Theoretical Physics

"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."

"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."

"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. 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."

"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."

"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 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."

"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."

"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 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."

"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."

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

"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?"

"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..."

"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. 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" λ was introduced into the field equations in order to enable us to account theoretically for the existence of a finite mean density in a static universe. It now appears that in the dynamical case this end can be reached without the introduction of λ."

"The general theory of relativity considers physical space-time as a four-dimensional manifold whose line element coefficients g_{\mu \nu} satisfy the differential equationsG_{\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 ."

"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 ".""

"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. ). ..."

"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."

"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."

"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."

"This circumstance of an expanding universe is irritating. ...To admit such possibilities seems senseless to me."

"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..."

"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 ?"

"[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.""

"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} + ...), A = \pi r^2 (1 - \frac{Kr^2}{12} + ...)"

"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."

"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..."

"[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."

"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. 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."

"Mach's profound critique of the foundations of Newtonian mechanics played a key role in Einstein's development of . Mach's principle has also guided other developments in gravitation theory such as the ... It has inspired interesting experiments, such as the ... and continues to be of current interest ... Mach identified the essential epistemological shortcoming of the Newtonian foundations of physics, namely, that the intrinsic state of a particle in Newtonian mechanics, i.e. its mass, has no immediate connection with its extrinsic state in space and time, i.e. its position and velocity. Mach's observation can be re-stated in terms of the a priori independence of position (x) and momentum (p) of a particle in Newtonian mechanics."

"Most terrestrial motions are of such brief duration and extent, that it is wholly unnecessary to take into account the earth's rotation and the changes of its progressive velocity with respect to the celestial bodies. This consideration is found necessary only in the case of projectiles cast great distances, or in the case of the vibrations of , and in similar instances. When now Newton sought to apply the mechanical principles discovered since Galileo's time to the planetary system, he found that, so far as it is possible to form any estimate at all thereof, the planets, irrespectively of dynamic effects, appear to preserve their direction and velocity with respect to bodies of the universe that are very remote and as regards each other apparently fixed, the same bodies moving on the earth do with respect to the fixed bodies of the earth. The comportment of terrestrial bodies with respect to the earth is reducible to the comportment of the earth with respect to the remote heavenly bodies. If we were to assert that we knew more of moving objects than this their last-mentioned, experimentally-given comportment with respect to the celestial bodies, we should render ourselves culpable of a falsity. When, accordingly, we say, a body preserves unchanged its direction and velocity in space, our assertion is nothing more or less than an abbreviated reference to the entire universe. The use of such an abbreviated expression is permitted the original author of the principle, because he knows, that as things are no difficulties stand in the way of carrying out its implied directions. But no remedy lies in his power, if difficulties of the kind mentioned present themselves; if, for example, the requisite, relatively fixed bodies are wanting."

"... the theory of relativity makes it appear probable that Mach was on the right road in his thought that inertia depends upon a mutual action of matter. For we shall show in the following that, according to our equations, inert masses do act upon each other in the sense of the relativity of inertia, even if only very feebly. What is to be expected along the line of Mach's thought? 1. The inertia of a body must increase when ponderable masses are piled up in the neighborhood. 2. A body must experience an accelerating force when neighbouring masses are accelerated, and, in fact, the force must be in the same direction as that acceleration. 3. A rotating hollow body must generate inside itself a "Coriolis field," which deflects moving bodies in the sense of the rotation, and a radial centrifugal field as well. We shall now show that these three effects, which are to be expected in accordance with Mach's ideas, are actually present according to our theory, although their magnitude is so small that confirmation of them by laboratory experiments is not to be thought of."

"The so-called Mach's Principle is surely one of the most elusive concepts in physics. On one hand, Machian aspects have been present either explicitly or implicitly in theoretical astronomy, general physics, and dynamics from their Greek infancy up the present day ... On the other hand, most of practical physics is done, and successfully done, without ever thinking of the 'deep questions' connected with Mach's Principle. (The situation is similar in quantum theory, which functions extremely well using established prescriptions notwithstanding deep and unresolved questions about its interpretation, its measuring process, and its classical limit.)"

