A Summary of Modern Cosmology

Traditionally, these are questions for philosophy, but philosophy is dead. Philosophy has not kept up with modern developments in science, particularly physics.

Stephen Hawking, and Leonard Mlodinow, The Grand Design [Bantam Book, 2010, 2012, p.5]

Modern cosmology begins with the heritage of ancient and mediaeval cosmology. The issues can be divided into essentially metaphysical questions about space and time and then observational issues about the content and structure of the universe. With space and time, the questions had always been whether they are finite or infinite, and what kind of thing, if anything, space was. The peculiarities of time, its directionality, and whether the past and the future can be said to exist, were sometimes noted but would remain a concern mainly of modern study. While the Atomists had believed in infinite space (and a vacuum), the consensus settled on the view of Plato and Aristotle that space was finite (and not a vacuum). The argument for this was essentially that infinite space was unimaginable and therefore, on the basis of the principle that the most knowable is the most real, impossible. Aristotle denied there could be actual infinities. On the other hand, Aristotle took time to be infinite since only the present time actually exists. The argument of Parmenides that there could not be a beginning of Being because there would be no reason for time to start at one moment rather than another, a violation of the Principle of Sufficient Reason, ruled out a beginning -- or end.

Subsequently, theological considerations modified these theories. With a God of arbitrary free will, and Revelation asserting the reality of a Creation, the objection to a beginning of time (or an end) could be overcome. The universe begins when it does just because God decides that will be the case. On the other hand, God being infinite, He can conceive an infinite quantity, so the objection against infinite space is removed. There had also been other objections against finite space. A finite universe ends at a boundary, and the Skeptics realized that this also violated the Principle of Sufficient Reason, since a larger or smaller space could always be conceived, making any given boundary arbitrary. Thus, we find Newton viewing space as infinite, indeed, "God's boundless uniform sensorium." Of course, the will of God could determine any arbitrary size for the universe, but the other issues seem to have predominated.

For the content and structure of the universe, all that was apparent to naked eye astronomy were the planets and the stars (with meteors and the occasional comet). While it is common now to say that the Ptolemaic universe, with the unmoving earth at the center, became the consensus view just because of human ego and arrogance, this overlooks the role of ancient physics in the debate. A moving Earth could not be generally accepted as long as rest was regarded as absolute and the concept of inertia was lacking. Greek physics held that an object could not move without being pushed. This was modified in the 6th century by John Philoponus with the idea that pushing could impart an immaterial force, an impetus, that would prolong motion beyond contact with its source. But the impetus still runs out, as is our experience with motion in daily life. Everything slows down and stops, unless you keep pushing it.

In introducing the concept of inertia Galileo fatefully also introduced the principle of the Relativity of motion. Moving at a constant velocity produces all the characteristics of rest in terms of any internal measurements for that moving frame of reference. This is very counter-intuitive and, indeed, rather bizarre, when you stop to consider it. The paradox of motion and rest appearing to be the same should remind us of the reasonableness, not the arrogance, of Mediaeval geocentrism. With his physics, and with his new telescopic observations, Galileo could vindicate the heliocentrism of Copernicus. Until then, nothing else could.

This is a point of crucial significance for the philosophy and history of science. Thomas Kuhn, realizing that Copernicus had not provided decisive evidence for heliocentrism, concluded that the triumph of that theory was a matter of social agreement rather than logic. This was then taken by many theorists, "deconstructionists" and the like, who had little or no interest in science, as proof that science was entirely "socially constructed," without any objective foundation in truth. Kuhn, however, had overlooked the circumstance that Galileo, a century after Copernicus, actually did provide the evidence that refuted geocentrism. The logical certainty was there, just not originally with Copernicus.

While Galileo and Newton continued to believe that space itself would determine an absolute frame of reference, they were aware of the difficulty that their own theories imposed on determining what that frame of reference was. We see the issues emerging in the debate of Samuel Clarke, representing Newton's ideas, with Leibniz, who did not believe that space even existed and so could provide no frame of reference at all. Curiously, Leibniz's best arguments are metaphysical. Space, if it existed, but which nevertheless is nothing (as the Atomists asserted), is supposed to underlie and support all substances. This would make something that is nothing more real than the things that are something. Leibniz found this incoherent. And Newton's idea that space was God's "sensorium" compromises God's omniscience with the requirement of a sense organ. Few theologians would be happy with that.

From caricatures of Newton, we might expect Clarke to offer some argument about how an absolute frame of reference for linear motion could be detected. That is not what we get. It would require a content of space, aether, for there to be hopes in the 19th century that absolute linear motion could be detected. What Clarke does instead is offer rotation as demonstrating the independent existence of space. A rotation is an acceleration, and within an inertial frame of reference we can determine whether we are spinning or not. Leibniz had argued, consistently, that rotation can only be detected with reference to external objects. But this is not the case.

This is a good enough point that some trouble has been taken since, with those sympathic to Leibniz, classically with Ernst Mach, to get around it. Mach argued that rotation can be detected only because of the gravity of all the other objects in the universe. This suggestion had the failing that it only existed to avoid a metaphysical embarrassment, that space exists, and so substituted ad hoc an obscure explanation for the simpler one, that space exists, violating Ockham's Razor [note]. On the other hand, Leibniz's argument, unanswered by Clarke, that there could be no physical differences between mirror images, was refuted by Kant with the observation that right and left hands cannot be transformed into each other by any rotations in space as we know it. They are physically different. The argument of Wittgenstein that mirror images could be rotated into each other through a fourth dimension serves to confirm rather than contradict Kant's point. A fourth dimension is clearly not available for physical processes, since handedness makes for distinctive interactions both in physics, where the Weak Force observes handedness, and in biology, where the mirror images of amino acids and proteins used by organisms do not occur. In Wittgenstein's day this feature of physics was not yet known. Nevertheless, as with rotation, there is a kind of hostility to Kant's argument whose motivation seems almost personal and can only be essentially metaphysical.

Meanwhile, there were new developments in the matter of the content and the structure of the universe. Copernicus, in denying that the stars moved around the earth every day, realized that they might be at very different distances from the Earth. Evidence for this might come from parallax, the apparent motion of nearby stars against more distant ones as the Earth moves around in its orbit. Unfortunately, there is no parallax evident to the naked eye, something obvious to ancient astronomy (and noted as evidence for geocentrism!), and even Galileo's use of the telescope did not reveal any. Observation would not be precise enough to reveal stellar parallax until the 19th century. Also, if the universe is infinite, and contains an infinite number of stars, another problem arises. This is now called "Olbers' Paradox," since it was described by Heinrich Olbers in 1823. But Kepler was already aware of the difficulty. An infinite number of stars means that there will be a star at every point on the celestial sphere. This leaves no blank space. The night sky will not be dark. Such a conclusion is inescapable given the Newtonian picture of an infinite and static universe.

