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- The Large Scale Structure of Space-Time - Stephen Hawking & G.F.R.Ellis
- The Nature of Space and Time
The Nature of Space and Time
These are usually expressed in the form ofdifferential equations. Secondly, there is the problem of the boundary conditions for these equations, and the global nature of their solutions. This involves thinking about the edge of space-time in some sense. These two parts may not be independent. Indeed it has been held that the local laws are determined by the large scale structure of the universe.
This view is generally connected with the name of Mach, and has more recently been developed by Dirac , Sciama , Dicke , Hoyle and Narlikar , and others. We shall adopt a less ambitious approach: we shall take the local physical laws that have been experi- mentally determined, and shall see what these laws imply about the large scale structure of the universe.
There is of course a large extrapolation in the assumption that the physical laws one determines in the laboratory should apply at other points of space-time where conditions may be very different. If they failed to hold we should take the view that there was some other physic,al field which entered into the local physical laws but whose existence had not yet bl'.
In fact most of our results will be independent of the detailed nature of the physical laws, but will merely involve certain general properties such as the description of space-time by a pseudo-Riemannian geometry and the positive definiteness of tlnt. The fundamental interactions at present known to physics can be divided into four classes: the strong and weak nuclear interactions, electromagnetism, and gravity. Nevertheless it plays the dominant role in shaping the large scale structure of the universe.
For reasons explained in chapters 1 and 3, we base our treatment on Einstein's General Theory of Relativity. This theory leads to two remarkable pre- dictions about the universe: first, that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singula.
Our discussion is principally aimed at developing these two results. They depend primarily on two areas of study: first, the theory of the behaviour of families of timelike and null curves in space-time, and secondly, the study of the nature of the various causal relations in any space-time.
We consider these subjects in detail. In addition we develop the theory of the time-development of solutions of Einstein's equations from given initial data. This book is based in part on an Adams Prize Essay by one of us S. Many of the ideas presented here are due to R. Penrose and R. Geroch, and we thank them for their help. We have benefited from dis- cussions and suggestions from many of our colleagues, particularly B.
Carter and D. Our thanks are due to them also. Oambridge S. Hawking January G. Gravity on the other hand appears to be always attractive. Thus the gravitational fields of all the particles in a body add up to produce a field which, for sufficiently large bodies, dominates over all other forces. Not only is gravity the dominant force on a large scale, but it is a force which affects every particle in the same way. This universality was first recognized by Galileo, who found that any two bodies fell with the same velocity.
This has been verified to very high precision in more recent experiments by Eotvos, and by Dicke and his collabo- rators Dicke It has also been observed that light is deflected by gravitational fields. Since it is thought that no signals can travel faster than light, this means that gravity determines the causal structure of the universe, i.
These properties of gravity lead to severe problems, for if a suffi- ciently large amount of matter were concentrated in some region, it could deflect light going out from the region so much that it was in fact dragged back inwards.
This was recognized in by Laplace, who pointed out that a body of about the same density as the sun but times its radius would exert such a strong gravitational field that no light could escape from its surface. That this should have been predicted so early is so striking that we give a translation of Laplace's essay in an appendix.
One can express the dragging back of light by a massive body more precisely using Penrose's idea of a closed trapped surface. Consider a sphere fT surrounding the body. At some instant letfT emit a flash of light. At some later time t, the ingoing and outgoing wave fronts from fT will form spheres. In a normal situa- tion, the area of. However if a suffi- ciently large amount of matter is enclosed within fT, the areas of.
The surfacefT is then said to be a closed trapped surface. As t increases, the area of S; will get smaller and smaller provided that gravity remains attractive, i. This suggests that something goes badly wrong.
We shall in fact show that in such a situation a space-time singularity must occur, if certain reasonable conditions hold. One can think of a singularity as a place where our present laws of physics breakdown. Alternatively, one can think of it as representing part of the edge of space-time, but a part which is at a finite distance instead ofat infinity. On this view, singularities are not so bad, but one still has the problem of the boundary conditions.
In other words, one does not know what will come out of the singularity. At some instant, the sphere 9" emits a flash of light. At a later time, the light from a point p forms a sphere sP around p, and the envelopes 51 and 9". If the areas of both 9"1 and 9". There are two situations in which we expect there to be a sufficient concentration of matter to cause a closed trapped surface.
The first is in the gravitational collapse of stars of more than twice the mass of the sun, which is predicted to occur when they have exhausted their nuclear fuel. In this situation, we expect the star to collapse to a singu- larity which is not visible to outside observers.
The second situation is that of the whole universe itself. Recent observations of the microwave background indicate that the universe contains enough matter to cause a time-reversed closed trapped surface. This implies the exist- ence of a singularity in the past, at the beginning of the present epoch of expansion of the universe.
This singularity is in principle visible to us. It might be interpreted as the beginning of the universe. The predic- tions of this theory are in agreement with all the experiments so far performed. However our treatment will be sufficiently general to cover modifications of Einstein's theory such as the Brans-Dicke theory.
While we expect that most of our readers will have some acquain- tance with General Relativity, we have endeavoured to write this book so that it is self-contained apart from requiring a knowledge of simple calculus, algebra and point set topology. We have therefore devoted chapter 2 to differential geometry. Our treatment is reason- ably modern in that we have formulated our definitions in a manifestly coordinate independent manner.