"Once one starts to admit anthropic interpretations of fine-tuning problems like the cosmological constant, it is clear that such a proposal might be made for other fine-tuning problems, such as the problem of the Higgs boson mass. Certainly, we would not be here if the Higgs boson mass, and hence also the and and and masses, were greatly bigger. If they were near the , for example, any collection of more than a few elementary particles would collapse into a Black Hole. More generally, if the elementary particle masses were scaled up by a factor N, the number of elementary particles in a star or planet would scale down like N–3, and for very modest N the stars would stop shining."

"The old cosmological constant problem is to understand why the is so small; the new problem is to understand why it is comparable to the present mass density. ... does not help with either; anthropic considerations offer a possibility of solving both. In theories with a that takes random initial values, the anthropic principle may apply to the cosmological constant, but probably to nothing else."

"There seems to be a vast landscape of possible universes. ... We live in one in which life is possible, but if the universe were only slightly different, beings like us could not exist. What are we to make of this fine-tuning? Is it evident that the universe, after all, was designed by a benevolent creator? Or does science offer a different explanation?"

"Extra spatial dimensions could be a good thing. ... In a nongravitational theory, the spacetime geometry is a rigid background on which the dynamics takes place. In that setup, the fact that we observe four-dimensional Minkowski spacetime is a compelling argument to formulate the theory in that background geometry. As you know very well, this is part of the story of the Standard Model. However, in a gravitational theory that abides by the general principles laid out by Einstein, the spacetime geometry is determined by the dynamical equations. In such a setup extra dimensions can make sense provided that the equations of the theory have a solution for which the geometry is the product of four-dimensional Minkowski spacetime and a compact manifold that is sufficiently small to have eluded detection. It turns out that there are many such solutions. Moreover, the details of the compact manifold play a crucial role in determining the symmetries and particle content of the effective low-energy theory in four dimensions, even when the compact dimensions are much too small to observe directly."

"Why try to unite the four forces in a single theory? Why not simply use Einstein’s theory of general relativity to govern big things and quantum mechanics for little ones? Some concepts, such as the Big Bang or how black holes form, live in both domains. When we combine equations of the four forces to describe these ideas, our answers usually end up being either zero or infinity. … Here’s where string theory comes to the rescue. By adding seven hidden dimensions to the familiar three and another for time, plus antiparticles and a mirror set of particles called superparticles, the math starts to make sense. The force of gravity is diluted because it permeates into one or more of the hidden dimensions. Dark matter and dark energy also may invisibly shape our universe from these phantom dimensions."

"... when non-perturbative phenomena are included, there is no problem from the string theory point of view in effecting continuous transitions between Calabi-Yau spaces of different topology. This shows that stringy ideas about geometry are really more general than those found in classical Riemannian geometry. The moduli space of Calabi-Yau manifolds should thus be regarded as a continuously connected whole, rather than a series of different ones individually associated with different topological objects ... Thus, questions about the topology of Calabi-Yau spaces must be treated on the same footing as questions about the metric on the spaces. That is, the issue of topology is another aspect of the the moduli fields. These considerations are relevant to understanding the ground state of the universe."

"Calabi-Yau manifolds admit Kähler metrics with vanishing Ricci curvatures. They are solutions of the Einstein field equation with no matter. The theory of motions of circles inside of a Calabi-Yau manifold provide a model of a conformal field theory. (It is called a σ-model in physics.) Because of this, Calabi-Yau manifolds are pivotal in superstring theory. ... It has long been argued that, in order to solve certain classic problems of unified gauge theories such as the gauge hierarchy problem, the 4-dimensional effective theory should admit an N = 1 supersymmetry. In a fundamental paper, Candelas-Horowitz-Strominger-Witten ... analyzed what the constraint of that N = 1 supersymmetry would mean for the geometry of the internal space X. They found that, for the most basic product models with N = 1 supersymmetry, the space X must be a Calabi-Yau manifold of complex dimension 3. Shortly afterwards, Strominger ... considered slightly more general models, allowing warped products. For these models, the N = 1 supersymmetry constraint results in a modification of the Ricci-flat equation of the earlier model."