Not much could be done about this until Thomas Wright suggested in 1750 that the physical universe was finite, that the Milky Way, the Galaxy, was a vast spinning disk, consisting of stars and everything else, and that the earth was part of this system. An observational confirmation of the structure of the Milky Way came from the great astronomer William Herschel in 1785. In a finite universe, especially one with the Newtonian dynamic of stars orbiting around the Galaxy, a problem like Olbers' Paradox doesn't arise. As a dynamic theory this was more sensible than a static universe of infinite unmoving stars (don't they feel the infinite range of the force of gravity?), but it also left the rather awkward situation of a finite structure in an infinite space. This would seem to violate Ockham's Razor, with much more space than is necessary, and the Principle of Sufficient Reason, since where the Galaxy happens to be would be completely arbitrary. The will of God could be invoked to resolve that, or some notion that every location is equivalent anyway; but it was becoming increasingly less appealing in science to resort to God where there seems to be some gap in a physical theory, and the overkill of infinite space for a finite body is palpable.

A threat to a Newtonian solution to Olbers' Paradox arose from Kant. In his General Natural History and Theory of the Heavens of 1755, Kant suggested that spiral nebulae, which were just being discovered, were not part of the Milky Way at all, but were external galaxies. The universe was filled, not with infinite stars as such, but with infinite galaxies. Thus, at every point on the celestial sphere, we would be looking at a galaxy -- unless there is simply a finite number of galaxies, which raises the previous awkward issues of dynamics and Sufficient Reason. Kant actually had no evidence for his surmise, so there was no particular reason for astronomers to court trouble by accepting it. Thus, the great mathematician Laplace (1749-1827), who accepted Kant's suggestion for the nebular origin of the Solar System, rejected the thesis of external galaxies and thought that the spiral nebulae were themselves primaeval solar systems. The 19th century did not improve the situation as it was left with Kant and Laplace.

On another front, the wave theory of light, for which evidence was discovered by Thomas Young, required that light have a medium by which the waves could be propagated. That came to be called the aether, after Aristotle's fifth element. In 1887 this inspired Albert Michelson and Edward Morley to design an experiment to measure the absolute velocity of the Earth by measuring the velocity of light along the direction of the Earth's motion and perpendicular to it. The motion of the Earth through the aether, while the fixed velocity of light in relation to the aether, should give different velocities for light along and perpendicular to the Earth's motion. Curiously, it didn't. Light has different velocities in different media (like glass, water, etc.), but its velocity in space turned out to be the same, however it was measured.

This is usually taken to mean that aether doesn't exist, and that light doesn't need a medium to propagate as a wave. However, the aether was postulated as a metaphysical proposition given the meaning of what a wave is. Waves are deformations of media. They do not exist independently. Surfers, even Zen surfers, at a beach without water are going to be disappointed. However, exactly what the medium of light was going to be like had not been thought out very carefully. Light is a transverse or shear wave (S or "secondary" in geology), i.e. it oscillates perpendicular to its direction of motion. A medium that can contain a transverse wave must be a solid because the wave requires elasticity for the deformation to right itself. Gases or liquids can only contain longitudinal or pressure waves (P or "primary" in geology), where the oscillation is along the direction of motion. A special case of a transverse wave in a liquid are the waves we see on the surface of water in the ocean. In this instance, water does not need to be elastic because gravity straightens the deformation. Without gravity, water would simply be scattered by a transverse wave.

Thus, the aether would need to be a solid, and the Earth could not move through it. There would be no "aether wind." This is rarely appreciated, as aether is commonly characterized as a gas, a liquid, or, closer to the truth, something like jello. Jello is at least an elastic solid, but it will not fit the requirements of the case. The velocity of a transverse wave is proportional to the rigidity of the solid. Jello is not very rigid. Transverse waves are not going to pass through it very quickly. Since light, as it happens, will have the highest velocity possible, aether will need to be the most rigid solid in nature. That is an extraordinary requirement for something that is supposed to fill all of space.

These conceptual or ontological problems with the aether, however, became moot in 1905 when Einstein introduced (1) Special Relativity, according to which aether, even if did exist, could not be used as a reference for absolute motion, and (2) Einstein's answer to the Photo-Electric Effect, according to which light consisted of particles, not waves, and thus could exist independent of any medium. For most people in physics and philosophy, this has settled the issue ever since, and Special Relativity has thus been taken as a decisive vindication of Leibniz's theory of space.

However, there was been a great deal of confusion involved in the evaluation of Einstein's theories. Einstein's Special Relativity does not depend on the relativity of motion, which was already a characteristic of the physics of Galileo and Newton, but on the requirement of an absolute, namely that the velocity of light in a vaccuum should be a constant, however measured. Relativistic effects, like time dilation and changes in length, are simply a function of preserving the velocity of light as a constant. Indeed, the formulae for these effects, the "Lorentz Transformations," had already been proposed by the Dutch physicist H.A. Lorentz. Einstein provided the reason and the context for the effects.

At the same time, the philosophers who celebrated the triumph of Leibniz over Newton don't always seem aware that Leibniz's thesis was that space doesn't exist. This probably isn't what most physicists would have in mind -- apart from the Positivists who don't think that science tells us anything about the nature of things anyway (and in which case they wouldn't care whether space has been shown to exist or not). More importantly, questions about the linear motion of light are completely irrelevant to the arguments offered by Clarke and Kant, concerning rotation and mirror images, against Leibniz. Thus, Relativity did not vindicate Leibniz; and the only real argument against the existence of a medium for the propagation of light is the awkward nature of that medium, aether, and of light as a particle rather than a wave.

After all this, however, modern cosmology does not really get started until Einstein's General Theory of Relativity in 1915. Einstein's theory constituted a revolution in cosmology unlike anything since the Presocratics first proposed naturalistic theories of the world. The critical feature of Einstein's theory was his use of space, which meant the use of geometry as a replacement for traditional physics. What made the difference in this respect was the nature of the geometry, which now for the first time in physical theory was going to be a non-Euclidean geometry.

