# Ezra NewmanUniversity of Pittsburgh | Pitt · Physics and Astronomy

Ezra Newman

PhD

## About

162

Publications

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5,104

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Citations since 2016

## Publications

Publications (162)

Considering the spin-coefficient version of the left-flat vacuum Einstein
equations, all but one of the fifty equations can be explicitly integrated via
the introduction of five spin-weight s=-2 complex potentials. The final
equation is a non-linear wave equation for the last of the potentials.
Solutions to this equation determine solutions for the...

Considering perturbations of the Reissner–Nordström metric while keeping the perturbations in the class of type II Einstein–Maxwell metrics, we perform a spherical harmonic expansion of all the variables up to the quadrupole term. This leads to rather surprising results. Referring to the source of the metric as a type II particle (analogous to refe...

Considering perturbations off the Reissner-Nordstrom metric while keeping the
perturbations in the class of type II Einstein-Maxwell metrics, we do a
spherical harmonic expansion of all the variables up to the quadrupole term.
This leads to a rather surprising results. Referring to the source of the
metric as a type II particle (analogous to referr...

We study geometric structures associated with shear-free null geodesic
congruences in Minkowski space-time and asymptotically shear-free null geodesic
congruences in asymptotically flat space-times. We show how in both the flat
and asymptotically flat settings, complexified future null infinity acts as a
"holographic screen," interpolating between...

The spin weighted spherical harmonic (SWSH) description of angular functions is typically associated with the Newman-Penrose (NP) null tetrad formalism. Recently, the SWSH description, but not the NP formalism, has been used in the study of the polarization anisotropy of the cosmic microwave background. Here we relate this application of SWSHs to a...

The properties of null geodesic congruences (NGCs) in Lorentzian manifolds are a topic of considerable importance. More specifically NGCs with the special property of being shear-free or asymptotically shear-free (as either infinity or a horizon is approached) have received a great deal of recent attention for a variety of reasons. Such congruences...

The general Lorentz transformation, the relation between observations made in different frames, involves the three components of the relative velocity of the two frames and three parameters describing the relative orientation of the frames. A simpler description is known in which the six parameters are encoded in four complex numbers subject to a s...

In connection with the study of shear-free null geodesics in Minkowski space, we investigate the real geometric effects in real Minkowski space that are induced by and associated with complex world-lines in complex Minkowski space. It was already known, in a formal manner, that complex analytic curves in complex Minkowski space induce shear-free nu...

We investigate the geometry of a particular class of null surfaces in space-time called vacuum Non-Expanding Horizons (NEHs). Using the spin-coefficient equation, we provide a complete description of the horizon geometry, as well as fixing a canonical choice of null tetrad and coordinates on a NEH. By looking for particular classes of null geodesic...

In classical electromagnetic theory, one formally defines the complex dipole moment (the electric plus 'i' magnetic dipole) and then computes (and defines) the complex center of charge by transforming to a complex frame where the complex dipole moment vanishes. Analogously in asymptotically flat spacetimes, it has been shown that one can determine...

\[\tag{2} l^{a}=o^{A}\overline{o}^{A^{\prime }},~\ m^{a}=o^{A}\overline{\iota } ^{A^{\prime }},~~~\overline{m}^{a}=\iota ^{A}\overline{o}^{A^{\prime }}, \ \ n^{a}=\iota ^{A}\overline{\iota }^{A^{\prime }}. \]Lower-case italic Latin letters from the beginning of the alphabet, \(a,b,c,\dots,h\ ,\) for tetrad indices, with the corresponding capital le...

A priori, there is nothing very special about shear-free or asymptotically shear-free null geodesic congruences. Surprisingly, however, they turn out to possess a large number of fascinating geometric properties and to be closely related, in the context of general relativity, to a variety of physically significant effects. It is the purpose of this...

A priori, there is nothing very special about shear-free or asymptotically shear-free null geodesic congruences. Surprisingly, however, they turn out to possess a large number of fascinating geometric properties and to be closely related, in the context of general relativity, to a variety of physically significant effects. It is the purpose of this...

