“We had become so accustomed to think of λ as an essentially positive quantity, and of a finite world with positive curvature, that the idea of investigating the possibility of solutions with negative or zero values of λ and of the curvature simply did not arise. But when this oversight was corrected, it appeared at once that in the non-static case both λ and the curvature need not be positive, but can be negative or zero quite as well.”

— Willem de Sitter

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

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Willem de Sitter 44

Dutch cosmologist 1872–1934

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“In February 1917, it was found that a static solution with a positive curvature—the solution A—was not possible without the λ. In fact the curvature is proportional to λ (in solution A, λ is equal to the curvature; in B… it is three times the curvature). Thus, at the time when we had only the two static solutions A and B, and thought that these were the only possible ones, here was a plausible physical interpretation of the meaning of λ: it was the curvature of the world, and the square root of its reciprocal, the radius of curvature, could be conceived as providing a natural unit of length.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

“If the curvature is small (as we know it must be, because it is imperceptible by ordinary geometric methods in our neighbourhood), then λ must be small, and if the curvature is very small, then λ must be very small. On the other hand, if λ is very small, or zero, then the curvature must be very small, and may even be zero.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

“In both the solutions A and B the curvature is positive, in both three-dimensional space is finite: the universe has a definite size, we can speak of its radius, and, in the case A, of its total mass. In the case A… the density is proportional to the curvature… Thus, if we wish to have a finite density in a static universe, we must have a finite positive curvature.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

“There is no direct observational evidence for the curvature [of space], the only directly observed data being the mean density and the expansion, which latter proves that the actual universe corresponds to the non-statical case. It is therefore clear that from the direct data of observation we can derive neither the sign nor that value of the curvature, and the question arises whether it is possible to represent the observed facts without introducing the curvature at all. Historically the term containing the 'cosmological constant λ' was introduced into the field equations in order to enable us to account theoretically for the existence of a finite mean density in a static universe. It now appears that in the dynamical case this end can be reached without the introduction of λ.”

Willem de Sitter (1872–1934) Dutch cosmologist

Joint memoir with Einstein (1932) as quoted by Gerald James Whitrow, The Structure of the Universe: An Introduction to Cosmology (1949)

“It sounds rather strange to talk of an infinite universe still expanding. If we were certain that the curvature was negative, we might still, as in the case of positive curvature, replace the phrase "the universe expands" by the equivalent one "the curvature of the universe decreases." But if the curvature is zero, and remains zero throughout, what sort of meaning are we to attach to the "expansion"? The real meaning is, of course, that the mutual distances between the galactic systems, measured in so-called natural measure, increase proportionally to a certain quantity R appearing in the equations, and varying with the time. The interpretation of R as the "radius of curvature" of the universe, though still possible if the universe has a curvature, evidently does not go down to the fundamental meaning of it.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

“Observations give us two data, viz. the rate of expansion and the average density, and there are three unknowns: the value of λ, the sign of the curvature, and the scale of the figure, i. e. the units of R and of the time. The problem is indeterminate.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

“To help us to understand three-dimensional spaces, two-dimensional analogies may be very useful… A two-dimensional space of zero curvature is a plane, say a sheet of paper. The two-dimensional space of positive curvature is a convex surface, such as the shell of an egg. It is bent away from the plane towards the same side in all directions. The curvature of the egg, however, is not constant: it is strongest at the small end. The surface of constant positive curvature is the sphere… The two-dimensional space of negative curvature is a surface that is convex in some directions and concave in others, such as the surface of a saddle or the middle part of an hour glass. Of these two-dimensional surfaces we can form a mental picture because we can view them from outside… But… a being… unable to leave the surface… could only decide of which kind his surface was by studying the properties of geometrical figures drawn on it. …On the sheet of paper the sum of the three angles of a triangle is equal to two right angles, on the egg, or the sphere, it is larger, on the saddle it is smaller. …The spaces of zero and negative curvature are infinite, that of positive curvature is finite. …the inhabitant of the two-dimensional surface could determine its curvature if he were able to study very large triangles or very long straight lines. If the curvature were so minute that the sum of the angles of the largest triangle that he could measure would… differ… by an amount too small to be appreciable… then he would be unable to determine the curvature, unless he had some means of communicating with somebody living in the third dimension…. our case with reference to three-dimensional space is exactly similar. …we must study very large triangles and rays of light coming from very great distances. Thus the decision must necessarily depend on astronomical observations.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932)

“In 1922, Friedmann… broke new ground by investigating non-static solutions to Einstein's field equations, in which the radius of curvature of space varies with time. This Possibility had already been envisaged, in a general sense, by Clifford in the eighties.”

Gerald James Whitrow (1912–2000) British mathematician

The Structure of the Universe: An Introduction to Cosmology (1949)
Context: The models of Einstein and de Sitter are static solutions of Einstein's modified gravitational equations for a world-wide homogeneous system. They both involve a positive cosmological constant λ, determining the curvature of space. If this constant is zero, we obtain a third model in classical infinite Euclidean space. This model is empty, the space-time being that of Special Relativity.
It has been shown that these are the only possible static world models based on Einstein's theory. In 1922, Friedmann... broke new ground by investigating non-static solutions to Einstein's field equations, in which the radius of curvature of space varies with time. This Possibility had already been envisaged, in a general sense, by Clifford in the eighties.<!--p.82

“The field equations, in their most general form, contain a term multiplied by a constant, which is denoted by the Greek letter λ… sometimes called the "cosmical constant." This is a name without any meaning… We have, in fact, not the slightest inkling of what its real significance is. It is put in the equations in order to give them the greatest possible degree of mathematical generality, but, so far as its mathematical function is concerned, it is entirely undetermined: it may be positive or negative, it might also be zero.”

Willem de Sitter (1872–1934) Dutch cosmologist

Kosmos (1932), Above is Beginning Quote of the Last Chapter: Relativity and Modern Theories of the Universe -->

“What appears as the positive is essentially the negative, i. e. the thing that is to be criticized.”

Theodor W. Adorno (1903–1969) German sociologist, philosopher and musicologist known for his critical theory of society

Source: Lectures on Negative Dialectics (1965-66), p. 18

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Willem de Sitter 44

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