Abstract
We argue that, in a theory of quantum gravity in a four dimensional asymptotically flat spacetime, all information about massless excitations can be obtained from an infinitesimal neighbourhood of the past boundary of future null infinity and does not require observations over all of future null infinity. Moreover, all information about the state that can be obtained through observations near a cut of future null infinity can also be obtained from observations near any earlier cut although the converse is not true. We provide independent arguments for these two assertions. Similar statements hold for past null infinity. These statements have immediate implications for the information paradox since they suggest that the fine-grained von Neumann entropy of the state defined on a segment (-infty,u) of future null infinity is independent of u. This is very different from the oft-discussed Page curve that this entropy is sometimes expected to obey. We contrast our results with recent discussions of the Page curve in the context of black hole evaporation, and also discuss the relation of our results to other proposals for holography in flat space.
Highlights
We review the basic ideas that underlie the physics of 4-dimensional asymptotically flat spacetimes
These results imply that the oft-discussed Page curve for the von Neumann entropy of the black hole radiation at future null infinity is not the right expectation
We have focused on a neighbourhood near I−+, the same argument above shows that any sector of the Hilbert space can be generated by acting with operators from any infinitesimal neighbourhood of future null infinity
Summary
The principle of holography has been understood for asymptotically AdS spacetimes for more than twenty years, but some of its implications for how quantum information is stored in theories of quantum gravity are still not widely appreciated. Result (1) follows as a special limit of result (2) This is because the assumptions that go into result (2) are stronger: this result requires us to assume that certain commutation relations that can be derived at null infinity in the semiclassical theory are corrected only by local terms in the full theory of quantum gravity. 3. The von Neumann entropy of any pure or mixed state of massless excitations, defined on a segment (−∞, u) of future null infinity, is independent of the upper limit u. These results imply that the oft-discussed Page curve for the von Neumann entropy of the black hole radiation at future null infinity is not the right expectation.
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