我们在前几章中看到了，长期以来人们关于时间性质 的观点是如何变化的。直到本世纪初，人们还相信绝对时间。也就是说，每一事件可由一个称为“时间“的数以唯一的方式来标记，所有好的钟在测量两个事件之间 的时间间隔上都是一致的。然而，对于任何正在运动的观察者光速总是一样的这一发现，导致了相对论；而在相对论中，人们必须抛弃存在一个唯一的绝对时间的观 念。代之以每个观察者携带的钟所记录的他自己的时间测量——不同观察者携带的钟不必要读数一样。这样，对于进行测量的观察者而言，时间变成一个更主观的概 念。
当 人们试图统一引力和量子力学时，必须引入“虚“时间的概念。虚时间是不能和空间方向区分的。如果一个人能往北走，他就能转过头并朝南走；同样的，如果一个 人能在虚时间里向前走，他应该能够转过来并往后走。这表明在虚时间里，往前和往后之间不可能有重要的差别。另一方面，当人们考察“实“时间时，正如众所周 知的，在前进和后退方向存在有非常巨大的差别。这过去和将来之间的差别从何而来？为何我们记住过去而不是将来？
科 学定律并不区别过去和将来。更精确地讲，正如前面所解释的，科学定律在称作C、 P和T的联合作用（或对称）下不变。（C是指将反粒子来替代粒子；P的意思是取镜象， 这样左和右就互相交换了；T是指颠倒所有粒子的运动方向，也就是使运动倒退回去。）在所有正常情形下，制约物体行为的科学定律在CP联合对称下不变。换言 之，对于其他行星上的居民，若他们是我们的镜像并且由反物质而不是物质构成，则生活会刚好是同样的。
如 果科学定律在CP联合对称以及CPT联合对称下都不变，它们也必须在单独的T对称下不变。然而，在日常生活的实时间中，前进和后退的方向之间还是有一个大 的差异。想像一杯水从桌子上滑落到地板上被打碎。如果你将其录像，你可以容易地辨别出它是向前进还是向后退。如果将其倒回来，你会看到碎片忽然集中到一起 离开地板，并跳回到桌子上形成一个完整的杯子。你可断定录像是在倒放，因为这种行为在日常生活中从未见过。如果这样的事发生，陶瓷业将无生意可做。
为 何我们从未看到碎杯子集合起来，离开地面并跳回到桌子上，通常的解释是这违背了热力学第二定律所表述的在任何闭合系统中无序度或熵总是随时间而增加。换言 之，它是穆菲定律的一种形式：事情总是趋向于越变越糟：桌面上一个完整的杯子是一个高度有序的状态，而地板上破碎的杯子是一个无序的状态。人们很容易从早 先桌子上的杯子变成后来地面上的碎杯子，而不是相反。
无 序度或熵随着时间增加是一个所谓的时间箭头的例子。时间箭头将过去和将来区别开来，使时间有了方向。至少有三种不同的时间箭头：第一个，是热力学时间箭 头，即是在这个时间方向上无序度或熵增加；然后是心理学时间箭头，这就是我们感觉时间流逝的方向，在这个方向上我们可以记忆过去而不是未来；最后，是宇宙 学时间箭头，在这个方向上宇宙在膨胀，而不是收缩。
我 将在这一章论断，宇宙的无边界条件和弱人择原理一起能解释为何所有的三个箭头指向同一方向。此外，为何必须存在一个定义得很好的时间箭头。我将论证心理学 箭头是由热力学箭头所决定，并且这两种箭头必须总是指向相同的方向。如果人们假定宇宙的无边界条件，我们将看到必然会有定义得很好的热力学和宇宙学时间箭 头。但对于宇宙的整个历史来说，它们并不总是指向同一方向。然而，我将指出，只有当它们指向一致时，对于能够发问为何无序度在宇宙膨胀的时间方向上增加的 智力生命的发展，才有合适的条件。
