Now if all moving clocks run slower,if no way of measuring time gives anything but a slower rate, we shall justhave to say, in a certain sense, that time itself appears to be slowerin a space ship. All the phenomena there—the man’s pulse rate, his thoughtprocesses, the time he takes to light a cigar, how long it takes to grow up andget old—all these things must be slowed down in the same proportion, because hecannot tell he is moving. The biologists and medical men sometimes say it isnot quite certain that the time it takes for a cancer to develop will be longerin a space ship, but from the viewpoint of a modern physicist it is nearlycertain; otherwise one could use the rate of cancer development to determinethe speed of the ship!
现在，如果所有移动着的表，都走得更慢了，如果任何测量实践的方法，所能给出的，只是更慢的速率，那么，我们在某种义上，就只能说：在一个太空飞船中，时间本身，表现出是更慢了。在那里，所有的现象--人的脉搏率，他的思想处理，他点烟所需的时间，长大和变老需要多久等等，所有这些事情，都应该是以同样的比例，变慢了，因为他无法得知，他在运动。生物学家和医务人员们，有时会说：无法完全确定，在太空飞船中，癌症发展的时间，可能会更长；但是，从一个现代物理学家的角度看，这几乎是肯定的，否则的话，有人就可以使用这个癌症发展的速率，来规定飞船的速度了！

A very interesting example of the slowingof time with motion is furnished by muons, which are particles thatdisintegrate spontaneously after an average lifetime of 2.2×10−6 sec. They come to the earth in cosmic rays, and can also beproduced artificially in the laboratory. Some of them disintegrate in midair,but the remainder disintegrate only after they encounter a piece of material andstop. It is clear that in its short lifetime a muon cannot travel, even at thespeed of light, much more than 600 meters. But although the muons are created at the top of theatmosphere, some 10 kilometers up, yet they are actually found in a laboratory downhere, in cosmic rays. How can that be? The answer is that different muons moveat various speeds, some of which are very close to the speed of light. Whilefrom their own point of view they live only about 2 μ sec, from our point of view they live considerably longer—enoughlonger that they may reach the earth. The factor by which the time is increasedhas already been given as 1/(1−u2/c2)1/2. The average life has been measured quite accurately for muons of differentvelocities, and the values agree closely with the formula.
关于随着运动，时间变慢，有个例子，非常有趣；它由U介子提供，U介子是一种粒子，平均生命周期为2.2×10−6 sec ，随后就会自发分解。它们跟着宇宙射线，来到地球，也可在实验室中，人工产生。它们中的一些，在半空中分解，剩下的，只有在遇到物质材料，停下后，才分解。很清楚，在其短暂的生命中，一个U介子，即便速度为光速，所走路程，也不能超过600米。但是，虽然U介子，是在大气层的上部，约10公里处，被造出来，尽管如此，在下面的实验室中，在宇宙射线中，也能找到它们。怎么会这样呢？答案是，不同的U介子，移动的速度不同，有些非常接近光速。从它们的观点，它们只生存了2 μ sec ，但从我们的观点，它们生存地相当长—长到足以达到地面。时间增长的因子，已经通过1/(1−u2/c2)1/2而给予。对于不同矢速的U介子，其平均生命周期，已经测量的非常准确了，其值与公式，高度吻合。

We do not know why the muon disintegratesor what its machinery is, but we do know its behavior satisfies the principleof relativity. That is the utility of the principle of relativity—it permits usto make predictions, even about things that otherwise we do not know muchabout. For example, before we have any idea at all about what makes the muondisintegrate, we can still predict that when it is moving at nine-tenths of thespeed of light, the apparent length of time that it lasts is (2.2×10−6)/(1−92/102)1/2sec; and our prediction works—that is the good thing about it.
U介子为什么会分裂，其机制为何，我们并不知道，但我们确实知道，它的表现，满足相对论原理。这就是相对论原理的用途—它允许我们，做出预测，即便那些东西，我们并不太了解。例如，究竟什么，使得U介子分裂，尽管对此，我们毫无头绪，但我们仍能预测：当它以光速的十分之九运动时，它所明显延长的时间，就是(2.2×10−6)/(1−92/102)1/2 sec；我们的预报成功了，这就是事情好的一面。

