The Influence of the Expansion of Space on the Gravitation Fields Surrounding the Individual Stars. WITH: Corrections and Additional Remarks to Our Paper . Offprints from: Reviews of Modern Physics, vols. 17, no. 2/3, April-June, 1945 & vol. 18, no. 1, January-March, 1946

EINSTEIN'S 'SWISS-CHEESE' MODEL OF THE UNIVERSE - OFFPRINT ISSUE. First edition, extremely rare offprints, of Einstein & Straus's introduction of the 'Swiss-cheese' model of the universe. "After [a] decade and a half of sometimes intense work on cosmology, Einstein returned to the subject only occasionally in his later years. His most significant later contribution was a discussion of the impact of cosmological expansion on the gravitational field surrounding a star [i.e., the offered papers] . This was an important first step in understanding the impact of global cosmological expansion on local physics" (Janssen & Lehner, pp. 257-8). In the 1920s and 1930s a general relativistic model of the universe was developed, called the Friedmann-Lemaître-Robertson-Walker (FLRW) model, which correctly described the expansion of the universe, discovered by Edwin Hubble. But the FLRW model was 'homogeneous' - it described a universe which looks the same wherever the observer is located. The actual universe, however, is manifestly inhomogeneous - it contains stars, galaxies, and clusters of galaxies. Einstein and Straus's papers represent the first serious attempt to model an inhomogeneous universe. "By the spring of 1945, Einstein and Straus had found a new type of possible universe using Einstein's equations. It described a universe which looked largely like one of the simple expanding universes of Friedmann and Lemaître containing material (like galaxies) which exerted no pressure. But it had spherical regions removed from it, like bubbles in a Swiss cheese. Each empty hole then had a mass placed at its centre. The mass was equal in magnitude to what had been excavated to create the hole. This was a step towards a more realistic universe in which the matter was not smoothly spread with the same density everywhere but gathered up into lumps, like galaxies, which were spread about in empty space" (Barrow, pp. 106-107). Not on OCLC; no copies in auction records. In 1916, just a few months after Einstein had formulated his general theory of relativity, Karl Schwarzschild had found a solution of Einstein's equations which described the gravitational field in the vicinity of a spherical distribution of mass, such as a star (or, to anticipate later developments, a black hole). This was, in fact, the first exact solution of Einstein's equations to be found - Einstein had earlier calculated an approximate solution which was enough to show that his theory correctly accounted for the advance of Mercury's perihelion. However, at great distances from the star, Schwarzschild's solution approaches the flat Minkowski spacetime, with zero curvature, and not the FLRW solution that represented an expanding universe. It seemed, therefore, that Schwarzschild's solution could not correctly describe the gravitational field of a star in an expanding FLRW universe. If the expansion of the universe meant that Schwarzschild's solution had to be modified, this could make it possible to detect and measure the expansion of the universe by making local observations, rather than by observing the motion of distant galaxies, as Hubble had done. The problem of describing the gravitational field of a star in an expanding FLRW universe was addressed by Einstein and Straus in the offered papers. "In the early 1930s theorists began to develop a richer account of the evolution of the universe based on expanding models. Hubble's results qualitatively agreed with the redshift effect calculated in these models, but the utility of the simple dynamical models depends on whether the universe is approximately uniform. The status of this assumption was the focus of lively debate . Relativistic cosmologists regarded the idealized uniformity of the FLRW models as a simplifying assumption . The unrelenting uniformity built into the FLRW models conflicts with the clear non-uniformity of the stars, star clusters, and galaxies of the local universe, but the models might still serve as a us.…