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爱因斯坦 摄于1921年
lbert Einstein 于 1879 年 3 月 14 日出生于德国符腾堡州的乌尔姆。不久后他们全家便搬到了慕尼黑，后来他开始在路易波特中学上学。再后来，他们搬到了意大利，阿尔伯特在瑞士阿劳继续接受教育，并于 1896 年进入苏黎世的瑞士联邦理工学院接受物理和数学教师培训。1901 年，也就是他获得文凭的那一年，他获得了瑞士公民身份，由于找不到教职，他接受了瑞士专利局的技术助理职位。1905 年，他获得了博士学位。在专利局工作期间和业余时间，他创作了许多出色的作品，并于 1908 年被任命为伯尔尼的私人助理。1909 年，他成为苏黎世的特命教授，1911 年成为布拉格的理论物理学教授，次年返回苏黎世担任类似职位。1914 年，他被任命为威廉皇帝物理研究所所长和柏林大学教授。他于 1914 年成为德国公民，并一直留在柏林，直到 1933 年他因政治原因放弃公民身份并移民到美国，在普林斯顿大学担任理论物理学教授。他于 1940 年成为美国公民，并于 1945 年退休。二战后，爱因斯坦是世界政府运动的领军人物，他被邀请担任以色列国总统，但他拒绝了，他与 Chaim Weizmann 博士合作建立了耶路撒冷希伯来大学。爱因斯坦似乎总是对物理学问题有清晰的看法，并始终怀揣着解决这些问题的决心。他有自己的策略，能天马行空般想象出通往目标的道路上的主要阶段。在爱因斯坦开始科学工作时，他就意识到牛顿力学的不足，他的狭义相对论源于试图调和力学定律与电磁场定律。他处理了统计力学的经典问题以及它们与量子理论融合的问题：这导致了对分子布朗运动的解释。他研究了低辐射密度光的热特性，这个观察为光的光子理论奠定了基础。早年在柏林，爱因斯坦假设对狭义相对论的正确解释也必须提供引力理论，并于 1916 年发表了关于广义相对论的论文。在此期间，他还为辐射理论和统计力学问题做出了贡献。在 1920 年代，爱因斯坦开始构建统一场论，尽管他继续致力于量子理论的概率解释，并且他在美国坚持不懈地从事这项工作。他通过发展单原子气体的量子理论为统计力学做出了贡献，他还在原子跃迁概率和相对论宇宙学方面完成了有价值的工作。退休后，他继续致力于统一物理学的基本概念，采取与大多数物理学家相反的方法，即几何化。当然，爱因斯坦的研究有据可查，他更重要的著作包括《狭义相对论》（1905 年）、《相对论》（英文译本，1920 年和 1950 年）、《广义相对论》（1916 年）、《布朗运动理论研究》（1926 年）和《物理学的进化》（1938 年）。在他的非科学著作中，《关于犹太复国主义》（About Zionism，1930）、《为什么战争？》（Why War？，1933）、《我的哲学》（My Philosophy，1934）和《走出我的晚年》（Out of My Later Years，1950）可能是最重要的。阿尔伯特·爱因斯坦 （Albert Einstein） 获得了许多欧洲和美国大学的科学、医学和哲学荣誉博士学位。在 1920 年代，他在欧洲、美洲和远东讲学，并被授予世界各地所有领先科学院的奖学金或会员资格。为了表彰他的工作，他获得了无数奖项，包括 1925 年伦敦皇家学会的科普利奖章和 1935 年富兰克林研究所的富兰克林奖章。爱因斯坦的天赋不可避免地导致他大部分时间生活在知识上的孤独中，为了放松，音乐在他的生活中扮演了重要的角色。他于 1903 年与 Mileva Maric 结婚，育有一女两子;他们的婚姻于 1919 年解除，同年他与他的表妹 Elsa Löwenthal 结婚，后者于 1936 年去世。他于 1955 年 4 月 18 日在新泽西州普林斯顿去世。
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From left to right: Nobel Laureates Walther Nernst, Albert Einstein, Max Planck, Robert A. Millikan and Max von Laue at a dinner given by Professor von Laue in Berlin, 11 November 1931.
Source: Nationaal Archief Photographer unknown Public domain via Wikimedia Commons
爱因斯坦形容自己为不可知论者，但并不具有像专业无神论者般的十字军精神；更仔细解释，由于人类对于大自然与自己本身的了解可能有缺失，因此应该采取谨慎谦卑的态度。美国犹太领袖拉比赫伯特·戈尔茨坦曾经问他是否相信神？他回答说：“我相信斯宾诺莎的神，一个通过存在事物的和谐有序体现自己的神，而不是一个关心人类命运和行为的神。” 换句话说，爱因斯坦认为，从宇宙世界的存在，可以感觉到神的伟大工作，但神并不会干预人们的日常生活，神是非人格化的神。爱因斯坦曾经在书信里表示：“我不相信人格化的神，我从未否认这一点，而且表达得很清楚。如果在我的内心里有什么能被称之为宗教，那就是，对于我们的科学所能够揭示的世界结构，对于这世界结构的无垠的敬仰。”
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The fifth Solvay International Conference on Electrons and Photons, was held in October 1927. Prominent physicists from all the world met to discuss the newly formulated quantum theory. 17 of the 29 participants were or became Nobel Laureates.
