Initial value formulation

Abstract: In this paper, we explore the questions of time, locality and causality in the framework of covariant open bosonic string field theory. We show that if an open string field is expressed as a certain local function on spacetime--in particular, a function of the lightcone component of the midpoint and the transverse center of mass degrees of freedom--that cubic string field theory is nonsingular and local in lightcone time. In particular, the theory has a well defined initial value formulation resembling that of an ordinary second order relativistic field theory in lightcone frame. This description can be achieved by a nonsingular unitary transformation on the Fock space, and we demonstrate explicitly that the theory is gauge invariant and the interaction vertex is local in this basis. With an initial value formulation at hand, we are able to construct an explicit second quantized operator formalism for the theory using the Hamiltonian BRST formalism. We also explore issues of causality by considering a singular limit of the theory where all spacetime coordinates are taken to the midpoint. At any stage in this limit, the theory is well-defined and arbitrarily close to being completely local and manifestly causal. We argue that the this limit must account for the macroscopic causality of the string S-matrix.

Abstract: We consider dissipative relativistic fluid theories on a fixed flat, globally hyperbolic, Lorentzian manifold (R×T3,gab). We prove that for all initial data in a small enough neighborhood of the constant equilibrium states (in an appropriate Sobolev norm), the solutions evolve smoothly in time forever and decay exponentially to some, in general undetermined, constant equilibrium state. To prove this, three conditions are imposed on these theories. The first condition requires the system of equations to be symmetric hyperbolic, a fundamental requisite to have a well posed and physically consistent initial value formulation. For the flat space-times considered here the equilibrium states are constant, which is used in the proof. The second condition is a generic consequence of the entropy law, and is imposed on the non-principal part of the equations. The third condition is imposed on the principal part of the equations and it implies that the dissipation affects all the fields of the theory. With these req...

Abstract: We present the details of an algorithm for the global evolution of asymptotically flat, axisymmetric spacetimes, based upon a characteristic initial value formulation using null cones as evolution hypersurfaces. We identify a new static solution of the vacuum field equations which provides an important test bed for characteristic evolution codes. We also show how linearized solutions of the Bondi equations can be generated by solutions of the scalar wave equation, thus providing a complete set of test beds in the weak field regime. These tools are used to establish that the algorithm is second order accurate and stable, subject to a Courant-Friedrichs-Lewy condition. In addition, the numerical versions of the Bondi mass and news function, calculated at scri on a compactified grid, are shown to satisfy the Bondi mass loss equation to second order accuracy. This verifies that numerical evolution preserves the Bianchi identities. Results of numerical evolution confirm the theorem of Christodoulou and Klainerman that in vacuum, weak initial data evolve to a flat spacetime. For the class of asymptotically flat, axisymmetric vacuum spacetimes, for which no nonsingular analytic solutions are known, the algorithm provides highly accurate solutions throughout the regime in which neither caustics nor horizons form.