Non-Euclidean geometry had previously been no more than a theoretical curiosity in mathematics. It began with attempts to prove Euclid's Fifth Postulate, the Parallel Postulate. No one really doubted the truth of the Postulate; but it was complicated, and to many (like the Neoplatonist Proclus) it looked more like a Theorem of geometry than one of the Axioms or Postulates. Centuries of attempts to derive the Fifth Postulate from the other Axioms and Postulates had all failed. Then Gerolamo Saccheri (1667-1733) attempted to prove it through a reductio ad absurdum proof. This requires assuming the contradiction of what is to be proven and then deriving an explicit contradiction. Saccheri realized that there were two ways to contradict Euclid. The Parallel Postulate itself holds, in modern restatements, that given a line and a point not on the line, only one line can be drawn in the same plane through the point that is parallel to the line, i.e. both lines can then be extended indefinitely without ever intersecting. Thus, this would be contradicted if either there were no parallel lines that could be drawn, or many. The result that Saccheri got that would be of later significance (since otherwise he failed to derive any contradictions) was that in the geometry without any parallels, lines were not infinite. Saccheri thought that this was good enough as a contradiction, but then it now looks like the requirement that (straight) lines be infinite follows from the Parallel Postulate itself. This one result, however, would contain all that would be needed for Einstein's theory of both gravity and cosmology.

Today, discussion of non-Euclidean geometry is hopelessly complicated with confusion over Kant's theory of geometry. It is often said that the very existence of non-Euclidean geometry contradicts and refutes Kant's theory. Only someone with a poor understanding of Kant (all too many) could ever say such a thing. Instead, Kant's theory implies a prediction of the existence of non-Euclidean geometry. That is because Kant says that the axioms and postulates of geometry are synthetic propositions, which means they can be denied without contradiction. Thus, Saccheri's attempts to derive contradictions from his pseudo-Euclidean premises failed. When William Dunham says, "Much to the surprise of virtually everyone, it turned out that the parallel postulate was not mandated by logic" [Journey Through Genius, The Great Theorems of Mathematics, Penguin, 1990, p.60], we don't know who he means by "virtually everyone," but that would definitely not include Kant, who affirmed the extra-logical character of geometry. "But," scholars object, "Kant says that the axioms and postulates are synthetic a priori propositions, which means it is impossible to conceive of them being false!" No, a priori simply means that they are known to be true independent of experience, not that they are inconceivable. That they are synthetic means that they are conceivable. But then it is fair to ask what is it, independent of experience, that Kant thought grounded geometry? That is what Kant called "pure intuition." Our own imagination of space grounded Euclidean geometry. It was not an empirical inquiry. Kant would say, not that non-Euclidean spaces are inconceivable or logically impossible, but that they are unimaginable.

There we get into a very curious issue. Whether something is unimaginable may be an entirely subjective matter. It is not uncommon to find people running around who seem to take for granted that they are in fact imagining non-Euclidean or even multidimensional spaces. When we get down to examples, however, matters look a little different. What people are often referring to as the genuine non-Euclidean (or multidimensional) spaces that they imagine are usually no more than models or projections of those spaces, accompanied with the analytical understanding, i.e. the algebraic equations, whose consistency and possibility are not at issue. Thus, the first non-Euclidean geometry that was constructed, by Gauss (1777-1855), Lobachevskii (1792-1856), and Bolyai (1802-1860), was based in effect on Saccheri's version of the Parallel Postulate that there are an infinite number of lines through a point parallel to a given line. Lobachevskian space (named after Lobachevski, who first published his work) has less space in it than a volume of Euclidean space with identical dimensions. We can see what this means by taking a surface and then shrinking it up and down into hills and valleys. If we draw perpendicular lines on the surface, they will curve away from each other in opposite directions, one up, the other down. This is especially evident where we have a shape of a saddle or a mountain pass. One line can go up the sides the pass (the mountains), the other goes down over the pass (into the valleys). This is called "negative curvature," which characterizes Lobachevskian space. In the same way, if we draw a square than then shink its area, we can imagine this in terms of the sides being pulled in. They take on the forms of hyperbolas, why is why Lobachevskian geometry can also be called "hyperbolic." As the sides are drawn in, we also observe that the angles in the corners of the square become more acute. This is also characteristic of Lobachevskian geometry, where the sum of the interior angles of a square will be less than four right angles.

Lobachevskian space is actually one of the least intuitive of the non-Euclidean geometries. Our stretched surface does not have the same shape everywhere on the plane. The properties of the surface should be those of a saddle shape at every point, but of course a saddle shape is a saddle just at one place. The saddle or the mountain pass, or for that matter the stretched surface, is simply a model, and not a very good one -- though it is common for people to think that the existence of any model, however defective, is sufficient to prove, not just the possibility, but the reality of the object it models. Indeed, there are no models of a Lobachevskian space that do not distort shapes and sizes or that can represent the appropriate properties of the space everywhere equally. A famous model of Lobachevskian space is that drawn by M.C. Escher at right. In this case, unlike the saddle model, the center of the figure, with straight, perpendicular lines intersecting, displays no non-Euclidean features. It is towards the edges of the diagram that we get distorting effects and where what are supposed to be straight lines begin to appear curved. All the strange little fish are actually same size, but the model makes them look like they are vanishing into the distance. This is especially deceptive when we realize that Lobachevskian space is infinite, as in Euclidean geometry. The model cannot show that -- as to the eye, infinity is simply a finite "vanishing point" in perspective. For our topic here, as it happens, a Lobachevskian geometry in its infinitude would not accomplish any useful purpose for Einstein, either in physics or in cosmology.

Now, if Kant is right that a non-Euclidean geometry would be unimaginable, does that mean that a non-Euclidean geometry cannot exist in reality? Only if the conditions of our experience are identical to the conditions of reality, the conditions of, as Kant says, "things in general." But this will certainly not be the case in Kantian philosophy. The phenomenal world will be Euclidean, because that is the way we are wired to see things, but this imposes no obligation on things in themselves. Now, Kant himself did not believe that space or time applied at all to things in themselves, but nothing is lost by allowing that he might be wrong in that, especially with regard to the universe itself, which is both a phenomenal object and, conceived as a whole with limits beyond any possible experience, a transcendent object (like Kant's "Ideas" of God, freedom, and immortality). This is the critical issue, of course, for cosmology, where Kant himself held that the age old question of the finitude or infinitude of space and time could not be answered -- the arguments form an Antinomy, equally good either way.

