An aspect of holography is demystified. Perhaps too much.
In the early 1970s, Jacob Bekenstein realized that the black hole event horizons have to carry some entropy. And in fact, it's the highest entropy among all localized or bound objects of a given size or mass. This "hegemony" of the black holes is understandable for a simple reason: in classical physics, black holes are the ultimate phase of a stellar collapse and the entropy has to increase by the second law which means that it is maximized at the end – for black holes.
The entropy \[
S = k \frac{A}{4G\hbar}
\] (where we only set \(c=1\)) is the maximum one that you can squeeze inside the surface \(A\), kind of. This universal Bekenstein-Hawking entropy applies to black holes – i.e. static spacetimes. The term "Bekenstein bound" is often used for inequalities that may involve other quantities such as the mass or the size (especially one of them that I don't want to discuss) but they effectively express the same condition – black holes maximize the entropy.
Is there a generalization of the inequality to more general time-dependent geometries? The event horizons are null hypersurfaces so in the late 1990s, Raphael Bousso proposed a generalization of the inequality that says that the entropy crossing a null hypersurface that is shrinking everywhere into the future (and may have to be truncated to obey this condition) is also at most \(kA/4G\); yes, \(k\) is always the Boltzmann constant that I decided to restore. I remember those days very well – my adviser Tom Banks was probably the world's most excited person when Raphael Bousso published those papers.
Various classical thermodynamic proofs were given for this inequality. I suppose that they would use Einstein's equations as well as some energy conditions (saying that the energy density is never negative, or some more natural cousins of this simple condition). Finally, there is also a quantum proof of the statement.
Today, Raphael Bousso, Horacio Casini, Zachary Fisher, and Juan Maldacena released a new preprint called
S_{\rm state} - S_{\rm vacuum} \leq k \frac{A-A'}{4G\hbar}
\] where the entropies \(S\) are measured as the entropies (of the actual state and/or the vacuum state, as indicated by the subscript) crossing the light sheet, \(A,A'\) are initial and final areas on the boundary of the light sheet, and the light sheet is a shrinking null hypersurface connecting these two areas.
One must be aware of the character of their proof. The entropies are computed as the von Neumann entropies\[
S = -k\,{\rm Tr}\, \ln (\rho \ln \rho)
\] so the proof uses the methods of quantum statistical physics. Also, they assume that the entropy is carried by free i.e. non-interacting (a quadratic action obeying) non-gravitational fields propagating on a curved gravitational background. The backreaction is neglected, too. Some final portions of the paper are dedicated to musings about possible generalizations to the case of significant backreaction; and the interacting fields.
They are not using any energy conditions which makes the proof "strong". Also, they say that they are not using any relationship between the energy and entropy. I think that this is misleading. They are and must be using various types of the Hamiltonian to say something about the entropy. Otherwise, Newton's constant couldn't possibly get to the inequality at all! After all, the evolution is dictated by the Hamiltonian and they need to know it to make the geometry relevant. Moreover, I think that the proof must be a rather straightforward translation of a classical or semiclassical proof to the quantum language.
Under some conditions, the inequality has to be right even in the interacting and backreacting cases. I haven't understood the proof in detail but I feel that it's a technical proof that had to exist and one isn't necessarily learning something conceptual out of it. By this claim, I am not trying to dispute that holography plays a fundamental role in quantum gravity. It undoubtedly does. But particular "holographic inequalities" such as this one are less canonical or unique or profound than the original Heisenberg uncertainty principle in quantum mechanics\[
\Delta x \cdot \Delta p \geq \frac{\hbar}{2}.
\] This inequality more or less "directly inspires" the commutator\[
[x,p] = xp - px = i\hbar
\] which conveys pretty much all the new physics of quantum mechanics. While the upper bounds for the entropy are the quantum gravity analogues of the Heisenberg inequality above, they are less unique and they don't seem to directly imply any comprehensible equation similar to one for the commutator – an equation that could be used to directly "construct" a theory of quantum gravity. At least it looks so to me. So quantum gravity is a much less "constructible" theory than quantum mechanics of one (or several) non-relativistic particles.
On the other hand, I still think that the power of Bousso-like inequalities hasn't been depleted yet.
Note that similar Bousso-like inequalities and similar games talk about the areas in the spacetime so they depend on the isolation of the metric tensor degrees of freedom from the rest of physics. This is why they are pretty much inseparably tied to the general relativistic approximation of the physics. String/M-theory unifies the spacetime geometry with all other matter fields in physics but this unification has to be cut apart before we discuss the geometric quantities which we have to do before we formulate things like the Bousso inequality and many other results. In other words, it seems likely that there cannot be any "intinsically stringy" proof of this inequality because the inequality seems to depend on some common non-stringy approximations of physics.