假 设一个系统从这少数的有序状态之中的一个出发。随着时间流逝，这个系统将按照科学定律演化，而且它的状态将改变。到后来，因为存在着更多的无序状态，它处 于无序状态的可能性比处于有序状态的可能性更大。这样，如果一个系统服从一个高度有序的初始条件，无序度会随着时间的增加而增大。
假 定拼板玩具盒的纸片从能排成一幅图画的有序组合开始，如果你摇动这盒子，这些纸片将会采用其他组合，这可能是一个不能形成一幅合适图画的无序的组合，就是 因为存在如此之多得多的无序的组合。有一些纸片团仍可能形成部份图画，但是你越摇动盒子，这些团就越可能被分开，这些纸片将处于完全混乱的状态，在这种状 态下它们不能形成任何种类的图画。这样，如果纸片从一个高度有序的状态的初始条件出发，纸片的无序度将可能随时间而增加。
然 而，假定上帝决定不管宇宙从何状态开始，它都必须结束于一个高度有序的状态，则在早期这宇宙有可能处于无序的状态。这意味着无序度将随时间而减小。你将会 看到破碎的杯子集合起来并跳回到桌子上。然而，任何观察杯子的人都生活在无序度随时间减小的宇宙中，我将论断这样的人会有一个倒溯的心理学时间箭头。这就 是说，他们会记住将来的事件，而不是过去的事件。当杯子被打碎时，他们会记住它在桌子上的情形；但是当它是在桌子上时，他们不会记住它在地面上的情景。
由 于我们不知道大脑工作的细节，所以讨论人类的记忆是相当困难的。然而，我们确实知道计算机的记忆器是如何工作的。所以，我将讨论计算机的心理学时间箭头。 我认为，假定计算机和人类有相同的箭头是合理的。如果不是这样，人们可能因为拥有一台记住明年价格的计算机而使股票交易所垮台。
大 体来说，计算机的记忆器是一个包含可存在于两种状态中的任一种状态的元件的设备，算盘是一个简单的例子。其最简单的形式是由许多铁条组成；每一根铁条上有 一念珠，此念珠可呆在两个位置之中的一个。在计算机记忆器进行存储之前，其记忆器处于无序态，念珠等几率地处于两个可能的状态中。（算盘珠杂乱无章地散布 在算盘的铁条上）。在记忆器和所要记忆的系统相互作用后，根据系统的状态，它肯定处于这种或那种状态（每个算盘珠将位于铁条的左边或右边。）这样，记忆器 就从无序态转变成有序态。然而，为了保证记忆器处于正确的状态，需要使用一定的能量（例如，移动算盘珠或给计算机接通电源）。这能量以热的形式耗散了，从 而增加了宇宙的无序度的量。人们可以证明，这个无序度增量总比记忆器本身有序度的增量大。这样，由计算机冷却风扇排出的热量表明计算机将一个项目记录在它 的记忆器中时，宇宙的无序度的总量仍然增加。计算机记忆过去的时间方向和无序度增加的方向是一致的。
所 以，我们对时间方向的主观感觉或心理学时间箭头，是在我们头脑中由热力学时间箭头所决定的。正像一个计算机，我们必须在熵增加的顺序上将事物记住。这几乎 使热力学定律变成为无聊的东西。无序度随时间的增加乃是因为我们是在无序度增加的方向上测量时间。拿这一点来打赌，准保你会赢。
在 经典广义相对论中，因为所有已知的科学定律在大爆炸奇点处失效，人们不能预言宇宙是如何开始的。宇宙可以从一个非常光滑和有序的状态开始。这就会导致正如 我们所观察到的、定义很好的热力学和宇宙学的时间箭头。但是，它可以同样合理地从一个非常波浪起伏的无序状态开始。在那种情况下，宇宙已经处于一种完全无 序的状态，所以无序度不会随时间而增加。或者它保持常数，这时就没有定义很好的热力学时间箭头；或者它会减小，这时热力学时间箭头就会和宇宙学时间箭头相 反向。任何这些可能性都不符合我们所观察到的情况。然而，正如我们看到的，经典广义相对论预言了它自身的崩溃。当空间——时间曲率变大，量子引力效应变得 重要，并且经典理论不再能很好地描述宇宙时，人们必须用量子引力论去理解宇宙是如何开始的。
