Revisiting the Darmois and Lichnerowicz junction conditions

What have become known as the “Darmois” and “Lichnerowicz” junction conditions are often stated to be equivalent, “essentially” equivalent, in a “sense” equivalent, and so on. One even sees not infrequent reference to the “Darmois–Lichnerowicz” conditions. Whereas the equivalence of these conditions is manifest in Gaussian-normal coordinates, a fact that has been known for close to a century, this equivalence does not extend to a loose definition of “admissible” coordinates (coordinates in which the metric and its first order derivatives are continuous). We show this here by way of a simple, but physically relevant, example. In general, a loose definition of the “Lichnerowicz” conditions gives additional restrictions, some of which simply amount to a convenient choice of gauge, and some of which amount to real physical restrictions, away from strict “admissible” coordinates. The situation was totally confused by a very influential, and now frequently misquoted, paper by Bonnor and Vickers, that erroneously claimed a proof of the equivalence of the “Darmois” and “Lichnerowicz” conditions within this loose definition of “admissible” coordinates. A correct proof, based on a strict definition of “admissible” coordinates, was given years previous by Israel. It is that proof, generally unrecognized, that we must refer to. Attention here is given to a clarification of the subject, and to the history of the subject, which, it turns out, is rather fascinating in itself.     

An approximate global solution of Einstein’s equation for a rotating compact source with linear equation of state

We use analytic perturbation theory to present a new approximate metric for a rigidly rotating perfect fluid source with equation of state (EOS) $\epsilon +(1-n)p=\epsilon _0$ . This EOS includes the interesting cases of strange matter, constant density and the fluid of the Wahlquist metric. It is fully matched to its approximate asymptotically flat exterior using Lichnerowicz junction conditions and it is shown to be a totally general matching using Darmois–Israel conditions and properties of the harmonic coordinates. Then we analyse the Petrov type of the interior metric and show first that, in accordance with previous results, in the case corresponding to Wahlquist’s metric it can not be matched to the asymptotically flat exterior. Next, that this kind of interior can only be of Petrov types I, D or (in the static case) O and also that the non-static constant density case can only be of type I. Finally, we check that it can not be a source of Kerr’s metric.     

On some involution theorems on twofold Poisson manifolds

We point out some involution theorems which are consequences of the existence of two compatible Poisson structures on a manifold. Using a theorem of Lichnerowicz on local triviality of the Schouten-Nijenhuis cohomology, we show that local exactness of the second Poisson structure with respect to the ground one is equivalent to involutivity of the algebra of invariant functions of the ground structure. Then an involution theorem of Mishchenko and Fomenko is given, founded on global exactness of the second structure. Finally a generalization of a recurrence operator is given to obtain a set of traces which are in involution.     

Eigentensors of the Lichnerowicz operator in Euclidean Schwarzschild metrics

Properties of the eigentensors of the Lichnerowicz Laplacian for the Euclidean Schwarzschild metric are discussed together with possible applications to the linear stability of higher‐dimensional instantons. The main statement of the article is that any eigentensor of the Lichnerowicz operator in a Euclidean (possibly higher‐dimensional) Schwarzschild metric is essentially singular at infinity.     

Stability of the charged radiating cylinder

We discuss the dynamical instability of cylindrically symmetric isotropic geometry under the effect of electromagnetic field. The interior geometry of the dynamical collapse is matched with an exterior geometry through Darmois junction conditions. The perturbation scheme is used to describe the collapse equation and categorize the Newtonian and post-Newtonian regions in radiating as well as non-radiating eras. It is concluded that energy density, pressure, radiation density and electromagnetic field control the stability of the cylinder leading to more unstable configuration.     

Model of a gravitating sphere set in rotation by internal stress

The systematic approximation technique of Synge is employed to construct a model of a spherical body at rest in the distant past and gradually attaining an angular velocity due to a modification of the internal stress. The calculations are carried out to include the second approximation so that, roughly speaking, there is an error of order m³ in Einstein's field equations wherem is the mass of the sphere.     

Schwarzschild and Synge once again

We complete the historical overview about the geometry of a Schwarzschild black hole at its horizon by emphasizing the contribution made by Synge in [6] to its clarification.     

Republication of: On the deviation of geodesics and null-geodesics,particularly in relation to the properties of spaces of constant curvature and indefinite line-element

This Golden Oldie is a reprinting of a paper by J. L. Synge first published in 1934. It is accompanied by a reprinting of a paper by F. A. E. Pirani first published in 1956. Together these papers pointed the way to the interpretation of geodesic deviation and its relation to the curvature tensor. These two Golden Oldies are accompanied by an Golden Oldie Editorial containing an editorial note written by A. Trautman, and by the biography of F. Pirani written by himself and commented by A. Trautman. An editorial note to this paper can be found in this issue preceding this Golden Oldie and online via doi:. Original paper: J. L. Synge, Annals of Mathematics 35, 705–713 (1934). Reprinted with the kind permission of the Editors of Annals of Mathematics. J. L. Synge: Deceased 30th March, 1995     

Stress-energy tensors and the Lichnerowicz Laplacian

To a critical point of a variational problem, we associate a divergence-free symmetric 2-tensor, called the stress-energy tensor. We calculate the Laplacian of this object as defined by Lichnerowicz. This has the property that it commutes with the divergence provided the Ricci curvature is covariantly constant. We deduce relations between different stress-energy tensors, discuss growth formulae and harmonic maps between spheres.     