However for computational con- venience we do use indices at times, and we have for the most part avoided the use of fibre bundles. The reader with some knowledge of differential geometry may wish to skip this chapter. In chapter 3 a formulation of the General Theory of Relativity is given in terms of three postulates about a mathematical model for space-time. This model is a manifold J with a metric g of Lorentz signature. The physical significance of the metric is given by the first two postulates: those of local causality and of local conservation of energy-momentum.
These postulates are common to both the General and the Special Theories of Relativity, and so are supported by the experimental evidence for the latter theory. The third postulate, the field equations for the metric g, is less well experimentally established. However most of our results will depend only on the property of the field equations that gravity is attractive for positive matter densities. This propertyis common to General Relativity and some modifications such as the Brans-Dicke theory.
In chapter 4, we discuss the significance of curvature by considering its effects on families of timelike and null geodesics. These represent the paths of small particles and of light rays respectively. The curva- ture can be interpreted as a differential or tidal force which induces relative accelerations between neighbouring geodesics.
If the energy- momentum tensor satisfies certain positive definite conditions, this differential force always has a net converging effect on non-rotating , families ofgeodesics. One can show by use ofRaychaudhuri's equation 4. To see the significance of these focal points, consider a one-dimen- sional surface.
Then there will be some curve from. Clearly this curve will be a geodesic, Le. In the situation shown in figure 2, there are in fact three geodesics orthogonal to. The geodesic through the point r is clearly not the shortest curve from.
The line pr cannot be the shortest line from p to.
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Einstein said that the most incomprehensible thing about the universe is that it is comprehensible. But was he right? Can the quantum theory of fields and Einstein's general theory of relativity, the two most accurate and successful theories in all of physics, be united into a single quantum theory of gravity? Can quantum and cosmos ever be combined? Though much progress has been made, Hawking and Penrose stress that physicists still have further to go in their quest for a quantum theory of gravity. Praise for Princeton's previous editions:: "If there were such a thing as the World Professional Heavyweight Theory Debating Society, this would be the title bout.
In Stephen W. Hawking and Roger Penrose gave a series of public lectures on general relativity at the Isaac Newton Institute for Mathematical Sciences at the University of Cambridge. From these lectures, published this year by Princeton University Press as The Nature of Space and Time, Scientific American has culled excerpts that serve to compare and contrast the perspectives of the two scientists. In particular, Hawking and Penrose disagree on what happens to the information stored in a black hole and on why the beginning of the universe differs from the end. The black hole will evaporate in the process, so that ultimately perhaps nothing of the original mass will be left. But during their formation, black holes swallow a lot of data—the types, properties and configurations of the particles that fall in.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs and how to get involved. High Energy Physics - Theory. Authors: S. Bibliographic Explorer What is the Explorer? Connected Papers What is Connected Papers?
file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of consider space and time to be a 4-dimensional continuum, called this book (at knutsfordlitfest.org~smarandache/knutsfordlitfest.org), it has.
The Large Scale Structure of Space-Time - Stephen Hawking & G.F.R.Ellis
Learn about our response to COVID , including freely available research and expanded remote access support. Hawking, Malcolm J. Perry, and Andrew Strominger Phys. Hawking, and Thomas Hertog Phys. D 82 , — Published 8 September
Einstein said that the most incomprehensible thing about the universe is that it is comprehensible.
The Nature of Space and Time
Copy the HTML code below to embed this book in your own blog, website, or application. An uncorrected copy, or prepublication, is an uncorrected proof of the book. We publish prepublications to facilitate timely access to the committee's findings.
Hawking wrote the book for readers who had no prior knowledge of physics and people who are interested in learning something new about interesting subjects. In A Brief History of Time , Hawking writes in non-technical terms about the structure, origin, development and eventual fate of the Universe , which is the object of study of astronomy and modern physics. He talks about basic concepts like space and time , basic building blocks that make up the Universe such as quarks and the fundamental forces that govern it such as gravity. He writes about cosmological phenomena such as the Big Bang and black holes. He discusses two major theories, general relativity and quantum mechanics , that modern scientists use to describe the Universe. Finally, he talks about the search for a unifying theory that describes everything in the Universe in a coherent manner.
OF SPACE-TIME. S. W. HAWKING, F.R.S.. Lucasian Professor of Mathematics in the University ofCambridge and. Fellow of Conville and Caius College. AND.
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Беккер заговорил по-испански с сильным франко-американским акцентом: - Меня зовут Дэвид Беккер. Я из канадского посольства. Наш гражданин был сегодня доставлен в вашу больницу. Я хотел бы получить информацию о нем, с тем чтобы посольство могло оплатить его лечение. - Прекрасно, - прозвучал женский голос.
Блестящий замысел. Выходит, Стратмор был зрителем теннисного матча, следящим за мячом лишь на одной половине корта. Поскольку мяч возвращался, он решил, что с другой стороны находится второй игрок. Но Танкадо бил мячом об стенку. Он превозносил достоинства Цифровой крепости по электронной почте, которую направлял на свой собственный адрес.