Academics From The United States

"[T]he distance at present isd_{\mathrm{max}}(t_0) = \frac{1}{H_0} \int_{0}^{1} \frac{dx}{x^2 \sqrt{\Omega_\Lambda+\Omega_K x^{-2}+\Omega_M x^{-3}}}...[T]here may have been a time before the radiation-dominated era in which there was nothing in the universe but , in which case the particle horizon distance would... be infinite. But as far as telescopic observations... [d_{max}(t_0)] gives the proper distance beyond which we cannot now see."

"Modern cosmological theories can exhibit horizons of two... types. which limit the distances at which past events can be observed or at which it will ever be possible to observe future events. These are called Rindler s and s, respectively."

"In 1929 Hubble announced... a "roughly linear" relation between and distance. ...His data points ...did not really support a linear relation. But in the early 1930s he had measured redshifts and distances out to the , with a redshift z \eqsim 0.02, corresponding to... 7,000 km/sec and a linear relation... was evident. The conclusion... the universe is really expanding. ...At the time of writing, the largest... z=6.96. It may eventually become possible to measure the expansion rate H(t) \equiv \dot{a}(t)/a(t) at times t earlier than the present, by observing the change in very accurately measured redshifts of individual galaxies over times as short as a decade."

"[T]he property of being a geodesic is invariant under coordinate transformations... so the path of the photon or particle will... be a geodesic in any spatial coordinate system..."

"Consider... [the formula given by special relativity for the magnitude of the ]P \equiv m_0 \sqrt{g_{ij}\frac{dx^i}{d\tau}\frac{dx^j}{d\tau}}...where d\tau^2 = dt^2 - g_{ij} dx^i dx^j. [This holds because in] a locally inertial Cartesian coordinate system, for which g_{ij} = \delta_{ij}, we have d\tau = dt\sqrt{1 - \mathbf {v}^2} where v^i = \frac{dx^i}{dt}... [The P] is evidently invariant under arbitrary changes in the spatial coordinates, so we can evaluate it... in Robertson-Walker coordinates. ...[T]o save work ...adopt a spatial coordinate system in which the particle position is near the origin x^i = 0, where \tilde{g}_{ij} = \delta_{ij} + \mathit0(\mathbf{x}), and we can therefore ignore the purely spatial components of \Gamma_{jk}^i of the . General relativity gives [the momentum]... with a metric g_{ij} = a^2(t)\delta_{ij}...P(t) \propto 1/a(t)... for any non-zero mass, however small... Hence, although for photons both m_0 and d\tau vanish... [the momentum relation] is still valid."

"[T]o extend this to the geometry of spacetime... include a term... in the spacetime line element, with a now an arbitrary function of time (known as the Robertson-Walker scale factor):d\tau^2 \equiv -g_{\mu\nu}(x) dx^\mu dx^\nu = dt^2-a^2(t)[d\mathbf{x}^2 + K \frac{(\mathbf{x} \cdot d\mathbf{x}^2)}{1-K\mathbf{x}^2}]"

"[In] the case of Euclidean space...ds^2 = a^2[d\mathbf{x}^2 + K \frac{(\mathbf{x} \cdot d\mathbf{x}^2)}{1-K\mathbf{x}^2}]...where K ="

"Consider the geometry of a three-dimensional homogeneous and isotropic space. ...[G]eometry is encoded in a metric g_{ij}(\mathbf{x}) (with i and j running over the three coordinate directions), or equivalently a line element ds^2 \equiv g_{ij} dx^i dx^j, with summation over repeated indices... ds is the proper distance between \mathbf{x} and \mathbf{x}+\mathbf{dx}, meaning... the distance measured by a surveyor who uses a... Cartesian [coordinate system] in a small neighborhood of... point \mathbf{x}.) One... homogeneous isotropic three-dimensional space with positive definite lengths is flat space, with line elementds^2=d\mathbf{x}^2...The coordinate transformations that leave this invariant are... ordinary three-dimensional rotations and translations. ...Another ...possibility is a four-dimensional with some radius a, with line elementds^2=d \mathbf{x}^2+dz^2,\;\;z^2 + \mathbf{x}^2 = a^2,...Here the transformations that leave the line element invariant are four-dimensional rotations; the direction of \mathbf{x} can be changed to any other direction by a four-dimensional rotation that does not change z. ...[T]he only other possibility (up to a coordinate transformation) is a hyperspherical surface in four-dimensional , with line elementds^2 = d\mathbf{x}^2 - dz^2,\;\;z^2 - \mathbf{x}^2 = a^2,...where a^2 is (so far) an arbitrary positive constant. The coordinate transformations that leave this invariant are four-dimensional pseudo-rotations, just like s, but with z instead of time."