A non-Euclidean geometry with the potential for a useful purpose was described by Bernhard Riemann (1826-1866). Riemann generalized the issue of non-Euclidean geometries but then in particular produced one based on Saccheri's Postulate that there are no parallel lines. This Riemannian space contains more space for the same dimensions than a volume of Euclidean space. If we take a surface, and stretch it, this will curve perpendicular lines in the same direction, giving the surface a "positive" curvature. This why there are no parallels. In a Lobachevskian space, lines that would eventually intersect on a Euclidean plane keep getting spread out. On a positively curved plane, lines that would be parallel on a "flat" (Euclidean) plane are brought together by the expanding surface. The best model of this, indeed the best model of any non-Euclidean geometry, is the surface of a sphere as a two-dimensional non-Euclidean space. If we were two-dimensional beings on the surface of a sphere, all we would know is that if we set off on a straight line (a "geodesic") in any direction, we would end up coming back around to where we began. This is what is required for a positively curved space. You follow a straight line, and it doesn't go on forever. Instead it comes back around to where you started. We can only imagine this in terms of a circle, which is why this kind of geometry is also called "elliptical," i.e. in that a circle is a kind of ellipse -- the class of conic sections that are closed curves. Again if we imagine a square, and expand its area, the sides will bow out, and the boundary lines will curve such that, if extended, they will intersect, meaning that they are no longer parallel. Also, the angles at the corner of the square will become more obtuse and add up to more than four right angles. On the surface of a sphere, it is not difficult to draw a triangle where each corner is itself 90o, which, with a sum of 270o, blows away the Euclidean theorem that the interior angles of a trangle equal 180o. Saccheri's result that, without parallels, lines are finite was so strange that he felt justified in rejecting the geometry. As it happened, even Gauss and Bolyai thought the same thing, which is why the first non-Euclidean geometries did not use the no parallels postulate.

But this is precisely what does the job for Einstein. The orbit of a planet is a closed curve. In Newtonian physics, we need a "force," gravity, to bring the planet around and close the curve. Einstein realized that if the path of the planet is a geodesic in space-time, then no force is necessary. This is a stunning insight. We can substitute geometry for mechanics. On the cosmological scale it is even more impressive. A positively curved space has a finite volume, but if we head off looking for the edge, the boundary, we just end up coming back around to where we started. This suggests a remarkable resolution to Kant's Antinomy of Space and Time. Finite universes were troubled by the arbitrary boundary they required, while infinite universes were, well, infinite. Now Einstein's theory holds out another possibility:  a finite universe with no boundary, avoiding the objections to both finite and infinite.

All this left more than one generation of philosophers, astronomers, and physicists dizzy with the novel perspective it provided. It is still the only theory ever suggested that so neatly resolves Kant's Antinomy. It is a shame that it hasn't quite worked out. There would be difficulties.

First of all, let's consider a basic requirement of this kind of non-Euclidean geometry, that a straight line will close on itself. An essential feature of the modern treatment of non-Euclidean geometries is that curvature can be "intrinsic" to a space, which means we do not need an extra dimension for it to exist. Intuitively, if we take a one-dimensional, straight line and curve it, we will need two dimensions for the curved line to exist. If we take a flat, two-dimensional plane and warp it, we will need three dimensions for this curved surface to exist. And, by analogy, if we take three-dimesional space and somehow curve it in the same way, it will warp into a fourth dimension. Nevertheless, a curvature of the surface or the three-dimensional volume can be analytically characterized, with equations, without the need for a term representing a fourth dimension. This is an "intrinsic" curvature, while warping our flat plane into a bowl involves an "extrinsic" curvature.

Now, just because we can characterize an intrinsic curvature, does this mean that such curvature can exist without the higher dimension nevertheless being necessary? This may be more of a metaphysical than a mathematical question, rather like Descartes asking whether he can be thinking if he were not existing. It is a question that generally does not get asked in treatments of this geometry. The assumption seems to be that if we don't need an algebraic term for a higher dimension to represent curvature, then curvature ontologically does not need a higher dimension. But this looks more like an open question.

With our spherical model of a positively curved space, it is obvious to us, in three dimensions, that the "straight lines," the geodesics, which on spheres are called "great circles," are indeed circles. What about for two-dimensional beings on the surface of the sphere, who have no idea what a third dimension would be or how it could exist? To them, the curvature of their surface is indeed intrinsic. But even for them, it would be obvious that great circles are indeed circles. We can undestand that this would be the case by beginning with the example of any circle on a sphere. It can be made arbitrarily small and will have the features we would expect of a circle, such as a center and a radius. Indeed, the smaller it gets, the flatter the surface, and the closer the circle will be to having Euclidean properties (i.e. where the circumference is 2π times the radius -- otherwise the circumference is smaller than 2π times the radius). We take any such circle on the surface of a sphere and we begin expanding it. The circle gets larger and larger. Eventually it becomes large enough that it is a great circle. It is now, in the non-Euclidean geometry, a straight line, but it also a straight line with the properties of a circle, i.e. a center (indeed the same center with which it began) and a radius. Indeed, a great circle has two centers, one on each side of the line, at equal radii from the line. A great circle cannot be further enlarged, and moving it towards the opposite center both makes the circle smaller again and produces a line that is extrinsically curved even to two-dimensional observers.

Thus, although the surface of a sphere is an excellent model of a non-Euclidean geometry in the sense that it does not distort the shapes or sizes of figures that we move around on it, nevertheless does not make for a persuasive case that intrinsic curvature is actually unrelated to extrinsic curvature. A great circle is a member of a two different sets of lines, which on a globe of the world we call in general either meridians of longitude or parallels of latitude. Meridians are all Great Circles. But with the parallels, all but one (the equator) obviously consist of curved lines even within the perspective of two-dimensional space. If by analogy we expand our example to a three-dimensional space which has a positively curved, non-Euclidean geometry, the analogous test would be if we begin, again, with obvious circles of arbitrary size and then begin to enlarge them. Is there a point where the circle gets large enough and the line intuitively becomes straight? I don't think so. Is there a point where the circle reaches a maximum size and we cannot imagine it getting any larger? I don't think so. But those would be the characteristics of a 3-D Riemannian space. Great circles in our imagination obviously have extrinsic curvature. In three-dimensional space the acid test of a non-Euclidean geometry would be our ability to imagine a straight line that nevertheless returns upon itself. I suggest that this cannot be imagined without making the line a circle or other closed (extrinsic) curve. This is all consistent with the Kantian thesis of the Euclidean nature of our spatial intuition.