15–5The Lorentz contraction 15-5 洛伦兹收缩
Now let us return to the Lorentztransformation (15.3)and try to get a better understanding of the relationship between the (x,y,z,t)and the (x′,y′,z′,t′) coordinate systems, which we shall call the S and S′ systems, or Joe and Moe systems, respectively. We have alreadynoted that the first equation is based on the Lorentz suggestion of contractionalong the x -direction; how can we prove that a contraction takes place? In theMichelson-Morley experiment, we now appreciate that the transversearm BC cannot change length, by the principle of relativity; yet the nullresult of the experiment demands that the times must be equal. So, inorder for the experiment to give a null result, the longitudinal arm BEmust appear shorter, by the square root (1−u2/c2)1/2. What does this contraction mean, in terms of measurements made byJoe and Moe? Suppose that Moe, moving with the S′ system in the x -direction, is measuring the x′ -coordinate of some point with a meter stick.用Joe和Moe He lays the stick downx′ times, so he thinks the distance is x′ meters. From the viewpoint of Joe in the S system, however, Moe is using a foreshortened ruler, so the“real” distance measured is x′(1−u2/c2)1/2 meters. Then if the S′ system has travelled a distance ut away from the S system, the S observer would say that the same point, measured in hiscoordinates, is at a distance x=x′(1−u2/c2)1/2+ut, or
x′=(x−ut) / (1−u2/c2)1/2
which is the first equation of the Lorentz transformation.
现在，让我们返回洛伦兹变换（15.3），尝试更好地理解坐标系 (x,y,z,t)与(x′,y′,z′,t′)之间的关系，我们把它们分别称为S和S′ ，或Joe和Moe 系统。我们已经注意到，第一个方程，是基于洛伦兹的沿着x方向的收缩建议；我们如何能证明，这个收缩发生了呢？在迈克耳孙--莫雷实验中，我们现在根据相对论原理，赞成：横向的臂BC，不能改变长度；尽管，此实验的结果为零，要求时间应该相等。于是，为了让实验给出零结果，纵向的臂BE，就应该表现的短一些，其量为(1−u2/c2)1/2。所做的测量术语来说，这个收缩的意思是什么呢？假设Moe，在x方向，与S′系统，一起运动，他用一个米尺，测量某些点的x′ 坐标。他用米尺量了x′次，所以他想，距离就是x′米。然而，Joe在S系统中，从他的观点看，Moe用的，是一个缩短了的尺子，所以，所测的“真正”距离，应是x′(1−u2/c2)1/2 米。因此，如果S′ 系统，与S系统的距离，为ut，那么，S上的观察者应该说，对于同一个点，在他的坐标系中测量，距离就是 x=x′(1−u2/c2)1/2+ut，或者
x′=(x−ut) / (1−u2/c2)1/2
它就是洛伦兹变换的第一个方程。

15–6Simultaneity 同时发生性（同发性）
In an analogous way, because of thedifference in time scales, the denominator expression is introduced into thefourth equation of the Lorentz transformation. The most interesting term inthat equation is the ux/c2 in the numerator, because that is quite new and unexpected. Now whatdoes that mean? If we look at the situation carefully we see that events thatoccur at two separated places at the same time, as seen by Moe in S′, do not happen at the same time as viewed by Joe in S. If one event occurs at point x1 at time t0 and the other event at x2 and t0 (the same time), we find that the two corresponding times t′1 and t′2 differ by an amount
This circumstance is called “failure of simultaneity at a distance,”and to make the idea a little clearer let us consider the following experiment.
在一种类比的方式中，因为时间尺度的不同，在洛伦兹变换的第四个方程中，引入了分母表达式。在这个方程中，最有趣的项，就是分子中的ux/c2，因为它是全新的，且是没有预料到的。那么，其意为何？如果我们仔细的看这个情况，我们就会看到，在同一时间，在两个分开的地方，所发生的事件，如由Moe 在 S′中看到的，与Joe 在S中观察到的，并不是同时发生的。如果在时间 t0，一个事件，发生在点x1，另外一个事件，发生在点x2，也在时间 t0（同一时间），我们发现，这两个相关的{观测}时间t′1和t′2之间，相差一个量：

