Expansion of the universe

 is the increase in distance between gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion, so it does not mean that the universe expands "into" anything or that space exists "outside" it. To any observer in the universe, it appears that all but the nearest galaxies (which are bound to each other by gravity) recede at speeds that are proportional to their distance from the observer, on average. While objects cannot move faster than light, this limitation applies only with respect to local reference frames and does not limit the recession rates of cosmologically distant objects.Cosmic expansion is a key feature of Big Bang cosmology. It can be modeled mathematically with the (FLRW), where it corresponds to an increase in the scale of the spatial part of the universe's spacetime metric tensor (which governs the size and geometry of spacetime). Within this framework, the separation of objects over time is associated with the expansion of space itself. However, this is not a generally covariant description but rather only a choice of coordinates. Contrary to common misconception, it is equally valid to adopt a description in which space does not expand and objects simply move apart while under the influence of their mutual gravity. Although cosmic expansion is often framed as a consequence of general relativity, it is also predicted by Newtonian gravity.According to inflation theory, during the about 10−32 of a second after the Big Bang, the universe suddenly expanded, and its volume increased by a factor of at least 1078 (an expansion of distance by a factor of at least 1026 in each of the three dimensions). This would be equivalent to expanding an object 1 nanometer(10−9 m, about half the width of a molecule of DNA) across to one approximately 10.6 s across (about 1017 m, or 62 trillion miles). Cosmic expansion subsequently decelerated to much slower rates, until around 9.8 billion years after the Big Bang (4 billion years ago) it began to gradually expand more quickly, and is still doing so. Physicists have postulated the existence of, appearing as a cosmological constant in the simplest gravitational models, as a way to explain this late-time. According to the simplest of the currently favored cosmological model, the, this acceleration becomes dominant in the future.

Quotes

 * This circumstance of an expanding universe is irritating. ...To admit such possibilities seems senseless to me.
 * Albert Einstein, as quoted by Robert Jastrow, "Have Astronomers" The New York Times (June 25, 1978)


 * If a distant galaxy is moving relative to us, its entire is Doppler-shifted in frequency. Its s are displaced relative to those of stationary light sources. Thanks to this effect, we know that distant galaxies recede from the solar system at speeds proportional to their distances from us. That's the effect that told us of the expanding universe, and of its birth, long ago, in the Big Bang.
 * Sheldon Lee Glashow, From Alchemy to Quarks (1994)


 * All kinds of questions remain. Many have to do with cosmology. How did the universe originate? How did the galaxies become distributed in space like the suds in the kitchen sink..? Why is the cosmological constant apparently very tiny but non-zero and has a peculiar value that leads the universe to expand more rapidly?
 * Sheldon Lee Glashow, The Elegant Universe (2003) NOVA Interview

Not speaking of quasars, the first indications for non-Doppler redshifts for a galaxy have been provided... What if not all galaxies were formed at the dawn of the Big Bang; what if some are being formed now? Then, at least, the can be anything larger than the age of our own Galaxy...
 * All of this picture of the expansion is exciting, pleasant, coherent, well in order. But what if the s are not to be interpreted by the Doppler-Fizeau law in the classical mechanical view, or general relativistically, by the fact that the ratio of the of a photon (as measured by a co-moving observer) to the space radius of curvature is independent of ?
 * Jean Heidmann, "The Expansion of the Universe in the Frame of Conventional General Relativity" Cosmology, History and Theology (1977) Ch. 5, ed., Wolfgang Yourgrau, Allen D. Breck Based on the 3rd unternational colloquium, University of Denver Nov 5-8, 1974.


 * Red-shifts are produced either in the nebulae, where the light originates, or in the intervening space through which the light travels. If the source is in the nebulae, then red-shifts are probably velocity-shifts and the nebulae are receding. If the source lies in the intervening space, the explanation of red-shifts is unknown, but the nebulae are sensibly stationary.
 * Edwin Hubble, as quoted by Gerald James Whitrow, The Structure and Evolution of the Universe: an Introduction to Cosmology (1959)


 * A book, too, can be a star, explosive material, capable of stirring up fresh life endlessly, a living fire to lighten the darkness, leading out into the expanding universe.
 * Madeleine L'Engle, "The Expanding Universe" (August, 1963) Newbery Award Acceptance Speech republished in A Wrinkle in Time: 50th Anniversary Commemorative Edition (2012).


