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Ideas and Phenomena of General Relativity Not everyone who reads this report will be familiar with the beautiful and simple ideas that underlie Einstein's general relativity or with the vast range of phenomena for which gravitational physics is important. In this chapter the Committee on Gravitational Physics briefly describes some key ideas in general relativity and some key phenomena of gravitational physics that are discussed in Chapter 3. A. .~.~Y .~.~./hS IN ~ EN [~ ~./h.~. ~ EL.~,4.~.! \71 r.~\r Gravity Is Geometry. Gravity is the geometry of four-dimensional spacetime. That is the central idea of Einstein's 1915 general theory of relativity the clas- sical theory of relativistic gravitation. It is not difficult to imagine a curved space. The curved surface of a sphere or a car fender are two-dimensional examples. But gravitational effects arise from the curvature offour-dimensional spacetime with three space dimensions and one time dimension. It is more difficult to imagine a notion of curvature involving time, but the Global Position- ing System (described in Box 2.1) provides an everyday practical example of its implications. In Newtonian physics two identically constructed clocks run at the same rate no matter what their positions in space. But in relativity a stationary clock above Earth's surface runs fast compared to a clock at the surface by 1 part in ten thousand billion for each kilometer in height. That tiny difference is the result of the curvature of spacetime produced by the mass of Earth a small effect indeed, but large enough that the Global Positioning System would fail in a few minutes 24
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IDEAS AND PHENOMENA OF GENERAL RELATIVITY 25 BOX 2.1 General Relativity and Daily Life There is no better illustration of the unpredictable application of fundamental science in daily life than the story of general relativity and the Global Positioning System (GPS). Built at a cost of more than $10 billion mainly for military naviga- tion, the GPS has been rapidly transformed into a thriving, multibillion-dollar com- mercial industry. GPS is based on an array of 24 Earth-orbiting satellites, each carrying a precise atomic clock. With a hand-held GPS receiver that detects radio emissions from any of the satellites that happen to be overhead, a user can deter- mine latitude, longitude, and altitude to an accuracy that currently can reach 50 feet, and local time to 50 billionths of a second. Apart from the obvious military uses, the GPS is finding applications in airplane navigation, wilderness recreation, sailing, and interstate trucking. Even Hollywood has met the GPS, pitting James Bond in "Tomorrow Never Dies" against an evil genius able to insert deliberate errors into the system and send British ships into harm's way. Because the satellite clocks are moving in high-speed orbits and are far from Earth, they tick at different rates than clocks on the ground. Gravity and speed contribute comparable amounts to the total discrepancy. The offset is so large that, if left uncompensated, it would lead to navigational errors that would accumu- late at a rate greater than 6 miles per day. In GPS, the relativity is accounted for by electronic adjustments to the rates of the satellite clocks, and by mathematical corrections built into the computer chips that solve for the user's location. Schematic illustration of segments used in operation of the Global Positioning System. (Adapted from a figure courtesy of the Aerospace Corporation.)
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26 GRAVITATIONS PHYSICS: E~LOHNG THE STRUT OF SPACE ^D TIME BOX 2.2 Newtonian and Einstein Gravity Compared In Newton's 300-year-old theory of gravity, a mass attracts other masses with a force of gravity that decreases as the inverse of the square of the distance be- tween them. Masses move in response to the forces acting on them, including gravitational forces, according to Newton's laws of motion. In Einstein's 1915 general theory of relativity, a mass curves the one time di- mension and three space dimensions of spacetime according to Einstein's equa- tion. The spacetime curvature is greatest near the mass and vanishes at a dis- tance. Other masses move along the straightest possible paths in this curved spacetime. Einstein's theory thus expresses both the gravitational effect of mass and the response of mass to that effect in terms of the geometry ofspacetime. The Newtonian idea of a gravitational force acting at a distance between bodies was replaced by the idea of a body moving in response to the curvature of spacetime. In relativity, mass and energy are the same thing according to Einstein's fa- mous E= mc2 relation. Not only mass but also any form of energy will curve spacetime. Gravity itself carries energy, and even small propagating ripples in spacetime cause further curvature. The equations of Einstein's theory keep track of this complex feedback interrelationship between energy and curvature. Newton's theory of gravity is not wrong. It is a correct approximation to Ein- stein's theory when spacetime curvature is small and the velocities of masses are much smaller than the velocity of light. The first general relativistic corrections beyond Newtonian theory (called "post-Newtonian") are responsible for small devi- ations to the motion of light and to the orbits of the planets from those predicted by Newton. Measurements of these deviations are among the most precise tests of general relativity. The founders of gravitational physics Isaac Newton (1642-1727) and Albert Einstein (1879- 1955). (Courtesy of the American Institute of Physics Emilio Segre Visual Archives.)
