Chapter16. RelativisticEnergy and Momentum第16章 相对论的能量和动量
16–1Relativity and the philosophers 16-1相对论与哲学家
In this chapter we shall continue todiscuss the principle of relativity of Einstein and Poincaré, as it affects ourideas of physics and other branches of human thought.
爱因斯坦的庞加莱的相对论原理，对我们的物理想法和人类思想的其他分支，产生了深刻的影响，本章我们将继续讨论它们。

Poincaré made the following statement ofthe principle of relativity: “According to the principle of relativity, thelaws of physical phenomena must be the same for a fixed observer as for anobserver who has a uniform motion of translation relative to him, so that wehave not, nor can we possibly have, any means of discerning whether or not weare carried along in such a motion.”
关于相对论的原理，庞加莱做了如下陈述：“依据相对论的原理，物理现象的规律，对于下面两个观察者，应该是同样的，一个，是固定的观察者，另一个，相对于他{指前一个？}做匀速平移运动，因此，对于我们是否处在这样一种运动中，我们没有、也不可能有，任何识别的方法，”

When this idea descended upon the world, itcaused a great stir among philosophers, particularly the “cocktail-partyphilosophers,” who say, “Oh, it is very simple: Einstein’s theory says all isrelative!” In fact, a surprisingly large number of philosophers, not only thosefound at cocktail parties (but rather than embarrass them, we shall just callthem “cocktail-party philosophers”), will say, “That all is relative is a consequenceof Einstein, and it has profound influences on our ideas.” In addition, theysay “It has been demonstrated in physics that phenomena depend upon your frameof reference.” We hear that a great deal, but it is difficult to find out whatit means. Probably the frames of reference that were originally referred towere the coordinate systems which we use in the analysis of the theory ofrelativity. So the fact that “things depend upon your frame of reference” issupposed to have had a profound effect on modern thought. One might well wonderwhy, because, after all, that things depend upon one’s point of view is so simplean idea that it certainly cannot have been necessary to go to all the troubleof the physical relativity theory in order to discover it. That what one seesdepends upon his frame of reference is certainly known to anybody who walksaround, because he sees an approaching pedestrian first from the front and thenfrom the back; there is nothing deeper in most of the philosophy which is saidto have come from the theory of relativity than the remark that “A person looksdifferent from the front than from the back.” The old story about the elephantthat several blind men describe in different ways is another example, perhaps,of the theory of relativity from the philosopher’s point of view.
这个想法，面世之后，在哲学家中，引起巨大轰动，特别是在“鸡尾酒会哲学家”中，他们说：“这事很简单，爱因斯坦的理论说了，所有都是相对的！”事实上，不仅是那些在鸡尾酒会上出现的哲学家（但是，与其说是要让他们尴尬，我们将只称他们为“鸡尾酒会哲学家”），还有数量惊人的哲学家，他们会说“所有都是相对的，是爱因斯坦理论的一个后果，它对我们的想法，有着深刻的影响”。另外，他们说：“在物理学中，已经演证了，现象，依赖于我们的参照系”。关于这个，我们听过很多，但要弄清楚，其意究竟为何，依然困难。或许，最初引用的参照系，就是我们在相对论理论的分析中，所用的参照系。于是，“事物依赖于你的参照系”这一事实，就被认为，对于现代思想，有着深刻影响。人们想知道为什么，这很正常，因为，事物依赖于我们的观点，是如此简单的一个想法，以至于，为了发现它，完全没有必要，去受如下麻烦：即用物理学的相对理论来解释。一个人所看到的，依赖于其参照系这件事，对于任何行人，当然都是知道的，因为，对于一个迎面走来的人，他是先看正面，然后看背面；对于“一个人，从前面看，与从后面看，是不一样的”这种说法，在大多数哲学中，并没有什么所谓来自相对论的东西，能比它更深刻了。古老故事，盲人摸象，就是另一例子，或许，就是从哲学家的观点出发，所能得到的相对论的例子。

But certainly there must be deeper thingsin the theory of relativity than just this simple remark that “A person looksdifferent from the front than from the back.” Of course relativity is deeperthan this, because we can make definite predictions with it. Itcertainly would be rather remarkable if we could predict the behavior of naturefrom such a simple observation alone.
然而，在相对论中，肯定有更深的事情，比“一个人，从前面看，与从后面看相比，是不一样的”这种说法要深。当然，相对论比这个要深，因为，我们可以用它，做出确定的预测。如果我们只从这样一个简单的观察出发，就能预测自然的表现，那么，它肯定非常值得说明。