正 如我们在上一章看到的，在量子引力论中，为了指定宇宙的态，人们仍然必须说清在过去的空间—时间的边界的宇宙的可能历史是如何行为的。只有如果这些历史满 足无边界条件，人们才可能避免这个不得不描述我们不知道和无法知道的东西的困难：它们在尺度上有限，但是没有边界、边缘或奇点。在这种情形下，时间的开端 就会是规则的、光滑的空间—时间的点，并且宇宙在一个非常光滑和有序的状态下开始它的膨胀。它不可能是完全均匀的，否则就违反了量子理论不确定性原理。必 然存在密度和粒子速度的小起伏，然而无边界条件意味着，这些起伏又是在与不确定性原理相一致的条件下尽可能的小。
宇 宙刚开始时有一个指数或“暴涨“的时期，在这期间它的尺度增加了一个非常大的倍数。在膨胀时，密度起伏一开始一直很小，但是后来开始变大。在密度比平均值 稍大的区域，额外质量的引力吸引使膨胀速度放慢。最终，这样的区域停止膨胀，并坍缩形成星系、恒星以及我们这样的人类。宇宙开始时处于一个光滑有序的状 态，随时间演化成波浪起伏的无序的状态。这就解释了热力学时间箭头的存在。
如 果宇宙停止膨胀并开始收缩将会发生什么呢？热力学箭头会不会倒转过来，而无序度开始随时间减少呢？这为从膨胀相存活到收缩相的人们留下了五花八门的科学幻 想的可能性。他们是否会看到杯子的碎片集合起来离开地板跳回到桌子上去？他们会不会记住明天的价格，并在股票市场上发财致富？由于宇宙至少要再等一百亿年 之后才开始收缩，忧虑那时会发生什么似乎有点学究气。但是有一种更快的办法去查明将来会发生什么，即跳到黑洞里面去。恒星坍缩形成黑洞的过程和整个宇宙的 坍缩的后期相当类似；这样，如果在宇宙的收缩相无序度减小，可以预料它在黑洞里面也会减小。所以，一个落到黑洞里去的航天员能在投赌金之前，也许能依靠记 住轮赌盘上球儿的走向而赢钱。（然而，不幸的是，玩不了多久，他就会变成意大利面条。他也不能使我们知道热力学箭头的颠倒，或者甚至将他的赢钱存入银行， 因为他被困在黑洞的事件视界后面。）
这 个观念是吸引人的，因为它表明在膨胀相和收缩相之间存在一个漂亮的对称。然而，人们不能置其他有关宇宙的观念于不顾，而只采用这个观念。问题在于：它是否 由无边界条件所隐含或它是否与这个条件不相协调？正如我说过的，我起先以为无边界条件确实意味着无序度会在收缩相中减小。我之所以被误导，部分是由于与地 球表面的类比引起的。如果人们将宇宙的开初对应于北极，那么宇宙的终结就应该类似于它的开端，正如南极之与北极相似。然而，北南二极对应于虚时间中的宇宙 的开端和终结。在实时间里的开端和终结之间可有非常大的差异。我还被我作过的一项简单的宇宙模型的研究所误导，在此模型中坍缩相似乎是膨胀相的时间反演。 然而，我的一位同事，宾夕凡尼亚州立大学的当·佩奇指出，无边界条件没有要求收缩相必须是膨胀相的时间反演。我的一个学生雷蒙·拉夫勒蒙进一步发现，在一 个稍复杂的模型中，宇宙的坍缩和膨胀非常不同。我意识到自己犯了一个错误：无边界条件意味着事实上在收缩相时无序度继续增加。当宇宙开始收缩时或在黑洞中 热力学和心理学时间箭头不会反向。
当 你发现自己犯了这样的错误后该如何办？有些人从不承认他们是错误的，而继续去找新的往往互相不协调的论据为自己辩解——正如爱丁顿在反对黑洞理论时之所 为。另外一些人首先宣称，从来没有真正支持过不正确的观点，如果他们支持了，也只是为了显示它是不协调的。在我看来，如果你在出版物中承认自己错了，那会 好得多并少造成混乱。爱因斯坦即是一个好的榜样，他在企图建立一个静态的宇宙模型时引入了宇宙常数，他称此为一生中最大的错误。