Abstract: In 1952, Mme Yvonne Choquet-Bruhat published a major paper, Th?or?me d'existence pour certains syst?mes d'?quations aux d?riv?es partielles non lin?aires (Acta Math. 88 141-225), which laid the foundation for modern studies of the Cauchy problem in general relativity. The fiftieth anniversary of this event was celebrated with an eponymous Carg?se Summer School in 2002. The proceedings of that summer school are summarized electronically (as audio, video, transparencies and lecture notes, where available) on a DVD archive included with this volume, and are also available on the internet. However the organizers decided that a separate volume describing the 'state of the art in mathematical general relativity' would be useful, and this book is the result. It includes some material not covered in the school and excludes some school material which has been covered adequately elsewhere. Unfortunately, I was unable to find, electronically, a table of contents, which every prospective purchaser would wish to see, and so this review does in fact list all the articles, ordered, roughly, by length. About one fifth of the book is devoted to a survey of Smoothness at Null Infinity and the Structure of Initial Data by Helmut Friedrich. This is a modern study of gravitational radiation, and the analysis of Einstein's equations. It is extremely helpful to survey all of this material, including some of the latest developments, using a consistent notation. This article is strongly recommended to anyone hoping to gain a foothold in this area. Note also that 47 pages of transparencies have become 84 book pages. Lars Andersson has surveyed, in The Global Existence Problem in General Relativity, some results and conjectures about the global properties of 3+1-dimensional spacetimes with a compact Cauchy surface. Again it is very useful to have essentially all of the known results presented in a consistent notation. This material is not on the DVD. Yvonne Choquet-Bruhat has contributed a long research paper, Future Complete U(1) Symmetric Einsteinian Spacetimes, the Unpolarized Case. There is a non-linear stability theorem due to her and Vincent Moncrief in which spacetime is of the form M ? R where M is a circle bundle over a compact orientable surface of genus >1 and the 4-metric admits a Killing symmetry along the spacelike circular fibres. The new result removes the polarization condition, i.e., the orthogonality of the fibres to quotient 3-manifolds. This is a classic example of how to derive results in this field. It is also available, in full, on the DVD. The article Group Actions on Lorentz Spaces, Mathematical Aspects: A Survey, contributed by Thierry Barbot and Abdelghani Zeghib, extends the second author's school presentation on the classification of large isometry groups of Lorentz manifolds, which brings together comprehensively material of surprising diversity, well worth a perusal. Robert Bartnik and Jim Isenberg have contributed a brilliant review of The Constraint Equations, greatly extending the first author's one hour school presentation. This is a worthy successor to the landmark survey of James W York, 26 years earlier. There is a survey of systems of partial differential equations which admit an initial value formulation up to gauge diffeomorphisms by Robert Geroch entitled Gauge, Diffeomorphisms, Initial-Value Formulation, Etc, which appears to be a reworking and update of an earlier review by the author. The school presentation is very nearly the same as the book version. Hubert Bray presented a trio of lectures on global inequalities at the summer school. These have been expanded into a review The Penrose Inequality, with a second author Piotr Chrusciel, which deals with its generalization as the Riemannian Penrose inequality. It is very satisfying to see an information flow from general relativity to Riemannian geometry! In Future Complete Vacuum Spacetimes, Lars Andersson and Vincent Moncrief contribute a global existence theorem for small perturbations of K = -1 vacuum Friedmann-Robertson-Walker spacetimes. The novelty here is the use of the Bel-Robinson energy and its generalizations. At a more specialized level, Michael Anderson has contributed a review of Cheeger-Gromov Theory and Applications to General Relativity, which is an update of the lectures he gave at the summer school. This reviewer is unfamiliar with the material presented here, but it looks to be potentially important. Luis Lehner and Oscar Reula have presented a rather useful but concise review of numerical relativity for mathematical relativists, entitled Status Quo and Open Problems in the Numerical Construction of Spacetimes, which is very different to the first author's school presentation. This should perhaps be worked up to a more encyclopaedic review. Gregory Galloway has contributed Null Geometry and the Einstein Equations, an extended version of his summer school lectures, which surveyed how techniques from global Lorentzian geometry and causality theory can be used to obtain results about the global behaviour of solutions of the Einstein equations, thus generalizing and extending the classic Hawking-Penrose results. The reader should recall that the articles have been ordered by objective length rather than subjective quality. Alan Rendall's is a masterly if terse review of The Einstein-Vlasov System, a laudable attempt to bring the mathematical status of results on the non-vacuum field equations up to that of their vacuum counterparts. This duplicates precisely his school presentation. Last but not least, John Friedman has given an excellent succinct review of what can go wrong if one abandons the requirement of time-orientability, The Cauchy problem on Spacetimes That Are Not Globally Hyperbolic, which seems to be identical to his school presentation. As always, this experienced author's presentation has exemplary clarity. This is an impressive volume presenting clearly the current state of the art. No relativity group should be without a copy.