这种情形，被称为“一定距离外的同发性的失败”，为了让此想法，更加清晰，我们考虑下面的实验。

Suppose that a man moving in a space ship(system S′ ) has placed a clock at each end of the ship and is interested inmaking sure that the two clocks are in synchronism. How can the clocks besynchronized? There are many ways. One way, involving very little calculation,would be first to locate exactly the midpoint between the clocks. Then fromthis station we send out a light signal which will go both ways at the samespeed and will arrive at both clocks, clearly, at the same time. Thissimultaneous arrival of the signals can be used to synchronize the clocks. Letus then suppose that the man in S′ synchronizes his clocks by this particular method. Let us see whetheran observer in system S would agree that the two clocks are synchronous. The man in S′has a right to believe they are, because he does not know that he ismoving. But the man in S reasons that since the ship is moving forward, the clock in the frontend was running away from the light signal, hence the light had to go more thanhalfway in order to catch up; the rear clock, however, was advancing to meetthe light signal, so this distance was shorter. Therefore the signal reachedthe rear clock first, although the man in S′ thought that the signals arrived simultaneously. We thus see that whena man in a space ship thinks the times at two locations are simultaneous, equalvalues of t′ in his coordinate system must correspond to different valuesof t in the other coordinate system!
假设在一个运动的太空飞船中（系统S′），有一个人，他在飞船两端，各放一个表，而且，保证让两个表是同步的，也是他感兴趣的事情。这两个表如何才能同步呢？有很多方式，一种方式，牵扯到非常少的计算，首先，在两个表之间，找到准确的中点。然后，从这个位置，我们发出一个光信号，它将以同样的速度，向两边走，显而易见，它将同时到达两个表。这个信号的同时到达，可被用来，同步这些表。然后，让我们假设，S′中的人，通过这个具体的方法，同步了他的表。下面让我们看看，系统S中的一个观测者，是否同意，这两个表是同步的。S′中的人，有权相信，它们是同步的，因为他并不知道，自己在运动。但是，S中的人，则推断说，由于此船，正在向前运动，所以，前端的表，就是在逃离光信号，因此，光走的距离，要比一半多，才能到达表；而后面的表，迎着光信号走，所以，这个距离，应该短些。因此，信号应该先到达后面的表，虽然S′中的人，认为信号是同时到达的。这样，我们就看到，飞船中的人，认为到达两个位置的时间，是同时的，然而此时，他的坐标系中的两个相等的t值，却应该相应于，另外一个坐标系中的两个不同的t值。

15–7Four-vectors 15-7 四个矢量
Let us see what else we can discover in theLorentz transformation. It is interesting to note that the transformationbetween the x ’s and t ’s is analogous in form to the transformation of the x ’s and y ’s that we studied in Chapter 11for a rotation of coordinates. We then had
x′=xcosθ+y sinθ ,
y′=ycosθ−xsinθ , (15.8)
in which the new x′ mixes the old x and y , and the new y′ also mixes the old x and y ; similarly, in the Lorentz transformation we find a new x′which is a mixture of x and t , and a new t′ which is a mixture of t and x . So the Lorentz transformation is analogous to a rotation, only it isa “rotation” in space and time, which appears to be a strange concept. Acheck of the analogy to rotation can be made by calculating the quantity
x′2+y′2+z′2−c2t′2=x2+y2+z2−c2t2. (15.9)
In this equation the first three terms on each side represent, inthree-dimensional geometry, the square of the distance between a point and theorigin (surface of a sphere) which remains unchanged (invariant) regardless ofrotation of the coordinate axes. Similarly, Eq. (15.9)shows that there is a certain combination which includes time, that isinvariant to a Lorentz transformation. Thus, the analogy to a rotation iscomplete, and is of such a kind that vectors, i.e., quantities involving“components” which transform the same way as the coordinates and time, are alsouseful in connection with relativity.
让我们看看，从洛伦兹变换，我们还能得到什么。我们在第11章，对于坐标系的旋转，研究过x ’和y ’之间的变换；我们非常感兴趣地注意到，在x ’和t ’之间的变换，在形式上，可与之类比。在第11章，我们有：
x′=xcosθ+y sinθ ,
y′=ycosθ−xsinθ , (15.8)
在其中，新的x′，混合了老的x和y，新的y′，也混合了老的x和y；类似地，在洛伦兹变换中，我们发现，新的x′，是x和t的混合，而新的t′，是t和x的混合。所以，洛伦兹变换，可类比于旋转，只是，它这个“旋转”，是空间和时间中的，这似乎是一个奇怪的概念。这个对旋转的类比，可以通过计算下面的量，来检查：
x′2+y′2+z′2−c2t′2=x2+y2+z2−c2t2. (15.9)
在这个方程中，方程两边前面的三项，在三维的几何中，代表着：一个点与原点之间距离的平方（一个球面），无论坐标轴怎么旋转，它都保持不变（恒定的）。这样，对于旋转的类比，就是完整的，并且，就是这样一种矢量，亦即，这种量，所牵扯到的“分量”，作为坐标和时间，会以同样的方式，发生变换；此矢量，在于相对论相联时，同样有用。