回 头再说时间箭头，余下的问题是；为何我们观察到热力学和宇宙学箭头指向同一方向？或换言之，为何无序度增加的时间方向正是宇宙膨胀的时间方向？如果人们相 信，按照无边界假设似乎所隐含的那样，宇宙先膨胀然后重新收缩，那么为何我们应在膨胀相中而不是在收缩相中，这就成为一个问题。
人 们可以在弱人择原理的基础上回答这个问题。收缩相的条件不适合于智慧人类的存在，而正是他们能够提出为何无序度增加的时间方向和宇宙膨胀的时间方向相同的 问题。无边界假设预言的宇宙在早期阶段的暴涨意味着，宇宙必须以非常接近为避免坍缩所需要的临界速率膨胀，这样它在很长的时间内才不至坍缩。到那时候所有 的恒星都会烧尽，而在其中的质子和中子可能会衰变成轻粒子和辐射。宇宙将处于几乎完全无序的状态，这时就不会有强的热力学时间箭头。由于宇宙已经处于几乎 完全无序的状态，无序度不会增加很多。然而，对于智慧生命的行为来说，一个强的热力学箭头是必需的。为了生存下去，人类必须消耗能量的一种有序形式——食 物，并将其转化成能量的一种无序形式——热量，所以智慧生命不能在宇宙的收缩相中存在。这就解释了，为何我们观察到热力学和宇宙学的时间箭头指向一致。并 不是宇宙的膨胀导致无序度的增加，而是无边界条件引起无序度的增加，并且只有在膨胀相中才有适合智慧生命的条件。
总 之，科学定律并不能区分前进和后退的时间方向。然而，至少存在有三个时间箭头将过去和将来区分开来。它们是热力学箭头，这就是无序度增加的时间方向；心理 学箭头，即是在这个时间方向上，我们能记住过去而不是将来；还有宇宙学箭头，也即宇宙膨胀而不是收缩的方向。我指出了心理学箭头本质上应和热力学箭头相 同。宇宙的无边界假设预言了定义得很好的热力学时间箭头，因为宇宙必须从光滑、有序的状态开始。并且我们看到，热力学箭头和宇宙学箭头的一致，乃是由于智 慧生命只能在膨胀相中存在。收缩相是不适合于它的存在的，因为那儿没有强的热力学时间箭头。
人 类理解宇宙的进步，是在一个无序度增加的宇宙中建立了一个很小的有序的角落。 如果你记住了这本书中的每一个词，你的记忆就记录了大约200万单位的信息——你头脑中的有序度就增加了大约200万单位。 然而，当你读这本书时，你至少将以食物为形式的1千卡路里的有序能量， 转换成为以对流和汗释放到你周围空气中的热量的形式的无序能量。这就将宇宙的无序度增大了大约20亿亿亿单位，或大约是你头脑中有序度增量——那是如果你 记住这本书的每一件事的话——的1干亿亿倍。我试图在下一章更增加一些我们头脑的有序度，解释人们如何将我描述过的部分理论结合一起，形成一个完整的统一 理论，这个理论将适用于宇宙中的任何东西。
In previous chapters we have seen how our views of the nature of time have changed over the years. Up to the beginning of this century people believed in an absolute time. That is, each event could be labeled by a number called “time“ in a unique way, and all good clocks would agree on the time interval between two events.
