incidentally), the two effects balance; such a substance feels no self-gravity. Finally, with larger negative pressure, gravity’s effect is such as to cause the substance, or the spacetime full of such a substance, to expand.

No, it’s not intuitive. But Einstein’s gravity is not Newton’s, and one has to build a new intuition to work with it.

1. Thanks for the explanation Matt.

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