The Large Scale Structure of Space-Time (Cambridge Monographs on Mathematical Physics) (英語) ペーパーバック – 1975/2/27
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Einstein's General Theory of Relativity leads to two remarkable predictions: first, that the ultimate destiny of many massive stars is to undergo gravitational collapse and to disappear from view, leaving behind a 'black hole' in space; and secondly, that there will exist singularities in space-time itself. These singularities are places where space-time begins or ends, and the presently known laws of physics break down. They will occur inside black holes, and in the past are what might be construed as the beginning of the universe. To show how these predictions arise, the authors discuss the General Theory of Relativity in the large. Starting with a precise formulation of the theory and an account of the necessary background of differential geometry, the significance of space-time curvature is discussed and the global properties of a number of exact solutions of Einstein's field equations are examined. The theory of the causal structure of a general space-time is developed, and is used to study black holes and to prove a number of theorems establishing the inevitability of singualarities under certain conditions. A discussion of the Cauchy problem for General Relativity is also included in this 1973 book.
'The book is a masterpiece, written by sure hands.' Science
'Certain to rank as an outstanding classic of the current advance of relativity theory.' Contemporary Physics
'La parution de ce livre est un événement important …' La Recherche
It covers in brilliant form the gravitational collapse of a star, the theory of black holes, the space-time singularities, the causal structure of space-time, and in its end the initial singularity of the universe, popularly known as the Big Bang. The book is highly mathematical, and is presupposed that the reader have studied basic abstract algebra and point set topology. But, for the readers highly interested in these subjects, this is not an obstacle. All theoretical physicists interested in modern General Relativity should have this book, a testimony of the Genius of Stephen Hawking. Definitively, a magnificent book!
Therefore, it behooves me to add anything of substance, as the terrain has been expatiated in those academic journals.
Be that as it may, I do hope to add my personal perspective on a book which has taken me many a year to digest.
The pertinent word, above, being "many," as this is not a monograph to be taken lightly---it demands undivided attention.
If one encounters this book with (at minimum) a prior exposure to General Relativity, it opens up an entirely new world.
There is so much to say, so much to cover in this review, that I will most certainly forget to say something of value.
Thus, regardless of what I do---or, do not--write, I will say without further ado that a serious researcher can hardly afford to ignore
this Classic. As with many other monographs, the term "Classic" is much hyped, yet, rarely defined. One definition would be--
this book ! The authors state that the book can be assimilated with prerequisites of "..simple calculus, algebra and point set topology."
I am not sure that the vast majority of readers would concur with that statement, but, at least it does hint at some guidance.
As many have noted, Chapter Two, of mathematical preliminaries ( Differential Geometry) is terse.
However, I urge the prospective student to assimilate this chapter, for, at least, these five reasons:
(1) Page 16: will give one a simple means to visualize one-forms.
(2) Page 23: will define Diffeomorphism-- and, that is a definition which must become a part of one's vocabulary---as it is used
everywhere, and often, in this text !
(3) Page 37: we read, "We shall in general regard such associated covariant and contravariant tensors as representations of the
same geometric object."
(4) Page 39: "A line element field is like a vector field, but with undetermined sign."
(5) Finally, the exposition of Fibre Bundles, which ends the mathematical survey (Pages 50-55), is as lucid as one is able to find !
Mathematics behind us, let us look forward. General Relativity, next. And, what a feast awaits; Hawking and Ellis stamp their
own imprint upon the subject.Three Postulates: Local Causality, Local conservation of energy and momentum, Field equations.
Already, Page 56: "the model for space-time is not just one pair (M,g), but a whole equivalence class of all pairs (M',g') which are
equivalent to (M,g). (Here M represents Manifold, g is Metric). Also, we read:
(1) "observation of local causality enables one to measure the metric up to a conformal factor "
(2) "knowledge of the energy-momentum tensor determines the conformal factor"
(3) " a small, isolated, body moves approximately on a timelike geodesic curve, independent of its internal constitution,
provided that the energy-density of matter in it is non-negative."
Field Equations then surmised, Page 71; It is this discussion which keeps one spellbound (summoning, too,the "Newtonian Limit").
We read: "...the field equations should define the metric only up to an equivalence class under diffeomorphisms..."
Another highlight: Lagrangian Formulation (Pages 64-70). Keeping handy paper and pencil. The steps in the derivation
demands attention from the astute reader. You will utilize formula 2.26 (Page 34) and much machinery from the second
chapter to arrive at these results of the Lagrangian formulation. (And, recall: the speed of light is set to unity in discussions).
Chapter One : it surveyed the landscape. Chapter Two: the Mathematical Preliminaries. Chapter Three: Physics of General Relativity.
Now, in the fourth chapter, the interplay between chapter two and chapter three, entitled: The Physical Significance of Curvature.
We learn "the Riemann Tensor term is analogous to the Newtonian Potential..." (80), Raychaudhuri Equation (84), and, we learn:
"...in a sense one could regard the Bianchi Identities as field equations for the Weyl Tensor, giving that part of the curvature at
a point that depends on the matter distribution at other points." Beautifully enunciated, thus ! The Raychaudhuri Equations are
given for timelike geodesics "...one sees again that vorticity causes expansion while shear causes contraction." (Page 88).
