## 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.

Lancaster, PA: American Physical Society, 1945-46.

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 useful approximation if the non-uniformities are negligible at larger scales” (Janssen & Lehner, p. 256).

“By 1944, Einstein had recruited a new assistant at Princeton. His assistants were always talented young mathematicians who could make up for Einstein’s self-confessed weakness in this area. Ernst Straus (1922-1983) was something of a mathematical prodigy … He was born in Munich but after the Nazis came to power in 1933 his family fled to Palestine, where he was educated at high school and at the Hebrew University in Jerusalem. Straus didn’t stay to take an undergraduate degree and instead, while still a teenager, moved to New York’s Columbia University in 1941 to begin graduate research. In 1944, he found himself recruited as Einstein’s new research assistant at the Institute for Advanced Study in Princeton. The young Straus had no background in physics and his mathematical inclinations were towards number theory and ‘pure’ mathematical topics but he lost no time in filling the gap left by the departures of Nathan Rosen (1935-45) and Leopold Infeld (1936-38)” (Barrow, pp. 105-6).

Einstein and Straus found an exact solution of the equations of general relativity in which a spherical ‘hole’ is cut out of an FLRW universe, and the hole is replaced by a single mass point (e.g., a star) surrounded by a spherical cavity. The initially homogeneous matter within the cavity can be thought of as having been “condensed into the star”. Einstein and Straus found that the interior of the cavity is described by the standard Schwarzschild solution. The radius of the hole is such that at its spherical boundary the outward pull from the cosmological masses is just balanced by the inward pull from the star. The cavity boundary expands according to the Hubble expansion of the whole universe. The universe outside the cavity is described by the standard expanding, homogeneous FLRW solution. The possibility of exactly matching the Schwarzschild solution near the star to the FLRW solution outside it showed that it was not, in fact, possible to detect the expansion of the universe by making observations close to the star. There were similar solutions with more than one hole – Einstein said that this reminded him of the holes in Swiss cheese. The static vacuum region inside the cavity is now called an ‘Einstein-Straus vacuole’.

The methods introduced by Einstein and Straus in these papers have been used extensively to model inhomogeneities in the universe. For example, in 1968 Martin Rees and Dennis Sciama investigated the effects of large-scale inhomogeneities (such as superclusters of galaxies) on the cosmic microwave background (the so-called ‘Rees-Sciama effect’). The Swiss-cheese model has also been used in the study of inflationary models of the early universe. According to this theory, the universe expanded exponentially in the first tiny fraction of a second after the big bang, with some parts of space-time expanding more quickly than others. This created ‘bubbles’ in space-time. The Swiss-cheese “embodies a natural way to model physical problems, such as describing the boundary between a galaxy and intergalactic space, or the relation between bubbles at the end of an inflationary era, by taking two different regions where the behaviour is smooth and joining them at a surface of discontinuity” (Ellis et al., p. 426).

In the last decade, several authors have suggested that Swiss-cheese models might solve a long-standing problem on the rate of expansion of the universe. Distant supernovas have been observed to be dimmer than expected on the basis of standard cosmological theories, indicating that the universe is expanding faster than these theories predict. This has been explained by hypothesizing the existence of ‘dark energy’, although exactly what dark energy might be is a mystery. But if the light from distant supernovas had to cross large vacuoles in reaching an observer on the earth, these would act like concave lenses making the supernovas appear dimmer and further away than they really are. Other authors have noted that the Milky Way is near the centre of a region that has fewer galaxies than other parts of the universe, and that we might be living near the centre of a particularly large vacuole perhaps more than a billion light years in diameter (see, for example, Bonnor).

Weil 216. Barrow, *The Book of Universes*, 2011. Bonnor, ‘A generalisation of the Einstein-Straus vacuole,’ *Classical and Quantum Gravity* 17 (2000), pp. 2739-2748. Ellis, Maartens & MacCallum, *Relativistic Cosmology*, 2012. Janssen & Lehner (eds.), *The Cambridge Companion to Einstein*, 2014.

Together two offprints, 4to (267 x 200 mm), pp. 120-124 & 148-149. Stapled as issued in original self-wrappers.

Item #5054

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Price:
$4,500.00
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