Cosmologists

"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."

"De Sitter's redshift phenomenon is not caused by the Doppler effect of stars moving away. It is a property of space-time, which appears when these are forced into the bitter conditions of the empty universe with the lambda-term. ...de Sitter was the first to suggest, in 1917 when galaxies were not yet known, that one should try to find a redshift-distance relation for very remote celestial bodies. ...Even Friedmann, who five years later demonstrated the possibility of an expanding universe, failed to point out the redshift phenomenon as a property of his own model."

"The triangles that we can measure are not large enough... to detect the curvature. Fortunately, however, we are, in a way, able to communicate with the fourth dimension. The theory of relativity has given us an insight into the structure of the real universe: ...a four-dimensional structure. The study of the way in which the three space-dimensions are interwoven with the time-dimension affords a kind of outside point of view of the three-dimensional space... from this outside point of view we might be able to perceive the curvature of the three-dimensional world."

"Matter is actually distributed very unevenly... conglomerated into stars and galactic systems. The average density is the density that we should get if all... could be evaporated into atoms of hydrogen, or protons, and... distributed evenly over the whole of space. ...three or four protons in every cubic foot. ...a million million times less than that of the most perfect vacuum that we can produce... The universe thus consists mostly of emptiness... consider a universe without any matter at all, an empty universe, as a good approximation. But we may also take as our first approximation a universe containing... three or four protons per cubic foot. The local deviations from the average, caused by the conglomeration of matter into stars and stellar systems, are then disregarded in the grand scale model, and are only taken into account when we come to study details."

"In the beginning of 1917, two solutions of the field equations for a homogeneous isotropic universe had been found, which I... call the solutions "A" and "B." ...at that time only static solutions were looked for. It was thought that the universe must be a stable structure...In one of these solutions (B) the average density was zero, it was empty; the other one (A) had a finite density. ...In B, to get the real universe, we should have to put in a few galactic systems, in A we should have to condense the evenly distributed matter into galactic systems. The universe A... has an average density, but no expansion. It is therefore called the static universe. B, on the other hand... expands, and it could only parade in the garb of a static universe because there is nothing in it to show the expansion. B is therefore called the empty universe. Thus we had two approximations : the static universe with matter and without expansion, and the empty one without matter and with expansion.The actual universe... has both matter and expansion... In 1917... the actual value of the density was still entirely unknown, and the expansion had not yet been discovered."

"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."

"Purely mathematical symbols have no meaning by themselves; it is the privilege of pure mathematicians, to quote Bertrand Russell, not to know what they are talking about. ...It is the physicist, and not the mathematician, who must know what he is talking about."

"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."

"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."

"Since we only consider the universe on a very large scale, and make abstraction of all details and local irregularities, our universe must be homogeneous and isotropic. It follows... that the three-dimensional space of it must be what mathematicians call a space of constant curvature."

"In 1916 and 1917 de Sitter presented to the Royal Astronomical Society a series of three papers on "Einstein's Theory of Gravitation and its Astronomical Consequences." Because there was no communication between Germany and England during World War I, these papers were instrumental in introducing general relativity to the English scientific community, and they played an important part in the decision of Arthur Eddington and others to send expeditions to observe the solar eclipse of 1919..."

"There is another side to the theory of relativity. We have pointed out in the beginning how the development of science is in the direction to make it less subjective, to separate more and more in the observed facts that which belongs to the reality behind the phenomena, the absolute, from the subjective element, which is introduced by the observer, the relative. Einstein's theory is a great step in that direction. We can say that the theory of relativity is intended to remove entirely the relative and exhibit the pure absolute."

"The sequence of different positions of the same particle at different times forms a one-dimensional continuum in the four-dimensional space-time, which is called the world-line of the particle. All that physical experiments or observations can teach us refers to intersections of world-lines of different material particles, light-pulsations, etc., and how the course of the world-line is between these points of intersection is entirely irrelevant and outside the domain of physics. The system of intersecting world-lines can thus be twisted about at will, so long as no points of intersection are destroyed or created, and their order is not changed. It follows that the equations expressing the physical laws must be invariant for arbitrary transformations."

"All systems are receding, not from any particular centre, but from each other: the whole system of galactic systems is expanding."

"In Einstein's general theory of relativity the identity of these two coefficients, the gravitational and the inertial mass, is no longer a miracle, but a necessity, because gravitation and inertia are identical."

"Gravitation is entirely independent of everything that influences other natural phenomena. It is not subject to absorption or refraction, no velocity of propagation has been observed. You can do whatever you please with a body, you can electrify or magnetise it, you can heat it, melt or evaporate it, decompose it chemically, its behaviour with respect to gravitation is not affected. Gravitation acts on all bodies in the same way, everywhere and always we find it in the same rigorous and simple form, which frustrates all our attempts to penetrate into its internal mechanism."

