# Being pushed around by empty space: The Casimir Effect

It is the view of modern physics that there is no such thing as *truly* empty space. When I first heard this, I thought that the person saying it was some kind of crackpot. Didn’t we move past the aether theory in the 19th century? But apparently it is the honest belief of most professional physicists that what we call empty space, or “vacuum”, is really some kind of infinite, space-filling “fabric” upon which ripples can be created that carry force from one object to another. This is the idea of the quantum field. And in a sense, it’s a ridiculous idea, but it’s one that developed slowly through many painful years of puzzling over strange experimental results. The properties of the vacuum — of this strange quantum fabric — are confusing and hard to decipher. A lot of nearly-impenetrable (to me) mathematics has been created for the purpose of their description.

What makes everything so confusing is that apparently the quantum field is always “boiling”. It is filled with a certain dense energy, one which allows particles to spontaneously pop in and out of existence. In fact, our most detailed and accurate description of forces is that they involve the transfer of energy across the quantum field via short-lived excitations which can be called “particles” or “virtual particles”. It all sounds a little like black magic.

The difficulty physicists have in describing empty space is actually the subject of a running joke, which goes like this: In the days of Isaac Newton, people struggled to understand how to predict the combined motions of three interacting objects: the “three-body problem”. However, scientists were pretty sure that they had definite predictive power in the “two-body problem”. Over time, as relativity and quantum field theory were developed, we realized that the two-body and even the one-body problems were much harder than we expected, and we lost the ability to perfectly predict what was happening. Currently, we have to say that we don’t even understand the zero-body problem! We can’t say for sure anymore what “empty space” is like, so apparently we have been making negative progress over the past 300 years.

Joking aside, talking about the boiling of empty space brings up a serious question. If the space around me is filled with bubbling energy, why doesn’t it push me around?

Well, the short answer to that question is the same as the answer to “why don’t I get pushed around by the energetic air molecules surrounding me?” Namely, the air molecules are all very small and they surround me equally, so that on the whole I don’t get pushed in any particular direction.

But there *is* such a thing as air pressure, and it actually can actually push you around under the right conditions. So is there such a thing as “vacuum pressure”? Can something be pushed around by empty space?

The answer, surprisingly, is yes. It’s called the Casimir Effect, and it usually gets stated like this: two metal plates, sitting next to each other in an absolute vacuum, will be attracted to each other as they are pushed by the “boiling” vacuum energy.

**The quantum field in one dimension**

Just to make things easier to visualize, I’m going to reduce everything down to one dimension. That is, I’m going to pretend that all of space consists of a single line, along which objects can move. I imagine it like a tightly-stretched string that objects can slide across. Objects living in this string-world can exert a force on each other by making vibrations of the string, which propagate down the line and disturb the motion of others.

In a sense, this is how I think about the electric field in the real world. Consider the repulsion between electrons, which we normally say is mediated by the electric field. In our analogy, two electrons would be dwellers on the “quantum string”. As they sit on the string, the continually disturb it, sending out packets of vibration in either direction. Something like this:

As the disturbance created by each electron hits the other, the electrons end up pushing each other in opposite directions. In our analogy, the disturbance of the string *is *what we call the electric field.

The picture is complicated by the fact that even in the absence of electrons, the quantum string is always vibrating. In fact, to the best of our knowledge, it is vibrating quite violently, in a way that can best be described as “white noise“: all possible frequencies of oscillation are equally represented. If you imagined the quantum string to have some finite length, its disturbance in the absence of electrons (its “vacuum oscillations”) might look like this:

When electrons are present, the electric field they create — the disturbances they send down the line — must travel on top of all this noise. But as long as the string is well-behaved this isn’t a problem: the extra disturbance just propagates down the line without any trouble.

One thing should bother you, though: when I say that the string vibrates with all possible frequencies, it implies that the string has infinite energy. For every mode of oscillation, there is a certain energy corresponding to that mode. The “white noise” picture above is actually a combination of many different oscillation frequencies, each with its own contribution to the energy. Like this:

The vacuum oscillations of our quantum string somehow have contributions from *all* of these modes, which means the string contains the sum of all their energies, plus infinitely more. If you think it sounds silly to say that vacuum oscillations have infinite energy, so that all of space is filled with an infinite density of energy, then I would agree with you. But that is precisely the case in our best description of the vacuum.

**What happens when you block the field?**

Apparently creating an electric field between two objects is a matter of disturbing the vacuum that separates them. But there are materials that can block electric field. What effect do such materials have on the vacuum itself?