Thus we contemplate an extension of theidea of vectors, which we have so far considered to have only space components,to include a time component. That is, we expect that there will be vectors withfour components, three of which are like the components of an ordinary vector,and with these will be associated a fourth component, which is the analog ofthe time part.
这样，我们就对矢量的想法，做了一个扩展，我们以前认为，矢量只有空间分量，现在，它将包括一个时间分量。也就是说，我们期待，有这种矢量，它有四个分量，其中三个，与普通矢量的分量一样，另外，还有第四个分量，它就是对时间的类比，且要与前三个相联合。

This concept will be analyzed further inthe next chapters, where we shall find that if the ideas of the precedingparagraph are applied to momentum, the transformation gives three space partsthat are like ordinary momentum components, and a fourth component, the timepart, which is the energy.
下一章，将进一步分析这个概念，在那里，我们将找出，如果上面一段的想法，被应用于动量，那么，变换将会给出三个空间部分、即类似普通动量的分量，另外还有第四个分量，即时间部分，也就是能量。

15–8Relativistic dynamics 15-8 相对论动力学
We are now ready to investigate, more generally,what form the laws of mechanics take under the Lorentz transformation. [We havethus far explained how length and time change, but not how we get the modifiedformula for m (Eq. 15.1).We shall do this in the next chapter.]]To see the consequences of Einstein’smodification of m for Newtonian mechanics, we start with the Newtonian law that force isthe rate of change of momentum, or
F=d(mv)/dt.
Momentum is still given by mv , but when we use the new m this becomes
(15.10)
This is Einstein’s modification of Newton’s laws. Under thismodification, if action and reaction are still equal (which they may not be indetail, but are in the long run), there will be conservation of momentum in thesame way as before, but the quantity that is being conserved is not theold mv with its constant mass, but instead is the quantity shown in (15.10),which has the modified mass. When this change is made in the formula formomentum, conservation of momentum still works.
我们现在准备更普遍地调研，究竟是什么，形成了洛伦兹变换下的力学规律。[到目前为止，我们已经解释了，长度和时间是如何变化的，但是，并未解释，我们是如何得到改变质量m的公式（方程15.1）的。这件事情，下一章做。爱因斯坦，对牛顿力学的m,做了修改，要看此修改的结果，我们从‘力是动量的变化率’这一牛顿的规律开始，即：
F=d(mv)/dt.
动量仍是通过mv给予，但是，当我们使用新的m时，这就变成：
(15.10)
这就是爱因斯坦，对牛顿规律的修改。在这个修改下，如果作用与反作用，仍然相等（在细节上，它们可能不等，但从长远看，是相等的），将有动量守恒，且与以前的方式一样，但是，那个被守恒的量，将不是旧的 mv，其质量为常数，而是方程（15.10）所示的量，其中的质量，已被修改。当对动量的方程，做了这个变化后，则动量守恒，依然成立。