However, the discovery that the speed of light appeared the same to every observer, no matter how he was moving, led to the theory of relativity – and in that one had to abandon the idea that there was a unique absolute time. Instead, each observer would have his own measure of time as recorded by a clock that he carried: clocks carried by different observers would not necessarily agree. Thus time became a more personal concept, relative to the observer who measured it.
When one tried to unify gravity with quantum mechanics, one had to introduce the idea of “imaginary“ time.
Imaginary time is indistinguishable from directions in space. If one can go north, one can turn around and head south; equally, if one can go forward in imaginary time, one ought to be able to turn round and go backward.
This means that there can be no important difference between the forward and backward directions of imaginary time. On the other hand, when one looks at “real“ time, there’s a very big difference between the forward and backward directions, as we all know. Where does this difference between the past and the future come from? Why do we remember the past but not the future? The laws of science do not distinguish between the past and the future. More precisely, as explained earlier, the laws of science are unchanged under the combination of operations (or symmetries) known as C, P, and T.
(C means changing particles for antiparticles. P means taking the mirror image, so left and right are interchanged. And T means reversing the direction of motion of all particles: in effect, running the motion backward.) The laws of science that govern the behavior of matter under all normal situations are unchanged under the combination of the two operations C and P on their own. In other words, life would be just the same for the inhabitants of another planet who were both mirror images of us and who were made of antimatter, rather than matter.
If the laws of science are unchanged by the combination of operations C and P, and also by the combination C, P, and T, they must also be unchanged under the operation T alone. Yet there is a big difference between the forward and backward directions of real time in ordinary life. Imagine a cup of water falling off a table and breaking into pieces on the floor. If you take a film of this, you can easily tell whether it is being run forward or backward. If you run it backward you will see the pieces suddenly gather themselves together off the floor and jump back to form a whole cup on the table. You can tell that the film is being run backward because this kind of behavior is never observed in ordinary life. If it were, crockery manufacturers would go out of business.
The explanation that is usually given as to why we don’t see broken cups gathering themselves together off the floor and jumping back onto the table is that it is forbidden by the second law of thermodynamics. This says that in any closed system disorder, or entropy, always increases with time. In other words, it is a form of Murphy’s law: things always tend to go wrong! An intact cup on the table is a state of high order, but a broken cup on the floor is a disordered state. One can go readily from the cup on the table in the past to the broken cup on the floor in the future, but not the other way round.
The increase of disorder or entropy with time is one example of what is called an arrow of time, something that distinguishes the past from the future, giving a direction to time. There are at least three different arrows of time. First, there is the thermodynamic arrow of time, the direction of time in which disorder or entropy increases. Then, there is the psychological arrow of time. This is the direction in which we feel time passes, the direction in which we remember the past but not the future. Finally, there is the cosmological arrow of time. This is the direction of time in which the universe is expanding rather than contracting.
In this chapter I shall argue that the no boundary condition for the universe, together with the weak anthropic principle, can explain why all three arrows point in the same direction – and moreover, why a well-defined arrow of time should exist at all. I shall argue that the psychological arrow is determined by the thermodynamic arrow,
and that these two arrows necessarily always point in the same direction. If one assumes the no boundary condition for the universe, we shall see that there must be well-defined thermodynamic and cosmological arrows of time, but they will not point in the same direction for the whole history of the universe. However, I shall argue that it is only when they do point in the same direction that conditions are suitable for the development of intelligent beings who can ask the question: why does disorder increase in the same direction of time as that in which the universe expands? I shall discuss first the thermodynamic arrow of time. The second law of thermodynamics results from the fact that there are always many more disordered states than there are ordered ones. For example, consider the pieces of a jigsaw in a box. There is one, and. only one, arrangement in which the pieces make a complete picture. On the other hand, there are a very large number of arrangements in which the pieces are disordered and don’t make a picture.
