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We consider the motion of small bodies in general relativity. The key result captures a sense in which such bodies follow timelike geodesics (or, in the case of charged bodies, Lorentz-force curves). This result clarifies the relationship between approaches that model such bodies as distributions supported on a curve, and those that employ smooth fields supported in small neighborhoods of a curve. This result also applies to "bodies" constructed from wave packets of Maxwell or Klein-Gordon fields. There follows a simple and precise formulation of the optical limit for Maxwell fields., 30 pages, forthcoming in Communications in Mathematical Physics

The familiar theories of physics have the feature that the application of the theory to make predictions in specific circumstances can be done by means of an algorithm. We propose a more precise formulation of this feature --- one based on the issue of whether or not the physically measurable numbers predicted by the theory are computable in the mathematical sense. Applying this formulation to one approach to a quantum theory of gravity, there are found indications that there may exist no such algorithms in this case. Finally, we discuss the issue of whether the existence of an algorithm to implement a theory should be adopted as a criterion for acceptable physical theories., Comment: 19 pages, a pre arXiv paper posted for accessibility

This chapter introduces general framework for systems of first-order, quasilinear partial differential equations for the description of physical systems. It analyses the structure of the partial differential equation describing a single physical system. The physical fields become cross-sections of an appropriate fibre bundle, and it is on these cross-sections that the differential equations are written. The chapter describes various structural features of the system of partial differential equations. A key feature of the partial differential equations of physics is their initial-value formulation, that is, their formulation in terms of initial data and ‘time’-evolution. It turns out that this formulation can be carried out in a rather general setting. The most direct way to prove local uniqueness of solutions of a partial differential equation is to show that it admits a hyperbolization.

There is proven a theorem, to the effect that a material body in general relativity, in a certain limit of sufficiently small size and mass, moves along a geodesic., 7 pages

The simple Navier–Stokes equations for dissipative fluids, translated into relativity, form a parabolic—and hence mathematically nonviable—system. There have been formulated numerous alternative theories, consisting instead of hyperbolic equations—theories that necessarily involve more dynamical variables and more free functions than does the simple Navier–Stokes theory. It is argued that, these mathematical differences notwithstanding, the physical content of these hyperbolic theories is in most cases precisely the same as that of Navier–Stokes.

For an asymptotically flat initial-data set in general relativity, the total mass-momentum may be interpreted as a Hermitian quadratic form on the complex, two-dimensional vector space of ``asymptotic spinors''. We obtain a generalization to an arbitrary initial-data set. The mass-momentum is retained as a Hermitian quadratic form, but the space of ``asymptotic spinors'' on which it is a function is modified. Indeed, the dimension of this space may range from zero to infinity, depending on the initial data. There is given a variety of examples and general properties of this generalized mass-momentum., 25 pages, LaTeX

A very wide class of theories for dissipative relativistic fluids is analyzed. General techniques for constructing explicit theories are discussed. The conditions under which these theories have causal evolution equations are determined. The general properties (including stability) of the equilibrium solutions of these theories are evaluated. The requirement that the theory have the appropriate number and kind of equilibrium solutions is a strong constraint on the structure of the fluid theory. The properties of the shock-wave solutions of these theories are briefly considered. Most causal fluid theories have no solutions capable of describing strong shock waves.

We investigate the theories of dissipative relativistic fluids in which all of the dynamical equations can be written as total-divergence equations. Extending the analysis of Liu, Mueller, and Ruggeri, we find the general theory of this type. We discuss various features of these theories, including the causality of the full nonlinear evolution equations and the nature and stability of the equilibrium states.

We introduce a large class of systems of partial differential equations on a base manifold M, a class that, arguably, includes most systems of physical interest. We then give general definitions — applicable to any system of equations in this class — of “having the diffeomorphisms on M as a gauge group”, and, for such a system, of “having an initial-value formulation, up to this gauge”. These definitions, being algebraic in the coefficients of the partial differential equations, are relatively easy to check in practice. The Einstein system, of course, satisfies our definitions.