In the early 1970s, Jacob Bekenstein realized that the black hole event horizons have to carry some entropy. And in fact, it's the highest entropy among all localized or bound objects of a given size or mass. This "hegemony" of the black holes is understandable for a simple reason: in classical physics, black holes are the ultimate phase of a stellar collapse and the entropy has to increase by the second law which means that it is maximized at the end – for black holes.
The entropy \[
S = k \frac{A}{4G\hbar}
\] (where we only set \(c=1\)) is the maximum one that you can squeeze inside the surface \(A\), kind of. This universal Bekenstein-Hawking entropy applies to black holes – i.e. static spacetimes. The term "Bekenstein bound" is often used for inequalities that may involve other quantities such as the mass or the size (especially one of them that I don't want to discuss) but they effectively express the same condition – black holes maximize the entropy.
Is there a generalization of the inequality to more general time-dependent geometries? The event horizons are null hypersurfaces so in the late 1990s, Raphael Bousso proposed a generalization of the inequality that says that the entropy crossing a null hypersurface that is shrinking everywhere into the future (and may have to be truncated to obey this condition) is also at most \(kA/4G\); yes, \(k\) is always the Boltzmann constant that I decided to restore. I remember those days very well – my adviser Tom Banks was probably the world's most excited person when Raphael Bousso published those papers.
Various classical thermodynamic proofs were given for this inequality. I suppose that they would use Einstein's equations as well as some energy conditions (saying that the energy density is never negative, or some more natural cousins of this simple condition). Finally, there is also a quantum proof of the statement.
Today, Raphael Bousso, Horacio Casini, Zachary Fisher, and Juan Maldacena released a new preprint called
Proof of a Quantum Bousso BoundThey prove that\[
S_{\rm state} - S_{\rm vacuum} \leq k \frac{A-A'}{4G\hbar}
\] where the entropies \(S\) are measured as the entropies (of the actual state and/or the vacuum state, as indicated by the subscript) crossing the light sheet, \(A,A'\) are initial and final areas on the boundary of the light sheet, and the light sheet is a shrinking null hypersurface connecting these two areas.
One must be aware of the character of their proof. The entropies are computed as the von Neumann entropies\[
S = -k\,{\rm Tr}\, \ln (\rho \ln \rho)
\] so the proof uses the methods of quantum statistical physics. Also, they assume that the entropy is carried by free i.e. non-interacting (a quadratic action obeying) non-gravitational fields propagating on a curved gravitational background. The backreaction is neglected, too. Some final portions of the paper are dedicated to musings about possible generalizations to the case of significant backreaction; and the interacting fields.
They are not using any energy conditions which makes the proof "strong". Also, they say that they are not using any relationship between the energy and entropy. I think that this is misleading. They are and must be using various types of the Hamiltonian to say something about the entropy. Otherwise, Newton's constant couldn't possibly get to the inequality at all! After all, the evolution is dictated by the Hamiltonian and they need to know it to make the geometry relevant. Moreover, I think that the proof must be a rather straightforward translation of a classical or semiclassical proof to the quantum language.
Under some conditions, the inequality has to be right even in the interacting and backreacting cases. I haven't understood the proof in detail but I feel that it's a technical proof that had to exist and one isn't necessarily learning something conceptual out of it. By this claim, I am not trying to dispute that holography plays a fundamental role in quantum gravity. It undoubtedly does. But particular "holographic inequalities" such as this one are less canonical or unique or profound than the original Heisenberg uncertainty principle in quantum mechanics\[
\Delta x \cdot \Delta p \geq \frac{\hbar}{2}.
\] This inequality more or less "directly inspires" the commutator\[
[x,p] = xp - px = i\hbar
\] which conveys pretty much all the new physics of quantum mechanics. While the upper bounds for the entropy are the quantum gravity analogues of the Heisenberg inequality above, they are less unique and they don't seem to directly imply any comprehensible equation similar to one for the commutator – an equation that could be used to directly "construct" a theory of quantum gravity. At least it looks so to me. So quantum gravity is a much less "constructible" theory than quantum mechanics of one (or several) non-relativistic particles.
On the other hand, I still think that the power of Bousso-like inequalities hasn't been depleted yet.
Note that similar Bousso-like inequalities and similar games talk about the areas in the spacetime so they depend on the isolation of the metric tensor degrees of freedom from the rest of physics. This is why they are pretty much inseparably tied to the general relativistic approximation of the physics. String/M-theory unifies the spacetime geometry with all other matter fields in physics but this unification has to be cut apart before we discuss the geometric quantities which we have to do before we formulate things like the Bousso inequality and many other results. In other words, it seems likely that there cannot be any "intinsically stringy" proof of this inequality because the inequality seems to depend on some common non-stringy approximations of physics.
A quantum proof of a Bousso bound
![A quantum proof of a Bousso bound]()
Reviewed by MCH
on
April 24, 2014
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