 * The definition of inflation is extraordinarily simple: it is any period of the Universe's evolution during which the scale factor, describing the size of the Universe, is accelerating. This leads to a very rapid expansion of the Universe, though perhaps a better way of thinking of this is that the characteristic scale of the Universe, given by the Hubble length, is shrinking relative to any fixed scale caught up in the rapid expansion. In that sense, inflation is actually akin to zooming in on a small part of the initial Universe.


 * One of the few authors to have explicitly connected the physical issue of the expansion of the universe with the philosophical topic of the metaphysical status of space is Gerald James Whitrow.
 * , "Philosophy of Space and Expanding Universe in G. J. Whitrow" (2014) Abstract


 * 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 . 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 ... 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. ). ...
 * , Philosophy and the Sciences for Everyone (2014)

This description of the substratum holds good in the scale of time in which the galaxies or fundamental particles are receding from one another with uniform velocities. This choice of the scale of time, together with the theory of equivalent time-keepers... makes possible the application of the Lorentz formulae to the private Euclidean spaces of the various observers. It thus brings the theory of the expanding universe into line with other branches of physics, which use the Lorentz formulæ and adopt Euclidean private spaces. ...[T]here is no more need to require a curvature for space itself in the field of cosmology than in any other department of physics. The observer at the origin is fully entitled to select a private Euclidean space in which to describe phenomena, and when he concedes a similar right to every other equivalent observer and imposes the condition of the same world-view of each observer, he is inevitably led to the model of the substratum which we have discussed.
 * This model of the expanding universe I shall call the substratum. It achieves in the private Euclidean space of each fundamental observer the objects for which Einstein developed his closed spherical space. Although it is finite in volume, in the measures of any chosen observer, it has all the properties of an infinite space in that its boundary is forever inaccessible and its contents comprise an infinity of members. It is also homogeneous in the sense that each member stands in the same relation to the rest.
 * , 15. "Gravitation Without General Relativity" Albert Einstein: Philosopher-Scientist (1949) ed., Paul Arthur Schilpp, Vol. 2.


 * The ideas that prove to be of lasting interest are likely to build on the framework of the now standard world picture, the hot big bang model of the expanding universe. The full extent and richness of this picture is not as well understood as I think it ought to be, even among those making some of the most stimulating contributions to the flow of ideas.
 * Phillip James Edwin Peebles, Principles of Physical Cosmology (1993) Preface.


 * We should, of course, expect that any universe which expands without limit will approach the empty de Sitter case, and that its ultimate fate is a state in which each physical unit—perhaps each nebula or intimate group of nebulae—is the only thing which exists within its own observable universe.
 * Howard P. Robertson, as quoted by Gerald James Whitrow, The Structure of the Universe: An Introduction to Cosmology (1949)


 * If the general picture, however, of a Big Bang followed by an expanding Universe is correct, what happened before that? Was the Universe devoid of all matter and then the matter suddenly somehow created? How did that happen? In many cultures, the customary answer is that a God or Gods created the Universe out of nothing. But if we wish to pursue this question courageously, we must of course ask the next question: where did God come from? If we decide that this is an unanswerable question, why not save a step and conclude that the origin of the Universe is an unanswerable question? Or, if we say that God always existed, why not save a step, and conclude that the Universe always existed? That there's no need for a creation, it was always here. These are not easy questions. Cosmology brings us face to face with the deepest mysteries, questions that were once treated only in religion and myth.
 * Carl Sagan, Cosmos: A Personal Voyage (1980), Ep. 10: "The Edge of Forever"