We study the physical consequences of two different but closely related perturbation schemes applied to the Einstein–Maxwell equations. In one case the starting spacetime is flat, while in the other case it is Schwarzschild. In both cases, the perturbation is due to a combined electric and magnetic dipole field. We can see within the Einstein–Maxwe...

The principle purpose of this paper is to analyze in the context of general relativity the effects of electromagnetic radiation on a gravitational field. The basic outline is that we start from flat-space with a time-dependent electric and magnetic dipole solution of Maxwell's equations. This field, treated as first order, then acts as the source f...

By a series of examples, we illustrate some rather remarkable and surprising (at least to some) relationships between (1)
all three-dimensional conformal Lorentzian manifolds and a wide class of 3rd order ODEs (all 3rd order ODEs with vanishing
Wunschmann invariant) and (2) all four dimensional conformal Lorentzian manifolds and a wide class of pai...

A problem in general relativity is how to extract physical information from solutions to the Einstein equations. Most often
information is found from special conditions, e.g., special vector fields, symmetries or approximate symmetries. Our concern
is with asymptotically flat space–times with approximate symmetry: the BMS group. For these spaces th...

We study the physical consequences of two diffferent but closely related perturbation schemes applied to the Einstein-Maxwell equations. In one case the starting space-time is flat while in the other case it is Schwarzschild. In both cases the perturbation is due to a combined electric and magnetic dipole field. We can see, within the Einstein-Maxw...

A major issue in general relativity, from its earliest days to the present, is how to extract physical information from any solution or class of solutions to the Einstein equations. Though certain information can be obtained for arbitrary solutions, e.g., via geodesic deviation, in general, because of the coordinate freedom, it is often hard or imp...

We argue that the well-known problem of the instabilities associated with the self-forces (radiation reaction forces) in classical electrodynamics are possibly stabilized by the introduction of gravitational forces via general relativity.

From the study of the asymptotic behavior of the Einstein or Einstein– Maxwell fields, a rather unusual new structure was found. This structure which is associated with asymptotically shearfree null congruences, appears to have significant physical interest or consequences. More specifically it allows us to define, at future null infinity, the cent...

We describe here what appears to be a new structure that is hidden in all asymptotically vanishing Maxwell fields possessing a non-vanishing total charge. Though we are dealing with real Maxwell fields on real Minkowski space nevertheless, directly from the asymptotic field one can extract a complex analytic world-line defined in complex Minkowski...

We show that, though they are rare, there are asymptotically flat space-times that possess null geodesic congruences that are both asymptotically shear- free and twist-free (surface forming). In particular, we display the class of space-times that possess this property and demonstrate how these congruences can be found. A special case within this c...

In analogy with classical electromagnetic theory, where one determines the total charge and both electric and magnetic multipole moments of a source from certain surface integrals of the asymptotic (or far) fields, it has been known for many years—from the work of Hermann Bondi—that the energy and momentum of gravitational sources could be determin...

We discuss the existence, arising by analogy to that in algebraically special space-times, of a unique asymptotically shear-free
congruence in any asymptotically flat space-time. Associated with it is a unique complex analytic curve in H-space. The surprising
potential physical significance of this curve is discussed.

We describe a natural relationship between all 3rd order ODEs
with a vanishing Wunschmann invariant, with all conformal Lorentzian
metrics on 3-manifolds and Cartan's normal O(3,2) conformal connections.
The generalization to pairs of second order PDEs and their relationship
to Cartan's normal O(4,2) conformal connections on four dimensional
manifo...

We study the Robinson-Trautman-Maxwell Fields in two closely related coordinate systems, the original Robinson-Trautman (RT) coordinates (in a more general context, often referred to as NU coordinates) and Bondi coordinates. In particular, we identify one of the RT variables as a velocity and then from the Bondi energy-momentum 4-vector, we find ki...

We discuss the existence, arising by analogy to that in algebraically special spacetimes, of a unique CR structure realized on null infinity for (almost) any asymptotically flat Einstein or Einstein–Maxwell spacetime.

We first review asymptotic twistor theory with its real subspace of null asymptotic twistors: a five-dimensional CR manifold. This is followed by a description of the Kerr theorem (the identification of shear-free null congruences, in Minkowski space, with the zeros of holomorphic functions of three variables) and an asymptotic version of the Kerr...