Energy Conditions, next. (Brush up on your understanding of Inequalities before proceeding). "...for every known form of matter,
the pressures are small when the density is small," and " matter always has a converging effect on congruences of null geodesics."
We keep in mind the difference between a positive-definite metric and a Lorentz metric where " there will not be any shortest curve."
Calculations which ensue--Variation of Arc Length--Pages 102-116, are straightforward and vital to the entire enterprise. Up Next:
Exact Solutions. Reading: "we shall discuss exact solutions with particular reference to their global properties." Pay attention to
the presentation of Penrose Diagrams ! That is, representations of the conformal structure of infinity via diagrams. We read:
"...we are not able to make cosmological models without some admixture of ideology" (Page 134). Godel's Universe given a nice
discussion: "...there can be no cosmic time coordinate in M which increases along every future-directed timelike or null curve."
At chapter's end, we read: "...a large number of exact solutions are known locally, relatively few have been examined globally."
Causal Structure (that is, sixth chapter):
(1) "the study of causal relationships is equivalent to that of the conformal geometry..."
(2) "If one assumes that space-time is time-orientable, then it must also be space-orientable."
(3) "In physically realistic solutions, the causality and chronology conditions are equivalent."
(4) Proposition 6.4.9. is remarkable: "...stable causality condition holds everywhere on M, if and only if there is a function, f, on M
whose gradient is everywhere timelike." The function, f, can be thought of as a sort of cosmic time. (Page 198). And, we read:
"....in General Relativity one's ability to predict the future is limited both by the difficulty of knowing data on the whole Cauchy
spacelike surface, and by the possibility that even if one did it would be insufficient." Also, "...if the whole of space-time were
globally hyperbolic, if there were a global Cauchy surface,its topology would be very dull." These quotes remind the reader that
Chapter Six needs to be studied with a fine-toothed comb.
The next (difficult) chapter demonstrates that the Einstein Field Equations satisfy the postulate that "....signals can only be
sent between points that can be joined by a non-spacelike curve." A nice explication of reduced Einstein equations (228-230)
serves,also, to remind one that "...a solution of the Einstein Field Equations can be unique only up to a diffeomorphism."
( A point which can not be too often stressed !) As is pointed out by the authors,chapter eight, Space-time singularities,
relies heavily on chapters four to six. Metric (m) and Geodesic (g) completeness are here defined. We learn :
" ...m and g completeness are equivalent for a positive-definite metric. Not so for the Lorentz Metric ! A Lorentz Metric
" does not define a topological metric....one is left with only g-completeness." Bundle completeness is utilized for the definition of
We read: "...singularities...involve infinite curvature in general." Closed Trapped Surfaces are prelude to Penrose Theorem and its proof;
"...in a collapsing star there will occur either a singularity or a Cauchy Horizon," and "...we have to take singularities seriously,
and, in general, causality breakdowns are not the way out." Also: "...in generic situations, one might expect the Boundary of the
manifold to be highly irregular and could not be given a smooth structure." And, "...the Singularities predicted...must be a more serious breakdown of the metric." Hopefully, these words excerpted from this extraordinary chapter will spur the reader into deeper study of
this important chapter. (Note Lemma 8.5.5, and its proof, a tour de force !).
If one has progressed this far, the final two chapters ( 9 &10: Gravitational Collapse & Initial Singularities) offer respite from the
heavily technical, preceding entries. Much here referenced in the famous Les Houches Lectures of 1972.
Take note:Utilization of Inequalities !
"When studying the asymptotic behavior, it is therefore convenient simply to forget about the star ...."
Take note: the definition of Apparent Horizon (Page 320).
Take note: the result, and proof, that a rotating Black Hole must be axisymmetric(Page 329).
Revel in the discussion of energy extraction from rotating Black Holes.(Inequalities,again ! A beautiful discussion of Proposition
9.3.2 "...each component of the horizon in a stationary, regular, predictable space is homeomorphic to a two-sphere..."(Page 335).
The last Chapter opens with these words:
"...the expansion of the universe is in many ways similar to the collapse of a star, except the sense of time is reversed."
Two arguments explicating the import of the cosmic microwave background:
(1) first utilizing the so-called Copernican Principle "...that we do not occupy a privileged position in space-time,"
(2) secondly, utilizing "the shape of the spectrum."
Finally, we read :" so far we have been exploring the mathematical consequences of taking a Lorentz manifold as the model for
space-time and requiring that the Einstein Field Equations hold...we have shown that according to this theory there should be
singularities in our past, associated with the collapse of the universe, and singularities in the future, associated with the collapse of stars."
By all means, peruse other available reviews ( for instance, Bryce Dewitt in Journal Science,1973 ).
By all means, peruse the 1966 Adams Prize Essay of Hawking (published Europhysics Journal H, November 2014)
on which essay this book is based.
By all means, prepare oneself with mathematical preliminaries before tackling all that is offered between these covers.
