貝肯斯坦上限
在物理學中,貝肯斯坦上限(英語:Bekenstein bound)是在一有限能量之有限空間內熵或資訊的上限。反過來說,該上限是要精確描述一物理系統至量子層級的最大需要資訊量[1]。這表示若要精確描述一個佔有有限空間之有限能量物理系統,只需要有限的資訊量。
![](http://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/BH_LMC.png/250px-BH_LMC.png)
緣起
編輯貝肯斯坦從涉及黑洞的啟發式觀點導出此上限式。如果存在系統違反此不等式,也就是有太多的熵,則貝肯斯坦認為這將違反熱力學第二定律。1995年,泰德·雅各布森證明了愛因斯坦場方程式[a]可以藉由假設貝肯斯坦上限和熱力學定律的真實性而導出[2][3]。然而,雖然一些理論已經表明某種形式的上限必須存在,以使熱力學和廣義相對論相互一致,但該上限的確切表述一直是人們爭論的一個問題[4][5][6][7][8][9][10][11][12][13][14]。
表示式
編輯此上限的普遍形式由雅各布·貝肯斯坦首次提出,以不等式表示之[1][4][5]。以熵表示之該不等式為:
其中 是熵、 是波茲曼常數、 是包圍整個系統的球殼半徑、 是包含任何靜止質量的總質能、 是約化普朗克常數、 則是真空中的光速。然而,雖然重力在此效應中扮演着很重要的角色,但該不等式中並未出現萬有引力常數 。
若以二進制資訊表示之,則該不等式為:
其中 是資訊含量,以位元數表示球殼中所含有的量子態。而式中ln2項則來自訂資訊量為量子狀態數目的自然對數值[15]。若使用質能等價定理,該資訊上限式可表示為:
貝肯斯坦-霍金方程式
編輯1972年,史蒂芬·霍金證明了黑洞視界的表面積永不會減少,兩個黑洞合併後的黑洞面積不會小於原先兩個黑洞面積之和。與此同時,雅各布·貝肯斯坦運用此理論提出了黑洞熵的概念。為了符合熱力學第二定律,黑洞必須擁有熵。如果黑洞沒有熵,則可以藉由將物質丟入黑洞中來違反熱力學第二定律。黑洞熵的增加必須超過被吞入物質所減少的熵。貝肯斯坦認為,黑洞的表面積與它的熵含量成正比,從而使其不違反熱力學第二定律。貝肯斯坦在他的論文中指出:
“ | 黑洞物理學和熱力學之間存在有很多相似之處,其中最顯著的是黑洞表面積和熵的行為的相似性:這兩個量都是不可逆地增加的[16]。 | ” |
——雅各布·貝肯斯坦:《黑洞和熵》Phys. Rev. D 7:2333-2346 (1973) |
貝肯斯坦認為,黑洞表面積與其熵含量的正比系數接近 。1974年,霍金提出了霍金輻射[17][18],並運用能量、溫度與熵之間的熱力學關係證實了貝肯斯坦的猜想,同時修正其正比系數為 [19][10]:
其中 是黑洞視界的表面積,利用 求得。 是波茲曼常數, 則是普朗克長度。此公式經常被稱為「貝肯斯坦-霍金方程式」(Bekenstein–Hawking formula),其中下標BH可指黑洞(black hole)或貝肯斯坦-霍金(Bekenstein-Hawking)的首字母縮寫。使用貝肯斯坦上限求得之最大熵含量正好等於由此方程式求得之黑洞熵,此結果促成了全像原理的發展[10]。
參見
編輯註釋
編輯參考資料
編輯- ^ 1.0 1.1 Jacob D. Bekenstein, "Universal upper bound on the entropy-to-energy ratio for bounded systems" (頁面存檔備份,存於互聯網檔案館), Physical Review D, Vol. 23, No. 2, (January 15, 1981), pp. 287-298, doi:10.1103/PhysRevD.23.287, Bibcode:1981PhRvD..23..287B. ().
- ^ 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, Bibcode:1995PhRvL..75.1260J. Also at , April 4, 1995. Also available here (頁面存檔備份,存於互聯網檔案館) and here (頁面存檔備份,存於互聯網檔案館). Additionally available as an entry (頁面存檔備份,存於互聯網檔案館)in the Gravity Research Foundation's 1995 essay competition..
