# Is the Stefan-Boltzmann law an upper limit?

The Stefan-Boltzmann law says that a black body radiates energy at a rate $$\sigma T^4$$ per unit surface area, where $$\sigma$$ is the Stefan-Boltzmann constant.

I have always assumed this is an upper limit, i.e. no body can ever lose heat faster than this due to radiation. However, the Wikipedia article linked above claims that this limit can be exceeded for some metamaterials. The reference cited for this seems not to be super-famous, and since it's such a surprising result I'd like to check I've understood it.

To make my question as concrete as possible, let me set up a hypothetical experiment: suppose I have an infinitely conductive metal sphere of radius $$r_1$$, which contains a nuclear reactor that keeps its surface always at a fixed temperature $$T$$. Surround this physical sphere with an imaginary sphere of radius $$r_2>r_1$$. Inside this imaginary sphere we can place any kind of apparatus we want, but it's not allowed to stick out of the imaginary sphere, and it must operate in steady state, i.e. it can't have any power source besides the temperature difference between the metal sphere and its surroundings.

This whole thing is to be placed in empty space, far away from any stars, and the goal is to extract energy from the metal sphere and radiate it away at as high a rate as possible. (Any spectrum of radiation is acceptable - it doesn't have to be black body radiation.)

The obvious solution is to surround the metal sphere with an infinitely conductive sphere of radius $$r_2$$, with a completely black surface. This will extract heat from the metal sphere and radiate it away at a rate $$4\pi r_2^2 \sigma T^4$$. The question is, is it possible, in principle, to do better than this?

If it's not possible, then is Wikipedia's reference wrong, or is it just claiming something else (and if so what)? If it is possible, then how can this be achieved, and is there any fundamental limit to how fast a body can radiate?

The Wikipedia article is correct, and moreover, its statements are relatively well known and uncontroversial. To see why, let's recap the physical intuition for why $$\sigma T^4$$ is an upper limit.

1. Begin with some object in the middle of free space. The object has some quantum energy levels, and the space itself has associated electromagnetic field modes. The object and the electromagnetic field can exchange energy by absorption and emission.
2. Assume that both the object and the surrounding electromagnetic field are at thermal equilibrium at temperature $$T$$. For each mode of the electromagnetic field, given a fixed shape for the object, there is a maximal rate of absorption, which is achieved when the object is completely black.
3. In thermal equilibrium, detailed balance is achieved: the average power absorbed by the object from each field mode precisely matches the average power the object puts into it by emission. Thus step 2 implies an upper bound on the emission rate per mode.
4. Summing over all of the electromagnetic field modes and setting the emission rate for each to the maximum quantitatively gives an emission intensity of $$\sigma T^4$$.
5. When the electromagnetic field isn't at temperature $$T$$, this bound on emission still applies, giving a maximum rate at which the object can cool off.

Almost all of this reasoning is correct in general. The only subtlety is in step 4. The point is that a metamaterial can have a different distribution of electromagnetic mode energies (i.e. density of states) than free space. When the sum over modes is performed, this can lead to an enhancement of the emission rate.

These kinds of effects are well known in applied physics. For example, in cavity QED, emission rates can be suppressed or enhanced by orders of magnitude, a result known as the Purcell effect.

Note that none of this discussion is about emission from metamaterials, it's about emission from objects embedded inside metamaterials, which see this modified density of states. If you insist your outer sphere is surrounded by vacuum, then as far as I know, there's no way to do better than $$\sigma T^4$$.

• Thanks, this I think must be the right answer. The key point is your last paragraph - Wikipedia's reference isn't making the claim it sounded like it was making. I'll probably accept this later on, though I'm still keen to know if the Stefan-Boltzmann law can be proven to be an upper bound if the apparatus is in empty space. Dec 14 '19 at 17:34
• Probably Wikipedia needs to be updated - the way it's worded strongly implies that metamaterials can radiate faster than $\sigma T^4$ into empty space. Would you update it (since you have greater knowledge), or should I? Dec 14 '19 at 17:37
• @Nathaniel Not me! I’d never be confident enough to edit Wikipedia, this isn’t my subfield. Dec 14 '19 at 19:28