In the end, be prepared to be dazzled by the breathtaking scenery--both Physical and Mathematical--of General Relativity.
A final note (pedagogic): there is very good reason--as the text shows-- to keep the hyphen in the word "space-time" !
This is not a book that you "read", it is a book that you "study"---certainly, to be studied more than once.
(As, too with: Dirac's Principles of Quantum Mechanics).
Study it time and time again, until it makes sense.
Once you have it, it will not let go of you.
Certainly a unique accomplishment--this book-- in the annals of research.
The authors begin the book by a discussion of the role of gravity in physics and its role as determining the causal structure of the universe. They introduce the idea of a closed trapped surface, setting the stage for the goal of the book, namely the study of the conditions under which a space-time singularity must occur. Black holes and the beginning of the universe are cited as examples of these singularities. The authors also outline briefly the content of each chapter. A neat argument is given for the significance of focal points via the use of Raychaudhari's equation.
The second chapter is an overview of the background in differential geometry needed in the rest of the book. Although complete from an axiomatic point of view, the approach is much too formal for readers who do not have a knowledge of differential geometry. Such a reader should gain the necessary background elsewhere.
General relativity as a theory of gravitation is discussed in chapter 3. Spacetime is assumed to be a connected 4-dimensional smooth manifold on which is defined a Lorentz metric. The topology is assumed to be Hausdorff. Some of the more interesting or well-written parts of this chapter include the example of a spacetime that is not inextendible, the determination of the conformal factor for the spacetime metric, and the discussion of alternative field equations.
The authors discuss the physicial significance of curvature in chapter 4, namely its effect on families of timelike and null curves. The most important part of this chapter is the discussion on certain inequalities tht the energy-momentum tensor should satisfy from a physical viewpoint. These inequalities, called the weak energy condition and the dominant energy condition, allow the authors to prove the existence of singularities in a later chapter. The reader can see clearly the role of the Jacobi equation, and its solution, the Jacobi field, in measuring the separation of nearby geodesics. The existence of conjugate points is proven, and shown to imply the existence of self-intersections in families of geodesics. As a warm-up to showing the non-existence of geodesics of maximal length, the authors employ variational calculus to study how to vary non-spacelike curves connecting points in convex normal neighborhoods in spacetime, and between points and hypersurfaces. In particular, it is shown that a timelike geodesic curve from a hypersurface to a point is maximal iff there is no conjugate point to the hypersurface along the curve. In addition, the authors prove that two points joined by a non-spacelike curve which is not a null geodesic can be joined by a timelike curve.
The authors consider the exact solutions of the Einstein field equations in chapter 5. Most of the "usual" spacetimes are considered, including Minkowski, De Sitter, Anti-de-Sitter, Robertson-Walker, Schwarzschild, Reissner-Nordstrom, Kerr, Taub-Nut, and Godel. The emphasis in on the global properties of the spacetimes and the existence of singularities in them. The famous Penrose diagrams are used to "compactify" spacetimes in order to study their behavior at infinity and their conformal properties. The authors first introduce the concept of a future (past) Cauchy development here, so important in later developments in the book. The reader can see the tools developed in chapter 4 in play here; for example, the existence of a singularity in a spatially homogeneous cosmology is shown to follow directly from the Raychaudhuri equation. The existence of the singularity is proved to be independent of any acceleration or rotation of matter in such cosmologies.
In chapter 5, the authors consider the causal structure of spacetime, namely the study of its conformal geometry. The consideration of the set of all metrics conformal to the physical metric allows one to discuss "geodesic completeness" of spacetime, this concept forming the basis of a later definition of a singularity in spacetime. The more interesting topics discussed in this chapter include the causality conditions (there are no closed non-spacelike curves), and the Alexandrov topology and its connection with the strong causality condition (every neighborhood of a point contains a neighborhood of the point no non-separable curve of which intersects it more than once). When strong causality does hold, the Alexandrov topology is equivalent to the usual manifold topology, and thus the topology of spacetime can be determined by the observation of causal relationships. The discussion on the role of global hyperbolicity in showing the existence of a maximal geodesic is also very well-written.
The next chapter is pretty much independent of the rest, and was put in no doubt for the mathematician who desires to understand the Einstein equations as a set of nonlinear second-order hyperbolic partial differential equations with initial data on a 3-dimensional manifold, the famous Cauchy problem in general relativity.
Chapter 8 is the most important in the book, for its uses the constructions of earlier chapters to define the notion of a singularity in spacetime. The authors argue that singularities are points where physical laws break down and thus to characterize them one attempts to find out whether any such points have been removed, making spacetime "incomplete" in some sense. Such a notion of incompleteness is very meaningful in topological spaces with a positive definite metric, since in that case one can define completeness in terms of the convergence of Cauchy sequences. In spacetimes with a Lorentz metric, the authors discuss the notion of geodesic completeness for null and timelike geodesics. A very detailed treatment of the now famous singularity theorems is given, these theorems involving an inequality of the Ricci tensor. The last two chapters of the book are more physical in nature wherein the singularity problem is shown to have physical relevance via the occurence of black holes at the endpoint of evolution of massive stars.
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