Now let us see how momentum varies withspeed. In Newtonian mechanics it is proportional to the speed and,according (15.10),over a considerable range of speed, but small compared with c , it is nearly the same in relativistic mechanics, because thesquare-root expression differs only slightly from 1 . But when v is almost equal to c , the square-root expression approaches zero, and the momentumtherefore goes toward infinity.
现在，让我们看，动量是如何随着速度变化的。在牛顿学的力学中，它正比于速度，且依据（15.10），覆盖一个相当可观的速度范围{？}，但是，这些速度，与c相比，则较小，在相对论力学中，它几乎是一样的，因为，平方根表达式，与1的差别，非常微小。但是，当 v几乎等于c时，平方根表达式将趋于零，因此，动量就将趋于无穷大。

？？？

15–9Equivalence of mass and energy 15-9 质量与能量的等价
The above observation led Einstein to thesuggestion that the mass of a body can be expressed more simply than by theformula (15.1),if we say that the mass is equal to the total energy content divided by c2. If Eq. (15.11)is multiplied by c2 the result is
mc2=m0c2+m0v2/2+⋯ (15.12)
Here, the term on the left expresses the total energy of a body, andwe recognize the last term as the ordinary kinetic energy. Einstein interpretedthe large constant term, m0c2 , to be part of the total energy of the body, an intrinsic energy knownas the “rest energy.”
上面的观察，导致爱因斯坦，提出建议：如果我们说质量等于总的能量，除以c2，那么，一个物体的质量，可以表达地比公式（15.1）更简单。如果方程（15.11）乘以c2，结果就是:
mc2=m0c2+m0v2/2+⋯ (15.12)
这里，左边的项，表达了物体的总的能量，我们可以把最后一项，认为是普通的动能。大的常数项m0c2，是物体的总的能量的一部分，爱因斯坦把它解释为一种内部的能量，称为“静止能量”。