Fundamental ideas and problems
of the theory of relativity
//相对论的基本思想和问题
“If we consider that part of the theory of relativity which may nowadays in
a sense be regarded as bona fide scientific knowledge, we note two aspects
which have a major bearing on this theory. The whole development of the
theory turns on the question of whether there are physically preferred states
of motion in Nature (physical relativity problem). Also, concepts and distinctions are only admissible to the extent that observable facts can be assigned to them without ambiguity (stipulation that concepts and distinctions
should have meaning). This postulate, pertaining to epistemology, proves to
be of fundamental importance.//“如果我们考虑到相对论的部分内容，在今天某种程度上可以被视为真正的科学知识，我们会注意到有两个方面对该理论有重要影响。该理论的发展完全围绕着一个问题：在自然界中是否存在物理上优先的运动状态（物理相对性问题）。此外，概念和区分只有在可以毫不含糊地与可观察的事实相联系时才是可接受的（即概念和区分必须具有意义的规定）。这个与认识论相关的假设被证明是至关重要的。”These two aspects become clear when applied to a special case, e.g. to classical mechanics. Firstly we see that at any point filled with matter there exists
a preferred state of motion, namely that of the substance at the point considered. Our problem starts however with the question whether physically
preferred states of motion exist in reference to extensive regions. From the
viewpoint of classical mechanics the answer is in the affirmative; the physically preferred states of motion from the viewpoint of mechanics are those of
the inertial frames.//“当将这两个方面应用于一个特例时，比如经典力学，它们变得更加清晰。首先，我们看到在任何被物质充满的点上，存在一个优先的运动状态，即该点上物质的运动状态。然而，我们的问题从是否存在与广泛区域相关的物理优先运动状态开始。根据经典力学的观点，答案是肯定的；从力学的角度来看，物理优先的运动状态是惯性参考系的运动状态。”This assertion, in common with the basis of the whole of mechanics as it
generally used to be described before the relativity theory, far from meets
the above "stipulation of meaning". Motion can only be conceived as the
relative motion of bodies. In mechanics, motion relative to the system of
coordinates is implied when merely motion is referred to. Nevertheless this
interpretation does not comply with the "stipulation of meaning" if the coordinate system is considered as something purely imaginary. If we turn our
attention to experimental physics we see that there the coordinate system is
invariably represented by a "practically rigid" body. //该论断与经典力学的基础（在相对论理论提出之前的一般描述方式）一致，但远未符合前述的“意义规定”。运动只能被理解为物体的相对运动。在力学中，当仅仅提到运动时，暗指的是相对于坐标系的运动。然而，如果将坐标系视为纯粹虚构的东西，这种解释就不符合“意义规定”。如果我们关注实验物理学，就会发现坐标系总是由一个“实际上刚性”的物体来表示。（* The Lecture was not delivered on the occasion of the Nobel Prize award, and did
not, therefore, concern the discovery of the photoelectric effect.）Furthermore it is assumed that such rigid bodies can be positioned in rest relative to one another in common with the bodies of Euclidian geometry. Insofar as we may think
of the rigid measuring body as existing as an object which can be experienced,
the "system of coordinates" concept as well as the concept of the motion of
matter relative thereto can be accepted in the sense of the "stipulation of
meaning". At the same time Euclidian geometry, by this conception, has been
adapted to the requirements of the physics of the "stipulation of meaning".
