# Bekenstein bound

In physics, the Bekenstein bound is an upper limit on the entropy S, or information I, that can be contained within a given finite region of space which has a finite amount of energy—or conversely, the maximum amount of information required to perfectly describe a given physical system down to the quantum level.[1] It implies that the information of a physical system, or the information necessary to perfectly describe that system, must be finite if the region of space and the energy is finite. In computer science, this implies that there is a maximum information-processing rate (Bremermann's limit) for a physical system that has a finite size and energy, and that a Turing machine with unbounded memory is not physically possible.

## Equations

The universal form of the bound was originally found by Jacob Bekenstein as the inequality[1][2][3]

$S \leq \frac{2 \pi k R E}{\hbar c}$

where S is the entropy, k is Boltzmann's constant, R is the radius of a sphere that can enclose the given system, E is the total mass-energy including any rest masses, ħ is the reduced Planck constant, and c is the speed of light. Note that while gravity plays a significant role in its enforcement, the expression for the bound does not contain Newton's Constant G.

In informational terms, the bound is given by

$I \leq \frac{2 \pi R E}{\hbar c \ln 2}$

where I is the information expressed in number of bits contained in the quantum states in the sphere. The ln 2 factor comes from defining the information as the logarithm to the base 2 of the number of quantum states.[4] The right-hand side of the foregoing relation is approximately equal to 2.5769087×1043×(mass in kilograms)×(radius in meters).

## Origins

Bekenstein derived the bound from heuristic arguments involving black holes. If a system exists that violates the bound, i.e. by having too much entropy, Bekenstein argued that it would be possible to violate the second law of thermodynamics by lowering it into a black hole. In 1995, Ted Jacobson demonstrated that the Einstein field equations (i.e., general relativity) can be derived by assuming that the Bekenstein bound and the laws of thermodynamics are true.[5][6] However, while a number of arguments have been devised which show that some form of the bound must exist in order for the laws of thermodynamics and general relativity to be mutually consistent, the precise formulation of the bound has been a matter of debate.[2][3][7][8][9][10][11][12][13][14][15]

## Examples

### Black holes

It happens that the Bekenstein-Hawking Entropy of three-dimensional black holes exactly saturates the bound

$S =\frac{kA}{4}$

where A is the two-dimensional area of the black hole's event horizon in units of the Planck area, $\hbar G/c^3$.

The bound is closely associated with black hole thermodynamics, the holographic principle and the covariant entropy bound of quantum gravity, and can be derived from a conjectured strong form of the latter.

## References

1. ^ a b Jacob D. Bekenstein, "Universal upper bound on the entropy-to-energy ratio for bounded systems", Physical Review D, Vol. 23, No. 2, (January 15, 1981), pp. 287-298, doi:10.1103/PhysRevD.23.287, Bibcode1981PhRvD..23..287B. Mirror link.
2. ^ a b Jacob D. Bekenstein, "How Does the Entropy/Information Bound Work?", Foundations of Physics, Vol. 35, No. 11 (November 2005), pp. 1805-1823, doi:10.1007/s10701-005-7350-7, Bibcode2005FoPh...35.1805B. Also at arXiv:quant-ph/0404042, April 7, 2004.
3. ^ a b Jacob D. Bekenstein, "Bekenstein bound", Scholarpedia, Vol. 3, No. 10 (October 31, 2008), p. 7374, doi:10.4249/scholarpedia.7374.
4. ^ Frank J. Tipler, "The structure of the world from pure numbers", Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964, doi:10.1088/0034-4885/68/4/R04, Bibcode2005RPPh...68..897T, p. 902. Mirror link. Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything", arXiv:0704.3276, April 24, 2007, p. 8.
5. ^ Ted Jacobson, "Thermodynamics of Spacetime: The Einstein Equation of State", Physical Review Letters, Vol. 75, Issue 7 (August 14, 1995), pp. 1260-1263, doi:10.1103/PhysRevLett.75.1260, Bibcode1995PhRvL..75.1260J. Also at arXiv:gr-qc/9504004, April 4, 1995. Also available here and here. Additionally available as an entry in the Gravity Research Foundation's 1995 essay competition. Mirror link.
6. ^ Lee Smolin, Three Roads to Quantum Gravity (New York, N.Y.: Basic Books, 2002), pp. 173 and 175, ISBN 0-465-07836-2, LCCN 2007-310371.
7. ^ Raphael Bousso, "Holography in general space-times", Journal of High Energy Physics, Vol. 1999, Issue 6 (June 1999), Art. No. 28, 24 pages, doi:10.1088/1126-6708/1999/06/028, Bibcode1999JHEP...06..028B. Mirror link. Also at arXiv:hep-th/9906022, June 3, 1999.
8. ^ Raphael Bousso, "A covariant entropy conjecture", Journal of High Energy Physics, Vol. 1999, Issue 7 (July 1999), Art. No. 4, 34 pages, doi:10.1088/1126-6708/1999/07/004, Bibcode1999JHEP...07..004B. Mirror link. Also at arXiv:hep-th/9905177, May 24, 1999.
9. ^ Raphael Bousso, "The holographic principle for general backgrounds", Classical and Quantum Gravity, Vol. 17, No. 5 (March 7, 2000), pp. 997-1005, doi:10.1088/0264-9381/17/5/309, Bibcode2000CQGra..17..997B. Also at arXiv:hep-th/9911002, November 2, 1999.
10. ^ Jacob D. Bekenstein, "Holographic bound from second law of thermodynamics", Physics Letters B, Vol. 481, Issues 2-4 (May 25, 2000), pp. 339-345, doi:10.1016/S0370-2693(00)00450-0, Bibcode2000PhLB..481..339B. Also at arXiv:hep-th/0003058, March 8, 2000.
11. ^ Raphael Bousso, "The holographic principle", Reviews of Modern Physics, Vol. 74, No. 3 (July 2002), pp. 825-874, doi:10.1103/RevModPhys.74.825, Bibcode2002RvMP...74..825B. Mirror link. Also at arXiv:hep-th/0203101, March 12, 2002.
12. ^ Jacob D. Bekenstein, "Information in the Holographic Universe: Theoretical results about black holes suggest that the universe could be like a gigantic hologram", Scientific American, Vol. 289, No. 2 (August 2003), pp. 58-65. Mirror link.
13. ^ Raphael Bousso, Éanna É. Flanagan and Donald Marolf, "Simple sufficient conditions for the generalized covariant entropy bound", Physical Review D, Vol. 68, Issue 6 (September 15, 2003), Art. No. 064001, 7 pages, doi:10.1103/PhysRevD.68.064001, Bibcode2003PhRvD..68f4001B. Also at arXiv:hep-th/0305149, May 19, 2003.
14. ^ Jacob D. Bekenstein, "Black holes and information theory", Contemporary Physics, Vol. 45, Issue 1 (January 2004), pp. 31-43, doi:10.1080/00107510310001632523, Bibcode2003ConPh..45...31B. Also at arXiv:quant-ph/0311049, November 9, 2003. Also at arXiv:quant-ph/0311049, November 9, 2003.
15. ^ Frank J. Tipler, "The structure of the world from pure numbers", Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964, doi:10.1088/0034-4885/68/4/R04, Bibcode2005RPPh...68..897T. Mirror link. Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything", arXiv:0704.3276, April 24, 2007. Tipler gives a number of arguments for maintaining that Bekenstein's original formulation of the bound is the correct form. See in particular the paragraph beginning with "A few points ..." on p. 903 of the Rep. Prog. Phys. paper (or p. 9 of the arXiv version), and the discussions on the Bekenstein bound that follow throughout the paper.