 * The most far-reaching implication of general relativity... is that the universe is not static, as in the orthodox view, but is dynamic, either contracting or expanding. Einstein, as visionary as he was, balked at the idea... One reason... was that, if the universe is currently expanding, then... it must have started from a single point. All space and time would have to be bound up in that "point," an infinitely dense, infinitely small "singularity." ...this struck Einstein as absurd. He therefore tried to sidestep the logic of his equations, and modified them by adding... a "cosmological constant." The term represented a force, of unknown nature, that would counteract the gravitational attraction of the mass of the universe. That is, the two forces would cancel... it is the kind of rabbit-out-of-the-hat idea that most scientists would label ad-hoc. ...Ironically, Einstein's approach contained a foolishly simple mistake: His universe would not be stable... like a pencil balanced on its point.
 * George Smoot, Keay Davidson, Wrinkles in Time (1993)


 * The cosmological constant['s]... most important consequence: the repulsive force, acting at cosmological distances, causes space to expand exponentially. There is nothing new about the universe expanding, but without a cosmological constant, the rate of expansion would gradually slow down. Indeed, it could even reverse itself and begin to contract, eventually imploding in a giant cosmic crunch. Instead, as a consequence of the cosmological constant, the universe appears to be doubling in size about every fifteen billion years, and all indications are that it will do so indefinitely.
 * Leonard Susskind, The Black Hole War: My Battle with Stephen Hawking to make the World Safe for Quantum Mechanics (2008)


 * 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...
 * Wolfgang Yourgrau, "On Some Cosmological Theories and Constants" Cosmology, History, and Theology (2012)


 * [W]e stress... the wide range of validity exhibited by s in theoretical physics. ...[I]t has ...been demonstrated how they can be employed to derive equations of optics, dynamics of particles and rigid bodies, and electromagnetism. In addition, physicists have succeeded in formulating the laws of elasticity and hydrodynamics as variational principles, and even Einstein's law of gravitation was included in this category by Hilbert, who found a scaler function... for which $$\partial\int\mathfrak{h}\,dx_0\,dx_1\,dx_2\,dx_3=0$$ is equivalent to Einstein's law. This function has been called the "curvature," an identification which induced Whittaker to describe Hilbert's principle in the laconic words, "gravitation simply represents a continual effort of the universe to straighten itself out."
 * Wolfgang Yourgrau, Variational Principles in Dynamics and Quantum Theory (1979) pp. 92-93.

"On Relativistic Cosmology" (1928)

 * by Howard P. Robertson, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science (1928) Series 7, Vol. 5, Issue 31: Supplement


 * The general theory of relativity considers physical space-time as a four-dimensional manifold whose line element coefficients $$g_{\mu \nu}$$ satisfy the differential equations $$G_{\mu \nu} = \lambda g_{\mu \nu} \qquad .\;.\;.\;.\;.\;.\; (1)$$ in all regions free from matter and electromagnetic field, where $$G_{\mu \nu}$$ is the contracted Riemann-Christoffel tensor associated with the fundamental tensor $$g_{\mu \nu}$$, and $$\lambda$$ is the.


 * An "empty world," i.e., a homogeneous manifold at all points at which equations (1) are satisfied, has, according to the theory, a constant Riemann curvature, and any deviation from this fundamental solution is to be directly attributed to the influence of matter or energy.


 * In considerations involving the nature of the world as a whole the irregularities caused by the aggregation of matter into stars and stellar systems may be ignored; and if we further assume that the total matter in the world has but little effect on its macroscopic properties, we may consider them as being determined by the solution of an empty world.


 * The solution of (1), which represents a homogeneous manifold, may be written in the form: $$ds^2 = \frac{d\rho^2}{1 - \kappa^2\rho^2} - \rho^2 (d\theta^2 + sin^2 \theta \; d\phi^2) + (1 - \kappa^2 \rho^2)\; c^2 d\tau^2, \qquad (2)$$ where $$\kappa = \sqrt \frac{\lambda}{3}$$. If we consider $$\rho$$ as determining distance from the origin... and $$\tau$$ as measuring the proper-time of a clock at the origin, we are led to the de Sitter spherical world...