It is well-known that the equations for a simple fluid can be cast into what is called their Lagrange formulation. We introduce a notion of a generalized Lagrange formulation, which is applicable to a wide variety of systems of partial differential equations. These include numerous systems of physical interest, in particular, those for various material media in general relativity. There is proved a key theorem, to the effect that, if the original (Euler) system admits an initial-value formulation, then so does its generalized Lagrange formulation., 34 pages, no figures, accepted in J. Math. Phys

We prove that an m-dimensional unit ball D^m in the Euclidean space {\mathbb R}^m cannot be isometrically embedded into a higher-dimensional Euclidean ball B_r^d \subset {\mathbb R}^d of radius r < 1/2 unless one of two conditions is met -- (1)The embedding manifold has dimension d >= 2m. (2) The embedding is not smooth. The proof uses differential geometry to show that if d, 20 Pages, 3 Figures

Consider an asymptotically simple solution of Einstein’s equation. It is shown that any internally generated, first‐order perturbation of the metric, as a consequence of the linearized Einstein equation, preserves asymptotic simplicity to first order.

We give an example of a causally well‐behaved, singular space–time for which all singular‐boundary constructions which fall in a certain wide class—a class which includes both the g‐boundary and b‐boundary—yield pathological topological properties. Specifically, for such a construction as applied to this example, a singular boundary point fails to be T 1‐related to an event of the original space–time. This example suggests that there may not exist any useful, generally applicable notion of the singular boundary of a space–time.

An example is given, in dimension six, of a curvature tensor having positive sectional curvatures and negative Gauss-Bonnet integrand. A large class of questions in differential geometry involves the relationship between the topology and the geometry of a compact Riemannian manifold. One of these is the Hopf conjecture: If, in even dimensions, the sectional curvatures of such a manifold are positive, then so is the Euler number. The Hopf conjecture is known to be true in dimensions two and four by the following argument (Milnor, unpublished; [2]). One first writes down the Gauss-Bonnet formula, which, in every even dimension, equates the Euler number of the manifold to a certain integral over the manifold, where the integrand involves only the curvature tensor, and that only algebraically. One then shows (in dimensions two and four) that, at each point, positivity of the sectional curvatures implies positivity of this integrand. Most attempts to prove the full Hopf conjecture have been attempts to generalize this argument [1], [3], [4], [5], [6]. Thus, there arises the following, purely algebraic, question: Over a vector space of any even dimension, does a tensor having the symmetries of a curvature tensor and having positive sectional curvatures necessarily have positive Gauss-Bonnet integrand? We here answer this question in the negative. Fix a real, six-dimensional vector space V. A wedge denotes the wedge product, and a star a Hodge star operator.2 Denote by V2 the vector space of second-rank, antisymmetric tensors over V, by V2 its dual (the space of 2-forms over V), and by V22 the vector space of symmetric linear mappings from V2 to V2. We shall make use of the following fact: For any element A of V2, (1) ((A A A)* A (A A A)*)* 9(A A A A A)*A. For A any element of V2, denote by TA the element of V22 with action Received by the editors September 3, 1974. AMS (MOS) subject classifications (1970). Primary 53B20. i Supported in part by the National Science Foundation under contract GP-34721Xi, and by the Sloan Foundation. 2 Our conventions for the star operation are these: For any form A, A* = A; for B a 2-form and C a 4-form, B(C*) = C(B*) = (B A C)* = (B* A C*)*. Note that we introduce no metric on V. ? American Mathematical Society 1976 267 This content downloaded from 157.55.39.17 on Wed, 31 Aug 2016 04:16:18 UTC All use subject to http://about.jstor.org/terms

Two senses in which the perturbation equations of general relativity can be misleading are explored. (i) Under certain circumstances there exist solutions of the perturbation equations that appear to be gauge, in that the metric perturbation is the symmetrized derivative of a vector field, but which nonetheless are not true gauge. (ii) Under certain circumstances there exist solutions of the perturbation equations that cannot, even locally, be extended to higher order in perturbation theory. The latter is a local version of the well‐known phenomenon of ‘‘linearization instability.’’

