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.

From Richard Mattuck’s “Guide to Feynman Diagrams in the Many-Body Problem”

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:

the indelible oscillations of the quantum field

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:

Oscillations with shorter wavelength (higher frequency) have higher energy

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):