In the Casimir effect, the region between two parallel conducting plates is said to have negative energy density. One of the reason this is interesting is because it seems to violate some energy conditions. Indeed, as shown in an answer to another question, the result for the stress-energy tensor is:

$$ \langle T_{\mu\nu} \rangle_0 = \mathrm{diag}(-1,1,1,-3) \frac{\hbar c}{a^4}\frac{\pi^2}{720} $$

On one hand, the Casimir effect setup produces something relatively simple - just some negative values in the stress-energy tensor. The violation of energy conditions is a rather simple and generic thing. On the other hand, the physical realization of such a setup is rather complicated in comparison - the conducting plates are made out of many particles, which are non-trivially arranged to function as a building blocks of conducting plates. There is a lot of stuff going on.

This seems to indicate that there might be some much simpler setup that could do the same thing.

So the question is: What would be a physical "minimal working example" to achieve this "negativity" (i.e. to achieve $\langle T_{\mu\nu} \rangle_0$ which violates some energy conditions)?


1 Answer 1


Negative value of energy density turns up in vacuum energy theory of Casimir's effect only after subtraction of a convenient number by hand. This introduces a new definition of EM energy density different from Poynting's in order to make it zero outside the plates => a questionable formal operation. Negative energy is not a prediction about Poynting's density of EM energy, this one is always positive or zero.

The vacuum energy theory of the Casimir force between the two plates assumes that outside the plates, there is a background EM radiation with zero-point spectrum with a possible cutoff, and inside there is a similar radiation but the spectral function of wavevectors is modified in one dimensions for small wavevectors because "some long waves do not fit between the plates". It also assumes that energy inside the plates is given by a sum over positive zero point energies $\frac{1}{2}\hbar \omega_k$ over all modes $k$, so EM energy density can't be negative in this theory.

Naive calculation shows that both sides of the plate have positive values of energy density, but the energy density between the plates is lower than the outside energy density. Then if we subtract some convenient number from both energy densities, we can make the outer one zero and the inner one negative. But this is not a prediction about Poynting's energy density.

The simplest case really is two infinite plates made of perfect conductor, a macroscopic model. For microscopic systems, this method of calculation does not make sense, because in a microscopic model, nothing can stop presence of long waves, and the zero point radiation should have the same spectrum everywhere. Even in a macroscopic model of Casimir's, rejection of the long waves is highly artificial. In reality some long waves penetrate even highly conductive plates. It is not clear how plates made of real conductive matter can filter out waves longer than plate separation but not the shorter waves.

A more physical way to calculate Casimir-like forces is to analyze mutual EM interaction of the plates due to fluctuations of charge in them (van der Waals forces, Lifshitz's theory).

  • $\begingroup$ The last paragraph seems to be the most illuminating and in the spirit of some papers like arXiv:1605.04143 Proof that Casimir force does not originate from vacuum energy. However, if that is the case it raises question regarding the recent attempts to use the Casimir effect to violate energy conditions as needed for a warp drive metric (e.g. the recent work by Harold G. White), but that would be material for another question. $\endgroup$
    – Danijel
    Dec 19, 2021 at 8:59

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