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IDEAS AND PHENOMENA OF GENERAL RELATIVITY 27 if this effect of the spacetime curvature implied by general relativity were not taken into account. Mass Produces Spacetime Curvature, and Spacetime Curvature Determines the Motion of Mass. Einstein's equation makes a quantitative connection be- tween mass and the amount of curvature of spacetime it produces. (See Box 2.2.) Just as Earth curves spacetime near its surface, so too does the Sun produce a slight curvature of spacetime in its vicinity. The curvatures produced near the surface of a black hole or a neutron star, or at the beginning of the universe, are much greater. These are realms of strong gravitational physics. According to general relativity, Earth follows an elliptical orbit about the Sun, not because it is attracted to the Sun by a gravitational force, but because it is following the straightest possible path through the spacetime that has been curved by the Sun. The Principle of Equivalence. General relativity predicts that a tiny asteroid, or indeed any other body, could follow the same path around the Sun as Earth does. Each body is following a path determined by the geometry of spacetime, not by its mass. This universality of free fall called the principle of equiva- lence is one of the foundations of general relativity. It is one of the most accurately tested predictions in all of physics. The equality of accelerations of different bodies in the curved spacetime of the Sun has been verified to a few parts in a thousand billion. Were a violation of this equality ever detected it would signal either new physical interactions or a revision in our ideas about the nature of space, time, and gravity. Described below are some important phenomena in gravitational physics. Strong gravitational physics plays a central role in all these examples. The essential features of general relativity are present, and the Newtonian approxima- tion is inadequate. Gravitational Waves. Einstein's theory predicts that ripples in spacetime curvature can propagate with the speed of light through otherwise empty space- a gravitational wave. Mass in motion is the source of a gravitational wave. In turn, gravitational waves can be detected through the motion of masses produced as the ripple in spacetime curvature passes by. The weak coupling of mass to spacetime curvature means that an extraordinarily energetic, strong-gravity event, such as the coalescence of two massive stars, is required to produce gravitational waves copious enough to be detected by gravitational wave receivers now under construction. By contrast, the indirect detection of gravitational waves from the Hulse-Taylor binary pulsar system resulted from the observation of the minus- cule shortening of the period of a pair of neutron stars orbiting about each other.