In this work we show that on the space of solutions of a certain class of fourth-order ODEs, u'''' = Λ(s, u, u', u'', u'''), a four-dimensional conformal metric, gab, can be constructed such that the four-dimensional eikonal equation, gabu,au,b = 0, holds. Furthermore, we remark that this structure is invariant under contact transformations. Our ge...

The purpose of the present work is to extend the earlier results for
asymptotically flat vacuum space-times to asymptotically flat solutions of the
Einstein-Maxwell equations. Once again, in this case, we get a class of
asymptotically shear-free null geodesic congruences depending on a complex
world-line in the same four-dimensional complex space....

We show how to define and go from the spin-s spherical harmonics to the tensorial spin-s harmonics. These quantities, which are functions on the sphere taking values as Euclidean tensors, turn out to be extremely useful for many calculations in General Relativity. In the calculations, products of these functions, with their needed decompositions wh...

The purpose of this work is to return, with a new observation and rather unconventional point of view, to the study of asymptotically flat solutions of Einstein equations. The essential observation is that from a given asymptotically flat space-time with a given Bondi shear, one can find (by integrating a partial differential equation) a class of a...

We show that for asymptotically vanishing Maxwell fields in Minkowski space with non-vanishing total charge, one can find a unique geometric structure, a null direction field, at null infinity. From this structure a unique complex analytic world-line in complex Minkowski space that can be found and then identified as the complex center of charge. B...

We show that certain structures defined on the complex four dimensional space known as H-Space have considerable relevance for its closely associated asymptotically flat real physical space-time. More specifically for every complex analytic curve on the H-space there is an asymptotically shear-free null geodesic congruence in the physical space-tim...

The aim of this work is to present a formulation to general relativity, which is analogous to the null surface formulation, but now instead of starting with a complete integral of the eikonal equation we start with a complete integral of the Hamilton–Jacobi equation. In the first part of this work we show that on the space of solutions of a certain...

We explore the different geometric structures that can be constructed from the class of pairs of second-order PDEs that satisfy the condition of a vanishing generalized Wünschmann invariant. This condition arises naturally from the requirement of a vanishing torsion tensor. In particular, we find that from this class of PDEs we can obtain all four-...

In the first part of this work we show that on the space of solutions of a certain class of systems of three second-order PDE’s, uαα = Υ(α,β,u,uα,uβ), uββ = Ψ(α,β,u,uα,uβ) and uαβ = Ω(α,β,u,uα,uβ), a three-dimensional definite or indefinite metric, gab, can be constructed such that the three-dimensional Hamilton–Jacobi equation, gabu,au,b = 1 holds...

We study and report on the class of vacuum Maxwell fields in Minkowski space that possess a non-degenerate, diverging, principle null vector field (null eigenvector field of the Maxwell tensor) that is tangent to a shear-free null geodesics congruence. These congruences can be either surface forming (the tangent vectors proportional to gradients) o...

By using two different procedures we show that on the space of solutions of a certain class of second-order ordinary differential equations, u″ = Λ(s,u,u′), a two-dimensional definite or indefinite metric, gab, can be constructed such that the two-dimensional Hamilton–Jacobi equation, gabu,au,b = 1 holds. Furthermore, we show that this structure is...

We show how, beginning with the space of curves in R 2 and the space of 2-surfaces in R 3 , one can define conformal Lorentzian three-and four-dimensional metrics on certain special subspaces. It is conjectured that for the case of curves in R 2 this is how Wunschmann obtained his well-known equation. Generalizing the argument to the case of 2-surf...

We begin with an arbitrary but given conformal Lorentzian metric on an open neighbourhood, U, of a four-dimensional manifold (spacetime) and study families of solutions of the eikonal equation. In particular, the families that are of interest to us are the complete solutions. Their level surfaces form a two-parameter (points of S2) family of foliat...

We show that every 2nd order ODE defines a 4-parameter family of projective connections on its 2-dimensional solution space. In a special case of ODEs, for which a certain point transformation invariant vanishes, we find that this family of connections always has a preferred representative. This preferred representative turns out to be identical to...

