由於內文跟物理電腦也有相關，所以就選在電腦板上與大家分享
這是一篇論文，述說著我們所在的世界物理理論在宇宙都是差不多相通的
內行人看門到，外行人看熱鬧。當然我也是半個外行人...
但是看著這些訊息也能略知我們的世界有多麼奇妙，而同樣頭上都是一顆腦袋
但是奇人們的腦似乎想的都是世界外的知識...只能說這些奇人們好神阿...
以下就是內文~大家慢慢觀賞吧。
原著
Information in the Holographic Universe
Theoretical results about black holes suggest that the universe could be like a gigantic hologrram.
Scientific American August 2003
編譯
何毓嵩/譯
原文連結
http://sufizmveinsan.com/fizik/holographic.html
編譯者連結
http://blog.xuite.net/wellsli/003/32382565
全文
如果你問別人物理世界是由什麼構成的，他很可能告訴你是“物質和能量”。但只要我們學過一點工程、生物和物理的話，就知道訊息同樣是一個不可或缺的組成部分。
只給汽車廠的機器人金屬和塑料，它們不可能做出任何有用的東西，只有給它們下達如何焊接的指令它們才能組裝出汽車。我們身體細胞中的核醣體擁有阿米諾酸組建模塊和ATP合成為ADP過程中釋放的能量，但如果沒有細胞核中DNA所攜帶的信息，同樣無法合成任何蛋白質。類似地，一個世紀以來物理學的進展告訴我們，訊息在物理系統和物理過程中起著關鍵的作用。實際上，現在就有一個學派認為物理世界是由訊息構成的，它的創始人是美國普林斯頓大學的John A. Wheeler。該理論認為訊息才是最重要的，物質和能量不過是附屬物而已。
這種觀點引發了對許多古老問題的重新審視。硬碟之類存儲設備的訊息存儲容量獲得了高速發展。這樣的進展什麼時候會終止？一個重量小於1公克，體積小於1立方公分（這大約是計算機晶片的尺寸）的設備的終極訊息存儲容量是多少？描述整個宇宙需要多少資訊？這種描述能被裝入計算機的記億體中嗎？我們真的能像William Blake說的那樣“透過一粒沙看世界”嗎？抑或這種說法只不過是詩人的狂想？
值得注意的是，近期理論物理學的進展解答了上面的部分問題，而這些回答很有可能是找到客觀的最終理論的重要線索。通過研究黑洞的那些神秘特性，物理學家已經推導出了某一部分空間或一定量的物質和能量所能包含訊息量的絕對限度。相關的研究結果表明，我們的宇宙也許並不是一個我們所認為的那種三維空間，它很有可能是某種“寫”在二維表面上的全像圖形(Holograms)。我們對日常世界的三維認知要麼是一種玄奧的幻覺，要麼就是觀照現實的兩種方式之一而已。一粒沙也許不能包含整個宇宙，但是一個平面顯示器卻有可能做到。兩種熵正統訊息論的創始人是美國應用數學家夏農。他於1948年發表了一系列開創性的論文，所引入的熵這一概念如今被廣泛用於訊息的度量。長久以來，熵就是熱力學（研究熱的一個物理學分支）的中心概念。熱力學中的熵通常被用於表徵一個物理系統的無序程度。 1877年，奧地利物理學家玻爾茲曼提出了一種更為精確的描述：一團物質在保持宏觀特性不變的情況下，其中所包含的粒子所有可能具有的不同微觀狀態數就是熵。例如，對於包圍你的室內空氣而言，就可以計算單個空氣分子所有可能的分佈方式及其所有可能的運動方式。
當夏農設法量化一條消息中的訊息時，他自然而然地得出了一條和玻爾茲曼一樣的公式。一條消息的夏農熵就是編碼這條消息所需二進制位即bit的個數。夏農熵並不能告訴我們一條消息的價值，因為後者主要取決於上下文。然而作為對信息量的一種客觀度量，夏農熵還是在科學技術中獲得了廣泛的應用。例如，任一現代通信設施――蜂巢電話、調製解調器、CD播放器等等――的設計都離不開夏農熵。
從概念上來說，熱力學熵和夏農熵是等價的：玻爾茲曼熵所代表的不同組成方式的數目反映了為實現某種特定組成方式所必須知道的夏農信息量。但這兩種熵還是存在著某些細微的差別。首先，一名化學家或製冷工程師所使用的熱力學熵的表示單位是能量除以溫度，而通信工程師所使用的夏農熵則表示為bits數，後者在本質上是無單位的。這一差別完全屬於習慣問題。
即使採用同樣的表示單位，兩種熵值的量級還存在著巨大的差異。例如，帶有1GB數據的晶片的夏農熵約為10*10個bits（1個英文字母相當於8個bits），這比該晶片的熱力學熵可小多了，後者在室溫下的取值約為10*23bits。這種差異來源於兩種熵在計算時所考慮的不同自由度。自由度指的是某一可變化的量，例如表示一個粒子位置或速度分量的座標。上述晶片的夏農熵關心的只是蝕刻在矽晶上所有晶體管的狀態。晶體管到底是開還是關；它要麼為0，要麼為1，是單一的二進制自由度。熱力學熵則不同，它取決於每一個晶體管所包含的數十億計的原子（以及圍繞它們的電子）的狀態。隨著小型化工藝的發展，不久的將來我們就能用一個原子來存儲一比特的信息，到那時，微晶片的夏農熵將在量級上迫近其材料的熱力學熵。當用同樣的自由度計算這兩種熵時，它們將是完全相同的。
那麼自由度是否存在極限？原子由原子核和電子組成，原子核又由質子和中子組成，質子和中子又由夸克組成。今天有許多物理學家認為電子和夸克不過是超弦的激發態而已，他們認為超弦才是最基本的實體。然而一個世紀以來物理學的興衰變遷告訴我們不能這樣武斷。宇宙的結構層次有可能比今天的物理學所夢想的還要多得多。
不知道一團物質的終極組成部分或其最深層次的結構，我們就無法計算其終極訊息容量，同樣也無法計算其熱力學熵。我把這種最深的結構層次稱為第X層。 （這種不確定的描述在實際的熱力學分析中毫無問題，例如當我們分析一個汽車引擎，原子中的夸克就可以被忽略掉，因為在引擎這樣一種相對溫和的環境下，它們是不會改變狀態的。）按照微型化技術目前這樣快的發展速度，我們可以設想將來某日夸克能被用來儲存信息，也許是一個夸克一個bit。到那時一立方厘米能存儲多少信息？假如我們能進一步利用超弦或者更深層次的結構來儲存訊息呢？令人吃驚的是，近30年來引力物理學領域的成果對這些看似深奧的問題提供了一些明確的答案。
黑洞熱力學
這些成果的一個中心角色就是黑洞。黑洞是廣義相對論（愛因斯坦1915年提出的引力幾何理論）的產物。根據這一理論，引力來源於時空的扭曲，它使得物體發生移動，就像有一個力在推動一樣。與之可逆的是，物質和能量的存在導致了時空的扭曲。根據愛因斯坦的方程式，一團足夠緻密的物質或能量能將時空彎曲到撕裂的極端程度，這時黑洞就形成了。至少在經典（非量子的）物理學範疇內，相對論決定了任何進入黑洞的物質都無法再從中逃脫。這個有去無回的點被稱為黑洞的視界。在最簡單的情況下，視界是一個球面，黑洞越大，這個球體的表面積就越大。
