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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 ?

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What do you mean by "non-gravitationnal energy?" I don't see more than one type of energy in a Schwarzschild spacetime. –  Ben Crowell Aug 17 '13 at 16:19
@BenCrowell : Well, from my point of view, the total energy of a black hole is zero, that is : the positive non-gravitationnal energy is compensated by the negative gravitationnal energy. –  Trimok Aug 17 '13 at 16:30
Sorry, I don't understand what you mean by that. What definition of energy are you using? For example, the Komar energy is simply equal to the mass of the black hole (the $m$ appearing in the Schwarzschild metric). –  Ben Crowell Aug 17 '13 at 16:39
@BenCrowell : I agree this is a naive definition, but I like it. The black hole is a limit case. Why? Because, in a classical point of view, the total energy of a isolated physical object cannot be negative. The black hole is a limit case, because the classical total energy is zero, negative gravitationnal energy is compensating positive non-gravitationnal energy. –  Trimok Aug 17 '13 at 16:45
I agree this is a naive definition, but I like it. What definition? You still haven't explained what you mean by gravitational versus nongravitational energy. Because, in a classical point of view, the total energy of a isolated physical object cannot be negative. By "classical" do you mean nonrelativistic? Either way, this statement is false. –  Ben Crowell Aug 17 '13 at 18:18
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1 Answer 1

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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"It is meaningless for energy to be "encoded"; only information may be "encoded". Yes, this is certainly the correct answer, but I think it is very surprising. –  Trimok Aug 17 '13 at 9:15
Dear @Trimok, it's a matter of terminology but I personally only use the word "coding" for "rewriting the information in some new or obscure way". So if I used the term "encoding the energy", I would mean some difficult da Vinci code by which the value of the energy is encoded. But the black hole clearly doesn't "encode" the energy in this sense. Its radius is literally equal to the energy with some fixed coefficient. When we talk about coding in holography, we mean that all the microscopic details about the state of objects are encoded, not just one overall energy. Unsure what's surprising. –  Luboš Motl Aug 17 '13 at 9:46
Also, I think that you wanted to say that the energy was encoded in the "length of the black hole" because it was proportional - in $D$ spacetime dimensions, the relationship is $E\sim R^{D-3}$, by the way. But if you were thinking about this proposed "new holography" just for extra 1 minute, you would have seen that it makes no sense because there's no "curve inside the black hole" with the right length where the information could be encoded. And the scaling would be far from giving you a reason to think that any such "curve" should be special in the first place. –  Luboš Motl Aug 17 '13 at 9:49
Holography is actually much more than just the scaling between the area and entropy. It is a claim that the equivalent or more precise laws that exist at the area are local or quasilocal or otherwise natural and they describe all the dynamics of the theory properly. This precise matching of the dynamical laws is much more than one scaling law for overall quantities. I suspect that you heavily underestimate what holography means and its depth. –  Luboš Motl Aug 17 '13 at 9:51
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