TITLE: Scaling of non-gravitational energy in a black hole
QUESTION [1 upvotes]: When looking at a Schwarzschild black hole, for instance, we know that we may apply black hole thermodynamics. We may define a entropy of the black hole which scales like the area of the horizon : $$S \sim R_s^2$$. 
It is understood in the more general context of the holographic principle which states that " the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon"
Now, the non-gravitationnal energy $E_{ng}$, so the mass $M$ for the Schwarzschild black hole, has a different scaling : 
$$E_{ng} \sim R_s$$
So, does that mean that the energy is encoded in a one-dimensional object (perimeter, loop, string, radius), and is it a different "holographic" principle ?

REPLY [4 votes]: You can't say whether the scalings $S\sim R^2$ and $E\sim R$ are the same or different because they are relationships between different pairs of physical quantities! It's like comparing apples and oranges. Well, you could say that the scalings are different already because they contain different quantities but if you defined "different" in this way, $S\sim R^2$ and $E\sim R^2$ (which are true for 5D black holes, by the way) would also be different!
The second relationship, one between the energy and radius of a black hole, has nothing to do with the holographic principle so the answer to your last question is No. It is meaningless for energy to be "encoded"; only information may be "encoded". The energy is just "equal" to what it is equal to. 
The holographic principle postulates some (maximum) information density per unit surface area. But if there were a law that postulated a constant energy density per unit length, area, or volume, it would have nothing to do with the holographic principle. Various objects may have constant densities; for example, the linear energy density of a fundamental string is known as the string tension. But these relationships hold for particular objects only; they are not universal relationships that hold for whole theories and everything in them.
The holographic principle is such a universal relationship, however, and it has to talk about the information for it to be this universal.
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