要探究黑洞內部是不可能的。沒有任何具體的信息能穿過視界逃離到外部世界中。然而，在進入黑洞並永久消失之前，一團物質還是能留下一些線索的。它的能量（按照愛因斯坦方程E=mc*2，可以將任意質量換算成能量）將不變地反映為黑洞質量的增量。如果在被黑洞捕獲前它正在圍繞黑洞旋轉，那麼它的角動量將被加到黑洞的角動量之中。黑洞的質量和角動量都可以通過黑洞對周圍時空的作用而獲得測量。這樣，黑洞也遵守能量和角動量守恆準則。但另一個基本定律，即熱力學第二定律，看起來是被破壞了。
熱力學第二定律是對慣常觀測現象的一個總結：自然界中絕大部分過程都是不可逆的。茶杯從桌上摔碎後，沒有人看到碎片自己按原路蹦回又組成一隻完整的杯子。熱力學第二定律禁止這些逆過程的發生。它指出，孤立系統的熵永遠不可能減少；熵最多保持不變，大部分情況下，熵值是增加的。這條定律是物理化學和工程學的核心；它被認為是對物理學之外其他領域產生影響最多的一條定律。
就像Wheeler首先指出的那樣，當物質消失於黑洞時，它的熵似乎永久消失了，熱力學第二定律這時看起來也失效了。解決這一謎題的線索首先出現於1970年。 Demetrious Christodoulou（當時他在普林斯頓大學做Wheeler的研究生）和英國劍橋大學的史蒂芬霍金各自獨立證明了，在多種不同的過程中（例如黑洞的合併等），最終的視界總表面積不會減少。通過將這一性質和熵值趨向於增加的特性相類比，我於1972年提出了黑洞熵值正比於其視界表面積的理論。根據我的推測，物質落入黑洞後，黑洞熵值的增加總能補償或者過補償該物質所“喪失”的熵。更廣泛地來說，黑洞的熵值及其外面的普通熵值之和永遠不會變小。這就是廣義第二定律（簡稱GSL）。
GSL已經通過了大量嚴格（如果僅從理論上來看的話）的驗證。當一顆恆星坍塌稱為一個黑洞時，黑洞的熵值將大大超過該恆星的熵值。 1974年霍金證明了黑洞必然會通過一個量子過程釋放我們現今稱之為霍金輻射的熱輻射。對於這種現象（黑洞的質量及其視界表面積都減少了），Christodoulou-Hawking定理就失效了，然而GSL卻能適用：黑洞散發出去的熵值超過了其本身熵值的減少，所以GSL仍然成立。 1986年，美國雪城大學的Rafael D. Sorkin研究了視界在阻止黑洞內部信息影響外部事件時起到的作用，他因此得出結論：對於黑洞發生的任何可能的過程，GSL（或與之非常相似的理論）必然是成立的。他的深入研究明確指出，無論X取值多少，GSL中的熵對層次X都是成立的。
霍金對輻射過程的處理使他得到了黑洞熵值和視界表面積之間的比例關係：黑洞的熵值恰恰是按照普朗克表面積丈量的視界表面積的1/4。 （普朗克長度，約為10*－33公分，是萬有引力和量子理論中的基本長度單位。普朗克表面積即它的平方。）即使是從熱力學熵的角度來看，這個值也是非常巨大的。一個直徑為1厘米的黑洞的熵值約為10*66bits，這大致和一個邊長為100億公里的立方水柱所含的熱力學熵相當。
GSL讓我們有可能為任何孤立的物理系統設定信息容量的限度。這些限度對於直到X層的任何結構層次都將成立。 1980年我開始研究第一個這樣的界。它被稱為通用熵界，它確定了特定尺寸特定質量的物質所能包含信息量的界限。美國斯坦福大學的Leonard Susskind於1995年提出了一個與之相關的稱為全息界的概念。它確定了佔據一定空間體積的物質或能量所能包含信息量的界限。
在研究全息界的過程中，Susskind考察的是一團近乎球體的孤立物質，它並非黑洞，而是被緊密地裝入到一個表面積為A的表面中。如果該物質能坍塌為黑洞，則最終形成的黑洞的視界表面積將小於A。這樣黑洞熵將小於A/4。按照GSL，該系統的熵不能減少，因而物質的初始熵不能大於A/4。這樣我們就可以得出結論：邊界表面積為A的孤立物理系統的熵值必然小於A/4。然而如果該物質無法自行坍塌又如何呢？我在2000年證明了，一個小的黑洞可以將一個和Susskind論證過程中那個沒什麼大區別的系統轉變為黑洞。因而這個界是獨立於系統的組成或者層次X的特性的。它僅僅依賴於GSL。
現在我們可以回答某些關於信息存儲量最終限度的深奧問題了。一個直徑為1厘米的裝置理論上可以存儲高達10*66比特的信息量，這可是一個令人難以置信的數量。可見的宇宙最少包含了10*100比特的熵，理論上可將之裝入到一個直徑為十分之一光年的球體之中。要估計宇宙的熵很困難，然而對於特別大的數值，例如一個幾乎與宇宙本身一樣大小的球體，則是完全可行的。
但是全息界的另一方面卻真正讓人大吃一驚。就是說，最大可能的熵值取決於邊界面積而不是體積。讓我們設想將計算機內存芯片堆成一個大堆。晶體管的數目（即總的數據存儲容量）隨著堆體積的增加而增大。所有芯片的熱力學熵之和也同樣增大。然而值得注意的是，這堆芯片所佔據空間的理論終極信息容量僅僅隨表面積的增加而增加。因為體積的增長遠遠快於表面積的增長，到某一程度，所有芯片的熵值之和將超過全息界。看起來無論是GSL還是我們通常意義上的熵和信息容量都失效了。實際的情況上，真正失效的是堆積過程本身：在上述情況出現之前，它就將因為本身的引力而坍塌並形成一個黑洞。在此之後每增加一個芯片都將增大黑洞的質量和表面積，但這都將遵循GSL。
如果全息原理（由諾貝爾得主、荷蘭烏得勒支大學的Gerarad't Hooft於1993年提出，並得到了Susskind的進一步闡述）是正確的話，信息容量取決於表面積這一令人吃驚的結論就將得到自然的解釋。在日常世界裡，全息圖形是一種特殊的膠片，當用合適的方法將它曝光時，它就將產生一個真正3維的影像。描述3維圖景的所有信息都被編碼到2維膠片上的明暗相間的圖樣上。用這個膠片隨時都可以復現該3維圖景。全息原理指出，這一視覺魔術的原理可以類推到對任何一個佔據3維區域的系統的所有物理學描述之中，另一個在該區域的2維邊界上定義的物理學理論能完全描述該3維區域的物理學。如果一個3維繫統能被運作於其2維邊界上的物理理論所完全描述，我們就有理由推測該系統的信息容量不可能超越其邊界上的描述。
“畫”在邊界上的宇宙
我們能把全息原理推廣到宇宙這樣大的範圍嗎？真正的宇宙是一個4維繫統：它有體積並隨著時間軸延伸。如果我們這個宇宙的物理學具有全息性，那麼就會存在另外一套運作在某個時空的三維邊界上的物理學定律，它們將和我們現在所知的4維物理學完全等效。到目前為止我們還沒有發現任何這樣的3維定律。事實上，我們拿哪個界面做為宇宙的邊界呢？要實現這些想法我們需要首先邁出的一步就是研究比真實宇宙更簡單的那些模型。
所謂的反德西特時空就是一類全息原理能成立的具體例子。原始的德西特時空是荷蘭天文學家威廉·德西特於1917年根據愛因斯坦方程式導出的一個解，其中包括了被稱為宇宙常量的斥力。德西特時空是空曠的，以一定的加速度膨脹並且是高度對稱的。 1997年，宇宙學家在研究遙遠的超新星爆發時得出結論：我們的宇宙正在加速膨脹，未來它有可能變得越來越像一個德西特時空。如果我們將愛因斯坦方程式中的斥力換成引力，那麼德西特解將變成一個反德西特時空，它和德西特時空具有相同的對稱性。對於全息概念來說，反德西特時空的重要性就在於它擁有一個位於“無限”處的邊界，這一點和我們的日常時空非常相似。
