# Negative energy density in a Casimir vacuum

1. Jul 15, 2004

### hellfire

I have read often that the vacuum energy density in a Casimir vacuum is negative, but I do not understand this.

My understanding is the following.

The Hamiltonian can be written as a sum over all spatial modes. Calculating the energy density for the vacuum state H|0> = ∫dV ρ_o |0>, leaves an integral ρ_o ~ ∫ w dk, which is divergent.

As far as I know one has two options now.

The first option is to take ρ_o as a normalization constant and assume the ground state energy is zero. In my oppinion this is only justified for cases where vacuum energy is negligible or not observable.

The second option is to calculate the integral with some cut-off for low wavelengths. This procedure is used for cases where the vacuum energy density has to be taken into account, like cosmology (additionally, other assumptions are made in order to make the intergral smaler to fit with the cosmological constant).

Now, to calculate the value of the energy density in a Casimir vacuum, one takes the same integral but ignores some modes, which are not possible due to geometrical constraints.

Thus, for cosmological or gravitational purposes the Casimir vacuum should have positive energy density according to my reasoning above.

But reading about metric engineering based on negative energy density (Alcubierre warp drive and so on,...) the Casimir vacuum is often mentioned.

Obviously I am missing something.

Regards.

2. Jul 18, 2004

### hellfire

May be I should try in the relativity subforum?

3. Jul 18, 2004

### marcus

no, quantum forum seems IMO like the right place to get some help understanding Casimir effect.

I will try to help make some discussion and we can see if someone knowledgeable appears.

what you say here is unquestionably right: "Thus, for cosmological or gravitational purposes the Casimir vacuum should have positive energy density according to my reasoning above."

It is indeed positive energy relative to no energy at all.

But compared with the unrestricted space outside it is less.
So compared with empty space energy density it could be said to be
negative.

A cubic meter of empty space with no conducting plates making geometrical restrictions on the waves in it must have more energy, so relative to that, what is between the plates is negative.

One can choose how one thinks, negative or positive. I shall think of it as positive both because that is how I like to and also that seems to be your take on it as well. So there is this positive energy density between the plates (and it is a little bit less than what the density is outside)

but it is not the pressure of the energy density that causes the force

What causes the force is that the energy density between the plates
depends on some power of the separation.
So when you pull the plates apart a little
you not only increase the volume between them
but you also increase the energy density in that volume
and so
when you pull the plates apart just a little you are actually
creating energy----there is quite a bit more between now.

and so you must have done work by pulling the plates apart
and that means there was a force

I think you know all this hellfire, but it should be said.

Now we come to the interesting part---the dependence of the
the energy density on the separation. Does it go as the square
of the sep, or as the cube, or what? this is important in being able
to calculate the casimir force.

I forget. maybe you know the dependence, or can guess, or someone
else will supply this information.

You probably have other questions as well, what are they? I may be
unable to answer but would like to know

Last edited: Jul 18, 2004
4. Jul 18, 2004

### robousy

The force of attraction goes as 1/a^4 so I guess that the energy density goes as the cube.

(a is the separation)

5. Jul 19, 2004

### hellfire

Thanks for the answers. I think the arguments and facts you listed are known to me, at least qualitatively. What I want to know is the error in my reasoning. I just summarize it again:

The vacuum (not the Casimir vacuum) gravitates with a positive energy density (the cosmological constant). If one takes energy from the vacuum it should remain a positive energy.

A negative energy density arises only if one normalizes the mentioned integral for the Casimir vacuum taking the energy in the vacuum to be zero. This would mean that the force between plates depends on the vacuum outside the plates. This would be a negative energy density ‘relative’ to outside, appropiate to calculate forces, but not appropiate to account with gravitational effects.