The question whether Euclidian geometry is valid becomes physically significant; its validity is assumed in classical physics and also later in the special
theory of relativity.//此外，还假设这些刚体能够彼此相对静止地放置，就像欧几里得几何中的物体一样。只要我们认为刚性测量物体作为一个可以被经验的对象存在，“坐标系”这一概念以及物质相对于该坐标系的运动概念，就可以依据“意义规定”来接受。同时，通过这种观念，欧几里得几何已经被调整以适应符合“意义规定”的物理学的要求。欧几里得几何是否有效的问题因此具有了物理学上的意义；在经典物理学中，欧几里得几何的有效性被假定为成立，并且在后来的狭义相对论中也是如此。In classical mechanics the inertial frame and time are best defined together
by a suitable formulation of the law of inertia: It is possible to fix the time
and assign a state of motion to the system of coordinates (inertial frame) such
that, with reference to the latter, force-free material points undergo no acceleration; furthermore it is assumed that this time can be measured without
disagreement by identical clocks (systems which run down periodically) in
any arbitrary state of motion. There are then an infinite number of inertial
frames which are in uniform translational motion relative to each other, and
hence there is also an infinite number of mutually equivalent, physically preferred states of motion. Time is absolute, i.e.independent of the choice of
the particular inertial frame; it is defined by more characteristics than logically necessary, although - as implied by mechanics - this should not lead
to contradictions with experience. Note in passing that the logical weakness
of this exposition from the point of view of the stipulation of meaning is
the lack of an experimental criterion for whether a material point is forcefree or not; therefore the concept of the inertial frame remains rather problematical. This deficiency leads to the general theory of relativity. We shall
not consider it for the moment. //在经典力学中，惯性系与时间的最佳定义是通过适当的惯性定律表述共同定义的：可以确定时间，并给坐标系（惯性系）分配一个运动状态，这样相对于惯性系，无外力作用的物质点不会产生加速度。此外，假设这种时间可以通过任意运动状态下的相同时钟（周期性运行的系统）来一致地测量。于是，相对彼此作匀速直线运动的惯性系有无穷多个，因此存在无穷多个相互等效、在物理上首选的运动状态。时间是绝对的，即独立于特定惯性系的选择；尽管在力学中这种定义具有比逻辑上必要更多的特征，但这不应导致与经验的矛盾。顺便提及，这种表述从“意义规定”的角度来看，其逻辑弱点在于缺乏实验标准来判断物质点是否受力，因此惯性系的概念仍然存在一定的问题。这一缺陷引发了广义相对论的发展，但我们暂时不讨论它。The concept of the rigid body (and that of the clock) has a key bearing
on the foregoing consideration of the fundamentals of mechanics, a bearing
which there is some justification for challenging. The rigid body is only approximately achieved in Nature, not even with desired approximation; this
concept does not therefore strictly satisfy the "stipulation of meaning". It is
also logically unjustifiable to base all physical consideration on the rigid or
solid body and then finally reconstruct that body atomically by means of
elementary physical laws which in turn have been determined by means of
the rigid measuring body. I am mentioning these deficiencies of method
because in the same sense they are also a feature of the relativity theory in
the schematic exposition which I am advocating here. Certainly it would be logically more correct to begin with the whole of the laws and to apply the
"stipulation of meaning" to this whole first, i.e. to put the unambiguous relation to the world of experience last instead of already fulfilling it in an imperfect form for an artificially isolated part, namely the space-time metric.
We are not, however, sufficiently advanced in our knowledge of Nature’s
elementary laws to adopt this more perfect method without going out of our
depth. At the close of our considerations we shall see that in the most recent
studies there is an attempt, based on ideas by Levi-Civita, Weyl, and Eddington, to implement that logically purer method.//刚性物体（以及时钟）的概念在前述力学基础的思考中起到了关键作用，但这种作用存在某些质疑的合理性。自然界中刚性物体只是近似存在，甚至不能达到所期望的近似程度，因此这个概念严格来说并不满足“意义规定”。此外，从逻辑上来说，将所有物理思考建立在刚性或固体物体的基础上，并最终通过基本物理定律的原子结构来重构刚体，也是站不住脚的。这些基本物理定律本身又是通过刚性测量物体来确定的。我提及这些方法上的缺陷，是因为同样的问题也出现在我在此所倡导的相对论的示意性表述中。当然，逻辑上更为正确的做法是从所有定律出发，首先将“意义规定”应用于这个整体，即将与经验世界的明确关系放在最后，而不是在一个被人为隔离的部分（即时空度量）中以不完善的形式实现它。然而，我们对自然界基本定律的认识还不够深入，无法采用这种更完美的方法而不陷入困境。在我们讨论结束时，我们将看到，最近的研究基于Levi-Civita、Weyl和Eddington的思想，尝试采用这种逻辑上更为纯粹的方法。It more clearly follows from the above what is implied by "preferred states
of motion". They are preferred as regards the laws of Nature. States of motion are preferred when, relative to the formulation of the laws of Nature,
coordinate systems within them are distinguished in that with respect to them
those laws assume a form preferred by simplicity. According to classical mechanics the states of motion of the inertial frames in this sense are physically
preferred. Classical mechanics permits a distinction to be made between (absolutely) unaccelerated and accelerated motions; it also claims that velocities