利用反德西特時空，理論家設計出了一個全息原理起作用的具體例子：一個在反德西特時空內運作的宇宙可以用超弦理論完全描述，這套描述和在該時空邊界上起作用的量子場論完全等效。這樣，上述反德西特時空內部超弦理論的全部奧秘就都被畫在了該宇宙的邊界上。 1997年，Juan Maldacena（那時他還在哈佛大學）首先推測，在5維反德西特時空上存在這種關係。此後，美國新澤西州普林斯頓大學高級研究院的Edward Witten及普林斯頓大學的Steven S. Gubser、Igor R. Klebanov和Alexander M. Polyakov在多種情況下證實了該推測。現在我們已經知道在多種不同維數的時空上都存在著這樣的全息對應關係。
這個結論意味著，兩個表面上看來非常不同的理論（它們甚至是各自生效在不同維數的時空裡）是完全等效的。生存在這些宇宙中的生物將無法確定它們是棲息於一個由弦論描述的5維時空還是一個由量子場論描述的4維時空中。 （當然，這些生物的大腦結構也許會給它們一種“常識”，讓它們以為自己是生存於某一種宇宙中。就像我們的大腦結構讓我們有一種內在的感覺，我們的宇宙具有3維空間結構；參見下頁圖示）全息等價使得一個在某一時空中難以計算的問題可以用另一種方式解決。比如，4維邊界時空上夸克和膠子特性的計算，就可以轉化為在高度對稱的5維反德西特時空上更簡易的計算。這種對應關係還有其他的表現方式。 Witten就曾證明，反德西特時空上的黑洞等價於其邊界時空上的熱輻射體。黑洞這個神秘概念的熵就等於該輻射體的熵，顯然後者要容易理解得多。
不斷膨脹的宇宙
度對稱且空曠的5維反德西特時空和我們這個充斥著物質和輻射且不斷受劇烈事件擾動的4維宇宙似乎有很大不同。即使把我們的宇宙近似為一個物質和輻射體均勻分佈的系統，我們得到的也不是一個反德西特宇宙，我們得到的將是一個“弗里德曼-羅伯遜-沃克”宇宙。今天絕大部分的天文學家都認為我們的宇宙是一個無限的、無邊界的並將永遠膨脹的“弗里德曼-羅伯遜-沃克”宇宙。
這樣的一個宇宙還遵守全息原理或具有全息界嗎？ Susskind基於坍塌至黑洞的推斷在這裡毫無作用。實際上，由黑洞所導出的全息界必然在我們這個單調膨脹的宇宙中失效。一塊均勻分佈著物質和輻射的區域的熵確實將和它的體積成正比。這樣的話，一塊足夠大的區域（所包含的熵）就會突破全息界。
Raphael Bousso於1999年（當時在斯坦福大學）提出了一個改進的全息界，後來發現這個界在上面所述的那些原全息界遇到問題的地方還能適用。 Bousso這個全息界的構成起始於任意合適的2維界面；它可以像一個球面一樣是封閉的，也可以像一張紙那樣是開放的。現在讓我們來想像一束短暫的光線同時從這個界面的一邊垂直射入。這裡唯一的要求就是這些虛擬的光線都是從同一點發射出來的。例如說，從一個球面的內部透射出來的光線就符合這一要求。現在讓我們來看這些光線所經過的物質和輻射體的熵。 Bousso推測說這個熵值不能超過由初始界面所代表的熵——表面積的1/4（以普朗克面積為單位）。這種計算熵的方法和原來那種全息界的計算方法有所不同。 Bousso界並非只考慮某一時刻某一區域的熵值，它計算的是不同時間不同位置的熵值之和：那些被從表面來的光線所“照亮”的熵。
Bousso界在繼承其他熵界的基礎上又避免了它們的局限性。只要所涉及的孤立系統變化不是很快，引力場不是很強，無論是通用熵界還是全息界的't Hooft-Susskind形式都可以從Bousso界中推導得出。如果這些條件都不滿足——例如涉及的物質已經落入了黑洞之中，那麼這些界就將失效，但Bousso界卻能繼續適用。 Bousso還證明了，他的這一方法能用於定位建立世界全息圖形的2維界面。 革命性的前夜
研究人員已經提出了各種各樣的熵界。對於全息這一課題，存在那麼多的流派，這證明它還沒有上升到物理定律的高度。雖然全息的思想還沒有完全被我們所理解，但它看起來確確實實是對的。隨之而來的是，人們開始認識到，盛行了50年的那個基本信仰，即場論是物理學的最終語言的看法，必須拋棄了。場，比如說電磁場，不同點之間是連續變化的，因而它們描述的自由度是無限的。超弦理論也支持無限多的自由度。全息論則將一個封閉界面裡的自由度限製到一個有限的數目上；場論因為其自由度的無限所以不可能是最終理論。此外，即使自由度無限的問題得到了解決，信息量和表面界之間那種神秘的對應關係也應該得到解決。
全息論也許為另一個更好的理論指明了方向。基本理論應該是什麼樣子的？全息論發展過程中的一系列論證推理讓某些科學家（其中最著名的是加拿大沃特盧理論物理Perimeter學院的Lee Smolin）提出，最終理論考慮的不是場，甚至不是時空，而應該是物理過程之間的信息交換。如果真是這樣的話，把信息看成世界的組成部分的觀點就體現了它的價值。
何毓嵩/譯
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Information in the Holographic Universe
Theoretical results about black holes suggest that the universe could be like a gigantic hologrram.
Scientific American August 2003
An astonishing theory called the holographic principle holds that the universe is like a hologram: just as a trick of light allows a fully three-dimensional image to be recorded on a flat piece of film, our seemingly three-dimensional universe could be completely equivalent to alternative quantum fields and physical laws "painted" on a distant, vast surface.
The physics of black holes--immensely dense concentrations of mass--provides a hint that the principle might be true. Studies of black holes show that, although it defies common sense, the maximum entropy or information content of any region of space is defined not by its volume but by its surface area.