"On the relation between the expansion and the mean density of the universe" (Mar 15, 1932)

 * by Albert Einstein, Willem de Sitter, Proceedings of the National Academy of Sciences Vol. 18, No. 3, pp. 213–214.

There is no direct observational evidence for the curvature, 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" &lambda; 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 &lambda;.
 * O. Heckmann has pointed out that the non-static solutions of the field equations of the general theory of relativity with constant density do not necessarily imply a positive curvature of three-dimensional space, but that this curvature may also be negative or zero.


 * The determination of the coefficient of expansion h depends on the measured red-shifts, which do not introduce any appreciable uncertainty, and the distances of the extra-galactic nebulae, which are still very uncertain. The density depends on the assumed masses of these nebulae and on the scale of distance, and involves, moreover, the assumption that all the material mass in the universe is concentrated in the nebulae. It does not seem probable that this latter assumption will introduce any appreciable factor of uncertainty.


 * Although... the density... corresponding to the assumption of zero curvature and to the coefficient of expansion... may perhaps be on the high side, it... is of the correct order of magnitude, and we must conclude that... it is possible to represent the facts without assuming a curvature of three-dimensional space. The curvature is, however, essentially determinable, and an increase in the precision of the data derived from observations will enable us in the future to fix its sign and to determine its value.

The Expanding Universe (1933)

 * by Arthur Eddington


 * Why should not the space be there already, and the material system expand into it..? ...[I]f the speed of recession continues to increase outwards, it will ere long approach the speed of light, so that something must break down. The result is that the system becomes a ... such a system cannot expand without the space also expanding. ...[E]xpansion of space has often been given too much prominence ...and readers have been led to think that it is more directly concerned in the explanation of the motions of the nebule than is... the case. ...If we adopt open space we encounter certain difficulties (not necessarily insuperable) which closed space entirely avoids; and we do not want... speculation as to the solution of difficulties which need never arise. If we wish to be noncommittal, we shall naturally work in terms of a closed universe of finite radius R, since we can at any time revert to an infinite universe by making R infinite.


 * The immediate results of introducing the cosmical term into the law of gravitation was the appearance... of two universes—the Einstein universe and the de Sitter universe. Both were closed spherical universes; so that a traveller going on and on in the same direction would at last find himself back at the starting-point... Both claimed to be static universes... thus they provided a permanent framework within which the small-scale systems—galaxies and stars—could change and evolve. ...[H]owever ...in de Sitter's universe there would be an apparent recession of remote objects ...At that time only three radial velocities were known, and these ...lamely supported de Sitter ...2 to 1. ...But in 1922 ...V. M. Sipher furnished me ...measures of 40 spiral nebulæ for ...my book Mathematical Theory of Relativity. ...[T]he majority had become 36 to 4 ...


 * The situation has been summed up in the statement that Einstein’s universe contains matter but no motion and de Sitter’s contains motion but no matter. ...[T]he actual universe containing both matter and motion does not correspond exactly to either... Which is the better choice for a first approximation? Shall we put a little motion into Einstein’s world of inert matter, or... a little matter into de Sitter’s ?


 * The choice between Einstein’s and de Sitter’s models... [W]e are not now restricted to these... extremes; we have... the whole chain of intermediate solutions between motionless matter and matterless motion... [W]e can pick... the right proportion of matter and motion to correspond with what we observe. ...[E]arlier... it was the preconceived idea that a static solution was a necessity... an unchanging background of space. ...[T]his ...should strictly have barred... de Sitter’s solution, but ...it was the precursor of the other non-static solutions...


 * [I]nvestigation of non-static solutions was carried out by A. Friedmann in 1922. His solutions were rediscovered in 1927 by Abbé G. Lemaître, who brilliantly developed the astronomical theory... and... remained unknown until 1930... In the meantime the solutions had been discovered... by H. P. Robertson, and through him... interest was... realised. The astronomical application, stimulated by Hubble and Humason’s observational work on the spiral nebule, was also being rediscovered, but it had not been carried so far as in Lemaître’s paper.