have only a relative existence (dependent on the selection of the inertial
frame), while accelerations and rotations have an absolute existence (independent of the selection of the inertial frame). This state of affairs can be
expressed thus: According to classical mechanics "velocity relativity" exists,
but not "acceleration relativity". After these preliminary considerations we
can pass to the actual topic of our contemplations, the relativity theory, by
characterizing its development so far in terms of principles.//从上述内容中可以更清楚地得出“优选运动状态”的含义。它们是就自然规律而言的优选状态。运动状态之所以被认为是优选的，是因为相对于自然规律的表述来说，处于这些状态中的坐标系统有一个特性：在这些坐标系统中，这些规律呈现出以简洁为特点的优选形式。根据经典力学，惯性系的运动状态在这个意义上是物理上优选的。经典力学允许我们区分（绝对）无加速的运动和加速的运动；同时它还认为速度只是相对存在的（依赖于惯性系的选择），而加速度和旋转是绝对存在的（独立于惯性系的选择）。这一状况可以这样表达：根据经典力学，“速度的相对性”存在，但“加速度的相对性”不存在。在这些初步考虑之后，我们可以通过对其迄今为止的发展进行原则上的描述，进入我们真正的讨论主题——相对论。The special theory of relativity is an adaptation of physical principles to
Maxwell-Lorentz electrodynamics. From earlier physics it takes the assumption that Euclidian geometry is valid for the laws governing the position of
rigid bodies, the inertial frame and the law of inertia. The postulate of equivalence of inertial frames for the formulation of the laws of Nature is assumed
to be valid for the whole of physics (special relativity principle). From Maxwell-Lorentz electrodynamics it takes the postulate of invariance of the velocity of light in a vacuum (light principle). //狭义相对论是物理原则对麦克斯韦-洛伦兹电动力学的一种适应。它从早期物理学中继承了欧几里得几何学对于刚体位置、惯性系以及惯性定律的有效性假设。还假设惯性系的等效性在整个物理学中成立（即狭义相对性原理）。同时，从麦克斯韦-洛伦兹电动力学中，它采用了真空中光速不变性的假设（即光速不变原理）。To harmonize the relativity principle with the light principle, the assumption that an absolute time (agreeing for all inertial frames) exists, had to
be abandoned. Thus the hypothesis is abandoned that arbitrarily moved and
suitably set identical clocks function in such a way that the times shown by
two of them, which meet, agree. A specific time is assigned to each inertial
frame; the state of motion and the time of the inertial frame are defined, in accordance with the stipulation of meaning, by the requirement that the
light principle should apply to it. The existence of the inertial frame thus
defined and the validity of the law of inertia with respect to it are assumed.
The time for each inertial frame is measured by identical clocks that are stationary relative to the frame. //为了调和相对性原理与光速不变原理，必须放弃绝对时间（即所有惯性系都共享相同时间）的假设。因此，放弃了以下假设：无论如何移动或设置的相同的时钟，总是能够在它们相遇时显示相同的时间。每个惯性系都有特定的时间，根据意义规定，这个时间通过光速不变原理来定义惯性系的运动状态和时间。假设如此定义的惯性系存在，并且惯性定律在其上有效。对于每个惯性系的时间测量，是通过相对于该系静止的相同时钟进行的。The laws of transformation for space coordinates and time for the transition from one inertial frame to another, the Lorentz transformations as they
are termed, are unequivocally established by these definitions and the hypotheses concealed in the assumption that they are free from contradiction. Their
immediate physical significance lies in the effect of the motion relative to the
used inertial frame on the form of rigid bodies (Lorentz contraction) and on
the rate of the clocks. According to the special relativity principle the laws of
Nature must be covariant relative to Lorentz transformations; the theory
thus provides a criterion for general laws of Nature. It leads in particular to
a modification of the Newtonian point motion law in which the velocity of
light in a vacuum is considered the limiting velocity, and it also leads to the
realization that energy and inertial mass are of like nature.//根据这些定义和假设（假设它们不自相矛盾），空间坐标和时间从一个惯性系到另一个惯性系的变换法则，即所谓的洛伦兹变换，得到了明确的建立。这些变换的直接物理意义在于运动相对于所用惯性系对刚体形状的影响（洛伦兹收缩）以及对时钟速率的影响。根据特殊相对性原理，自然法则必须对洛伦兹变换具有协变性；因此，该理论为一般自然法则提供了一个标准。这尤其导致了牛顿点运动定律的修正，在这一修正中，真空中的光速被认为是极限速度，同时也导致了能量和惯性质量本质上的类似性。The special relativity theory resulted in appreciable advances. It reconciled
mechanics and electrodynamics. It reduced the number of logically independent hypotheses regarding the latter. It enforced the need for a clarification of the fundamental concepts in epistemological terms. It united the momentum and energy principle, and demonstrated the like nature of mass and
energy. Yet it was not entirely satisfactory - quite apart from the quantum
problems, which all theory so far has been incapable of really solving. In
common with classical mechanics the special relativity theory favours certain
states of motion - namely those of the inertial frames - to all other states of
motion. This was actually more difficult to tolerate than the preference for
a single state of motion as in the case of the theory of light with a stationary
ether, for this imagined a real reason for the preference, i.e. the light ether.
