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A quantum proof of a Bousso bound

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

Proof of a Quantum Bousso Bound
They 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.

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reader edward said...

Proper time dilation and constant time

Here is a typical image of an out of
phase chart. Supposing the wave to the left is at value C and the
line to the right is proper time less than C. Both are vehicles and in
motion. Supposing the left line has a Time = 0 constant and the line to
the right = Proper time and moving <value C. This line is accelerating
and at a coincident point to the far left. At this point the velocity of
the proper time vehicle is matched to that of the vehicle at C where upon one
may say they are in phase with each other. The outcome is that the proper
time vehicle now experiences a Time = 0 the same as the time on the left
vehicle. The velocity of both Left and Right are at value C and they both
have a Time = 0. If the right hand vehicle line reduces velocity
it’s proper time starts to advance i.e. 0 + Tx1, Tx2 etc etc as it slows.
Proper time starts to accelerate until it reaches a point where its fundamental
velocity = 60secs/Min etc.

In order to reach the point where its
time is 0 it has to be moving with the velocity C. But in this scenario
it is not light that is moving it is the Space itself. But also in this
scenario the Space is not moving it is constantly emerging.

If one thinks on this situation is sort
of explains the Time dilation consideration, and also opens the door to many
other similar paradoxical misunderstanding including the ability of any
movement available in the Universe.

reader Alfred E. Popper said...

Hey Lubos,

You're so gifted why waste it on this blog and instead work on physics? Are you doing any work? You can do both obviously.

reader Gary Ehlenberger said...

Have you read this?

reader Gary Ehlenberger said...

This may be out of place, but is interesting. Have you read this?

reader Gary Ehlenberger said...

Have you read this?

reader JollyJoker said...

"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."

Any chance there's some real new insight buried there? Finding a new proof for the more general quantum case (if I understood correctly) sounds like a real "foundations of quantum mechanics" advance that simply has to imply something important! Can you translate any of this to "if this inequality held for string theory" proposals in any (however roundabout) way?

reader Toon Pepermans said...

there is something about that, even christian philosophers (like Thomas Aquinas) make a clear distinction between 'reason' and 'faith', in eastern religions this distinction is not so clear at all

reader Eclectikus said...

Yeah, I think this is a key distinction, not only between west and oriental religions, but also between muslim and christians. Pope Benedict XVI points some interesting points in his Regensburg lecture: "Faith, Reason and the University Memories and Reflections". Worth taking a look:

(I recommend reading the conference far beyond the controversy on which focuses Wikipedia)

reader Peter F. said...

Am glad you keep sensing the superior strength of the stringy way of ultimately weaving all things super-tiny and super-massive mathematically together;
This because
1. my fluffy 'evolutionary psychology type' FOOThold on What Is going on is somehow felt to have enhanced philosophical traction by the 'peripheral' support that your stringy TOEhold (on same) provides ;-)
2. without this and other TRFic support I would have a harder or less easy time (definitely a less interest-arousing, pleasing, or less distress deflecting time) to put up with the philosophical and other potholes that are part of any sufficiently reflective animal's road to absolute peace or oblivion. %-]

reader peregrine said...

Hi Lubos,
A critic on string is that if the world is 11d then at Big Bang it should have had 10 space dimensions too. Why only three dimensions expanded (about 14 billion light years until now) and the other remained so tiny (i.e. remained at the Planck length)? What is your comment?

reader Luboš Motl said...

Dear peregrine, I don't think that this question is a "critique" of string theory.

Why only 3+1 dimensions expanded is a question on early string cosmology no one fully understands but the first potential mechanism that explains exactly that - and computes d=4 essentially as twice the world volume dimension D=2 of a string - was proposed in the late 1980s by Brandenberger and Vafa under the name "string gas cosmology".

reader Dilaton said...

Yes I already liked this mechanism when reading the string gas cosmology article ... :-)

reader scooby said...