Since the Bekenstein-Hawking black hole entropy made its appearance in the seventies, the idea that the ultimate number of degrees of freedom in a region depends on the area of its boundary began to gain currency. The idea sounds surprising. Entropy measures the number of degrees of freedom (“the stuff”). Normally we expect its amount to grow with volume, as with the amount of stuff we can squeeze into a bag or a box.  The research in string theory and loop quantum gravity allowed to derive the “area laws” for black hole entropy  from the first principles, with only a small amount of additional assumptions. At the same time the black hole thermodynamics was developed. It is very much  like a usual thermodynamics, but in addition to heat engines, heat baths and other nineteenth century devices, black holes are  used as  well. Using this gudget allows to argue for some sort of “holography” [another term to describe the situation when what you get is determined by the boundary]  law for general space-time region.

In our work we consider a simple model, which in two spatial dimensions represents gravity and in our three-plus-one (space+time) spacetime is part of the description of general relativity. Unlike its more sophisticated counterparts, this  model is exactly solvable. Two-dimensional space is rather dull: it is flat, there are no gravitational waves, and anything  interesting  happens only if you add particles. We used this dullness to our advantage: it was possible to play Fates and add matter in a fully controlled way.   Without any matter we obtained a clean area law, but really interesting things happened when we started to add particles. They contribute to the entropy only if they sit on the boundary between the regions, so we again have holography.

The preprint of the paper is on the arxiv: [Etera R. Livine, Daniel R. Terno, The entropic boundary law in BF theory, arXiv:0805.2536]