A theory which from the outset prefers no state of motion should appear more
satisfactory. Moreover the previously mentioned vagueness in the definition
of the inertial frame or in the formulation of the law of inertia raises doubts
which obtain their decisive importance, owing to the empirical principle for
the equality of the inertial and heavy mass, in the light of the following consideration.//特殊相对论带来了显著的进展。它调和了力学和电动力学，并减少了与后者相关的逻辑独立假设的数量。它加强了从认识论角度澄清基本概念的必要性。它统一了动量和能量原则，证明了质量和能量的本质相似。然而，它并不完全令人满意——不仅仅是因为量子问题，而所有的理论迄今为止都无法真正解决这一问题。与经典力学一样，特殊相对论偏向于某些运动状态，即惯性参考系，而不是所有其他运动状态。这实际上比光的理论中以静止以太为基础的单一运动状态的偏好更难以容忍，因为后者想象了一个真实的偏好理由，即光以太。一个从一开始就不偏好任何运动状态的理论应该显得更为令人满意。此外，惯性参考系定义的不确定性或惯性定律表述的不明确性引发了疑虑，这些疑虑由于关于惯性质量与重力质量相等的经验原则而变得更加重要，这一点在以下的考虑中尤为显著。Let K be an inertial frame without a gravitational field, K’ a system of coordinates accelerated uniformly relative to K. The behaviour of material
points relative to K’ is the the same as if K’ were an inertial frame in respect of which a homogeneous gravitational field exists. On the basis of the empirically known properties of the gravitational field, the definition of the
inertial frame thus proves to be weak. The conclusion is obvious that any
arbitrarily moved frame of reference is equivalent to any other for the formulation of the laws of Nature, that there are thus no physically preferred
states of motion at all in respect of regions of finite extension (general relativity principle). //设 K 为没有引力场的惯性参考系，K’ 为相对于 K 以均匀加速度运动的坐标系统。相对于 K’ 的物质点行为与 K’ 在存在均匀引力场的情况下作为惯性参考系的情况相同。因此，基于经验上已知的引力场特性，惯性参考系的定义显得较为薄弱。由此得出的结论很明显，任何任意移动的参考系在制定自然法则时与其他参考系是等效的，因此，在有限范围内并不存在物理上优先的运动状态（广义相对论原则）。The implementation of this concept necessitates an even more profound
modification of the geometric-kinematical principles than the special relativity theory. The Lorentz contraction, which is derived from the latter,
leads to the conclusion that with regard to a system K’ arbitrarily moved relative to a (gravity field free) inertial frame K, the laws of Euclidian geometry
governing the position of rigid (at rest relative to K’) bodies do not apply.
Consequently the Cartesian system of coordinates also loses its significance
in terms of the stipulation of meaning. Analogous reasoning applies to time;
with reference to K’ the time can no longer meaningfully be defined by the
indication on identical clocks at rest relative to K’, nor by the law governing
the propagation of light. Generalizing, we arrive at the conclusion that gravitational field and metric are only different manifestations of the same physical
field.//这一概念的实施需要比特殊相对论理论更深刻的几何运动学原则的修改。从特殊相对论中推导出的洛伦兹收缩得出结论：对于相对于（无引力场的）惯性参考系 K 任意移动的系统 K’，统治静止相对于 K’ 的刚体位置的欧几里得几何定律不再适用。因此，笛卡尔坐标系统在意义的规定方面也失去了其意义。类似的推理适用于时间；相对于 K’ 的时间不能再通过相对于 K’ 静止的相同钟表的指示或光的传播规律来有意义地定义。概括而言，我们得出结论，引力场和度量只是同一物理场的不同表现。We arrive at the formal description of this field by the following consideration. For each infinitesimal point-environment in an arbitrary gravitational field a local frame of coordinates can be given for such a state of motion that relative to this local frame no gravitational field exists (local inertial
frame). In terms of this inertial frame we may regard the results of the special
relativity theory as correct to a first approximation for this infinitesimally
small region. There are an infinite number of such local inertial frames at
any space-time point; they are associated by Lorentz transformations. These
latter are characterised in that they leave invariant the "distance" ds of two
infinitely adjacent point events - defined by the equation://通过以下考虑，我们得出了对这一场的形式描述。在任意引力场中的每一个无穷小点环境，都可以为这样的运动状态提供一个局部坐标系，使得相对于该局部坐标系不存在引力场（局部惯性参考系）。在这个惯性参考系中，我们可以认为特殊相对论的结果在这个无穷小的区域内是正确的。有无穷多个这样的局部惯性参考系位于任意时空点；它们通过洛伦兹变换相互关联。这些洛伦兹变换的特征在于，它们保持两个无穷邻近点事件之间的“距离” ds 不变，定义为以下方程：![]()
which distance can be measured by means of scales and clocks. For, x, y, z, t
represent coordinates and time measured with reference to a local inertial
frame. To describe space-time regions of finite extent arbitrary point coordinates
in four dimensions are required which serve no other purpose than to provide an unambiguous designation of the space-time points by four numbers
each, x1, x2, x3 and x4, which takes account of the continuity of this fourdimensional manifold (Gaussian coordinates). The mathematical expression
of the general relativity principle is then, that the systems of equations expressing the general laws of Nature are equal for all such systems of coordinates.//为了描述有限范围的时空区域，需要在四维中使用任意的点坐标，这些坐标的唯一作用是通过四个数字（x1, x2, x3 和 x4）为时空点提供明确的标识，这样做考虑到了这个四维流形的连续性（高斯坐标）。广义相对论原理的数学表达是，表示自然规律的一般方程组在所有这些坐标系中是相等的。Since the coordinate differentials of the local inertial frame are expressed