Even if we cannot imagine curvature without adding a dimension to that figure which is to be curved, there is of course the circumstance that Einstein is not just talking about space, but about space-time. Time has been added to the three dimensions of space. Properly speaking, we have a curvature of "space-time," and not just space, though that is the language we often see. Indeed, the idea that time is really an extensive dimension exactly like space appeals to many people, but it passes over the difficulty that both intutively and mechanically time looks and works very differently indeed from space. Metaphysical questions about time sometimes seem more an annoyance than anything else in the context of the physics. That the equations of both Newton and Einstein do not distinguish between the direction of time, from past to future, persuades many that the directionality of time is not a fundamental feature of the world. Directionality (the "arrow" of time) might be happily dismissed altogether if it were not obvious that the world in fact observes it, and that this is incorporated into science through the Second Law of Thermodynamics, the law of Entropy, ἐντροπία,. Nobody is happy with the situation the way its stands, and there are probably many who wish that Entropy (and a hierarchy of Order) would go away and stop bothering them.

If we examine a simplified version of space-time, with only one spatial dimension, what is going on becomes a little clearer. Whatever time may be in ultimate ontology, what it allows to happen physically is motion. No time, no motion. Now, just as Einstein's postulate for Special Relativity was that the velocity of light is a constant from any inertial frame of reference, the postulate for General Relativity is that the characteristics of motion for inertial mass and gravitating mass are due to physically identical circumstances -- the "Equivalence Principle." For instance, in free fall, we are weightless. It doesn't matter whether we are in the presence of an external gravitational field or not. If we did not look around, we would have no way of knowing what was going on in relation to other things. In the absence of gravity, the object in free fall does not change its velocity or position. However, with a gravitating object nearby, the object in free fall does change its velocity and position, but only in relation to the external gravitating object. This can be represented by curving the lines on the axis of time in the diagram. The curvature is of time because it is the passage of time that offsets the position of our falling object in space. The other cases for Equivalence are examined elsewhere. That the curvature of space-time is due to curvature in the temporal dimension is of considerable importance. It means that a true non-Euclidean geometry of space is not even necessary for General Relativity; and if time as a dimension is not like space, then a non-Euclidean four-dimensional geometry is mathematically elegant as a model for space-time but does not constitute a threat to Euclidean geometry as physically appropriate for space alone.

Now with some of these issues unpacked for Einstein's use of non-Euclidean geometry, it is time to look at the difference between the mechanical and the cosmological aspects of Special Relativity. What is now often forgotten is that Einstein, like Newton, expected the universe to be static. Well, perhaps it is not exactly forgotten, but it is then overlooked that Einstein expected the universe to be both static and globally possess the positively curved non-Euclidean geometry. This means that the structure of space was divorced from the dynamics of the universe. Since gravity would tend to collapse a static universe, Einstein realized that a universe with gravity must either be expanding, to overcome the attraction, or collapsing already. He introduced a repulsive force, the Cosmological Constant, into his equations to counterbalance gravity.

When Edwin Hubble (1889-1953) proved in 1923-1924 that spiral nebulae were external galaxies and discovered that the galaxies were all moving away from us, at velocities proportional to their distances:  Hubble's Law [note], this took care of Einstein's static universe. Einstein's himself then said that the Cosmological Constant was the biggest mistake he ever made. Not so fast. At the time, however, this all produced a classic vision of the universe. The place is expanding. The dynamics and structure of the universe could now be linked, and we get the classic model of the expanding balloon to represent the cosmology. A spherical balloon, of course, models a positively curved non-Euclidean space. If we then draw little galaxies on the surface of the balloon, we get the distribution of galaxies in space. If we then blow up the balloon, the area of space expands, the galaxies all recede from each other, and we see what Hubble observed: the homogeneous (the same everywhere) and isotropic (looks the same in every direction) expansion of the universe.

This image of the expanding balloon was so compelling and so satisfying that it was simply repeated endlessly for decades without much reflection. There are, however, a couple of very significant features of the model:  (1) it left open the question whether the universe (the balloon) would expand forever or perhaps slow down, stop, and begin to collapse; and (2) the form of the model still divorced the structure of space from cosmic dynamics, that is, the balloon represents a non-Euclidean geometry regardless of whether the balloon expands forever or not. This came to misrepresent everyone's expectations, since the structure of space was soon linked to the dynamics. Thus, a properly positive-curved, finite but unbounded universe, was linked to closed dynamics, i.e. that the universe eventually ceases to expand and will in fact collapse. An open universe, which expands forever, was linked to the infinite geometries, either Lobachevskian or actually, after all this, Euclidean. Linking the structure to the dynamics was a step not always taken in an open and self-conscious way -- it was not necessary either in terms of Einstein's original static cosmology or on the basis of the balloon model. This would not have made much difference if the dynamics were going to come out the "right" way. In his Brief History of Time [1988], Stephen Hawking barely considered the possiblity of an open universe (it would just be so philosophically unsatifying), but by the time he wrote, the evidence was already beginning to weigh heavily against it.

The dynamics of a closed universe required that gravity be strong enough to overcome the expansion of the universe, and this required a great enough density of matter to supply the gravity. As noted elsewhere, it was already becoming obvious by the mid-1970's that the matter just wasn't there -- at the time only about 1/40th of the density required for closure was evident in the visible matter of the universe. The philosophical threat posed by that circumstance, however, was little noted. If the universe is open, with perhaps a Lobachevskian geometry (which would have been the case with 1/40 the required density), then space would be infinite, Kant's Antinomy would be back, and all the conceptual elegance of Einstein's finite and unbounded universe would be out the window. This was perhaps a fate worse than death, and it is remarkable how persistent the expanding balloon still is when the foundational rug has been pulled out from under it (though that sounds a bit like a mixed metaphor).

Meanwhile, we get other developments. The expanding universe, especially with a finite volume, had very early been taken to imply that there was an origin of this expansion from an early very dense and very hot beginning. Not everyone liked this, and the theory, of the Big Bang, even got named with a derisive connotation. There wasn't much in the way of serious alternatives, however. In 1948 Fred Hoyle and others proposed a "Steady State" universe, which, although expanding, remained static in content because of a continuous creation of matter from empty space. There wasn't a proposal of the mechanism for this, and it was a lot to swallow. These days, when the idea that matter, or entire universes, can come out of nothing because of quantum effects is actually rather popular, there might be less objection to the Steady State theory. But something out of nothing turned out to be the least of the problems.

If the spontaneous creation of matter kept the universe looking the same, then it would always look the same. As we look out in space, we are also looking back in time -- a year for each light year. Thus, if the universe looks different at cosmological distances, it has not been in a Steady State. An awkward reality for the Steady State universe was thus the existence of Quasars. Quasars (Quasi-Stellar Objects, or QSO's) are among the most mysterious objects in the universe. Small, distant, and incredibly bright, the only mechanism proposed for their energy output (a mass the size of the moon may be turned into energy every second) involves the high energy effects of matter falling into Black Holes. It is not at all impossible that some other exotic effects may be involved, however, especially when we consider that no quasars exist in nearby space or recent cosmic history. Quasar 3C 273, one of the closest quasars, is no less than 1.90 billion Light Years away, with a Red Shift at 14.6% of the velocity of light. Such objects do not look like the citizens of a Steady State universe.