 * The intermediate solutions of Friedmann and Lemaitre are "expanding universes." Both the material system and the closed space, in which it exists, are expanding. At one end we have Einstein’s universe with no motion and therefore in equilibrium. Then... we have model universes showing more and more rapid expansion until we reach de Sitter’s... The rate of expansion increases all the way along the series and the density diminishes; de Sitter’s universe is the limit when the average density of celestial matter approaches zero. The series of expanding universes then stops... but because there is nothing left to expand.

As the matter drew closer... in the condensations... evolutionary processes followed—evolution of stars... of... more complex elements... of planets and life.
 * [T]he most satisfying theory would be one which made the beginning not too unæsthetically abrupt. This... can only be satisfied by an Einstein universe with all... major forces balanced. Accordingly, the primordial state of things... is an even distribution of s and electrons, extremely diffuse and filling all (spherical) space, remaining nearly balanced for an exceedingly long time until its inherent instability prevails. ...[T]he density of this distribution can be calculated ...[at] about one proton and electron per litre. ...[S]mall irregular tendencies accumulate, and evolution gets under way. ...[T]he formation of condensations ultimately ...become the galaxies; this ...started off an expansion, which ...automatically increased in speed until ...now manifested ...in the recession of the spiral nebulae.

The result... only the intergalactic distances expand. The galaxies... are unaffected... —s, stars, human observers and their apparatus, atoms—are entirely free from expansion. Although the cosmical repulsion or expansive tendency is present in all of these... it is checked by much larger forces... [T]he demarcation between permanent and dispersing systems is... abrupt. It corresponds to the distinction between periodic and aperiodic phenomena.
 * Within the galaxy the average world-curvature is... thousands of times greater than Lamaître's average for the universe... his formulæ are inapplicable.


 * If you think... the shattering of the bubble universe is... tragic... [W]hen the worst has happened our galaxy... will be left intact. ...not so bad a prospect.


 * All change is relative. The universe is expanding relatively to our common material standards; our material standards are shrinking relatively to the size of the universe. The theory of the "expanding universe" might also be called the theory of the "shrinking atom". ...[T]ake the... universe as our standard of constancy... he sees us shrinking... only the intergalactic spaces remain the same. The earth spirals round the sun in an ever‑decreasing orbit. ...Our years will ...decrease in geometrical progression in the cosmic scale of time. ... Owing to the property of geometrical progressions an infinite number of our years will add up to a finite cosmic time; so that what we should call the end of eternity is an ordinary finite date in the cosmic calendar. But on that date the universe has expanded to infinity in our reckoning, and we have shrunk to nothing in the reckoning of the cosmic being. ...When the last act opens the curtain rises on midget actors rushing through their parts at frantic speed. Smaller and smaller. Faster and faster. One last microscopic blurr of intense agitation. And then nothing.

New pathways in science (1935)

 * by Arthur Eddington, (, 1934)

So in starting to theorise about the expanding universe I am not taking it for granted that the observational evidence which we have been considering is entirely certain.
 * If the astronomers are right, it is a straightforward conclusion from the observational measurements that the system of galaxies is expanding—or, since the system of the galaxies is all we know—that the universe is expanding. There is no subtlety or metaphysics about it ...But are we sure of the observational facts? Scientific men are rather fond of saying pontifically that one ought to be quite sure of one's observational facts before embarking on theory. Fortunately those who give this advice do not practice what they preach. Observation and theory get on best when they are mixed together, both helping one another in the pursuit of truth. It is a good rule not to put overmuch confidence in a theory until it has been confirmed by observation. I hope I shall not shock the experimental physicists too much if I add that it is also a good rule not to put overmuch confidence in the observational results that are put forward until they have been confirmed by theory.


 * It is scarcely true... that we observe these velocities of recession. We observe a shift of the spectrum to the red; and although such... is usually due to recession... it is not inconceivable that it should arise from another cause.