In the context of general relativity, there is a sense in which certain objects cannot be constructed, using reasonable matter, from normal initial conditions. An attempt is made to capture this sense as a definition. The implications of such a definition, along with some related results and open questions, are discussed.

This paper deals with two broad issues: the formulation of a mathematical framework for concentrated sources in general relativity and its application to strings. We isolate a class of those metrics whose curvature tensors are well defined as distributions. It is shown that shells of matter---but neither point particles nor strings---can be described by metrics in this class. This conclusion is examined in more detail for the case of strings. We estimate the errors inherent in certain determinations of the mass per unit length of a cosmic string, and in certain calculations of the gravitational radiation from such a string.

For an asymptotically flat space–time in general relativity there exist certain integrals, called linkages, over cross sections of null infinity, which represent the energy, momentum, or angular momentum of the system. A new formulation of the linkages is introduced and applied. It is shown that there exists a flux, representing the contribution of gravitational and matter radiation to the linkage. A uniqueness conjecture for the linkages is formulated. The ambiguities due to the possible presence of supertranslations in asymptotic rotations are studied using the behavior of the linkages under first‐order perturbations in the metric. While in certain situations these ambiguities disappear in the first‐order treatment, an example is given which suggests that they are an essential feature of general relativity and its asymptotic structure.

All exact solutions of Einstein’s equation that represent static, axisymmetric black holes distorted by an external matter distribution are obtained. Their structure—local and global—is examined. The Hawking temperature is derived and laws of thermodynamics given for both the total system of black hole and external matter and the black hole considered as a single system. The evolution, induced by Hawking radiation, of distorted black holes is discussed.

Let H be a closed subgroup of compact Lie group G. A necessary and sufficient condition is obtained for the existence of a left-invariant Riemannian metric on G such that the subduced metric on the quotient H G has strictly positive sectional curvatures.

It is proven that, for a certain class of hyperbolic systems (a class which includes Einstein’s equation), sufficiently small initial data on a bounded patch of initial surface generates a solution nonsingular in the region determined by that initial data. This theorem is virtually a corollary of the boost theorem. Various consequences and possible generalizations are discussed.

An argument is given in favor of the conjecture that an isolated system in general relativity must have nonnegative total energy.

A scheme is introduced which yields, beginning with any source‐free solution of Einstein's equation with two commuting Killing fields for which a certain pair of constants vanish (e.g., the exterior field of a rotating star), a family of new exact solutions. To obtain a new solution, one must specify an arbitrary curve (up to parametrization) in a certain three‐dimensional vector space. Thus, a single solution of Einstein's equationgenerates a family of new solutions involving two arbitrary functions of one variable. These transformations on exact solutions form a non‐Abelian group. The extensive algebraic structure thereby induced on such solutions is studied.

The various properties of the domain of dependence (Cauchy development) which have been found particularly useful in the study of gravitational fields are reviewed. The basic techniques for constructing proofs and counterexamples are described. A new tool—the past and future volume functions—for treating certain global properties of space‐times is introduced. These functions are used to establish two new theorems: (1) a necessary and sufficient condition that a space‐time have a Cauchy surface is that it be globally hyperbolic; and (2) the existence of a Cauchy surface is a stable property of space‐times.

