Preamble
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| Bekenstein's Information (Wikipedia) |
Black holes are not too esoteric anymore after LIGO's success and successful imaging efforts. Their entropy behaves much different than the entropy of an ordinary matter. This leads to incredible discovery of so called holographic principle. The principle stating that we live on a projection of higher-dimensional manifestation of universe. On the other hand, Landauer made silently a discovery on energy expenditure of keeping information processing reversible. Similar bound put forward by Bekenstein for black holes made Landauer-Bekenstein bound for quantum gravity a natural avenue to study.
What is the Bekenstein Bound?
Basically, this puts limits the size of a black hole, via its entropy bound. Essentially it states that, entropy of a black hole $S$ is bounded with the radius of the black hole $R$ and Energy $E$, other constants being 1,
$$S \le R \dot E$$
What is Landauer Principle?
A limit on any process wants to delete 1 bit of information, has to dissipate energy proportional to its temperature,
$$ E \ge T ln 2$$
Again we made the constants to 1.
Physical limit of data storage: 1 BekensteinBytes
Using both Bekenstein bound and Landauer principle one can compute the a physical limit for a data storage on a unit sphere,
$$S \sim T$$
If we scaled this with inverse of Planck area $\ell_{p}$; One bit of information proportional to Planck area and maximum attainable temperature $10^{32}$ K is scaled with this. The final value corresponds to $\approx 10^{100}$ bits. This again corresponds to about 10 Giga-Quetta-Quatta-Quatta bytes (1 Quetta byte is $10^{30}$ bytes).
At this point in time, we can propose that $10^{100}$, would be to call 1 BekensteinBytes. However with more fine-grain computations, the number may change. Our purpose here is to give a very rough idea about the scale of ultimate physical limitations of data storage in BekensteinBytes.
Bekenstein Information Conjecture : One cannot compress more than 1 BekensteinBytes on a smallest patch of space.
Outlook
An interesting connections in quantum gravity and computation, provides certain physical limitations on how much information we can be stored at a given unit space. We called this Bekenstein Information Conjecture.
Further reading
- Insights into Bekenstein entropy with an intuitive mathematical definitions: A look into Thermodynamics of Black-holes (2023)
- Bekenstein Bound (Wikipedia)
- Landauer's Principle (Wikipedia)
- On the relation between Bekenstein entropy and Landauer’s principle, D. Song (2024) DOI
- Planck Scales (Wikipedia)
- Metric Prefixes (Wikipedia)
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