6. Jul 19, 2004

### vanesch

Staff Emeritus
True, but the Casimir force is not a gravitational effect. In fact, the QFT calculation of the energy density of the vacuum, with a reasonable cutoff, gives such a big number, that it cannot possibly be true. If the cosmological constant were 0, and this energy density were true, there would be very strong gravitational effects which are not observed. So the cosmological constant, in this view, must compensate to a high degree this QFT energy density. This fine tuning is a mystery for the moment ; at least it was a few years ago, maybe people made progress in that area, I'm no expert.

cheers,
Patrick.

7. Jul 19, 2004

### hellfire

Yes, but my confusion arises from the fact that the Casimir energy is often mentioned in relation to exotic matter and the posibility of creating wormholes or Alcubierre warps (just make a search in google with casimir and alcubierre or “metric engineering” and casimir).

This means that the energy density in a Casimir vacuum is supposed to warp spacetime in, lets say, an inverse manner as ordinary energy does, i.e., it is assumed that a Casimir vacuum creates indeed a negative energy density.

But as far as I understand (as I argued above) the Casimir energy is always positive wrt its influence on spacetime (although it may be negative wrt to the energy density of a vacuum outside plates).

8. Jul 19, 2004

### vanesch

Staff Emeritus
I'm out of my depth here. But I'd like to make a guess. Obviously, the gravitational effect of the QFT vacuum energy is absent, so there must be something working against it. The most obvious thing is a cosmological constant that matches it exactly. But if it matches exactly the VACUUM density, then it will be a bit too strong in a cavity where the QFT vacuum energy density is slightly less. So this could then give your "negative energy density" effect.
But I'm simply guessing here, I have no serious knowledge of these matters.

cheers,
Patrick.

9. Jul 19, 2004

### marcus

well hellfire, did you or did you not get a discussion!
far more than Patrick I should assure you of my nonexpertise
however I will tell you where I non-expertly stand on this

I didnt ever hear of a "Casimir energy" only a "Casimir effect" that happens between two conducting plates very close together

in the actual experiments they sometimes use a sphere and a plate
but same idea as two parallel plates

there is the QFT vacuum energy and the cosmological constant or dark energy which is also an energy density and the Casimir effect is invoked as evidence that the QFT vacuum energy arising from virtual particles actually exists.

So maybe some writers who are in a hurry say "Casimir energy" when they should say "Vacuum energy associated in people's minds with Casimir effect"

I just looked at Wikipedia "Casimir effect" to see what common knowledge is-----they have nothing about "Casimir energy" which I suspect has no technical meaning although of course it may have only was not picked up by Wiki

The wikipedia article on Casimir effect is very nice

http://en.wikipedia.org/wiki/Casimir_effect

en.wikipedia.org/wiki/Casimir_effect

From wiki one gets a beautiful formula which IMHO is what everybody should know about Casimir effect. It is a formula for the small amount by which the vacuum energy density is less, between the plates, than it is outside

If Rho is the vacuum energy density in normal open space (astronomers tell us it is something but the QFT people have not assimilated this yet)

then the vacuum energy density between the plates is

$$Rho - \frac{\hbar c \pi^2}{720 d^4}$$

this is where d is the distance between plates, and therefore if A is area of plates then the volume between the plates is Ad. So multiply the density up there by Ad and you get the total amount of energy in the volume between the plates. then if you differentiate that you can derive a formula for the Casimir force F divided by the area A of the plates

$$\frac{F}{A} = \frac{\hbar c \pi^2}{240 d^4}$$

this is the formula they give in Wikipedia (they give only formula for the force not the energy density that produces the force)

This formula has been verified in the lab to within 1 percent accuracy---a great triumph----by someone with a name like Lamoreaux and others.

then what one asks is why is it that the energy density is less between the plates by some amount which is proportional to the reciprocal fourth power
of the separation.

maybe someone else wants to explain why it should be less by the reciprocal fourth power and not some other function of separation

Last edited: Jul 19, 2004
10. Jul 20, 2004

### hellfire

I think the term "Casimir energy" means the energy of the ground state in a Casimir vacuum.