There is another school of philosophers whofeel very uncomfortable about the theory of relativity, which asserts that wecannot determine our absolute velocity without looking at something outside,and who would say, “It is obvious that one cannot measure his velocity withoutlooking outside. It is self-evident that it is meaningless to talk aboutthe velocity of a thing without looking outside; the physicists are ratherstupid for having thought otherwise, but it has just dawned on them that thisis the case. If only we philosophers had realized what the problems were thatthe physicists had, we could have decided immediately by brainwork that it isimpossible to tell how fast one is moving without looking outside, and we couldhave made an enormous contribution to physics.” These philosophers are alwayswith us, struggling in the periphery to try to tell us something, but theynever really understand the subtleties and depths of the problem.
还有另外一派哲学家，对于相对论，它们感到很不舒服，因为相对论断言，如果不看外面的某物，我们无法得到我们的绝对矢速，且他们会说：“一个人，不往外看，就不能测量他的矢速，这一点很明显。不用往外看而谈论一个事物的矢速，是无意义的，这是自证明的；如果物理家们往其他方向想，反倒是愚蠢的，他们只是刚刚理解到，情况正是这样。对于物理学家们所要面对的那些问题，假如只是我们哲学家，曾经意识到它们是什么，那么，我们可以立即通过脑力劳动，来决定，不往外看，而能决定一个人的移动有多快，是不可能的，这样，我们就可以对物理学，做出巨大的贡献。”这些哲学家们，总是和我们在一起，他们在外围，努力奋斗，想要诉我们一些事情，但是，他们永远也不能真正理解这个问题的精妙和深度。

Now, is it absolutely, definitely,philosophically necessary that one should not be able to tell how fasthe is moving without looking outside? One of the consequences of relativity wasthe development of a philosophy which said, “You can only define what you canmeasure! Since it is self-evident that one cannot measure a velocity without seeingwhat he is measuring it relative to, therefore it is clear that there is no meaningto absolute velocity. The physicists should have realized that they can talkonly about what they can measure.” But that is the whole problem:whether or not one can define absolute velocity is the same as theproblem of whether or not one can detect in an experiment, withoutlooking outside, whether he is moving. In other words, whether or not a thingis measurable is not something to be decided a priori by thought alone,but something that can be decided only by experiment. Given the fact that thevelocity of light is 186,000 mi/sec, one will find few philosophers who will calmly statethat it is self-evident that if light goes 186,000 mi/sec inside a car, and the car is going 100,000 mi/sec, that the light also goes 186,000 mi/sec past an observer on the ground. That is a shocking factto them; the very ones who claim it is obvious find, when you give them aspecific fact, that it is not obvious.
现在，对于一个人不往外看，就不能知道他走的有多快，这一点，是不是绝对地、确定地、和哲学性地必然的呢？相对论的一个后果，就是某种哲学的发展，该哲学说：“你只能定义你能测的东西。一个人，要测量某物的速度，如果他不看此物所相对的东西，那么，他是无法测量此物的矢速的，由于这一点，是自证明的，因此，很清楚，绝对矢速，没有意义。物理学家们应该已经意识到，他们只能谈论：他们能测量的东西”。但是，这就是全部的问题：一个人能否定义绝对矢速，这是第一个问题；一个人在实验中，不用往外看，能否探测出：他是否在移动；这是第二个问题；这两个问题，是一样的。换句话说，一个事物，是否可测，这种事情，并不能通过思想，预先决定，而是只能通过实验，来决定。光速是186,000 mi/sec，给定这个事实，那么，一个人就可以找出一些哲学家，他们可以平静地声明，下面所说，是自证明的：如果光在一个汽车中，以186,000 mi/sec走，而汽车速度是100,000 mi/sec，那么，对于地面上的一个观察者来说，光也走了186,000 mi/sec。这个事实，对他们来讲，是震惊的；因为这些人号称，找出这一点，很明显，而你给出的这个具体事实，{则说明}，{这一点}并非很明显。{？}