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28 GRAVITATIONS PHYSICS: E~LOHNG THE STRUT OF SPACE ^D TIME This weak coupling of gravity to matter is the reason that gravitational waves have not yet been detected directly. But this weak coupling also means that the universe is largely transparent to gravitational waves. Once produced, little is absorbed. A gravitational wave receiver could therefore enable researchers to see phenomena in the universe that are visible in no other way. Black Holes. Perhaps no other concept in physics has made as deep an impact on public consciousness as has the black hole. General relativity predicts that a black hole is created whenever mass is compressed into a volume small enough that the gravitational pull at the surface is too large for anything to escape, no matter how fast it accelerates. The surface of a black hole called its event horizon is like a one-way membrane. Mass, information, and observers can fall into it, but nothing can emerge from it. Although black holes in nature are typically produced by complex gravitational collapse, such as gave rise to binary x-ray sources or as occurred at the centers of galaxies, general relativity predicts that they are remarkably simple objects completely characterized by just a few parameters. Black holes exhibit many properties of ordinary objects: they have mass and spin and can have electric charge; they can oscillate, change shape, show tides, and emit gravitational radiation; they can exhibit electric polarizability, resistiv- ity, eddy currents, and threaded magnetic fields; they can act as generators and engines for the most energetic phenomena in the universe. Yet all this richness of physics is described cleanly by the Einstein equation coupled to ordinary matter. As S. Chandrasekhar put it: "The black holes of nature are the most perfect macroscopic objects there are in the universe: the only elements in their con- struction are our concepts of space and time. And since the general theory of relativity provides only a single unique family of solutions for their descriptions, they are the simplest objects as well." (The Mathematical Theory of Black Holes, Oxford University Press, New York, 1992, p. 1.) According to quantum mechanics, black holes exhibit yet more remarkable properties as participants in the second law of thermodynamics. They possess an entropy proportional to their area whose statistical mechanical origin is begin- ning to be understood. They radiate like blackbodies with a temperature in- versely proportional to their mass. Thus, as they radiate they heat up and radiate even more. They may radiate completely away, producing for a brief explosive moment the strongest spacetime curvatures since the big bang. The Universe and the Big Bang. Gravity governs the structure and evolution of the universe on the largest scales of space and time. This is true even though gravity is the weakest of the four fundamental forces. Gravitation is universal, acts at long range, and cannot be canceled since it has no negative "charges." Cosmology and gravitational physics are thus inextricably linked. From cosmo
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IDEAS AND PHENOMENA OF GENERAL RELATIVITY 29 logical observations and Einstein's theory, the future fate of the universe can be extrapolated and its origins reconstructed. Galaxies the basic building blocks of the present universe are distributed uniformly on the largest distance scales. They recede from one another in a way that shows that the universe is expanding. Observations show that the universe was even simpler earlier that it is now. The origin of the universe was an initial state of extremely high density, pressure, and spacetime curvature about 13 bil- lion years ago the big bang. Although extreme in these measures, the big bang was remarkably regular. It was an explosive event that happened everywhere in space at the same time, producing matter in nearly perfect thermal equilibrium. Such a uniform, expanding universe is described by solutions of Einstein's equa- tion known as the Friedmann-Robertson-Walker (FRW) cosmological models. They are characterized by a few cosmological parameters whose values are the subject of ever refined observational searches. The simplest FRW models come in two varieties: models in which space is closed like the surface of a sphere, and models in which space is unlimited or open. The closed models end in a finite time in a "big crunch," whereas the open models expand forever. The closed models have a higher density than the open ones, and the dividing density be- tween them is called the critical density to close the universe. The real universe cannot exhibit exactly the perfect uniformity of the FRW models, since small quantum fluctuations away from uniformity must have occurred. These tiny "seeds" grew by the action of gravitational attraction to form the galaxies and stars we see today. Cosmic Backgrounds. Matter cooled as the universe expanded from its initial hot beginning. Protons and neutrons formed in the first microsecond after the start of the big bang; during the first few minutes they combined to form primordial nuclei, chiefly hydrogen and helium. About 300,000 years after the big bang, nuclei combined with electrons to make atoms. Once most of the electrons combined into atoms, matter was cool enough to be transparent to light. This light from the big bang has been propagating to us ever since. The subse- quent expansion has cooled it to a temperature of only 2.73 degrees above abso- lute zero, but it still comes toward us from every direction, forming the cosmic background radiation. This light from the early universe is detectable by sensi- tive instruments on the ground and in space, giving the most compelling evidence for the big bang. Small variations of a few microdegrees that are observed in the temperature of the cosmic background radiation are evidence for the slight initial concentrations of density that grew to be the galaxies today. From the details of these fluctuations their amplitude, angular distribution, and spectrum we can learn a great deal about the universe. If we could observe the similar background of gravitational waves, we could see back to the earliest moments of the big bang.
