In the period covered by this volume, Einstein's life and career entered a new phase, which we characterize as the Berlin years. His appointment as salaried member of the Prussian Academy of Sciences placed him at the focus of the scientific community of his time-a major change from his professorship at the ETH in more peripheral Zurich. This new appointment also had major consequences for his personal life: during the Berlin years Einstein became more and more of a public figure, whose opinion on nonscientific issues was sought with increasing frequency. Indeed, this first volume of the Berlin years already contains some nonscientific items, two relating to the First World War (Docs. 8 and 20), and a brief statement on the deleterious effect of the traditional final secondary school exam in the German school system (Einstein 1917h [Doc. 49]). The early months of the Berlin years also mark the culmination of an unhappy period in Einstein's private life: the estrangement from his wife Mileva, which had started in Zurich, finally led to a separation. After only a few months spent with Einstein in Berlin, Mileva and the two boys, Hans Albert and Eduard, returned to Zurich. During the war years Einstein's relationship with his cousin Elsa Löwenthal, whose presence there had helped attract him to Berlin, continued and strengthened. After his divorce from Mileva in February 1919 he married Elsa in June of the same year.

The first Berlin years saw the conclusion of a task that had occupied much of Einstein's time and energy since his return from Prague to Zurich in the late summer of 1912: the work on a theory of gravitation. With the publication in November 1915 of Einstein 1915f, 1915g, and 1915i (Docs. 21, 22, and 25), in which generally covariant field equations were derived, the theory was completed. Its successful explanation of the observed anomaly in the motion of the perihelion of Mercury in Einstein 1915h (Doc. 24) gave the theory crucial empirical support. Although Einstein continued to work out further consequences of the theory, and wrote a major review paper (Einstein 1916e [Doc. 30]) as well as a popular exposition (Einstein 1917a [Doc. 42]), he now had time once again to pursue other interests, returning, in particular, to quantum theory. His two papers on the emission and absorption of radiation in quantum theory (Einstein 1916j and Einstein 1916n [Docs. 34 and 38]) represent a major step forward in the field. Other features of this volume are the papers resulting from Einstein's collaboration with the Dutch physicist Wander Johannes de Haas on an experimental investigation of the existence of Ampère's molecular currents, as well as his first appearance as a technical expert in a patent dispute. The volume concludes with two appendixes: the first one summarizes student notes for two lecture courses given by Einstein at the University of Berlin (in 1916/1917 and 1917/1918, respectively), supplementing his own lecture notes for a course on relativity from 1914/1915, which are presented here as Doc. 6. The second appendix presents notes by an auditor of part of a set of lectures on relativity given by Einstein in Göttingen in early summer 1915.

When Einstein left Zurich for Berlin in March 1914, general relativity was in a less than satisfactory state. In the year that had passed since the publication of Einstein and Grossmann's "Entwurf" theory (see Einstein and Grossmann 1913 [Vol. 4, Doc. 13]), not much progress had been made. In particular the lack of general covariance was still a problematic feature of the theory. Although it appears that Einstein was by now convinced of the impossibility and even undesirability of a generally covariant theory-his "hole argument" had been instrumental in this respect-it was still unclear to what extent the theory was covariant. This question was the topic of a final collaborative effort of Einstein and Grossmann that led to the publication of Einstein and Grossmann 1914b (Doc. 2). Taking a Hamiltonian formalism of their theory as a starting point, they tried to establish more precisely than in their previous paper which transformations were allowed by the theory. Introducing the terminology of "adapted coordinate systems" and "justified transformations" for the frames in which the theory was valid and for the transformations connecting them, they succeeded in deriving a condition which all justified transformations had to fulfill. They also claimed without explicit proof that the allowed transformations included accelerations.

In a lengthy and complicated paper published in November 1914 (Einstein 1914o [Doc. 9]), the theory is developed in a systematic way, starting with a detailed exposition of tensor calculus. The derivation of the field equations is preceded by a new version of the "hole argument" to show that the equations describing the processes in a gravitational field "cannot possibly be generally covariant." As in Einstein and Grossmann 1914b (Doc. 2), a variational principle is the starting point for the derivation of the field equations, but in contrast to the earlier paper, in which the form of the Hamiltonian was specified from the outset, Einstein now formulates general conditions from which the explicit form of the Hamiltonian is then derived. It turns out that the Hamiltonian obtained in this way gives rise to the "Entwurf" field equations. As Einstein concludes: "We now have arrived at very definite field equations in a purely formal way, i.e., without drawing directly on our physical knowledge of gravitation."

