Holographic principle in reverse

[spenserf]

I'm getting a rough idea of the holographic principle relating the shannon entropy of a boundary surface to the thermodynamic entropy contained within the bounded volume. So far as I understand the primary claim is that the total information needed to describe the entirety of the internal volume if proportional to the area, borrowing the equation for black hole entropy, by the equation S = kA/4. What I'm wrestling with is a sort of application problem. Could this encoded surface be retopologized and fit within the volume which it described? How could this be? What about going the other direction? If we can fully describe space of D+1 dimensions using only D dimensions, would it be possible to take a region of space and have an accurate 2d representation of our entire universe? I have a feeling the answer lies somewhere in the Bekenstein bound.

Can anyone help to clarify the situation to me?

Danke.

 

[Chalnoth]

Could this encoded surface be retopologized and fit within the volume which it describes? How could this be? What about going the other direction? If we can fully describe space of D+1 dimensions using only D dimensions, would it be possible to take a region of space and have an accurate 2d representation of our entire universe?

The way holography works is you have two different descriptions of the same physical system. In one description, the system has D dimensions and follows some set of physical laws. In the other description, the system has D+1 dimensions and follows a different (but related) set of physical laws.

We already know this is possible with the electromagnetic force: if you know the electromagnetic four-potential at every point along the boundary of a location in space, then you can calculate (in principle) the electromagnetic field at every three-dimensional point within that space. I don't think it's proven that this is possible for all of the other forces as well, but there are some entropy arguments that seem to suggest it may be the case.

These two descriptions are just two sides of the same coin. If you have one description of the system, and know the relationship between these two descriptions, then you can, in principle, translate from one way of describing the system to the other. There's no reason to believe that you could describe the system properly with D-1 or D+2 dimensions, however. The fact that holography exists at all is a very peculiar fact of the specific mathematics of the physical laws in question.

 

[spenserf]

I see. So more specifically, is it possible to contain the D description of the D+1 system within the D+1 system it's describing? I'm trying to get at the implications to the simulation argument. Can a complete description of the universe be contained within the universe? Or, I guess more practically, can any universe contain a complete description of another universe of the same volume?

 

[Chalnoth]

I see. So more specifically, is it possible to contain the D description of the D+1 system within the D+1 system it's describing? I'm trying to get at the implications to the simulation argument. Can a complete description of the universe be contained within the universe? Or, I guess more practically, can any universe contain a complete description of another universe of the same volume?

No. A complete description of the universe cannot be contained within the universe.

All of the information that is possible to store within a horizon is encoded on the horizon. Holography doesn't get you out of the impossibility of a set of finite size to include itself as a member.

The only way out of this is to say that the "description" of the universe is the universe itself. But that's sort of trivial.

 

[sardar]

The holographic principle is a concept in physics that suggests the information contained within a bounded volume can be fully described by the information on its boundary surface. This is analogous to a hologram, where a 2D surface contains all the information to create a 3D image. The equation you mentioned, S = kA/4, is known as the Bekenstein-Hawking entropy formula and relates the entropy of a black hole to its surface area.

Your question about retopologizing the encoded surface is an interesting one. It is possible that the information on the boundary surface could be rearranged and fit within the bounded volume, but it would likely require complex mathematical transformations. As for going the other direction, it is possible that a 2D representation of our universe could contain all the information needed to fully describe it in higher dimensions, but this is still a topic of ongoing research.

The Bekenstein bound, which states that the entropy of a bounded region cannot exceed a certain limit, is closely related to the holographic principle. It suggests that there is a maximum amount of information that can be contained within a bounded region, which supports the idea that the information on the boundary surface is sufficient to describe the internal volume.

Overall, the holographic principle and its implications are still being explored and understood by scientists. It is a fascinating concept that has the potential to revolutionize our understanding of the universe.