"Almost all of modern cosmology is based on this Robert-Walker metric, at least as a first approximation."

"The visible universe seems the same in all directions... larger than about 300 million light years. The ... (...[~]one part in 105) in the cosmic microwave background... radiation... traveling... [~]14 billion years, supporting the conclusion..."

"I hope the readers will let me know of any mistakes... I will post them on... zippy.ph.utexas.edu/~weinberg/corrections.html."

"Where I knew them I have included references to... the Cornell archive, ...arxiv.org, as well as to the published literature. ...I have quoted the latest measurements of cosmological s known to me, in part... to give the reader a sense of what is... observationally possible. ...I have not tried to combine measurements from observations of different types because... it would [not] add... additional physical insight and any... cosmological concordance would... soon be out of date."

"I decided to exclude material that was highly speculative... cosmological theory in higher dimensions... anthropic reasoning... holographic cosmology... conjectures about the details of inflation, or many other new ideas. ...The present book is largely concerned with ...mainstream cosmology: ...inflation driven by one or more scalar fields ...followed by a big bang dominated by radiation, , baryonic matter, and ."

"So much has happened in cosmology since the 1960s that this book... bears little resemblance to... Gravitation and Cosmology. On occasion I refer back to (..."G&C") for material that does not seem worth repeating... Classical general relativity has not changed much since 1972 (apart from a great strengthening of its experimental verification) so it did not seem necessary to cover gravitation... I provide a brief introduction in Appendix B. Other appendices deal with technical material... I have also supplied a glossary of symbols..."

"I have tried... to present analytic calculations of cosmological phenomena, and not just report results obtained elsewhere by numerical computation. The calculations... in the literature... necessarily take many details into account, which either make an analytic treatment impossible, or obscure the main physical features of the calculation. Where this is the case, I have not hesitated to sacrifice some degree of accuracy for greater transparency."

"In 1999 I finished my three volume book on the quantum theory of fields (..."QTF"), and... set... the task of learning... the theory underlying the great progress in cosmology in the previous two decades. ...Review articles ...gave good summaries of the data, but ...often quoted formulas without ...derivation, and sometimes ...without reference to the original derivation. Occasionally the formulas were wrong, and extremely difficult for me to rederive. ...[O]riginal ...articles sometimes had gaps in their arguments, or relied on hidden assumptions, or used unexplained notation. Often massive computer programs had taken the place of analytic studies. In many cases... it was easiest to work out the relevant theory myself. This book is the result. Its aim... self-contained explanations of the ideas and formulas... used and tested in modern cosmological observations."

"I'm offended by the kind of smarmy religiosity that's all around us, perhaps more in America than in Europe, and not really that harmful because it's not really that intense or even that serious, but just... you know after a while you get tired of hearing clergymen giving the invocation at various public celebrations and you feel, haven't we outgrown all this? Do we have to listen to this?"

"There is one constant that seems to be fine tuned...and that is dark energy."

"I have a friend — or had a friend, now dead — Abdus Salam, a very devout Muslim, who was trying to bring science into the universities in the Gulf states and he told me that he had a terrible time because, although they were very receptive to technology, they felt that science would be a corrosive to religious belief, and they were worried about it... and damn it, I think they were right. It is corrosive of religious belief, and it's a good thing too."

"Maybe at the very bottom of it... I really don't like God. You know, it's silly to say I don't like God because I don't believe in God, but in the same sense that I don't like Iago, or the Reverend Slope or any of the other villains of literature, the god of traditional Judaism and Christianity and Islam seems to me a terrible character. He's a god who will... who obsessed the degree to which people worship him and anxious to punish with the most awful torments those who don't worship him in the right way. Now I realise that many people don't believe in that any more who call themselves Muslims or Jews or Christians, but that is the traditional God and he's a terrible character. I don't like him."

"Many people do simply awful things out of sincere religious belief, not using religion as a cover the way that Saddam Hussein may have done, but really because they believe that this is what God wants them to do, going all the way back to Abraham being willing to sacrifice Isaac because God told him to do that. Putting God ahead of humanity is a terrible thing."