A year later Einstein had changed his mind completely. The story of how he struggled with the theory during this year until he came to the conclusion that it had to be abandoned has been told many times and will only be recapitulated here. Not much contemporary material is available to reconstruct Einstein's thinking during that year, and we have to rely to a large extent on Einstein's own statements made on various later occasions. From the available documents it appears that by the fall of 1915 Einstein had become aware of three major flaws in the theory: the argument used in Einstein 1914o (Doc. 9) to fix the Hamiltonian was wrong; the theory was not covariant for rotations; and the perihelion motion of Mercury came out too small. For all three points some background can be given.

The first problem, the realization that Einstein's method for determining the form of the Hamiltonian in fact did no such thing, perhaps has its roots in a discussion with the Italian mathematician Tullio Levi-Civita on an essential point in Einstein's derivation of the field equations in Einstein 1914o (Doc. 9). The second point is connected with Einstein's conviction that the justified transformations should include rotations because the theory had to incorporate Mach's principle, understood as the relativity of rotation and the determination of inertia by distant masses. His Machian views had in fact guided him in his search for generally covariant field equations. A letter Einstein wrote to Erwin Freundlich sheds some interesting light on the way Einstein came to the conclusion that rotations were not included in the allowed transformations. The third problem, the theory's failure to account for the motion of the perihelion of Mercury, can be connected to the calculations Einstein had already done earlier in collaboration with his friend Michele Besso on this motion of Mercury on the basis of the "Entwurf" theory. These calculations, previously unknown and presented as Doc. 14 in Vol. 4, lead to a result that is too small by a factor of about two.

Once Einstein had realized the seriousness of the three flaws, he decided to make a fresh start: "For these reasons I completely lost confidence in the field equations drawn up by me and looked for a way to limit the possibilities in a natural manner." He returned to general covariance and in rapid succession published the papers that contained the essentials of the final theory. Initial errors, made in the first two papers concerning the form of the field equations and the trace of the energy-momentum tensor, were corrected in the third and final paper. Interestingly, none of the papers contains any reference to the "hole argument" which had played such a crucial role in Einstein's earlier thinking. It is only through Einstein's correspondence that we know that he came to reject it when he realized that not the metric field, but only the totality of space-time coincidences has physical meaning.

Even before the last of the three papers of November 1915 had been published, Einstein used his generally covariant field equations to calculate the perihelion motion of Mercury, arriving at the result of 43 seconds of arc per century, in very satisfactory agreement with observations. When he saw the result, Einstein later told his former collaborator Adriaan Fokker, he was so excited that he had heart palpitations.

A few months after the final papers had appeared, Einstein was ready to spend some time writing a review paper in which the whole theory was presented and explained in a consistent and accessible way. The paper appeared in the Annalen der Physik (Einstein 1916e [Doc. 30]), but was also widely sold as a separate booklet (Einstein 1916f). It gives an excellent overview of the theory. In the same year Einstein also completed a book-length popular exposition of both the special and the general theory, Einstein 1917a (Doc. 42). It was an instant success and remains a classic to this day. The new theory also gave rise to other elaborations and consequences: a paper on the Hamiltonian formulation of the theory (Einstein 1916o [Doc. 41]), an earlier manuscript version of which is presented in this volume as Doc. 31, and a paper on gravitational waves (Einstein 1916g [Doc. 32]), which had to be retracted in 1918 because of a serious error.

In the spring of 1917 Einstein published his first paper on cosmology, Einstein 1917b (Doc. 43). It may be said to mark the birth of modern cosmology. Einstein's interest in cosmology derived from his conviction, already mentioned above, that a theory of gravitation should include in some way or another Mach's principle. That Einstein turned to cosmology so soon after the completion of general relativity can also be understood from the fact that cosmology is an integral part of general relativity, in the sense that the geometric structure of the universe is not given a priori, as in Newtonian cosmology, but must fit into the framework of the general theory. Cosmological considerations appear already in Einstein 1916e (Doc. 30). Indeed, the argument given in the introductory part of this paper concerning two spheres, one of which is rotating, touches directly on Mach's principle. The physical cause needed to explain the difference in shape between the rotating sphere and the non-rotating one is found in the presence of distant masses; making absolute space responsible, as Newton did, is "a purely fictitious cause, not something observable." According to Einstein, empty space cannot have a geometrical structure, and a single isolated mass cannot have inertia or impose a structure on space at infinity.