Physicists hope that this surprising finding is a clue to the ultimate theory of reality.
Ask anybody what the physical world is made of, and you are likely to be told "matter and energy."
Yet if we have learned anything from engineering, biology and physics, information is just as crucial an ingredient. The robot at the automobile factory is supplied with metal and plastic but can make nothing useful without copious instructions telling it which part to weld to what and so on. A ribosome in a cell in your body is supplied with amino acid building blocks and is powered by energy released by the conversion of ATP to ADP, but it can synthesize no proteins without the information brought to it from the DNA in the cell's nucleus. Likewise, a century of developments in physics has taught us that information is a crucial player in physical systems and processes. Indeed, a current trend, initiated by John A. Wheeler of Princeton University, is to regard the physical world as made of information, with energy and matter as incidentals.
This viewpoint invites a new look at venerable questions. The information storage capacity of devices such as hard disk drives has been increasing by leaps and bounds. When will such progress halt? What is the ultimate information capacity of a device that weighs, say, less than a gram and can fit inside a cubic centimeter (roughly the size of a computer chip)? How much information does it take to describe a whole universe? Could that description fit in a computer's memory? Could we, as William Blake memorably penned, "see the world in a grain of sand," or is that idea no more than poetic license?
Remarkably, recent developments in theoretical physics answer some of these questions, and the answers might be important clues to the ultimate theory of reality. By studying the mysterious properties of black holes, physicists have deduced absolute limits on how much information a region of space or a quantity of matter and energy can hold. Related results suggest that our universe, which we perceive to have three spatial dimensions, might instead be "written" on a two-dimensional surface, like a hologram. Our everyday perceptions of the world as three-dimensional would then be either a profound illusion or merely one of two alternative ways of viewing reality. A grain of sand may not encompass our world, but a flat screen might.
The Entropy of a Black Hole
The Entropy of a Black Hole is proportional to the area of its event horizon, the surface within which even light cannot escape the gravity of the hole. Specifically, a hole with a horizon spanning A Planck areas has A/4 units of entropy. (The Planck area, approximately 10-66 square centimeter, is the fundamental quantum unit of area determined by the strength of gravity, the speed of light and the size of quanta.) Considered as information, it is as if the entropy were written on the event horizon, with each bit (each digital 1 or 0) corresponding to four Planck areas.