in the ea:ly years of general relativity, the most pressing questions ~vcre local in character, e.g., the suitability of de~'ribing the gra~ itational ticld by a metric, the structure and consequences of Einstein's eqttalions, etc. It was presumably felt that global questions, while possibly of some physical interest, could safely be deferred until a litter stage. We are now involved in this later stage. Perhaps the most important reason for the rcceilt emphitsis on global properties was tile realization that to establish certain results ~ i n particular, the theorems on singularities ~it is neces,;ary at the outset to exercise some control over the admissible global bchavi()r of the spacetime. The .scope of global work his gradually I~'en extended beyond attempts to control misbehavior until t t~ay global m,:thods comprise a small but vigorous brat.oh of relativity. "What have we gained from these efforts ? We h:|ve becolne nlorc sensitive to glob:d possibilities and pitfalls. (When will a global condition be needed? Which intuitive ideas have been formulalcd precisely? What i, re the consequences of imposing various conditions ? Ilow restrictive are the conditions ?j We hitve reached a lx.ttcr understanding of the various levels of structure of a space-time and their interaction. (Which properties of a .~pacetintc involve only the c:.tusal .,,|ructurc, which only the toi~t~logy, etc. ? Do the causal relations determine the topology?) We have becll led to deeper and more r trealments of kno~.~, n result.,;. (]'lie IIOlit)ll Ol,"l 'generic' property of space-times, for example, offers a potentially v;,luablc approach to singularities.) Finally. ~e h:,ve come to appreciate that the structure of a space-time in tile large may have tme,~pectcd physical signiticante. (I)o the homoh)gy or homot,~py groups, the Siicfcl Whitney cL,sscs, or the spinor structure have implic:ttions for clcmclltary particle physics ?) This pap~;r con,,ists of a collecti,m r162 ;tl~otit the present status of glob;,I work plus ;l fi:w remarks about ~hcre t~c might go from here. (]'he usual caution is in force: many of the impt~rt;mt ildvallccs ill physics have been a result of asking new tlue.~tions which did not appear on ii~ts of

A formalism is introduced for analyzing the structure of the gravitational field in the asymptotic limit at spatial infinity. Consider a three‐dimensional surface S in the space‐time such that the initial data on S is asymptotically flat in an appropriate sense. Using a conformal completion of S by a single point A ``at spatial infinity,'' the asymptotic behavior of fields on S can be described in terms of local behavior at A. In particular, the asymptotic behavior of the initial data on S defines four scalars which depend on directions at A. Since there is no natural choice of a surface S in a space‐time, the dependence of these scalars on S is essential. The asymptotic symmetry group at spatial infinity, whose elements represent transformations from S to other asymptotically flat surfaces, is introduced. It is found that this group, which emerges initially as an infinite‐dimensional generalization of the Poincare group, can be reduced to the Lorentz group. A set of evolution equations is obtained: These equations describe the behavior of the four scalars under the action of the asymptotic symmetry group. The four scalars can thus be considered as fields on a three‐dimensional manifold consisting of all points at spatial infinity. The notion of a conserved quantity at spatial infinity is defined, and, as an example, the expression for the energy‐momentum at spatial infinity is obtained.

It is shown that, given any set of initial data for Einstein's equations which satisfy the constraint conditions, there exists a development of that data which is maximal in the sense that it is an extension of every other development. These maximal developments form a well-defined class of solutions of Einstein's equations. Any solution of Einstein's equations which has a Cauchy surface may be embedded in exactly one such maximal development.

A method is described for constructing, from any source‐free solution of Einstein's equations which possesses a Killing vector, a one‐parameter family of new solutions. The group properties of this transformation are discussed. A new formalism is given for treating space‐times having a Killing vector.

A number of theorems and definitions which are useful in the global analysis of relativistic world models are presented. It is shown in particular that, under certain conditions, changes in the topology of spacelike sections can occur if and only if the model is acausal. Two new covering manifolds, embodying certain properties of the universal covering manifold, are defined, and their application to general relativity is discussed.

There is an intimate connection between multipole moments and the conformal group. While this connection is not emphasized in the usual formulation of moments, it provides the starting point for a consideration of multipole moments in curved space. As a preliminary step in defining multipole moments in general relativity (a program which will be carried out in a subsequent paper), the moments of a solution of Laplace's equation in flat 3‐space are studied from the standpoint of the conformal group. The moments emerge as certain multilinear mappings on the space of conformal Killing vectors. These mappings are re‐expressed as a collection of tensor fields, which then turn out to be conformal Killing tensors (first integrals of the equation for null geodesics). The standard properties of multipole moments are seen to arise naturally from the algebraic structure of the conformal group.