 * [I]t was theory that first suggested a systematic recession of the spiral nebulae and so led to a search for this effect. The theoretical possibility was first discovered by W. de Sitter in 1917. Only three radial velocities were known at that time, and they... lamely supported his theory by... 2 to 1. Since then... support is far more unanimous... mainly due to V. M. Slipher... and M. L. Humason... The linear law of proportionality between speed and distance was found by E. H. Hubble. Meanwhile the theory has also developed, and... taken the form... associated with... A. Friedman and G. Lemaître.


 * The theory of relativity predicts... a... force... we call the cosmical repulsion... directly proportional to the distance... It is so weak... we can leave it out of... motions of the planets... or any motion within... our... galaxy. ...[S]ince it increases... to the distance we... if we go far enough, find it significant.


 * I have said the repulsion is proportional to the distance... Distance from what? From anywhere you like. ...Cosmical repulsion is a dispersing force tending to make a system expand uniformly—not diverging from any centre in particular, but such that all internal distances increase at the same rate. That corresponds precisely to the kind of expansion we observe in the system of the galaxies.


 * I have said that relativity theory predicts a force of cosmical repulsion. ...[R]elativity theory does not talk of anything so crude as force; it describes... curvature of space-time. But for practical purposes... nearly equivalent to the Newtonian force of gravitation... [T]he actual relativity effect is represented with sufficient accuracy by a force of cosmical repulsion... up to the greatest distances... we... observe.


 * The galaxies exert on one another their ordinary gravitational attraction approximately according to Newton's law. This makes them tend to cling together. So we... have a contest of two forces, Newtonian attraction... and cosmical repulsion... If our theory is right cosmical repulsion must have got the upper hand... Having got the advantage, cosmical repulsion will keep it; because, as the nebulae become further apart, their mutual attraction will become weaker...

Geometry as a Branch of Physics (1949)

 * by Howard P. Robertson, included in Albert Einstein: Philosopher-Scientist, ed..


 * is a congruence geometry, or equivalently the space comprising its elements is homogeneous and isotropic; the intrinsic relations between... elements of a configuration are unaffected by the position or orientation of the configuration. ...[M]otions of are the familiar translations and rotations... made in proving the theorems of Euclid.


 * [O]nly in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained.


 * That the existence of these motions (the "axiom of free mobility") is a desideratum, if not... a necessity, for a geometry applicable to physical space, has been forcefully argued on a priori grounds by von Helmholtz, Whitehead, Russell and others; for only in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained.


 * Euclidean geometry is only one of several congruence geometries... Each of these geometries is characterized by a real number $$K$$, which for Euclidean geometry is 0, for the hyperbolic negative, and for the spherical and elliptic geometries, positive. In the case of 2-dimensional congruence spaces... $$K$$ may be interpreted as the  of the surface into the third dimension—whence it derives its name...


 * [W]e propose... to deal exclusively with properties intrinsic to the space... measured within the space itself... in terms of... inner properties.

$$A = \pi r^2 (1 - \frac{Kr^2}{12} + ...)$$
 * Measurements which may be made on the surface of the earth... is an example of a 2-dimensional congruence space of positive curvature $$K = \frac{1}{R^2}$$... [C]onsider... a "small circle" of radius $$r$$ (measured on the surface!)... its perimeter $$L$$ and area $$A$$... are clearly less than the corresponding measures $$2\pi r$$ and $$\pi r^2$$... in the Euclidean plane. ...for sufficiently small $$r$$ (i.e., small compared with $$R$$) these quantities on the sphere are given by 1): $$L = 2 \pi r (1 - \frac{Kr^2}{6} + ...)$$,


 * In the sum $$\sigma$$ of the three angles of a triangle (whose sides are arcs of s) is greater than two right angles [180&deg;]; it can... be shown that this "spherical excess" is given by 2) $$\sigma - \pi = K \delta$$ where $$\delta$$ is the area of the spherical triangle and the angles are measured in s (in which 180&deg; = $$\pi$$ [radians]). Further, each full line (great circle) is of finite length $$2 \pi R$$, and any two full lines meet in two points—there are no parallels!


 * [T]he space constant $$K$$... "" may in principle at least be determined by measurement on the surface, without recourse to its embodiment in a higher dimensional space.