Finally, there is even a philosophy whichsays that one cannot detect any motion except by looking outside. It issimply not true in physics. True, one cannot perceive a uniform motionin a straight line, but if the whole room were rotating we wouldcertainly know it, for everybody would be thrown to the wall—there would be allkinds of “centrifugal” effects. That the earth is turning on its axis can be determinedwithout looking at the stars, by means of the so-called Foucault pendulum, forexample. Therefore it is not true that “all is relative”; it is only uniformvelocity that cannot be detected without looking outside. Uniform rotationabout a fixed axis can be. When this is told to a philosopher, he isvery upset that he did not really understand it, because to him it seemsimpossible that one should be able to determine rotation about an axis withoutlooking outside. If the philosopher is good enough, after some time he may comeback and say, “I understand. We really do not have such a thing as absoluterotation; we are really rotating relative to the stars, you see. And sosome influence exerted by the stars on the object must cause the centrifugal force.”
最后，甚至有这样一种哲学，它说：一个人，如果不往外看的话，就不能探测到任何运动。不过，在哲学中，这并不正确。确实，人不能感知到匀速直线运动，但是，如果整个屋子都在旋转，我们当然会知道，因为每个人都会被扔到墙上—会有各种“离心的”效果。地球绕着其轴转动，不用看星星，也能确定，例如，凭借所谓的傅科摆。因此，“所有都是相对的”这一说法，并不正确；不往外看时，只有匀速才无法被探测到。绕着一个轴的匀速转动，则可以。当把这事，告诉一位哲学家，由于他并不真正理解此事，所以他会非常生气，因为，对他来说，一个人不用往外看，也可以确定绕轴旋转，应该是不可能的。如果这位这些家足够好，那么，过段时间后，他会回来说：“我理解了。你看，我们确实没有绝对转动这种事；我们确实是相对于恒星在旋转。所以，由恒星对对象所产生的某些影响，应该引起离心力。”

Now, for all we know, that is true; we haveno way, at the present time, of telling whether there would have been centrifugalforce if there were no stars and nebulae around. We have not been able to dothe experiment of removing all the nebulae and then measuring our rotation, sowe simply do not know. We must admit that the philosopher may be right. Hecomes back, therefore, in delight and says, “It is absolutely necessary thatthe world ultimately turn out to be this way: absolute rotation means nothing;it is only relative to the nebulae.” Then we say to him, “Now, myfriend, is it or is it not obvious that uniform velocity in a straight line, relativeto the nebulae should produce no effects inside a car?” Now that the motionis no longer absolute, but is a motion relative to the nebulae, it becomesa mysterious question, and a question that can be answered only by experiment.
现在，就我们所知，这是对的；目前，我们没有其他方法来告知：如果没有恒星或星云，是否会有离心力。我们无法把所有星云移开，然后做实验，以测我们的旋转，所以，我们不知道。我们应该承认，哲学家或许是对的。因此，他回来，高兴地说：“世界最终，实际是这样：绝对的旋转，毫无意义；它只是相对于星云的；这是绝对必然的。”然后，我们跟他说：“现在，我的朋友，匀速直线运动，相对于星云的，不应该在一辆汽车内产生任何结果，这一点，是否很明显？”现在，这个运动，已经不再是绝对运动了，而是一个相对于星云的运动，这个问题，就变成了一个神秘的问题，一个只能通过实验来回答的问题。

What, then, are the philosophic influencesof the theory of relativity? If we limit ourselves to influences in the senseof what kind of new ideas and suggestions are made to the physicist bythe principle of relativity, we could describe some of them as follows. Thefirst discovery is, essentially, that even those ideas which have been held fora very long time and which have been very accurately verified might be wrong.It was a shocking discovery, of course, that Newton’s laws are wrong, after allthe years in which they seemed to be accurate. Of course it is clear, not that theexperiments were wrong, but that they were done over only a limited range ofvelocities, so small that the relativistic effects would not have been evident.But nevertheless, we now have a much more humble point of view of our physicallaws—everything can be wrong!
那么，相对论的哲学影响，又是什么呢？影响的意义，可以是：通过相对论原理，给物理学家，造成了什么样的新想法和建议；如果我们如此取影响的意义，那么，我们就可以按如下方式，来描述其中的一些想法和建议。第一个发现，是本质性的发现，即很多想法，被当真了很多年，还有很多想法，被准确地验证过，但现在，它们可能都是错的。牛顿规律是错的，这个发现，当然令人震惊，因为，这么多年来，它们似乎都是准确的。当然，很清楚，并非实验有错，而是，实验只是在一个有限的速度范围内做的，这个范围，如此之小，以至于相对论的效果，并不明显。但尽管如此，对于我们的物理规律，我们现在有了一个更谦逊的观点--每件事情，都可能是错的！