A prescription is given for attaching to a space-time M , subject only to a causality condition, a collection of additional ‘ideal points’. Some of these represent ‘points at infinity’, others ‘singular points’. In particular, for asymptotically simple space-times, the ideal points can be interpreted as the boundary at conformal infinity. The construction is based entirely on the causal structure of M , and so leads to the introduction of ideal points also in a broad class of causal spaces. It is shown that domains of dependence can be characterized in terms of ideal points, and this makes possible an extension of the domain-of-dependence concept to causal spaces. A suggestion is made for assigning a topology to M together with its ideal points. This specifies some singular-point structure for a wide range of possible space-times.

A formalism is presented for the treatment of space‐times, which is intermediate between a fully covariant approach and the spin‐coefficient method of Newman and Penrose. With the present formalism, a pair of null directions only, rather than an entire null tetrad, is singled out at each point. The concept of a spin‐ and boost‐weighted quantity is defined, the formalism operating entirely with such quantities. This entails the introduction of modified differentiation operators, one of which represents a natural extension of the definition of the operator ð which had been introduced earlier by Newman and Penrose. For suitable problems, the present formalism should lead to considerable simplifications over that achieved by the standard spin‐coefficient method.

In order to define spinor fields on a space‐time M, it is necessary first to endow M with some further structure in addition to its Lorentz metric. This is the spinor structure. The definition and the elementary implications of the existence of a spinor structure are discussed. It is proved that a necessary and sufficient condition for a noncompact space‐time M to admit a spinor structure is that M have a global field of orthonormal tetrads.

Multipole moments are defined for static, asymptotically flat, source‐free solutions of Einstein's equations. The definition is completely coordinate independent. We take one of the 3‐surfaces V, orthogonal to the timelike Killing vector, and add to it a single point Λ at infinity. The resulting space inherits a conformal structure from V. The multipole moments of the solution emerge as a collection of totally symmetric, trace‐free tensors P, Pa, Pab, ⋯ at Λ. These tensors are obtained as certain combinations of the derivatives of the norm of the timelike Killing vector. (For static space‐times, this norm plays the role of a ``Newtonian gravitational potential.'') The formalism is shown to yield the usual multipole moments for a solution of Laplace's equation in flat space, the dependence of these moments on the choice of origin being reflected in the conformal behavior of the P's. As an example, the moments of the Weyl solutions are discussed.

A simple result, restricting the behavior of a timelike curve under limitations on its integrated acceleration, is obtained and discussed.

Do there exist topologies U \mathcal {U} and V \mathcal {V} for the set R R of real numbers such that a function f f from R R to R R is smooth in some specified sense (e.g., differentiable, C n {C^n} , or C ∞ {C^\infty } ) with respect to the usual structure of the real line if and only if f f is continuous from U \mathcal {U} to V \mathcal {V} ? We show that the answer is no.

A large class of questions in differential geometry involves the relationship between the geometry and the topology of a Riemannian (= positive-definite) manifold. We briefly review the status of the following question from this class: given that a compact, even-dimensional manifold admits a Riemannian metric of positive sectional curvatures, what can one say about its topology? Very few manifolds are known to admit such metrics. For example, is it not known whether or not the product of then-sphere with itself (n ≥ 2) does. One answer to the question above is provided by Synge's theorem: if the manifold is orientable, then it is simply connected. Another possible answer is given by the Hopf conjecture: such a manifold necessarily has positive Euler number. The Hopf conjecture is known to be true for homogeneous manifolds, and for arbitrary manifolds in dimensions two and four. This last result has two, apparently entirely different, proofs, one using Synge's theorem and the other the Gauss-Bonnet formula. Neither, it is shown, can be generalized directly to dimensions six or greater. The Hopf conjecture in these higher dimensions remains open.