 * These formulae [in (1) and (2) above] may be shown to be valid for a circle or a triangle in the hyperbolic plane... for which $$K < 0$$. Accordingly here the perimeter and area of a circle are greater, and the sum of the three angles of a triangle are less, than the corresponding quantities in the Euclidean plane. It can also be shown that each full line is of infinite length, that through a given point outside a given line an infinity of full lines may be drawn which do not meet the given line (the two lines bounding the family are said to be "parallel" to the given line), and that two full lines which meet do so in but one point.

$$V = \frac{4}{3} \pi r^3 (1 - \frac{Kr^2}{5} + ...)$$.
 * The value of the intrinsic approach is especially apparent in considering 3-dimensional congruence spaces... The intrinsic geometry of such a space of curvature $$K$$ provides formulae for the surface area $$S$$ and the volume $$V$$ of a "small sphere" of radius $$r$$, whose leading terms are 3) $$S = 4 \pi r^2 (1 - \frac{Kr^2}{3} + ...)$$,


 * In all these congruence geometries, except the Euclidean, there is at hand a natural unit of length $$R = \frac{1}{K^\frac{1}{2}}$$; this length we shall, without prejudice, call the "radius of curvature" of the space.


 * We have merely (!) to measure the volume $$V$$ of a sphere of radius $$r$$ or the sum $$\sigma$$ of the angles of a triangle of measured are $$\delta$$, and from the results to compute the value of $$K$$.


 * What is needed is a homely experiment which could be carried out in the basement with parts from an old sewing machine and an Ingersoll watch, with an old file of Popular Mechanics standing by for reference! This I am, alas, afraid we have not achieved, but I do believe that the following example... is adequate to expose the principles...


 * Let a thin, flat metal plate be heated... so that the temperature T is not uniform... clamp or otherwise constrain the plate to keep it from buckling... [and] remain [reasonably] flat... Make simple geometric measurements... with a short metal rule, which has a certain coefficient of expansion c... What is the geometry of the plate as revealed by the results of those measurements? ...[T]he geometry will not turn out to be Euclidean, for the rule will expand more in the hotter regions... [T]he plate will seem to have a negative curvature $$K$$... the kind of structure exhibited... in the neighborhood of a "."


 * What is the true geometry of the plate? ...Anyone examining the situation will prefer Poincaré's common-sense solution... to attribute it Euclidean geometry, and to consider the measured deviations... as due to the actions of a force (thermal stresses in the rule). ...On employing a brass rule in place of one of steel we would find that the local curvature is trebled—and an ideal rule (c = 0) would... lead to Euclidean geometry.


 * In what respect... does the general theory of relativity differ...? The answer is: in its universality; the force of gravitation in the geometrical structure acts equally on all matter. There is here a close analogy between the gravitational mass M...(Sun) and the inertial mass m... (Earth) on the one hand, and the heat conduction k of the field (plate)... and the coefficient of expansion c... on the other. ...The success of the general relativity theory... is attributable to the fact that the gravitational and inertial masses of any body are... rigorously proportional for all matter.


 * The field equation may... be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean curvature of the space vanish at any point at which no heat is being applied to the medium—in complete analogy with... the general theory of relativity by which classical field equations are replaced by the requirement that the Ricci contracted curvature tensor vanish.
 * Footnote


 * Now it is the practice of astronomers to assume that brightness falls off inversely with the square of the "distance" of an object—as it would do in Euclidean space, if there were no absorption... We must therefore examine the relation between this astronomer's "distance" $$d$$... and the distance $$r$$ which appears as an element of the geometry.

$$r = d (1 + \frac{K d^2}{6} + ...).$$
 * All the light which is radiated... will, after it has traveled a distance $$r$$, lie on the surface of a sphere whose area $$S$$ is given by the first of the formulae (3). And since the practical procedure... in determining $$d$$ is equivalent to assuming that all this light lies on the surface of a Euclidean sphere of radius $$d$$, it follows... $$4 \pi d^2 = S = 4 \pi r^2 (1 - \frac{K r^2}{3} + ...);$$ whence, to our approximation 4) $$d = r (1- \frac{K r^2}{6} + ...),$$ or