Secondly, if we have a set of “strange”ideas, such as that time goes slower when one moves, and so forth, whether we likethem or do not like them is an irrelevant question. The only relevantquestion is whether the ideas are consistent with what is found experimentally.In other words, the “strange ideas” need only agree with experiment, andthe only reason that we have to discuss the behavior of clocks and so forth isto demonstrate that although the notion of the time dilation is strange, it is consistentwith the way we measure time.
第二，如果我们有一系列“奇怪的”想法，例如，当我们运动时，时间走得慢，等等，我们是否喜欢它们，则是一个不相关的问题。唯一相关的问题就是，这些想法，与实验所发现的，是否一致？换句话说，这些“奇怪的”想法，只需与实验一致，对于表的表现等，我们讨论它们的唯一原因，就是要验证：虽然时间放大的观念，比较奇怪，但它与我们测量时间的方式，是一致的。

辛苦了辛苦了

Finally, there is a third suggestion whichis a little more technical but which has turned out to be of enormous utility inour study of other physical laws, and that is to look at the symmetry of thelaws or, more specifically, to look for the ways in which the laws can betransformed and leave their form the same. When we discussed the theory ofvectors, we noted that the fundamental laws of motion are not changed when werotate the coordinate system, and now we learn that they are not changed whenwe change the space and time variables in a particular way, given by the Lorentztransformation. So this idea of studying the patterns or operations under whichthe fundamental laws are not changed has proved to be a very useful one.
最后，是第三个建议，它技术性稍微多一点，但是，最终证明，在我们研究其他物理规律时，它有巨大的用途，它就是：查看规律的对称性，或更具体地说，寻找某种方式，在其中，规律可以被变换，以让它们的形式，是一样的。当我们讨论矢量理论时，我们注意到，当我们旋转坐标系时，基本运动规律，并未改变，现在，我们知道，当我们以一种具体的方式，例如通过洛伦兹变换，来改变空间和时间的变量时，这些规律，并未被改变。在有些模式或操作下，基础规律，并未被改变；研究这些模式或操作的想法，已经被证明，非常有用。

16–2The twin paradox 16-2 双生子悖论
To continue our discussion of the Lorentztransformation and relativistic effects, we consider a famous so-called“paradox” of Peter and Paul, who are supposed to be twins, born at the same time.When they are old enough to drive a space ship, Paul flies away at very highspeed. Because Peter, who is left on the ground, sees Paul going so fast, allof Paul’s clocks appear to go slower, his heart beats go slower, his thoughts goslower, everything goes slower, from Peter’s point of view. Of course, Paulnotices nothing unusual, but if he travels around and about for a while andthen comes back, he will be younger than Peter, the man on the ground! That isactually right; it is one of the consequences of the theory of relativity whichhas been clearly demonstrated. Just as the muons last longer when they aremoving, so also will Paul last longer when he is moving. This is called a “paradox”only by the people who believe that the principle of relativity means that allmotion is relative; they say, “Heh, heh, heh, from the point of view ofPaul, can’t we say that Peter was moving and should therefore appear toage more slowly? By symmetry, the only possible result is that both should bethe same age when they meet.” But in order for them to come back together andmake the comparison, Paul must either stop at the end of the trip and make a comparisonof clocks or, more simply, he has to come back, and the one who comes back mustbe the man who was moving, and he knows this, because he had to turn around. Whenhe turned around, all kinds of unusual things happened in his space ship—therockets went off, things jammed up against one wall, and so on—while Peter feltnothing.
要继续讨论洛伦兹变换和相对论的影响，我们考虑著名的保罗（Paul）--彼得（Peter）“悖论”，他们是双生子，同时出生。当他们长大，足以开太空飞船时，保罗驾船，高速飞走。彼得则被留在地上，因为彼得看到保罗，走的飞快，从彼德的观点看，所有保罗的表，都变慢了，他的心跳也慢了，他的思想也慢了，所有都慢了。当然，保罗没有注意到任何异常，但是，如果他转了一圈，然后回来，他就会比留在地上的保罗要年轻。这完全正确；它就是相对论的一个后果，已经被清楚地演证了。正如U介子，当其运动时，活得更长，于是，当保罗运动时，也活得更长。有些人，相信相对论的原理就是：所有运动都是相对的；这些人，把上述现象，称为“悖论”；他们说：“嘿、嘿、嘿，从保罗的观点看，我们难道我们不能说彼得在运动，且因此看上去，就显得变老较慢吗？通过对称，唯一可能的结果就是，当他俩相遇时，年龄应相同。”但是，为了让他们回来重聚，并进行比较，那么保罗，要么应该在旅行的终点，停下来，用表做比较，要么，更简单一点，他必须回来，而且，回来的那个人的，应该就是那个移动的人，并且，他知道这一点，因为，他必须要转一圈。当他在转时，在他的太空飞船中，所有异常之事，都发生了—火箭爆炸了，各种东西挤在同一面墙上，等等--，而此时的彼得，则什么也感觉不到。