Suppose a system starts out in one of the small number of ordered states. As time goes by, the system will evolve according to the laws of science and its state will change. At a later time, it is more probable that the system will be in a disordered state than in an ordered one because there are more disordered states. Thus disorder will tend to increase with time if the system obeys an initial condition of high order.
Suppose the pieces of the jigsaw start off in a box in the ordered arrangement in which they form a picture. If you shake the box, the pieces will take up another arrangement. This will probably be a disordered arrangement in which the pieces don’t form a proper picture, simply because there are so many more disordered arrangements. Some groups of pieces may still form parts of the picture, but the more you shake the box, the more likely it is that these groups will get broken up and the pieces will be in a completely jumbled state in which they don’t form any sort of picture. So the disorder of the pieces will probably increase with time if the pieces obey the initial condition that they start off in a condition of high order.
Suppose, however, that God decided that the universe should finish up in a state of high order but that it didn’t matter what state it started in. At early times the universe would probably be in a disordered state. This would mean that disorder would decrease with time. You would see broken cups gathering themselves together and jumping back onto the table. However, any human beings who were observing the cups would be living in a universe in which disorder decreased with time. I shall argue that such beings would have a psychological arrow of time that was backward. That is, they would remember events in the future, and not remember events in their past. When the cup was broken, they would remember it being on the table, but when it was on the table, they would not remember it being on the floor.
It is rather difficult to talk about human memory because we don’t know how the brain works in detail. We do, however, know all about how computer memories work. I shall therefore discuss the psychological arrow of time for computers. I think it is reasonable to assume that the arrow for computers is the same as that for humans. If it were not, one could make a killing on the stock exchange by having a computer that would remember tomorrow’s prices! A computer memory is basically a device containing elements that can exist in either of two states. A simple example is an abacus. In its simplest form, this consists of a number of wires; on each wire there are a number of beads that can be put in one of two positions. Before an item is recorded in a computer’s memory, the memory is in a disordered state, with equal probabilities for the two possible states.
(The abacus beads are scattered randomly on the wires of the abacus.) After the memory interacts with the system to be remembered, it will definitely be in one state or the other, according to the state of the system.
(Each abacus bead will be at either the left or the right of the abacus wire.) So the memory has passed from a disordered state to an ordered one. However, in order to make sure that the memory is in the right state, it is necessary to use a certain amount of energy (to move the bead or to power the computer, for example). This energy is dissipated as heat, and increases the amount of disorder in the universe. One can show that this increase in disorder is always greater than the increase in the order of the memory itself. Thus the heat expelled by the computer’s cooling fan means that when a computer records an item in memory, the total amount of disorder in the universe still goes up. The direction of time in which a computer remembers the past is the same as that in which disorder increases.
Our subjective sense of the direction of time, the psychological arrow of time, is therefore determined within our brain by the thermodynamic arrow of time. Just like a computer, we must remember things in the order in which entropy increases. This makes the second law of thermodynamics almost trivial. Disorder increases with time
because we measure time in the direction in which disorder increases You can’t have a safer bet than that! But why should the thermodynamic arrow of time exist at all? Or, in other words, why should the universe be in a state of high order at one end of time, the end that we call the past? Why is it not in a state of complete disorder at all times? After all, this might seem more probable. And why is the direction of time in which disorder increases the same as that in which the universe expands? In the classical theory of general relativity one cannot predict how the universe would have begun because all the known laws of science would have broken down at the big bang singularity. The universe could have started out in a very smooth and ordered state. This would have led to well-defined thermodynamic and cosmological arrows of time, as we observe. But it could equally well have started out in a very lumpy and disordered state. In that case, the universe would already be in a state of complete disorder, so disorder could not increase with time. It would either stay constant, in which case there would be no well-defined thermodynamic arrow of time, or it would decrease, in which case the thermodynamic arrow of time would point in the opposite direction to the cosmological arrow. Neither of these possibilities agrees with what we observe.