 * [T]he astronomical data give the number N of nebulae counted out to a given inferred "distance" $$d$$, and in order to determine the curvature... we must express N, or equivalently $$V$$, to which it is assumed proportional, in terms of $$d$$. ...from the second of formulae (3) and... (4)... to the approximation here adopted, 5) $$V = \frac{4}{3} \pi d^2 (1 + \frac{3}{10} K d^2 + ...);$$ ...plotting N against... $$d$$ and comparing... with the formula (5), it should be possible operationally to determine the "curvature" $$K$$.


 * This... is an outrageously over-simplified account of the assumptions and procedures...
 * Footnote


 * The search for the curvature $$K$$ indicates that, after making all known corrections, the number N seems to increase faster with $$d$$ than the third power, which would be expected in a Euclidean space, hence $$K$$ is positive. The space implied thereby is therefore bounded, of finite total volume, and of a present "radius of curvature" $$R = \frac{1}{K^\frac{1}{2}}$$ which is found to be of the order of 500 million light years. Other observations, on the "red shift" of light from these distant objects, enable us to conclude with perhaps more assurance that this radius is increasing...

The Structure and Evolution of the Universe (1959)

 * : an Introduction to Cosmology by Gerald James Whitrow


 * Hubble was inclined, from about 1936, to reject the Doppler-effect interpretation of the red shifts and to regard the nebulae as stationary; but theoretical cosmologists, notably McVittie... and Heckmann... severely criticized Hubble’s method...and disputed his conclusions. Although these criticisms... came to be generally accepted, it still seemed that the available data were open to rival interpretations, depending on the method of analysis...


 * At last, in 1949, the... ... was ready... Humason... succeeded in photographing the spectra of two remote galaxies in the . These exhibited red-shifts which, on the Doppler interpretation, indicated... one-fifth of the velocity of light. [I]n 1956, with... photoelectric equipment attached... [W. A.] Baum obtained a red-shift... recessional velocity of about two-fifths of the velocity of light.


 * [I]n... 1952, Baade... announced that Hubble’s entire distance scale was in error... According to Baade, the distances formerly assigned to all extragalactic objects must be multiplied by a factor of about two. Later it was generally accepted that this... was probably nearer three. ...[I]t followed that the sizes of all such objects had been underestimated. ...Therefore ...this nebula must be... twice as far away... [T]he average absolute magnitude at maximum brightness of novae... in the Milky Way attain on the average... 7.4, whereas those... in the Andromeda... 5.7... [T]he apparent anomaly could be removed by placing... Andromeda... rather more than twice as far as previously. ...[E]xtragalactic distances had ...been underestimated because of an error in converting... relative distances of s into an absolute scale. ...Baade's revision ...applied only to extragalactic objects... [and] had momentous consequences concerning the size and, for the scale of both was correspondingly increased.


 * An important new survey of the law relating red-shifts and magnitudes published in 1956 by Humason, Mayall and Sandage suggested... that the expansion of the universe may have been faster in the past... so that its age may be somewhat less than that estimated on the hypothesis of uniform expansion. But... caution, for a recent review (1958) by Sandage of Hubble's criteria for constructing the extragalactic distance-scale has revealed that, not only must his Cepheid criterion be corrected but also... the brightest star criterion...


 * As for Hubble’s brightest star criterion, Sandage... has shown that objects in the of galaxies which Hubble believed to be highly luminous stars are... regions of glowing  of intrinsic luminosity... two magnitudes brighter...  If Sandage’s result is accepted, then the distances of all galaxies beyond those in which Cepheids can be detected... must be augmented by a factor... between 5 and 10... with the result that the rate of increase of velocity with distance will be reduced to between 5O and 100 kilometres per second per megaparsec. Consequently, taking 80 as a rough average... the, if it has expanded uniformly, will have to be increased to about 13-5 thousand million years. If... it was expanding more rapidly in the past... this... might be reduced to about 9 thousand million years.