Necessary and sufficient conditions for a space-time to be conformal to an Einstein space-time are interpreted in terms of curvature restrictions for the corresponding Cartan conformal connection.

We want to discuss gravitational lensing as far as presently possible from the point of view of spacetime geometry without use of perturbation theory or a background metric. The intent is to construct a conceptual framework so that lensing theory fits covariantly into general relativity and then to see how the usual perturbation approach is related...

The gravitational lensing problem is discussed in an exact setting - namely, without the use of approximations such as weak fields or thin lenses. In particular, the distortion of the images of elliptical sources is described in terms of three parameters (the semiaxes ratio, the area and orientation of the ellipse), and evolution equations for thes...

We explore and show a natural relationship between all third-order ordinary differential equations that possess a vanishing Wunschmann invariant, with conformal metrics on 3-manifolds and Cartan's normal O(3, 2) conformal connections. The generalization to pairs of second-order PDEs and their relationship to Cartan's normal O(4, 2) conformal connec...

It has recently been proved [3] that the solution spaces of certain classes of differential equations whose local solutions are parametrized by three or four arbitrary constants can be endowed with conformal Lorentzian metrics in a natural way. We shall prove that these conformal structures are preserved when the differential equations are transfor...

The images of many distant galaxies are displaced, distorted and often multiplied by the presence of foreground massive galaxies near the line of sight; the foreground galaxies act as gravitational lenses. Commonly, the lens equation, which relates the placement and distortion of the images to the real source position in the thin-lens scenario, is...

We present a framework, based on the null-surface formulation of general relativity, for discussing the dynamics of Fermat potentials for gravitational lensing in a generic situation without approximations of any kind. Additionally, we derive two lens equations: one for the case of thick compact lenses and the other one for lensing by gravitational...

By treating the real Maxwell Field and real linearized Einstein equations as being imbedded in complex Minkowski space, one can interpret magnetic moments and spin-angular momentum as arising from a charge and mass monopole source moving along a complex world line in the complex Minkowski space. In the circumstances where the complex center of mass...

In a previous article concerning image distortion in nonperturbative gravitational lensing theory we described how to introduce shape and distortion parameters for small sources. We also showed how they could be expressed in terms of the scalar products of the geodesic deviation vectors of the source’s pencil of rays in the past light cone of an ob...

We introduce the idea of shape parameters to describe the shape of the pencil of rays connecting an observer with a source lying on his past light cone. On the basis of these shape parameters, we discuss a setting of image distortion in a generic (exact) spacetime, in the form of three distortion parameters. The fundamental tool in our discussion i...

We first show how, from the general 3rd order ODE of the form z'''=F(z,z',z'',s), one can construct a natural Lorentzian conformal metric on the four-dimensional space (z,z',z'',s). When the function F(z,z',z'',s) satisfies a special differential condition of the form, U[F]=0, the conformal metric possesses a conformal Killing field, xi = partial w...

We begin with a four-dimensional manifold, , that possesses a two parameter family of (local) foliations by three-surfaces, with the two parameters being the coordinates on the sphere of directions at each point of the manifold expressed via homogeneous coordinates πA and πA′. By then requiring that each foliation (of the two parameter set of folia...

We describe an approach to the issue of the singularities of null hypersurfaces, due to the focusing of null geodesics, in the context of the recently introduced formulation of GR via null foliations.

This is intended as an introduction to and review of the theory of Lagrangian and Legendrian submanifolds and their associated maps developed by Arnold and his collaborators. The theory is illustrated by applications to HamiltonJacobi theory and the eikonal equation, with an emphasis on null surfaces and wave fronts and their associated caustics a...

Predictions of the standard thin lens approximation and a new iterative approach to gravitational lensing are compared with an ``exact'' approach in simple test cases involving one or two lenses. We show that the thin lens and iterative approaches are remarkably accurate in predicting time delays, source positions and image magnifications for a sin...

We propose a definition of an exact lens equation without reference to a background spacetime, and construct the exact lens equation explicitly in the case of Schwarzschild spacetime. For the Schwarzschild case, we give exact expressions for the angular-diameter distance to the sources as well as for the magnification factor and time of arrival of...