A Tale of Two Entropies
Formal information theory originated in seminal 1948 papers by American applied mathematician Claude E. Shannon, who introduced today's most widely used measure of information content: entropy. Entropy had long been a central concept of thermodynamics, the branch of physics dealing with heat. Thermodynamic entropy is popularly described as the disorder in a physical system. In 1877 Austrian physicist Ludwig Boltzmann characterized it more precisely in terms of the number of distinct microscopic states that the particles composing a chunk of matter could be in while still looking like the same macroscopic chunk of matter. For example, for the air in the room around you, one would count all the ways that the individual gas molecules could be distributed in the room and all the ways they could be moving.
When Shannon cast about for a way to quantify the information contained in, say, a message, he was led by logic to a formula with the same form as Boltzmann's. The Shannon entropy of a message is the number of binary digits, or bits, needed to encode it. Shannon's entropy does not enlighten us about the value of information, which is highly dependent on context. Yet as an objective measure of quantity of information, it has been enormously useful in science and technology. For instance, the design of every modern communications device--from cellular phones to modems to compact-disc players--relies on Shannon entropy.
Thermodynamic entropy and Shannon entropy are conceptually equivalent: the number of arrangements that are counted by Boltzmann entropy reflects the amount of Shannon information one would need to implement any particular arrangement. The two entropies have two salient differences, though. First, the thermodynamic entropy used by a chemist or a refrigeration engineer is expressed in units of energy divided by temperature, whereas the Shannon entropy used by a communications engineer is in bits, essentially dimensionless. That difference is merely a matter of convention.
Limits of Functional Density
The thermodynamics of black holes allows one to deduce limits on the density of entropy or information in various circumstances. The holographic bound defines how much information can be contained in a specified region of space. It can be derived by considering a roughly spherical distribution of matter that is contained within a surface of area A. The matter is induced to collapse to form a black hole (a). The black hole's area must be smaller than A, so its entropy must be less than A/4 [see illustration]. Because entropy cannot decrease, one infers that the original distribution of matter also must carry less than A/4 units of entropy or information. This result--that the maximum information content of a region of space is fixed by its area--defies the commonsense expectation that the capacity of a region should depend on its volume.
The universal entropy bound defines how much information can be carried by a mass m of diameter d. It is derived by imagining that a capsule of matter is engulfed by a black hole not much wider than it (b). The increase in the black hole's size places a limit on how much entropy the capsule could have contained. This limit is tighter than the holographic bound, except when the capsule is almost as dense as a black hole (in which case the two bounds are equivalent).
The holographic and universal information bounds are far beyond the data storage capacities of any current technology, and they greatly exceed the density of information on chromosomes and the thermodynamic entropy of water (c).
Even when reduced to common units, however, typical values of the two entropies differ vastly in magnitude. A silicon microchip carrying a gigabyte of data, for instance, has a Shannon entropy of about 1010 bits (one byte is eight bits), tremendously smaller than the chip's thermodynamic entropy, which is about 1023 bits at room temperature. This discrepancy occurs because the entropies are computed for different degrees of freedom. A degree of freedom is any quantity that can vary, such as a coordinate specifying a particle's location or one component of its velocity.
The Shannon entropy of the chip cares only about the overall state of each tiny transistor etched in the silicon crystal--the transistor is on or off; it is a 0 or a 1--a single binary degree of freedom. Thermodynamic entropy, in contrast, depends on the states of all the billions of atoms (and their roaming electrons) that make up each transistor. As miniaturization brings closer the day when each atom will store one bit of information for us, the useful Shannon entropy of the state-of-the-art microchip will edge closer in magnitude to its material's thermodynamic entropy. When the two entropies are calculated for the same degrees of freedom, they are equal.
What are the ultimate degrees of freedom? Atoms, after all, are made of electrons and nuclei, nuclei are agglomerations of protons and neutrons, and those in turn are composed of quarks. Many physicists today consider electrons and quarks to be excitations of superstrings, which they hypothesize to be the most fundamental entities. But the vicissitudes of a century of revelations in physics warn us not to be dogmatic. There could be more levels of structure in our universe than are dreamt of in today's physics.
One cannot calculate the ultimate information capacity of a chunk of matter or, equivalently, its true thermodynamic entropy, without knowing the nature of the ultimate constituents of matter or of the deepest level of structure, which I shall refer to as level X. (This ambiguity causes no problems in analyzing practical thermodynamics, such as that of car engines, for example, because the quarks within the atoms can be ignored--they do not change their states under the relatively benign conditions in the engine.) Given the dizzying progress in miniaturization, one can playfully contemplate a day when quarks will serve to store information, one bit apiece perhaps. How much information would then fit into our one-centimeter cube? And how much if we harness superstrings or even deeper, yet undreamt of levels? Surprisingly, developments in gravitation physics in the past three decades have supplied some clear answers to what seem to be elusive questions.
The information content of a pile of computer chips increases in proportion with the number of chips or, equivalently, the volume they occupy. That simple rule must break down for a large enough pile of chips because eventually the information would exceed the holographic bound, which depends on the surface area, not the volume. The "breakdown" occurs when the immense pile of chips collapses to form a black hole. Black Hole Thermodynamics
A central player in these developments is the black hole. Black holes are a consequence of general relativity, Albert Einstein's 1915 geometric theory of gravitation. In this theory, gravitation arises from the curvature of spacetime, which makes objects move as if they were pulled by a force. Conversely, the curvature is caused by the presence of matter and energy. According to Einstein's equations, a sufficiently dense concentration of matter or energy will curve spacetime so extremely that it rends, forming a black hole. The laws of relativity forbid anything that went into a black hole from coming out again, at least within the classical (nonquantum) description of the physics. The point of no return, called the event horizon of the black hole, is of crucial importance. In the simplest case, the horizon is a sphere, whose surface area is larger for more massive black holes.