"Finally, learn something about the history of science, or at a minimum the history of your own branch of science. The least important reason for this is that the history may actually be of some use to you in your own scientific work."

"As you will never be sure which are the right problems to work on, most of the time that you spend in the laboratory or at your desk will be wasted. If you want to be creative, then you will have to get used to spending most of your time not being creative, to being becalmed on the ocean of scientific knowledge."

"Another lesson to be learned, to continue using my oceanographic metaphor, is that while you are swimming and not sinking you should aim for rough water."

"I managed to get a quick PhD — though when I got it I knew almost nothing about physics. But I did learn one big thing: that no one knows everything, and you don't have to."

"The best military historians in fact do recognize the difficulty in stating rules of generalship. They do not speak of a science of war, but rather of a pattern of military behavior that cannot be taught or stated precisely but that somehow or other sometimes helps in winning battles. This is called the art of war. In the same spirit I think that one should not hope for a science of science, the formulation of any definite rules about how scientists do or ought to behave, but only aim at a description of the sort of behavior that historically has led to scientific progress—an art of science."

"The last thirty years of Einstein's life were largely devoted to a search for a so-called unified field theory that would unify James Clerk Maxwell's theory of electromagnetism with the general theory of relativity, Einstein's theory of gravitation. Einstein's attempt was not successful, and with hindsight we can now see that it was misconceived. Not only did Einstein reject quantum mechanics; the scope of his effort was too narrow. ... Nevertheless Einstein's struggle is our struggle today. It is the search for a final theory."

"It is with Isaac Newton that the modern dream of a final theory really begins."

"The years since the mid-1970s have been the most frustrating in the history of particle physics. We are paying the price of our own success: theory has advanced so far that further progress will require the study of processes at energies far beyond the reach of existing facilities. In order to break out of this impasse, physicists began in 1982 to develop plans for a scientific project of unprecedented size and cost, known as the Superconducting Super Collider."

"It is... convenient to state Coulomb's law in modern terms, first used... by James Clerk Maxwell. The electric force... is always proportional to the electric charge... We call the factor of proportionality the electric field so...Electric force... = Electric charge... x Electric field"

"Coulomb used a... device of his own invention, the torsion balance, to measure forces between small pith balls. The inverse-square law was found to hold accurately..."

"(...Newton's theory... is now known only to be an approximation... for particles... not moving too fast and gravitational forces... not too strong. ...It is one of the consequences of General Relativity that gravitation is produced by and acts on energy as well as mass, so that it even affects particles of zero mass, like the photon."

"Early speculation on about electric forces relied... on an analogy with Newton's theory of gravitational forces. At the end of Principia, Newton described gravitation as a cause that acts on the sun and the planets "according to the quantity of solid matter which they contain and propagates on all sides to immense distances, decreasing always as the inverse square of the distances." ...It was irresistible to guess that the electric force might obey a similar law, also proportional to the inverse square of the distance... with charge playing the role that mass plays..."

"Thomson used Newton's Second Law to obtain a general formula... to interpret measurements of the cathode-ray deflection... produced by... electric or magnetic forces... In his cathode ray tube, the ray particles pass through... the deflection region... subjected to electric and magnetic forces... at right angles to their original direction... then through a much longer force-free... drift region... in which they drift freely until they hit the end of the tube... [a] glowing spot... The forces exerted on the cathode ray particles give them an acceleration at right angles to the axis of the tube, so... the particles have a small component of velocity at right angles to their original motion... equal to the product of the acceleration and the time... in the [very short] deflection region... [T]he downward displacement of the ray when it hits the end of the tube is the downward velocity produced in the deflection region times the length of time... in the drift region... [T]he electric force... on a particle is proportional to the [particle's] electric charge... [U]nlike the electric force, the magnetic force... on a particle is proportional to the particle's velocity as well as its charge. By measuring... deflections due to... [both] forces, Thomson... could determine both the ray-particle velocities and the ratio of their charge and mass."

"When several forces act... the total force is the sum... each component of the total force is the sum of the corresponding components of the individual forces."

"Velocity, acceleration, and force are vectors... they have direction as well as magnitude. It is often convenient to describe... [vectors] in terms of their components along specified directions. ...Components of vectors can be negative as well as positive ...Newton's Second Law applies separately to each component... it says... the component of force in any direction is equal to the mass times the corresponding component of acceleration."