"Gravitation is not only similar to inertia in its generality, it is also measured by the same number... the mass. The inertial mass is what Newton calls the "quantity of matter": it is a measure for the resistance offered by a body to a force trying to alter its state of motion. It might be called the "passive mass." The gravitational mass, on the other hand, is a measure of the force exerted by the body in attracting other bodies. We might call it the "active" mass. The equality of active and passive, or gravitational and inertial, mass was in Newton's system a most remarkable accidental co-incidence, something like a miracle. Newton himself decidedly felt it as such, and made experiments to verify it, by swinging a pendulum with a hollow bob which could be filled with different materials. The force acting on the pendulum is proportional to its gravitational mass, the inertia to its inertial mass: the period of its swing thus depends on the ratio between these two masses. The fact that the period is always the same therefore proves that the gravitational and inertial masses are equal."

"Gradually, during the eighteenth century, physicists and philosophers had become so accustomed to Newton's law of gravitation, and to the equality of gravitational and inertial mass, that the miraculousness of it was forgotten and only an acute mind like Bessel's perceived the necessity of repeating those experiments. By the experiments of Bessel about 1830 and of Eötvös in 1909 the equality of gravitational and inertial mass has become one of the best ascertained empirical facts in physics."

"Let the universe have only two dimensions, and let it be the surface of an india rubber ball. It is only the surface that is the universe, not the ball itself. ...Let there be specks of dust fixed to the surface to represent the different galactic systems. If the ball is inflated, the universe expands, and these specks of dust will recede from each other, their mutual distances, measured along the surface, will increase in the same rate as the radius of the ball. An observer in any one of the specks will see all the others receding from himself, but it does not follow that he is the centre of the universe. The universe (which is the surface of the ball, not the ball itself) has no centre."

"[Einstein's cosmological constant] is a name without any meaning. ...We have, in fact, not the slightest inkling of what it's real significance is. It is put in the equations in order to give the greatest possible degree of mathematical generality."

"In electromagnetism... the law of the inverse square had been supreme, but, as a consequence of the work of Faraday and Maxwell, it was superseded by the field. And the same change took place in the theory of gravitation. By and by the material particles, electrically charged bodies, and magnets which are the things that we actually observe come to be looked upon only as "singularities" in the field. So far this transformation from the force to the potential, from the action at a distance to the field, is only a purely mathematical operation."

"The inconsistency of first explaining matter by atoms and then explaining atoms by matter was only slowly realised, and it is only comparatively recently that we have come to see that there is nothing paradoxical in the fact that an atom or an electron, which are not matter, may have properties different from those of matter, and must be allowed to do things that a material particle could not do."

"It was early 1932, when Einstein and I both were at the California Institute of Technology in Pasedena, and we just decided to look for a simple relativistic model that agreed reasonably well with the known observational data, namely, the Hubble recession rate and the mean density of matter in the universe. So we took the space curvature to be zero and also the cosmological constant and the pressure term to be zero, and then it follows straightforwardly that the density is proportional to the square of the Hubble constant. It gives a value for the density that is high, but not impossibly high. That's about all there was to it. It was not an important paper, although Einstein apparently thought that it was. He was pleased to have a simple model with no cosmological constant. That's it."

"Hubble, who knew nothing of the work of Friedmann and Lamaître, had read de Sitter, whose sometimes comic absentmindedness masked a supple and inventive mind. Stimulated by one of Einstein's papers on general relativity penned in 1916, de Sitter entered into a fruitful correspondence with its author and soon produced three lengthy papers of his own on the subject. In the third article, published in 1917, he simplified his calculations by assuming that the universe is devoid of matter, a mathematical fiction that he defended on the basis that the real universe is composed mostly of space anyway. He then conjectured that if two stationary objects were introduced into this void, light passing between them with respect to one another would be redshifted. Curiously, the redshift is not due to either the expansion of intergalactic space or the Doppler effect. Instead, it is the effect of the mysterious slowing down of time at great distances. ...the amount of reshift in de Sitter's model was directly proportional to the distance between the emitting and receiving objects, a relationship that had only to be tested by someone with a telescope powerful enough..."

"These... are the two observational facts about our neighbourhood, which have to be accounted for by the theory: there is a finite density of matter, and there is expansion, i.e. the mutual distances are increasing, and therefore the density is decreasing."

"How does it come about that we have been able to find satisfactory hypotheses to explain electricity and magnetism, light and heat, in short all other physical phenomena, but have been unsuccessful in the case of gravitation?"

"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."

"If de Sitter's solution were valid everywhere, then it would be thereby shown that the purpose which I pursued with the introduction of the λ-term has not been reached. In my opinion the general theory of relativity only forms a satisfactory system if according to it the physical qualities of space are completely determined by matter alone. Hence no gμv-field must be possible, i.e., no space-time-continuum, without matter that generates it."