It is impossible to determine what is inside a black hole. No detailed information can emerge across the horizon and escape into the outside world. In disappearing forever into a black hole, however, a piece of matter does leave some traces. Its energy (we count any mass as energy in accordance with Einstein's E = mc2) is permanently reflected in an increment in the black hole's mass. If the matter is captured while circling the hole, its associated angular momentum is added to the black hole's angular momentum. Both the mass and angular momentum of a black hole are measurable from their effects on spacetime around the hole. In this way, the laws of conservation of energy and angular momentum are upheld by black holes. Another fundamental law, the second law of thermodynamics, appears to be violated.
Holographic Space-Time
Two universes of different dimension and obeying disparate physical laws are rendered completely equivalent by the holographic principle. Theorists have demonstrated this principle mathematically for a specific type of five-dimensional spacetime ("anti苓e Sitter") and its four-dimensional boundary. In effect, the 5-D universe is recorded like a hologram on the 4-D surface at its periphery. Superstring theory rules in the 5-D spacetime, but a so-called conformal field theory of point particles operates on the 4-D hologram. A black hole in the 5-D spacetime is equivalent to hot radiation on the hologram--for example, the hole and the radiation have the same entropy even though the physical origin of the entropy is completely different for each case. Although these two descriptions of the universe seem utterly unalike, no experiment could distinguish between them, even in principle.
The second law of thermodynamics summarizes the familiar observation that most processes in nature are irreversible: a teacup falls from the table and shatters, but no one has ever seen shards jump up of their own accord and assemble into a teacup. The second law of thermodynamics forbids such inverse processes. It states that the entropy of an isolated physical system can never decrease; at best, entropy remains constant, and usually it increases. This law is central to physical chemistry and engineering; it is arguably the physical law with the greatest impact outside physics.
As first emphasized by Wheeler, when matter disappears into a black hole, its entropy is gone for good, and the second law seems to be transcended, made irrelevant. A clue to resolving this puzzle came in 1970, when Demetrious Christodoulou, then a graduate student of Wheeler's at Princeton, and Stephen W. Hawking of the University of Cambridge independently proved that in various processes, such as black hole mergers, the total area of the event horizons never decreases. The analogy with the tendency of entropy to increase led me to propose in 1972 that a black hole has entropy proportional to the area of its horizon. I conjectured that when matter falls into a black hole, the increase in black hole entropy always compensates or overcompensates for the "lost" entropy of the matter. More generally, the sum of black hole entropies and the ordinary entropy outside the black holes cannot decrease. This is the generalized second law--GSL for short.
Our innate perception that the world is three-dimensional could be an extraordinary illusion.
Hawking's radiation process allowed him to determine the proportionality constant between black hole entropy and horizon area: black hole entropy is precisely one quarter of the event horizon's area measured in Planck areas. (The Planck length, about 10-33 centimeter, is the fundamental length scale related to gravity and quantum mechanics. The Planck area is its square.) Even in thermodynamic terms, this is a vast quantity of entropy. The entropy of a black hole one centimeter in diameter would be about 1066 bits, roughly equal to the thermodynamic entropy of a cube of water 10 billion kilometers on a side.
The World as a Hologram
The GSL allows us to set bounds on the information capacity of any isolated physical system, limits that refer to the information at all levels of structure down to level X. In 1980 I began studying the first such bound, called the universal entropy bound, which limits how much entropy can be carried by a specified mass of a specified size [see box on opposite page]. A related idea, the holographic bound, was devised in 1995 by Leonard Susskind of Stanford University. It limits how much entropy can be contained in matter and energy occupying a specified volume of space.
In his work on the holographic bound, Susskind considered any approximately spherical isolated mass that is not itself a black hole and that fits inside a closed surface of area A. If the mass can collapse to a black hole, that hole will end up with a horizon area smaller than A. The black hole entropy is therefore smaller than A/4. According to the GSL, the entropy of the system cannot decrease, so the mass's original entropy cannot have been bigger than A/4. It follows that the entropy of an isolated physical system with boundary area A is necessarily less than A/4. What if the mass does not spontaneously collapse? In 2000 I showed that a tiny black hole can be used to convert the system to a black hole not much different from the one in Susskind's argument. The bound is therefore independent of the constitution of the system or of the nature of level X. It just depends on the GSL.
We can now answer some of those elusive questions about the ultimate limits of information storage. A device measuring a centimeter across could in principle hold up to 1066 bits--a mind-boggling amount. The visible universe contains at least 10100 bits of entropy, which could in principle be packed inside a sphere a tenth of a light-year across. Estimating the entropy of the universe is a difficult problem, however, and much larger numbers, requiring a sphere almost as big as the universe itself, are entirely plausible.
But it is another aspect of the holographic bound that is truly astonishing. Namely, that the maximum possible entropy depends on the boundary area instead of the volume. Imagine that we are piling up computer memory chips in a big heap. The number of transistors--the total data storage capacity--increases with the volume of the heap. So, too, does the total thermodynamic entropy of all the chips. Remarkably, though, the theoretical ultimate information capacity of the space occupied by the heap increases only with the surface area. Because volume increases more rapidly than surface area, at some point the entropy of all the chips would exceed the holographic bound. It would seem that either the GSL or our commonsense ideas of entropy and information capacity must fail. In fact, what fails is the pile itself: it would collapse under its own gravity and form a black hole before that impasse was reached. Thereafter each additional memory chip would increase the mass and surface area of the black hole in a way that would continue to preserve the GSL.
This surprising result--that information capacity depends on surface area--has a natural explanation if the holographic principle (proposed in 1993 by Nobelist Gerard 't Hooft of the University of Utrecht in the Netherlands and elaborated by Susskind) is true. In the everyday world, a hologram is a special kind of photograph that generates a full three-dimensional image when it is illuminated in the right manner. All the information describing the 3-D scene is encoded into the pattern of light and dark areas on the two-dimensional piece of film, ready to be regenerated. The holographic principle contends that an analogue of this visual magic applies to the full physical description of any system occupying a 3-D region: it proposes that another physical theory defined only on the 2-D boundary of the region completely describes the 3-D physics. If a 3-D system can be fully described by a physical theory operating solely on its 2-D boundary, one would expect the information content of the system not to exceed that of the description on the boundary.