"We do not know to what extent and how MOND affects nongravitational phenomena such as electromagnetism (EM). For example, if there is a consistent way to extend and apply the basic tenets to nongravitational physics."

"... the interpretation of MOND as modified inertia (MI) ... ... Who is afraid of modified inertia? The interpretation of MOND as MI was on the table from the very inception of MOND ..."

"Viewed simply, MOND is an algorithm that, with one additional fundamental parameter having units of acceleration, allows calculation of the distribution of the effective gravitational force in astronomical objects from the observed distribution of baryonic dark matter — and it works remarkably well. This is evidenced primarily by the use of the MOND algorithm in the determination of rotation curves of disk galaxies where the agreement with observed rotation curves is often precise, even in details. The existence of such an algorithm is problematic for CDM because this is not something that dissipationless dark matter on the scale of galaxies can naturally do; it would seem to require a coupling between dark matter and baryonic matter which is totally at odds with the perceived properties of CDM."

"Mild failures aside, it is clear that there is a broad range of masses, 106 — 1011 M☉, in which systems adhere to MOND in their systematics. This must be telling us something; logically there are the following possibilities. a) MOND is merely an efficient summary of the way DM is distributed in the said systems? b) MOND reveals the dependence of inertia on acceleration for small accelerations? c) MOND betrays hitherto unknown forces particularly effective at astronomical scales? d) MOND encapsulates departures from standard Newtonian-Einsteinian gravity theory at the mentioned scales?"

"Ten years ago, it was perfectly respectable to speculate that there was no such thing as dark matter, just a modification of gravity. (It couldn’t have been MOND alone, which was ruled out by clusters, but it could have been some more elaborate modification.) That’s no longer true. The Bullet Cluster and the CMB both provide straightforward evidence that there is gravity pointing in the direction of something other than the ordinary matter. The source for that gravity is “dark matter.” It could be simple, like an axion or a thermal relic, or it could be quite baroque, like TeVeS + sprinkles of other dark matter as required, but it’s definitely there."

"... Milgrom and those few who work on it, are quite aware of the pressing need to have a fully consistent theory that goes beyond the Newtonian non-relativistic limit to a theory that can be applied to cosmology. They don't have one. They fully admit it and they agree that this is a big gap, big lack in the theory. There it is. They do insist that on the scales of galaxies and smaller where it is intended to apply it works remarkably well, and they're right. There are just a few people working on this theory. The most active of the young people is Stacy McGaugh at the University of Maryland. If you ever get a chance you might be amused to talk to him."

"MOND is so successful that, as a minimum, it is telling us the exact functional form of the force in galaxies. Any theory of galaxy and structure formation must therefore be able to reproduce the MOND phenomenology."

"I've had conversations about MOND with several of the most imaginative theorists I know. Often it went like this: We would be talking about some sober mainstream problem and one of us would mention galaxies. We would look at each other with a glint of recognition and one of would say, "So you worry about MOND, too," as if admitting a secret vice. Then we would share our crazy ideas — because all ideas about MOND that are not immediately wrong turn out to be crazy."

"Fundamental to the idea of MOND is that it is an 'effective' theory, playing a role similar to Kepler's laws (as stressed by Felten 1984). The proponents of MOND have yet to develop the analogue of Newtonian mechanics to explain the effective theory. The absence of a full theory seriously limits the predictive power of MOND, and leads various authors to disagree as to what the observational consequences of this revision will be."