Are there other types or versions of holographic principle?

Question

Holographic principle (https://en.wikipedia.org/wiki/Holographic_principle) establishes a correlation between a bulk and its boundary. It says, in layman terms, that the amount of information in a bulk or volume is proportional to the area of that volumes surface.

But are there any other types of correlations? Are there other types of holographic principles?

I found a statement on Wikipedia which says:

The holographic principle resolves the black hole information paradox within the framework of string theory. However, there exist classical solutions to the Einstein equations that allow values of the entropy larger than those allowed by an area law, hence in principle larger than those of a black hole. These are the so-called "Wheeler's bags of gold". The existence of such solutions conflicts with the holographic interpretation, and their effects in a quantum theory of gravity including the holographic principle are not yet fully understood.

Does this mean that these solutions change the correspondence/correlation and modify holographic principle? Could we have holographic principles where the amount of information in a volume is proportional to the the square of the surface area, or perhaps where there is not general correlation at all?. Does this create other types of holographic principles?

Answer 1

Conceptually, if elements of the volume are completely independent (e.g. random), then obviously a twice larger volume would contain twice more information (at least momentarily). In this case the amount of information is proportional to the volume, but not to the surface (or hyper-surface) area.

In fact, a twice larger volume may contain more than twice the amount of information, because elements of each part may also get the degrees of freedom of the other part.

However, often, the elements of the volume are not completely independent. For example, they may be organized in a crystalline lattice or obey some field equations that require them to be in a certain relation or symmetry. There are also may be general requirements (e.g. a differentiable manifold) or initial conditions. All these relations would reduce the amount of information compared to the random or independent case above.

In the opposite extreme case of, say, an empty geometrical cube, all its information is encoded in its linear dimension.

Thus the answer to your question depends on the underlying theory that equips the volume with specific relations among its elements. In the extreme abstract cases, the amount of information may be proportional to the volume or to the linear dimension and so on. However, in more realistic cases, such as a string theory or solid state physics, the information may be proportional to the surface area (or hyper-surface in higher dimensions).