A Universe Painted on Its Boundary
Can we apply the holographic principle to the universe at large? The real universe is a 4-D system: it has volume and extends in time. If the physics of our universe is holographic, there would be an alternative set of physical laws, operating on a 3-D boundary of spacetime somewhere, that would be equivalent to our known 4-D physics. We do not yet know of any such 3-D theory that works in that way. Indeed, what surface should we use as the boundary of the universe? One step toward realizing these ideas is to study models that are simpler than our real universe.
A class of concrete examples of the holographic principle at work involves so-called anti-de Sitter spacetimes. The original de Sitter spacetime is a model universe first obtained by Dutch astronomer Willem de Sitter in 1917 as a solution of Einstein's equations, including the repulsive force known as the cosmological constant. De Sitter's spacetime is empty, expands at an accelerating rate and is very highly symmetrical. In 1997 astronomers studying distant supernova explosions concluded that our universe now expands in an accelerated fashion and will probably become increasingly like a de Sitter spacetime in the future. Now, if the repulsion in Einstein's equations is changed to attraction, de Sitter's solution turns into the anti-de Sitter spacetime, which has equally as much symmetry. More important for the holographic concept, it possesses a boundary, which is located "at infinity" and is a lot like our everyday spacetime.
Using anti-de Sitter spacetime, theorists have devised a concrete example of the holographic principle at work: a universe described by superstring theory functioning in an anti-de Sitter spacetime is completely equivalent to a quantum field theory operating on the boundary of that spacetime [see box above]. Thus, the full majesty of superstring theory in an anti-de Sitter universe is painted on the boundary of the universe. Juan Maldacena, then at Harvard University, first conjectured such a relation in 1997 for the 5-D anti-de Sitter case, and it was later confirmed for many situations by Edward Witten of the Institute for Advanced Study in Princeton, N.J., and Steven S. Gubser, Igor R. Klebanov and Alexander M. Polyakov of Princeton University. Examples of this holographic correspondence are now known for spacetimes with a variety of dimensions.
This result means that two ostensibly very different theories--not even acting in spaces of the same dimension--are equivalent. Creatures living in one of these universes would be incapable of determining if they inhabited a 5-D universe described by string theory or a 4-D one described by a quantum field theory of point particles. (Of course, the structures of their brains might give them an overwhelming "commonsense" prejudice in favor of one description or another, in just the way that our brains construct an innate perception that our universe has three spatial dimensions; see the illustration on the opposite page.)
The holographic equivalence can allow a difficult calculation in the 4-D boundary spacetime, such as the behavior of quarks and gluons, to be traded for another, easier calculation in the highly symmetric, 5-D anti-de Sitter spacetime. The correspondence works the other way, too. Witten has shown that a black hole in anti-de Sitter spacetime corresponds to hot radiation in the alternative physics operating on the bounding spacetime. The entropy of the hole--a deeply mysterious concept--equals the radiation's entropy, which is quite mundane.
The Expanding Universe
Highly symmetric and empty, the 5-D anti-de Sitter universe is hardly like our universe existing in 4-D, filled with matter and radiation, and riddled with violent events. Even if we approximate our real universe with one that has matter and radiation spread uniformly throughout, we get not an anti-de Sitter universe but rather a "Friedmann-Robertson-Walker" universe. Most cosmologists today concur that our universe resembles an FRW universe, one that is infinite, has no boundary and will go on expanding ad infinitum.
Does such a universe conform to the holographic principle or the holographic bound? Susskind's argument based on collapse to a black hole is of no help here. Indeed, the holographic bound deduced from black holes must break down in a uniform expanding universe. The entropy of a region uniformly filled with matter and radiation is truly proportional to its volume. A sufficiently large region will therefore violate the holographic bound.
In 1999 Raphael Bousso, then at Stanford, proposed a modified holographic bound, which has since been found to work even in situations where the bounds we discussed earlier cannot be applied. Bousso's formulation starts with any suitable 2-D surface; it may be closed like a sphere or open like a sheet of paper. One then imagines a brief burst of light issuing simultaneously and perpendicularly from all over one side of the surface. The only demand is that the imaginary light rays are converging to start with. Light emitted from the inner surface of a spherical shell, for instance, satisfies that requirement. One then considers the entropy of the matter and radiation that these imaginary rays traverse, up to the points where they start crossing. Bousso conjectured that this entropy cannot exceed the entropy represented by the initial surface--one quarter of its area, measured in Planck areas. This is a different way of tallying up the entropy than that used in the original holographic bound. Bousso's bound refers not to the entropy of a region at one time but rather to the sum of entropies of locales at a variety of times: those that are "illuminated" by the light burst from the surface.
Bousso's bound subsumes other entropy bounds while avoiding their limitations. Both the universal entropy bound and the 't Hooft-Susskind form of the holographic bound can be deduced from Bousso's for any isolated system that is not evolving rapidly and whose gravitational field is not strong. When these conditions are overstepped--as for a collapsing sphere of matter already inside a black hole--these bounds eventually fail, whereas Bousso's bound continues to hold. Bousso has also shown that his strategy can be used to locate the 2-D surfaces on which holograms of the world can be set up.
Researchers have proposed many other entropy bounds. The proliferation of variations on the holographic motif makes it clear that the subject has not yet reached the status of physical law. But although the holographic way of thinking is not yet fully understood, it seems to be here to stay. And with it comes a realization that the fundamental belief, prevalent for 50 years, that field theory is the ultimate language of physics must give way. Fields, such as the electromagnetic field, vary continuously from point to point, and they thereby describe an infinity of degrees of freedom. Superstring theory also embraces an infinite number of degrees of freedom. Holography restricts the number of degrees of freedom that can be present inside a bounding surface to a finite number; field theory with its infinity cannot be the final story. Furthermore, even if the infinity is tamed, the mysterious dependence of information on surface area must be somehow accommodated.