Let us follow out the consequences ofassuming, with Einstein, that the energy of a body always equals mc2. As an interesting result, we shall find the formula (15.1)for the variation of mass with speed, which we have merely assumed up to now.We start with the body at rest, when its energy is m0c2. Then we apply a force to the body, which starts it moving and givesit kinetic energy; therefore, since the energy has increased, the mass hasincreased—this is implicit in the original assumption. So long as the forcecontinues, the energy and the mass both continue to increase. We have alreadyseen (Chapter 13) that the rate of change of energy with time equalsthe force times the velocity, or
dE/dt=F⋅v. (15.13)
We also have (Chapter 9,Eq. 9.1) that F=d(mv)/dt . When these relations are put together with the definition of E, Eq. (15.13)becomes
d(mc2)/dt=v⋅d(mv)/dt. (15.14)
We wish to solve this equation for m . To do this we first use the mathematical trick of multiplying bothsides by 2m , which changes the equation to
c2 (2m)dm/dt=2mv⋅d(mv)/dt. (15.15)
We need to get rid of the derivatives, which can be accomplished byintegrating both sides. The quantity (2m)dm/dt can be recognized as the time derivative of m2 , and (2mv)⋅d(mv)/dt is the time derivative of (mv)2 . So, Eq. (15.15)is the same as
c2d(m2)/dt=d(m2v2)/dt. (15.16)
If the derivatives of two quantities are equal, the quantitiesthemselves differ at most by a constant, say C . This permits us to write
m2c2=m2v2+C. (15.17)
We need to define the constant C more explicitly. Since Eq. (15.17)must be true for all velocities, we can choose a special case where v=0, and say that in this case the mass is m0 . Substituting these values into Eq. (15.17)gives
m20c2=0+C.
We can now use this value of C in Eq. (15.17),which becomes
m2c2=m2v2+m20c2. (15.18)
Dividing by c2 and rearranging terms gives
m2 (1−v2/c2)=m20,
from which we get
m=m0/(1−v2/c2)1/2 (15.19)
This is the formula (15.1),and is exactly what is necessary for the agreement between mass and energy inEq. (15.12).
爱因斯坦假定了，物体的能量，总是等于mc2，这会带来若干后果，下面我们看看。作为一个有趣的结果，我们将找出质量随速度的变化，即公式（15.1），因为到目前为止，我们只是假定了它。我们从静止的物体开始，此时。其能量为m0c2。然后，我们把一个力，应用于物体，力让物体开始运动，并赋予它动能；因此，由于能量已经增加了，质量也就增加了，这在原始的假定中，已经暗示了。只要力在继续，那么能量和质量，就都在继续增加。我们已经看到（在第13章），能量随时间的变化率，等于力乘以矢速，或：
dE/dt=F⋅v. (15.13)
我们还有 (第9章,方程 9.1) F=d(mv)/dt。当把这些关系，与E的定义放在一起，方程（15.13）就变为：
d(mc2)/dt=v⋅d(mv)/dt. (15.14)
我们希望解此方程，以求m。首先利用数学技巧，两边都乘以2m，则方程变为：
c2 (2m)dm/dt=2mv⋅d(mv)/dt. (15.15)
我们需要去掉导数，这可通过两边积分完成。量 (2m)dm/dt，可被认为 m2 对时间的导数, 而(2mv)⋅d(mv)/dt 则是 (mv) 2对时间的导数。 于是，方程 (15.15)与下式同：
c2d(m2)/dt=d(m2v2)/dt. (15.16)
如果这两个量的导数相等，那么，这两个量之间，最多差一个常数，比如说C。我们就可以这样写：
m2c2=m2v2+C. (15.17)
我们需要把常数 C ，定义地更明确些。由于方程 (15.17)，应该对于所有的矢速，都为真，我们可以选一种特殊情况，即 v=0 ，且说，在此情况下，质量是m0。把这些带入方程(15.17)，得出：
m20c2=0+C.
现在，我们可以把这个C的值，带入方程 (15.17)，它就变成：
m2c2=m2v2+m20c2. (15.18)
两边 除以c2，重新排列，得出：
m2 (1−v2/c2)=m20,
由此得出：
m=m0/(1−v2/c2)1/2 (15.19)
这就是公式 (15.1)，也正是在方程(15.12)中，让质量与能量能够一致，所必需的东西。

Ordinarily these energy changes representextremely slight changes in mass, because most of the time we cannot generatemuch energy from a given amount of material; but in an atomic bomb of explosiveenergy equivalent to 20 kilotons of TNT, for example, it can be shown that the dirtafter the explosion is lighter by 1 gram than the initial mass of the reacting material, because ofthe energy that was released, i.e., the released energy had a mass of 1 gram, according to the relationship ΔE=Δ(mc2). This theory of equivalence of mass and energy has been beautifullyverified by experiments in which matter is annihilated—converted totally toenergy: An electron and a positron come together at rest, each with a restmass m0 . When they come together they disintegrate and two gamma rays emerge,each with the measured energy of m0c2 . This experiment furnishes a direct determination of the energyassociated with the existence of the rest mass of a particle.
通常，这些能量变化，代表着极为轻微的质量变化，因为，大多数时间，我们不能从给定量的材料中，产生出很多能量；但是，在原子弹中，例如，爆炸出的能量，等价于20千吨的TNT，可以指出，原子弹爆炸后的废弃物，比最初的反应物质的质量，要轻一克，因为所释放的能量，亦即，依据关系ΔE=Δ(mc2)，所释放的能量，有一克的质量。这个质量与能量等价的理论，已经通过实验，被漂亮地证实了，在实验中，物质被湮灭了—全被转化成了能量：一个电子和一个正电子，相遇并静止，每个都有静止质量m0。当它们相遇时，它们分裂，显现出两个伽马射线，每个射线测出的能量，都是m0c2。一个粒子，是有静止质量的，这是一种存在{状态}，能量与这种存在，有某种关联，对此，这个实验，提供了直接的确认。
1. The electrons would actually win the race versus visiblelight because of the index of refraction of air. A gamma ray would make outbetter.
脚注1、在与可见光的速度竞赛中，电子实际上是可以胜出的，这是因为空气的折射系数。伽马射线，会赢的更多。