As it happens, in 1948 also, George Gamow and others predicted that if there had been a Big Bang, there should be visible evidence of it remaining as a Cosmic Background Radiation. The early universe would have been dense, hot, and opaque. As the universe expanded, it would reach a point where it had cooled and attenuated enough that it became transparent. Looking out in space and back in time that moment should appear as a wall of radiation -- the opaque image of the earlier universe. Since that wall of radiation will be redshifted, it would now be visible as the black body radiation (i.e. radiation produced only by heat and not by reflection) for an object with a temperature of about 3 Kelvins (see Max Planck's equation for black body radiation). This would be radiation in the microwave range. In 1965 just such a Cosmic Background Microwave Radiation was discovered by Arno Penzias and Robert Woodrow Wilson, working for Bell Laboratories in New Jersey. The Steady State theory could not explain quasars, and it really could not deal with such a dramatic confirmation of a prediction based on the Big Bang. The Big Bang had just blown away the opposition.

The Cosmic Background Radiation turned out to be almost too good to be true. It was too good. It seemed so uniform that it was difficult to relate it to the lumpy character of the present universe. It was also hard to see how it could be so uniform. The problems got solved with a combination of observation and theory. With more study, and especially with data from satellites, the requisite lumpiness was discovered. Indeed, a large scale anistrophy, an asymmetry, was discovered. After Galileo and Newton made it difficult to discover an absolute inertial frame of reference, and interpretations of Einstein positively affirmed that such a thing was impossible, we now discover it in fact. One direction of the Cosmic Background Radiation is redshifted significantly more than the Radiation in the opposite direction. This gives the absolute velocity of the planet Earth with respect to the universe as a whole.

Meanwhile, the smoothness of the Radiation was explained with a theoretical twist:  Inflation. The idea of inflation is that some time after the Big Bang, the rate of expansion of the universe suddenly increased, greatly but temporarily. This can have occurred because of a kind of phase transition as the forces of nature differentiate themselves in the cooling universe. Thus, in the earliest moments, there was really only one force of nature. Then, successively, gravity split off, then the Strong force, then the Weak force, and finally we get the differentiation between electrical and magnetic forces. In any case, something of the sort may be responsible for Inflation. With Inflation, any initial lumpiness of the universe gets smoothed out a good deal, producing the Background Radiation as observed. And there are other consequences. If Inflation is faster than the velocity of light (which, it seems, is allowed not to violate Special Relativity if space itself is doing the expanding), this removes much of the actual universe from what then becomes the "Observable" universe. This can Save the Balloon, since the Observable universe then becomes just part of the surface of the balloon; and if Inflation makes the balloon big enough, then the Observable universe will actually look flat and Euclidean [note]. So we can have an Einsteinian universe and Euclid too. In any case, this is how Inflation is often illustrated, using our old friend the expanding balloon.

The desire to Save the Balloon perhaps follows from the conceptual difficulties engendered by combining the structural and dynamic issues of cosmology. If space is Lobachevskian or Euclidean, and infinite, we get an unfortunate consequence for the Big Bang. The universe would not have begun as a finite thing and then suddenly become infinite. Infinite space was always infinite, but this simple truth seems to have been systematically overlooked or ignored. When he was a young Wunderkind, Stephen Hawking proved that the Big Bang must have begun as a singularity -- a single point in space-time. This was the flip side of another matter of interest to Hawking, that Black Holes collapse into a singularity. For a while, this provoked eager speculation, that perhaps the collapsing universe all became a single great Black Hole, which then transforms into the singularity of the Big Bang, a great White Hole out of which a new universe comes. There might even be an eternal and cyclical universe which goes through a succession of collapses and new Big Bangs. But this was not to be. The definition of a Black Hole is that gravity is so strong that nothing can come out of it (at least not abruptly), and certainly not an entire universe. Furthermore, as Roger Penrose points out, the singularities of Black Holes and the Big Bang are very different kinds of objects in regard to entropy. Black Holes have very high entropy, i.e. they destroy information, as objects fall into them, and so lose order. The Big Bang, however, is a very low entropy object, very orderly, and the orderliness of the universe is the consequence. Unless, the world-ending Black Hole could flip its entropy from positive to negative, it will not become a Big Bang. That the Big Bang is then a very extraordinary and exotic object, a White Hole in truth, does increase its mystery.

Meanwhile, the Big Bang in infinite space is no longer a singularity. It is an infinite volume of infinite density. This means infinite gravity and a very unlikely beginning for an expanding universe such as we observe and live in. I have yet to see this embarrassing result discussed in an open and perceptive way, but perhaps I have missed the relevant literature. Instead, we continue to get the balloon, with a small patch of Observable universe, and a Big Bang in which the balloon can still be run back to a finite point. But this will not do. In Inflationary universes, the density of matter approaches that which corresponds to a Euclidean space. If that is the density of the matter in the whole universe, then the Euclidean appearance of the Observable universe is not just a local effect but a global one. I detect a reluctance to acknoweldge this circumstance. Instead, we can Save the Balloon by postulating the structurally and dynamically closed universe and then obscure the evidence for it by having Inflation thin the density of matter down to, but not quite at, the Euclidean level. So we can have an Observable universe that is, as far as we can tell, Euclidean, while meanwhile we can continue believing that the (hidden) universe at large is finite but unbounded. At this point, things have become rather more a matter of metaphysics than of physics, of philosophy than of science. What is the evidence for the universe at large being dynamically closed? None. The move seems to be the dishonest one of putting falsifying evidence beyond the possibility of observation. That is not the way to do science. But if it is a matter of philosophical preference, then there should at least be some acknowledgement that cosmology has now been returned to metaphysics from its 20th century sojourn into physics and astronomy. I doubt, however, that anyone wants to openly acknowledge or advocate such a thing.