Holography may be a guide to a better theory. What is the fundamental theory like? The chain of reasoning involving holography suggests to some, notably Lee Smolin of the Perimeter Institute for Theoretical Physics in Waterloo, that such a final theory must be concerned not with fields, not even with spacetime, but rather with information exchange among physical processes. If so, the vision of information as the stuff the world is made of will have found a worthy embodiment.
Jacob D. Bekenstein has contributed to the foundation of black hole thermodynamics and to other aspects of the connections between information and gravitation. He is Polak Professor of Theoretical Physics at the Hebrew University of Jerusalem, a member of the Israel Academy of Sciences and Humanities, and a recipient of the Rothschild Prize. Bekenstein dedicates this article to John Archibald Wheeler (his Ph.D. supervisor 30 years ago). Wheeler belongs to the third generation of Ludwig Boltzmann's students: Wheeler's Ph.D. adviser, Karl Herzfeld, was a student of Boltzmann's student Friedrich Hasen鐬rl.
In the 1950s, while conducting research into the beliefs of LSD as a psychotherapeutic tool, Grof had one female patient who suddenly became convinced she had assumed the identity of a female of a species of prehistoric reptile. During the course of her hallucination, she not only gave a richly detailed description of what it felt like to be encapsuled in such a form, but noted that the portion of the male of the species's anatomy was a patch of colored scales on the side of its head.
What was startling to Grof was that although the woman had no prior knowledge about such things, a conversation with a zoologist later confirmed that in certain species of reptiles colored areas on the head do indeed play an important role as triggers of sexual arousal.
The woman's experience was not unique. During the course of his research, Grof encountered examples of patients regressing and identifying with virtually every species on the evolutionary tree (research findings which helped influence the man-into-ape scene in the movie Altered States). Moreover, he found that such experiences frequently contained obscure zoological details which turned out to be accurate.
Regressions into the animal kingdom were not the only puzzling psychological phenomena Grof encountered. He also had patients who appeared to tap into some sort of collective or racial unconscious. Individuals with little or no education suddenly gave detailed descriptions of Zoroastrian funerary practices and scenes from Hindu mythology. In other categories of experience, individuals gave persuasive accounts of out-of-body journeys, of precognitive glimpses of the future, of regressions into apparent past-life incarnations.
In later research, Grof found the same range of phenomena manifested in therapy sessions which did not involve the use of drugs. Because the common element in such experiences appeared to be the transcending of an individual's consciousness beyond the usual boundaries of ego and/or limitations of space and time, Grof called such manifestations "transpersonal experiences", and in the late '60s he helped found a branch of psychology called "transpersonal psychology" devoted entirely to their study.
Although Grof's newly founded Association of Transpersonal Psychology garnered a rapidly growing group of like-minded professionals and has become a respected branch of psychology, for years neither Grof or any of his colleagues were able to offer a mechanism for explaining the bizarre psychological phenomena they were witnessing. But that has changed with the advent of the holographic paradigm.
As Grof recently noted, if the mind is actually part of a continuum, a labyrinth that is connected not only to every other mind that exists or has existed, but to every atom, organism, and region in the vastness of space and time itself, the fact that it is able to occasionally make forays into the labyrinth and have transpersonal experiences no longer seems so strange.
The holographic prardigm also has implications for so-called hard sciences like biology. Keith Floyd, a psychologist at Virginia Intermont College, has pointed out that if the concreteness of reality is but a holographic illusion, it would no longer be true to say the brain produces consciousness. Rather, it is consciousness that creates the appearance of the brain -- as well as the body and everything else around us we interpret as physical.
Such a turnabout in the way we view biological structures has caused researchers to point out that medicine and our understanding of the healing process could also be transformed by the holographic paradigm. If the apparent physical structure of the body is but a holographic projection of consciousness, it becomes clear that each of us is much more responsible for our health than current medical wisdom allows. What we now view as miraculous remissions of disease may actually be due to changes in consciousness which in turn effect changes in the hologram of the body.
Similarly, controversial new healing techniques such as visualization may work so well because in the holographic domain of thought images are ultimately as real as "reality".
Even visions and experiences involving "non-ordinary" reality become explainable under the holographic paradigm. In his book "Gifts of Unknown Things," biologist Lyall Watson discribes his encounter with an Indonesian shaman woman who, by performing a ritual dance, was able to make an entire grove of trees instantly vanish into thin air. Watson relates that as he and another astonished onlooker continued to watch the woman, she caused the trees to reappear, then "click" off again and on again several times in succession.
Although current scientific understanding is incapable of explaining such events, experiences like this become more tenable if "hard" reality is only a holographic projection.
Perhaps we agree on what is "there" or "not there" because what we call consensus reality is formulated and ratified at the level of the human unconscious at which all minds are infinitely interconnected.
If this is true, it is the most profound implication of the holographic paradigm of all, for it means that experiences such as Watson's are not commonplace only because we have not programmed our minds with the beliefs that would make them so. In a holographic universe there are no limits to the extent to which we can alter the fabric of reality.
What we perceive as reality is only a canvas waiting for us to draw upon it any picture we want. Anything is possible, from bending spoons with the power of the mind to the phantasmagoric events experienced by Castaneda during his encounters with the Yaqui brujo don Juan, for magic is our birthright, no more or less miraculous than our ability to compute the reality we want when we are in our dreams.
Indeed, even our most fundamental notions about reality become suspect, for in a holographic universe, as Pribram has pointed out, even random events would have to be seen as based on holographic principles and therefore determined. Synchronicities or meaningful coincidences suddenly makes sense, and everything in reality would have to be seen as a metaphor, for even the most haphazard events would express some underlying symmetry.
Whether Bohm and Pribram's holographic paradigm becomes accepted in science or dies an ignoble death remains to be seen, but it is safe to say that it has already had an influence on the thinking of many scientists. And even if it is found that the holographic model does not provide the best explanation for the instantaneous communications that seem to be passing back and forth between subatomic particles, at the very least, as noted by Basil Hiley, a physicist at Birbeck College in London, Aspect's findings "indicate that we must be prepared to consider radically new views of reality".
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