Bousso's holographic bound

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The Bousso bound captures a fundamental relation between quantum information and the geometry of space and time. It appears to be an imprint of a unified theory that combines quantum mechanics with Einstein's general relativity.^[1] The study of black hole thermodynamics and the information paradox led to the idea of the holographic principle: the entropy of matter and radiation in a spatial region cannot exceed the Bekenstein-Hawking entropy of the boundary of the region, which is proportional to the boundary area. However, this "spacelike" entropy bound fails in cosmology; for example, it does not hold true in our universe.^[2]

Raphael Bousso showed that the spacelike entropy bound is violated more broadly in many dynamical settings. For example, the entropy of a collapsing star, once inside a black hole, will eventually exceed its surface area.^[3] Due to relativistic length contraction, even ordinary thermodynamic systems can be enclosed in an arbitrarily small area.^[1]

To preserve the holographic principle, Bousso proposed a different law, which does not follow from black hole physics: the covariant entropy bound^[3] or Bousso bound.^[4][5] Its central geometric object is a lightsheet, defined as a region traced out by non-expanding light-rays emitted orthogonally from an arbitrary surface B. For example, if B is a sphere at a moment of time in Minkowski space, then there are two lightsheets, generated by the past or future directed light-rays emitted towards the interior of the sphere at that time. If B is a sphere surrounding a large region in an expanding universe (an anti-trapped sphere), then there are again two light-sheets that can be considered. Both are directed towards the past, to the interior or the exterior. If B is a trapped surface, such as the surface of a star in its final stages of gravitational collapse, then the lightsheets are directed to the future.

The Bousso bound evades all known counterexamples to the spacelike bound.^[1][3] It was proven to hold when the entropy is approximately a local current, under weak assumptions.^[4][5][6] In weakly gravitating settings, the Bousso bound implies the Bekenstein bound^[7] and admits a formulation that can be proven to hold in any relativistic quantum field theory.^[8] The lightsheet construction can be inverted to construct holographic screens for arbitrary spacetimes.^[9]

A more recent proposal, the quantum focusing conjecture,^[10] implies the original Bousso bound and so can be viewed as a stronger version of it. In the limit where gravity is negligible, the quantum focusing conjecture predicts the quantum null energy condition,^[11] which relates the local energy density to a derivative of the entropy. This relation was later proven to hold in any relativistic quantum field theory, such as the Standard Model.^{[11][12][13][14]}

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