And then things get worse. The curiosity about whether the universe would expand forever or collapse suddenly was revealed as perhaps involving the wrong question. In 1998 Adam Riess and others reported data from observations of supernovas that indicated an acceleration in their velocity away from us. This result has been confirmed and generally accepted, and it means that the expansion of the universe is itself accelerating, not slowing as would be necessary given the influence of gravity alone. Something else is going on. There is a force that is opposed to gravity and that is becoming stronger as the universe ages and enlarges. This curve ball from the observational side of astronomy has now made for one of the greatest mysteries in science. One way to handle it is simply to reinstitute Einstein's Cosmological Constant. Now it doesn't need to be there to balance gravity but to overbalance it. Einstein's "biggest mistake" is now back at the cutting edge of physics. Exactly what the Constant means, however, is unanswered. A force operating against gravity is not something that is otherwise provided for in physics, except at the margins of speculation. It is now generally called "dark energy," a term coined by Michael Turner. One approach is that in quantum mechanics empty space (so space exists?) is not simply a vacuum. There are virtual particles that come out of nothing and pass back into nothing, thanks to the Uncertainty Principle. This has an influence and exerts a "vacuum pressure." That is a start, but it might seem that a force that is going to end up dominating the history of the universe should have a more integral place at the foundations of physics. What the problem may reveal are serious gaps in that foundation. Since anomalies have now arisen over General Relativity itself, that its predictions of the trajectories of spacecraft involve small errors, we may be seeing gaps opening up where a revolution in physics, and a new Einstein, may appear.

"Dark energy" is a mystery added to a previous mystery. As noted, it was already obvious in the 1970's that there wasn't nearly enough visible matter in the universe to effect a dynamic closure. There had already been some evidence that the universe, however, contained invisible mass. Some considerable hopes were riding on this. The "problem" of the "missing mass" had been a problem, of course, only because a closed universe was philosophically preferable and anything else was going to be a "problem." In 1975 Vera Rubin and others announced that studies of the rotation of galaxies showed that they violated Kepler's Laws and the simplest principles of Newtonian mechanics. Thus, as the outer planets orbit more slowly than the inner ones, it was always expected that the outer areas of galaxies would orbit more slowly than the inner ones. Not to be. The margins of galaxies are orbiting their centers just as fast as the interiors. The obvious meaning of this (unless the physics of Kepler, Newton, and Einstein was all wrong) was that there was a great deal of invisible mass outside the galaxies that was influencing the rotation. There had to be a lot of this, and it has come to be called "dark matter." What it might be is a subject, like dark energy, of pure speculation. It neither emits, absorbs, nor reflects energy nor interacts with visible kinds of matter. It does refract light, with its gravity, and this becomes another useful way to detect its presence. It might just be made of neutrinos, which were once thought to be massless, but now may not be so and are otherwise very inert. Whether neutrinos are massive enough and inert enough is the question, and we also are faced with a certain inelegant picture, as with the vacuum pressure, how a kind of byproduct of the rest of physics ends up dominating the structure and history of the universe. Unknown and mysterious entities that dominate the universe would seem to call for more fundamental kinds of answers.

Dark matter wasn't enough to close the universe. The numbers now, for the total mass-energy of the universe, are about 4% for visible matter, 22/23% for dark matter, and 73/74% for dark energy. Dark energy thus dominates the universe, and its influence, far from effecting a dynamic closure, is exactly the opposite. Considerable theory has thus been devoted to what will happen to the universe as it gets larger and larger, colder and colder. Eventually the force of dark energy may overcome even electromagnetic and nuclear forces, resulting in a universe where every atom and particle, with space and time itself, will blow apart. Thus would be realized the childhood fears of Woody Allen in Annie Hall, that the expanding universe will eventually just explode, "and that will be the end of everything." This has come to be called the "Big Rip." The neat picture of cosmology in the 1970's, to which Stephen Hawking still clung in A Brief History of Time, with everything wrapped up in a nice non-Euclidean ball, is itself exploded; and it is hard not to look at contemporary cosmology and simply see a shambles.

Meanwhile, a gap has finally appeared again between the structural and the dynamic geometry of the universe. It is a revelation to some that we could have various kinds of geometry regardless of the (now dismally pessimistic) dynamics. Part of this is the theory of "branes." The word derives from "membrane," which in common parlance is a two-dimensional surface between three-dimensional volumes. Now we can have a "3-brane," which is a three-dimensional volume between 4-dimensional or multi-dimensional spaces. Our universe might be the "surface" of a higher universe or the result of some interaction between multi-dimensional universes. There isn't a shred of evidence for this, but it does have an interesting pied à terre in serious and interesting science. As noted, Einstein's great breakthrough of the imagination was to replace forces with geometry. As a revolution, however, the job was left very incomplete. There were other forces besides gravity, and although suggestions were floated, even to Einstein himself, not much came of it. As long as expectations were that quantum mechanics would replace Einstein's treatment of gravity, there was little interest in the further use of geometry. When, as it happened, quantum mechanics had little luck with gravity, the tide began to set in the other way, and since the 1970's a geometrical theory of all the forces of gravity has become the popular expectation for the future. I have treated this in some detail elsewhere, and the treatment of the forces is not of immediate concern here. The idea, however, that fundamental physics may require more dimensions than are obvious in space and time had a certain liberating effect in cosmology. The previous narrow insistence on an intrinsic curvature of non-Euclidean space, with a sort of huffy "we don't need no higher dimensions," has given away to a free-for-all of infinite dimensions and exotic multi-dimensional geometries. This is almost purely speculative, and might be thought to represent a very serious case of overkill.

All that was ever needed was simply to Save the Balloon, a model where the perceptive might always have appreciated that the structure and the dynamics were independent of each other. The balloon can expand indefinitely, or expand and collapse, while retaining its finite and positively curved form. The roadblock may simply be the persistent philosophical preference that space not exist independently but be an epiphenomenon of something else in the physics. If the universe is a 3-D Riemannian space regardless of gravity or General Relativity, this posits an independent stucture of space. So, at the moment, the cutting edge in cosmology may still be the argument between Newton and Leibniz. The belief that Einstein (or Mach) had vindicated Leibniz was facile and made it look like people either didn't understand Liebniz or did not appreciate the implication that Einstein was reifying space-time. When I was a graduate student, the great physicist E.C. George Sudarshan (b.1931), who (with others) introduced the theory of tachyons, particles that travel faster than the velocity of light, was invited to give a talk to my class, Irwin Lieb's seminar on time. He said that Einstein's equations for space-time looked like the equations for fluid mechanics. Space-time flows. This is not surprising, but a great deal of effort on the philosophical side of cosmology has been expended on making this a flow of nothing. I find this attitude puzzling, perhaps even more so than that of the Positivists (with Hawking as one) who don't want to make positive ontological statements at all.