"In the "static" universe expansion is impossible, the "empty" universe does expand. Therefore we may be tempted to consider the empty universe as the most likely approximation; and we can proceed to compute the radius of curvature of the universe, supposing it to be of the empty type, from the observed rate of expansion."

"In 1917 de Sitter showed that Einstein's field equations could be solved by a model that was completely empty apart from the cosmological constant—i.e. a model with no matter whatsoever, just dark energy. This was the first model of an expanding universe. although this was unclear at the time. The whole principle of general relativity was to write equations for physics that were valid for all observers, independently of the coordinates used. But this means that the same solution can be written in various different ways... Thus de Sitter viewed his solution as static, but with a tendency for the rate of ticking clocks to depend on position. This phenomenon was already familiar in the form of gravitational time dilation... so it is understandable that the de Sitter effect was viewed in the same way. It took a while before it was proved (by Weyl, in 1923) that the prediction was of a redshifting of spectral lines that increased linearly with distance (i.e. Hubble's law). ..."

"If anything, de Sitter disliked Milne's program even more than Eddington's. In this dislike, he was soon joined by Herbert Dingle. ...de Sitter came to good knowledge of the genuine, actual, empirical basis of astronomical science. Secondly, reducing the observational data from the Jovian satellites practiced the young astronomer's mathematical skills in thorough and useful ways. Such a powerful combination of two skills—empirical, observational skill plus theoretical, mathematical skill—was lacking in the colleagues with whom de Sitter would later practice cosmology—with the interesting exception of Dingle."

"The theorists of the time were still debating whether the universe obeyed Einstein's static model, in which no red-shift is seen, or a model by Willem de Sitter in which the universe is expanding, and distant objects are red-shifted, but the universe contains no matter. ...Few astronomers were aware of the more general expanding universe solutions found by Georges Lemaître, and, especially, Alexander Friedmann, who had found in 1922-24 a complete set of solutions for Einstein's equations for the cosmological problem. Following Hubble's paper it was quickly realized that the expanding universe solutions were the ones that were needed to understand his observations."

"In "Kosmos", de Sitter shows how steps forward in the development of science can be associated with scientists, who considered the Universe from a non-static, dynamic point of view. Charles Darwin is cited as one of the examples. ...From early on, de Sitter was actively engaged in the debate between Einstein, LeMaître, Eddington and others in the significance of the General Theory of Relativity for cosmology. De Sitter found in his preferred solutions of Einstein's field equations that the Universe was expanding and the relative velocity V of recession was expected to be in proportion to the distance r. De Sitter was searching for observational evidence of such a relative recession and turned to Kapteyn for advice on the feasibility..."

"De Sitter pointed out that some... pairs of stars have orbits aligned so that each star is either coming towards us or going away from us as it goes round its orbit. If the speed of light is modified by the motion of the star then 'faster' light from positions when the star is approaching the Earth could catch up with 'slower' light emitted earlier when the star was moving away from us in its orbit. There would then arise the possibility of multiple ghost images. Such ghost images have not been observed... de Sitter concluded that Einstein was right... it has been argued that one needs to be careful in accepting de Sitter's binary star argument... because of a phenomenon called 'extinction.' ...To avoid this problem, the binary experiment has... been repeated with stars emitting X-rays... experiment conclusively confirmed Einstein's postulate."

"De Sitter made observations in 1913 of double-star systems and became the first astronomer to prove that the velocity of light had nothing to do with the velocity of its source. ...De Sitter ...published three papers entitled "On Einstein's Theory of Gravitation and Its Astronomical Consequences. The first two... explained Einstein's theories, and the third proposed his own interpretation of the astronomical consequences... de Sitter and Einstein published a joint paper in which they proposed there may be in the universe large amounts of matter that does not emit light and consequently has not been detected. This became known as "dark matter"..."

"The outstanding feature... is the possibility that the velocity-distance relation may represent the de Sitter effect, and hence that numerical data may be introduced into discussions of the general curvature of space. In the de Sitter cosmology, displacement of the spectra arises from two sources, an apparent slowing down of atomic vibrations and a general tendency of material particles to scatter. The latter involves an acceleration and hence introduces the element of time. The relative importance of these two effects should determine the form of the relation between distances and velocities; and in this connection it may be emphasized that the linear relation found in the present discussion is a first approximation representing a restricted range in distance."

"He [de Sitter] reminds us that in Einstein's paper of November 1915 the Λ term did not appear at all. Hence, Λ had obviously the value zero. Cosmologically speaking, this implies that the curvature of the Universe is proportional to Λ. In 1917, two solutions of Einstein's field equations for a homogeneous, isotropic universe were offered. ...in A the universe has a finite density, whereas in B, the average density is zero ...We are confronted with a static unverse containing matter and having no expansion on the one hand, and on the other an empty universe void of matter and expanding... when these two solutions were constructed, the expansion of the universe had not yet been discovered."