It was precisely this conviction that Einstein took as the starting point for his paper on cosmology (Einstein 1917a [Doc. 43]). The consequence of Einstein's version of Mach's principle is that at infinity the components of the metric tensor should degenerate: for an isotropic field the spatial components become zero, whereas the timelike component goes to infinity. It turned out to be impossible to realize these conditions for centrally symmetric static fields. Einstein's way out was to postulate a universe that is spatially finite, closed, and static, with a uniform mass distribution, a universe in which no boundary conditions are needed. In order to do so, however, Einstein had to modify his field equations to include what became known as the "cosmological constant." In this way Einstein had incorporated the Machian ideas as well as he could, without, however, completely solving the problem of relativity of rotation.

The paper marks the beginning of a discussion between Einstein and the Dutch astronomer Willem de Sitter on the relativity of rotation and the relativity of inertia, carried out in correspondence as well as in published papers. A major topic of discussion was De Sitter's own cosmological solution, which showed that an empty universe can exhibit global curvature. Although the existence of such a solution was a serious blow to Einstein's ideas on the relativity of inertia, he did not abandon his views on cosmology. Neither did he change his mind when first Friedmann and later Lemaitre found nonstatic cosmological solutions. It would take until 1931 before Einstein finally accepted the nonstatic character of the universe and rejected his cosmological constant as unnecessary and compromising the simplicity of his field equations.

Two documents in this volume, Docs. 12 and 19, are of a special character. They are opinions drawn up by Einstein as an expert witness in a court case involving a patent dispute. In the fall of 1914 Einstein became involved in a conflict between the German firm Anschütz & Co. and the American Sperry Gyroscope Company. The issue was the design of a gyrocompass. The episode resulted in a close friendship between Einstein and the owner of the German firm, Hermann Anschütz-Kaempfe. The gyrocompass goes back to the work of Léon Foucault in the nineteenth century on the stability of orientation of the axis of rotation of a spinning top with respect to the revolving earth. When the use of iron and the proliferation of electrical apparatus on board ships made the use of a magnetic compass more and more problematic, the gyrocompass made its appearance as an attractive alternative. The Dutchman Martinus Gerardus van den Bos obtained a patent on an early form of a gyrocompass in 1885, but his invention never worked properly. In 1903 Hermann Anschütz-Kaempfe designed a gyrocompass to be used on an expedition by submarine to the North Pole (which never took place). He obtained a patent on the design, and with support from the German Navy a factory was established at Kiel in 1905.

In the USA the inventor Elmer Ambrose Sperry had also developed a gyrocompass. Anschütz and Sperry competed in a market that was potentially very profitable because of the intensive rearmament in the years preceding the First World War. When Sperry sold a compass to the German Navy in May 1914, Anschütz decided to sue Sperry for patent infringement before the Königliches Landgericht I in Berlin. Sperry's defense was based on the claim that the Anschütz patent did not add anything to the old patent of Van den Bos (in the meantime purchased by Siemens and Halske) and was therefore void. Another point under discussion in the case was the claim that Sperry used a method of damping that had been patented by Anschütz.

The court proposed the appointment of an expert on whom both parties could agree. Einstein was appointed and made his first appearance in court on 5 January 1915. He had apparently not prepared himself well, because in his presentation he contradicted himself several times. The court then asked him to answer a number of questions in a written report, for which he was to be paid 1,000 marks. Einstein's report is presented as Doc. 12. In it he states that the Anschütz patent did not improve on the Van den Bos patent. Anschütz disagreed and the court was not convinced either. After a further session on 26 March 1915, at which Einstein was not present, he was asked personally to inspect the Sperry compass and write a further report on the differences between the compasses of Van den Bos, Anschütz, and Sperry. Einstein decided to perform a number of tests on the Sperry compass, which took place in Kiel on 10 July 1915. In his second report of 7 August 1915 (Doc. 19) Einstein corrects his earlier conclusion and states that the horizontal stabilization of the Anschütz compass does constitute an improvement over the compass of Van den Bos. He also concludes that Sperry's method of damping is the one described in Anschütz's earlier patent. The court decision of 16 November 1915 followed Einstein's report and prohibited Sperry from manufacturing and selling gyrocompasses that used the method patented by Anschütz.

Einstein wrote only a few papers on the quantum theory during this period, but several of these are of major importance. The only unpublished manuscript in this group, Doc. 26, deals with the theoretical calculations of the entropy constant of an ideal gas made independently by Otto Sackur and Hugo Tetrode several years earlier. Einstein lectured on this subject in February 1916, and the manuscript is related to that lecture. His goal was to bring out the fundamental aspects of the ways in which Sackur and Tetrode had used the concepts of the quantum theory, in order to gain more insight into the subject "without offering anything substantially new."