The Motion of a Pair of Charged Particles

We revisit the problem of two (oppositely) charged particles interacting electromagnetically in one dimension with retarded potentials and no radiation reaction. The specific quantitative result of interest is the time it takes for the particles to fall in towards one another. Starting with the nonrelativistic form, we answer this question while adding layers of complexity until we arrive at the full relativistic delay differential equation that governs this problem. That case can be solved using the Synge method, which we describe and discuss.     

Effects of electromagnetic field on the dynamical instability of expansionfree gravitational collapse

In this paper, we discuss the effects of electromagnetic field on the dynamical instability of a spherically symmetric expansionfree gravitational collapse. Darmois junction conditions are formulated by matching interior spherically symmetric spacetime to exterior Reissner–Nordström spacetime. We investigate the role of different terms in the dynamical equation at Newtonian and post Newtonian regimes by using perturbation scheme. It is concluded that instability range depends upon pressure anisotropy, radial profile of energy density and electromagnetic field, but not on the adiabatic index Γ. In particular, the electromagnetic field reduces the unstable region.     

Regular spherical dust spacetimes

Physical (and weak) regularity conditions are used to determine and classify all the possible types of spherically symmetric dust spacetimes in general relativity. This work unifies and completes various earlier results. The junction conditions are described for general non-comoving (and non-null) surfaces, and the limits of kinematical quantities are given on all comoving surfaces where there is Darmois matching. We show that an inhomogeneous generalisation of the Kantowski-Sachs metric may be joined to the Lema?tre-Tolman-Bondi metric. All the possible spacetimes are explicitly divided into four groups according to topology, including a group in which the spatial sections have the topology of a 3-torus. The recollapse conjecture (for these spacetimes) follows naturally in this approach.     

Review: Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge

We define the rest-frame instant form of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find a quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.     

Slowly, Rotating Non-Stationary, Fluid Solutions of Einstein's Equations and Their Match to Kerr Empty Space-Time

A general class of solutions of Einstein's equation for a slowly rotating fluid source, with supporting internal pressure, is matched using Lichnerowicz junction conditions, to the Kerr metric up to and including first order terms in angular speed parameter. It is shown that the match applies to any previously known non-rotating fluid source made to rotate slowly for which a zero pressure boundary surface exists. The method is applied to the dust source of Robertson-Walker and in outline to an interior solution due to McVittie describing gravitational collapse. The applicability of the method to additional examples is transparent. The differential angular velocity of the rotating systems is determined and theinduced rotation of local inertial frame is exhibited.     

Gregory–Laflamme analysis of MGD black strings

The minimal geometric deformation (MGD), associated with the 4D Schwarzschild solution of the Einstein equations, is shown to be a solution of the pure 4D Ricci quadratic gravity theory, whose linear perturbations are then implemented by the Gregory–Laflamme eigentensors of the Lichnerowicz operator. The stability of MGD black strings is hence studied, through the correspondence between their Lichnerowicz eigenmodes and the ones associated with the 4D MGD solutions. It is shown that there exists a critical mass driving the MGD black strings stability, above which the MGD black string is precluded from any Gregory–Laflamme instability. The general-relativistic limit shows the MGD black string to be unstable, as expected, corresponding to the standard Gregory–Laflamme black string instability.     

The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system,I

A systematic presentation of the quasi-linear first order symmetric hyperbolic systems of Friedrichs is presented. A number of sharp regularity and smoothness properties of the solutions are obtained. The present paper is devoted to the case ofR ⁿ with suitable asymptotic conditions imposed. As an example, we apply this theory to give new proofs of the existence and uniqueness theorems for the Einstein equations in general relativity, due to Choquet-Bruhat and Lichnerowicz. These new proofs usingfirst order techniques are considerably simplier than the classical proofs based onsecond order techniques. Our existence results are as sharp as had been previously known, and our uniqueness results improve by one degree of differentiability those previously existing in the literature.Partially supported by AEC Contract AT(04-3)-34.Partially Supported by NSF Contract GP-8257.     

On Synge's covariant conservation laws for general relativity

Following Synge, the covariant formulas for the total four-momentum and angular momentum of an isolated physical system in general relativity are derived. These formulas are first obtained in the weak-field approximation, for which they are shown to be expressible in surface integral form, to be unique, and to represent covariantly conserved quantities. The covariant expressions for the general case are then shown to be identical to those for the weak-field case. The uniquely determined and covariantly conserved quantities so obtained are found to agree with the corresponding canonical, noncovariant surface integral expressions.     

The path group construction of Lie group extensions

We present an explicit realization of abelian extensions of infinite dimensional Lie groups using abelian extensions of path groups, by generalizing Mickelsson’s approach to loop groups and the approach of Losev–Moore–Nekrasov–Shatashvili to current groups. We apply our method to coupled cocycles on current Lie algebras and to Lichnerowicz cocycles on the Lie algebra of divergence free vector fields.     

Admissible coordinate systems for static,spherically symmetric stellar models

For static, spherically symmetric stellar models it is shown that imposing the condition that the determinant of the metrical coefficients takes on its flat space-time value everywhere is sufficient to ensure that the coordinates are admissible in the sense of Lichnerowicz. The general method of solution is illustrated by integrating the equations for a star of constant, uniform density.