# Scaling of non-gravitational energy in a black hole

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 ?

• What do you mean by "non-gravitationnal energy?" I don't see more than one type of energy in a Schwarzschild spacetime. – user4552 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). – user4552 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. – user4552 Aug 17 '13 at 18:18

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!
• 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