Consider a slab of metal. As a rule, the electric field inside a piece of metal is zero. This is because of all the mobile charges the metal has: if electric field is applied to the interior of the metal, then all the electrons inside it are pushed around until they achieve a configuration that cancels the field. The idea behind the Casimir effect is this: maybe a piece of metal can block not just the electric field made by charges, but the vacuum oscillations themselves. Perhaps the mobile charges within the metal respond to oscillations in the vacuum in the same way that they responded to normal electric field: by rearranging themselves to cancel it out.

If that’s the case, then the vacuum between two pieces of metal should be somewhat different than the vacuum outside. Normally, the vacuum can have arbitrarily low-energy oscillations (if nothing is bounding it, it can have infinite wavelength). But between two pieces of metal, it cannot oscillate with a wavelength longer than twice the distance between the plates. So there is a certain amount of energy *outside* the two metal plates that is not present *inside*. This is the main idea behind the Casimir effect. The vacuum is oscillating with more energy outside the plates than it is inside, and as a result the plates get pushed together. It’s kind of like having an air-tight box with some of the air removed from the inside. There is a difference between the amount of energy outside the box (in the form of energetic air molecules) and the amount inside the box, so the walls of the box get pushed inward. The same thing is happening here: the surrounding “medium” (the vacuum) has more energy outside the plates than inside, so the plates get pushed together.

To make it concrete, you can think again about our one-dimensional “quantum string”, but this time we put three metal plates along the string. If you arrange them so that the distance between the left plate and the middle plate is half the distance between the middle plate and the right plate, you get a situation like this:

That extra mode on the right makes a difference. It means that there is a higher energy density on the right than there is on the left. As a result, the middle plate gets pushed to the left.

And of course, the closer the two plates on the left are to each other, the greater the difference between the energy densities. As a result, the apparent pressure pushing the plates together gets larger as the distance between them gets smaller. In 1D the pressure is proportional to , and in 3D it’s .

**Visualizing the Casimir effect**

On Wikipedia, the Casimir effect is illustrated like this:

I guess those big, blue bubbles stand for vacuum oscillations. It’s a pretty good picture; you can see that the longer-wavelength oscillations exist outside the plates but not inside. But I think an animation is in order, where you can actually watch the torturous twitching of the vacuum. I made my own attempt at a visualization in the video below.

The two plates on the left start out slightly closer together than the ones on the right, so there is an imbalance in pressure that causes the plates to come together. If you look closely, you can see that low-frequency oscillations start getting “squeezed out” near the end of the video.

**Update**: I just came across this beautiful video on wikimedia showing the “Casimir effect” for water waves (click to view):

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Your blog is GREAT.

Empty space (between galaxy clusters?) is filled with photons from every direction. Photons of light from stars 10 billion light years distant, neutrinos, photons of microwave energy, protons, hydrogen molecules, etc. The solar wind, for example, has a density of about one to six) protons per cubic centimeter at over 300 kilometers per second. There’s stuff from supernovae measured outside the Sun’s photosphere. Although not very dense, a few protons per cubic centimeter, plus cosmic waves, x-rays, and everything else from every direction, the inter-galactic “vacuum” can hardly be considered empty.

Hi Henry,

Your point is good; interstellar space is not really empty, but has lots of photons and other stuff. However, I should emphasize that the Casimir effect does /not/ require actual photons to be present. It arises from /virtual/ photons, which are intrinsic fluctuations of the quantum field.

So two metal plates in truly empty space are still pushed together, even if there were no stars or anything else in the universe emitting radiation.

I don’t think it’s correct to say that the vacuum energy is infinite: It’s zero point because while a virtual particle can be of any energy, it’s created with it’s opposite and the sum of the two is always zero. Hence, the vacuum energy is always at ground state. Am I wrong about this?

First, a clarification, just to make sure we’re speaking the same language:

A “particle” is the creation of a vibrating mode in the “quantum string” that is above the ground state. The quantum string is vibrating all the time in the way that I described above. It is only when additional vibrations are added that you have particles.

So the energy of the vacuum is completely independent of the existence of particles. And, by our best description, that energy is infinite because the quantum string is vibrating “infinitely violently”.

Now, about particles and antiparticles:

It is true that when particles spontaneously appear/disappear, they do so in pairs; electrons appear together with positrons, for example. But when they are created, both the electron and the positron have positive energy. It still requires a large amount of energy input to create them out of nothing. The concept of “negative energy” in a vacuum doesn’t really have a meaning.

En excellent blog, keep writing! and we’ll keep reading. Very well explained the Casimir effect.

Thank you so much. FASCINATING.

