The holographic principle by Susskind states that the maximum amount of information stored in a 3D volume can be dictated by its surface area. Thus, in some way the boundary encodes the information in the interior.
I ask whether such a principle can be generalized to n dimensions. To write in simple mathematical terms: "The maximum amount of information that can be stored in an n-dimensional volume is proportional to the surface area of the (n-1)-dimensional boundary of the region". Have people worked on such generalizations. I would like to thank the community for their contributions in advance.
This is a great question.
The holographic principle asserts that everything inside a region of [3D] space can be described by bits of information restricted to the boundary [of the region, coded on a distant 2D surface], from page 298 in
L. Susskind
The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics. Hachette Inc. ISBN 978-0-316-01640-7
https://en.wikipedia.org/wiki/The_Black_Hole_War
The conclusion by L. Susskind and G. 't Hooft presented in this book is that images in our 3D world (in a sort of sphere with a bounding surface) can be represented on flat 2D surface. This is really not surprising, since it was long ago observed that, although we live in a 3D world, we do all of our doodling in a flatland, introduced in
E.A. Abbott
FLATLAND: A Romance of Many Dimensions
https://ned.ipac.caltech.edu/level5/Abbott/paper.pdf
4D space postulate: More to the point, if we work inside a region of space-time [4D] space, then we can postulate that the information from a 4D space can be encoded on the boundary region of a 3D surface. The boundary region of 4D space would probably resemble the inner walls of either a torus (see the attached image) or a sphere.
Extending this postulate to higher dimensions to higher dimensional spaces is entirely possible.
Dear Zubair, the question is intriguing. However, we do not have to forget that the holographic principle holds for QUANTUM DYNAMICS surfaces, not for macroscopic ones. Indeed, a black hole and the cosmic horizon are considered as obeying to quantum dynamics laws. Therefore, tge sue, of this formula cannot be extended to macroscopic systems. Furthermore, the Beckenstein-Hawking formula talks about R^2, not about other dimensions.
This is a great question.
The holographic principle asserts that everything inside a region of [3D] space can be described by bits of information restricted to the boundary [of the region, coded on a distant 2D surface], from page 298 in
L. Susskind
The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics. Hachette Inc. ISBN 978-0-316-01640-7
https://en.wikipedia.org/wiki/The_Black_Hole_War
The conclusion by L. Susskind and G. 't Hooft presented in this book is that images in our 3D world (in a sort of sphere with a bounding surface) can be represented on flat 2D surface. This is really not surprising, since it was long ago observed that, although we live in a 3D world, we do all of our doodling in a flatland, introduced in
E.A. Abbott
FLATLAND: A Romance of Many Dimensions
https://ned.ipac.caltech.edu/level5/Abbott/paper.pdf
4D space postulate: More to the point, if we work inside a region of space-time [4D] space, then we can postulate that the information from a 4D space can be encoded on the boundary region of a 3D surface. The boundary region of 4D space would probably resemble the inner walls of either a torus (see the attached image) or a sphere.
Extending this postulate to higher dimensions to higher dimensional spaces is entirely possible.
Dear Jim, the problem with your account is that, when you go in dimensions higher and higher, the ratio surface/volume continuously change, is not a stable parameter: the volume encompassed in a 2D sphere is slightly different from the volume encompassed in a 6D sphere with the same radius. This makes unfeasible to achieve the Beckenstein-Hawking entropy in different dimensions. And my claim about the Beckenstein-Hawking entropy related just to quantistic systems prevents its use in our macroscopic world.
Yes-the holographic correspondence is defined for any spacetime dimension. Certain cases have attracted, of course, more attention than others. Indeed there's been considerable activity in the AdS3/CFT2 case, the AdS5/CFT4 correspondence, but, also, in the AdS7/CFT6 case.
Dear Zubair, do not forget that the holographic principle is not a physical theory, rather is just an hypothesis among thousands of others. The same concept of a framework based on information is misleading: what is information? The term information is as clear as the answer to the question: what is love? What is consciousness?
It's become considerably more than a hypothesis among others. The reason is because it provides the framework for performing concrete calculations.
Its relation with information theory is based on the idea that one way to describe the bulk dynamics in terms of boundary dynamics is through the entanglement entropy, between bulk and boundary, that can be described in terms of information theory.
All this is quite explicit, though, of course, does require some study, as this review: https://arxiv.org/pdf/hep-th/0203101.pdf shows. But anyone can learn it.
https://www.researchgate.net/publication/331731738_The_holographic_principle_just_an_empty_black_hole
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