linearly by the differentials dxv of a Gaussian system of coordinates, when
the latter is used, for the distance ds of two events an expression of the form//由于局部惯性框架的坐标微分可以通过高斯坐标系的微分 dxv 线性表示，因此当使用高斯坐标系时，两个事件之间的距离 ds 可以表示为一种形式：![]()
is obtained. The g v which arc continuous functions of xv, determine the
metric in the four-dimensional manifold where ds is defined as an (absolute)
parameter measurable by means of rigid scales and clocks. These same parameters g v however also describe with reference to the Gaussian system of
coordinates the gravitational field which we have previously found to be
identical with the physical cause of the metric. The case as to the validity of
the special relativity theory for finite regions is characterised in that when
the system of coordinates is suitably chosen, the values of g v for finite regions
are independent of xv.//其中 g_v 是 x_v 的连续函数，确定了在四维流形中度量的定义，ds 作为一个（绝对的）参数，可以通过刚性尺子和时钟测量。然而，这些同样的参数 g_v 也描述了相对于高斯坐标系的引力场，而我们之前发现该引力场与度量的物理原因是相同的。在合适选择坐标系的情况下，有限区域的 g_v 值独立于 x_v，从而表明特殊相对论理论在有限区域内的有效性。In accordance with the general theory of relativity the law of point motion in the pure gravitational field is expressed by the equation for the geodetic line. Actually the geodetic line is the simplest mathematically which
in the special case of constant g v becomes rectilinear. Here therefore we
are confronted with the transfer of Galileo’s law of inertia to the general
theory of relativity. //根据广义相对论，纯引力场中的点运动定律由测地线方程表示。实际上，测地线在特殊情况下，若 g_v 为常数，则表现为直线。因此，我们在这里面临着将伽利略的惯性定律转移到广义相对论的问题。In mathematical terms the search for the field equations amounts to ascertaining the simplest generally covariant differential equations to which the
gravitational potentials g v can be subjected. By definition these equations
should not contain higher derivatives of g v with respect to xv than the second, and these only linearly, which condition reveals these equations to be a
logical transfer of the Poisson field equation of the Newtonian theory of gravity to the general theory of relativity.//在数学上，寻找场方程意味着确定可以应用于引力势 g_v 的最简单的广义协变微分方程。根据定义，这些方程不应包含关于 x_v 的 g_v 的高于二阶的导数，并且这些导数仅应线性出现，这一条件揭示出这些方程实际上是牛顿引力理论中泊松方程向广义相对论的逻辑转移。The considerations mentioned led to the theory of gravity which yields
the Newtonian theory as a first approximation and furthermore it yields the
motion of the perihelion of Mercury, the deflection of light by the sun, and
the red shift of spectral lines in agreement with experience.//上述考虑导致了一种重力理论，该理论在一阶近似上还原为牛顿理论，并且还成功地解释了水星近日点的运动、光在太阳引力作用下的偏折以及光谱线的红移，这些结果均与经验相符。* As regards the red shift, the agreement with experience is not yet completely assured,
however. To complete the basis of the general theory of relativity, the electromagnetic field must still be introduced into it which, according to our present conviction, is also the material from which we must build up the elementary structures of matter. The Maxwellian field equations can readily
be adopted into the general theory of relativity. This is a completely unambiguous adoption provided it is assumed that the equations contain no
differential quotients of g v higher than the first, and that in the customary
Maxwellian form they apply in the local inertial frame. It is also easily possible to supplement the gravitational field equations by electromagnetic
terms in a manner specified by the Maxwellian equations so that they contain the gravitational effect of the electromagnetic field. //为了完成广义相对论的基础，还必须将电磁场引入其中，根据我们当前的信念，电磁场也是构建物质基本结构的材料。麦克斯韦场方程可以很容易地被纳入广义相对论中。这是一个完全明确的采纳，前提是这些方程中不包含高于一阶的 g_v 的微分商，并且以通常的麦克斯韦形式适用于局部惯性框架。还可以按照麦克斯韦方程的规定，以电磁项补充引力场方程，从而使它们包含电磁场的引力效应。These field equations have not provided a theory of matter. To incorporate the field generating effect of ponderable masses in the theory, matter
had therefore (as in classical physics) to be introduced into the theory in an
approximate, phenomenological representation. //这些场方程并没有提供物质的理论。因此，为了将可称量质量的场产生效应纳入理论中，物质必须像在经典物理中那样以近似的现象学表现形式引入理论。And that exhausts the direct consequences of the relativity principle. I shall
turn to those problems which are related to the development which I have
traced. Already Newton recognized that the law of inertia is unsatisfactory
in a context so far unmentioned in this exposition, namely that it gives no
real cause for the special physical position of the states of motion of the inertial frames relative to all other states of motion. It makes the observable
material bodies responsible for the gravitational behaviour of a material
