Robert Geroch

Bio: Robert Geroch is an academic researcher from University of Chicago. The author has contributed to research in topics: Theory of relativity & Four-force. The author has an hindex of 30, co-authored 52 publications receiving 7089 citations.

A Method for Generating Solutions of Einstein's Equations

Abstract: 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.

Domain of Dependence

Abstract: 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.

Global aspects of the Cauchy problem in general relativity

Abstract: 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

Spinor structure of space-times in general relativity. i

Abstract: 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.

A space‐time calculus based on pairs of null directions

Abstract: 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.

Numerical methods for conservation laws

Abstract: I Mathematical Theory- 1 Introduction- 11 Conservation laws- 12 Applications- 13 Mathematical difficulties- 14 Numerical difficulties- 15 Some references- 2 The Derivation of Conservation Laws- 21 Integral and differential forms- 22 Scalar equations- 23 Diffusion- 3 Scalar Conservation Laws- 31 The linear advection equation- 311 Domain of dependence- 312 Nonsmooth data- 32 Burgers' equation- 33 Shock formation- 34 Weak solutions- 35 The Riemann Problem- 36 Shock speed- 37 Manipulating conservation laws- 38 Entropy conditions- 381 Entropy functions- 4 Some Scalar Examples- 41 Traffic flow- 411 Characteristics and "sound speed"- 42 Two phase flow- 5 Some Nonlinear Systems- 51 The Euler equations- 511 Ideal gas- 512 Entropy- 52 Isentropic flow- 53 Isothermal flow- 54 The shallow water equations- 6 Linear Hyperbolic Systems 58- 61 Characteristic variables- 62 Simple waves- 63 The wave equation- 64 Linearization of nonlinear systems- 641 Sound waves- 65 The Riemann Problem- 651 The phase plane- 7 Shocks and the Hugoniot Locus- 71 The Hugoniot locus- 72 Solution of the Riemann problem- 721 Riemann problems with no solution- 73 Genuine nonlinearity- 74 The Lax entropy condition- 75 Linear degeneracy- 76 The Riemann problem- 8 Rarefaction Waves and Integral Curves- 81 Integral curves- 82 Rarefaction waves- 83 General solution of the Riemann problem- 84 Shock collisions- 9 The Riemann problem for the Euler equations- 91 Contact discontinuities- 92 Solution to the Riemann problem- II Numerical Methods- 10 Numerical Methods for Linear Equations- 101 The global error and convergence- 102 Norms- 103 Local truncation error- 104 Stability- 105 The Lax Equivalence Theorem- 106 The CFL condition- 107 Upwind methods- 11 Computing Discontinuous Solutions- 111 Modified equations- 1111 First order methods and diffusion- 1112 Second order methods and dispersion- 112 Accuracy- 12 Conservative Methods for Nonlinear Problems- 121 Conservative methods- 122 Consistency- 123 Discrete conservation- 124 The Lax-Wendroff Theorem- 125 The entropy condition- 13 Godunov's Method- 131 The Courant-Isaacson-Rees method- 132 Godunov's method- 133 Linear systems- 134 The entropy condition- 135 Scalar conservation laws- 14 Approximate Riemann Solvers- 141 General theory- 1411 The entropy condition- 1412 Modified conservation laws- 142 Roe's approximate Riemann solver- 1421 The numerical flux function for Roe's solver- 1422 A sonic entropy fix- 1423 The scalar case- 1424 A Roe matrix for isothermal flow- 15 Nonlinear Stability- 151 Convergence notions- 152 Compactness- 153 Total variation stability- 154 Total variation diminishing methods- 155 Monotonicity preserving methods- 156 l1-contracting numerical methods- 157 Monotone methods- 16 High Resolution Methods- 161 Artificial Viscosity- 162 Flux-limiter methods- 1621 Linear systems- 163 Slope-limiter methods- 1631 Linear Systems- 1632 Nonlinear scalar equations- 1633 Nonlinear Systems- 17 Semi-discrete Methods- 171 Evolution equations for the cell averages- 172 Spatial accuracy- 173 Reconstruction by primitive functions- 174 ENO schemes- 18 Multidimensional Problems- 181 Semi-discrete methods- 182 Splitting methods- 183 TVD Methods- 184 Multidimensional approaches

Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity

Abstract: It is shown that the global charges of a gauge theory may yield a nontrivial central extension of the asymptotic symmetry algebra already at the classical level. This is done by studying three dimensional gravity with a negative cosmological constant. The asymptotic symmetry group in that case is eitherR×SO(2) or the pseudo-conformal group in two dimensions, depending on the boundary conditions adopted at spatial infinity. In the latter situation, a nontrivial central charge appears in the algebra of the canonical generators, which turns out to be just the Virasoro central charge.

Black Hole Explosions

Abstract: QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries.

Abstract: The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general post-Newtonian sources. The exterior field of the source is investigated by means of a combination of analytic post-Minkowskian and multipolar approximations. The physical observables in the far-zone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the postNewtonian source in the near-zone we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many nonlinear multipole interactions, among them those associated with the tails (and tails-of-tails) of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third post-Newtonian (3PN) order beyond the Newtonian acceleration. The gravitational-wave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail eects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary’s orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument.

The Singularities of gravitational collapse and cosmology

Abstract: A new theorem on space-time singularities is presented which largely incorporates and generalizes the previously known results. The theorem implies that space-time singularities are to be expected if either the universe is spatially closed or there is an ‘object’ undergoing relativistic gravitational collapse (existence of a trapped surface) or there is a point p whose past null cone encounters sufficient matter that the divergence of the null rays through p changes sign somewhere to the past of p (i. e. there is a minimum apparent solid angle, as viewed from p for small objects of given size). The theorem applies if the following four physical assumptions are made: (i) Einstein’s equations hold (with zero or negative cosmological constant), (ii) the energy density is nowhere less than minus each principal pressure nor less than minus the sum of the three principal pressures (the ‘energy condition’), (iii) there are no closed timelike curves, (iv) every timelike or null geodesic enters a region where the curvature is not specially alined with the geodesic. (This last condition would hold in any sufficiently general physically realistic model.) In common with earlier results, timelike or null geodesic incompleteness is used here as the indication of the presence of space-time singularities. No assumption concerning existence of a global Cauchy hypersurface is required for the present theorem.