"The manner in which time and space are bound up with each other in the four-dimensional continuum is variable. It is difficult to express this variability of the cross-connections between space and time in simple language, and different interpretations of it are possible, corresponding to different mathematical transformations of the fundamental line-element, e.g. a different choice of the variable which we interpret as "time." Perhaps the best way we can express it is by saying that the solution of the field-equations of the theory of relativity shows that there is in the universe a tendency to change its scale, which at the present time results in an expansion, but may perhaps at other times become, or have been, a shrinking. This is true of the grand scale model of the universe."

"If we put in the details, the singularities of the field, viz. the galactic systems and the stars, we find that there is... a tendency, called gravitation, to decrease the mutual distances of these "singularities." At short distances, within the confines of a galactic system, this second tendency is by far the strongest, and the galactic systems retain their size independent of the expansion or contraction of the universe..."

"From 1916 Einstein and de Sitter corresponded extensively on exactly what kind of universe best fit the relativity equations. De Sitter initially developed a model of a spherical universe, in contrast to the cylindrical one Einstein had envisioned. De Sitter also tried to map out the shape of the spherical universe in absence of all matter. Einstein's reaction to de Sitter's model was strong and negative...de Sitter's sphere described a universe that changed in size instead of remaining nicely constant. ... Einstein saw matter—and its corresponding gravitational field—as what inherently created the shape of the universe. He cited what he dubbed "Mach's principle,"...the movements of any object ...were determined by all other bodies in the universe. ...how a body moves through space is tantamount to what shape space is, the concept of "shape" without matter, Einstein insisted, was meaningless."

"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."

"Shortly after Einstein published his original memoir on cosmology in 1917, de Sitter constructed an alternative static world-model, which satisfied the same laws of world-gravitation. In this model, unlike Einstein's, space-time has an intrinsic structure of its own, independent of the presence of matter. ...there is ...no matter nor radiation. Nebulae... must therefore be considered as 'test particles,' having no influence on the model as a whole. ...whereas a test particle in Einstein's universe will remain at rest if it has no intitial motion, a similar particle... in de Sitter's world will immediately acquire an ever-increasing velocity of recession from the observer. ...in de Sitter's model space-time is 'hyperbolic'. There is no absolute time, and each observer will perceive a horizon at which time will appear to stand still... This phenomenon... is only apparent, like a rainbow. At any point on the (relative) horizon the time-flux experienced by an observer there will be the same as at the original observer. Thus in de Sitter's world there will be an apparent slowing-down of distant atomic vibrations, if these keep standard time. Consequently the radiation from a distant nebula will appear to be shifted toward the red, due to an increase in wave length corresponding to the decrease in vibrational frequency. This effect... will be supplemented by the Doppler effect, due to the relative recession of the nebula regarded as a test particle."

"In 1913, Willem de Sitter suggested that fast-moving binary stars... could be used to measure the effect of a moving source on the speed of light. Various experiments of this sort over the past eight decades have verified that the speed of light received from a moving star is the same as that from a stationary star... a wealth of other detailed experiments have been carried out during the past century..."

"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."

"De Sitter proposed three types of nonstatic universes: the oscillating universes and the expanding universes of the first or second kiind. The main characteristic of the expanding "family" of the first kiind is that the radius is continually increasing from a definite initial time when it had the value zero. The universe becomes infinitely large after an infinite time. In the second kind... the radius possesses at the initial time a definite minimum value... in the Einstein model... the cosmological constant is supposed to be equal to the reciprocal of R2, whereas de Sitter computed for his interpretation the constant to be equal to 3/R2. Whitrow correctly points out the significant fact that in special relativity the cosmological constant is omitted..."

"It is easy to compute the density of a static universe of a radius of two thousand million lightyears, and it comes out only very little larger than the observed density. The actual universe is thus very far from empty, it is, on the contrary, nearly full."

"We... come to the conclusion... that the actual universe is neither the static nor the empty one. It differs so much from both of these that neither can be used as an appropriate grand scale model. We must thus look for other solutions of the general field-equations. On account of the expansion our solution must necessarily be a non-static one, and it must have a finite density. There is only one possible static solution possessing a finite density, viz. our old friend A, but of non-static solutions with finite density there exists a great variety."

"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."

"Is the density anywhere near that corresponding to the static universe, or is it so small that we can consider the empty universe as a good approximation?"

"Our own galaxy system is only one of a great many, and observations made from any of the others would show exactly the same thing: all systems are receding, not from any particular centre, but from each other: the whole system of galaxies is expanding."