I have to admit that I do not understand the derivation of this energy (making use of the Riemann zeta function and so on), but my understanding is that the negative result is wrt the ground state outside the plates. I may be wrong.

If I am right, then the question which originates this thread remains unsolved for me.

Last edited: Jul 20, 2004
11. Jul 20, 2004

### marcus

This sounds right to me. then the vacuum energy density between the plates (expressed positively) is

$$Rho - \frac{\hbar c \pi^2}{720 d^4}$$

And to express it negatively, by making the density outside the plates be our zero, simply means to set Rho = 0.

So then the vacuum energy or ground state energy density is just

$$- \frac{\hbar c \pi^2}{720 d^4}$$

This is the term which will give the right answer for the force--which has been experimentally measured. And it is the expression gotten by that stuff with the Zeta function.

I can see, in the -1/720, two things I recognize from other places I've met the Zeta. There is the -1/12 that comes from Zeta(-1) and there is a $$\inline \pi^2/60$$ which Ive seen in connection with the Planck black body and Stefan Boltzmann fourth power law.

so without going through the steps of the derivation I am pretty sure we have the right formula for the "Casimir energy" or groundstate vacuum energy between the places----expressed as a negative by making the outside vacuum energy our zero.

I thought we'd now taken care of the question. But maybe you should rephrase it.

"I have read often that the vacuum energy density in a Casimir vacuum is negative, but I do not understand this."

You dont ask a specific question in your post so I assume that the question is "why is it negative?"

To summarize what we just did, if you have a positive energy whose formula is made of a bigger positive piece (rho, the outside density) and a small negative piece (amount by which inside density is lacking)
then if you arbitrarily set the big positive piece equal to zero what you have left is the negative piece.

energies are often measured on a sliding scale and where you put the zero will decide what energies are positive and what ones are negative.

that's my take on it, at this point. (with the usual disclaimer of certainty)
how does it look to you now---hope the discussion has been helpful

Last edited: Jul 20, 2004
12. Jul 20, 2004

### hellfire

Yes marcus, it is helpful, thanks, but it still does not address the question.

It is clear for me that the energy density inside the plates will be negative if one sets the energy density outside the plates equal to zero.

But I think one is not always allowed to set arbitrarily the zero value in the energy scale. This is the case if one is taking into account gravitational effects.

My understanding is that this negative energy density shall behave wrt to spacetime as a positive energy density. This is because I assume that the intergral I mentioned in the first post (rho ~ ∫ w dk) leads always to a positive value: at least in cosmology, the result of the intergral taking into account all fields is the cosmological constant, which is a positive energy density.

Something must be wrong in this reasoning, otherwise I cannot understand a statement like e.g. this:

(From http://www.daviddarling.info/encyclopedia/A/Alcubdrive.html)

I hope my question is clear now.

Last edited by a moderator: Apr 21, 2017
13. Jul 20, 2004

### marcus

I think the trouble is with the "daviddarling.info"
It sounds like the author is confused
the casimir effect is about ordinary matter and
not "negative energy" matter
it is explained by ordinary stuff happening in the vacuum

you are right that the vacuum energy is positive, both between the plates and outside, just a little less in between
but not negative (unless you slide the scale to make it so)

but the daviddarling.info author sounds as if he may not understand

so by getting rid of the daviddarling.info stuff, maybe you can put things in order for yourself

another idea: keep the daviddarling quote but just erase the inappropriate and misleading reference to Casimir

let him have his exotic "negative energy" matter and say everything he wants to----just dont let him make the misleading reference to Casimir---then maybe everything will be all right. hope so. cheers

Last edited by a moderator: Apr 21, 2017
14. Jul 20, 2004

### hellfire

OK, this may be an explanation. But note that there are lots of sites in internet with similar claims as the mentioned above. Take for example:

from http://www.hawking.org.uk/lectures/warps3.html [Broken]

And lots more. This lead me to assume I was wrong.