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30 GRAVITATIONS PHYSICS: E~LOHNG THE STRUT OF SPACE ED TIME Binary Pulsars. Gravitational physics is central to some of the most dra- matic and large-scale phenomena in nature. The big bang, black holes, explosive gravitational collapse, quasars, pulsars, and x-ray sources are all examples. Yet because gravity couples universally to all matter, its effects are in principle ob- servable in any physical system. Just as remarkable, just as beautiful, and just as confirming as the dramatic phenomena mentioned above are the minute, pre- cisely observable predictions of relativistic gravity for the deviations of the paths of orbiting bodies from the laws of Newton. These effects have been observed with impressive accuracy in the solar system. They are observed even more cleanly in binary neutron stars pairs of neutron stars orbiting about each other. Neutron stars are extraordinarily compact somewhat more than the mass of the Sun in a radius of 10 kilometers. Spacetime in their vicinity is more highly curved than in any place in the universe other than the big bang and black holes, and binary neutron stars are therefore among the best laboratories for precision tests of general relativity. Their orbits can be observed when one of the neutron stars is a pulsar a magnetized object whose rotation can be observed from the radio waves it emits, received at Earth as precisely periodic signals. More than 1000 pulsars are known, and 50 are in binary systems with neutron star or white dwarf companions. Many of them are extraordinarily accurate clocks. The rotational (spin) period of the Hulse-Taylor binary pulsar PSR1913+16, for ex- ample, is 0.059029997929613 _ 0.000000000000007 seconds. By noting the changes in this period induced by the pulsar's orbital motion over decades, the effects of general relativity can be precisely observed. Singularities. Einstein's classical theory predicts the formation of a singu- larity in the interior of a massive body whose gravity collapses it to a sufficiently compact volume. A singularity is a region of the universe where a classical description breaks down because it predicts infinite spacetime curvatures or den- sities of matter. Singularities limit the predictive ability of classical general relativity and are therefore places where we can expect to find new physics. But there is considerable evidence that singularities produced in any realistic collapse are hidden inside the event horizons of black holes where they cannot interfere with the predictability of physics on the outside. The idea that this always happens is called the cosmic censorship conjecture. Proving or disproving it is one of the outstanding challenges of general relativity theory. The singularities of gravitational collapse may be hidden inside black holes, but Einstein's theory also predicts that the universe began in a singularity the big bang whose consequences are all about us. The Small-Scale Structure of Space and Time. The union of the two most significant developments of 20th-century physics general relativity and quan- tum theory is one of the greatest challenges of contemporary theoretical phys- ics. The result of this union a quantum theory of gravity will have implica
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IDEAS AND PHENOMENA OF GENERAL RELATIVITY 3 lions as profound for our understanding of spacetime on small scales as Einstein's theory did for that understanding on large scales. The frontier of small scales for quantum gravitational phenomena is marked by the combination of the quantum of action it, the velocity of light c, and Newton's gravitational constant G. called the Planck length: ~(hG / 34~/2 10-33 The corresponding Planck energy is about ten thousand trillion times greater than the energy reached by the world's largest accelerators. Yet these energies, with their accompanying enormous curvatures, occurred at the big bang and occur in the final stages of gravitational collapse. An understanding of spacetime (or whatever replaces it) on these extreme scales is necessary to understand these central phenomena of astrophysics. But the small-scale structure of spacetime is also central to the quest for a unified theory of the fundamental interactions, because it is only there, where gravity is as strong as any other force, that the full symmetry between these interactions is likely to emerge. There are currently two main approaches to constructing this new theory. In one, Einstein' s picture of gravity as spacetime geometry is fundamental so that a quantum theory of gravity brings with it a quantum theory of geometry. At the Planck scale, quantum excitations of geometry are structured like a branched polymer, and familiar quantities such as lengths and areas can assume only dis- crete values. It is only because the basic discrete unit of length the Planck length is so small that space can be approximated by a continuum under ordi- nary circumstances. The other approach to quantum gravity unifies gravity with all the other forces and matter in a natural way. The basic idea is that although elementary particles appear point-like, they are actually excitations of a one-dimensional extended object called a string. One mode of oscillation of the string is a gravi- ton, a quantum of gravity, while other modes are photons, electrons, quarks, and so on. Furthermore, the interactions between these particles are reproduced by a splitting and joining interaction between the strings. We thus obtain a compelling and beautiful unified picture of particles and their interactions, including gravity. Both of these approaches are still being developed, and the coming decade promises a much more detailed picture of space and time at small scales.
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