So the way to state the rule is to say thatthe man who has felt the accelerations, who has seen things fall againstthe walls, and so on, is the one who would be the younger; that is the differencebetween them in an “absolute” sense, and it is certainly correct. When wediscussed the fact that moving muons live longer, we used as an example theirstraight-line motion in the atmosphere. But we can also make muons in a laboratoryand cause them to go in a curve with a magnet, and even under this acceleratedmotion, they last exactly as much longer as they do when they are moving in astraight line. Although no one has arranged an experiment explicitly so that wecan get rid of the paradox, one could compare a muon which is left standingwith one that had gone around a complete circle, and it would surely be foundthat the one that went around the circle lasted longer. Although we have notactually carried out an experiment using a complete circle, it is really notnecessary, of course, because everything fits together all right. This may notsatisfy those who insist that every single fact be demonstrated directly, butwe confidently predict the result of the experiment in which Paul goes in a completecircle.
于是，陈述这个规则的方式，就是说，那个感觉到加速度的人，那个看到事物全挤到墙上的人，等等，就是那个会更年轻一点的人；这个就是，在一种“绝对”意义上的‘他们之间的差别’，这当然是正确的{？}。运动着的U介子，活得更久，这是事实，当我们讨论此事实时，我们是把它们在大气中的直线运动，当作例子来使用的。然而，我们也可以在实验室中，制造U介子，然后，利用磁铁，让其做曲线运动，甚至是做加速运动，这时，它们所延长的生命，与它们做直线运动时所延长的，完全一样。虽然没有人明确地安排实验，让我们可以摆脱悖论，但是，对于两个U介子，一个原地不动，一个在做完整的圆运动，我们还是可以比较它们的，而且，肯定会发现，后者活得更久一点。虽然我们并没有用一个完整的圆，来做这样一个实验，当然这并非必需，因为所有的事情，都符合得很好。这可能无法满足某些人，因为这些人，坚持所有单独的事实，都应直接被演证，但是，对于保罗在走一个完整的圆这种实验，我们可以非常自信地预测其结果。

16–3Transformation of velocities 16-3 矢速的变换
The main difference between the relativityof Einstein and the relativity of Newton is that the laws of transformationconnecting the coordinates and times between relatively moving systems aredifferent. The correct transformation law, that of Lorentz, is
(16.1)
These equations correspond to the relatively simple case in which therelative motion of the two observers is along their common x -axes. Of course other directions of motion are possible, but the mostgeneral Lorentz transformation is rather complicated, with all four quantitiesmixed up together. We shall continue to use this simpler form, since itcontains all the essential features of relativity.
有爱因斯坦的相对论，有牛顿的相对论，两者之间的主要区别，就在于，把相关移动系统的坐标与时间，联系在一起的变换规律，是不同的。正确的变换规律，即洛伦兹的变换规律，就是：
(16.1)
这些方程，相应于相关的简单情况，在其中，两个观察者的相关运动，就是沿着它们共同的x轴。当然，其他运动的方向，也是可能的，但是，最普遍的洛伦兹变换，所有四个量混在一起，相当复杂。我们将继续使用这个简单的形式，由于它包含了相对论的基本特性。