Thanks for this explanation. I came here looking for another one though and well, 5 years after this post was put online, I’m still going to ask :

Is ’empty’ space ‘filled’ with an electromagnetic field? Behind this question I’m trying to understand how else can light travel through ’empty’ space… as a wave of vibrating what, if not the field itself ?

I’ve searched around a confusing and contradicting bunch of explanations on the Web, so I thought I’d ask someone with good explaining skills!

And sorry if I post in a wrong place

Hi Sylvain,

Yes, I think it’s fair to say that the electromagnetic field fills all of empty space, and that electromagnetic waves are ripples or disturbances of this field. But one should be careful not to think of the electromagnetic field as literally identical to some physical liquid like water. For example, an electromagnetic wave moves at the same speed (the speed of light) with respect to any reference frame. This clearly wouldn’t be the case for a water wave.

So it’s probably best to think of all this language describing “ripples on a space-filling fluid” as a metaphor designed to give intuition about how the electromagnetic field works. But ultimately, if one is to understand the electromagnetic field at the most correct level, it should be understood on its own terms, rather than by analogy with something you can see.

Thank you very much for your answer. All in all, the universe is pretty cool I’d say, no pun intended 😉

Have a Happy New Sun Revolution!

But, for the contribution of each mode, don’t we have to take into consideration the coefficients of the Fourier expansion? They way I think it, is that there are contributions from each mode indeed, but the ones with higher energy have smaller coefficients and thus your integral converges (i.e. not infinite energy).

As an aside, when the plates get closer, does the entropy increases or decreases? (well, the answer is increases because of the 2nd law, isn’t?). How do you picture intuitively that the configuration with the plates closer to each other has higher entropy (A first, naive thought tells me otherwise!)? Or asked in another way, how are the plates minimizing energy by being closer? Should I think it as an “electromagnetic vacuum pressure”?

Good questions. I’ll second your request for a comment from Brian.

Well, now that the request is seconded, I guess my hand is forced. 🙂

Danyel, the way you imagined it would, hypothetically, make sense. But, as far as we can tell, there really is an infinite amount of energy in the vacuum. In other words, those Fourier coefficients do

notget smaller at larger frequencies. One way you can think about it is that the properties of the electromagnetic vacuum arise from the behavior of a whole bunch of coupled oscillators (like tiny springs tied together), one at every point in space. (see, for example, https://gravityandlevity.wordpress.com/2010/08/29/so-is-the-universe-made-of-tiny-springs-or-isnt-it/ ) Each oscillator has one quantum of energy. But since there are infinitely many points in a given region of space, there are also infinitely many oscillators, and therefore an infinite amount of energy.Of course, it might be true (and is even likely to be true) that at high enough frequency those Fourier coefficients

dostart to decrease, and the energy in the vacuum remains finite. But this is sort of equivalent to saying that there is a fundamental spatial resolution to free space: a minimum spacing between “springs”.As for your second question, you’re right that the energy of the vacuum is decreasing when the plates come together, and you definitely can think of it as an “electromagnetic vacuum pressure”. I don’t have a great intuitive way to see this, but you can arrive at the conclusion by looking at the three-plate setup above, and then counting the total amount of energy in all of the modes on the left side and on the right side. When you’re done counting (which is tricky, since the total energy on each side is infinity, and you have to compare two infinities), you’ll see that the total energy of the universe goes down when the middle plate comes closer to one of the other two.

what do you mean by “energy is lower”? shouldn’t the energy be conserved and never change in total?

I guess I mean that “the potential energy is lower”. By definition, a force always pushes in the direction of lower potential energy. So when a (net) force exists on some object, it necessarily means that the potential energy is lowered when the object moves in the direction of that force.

Fantastic. I have never followed a blog, but I will start now. You have a great way of simplifying concepts that make them reasonable to people without the heavy science background. I love it!

Thank you!

Thank you for this interesting blog Brian. My question is that do we know the reason of existence of vacuum oscillations which are electromagnetic in nature. I have always read that only charged particles can generate E-M waves. So there must be particles in the vacuum then and hence the cyclic argument? Could you please share your thought on this? Thanks.

Thus, all objects that are not repulsive that get close together, get pushed further together?

That seems to be the case, yes. Unless some other effect appears that overwhelms the Casimir effect, vacuum fluctuations will always draw objects together.

The Casimir effect is actually closely related to the Van der Waals force, and usually there is no meaningful distinction between the two phenomena.

I don’t understand the idea that having infinetly many frequencies makes space have infinite energy, because i think of a fourier transform, when representing any function I could write it as a sum of infinitely many modes of vibration, yet we have a finite function with finite energy