Last edited by a moderator: May 1, 2017
15. Jul 20, 2004

### marcus

Hellfire you have succeeded in teaching me something. I have been thinking about this (and what it said in the link) and have concluded there is something funny going on----it seems like if the plates are close enough together there could be negative energy density, even on a scale that pegs the cosmological constant at current estimates. Have to think more about this.

The link took me to some delightful Stephen Hawking pages, which
unfortunately went back to 1993---public lectures given I think then
and related to some engaging popular essays about black holes
spacetime warping time travel and other topics.

the page you linked to contained something which ten years later in 2004
seems to be an error:

"Because there are fewer virtual particles, or vacuum fluctuations, between the plates, they have a lower energy density, than in the region outside. But the energy density of empty space far away from the plates, must be zero. Otherwise it would warp space-time, and the universe wouldn't be nearly flat. So the energy density in the region between the plates, must be negative.

From a post-1998 perspective, this is wrong since cosmologists
started believing in 1998 that
the energy density of empty space far away from the plates, must NOT be zero.

In fact a positive dark energy density is what is required for spatial flatness. So it is the contrary of what S.H. says here.
It is not the current view that " Otherwise it would warp space-time, and the universe wouldn't be nearly flat."

BTW his explanation of the Casimir effect is just fine---and very nice pictures etc for general audience. The only trouble is the the conclusion he draws from it is false---because he assumes zero cosm. const. or zero dark energy (he hasnt heard of accel. expansion etc.)

But that doesnt matter! It is just the difference between 1993 and now!

the basic point still seems right to me---what hawking says about the negativity of the energy. Guests are coming. I may have to get back to this later

Last edited by a moderator: May 1, 2017
16. Jul 20, 2004

### marcus

the answer may be in something Vanesch said but before we look at that let us just proceed very naively. Astronomers are able to measure cosmological constant (they now think) and in Planck units it is right around E-123. If we use planck units Rho is E-123

The vacuum energy density between the plates (expressed positively) is

$$Rho - \frac{\hbar c \pi^2}{720 d^4}$$

And if we use Planck units then hbar and c are 1 so this is just

$$Rho - \frac{ \pi^2}{720 d^4}$$

and Rho is E-123 and pi-squared is about 10 so that

E-123 - (1/72) d-4

so for example if d is smaller than E30 then
this looks negative.

Aaarghhhh!

this is just a very quick estimate, and I dont know if it being negative really means anything---energy is on such a sliding scale. but
on the scale that make the dark energy be what astronomers say it is, on that scale at least, we got some negative energy density between thos plates. I am swampt with other stuff I have to do. will try to get back to this

17. Jul 21, 2004

### vanesch

Staff Emeritus
Yes, but this means that you need to introduce a cosmological constant which is ALMOST, but not completely, equal (but opposite) to the QFT vacuum energy density. That's the hard to swallow fine tuning I mentionned earlier. Note that Hawkin's reasoning is essentially right, if you replace "0" by "a very small quantity". Indeed, without cosmological constant correction, the bare QFT vacuum energy density, is WAY TOO STRONG.
So you still need to cancel the bulk of it!

cheers,
Patrick.

18. Jul 21, 2004

### hellfire

Excellent, I think it is clear now.
Thanks to both.

Regards.

19. Aug 23, 2004

### Xybot

Just had a quick scan on this thread, I think there may be a problem with the terms of reference being used for a zero energy state. From my understanding the energy constant of a vaccum is considered 0 energy, as the casmir effect excludes certain wavelengths of Electro-magnetic radiation it's energy is said to be less than the ground energy state of a vaccum, in effect a negetive energy density.

As a side note, theoretically a photon moving through this negetive energy density will travel faster than the speed of light in a vaccum, this may lead to the idea of spacetime being negetively curved within this area (this is a guess??!!).