point, yet indicates no material cause for the inertial behaviour of the material point but devises the cause for it (absolute space or inertial ether). This
is not logically inadmissible although it is unsatisfactory. For this reason
E. Mach demanded a modification of the law of inertia in the sense that the
inertia should be interpreted as an acceleration resistance of the bodies against
one another and not against "space".//这就是相对性原理的直接结果所带来的所有结论。我将转向与我所描述的发展相关的问题。牛顿已经认识到惯性定律在本次论述中尚未提及的背景下是不令人满意的，特别是它并没有对惯性参考系相对于其他运动状态的特殊物理位置给出真实的原因。它使可观察的物质体对物质点的引力行为负责，但并未指出物质点惯性行为的物质原因，而是为其构想了一个原因（绝对空间或惯性以太）。虽然这种做法在逻辑上并不不可接受，但仍然令人不满意。因此，E. 马赫要求以一种方式修正惯性定律，即将惯性解释为物体之间的加速度抵抗，而不是对“空间”的抵抗。 This interpretation governs the expectation that accelerated bodies have concordant accelerating action in the same
sense on other bodies (acceleration induction).
This interpretation is even more plausible according to general relativity
which eliminates the distinction between inertial and gravitational effects.
It amounts to stipulating that, apart from the arbitrariness governed by the
free choice of coordinates, the g v -field shall be completely determined by
the matter. Mach’s stipulation is favoured in general relativity by the circumstance that acceleration induction in accordance with the gravitational field
equations really exists, although of such slight intensity that direct detection
by mechanical experiments is out of the question.//这种解释支配了这样的期望：加速的物体在同一意义上对其他物体具有一致的加速作用（加速度诱导）。根据广义相对论，这种解释甚至更为合理，因为它消除了惯性效应和引力效应之间的区别。这意味着，除了由自由选择坐标所支配的任意性外，g_v场应完全由物质决定。马赫的规定在广义相对论中得到了支持，因根据引力场方程确实存在加速度诱导，尽管这种强度极其微弱，以至于通过机械实验无法直接检测到。Mach’s stipulation can be accounted for in the general theory of relativity
by regarding the world in spatial terms as finite and self-contained. This hypothesis also makes it possible to assume the mean density of matter in the
world as finite, whereas in a spatially infinite (quasi-Euclidian) world it should
disappear. It cannot, however, be concealed that to satisfy Mach’s postulate
in the manner referred to a term with no experimental basis whatsoever
must be introduced into the field equations, which term logically is in no
way determined by the other terms in the equations. For this reason this
solution of the "cosmological problem" will not be completely satisfactory
for the time being. //在广义相对论中，马赫的规定可以通过将世界视为有限且自给自足的空间来解释。这一假设还使得可以假定世界中的物质平均密度是有限的，而在空间无限（类欧几里得）的世界中，这种密度应当消失。然而，不可否认的是，为了以所述方式满足马赫的假设，必须在场方程中引入一个完全没有实验基础的项，这个项在逻辑上并不由方程中的其他项决定。因此，针对“宇宙学问题”的这一解决方案在现阶段将不完全令人满意。A second problem which at present is the subject of lively interest is the
identity between the gravitational field and the electromagnetic field. The
mind striving after unification of the theory cannot be satisfied that two
fields should exist which, by their nature, are quite independent. A mathematically unified field theory is sought in which the gravitational field and
the electromagnetic field are interpreted only as different components or
manifestations of the same uniform field, the field equations where possible
no longer consisting of logically mutually independent summands. //目前受到广泛关注的第二个问题是引力场和电磁场之间的同一性。追求理论统一的思维无法接受存在两个在本质上完全独立的场。人们寻求一个数学上统一的场理论，在这个理论中，引力场和电磁场仅被解释为同一均匀场的不同组成部分或表现形式，尽可能使场方程不再由逻辑上相互独立的项构成。The gravitational theory, considered in terms of mathematical formalism,
i.e. Riemannian geometry, should be generalized so that it includes the laws
of the electromagnetic field. Unfortunately we are unable here to base ourselves on empirical facts as when deriving the gravitational theory (equality
of the inertial and heavy mass), but we are restricted to the criterion of mathematical simplicity which is not free from arbitrariness. The attempt which
at present appears the most successful is that, based on the ideas of LeviCivita, Weyl and Eddington, to replace Riemannian metric geometry by
the more general theory of affine correlation. //在数学形式上考虑的引力理论，即黎曼几何，应该进行推广，以使其包括电磁场的法则。不幸的是，我们在此无法像推导引力理论时那样以经验事实为基础（即惯性质量和重力质量的相等），而只能依赖于不免带有任意性的数学简约性标准。目前看起来最成功的尝试是基于利维-奇维塔、魏尔和爱丁顿的思想，将黎曼度量几何替换为更一般的仿射关联理论。The characteristic assumption of Riemannian geometry is the attribution
to two infinitely adjacent points of a "distance" ds, the square of which is a
homogeneous second order function of the coordinate differentials. It follows from this that (apart from certain conditions of reality) Euclidian geometry is valid in any infinitely small region. Hence to every line element
(or vector) at a point P is assigned a parallel and equal line element (or vector) through any given infinitesimally adjacent point P’ (affine correlation).