"When the velocity and the acceleration are changing, we can define the instantaneous velocity or acceleration at any moment as the average values... over a vanishingly small time interval centered on that moment. Newton's Law actually relates the force to the instantaneous acceleration."

"As a special case of Newton's Second Law, a body... when acted on by zero force, will experience zero acceleration—that is, it will move with constant velocity. Newton listed this... as the First Law... The Third Law... action equals reaction: If one body exerts a force on another... the second... exerts an equal force in the opposite direction on the first."

"Acceleration is the rate of change of . ...The units are ...velocity per unit time, or distance-per-time per time. ...[F]alling bodies ...near ...earth fall with an acceleration or 9.8 meters-per-second per second ...after the first second ...falling at speed ...9.8 meters per second, after two seconds... 19.6 ...and so on. [T]he units of velocity are length/time... and units of acceleration... (distance/time)/time, or equivalently distance/time2 ...[T]he acceleration near... Earth would be written 9.8 m/sec2 for short."

"The laws of motion... were set out by... Newton at the beginning of... the Principia. ...[T]he key principle is ...in the Second Law... paraphrased as... the force to give an object a certain acceleration is proportional to the product of the and the ."

"Thomson... exerted electric and magnetic forces on the rays and measured the amount... deflected."

"[I]n 1897 Thomson... detected a deflection... by electric forces between the rays and the electrified metal plates. ...due largely to the use of better vacuum pumps ...to where the effects of residual gas ...became negligible. (Some evidence for... deflection was [also] found... by Goldstein.) [D]eflection was toward the positively charged plate... away from the negatively charged one, confirming Perrin... that the rays carry negative electric charge."

"Perrin... showed in 1895 that the rays deposit negative electrical charge on a charge collector... inside... the tube."

"Hertz showed... the... rays were not appreciably deflected by electrified metal plates. This seemed to rule out... electrically charged particles... Hertz concluded the rays were some sort of wave... the nature of light was... not well understood, and a magnetic deflection did not seem impossible. In 1891 Hertz made a further observation... to support the wave theory... The rays could penetrate thin foils of gold and other metals, much as light penetrates glass. ...We know now that... the... particles were traveling so fast, and the electric forces were so weak... the deflection was too small to observe."

"Plücker's student J. W. Hittorf... observed that solid bodies... near a small cathode would cast shadows... [and] deduced... the rays travel... in straight lines."

"In 1858 Johann Heinrich Geissler... invented a pump that used columns of mercury as pistons and consequently needed no gaskets. ...Geissler's pump was used... by ... [M]etal plates inside a glass tube were connected to a powerful source of electricity. ...[W]hen almost all of the air was evacuated ...the light disappeared through most of the tube, but a greenish glow appeared ...near the cathode. ...A few years later, ... introduced a name... s. We know now that these rays are streams of electrons. ...But this was far from obvious to nineteenth century physicists. ...Plücker ...observed that the position of the glow on the walls of the tube could be moved by ...a magnet ..."

"In 1709 Hauksbee observed that when air inside a glass vessel was evacuated... [to] 1/60 normal air pressure and the vessel was attached to... frictional electricity, a strange light would be seen... Flashes... similar... had... been noticed in the partial vacuum above... mercury in barometers. ...[T]oday we know ...[w]hen an electric current flows through a gas, the electrons knock into the gas atoms and give up some... energy... reemitted as as light. Today's fluorescent lights and neon signs are based on the same principle... but even at 1/60 atmospheric pressure the air interfered too much with the flow of electrons to allow their nature to be discovered. Real progress became possible only when the gas... could be removed..."

"The reader may... wonder why when amber is rubbed with fur the electrons go from the fur to the amber, but when glass is rubbed with silk the electrons go from the glass to the silk? ...[W]e still don't know. The question involves the physics of surfaces of complex solids... In a purely empirical way, there has been developed... the triboelectric sequence... The electrification is most intense for objects... well separated in the... sequence. ...It is ironic that we still do not have a detailed understanding of frictional electrification, even though it was the first... to be studied... But... often... science progresses... by selecting problems that are as free as possible from irrelevant complications and... provide opportunities to get at fundamental principles..."

"Electrons are believed to be absolutely stable, and protons and neutrons (when bound...) live at least 1030 years. With few exceptions, all other particles have very short lifetimes..."

"The s and s in nuclei, like the electrons surrounding the nuclei, can be excited to states of higher energy or... fall back to a state of lower energy, but the energies... are typically a million times those need to excite the electrons..."