However, as we have seen, classical general relativity predicts its own downfall. When the curvature of space-time becomes large, quantum gravitational effects will become important and the classical theory will cease to be a good description of the universe. One has to use a quantum theory of gravity to understand how the universe began.
In a quantum theory of gravity, as we saw in the last chapter, in order to specify the state of the universe one would still have to say how the possible histories of the universe would behave at the boundary of space-time in the past. One could avoid this difficulty of having to describe what we do not and cannot know only if the histories satisfy the no boundary condition: they are finite in extent but have no boundaries, edges, or singularities. In that case, the beginning of time would be a regular, smooth point of space-time and the universe would have begun its expansion in a very smooth and ordered state. It could not have been completely uniform, because that would violate the uncertainty principle of quantum theory. There had to be small fluctuations in the density and velocities of particles. The no boundary condition, however, implied that these fluctuations were as small as they could be, consistent with the uncertainty principle.
The universe would have started off with a period of exponential or “inflationary“ expansion in which it would have increased its size by a very large factor. During this expansion, the density fluctuations would have remained small at first, but later would have started to grow. Regions in which the density was slightly higher than average would have had their expansion slowed down by the gravitational attraction of the extra mass.
Eventually, such regions would stop expanding and collapse to form galaxies, stars, and beings like us. The universe would have started in a smooth and ordered state, and would become lumpy and disordered as time went on. This would explain the existence of the thermodynamic arrow of time.
But what would happen if and when the universe stopped expanding and began to contract? Would the thermodynamic arrow reverse and disorder begin to decrease with time? This would lead to all sorts of science-fiction-like possibilities for people who survived from the expanding to the contracting phase. Would they see broken cups gathering themselves together off the floor and jumping back onto the table? Would they be able to remember tomorrow’s prices and make a fortune on the stock market? It might seem a bit academic to worry about what will happen when the universe collapses again, as it will not start to contract for at least another ten thousand million years. But there is a quicker way to find out what will happen: jump into a black hole. The collapse of a star to form a black hole is rather like the later stages of the collapse of the whole universe. So if disorder were to decrease in the contracting phase of the universe, one might also expect it to decrease inside a black hole. So perhaps an astronaut who fell into a black hole would be able to make money at roulette by remembering where the ball went before he placed his bet. (Unfortunately, however, he would not have long to play before he was turned to spaghetti. Nor would he be able to let us know about the reversal of the thermodynamic arrow, or even bank his winnings, because he would be trapped behind the event horizon of the black hole.) At first, I believed that disorder would decrease when the universe recollapsed. This was because I thought that the universe had to return to a smooth and ordered state when it became small again. This would mean that the contracting phase would be like the time reverse of the expanding phase. People in the contracting phase would live their lives backward: they would die before they were born and get younger as the universe
This idea is attractive because it would mean a nice symmetry between the expanding and contracting phases.
However, one cannot adopt it on its own, independent of other ideas about the universe. The question is: is it implied by the no boundary condition, or is it inconsistent with that condition? As I said, I thought at first that the no boundary condition did indeed imply that disorder would decrease in the contracting phase. I was misled partly by the analogy with the surface of the earth. If one took the beginning of the universe to correspond to the North Pole, then the end of the universe should be similar to the beginning, just as the South Pole is similar to the North. However, the North and South Poles correspond to the beginning and end of the universe in imaginary time. The beginning and end in real time can be very different from each other. I was also misled by work I had done on a simple model of the universe in which the collapsing phase looked like the time reverse of the expanding phase. However, a colleague of mine, Don Page, of Penn State University, pointed out that the no boundary condition did not require the contracting phase necessarily to be the time reverse of the expanding phase. Further, one of my students, Raymond Laflamme, found that in a slightly more complicated model, the collapse of the universe was very different from the expansion. I realized that I had made a mistake: the no boundary condition implied that disorder would in fact continue to increase during the contraction. The thermodynamic and psychological arrows of time would not reverse when the universe begins to recontract, or inside black holes.