Riemannian metric determines an affine correlation. Conversely, however,
when an affine correlation (law of infinitesimal parallel displacement) is mathematically given, generally no Riemannian metric determination exists from
which it can be derived. //黎曼几何的特征假设是将两个无限邻近的点赋予一个“距离”ds，其平方是坐标微分的均匀二次函数。由此可以推导出（除了某些现实条件）在任何无限小的区域内欧几里得几何都是有效的。因此，在点P处的每一个线元素（或向量）都被赋予一个平行且相等的线元素（或向量），通过任何给定的无限邻近点P'（仿射关联）。黎曼度量确定了仿射关联。然而，当仿射关联（微小平行位移法则）在数学上给出时，通常不存在可以从中推导出的黎曼度量确定。 The most important concept of Riemannian geometry, "space curvature",
on which the gravitational equations are also based, is based exclusively on
the "affine correlation". If one is given in a continuum, without first proceeding from a metric, it constitutes a generalization of Riemannian geometry but which still retains the most important derived parameters. By
seeking the simplest differential equations which can be obeyed by an affine
correlation there is reason to hope that a generalization of the gravitation
equations will be found which includes the laws of the electromagnetic field.
This hope has in fact been fulfilled although I do not know whether the formal connection so derived can really be regarded as an enrichment of physics
as long as it does not yield any new physical connections. In particular a field
theory can, to my mind, only be satisfactory when it permits the elementary
electrical bodies to be represented as solutions free from singularities. //黎曼几何中最重要的概念是“空间曲率”，它也是引力方程的基础，这一概念完全基于“仿射相关性”。如果在一个连续体中给定了这种相关性，而没有首先从度量出发，它构成了黎曼几何的一个推广，但仍保留了最重要的导出参数。通过寻找仿射相关性可能满足的最简单微分方程，有理由希望找到一个包含电磁场定律的引力方程的推广。这一希望事实上已经实现，尽管我不知道这种形式的联系是否真的能被视为物理学的丰富性，尤其是在它没有带来任何新的物理联系的情况下。特别地，我认为，场论只有在它允许将基本电荷体作为无奇点的解来表示时才是令人满意的。Moreover it should not be forgotten that a theory relating to the elementary electrical structures is inseparable from the quantum theory problems.
So far also relativity theory has proved ineffectual in relation to this most
profound physical problem of the present time. Should the form of the general equations some day, by the solution of the quantum problem, undergo
a change however profound, even if there is a complete change in the parameters by means of which we represent the elementary process, the relativity
principle will not be relinquished and the laws previously derived therefrom
will at least retain their significance as limiting laws. ”//此外，不应忘记的是，与基本电荷结构相关的理论是与量子理论问题不可分割的。到目前为止，相对论在当今这个最深奥的物理问题上仍显得无效。不过，如果有一天通过解决量子问题，这些广义方程的形式发生了多么深刻的变化，即便是通过全新的参数来表示基本过程，仍然不会放弃相对性原理，并且从中导出的定律至少会作为极限定律保留其意义。——(Lecture delivered to the Nordic Assembly of Naturalists at Gothenburg* July 11, 1923)——“于1923年7月11日在哥德堡向北欧博物学家大会发表的讲座。”![]()
Albert Einstein delivering his Nobel Lecture to the Nordic Assembly of Naturalists in Gothenburg, Sweden, 11 July 1923.
Photo: Anders Wilhelm Karnell Source: Gothenburg Library Archive Public domain via Wikimedia Commons
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参考资料
1.Halmos, P.R. "The Legend of von Neumann", The American Mathematical Monthly, Vol. 80, No. 4. (April 1973), pp. 382–394
2.Corry, Leo. "The influence of David Hilbert and Hermann Minkowski on Einstein's views over the interrelation between physics and mathematics." Endeavour 22.3 (1998): 95-97.
3.Spekkens, Robert W. "The ontological identity of empirical indiscernibles: Leibniz's methodological principle and its significance in the work of Einstein." arXiv preprint arXiv:1909.04628 (2019).
4.Forman, Paul. “Einstein and Newton: Two Legacies.” The Wilson Quarterly (1976-), vol. 3, no. 1, 1979, pp. 107–14.