The Ontology and Cosmology of Non-Euclidean Geometry

The Clarke-Leibniz Debate (1715-1716)

Three Points in Kant's Theory of Space and Time

Note on the Metaphysic of Space

Historic Equations in Physics and Astronomy

Philosophy of Science, Space and Time

Philosophy of Science

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Copyright (c) 2008, 2009, 2010, 2012, 2013, 2018 Kelley L. Ross, Ph.D. All Rights Reserved

A Summary of Modern Cosmology, Note 1;
Monads, Particles, and Space

I have never thought that Leibniz's monads were elementary particles. I have never understood what they are, and I don't really care, because I can't see why anyone should take them seriously.

Steven Weinberg, Physics, The University of Texas at Austin, Nobel Laureate in Physics (1979), e-mail to K.L. Ross, November 30, 2015

A rotation is an acceleration in a significant way. A body may be rotating with a constant speed; and if speed were velocity, there would thus be no change in velocity and no acceleration. But speed is not velocity. Velocity is speed plus a vector, i.e. a direction of motion. In a rotating object, its parts (except for a point at the center) are constantly changing their direction of motion. It is not a change in speed but a change in direction -- hence an acceleration. The only factor that changes in this acceleration is the spatial one. So, if space did not exist, Leibniz would be perfectly justified in arguing that it would be impossible to detect rotation without reference to external objects.

But rotation remains a challenge. Would Mach's theory mean that a sphere in empty space, with a rocket on one side, directed on a tangent to the surface, would not rotate if the rocket were fired, just because the rotation would not be evident in relation to external referents? And if Mach's theory is that there is no inertia without external gravitiational fields, does this mean that firing the rocket would produce no discernable effect at all? Doesn't this violate Newton's Third Law as well as the First? How much of physics was Mach willing to sacrifice before simply admitting that space exists?

It is not clear to me whether Mach and his successors understand the full implication of Leibniz's argument. It is a postulate of Leibniz's metaphysics that space does not exist. If the purpose of removing Clarke's argument about rotation is to remove all arguments for the physical reality of space (with Wittgenstein's point about mirror images often taken to refute Kant), are they really going to completely agree with Leibniz? Space does not exist? Are they going to go for the monads and the Pre-Established Harmony was well? That is hard to credit. But without space, the monads lose their windows and there is truly no external reality at all. It is a "consciousness only" doctrine. But perhaps a 19th century German like Mach actually was a metaphysical Hegelian. There he would have a "consciousness only" doctrine also.

Failure to understand Leibniz's metaphysics is something I find in Great Feuds in Science, Ten of the Liveliest Disputes Ever, by the science writer Hall Hellman [John Wiley & Sons, 1998]. Hellman says,

Leibniz even argued against Newton's hard, "massy" particles as the constituents of matter. He replaced these particles with a set of monads -- entities without extension, parts, or configuration, yet which possess in infinitely varying degrees the power of perception. To the hardheaded realist, his monads sound impossibly metaphysical; Newton derisively called them "conspiring motions." Yet even "conspirating motions" come closer to the quantum mechanical conception of atoms than do the "massy" particles of Newton. [p.60]

Although Hellman has picked up some of the characteristics of monads, he clearly does not understand that they are not material particles at all and that they can have nothing to do with the quantum mechanical conception of atoms (see the Steven Weinberg quote above). Monads do not have causal interactions with other monads. But the particles in particle physics do. That is what forces are all about. Also, the monads do not really have perception, which implies the causal input of external objects, but only representation, which is each monad's innate internal record of the entire history of the universe. I don't think Hellman is aware of this.

And the particles of particle physics can have mass, as the monads do not. So what is the objection against Newton's "massy" particles? It is not that they have mass -- protons, neutrons, and electrons have mass -- but that they are extended and solid, just like Cartesian matter. What Hellman is thinking of, without saying so, is that electrons and quarks are Dirac Point Particles, which are as unextended as the monads. But monads are unextended just because there is no space. An electron is unextended because it only occupies a geometrical point. Hellman's discussion displays some very sloppy thinking, even before we realize that he does not understand Leibniz.

So how does the metaphysics of Leibniz actually compare to the "quantum mechanical conception of atoms"? Well, atoms are mostly (in fact, with point particles, entirely) empty space. But, since space doesn't exist in Leibniz, what would all that empty space be? Monads. Monads fill the extensive continuum that our imperfect, confused representation construes as space. So the volume of a hydrogen atom contains, not just an electron and three quarks (as a proton), but an infinite number of monads. We could now even give some physical meaning to the representation of those monads, since empty space in physics is now filled by fields, virtual particles, "vacuum pressure," and such esoterica.

So what is the lesson of Hellman's treatment? Well, we might comfort ourselves that his confusions arise because he is not a professional physicist, only a popular writer. But then, one does not need to be a professional physicist to understand Leibniz's metaphysics; and I am left with the suspicion that professonal physicists, like Ernst Mach, are perhaps no more likely than Hall Hellman to understand the metaphysics of a professional philosopher like Gottfried Leibniz. If Hellman is thinking of the monads as material particles, I bet he picked this up from reading or hearing the sorts of discussions and claims that physicists were making about them. But if their understanding was no better than his, then everyone, beginning with Mach, has been very much out of their reckoning in the matter.

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A Summary of Modern Cosmology, Note 3

The June 2010 edition of Sky & Telescope reports that space has been measured to be flat (1) to an accuracy of 1.0023 +/- 0.0055.

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A Summary of Modern Cosmology, Note 2

The Red Shift from the Doppler Effect is how we actually know about the radial velocity of recession, and distance, of galaxies. The Red Shift (z) is the change in a wavelength of radiation (Δλ) from the object divided by what the original wavelength was (Δλ/λ). This can be related to velocity (u) as: (z+1)2=(c+u)/(c-u) or u/c=(z2+2z)/(z2+2z+2), where c=1. Distance (s) then depends on Hubble's Law: u=sH, where the Hubble Constant (H) is taken to be 75 km/s/MPC (it is somewhere between 50 and 100 km/s/MPC). 1/H is the Hubble Time, which for the given H is 13.04 Gy. c/H, the Hubble Radius, is thus 13.04 GLY. At low values, z is virtually identical to u/c. The two values begin to diverge at about 0.023 of the velocity of light. u/c cannot be larger than 1, but z can be any number up to infinity (at the speed of light).

The June 2010 edition of Sky & Telescope reports the Hubble Constant to be 70.4 +/- 1.4 km/s/MPC [p.14]. They also report the age of the universe to be 13.75 +/- 0.11 years. It used to be that the Hubble Time would be larger than the age of the unvierse; but if the expansion of the universe in accelerating, then the Hubble Time is smaller than the age of the universe.

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A Summary of Modern Cosmology, Note 3

The June 2010 edition of Sky & Telescope reports that space has been measured to be flat (1) to an accuracy of 1.0023 +/- 0.0055.

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