What is the connection between the holographic principle in cosmology and fractal geometry?

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What is the connection between the holographic principle in cosmology and fractal geometry?

There is a connection between the holographic principle in cosmology and fractal geometry. Fractals are mathematical structures that exhibit self-similarity at different scales, meaning that they look the same at all levels of magnification. The holographic principle also suggests that the information contained in a region of space is spread out over its boundary, which has a fractal-like structure.

 

In particular, the holographic principle implies that the information content of a region of space is related to the number of degrees of freedom on its boundary. This number of degrees of freedom can be related to the area of the boundary, which has a fractal-like structure, and can be described by fractal geometry. This connection between the holographic principle and fractal geometry has led some researchers to propose that the universe itself may be a fractal structure.

 

One of the most prominent examples of the connection between the holographic principle and fractal geometry is the AdS/CFT correspondence. This is a duality between two seemingly different theories, one of which describes a higher-dimensional space with gravity (called anti-de Sitter space, or AdS), while the other describes a lower-dimensional space without gravity. The AdS/CFT correspondence suggests that these two theories are actually equivalent and describe the same physical system. This correspondence has been shown to have a fractal structure, with the higher-dimensional space exhibiting self-similarity at different scales.

 

Overall, the connection between the holographic principle and fractal geometry suggests that the structure of the universe may be intimately connected to the mathematics of fractals, which could have important implications for our understanding of the nature of space, time, and gravity.

 

Further reading: QFT on DeSitter space -

https://home.uni-leipzig.de/tet/?page_id=285

Is de Sitter space with non-zero curvature an acceptable model for the universe? -  https://physics.stackexchange.com/questions/353148/is-de-sitter-space-with-non-zero-curvature-an-acceptable-model-for-the-universe


 

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