- ^ Lee Smolin, Three Roads to Quantum Gravity (New York, N.Y.: Basic Books, 2002), pp. 173 and 175, ISBN 0-465-07836-2, .
- ^ 4.0 4.1 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, Bibcode:2005FoPh...35.1805B. Also at , April 7, 2004.
- ^ 5.0 5.1 Jacob D. Bekenstein, "Bekenstein bound" (頁面存檔備份,存於互聯網檔案館), Scholarpedia, Vol. 3, No. 10 (October 31, 2008), p. 7374, doi:10.4249/scholarpedia.7374.
- ^ 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, Bibcode:1999JHEP...06..028B. Mirror link. Also at , June 3, 1999.
- ^ 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, Bibcode:1999JHEP...07..004B. Mirror link. Also at , May 24, 1999.
- ^ 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, Bibcode:2000CQGra..17..997B. Also at , November 2, 1999.
- ^ 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, Bibcode:2000PhLB..481..339B. Also at , March 8, 2000.
- ^ 10.0 10.1 10.2 Bousso, Raphael. The Holographic Principle. Reviews of Modern Physics. 2002, 74 (3): 825–874. Bibcode:2002RvMP...74..825B. arXiv:hep-th/0203101 . doi:10.1103/RevModPhys.74.825 (英語).
- ^ Jacob D. Bekenstein, "Information in the Holographic Universe: Theoretical results about black holes suggest that the universe could be like a gigantic hologram" (頁面存檔備份,存於互聯網檔案館), Scientific American, Vol. 289, No. 2 (August 2003), pp. 58-65..
- ^ 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, Bibcode:2003PhRvD..68f4001B. Also at , May 19, 2003.
- ^ Jacob D. Bekenstein, "Black holes and information theory", Contemporary Physics, Vol. 45, Issue 1 (January 2004), pp. 31-43, doi:10.1080/00107510310001632523, Bibcode:2003ConPh..45...31B. Also at , November 9, 2003. Also at , November 9, 2003.
- ^ Frank J. Tipler, "The structure of the world from pure numbers" (頁面存檔備份,存於互聯網檔案館), Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964, doi:10.1088/0034-4885/68/4/R04, Bibcode:2005RPPh...68..897T.. Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything" (頁面存檔備份,存於互聯網檔案館), , 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.
- ^ Frank J. Tipler, "The structure of the world from pure numbers" (頁面存檔備份,存於互聯網檔案館), Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964, doi:10.1088/0034-4885/68/4/R04, Bibcode:2005RPPh...68..897T, p. 902. Mirror link. Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything" (頁面存檔備份,存於互聯網檔案館), , April 24, 2007, p. 8.
- ^ Jacob D. Bekenstein. Black Holes and Entropy. Phys. Rev. D. 1973-04-15, 7 (8): 2333–2346 [2015-09-08]. doi:10.1103/PhysRevD.7.2333. (原始內容存檔於2023-06-02) (英語).
- ^ Matson, John. Artificial event horizon emits laboratory analogue to theoretical black hole radiation. Sci. Am. Oct 1, 2010 [2015-09-08]. (原始內容存檔於2013-11-15) (英語).
- ^ A Brief History of Time, Stephen Hawking, Bantam Books, 1988.
- ^ Majumdar, Parthasarathi. Black Hole Entropy and Quantum Gravity 73: 14. 1999. Bibcode:1999InJPB..73..147M. arXiv:gr-qc/9807045 (英語).
外部連結
編輯- Jacob D. Bekenstein, "Bekenstein bound" (頁面存檔備份,存於互聯網檔案館) , Scholarpedia, Vol. 3, No. 10 (2008), p. 7374, doi:10.4249/scholarpedia.7374.
- Jacob D. Bekenstein, "Bekenstein-Hawking entropy" (頁面存檔備份,存於互聯網檔案館) , Scholarpedia, Vol. 3, No. 10 (2008), p. 7375, doi:10.4249/scholarpedia.7375.
- Jacob D. Bekenstein's website (頁面存檔備份,存於互聯網檔案館) at the Racah Institute of Physics, Hebrew University of Jerusalem, which contains a number of articles on the Bekenstein bound.