Three of the four published papers on the quantum theory in this volume are concerned, at least in part, with a recurrent theme in Einstein's work: what conditions and presuppositions are needed to derive Planck's radiation law? This is the primary subject of Einstein 1914n (Doc. 5), one of the first papers Einstein wrote in Berlin. Here he avoids the use of Boltzmann's principle relating entropy and probability and derives the Planck distribution by purely thermodynamic arguments with the help of the fundamental idea of quantum theory. His argument is particularly interesting for the way it brings out the basic similarity between physical and chemical changes when these are both considered from the point of view provided by the quantum theory. In the second part of this paper Einstein returns to another topic he had been thinking about for several years and had discussed on previous occasions: the validity of Nernst's heat theorem. During the summer of 1916, less than a year after he had completed the general theory of relativity, Einstein made a new, major contribution to the quantum theory. The two papers he wrote then, Einstein 1916j (Doc. 34) and Einstein 1916n (Doc. 38), deal with the quantum theory of radiation by arguments that do not depend on the classical electromagnetic theory, as had all earlier treatments of Planck's radiation law. The new theory also led to a startling conclusion: radiation emitted or absorbed by atoms in a radiation field has a specific direction, and does not consist of spherical waves.

In the first of these papers, Einstein 1916j (Doc. 34), Einstein considers a system of atoms in equilibrium with an external radiation field. An atom can change its internal energy state by absorbing or emitting radiation. Einstein introduces three basic assumptions about these exchanges of energy between matter and field. First, the probability of absorption of radiation is proportional to the radiation density. Second, there are two kinds of emission processes: one-spontaneous-following a law like that of radioactive decay; the other-stimulated-induced by the radiation field and with probability proportional to the radiation density. Third, at equilibrium the atoms are distributed among their internal states according to the Boltzmann distribution law. From these assumptions Planck's law follows in a simple way. Einstein was very pleased with his derivation, which he characterized in a letter to Besso: "An amazingly simple derivation of Planck's formula, I should like to say the derivation." As a bonus from his derivation Einstein found that the energy difference between two internal energy states of the atom had to be equal to hn, with n the frequency of the radiation absorbed or emitted in transitions between these two states, thus confirming one of the postulates of Niels Bohr's theory of spectra.

In Einstein 1916n (Doc. 38) the derivation of the first paper is repeated and augmented by a discussion of the momentum transfer that accompanies emission and absorption processes. Einstein points out that in addition to energy, momentum is transferred between atom and field if the radiation exchanged is directed; if energy is radiated in the form of spherical waves, there is no momentum transfer. He then makes the fundamental and nonclassical assumption that an amount of momentum hn/c is transferred in all instances of emission or absorption of radiation by atoms in a radiation field. He explores the quantitative consequences of this assumption, using a technique that he had already successfully applied to quantum phenomena on various earlier occasions: the technique of calculating fluctuations, in this case the momentum fluctuations experienced by a particle moving in a radiation field. For the actual calculation Einstein draws on methods developed earlier in his joint work with Ludwig Hopf. The final result is an equilibrium condition for a system of atoms and radiation which takes the form of an equation for the radiation density. It is satisfied by Planck's law but not by the Rayleigh-Jeans radiation law. This is interpreted as a strong confirmation that all radiation emitted by atoms, induced as well as spontaneous, is indeed directed: "There is no radiation in spherical waves." This very important conclusion constitutes a major step toward the photon concept. As Einstein put it: "With this, the existence of light quanta is practically assured."

The final paper on quantum theory in this volume, Einstein 1917d (Doc. 45), deals with a totally different topic: the Bohr-Sommerfeld quantum condition for periodic systems. This condition can only be put in its usual form if a separation of coordinates is possible; in all other cases the condition has the general form . In his paper, Einstein discusses a possible way to avoid the need of a separation of variables. He gives the general form of the condition a coordinate-independent meaning by interpreting it as a line integral in (coordinate) phase space along some closed contour. If phase space is structured in such a way that not all closed curves can be contracted to a single point, i.e., if the motion of the system is restricted to some invariant subspace, there must exist a finite number of topologically independent contours. For multiple periodic systems the integral has a finite value for these contours, each of which corresponds to a separate quantum condition. If there is no periodicity, however, quantization according to this method is not possible. In spite of its general and novel approach, Einstein's paper was ignored by most. One notable exception is Louis de Broglie, who used Einstein's phase space approach in 1924 in his dissertation. Only many decades later was it seen in retrospect that Einstein's approach foreshadowed the use of the concept of invariant tori in phase space in the analysis of integrable dynamical systems.