What should you do when you find you have made a mistake like that? Some people never admit that they are wrong and continue to find new, and often mutually inconsistent, arguments to support their case – as Eddington did in opposing black hole theory. Others claim to have never really supported the incorrect view in the first place or, if they did, it was only to show that it was inconsistent. It seems to me much better and less confusing if you admit in print that you were wrong. A good example of this was Einstein, who called the cosmological constant, which he introduced when he was trying to make a static model of the universe, the biggest mistake of his life.
To return to the arrow of time, there remains the question: why do we observe that the thermodynamic and cosmological arrows point in the same direction? Or in other words, why does disorder increase in the same direction of time as that in which the universe expands? If one believes that the universe will expand and then contract again, as the no boundary proposal seems to imply, this becomes a question of why we should be in the expanding phase rather than the contracting phase.
One can answer this on the basis of the weak anthropic principle. Conditions in the contracting phase would not be suitable for the existence of intelligent beings who could ask the question: why is disorder increasing in the same direction of time as that in which the universe is expanding? The inflation in the early stages of the universe, which the no boundary proposal predicts, means that the universe must be expanding at very close to the critical rate at which it would just avoid recollapse, and so will not recollapse for a very long time. By then all the stars will have burned out and the protons and neutrons in them will probably have decayed into light particles and radiation. The universe would be in a state of almost complete disorder. There would be no strong thermodynamic arrow of time. Disorder couldn’t increase much because the universe would be in a state of almost complete disorder already. However, a strong thermodynamic arrow is necessary for intelligent life to operate. In order to survive, human beings have to consume food, which is an ordered form of energy, and convert it into heat, which is a disordered form of energy. Thus intelligent life could not exist in the contracting phase of the universe. This is the explanation of why we observe that the thermodynamic and cosmological arrows of time point in the same direction. It is not that the expansion of the universe causes disorder to increase. Rather, it is that the no boundary condition causes disorder to increase and the conditions to be suitable for intelligent life only in the expanding phase.
To summarize, the laws of science do not distinguish between the forward and backward directions of time.
However, there are at least three arrows of time that do distinguish the past from the future. They are the thermodynamic arrow, the direction of time in which disorder increases; the psychological arrow, the direction of time in which we remember the past and not the future; and the cosmological arrow, the direction of time in which the universe expands rather than contracts. I have shown that the psychological arrow is essentially the same as the thermodynamic arrow, so that the two would always point in the same direction. The no boundary proposal for the universe predicts the existence of a well-defined thermodynamic arrow of time because the universe must start off in a smooth and ordered state. And the reason we observe this thermodynamic arrow to
agree with the cosmological arrow is that intelligent beings can exist only in the expanding phase. The contracting phase will be unsuitable because it has no strong thermodynamic arrow of time.
The progress of the human race in understanding the universe has established a small corner of order in an increasingly disordered universe. If you remember every word in this book, your memory will have recorded about two million pieces of information: the order in your brain will have increased by about two million units.
However, while you have been reading the book, you will have converted at least a thousand calories of ordered energy, in the form of food, into disordered energy, in the form of heat that you lose to the air around you by convection and sweat. This will increase the disorder of the universe by about twenty million million million million units – or about ten million million million times the increase in order in your brain – and that’s if you remember everything in this book. In the next chapter but one I will try to increase the order in our neck of the woods a little further by explaining how people are trying to fit together the partial theories I have described to form a complete unified theory that would cover everything in the universe.