We develop an iterative approach to gravitational lensing theory based on approximate solutions of the null geodesic equations. The approach can be employed in any space-time which is ``close'' to a space-time in which the null geodesic equations can be completely integrated, such as Minkowski space-time, Robertson-Walker cosmologies, or Schwarzsch...

This is intended as an introduction to and review of the work of V, Arnold and his collaborators on the theory of Lagrangian and Legendrian submanifolds and their associated maps. The theory is illustrated by applications to Hamilton-Jacobi theory and the eikonal equation, with an emphasis on null surfaces and wavefronts and their associated causti...

In an arbitrary Lorentzian manifold we provide a description for the construction of null surfaces and their associated singularities, via solutions of the Eikonal equation. In particular, we study the singularities of the past light-cones from points on null infinity, the future light-cones from arbitrary interior points and the intersection of th...

In this work the asymptotic solutions of massless spin- fields are studied. First, in a Minkowski space, the fields, their potentials, and the associated gauges are analysed in the Rarita - Schwinger description. We exhibit the explicit appearance of the conserved charges of the fields and their singular behaviour in the potentials. The charges of...

The null-surface formulation of general relativity - recently introduced - provides novel tools for describing the gravitational field, as well as a fresh physical way of viewing it. The new formulation provides `local' observables corresponding to the coordinates of points - which constitute the spacetime manifold - in a geometrically defined char...

We locate pairs of conjugate points on null geodesics along which there is a `barrier' of Weyl curvature. The existence of conjugate points in this case is predicted by general theorems. We also find that the same conjugate points can be obtained perturbatively off flat space, assuming the barrier to be weak. The conjugate points appear at second o...

It is the purpose of this note to point out (or perhaps, more accurately, to argue) that for asymptotically flat spacetimes that are sufficiently close to flat space, there are global vector fields and associated global transformations (arising from the existence of the asymptotic symmetries) that can be identified as nonlinear counterparts of the...

Normally the issue or question of the time of arrival of light rays at an observer coming from a given source is associated with Fermat's Principle of Least Time which yields paths of extremal time. We here investigate a related but different problem. We consider an observer receiving light from an extended source that has propagated in an arbitrar...

We first define what we mean by gravitational lensing equations in a general space-time. A set of exact relations are then derived that can be used as the gravitational lens equations in all physical situations. The caveat is that into these equations there must be inserted a function, a two-parameter family of solutions to the eikonal equation, no...

A method of solving the eikonal equation, in either flat or curved space-times, with arbitrary Cauchy data, is extended to the case of data given on a characteristic surface. We find a beautiful relationship between the Cauchy and characteristic data for the same solution, namely they are related by a Legendre transformation. From the resulting sol...

The level surfaces of solutions to the eikonal equation define null or characteristic surfaces. In this note we study, in Minkowski space, properties of these surfaces. In particular we are interested both in the singularities of these ``surfaces'' (which can in general self-intersect and be only piece-wise smooth) and in the decomposition of the n...

In this work we investigate aspects of light cones in a Schwarzschild geometry, making connections to gravitational lensing theory and to a new approach to general relativity, the null surface formulation. By integrating the null geodesics of our model, we obtain the light cone from every space-time point. We study three applications of the light c...

We reformulate the standard local equations of general relativity for asymptotically flat spacetimes in terms of two non-local quantities, the holonomy H around certain closed null loops on characteristic surfaces and the light cone cut function Z, which describes the intersection of the future null cones from arbitrary spacetime points, with futur...

In this work we explore further consequences of a recently developed alternate formulation of general relativity, where the metric variable is replaced by families of surfaces as the primary geometric object of the theory—the (conformal) metric is derived from the surfaces—and a conformal factor that converts the conformal metric into an Einstein m...

We define and discuss various quantum operators that describe the geometry of spacetime in quantum general relativity. These are obtained by combining the null-surface formulation of general relativity, recently developed, with asymptotic quantization. One of the operators defined describes a “fuzzy” quantum light cone structure. Others, denoted “s...

For a rich class of asymptotically flat vacuum space-times, we show that it is possible to introduce a global coordinate system in a canonical fashion that is analogous to the standard Minkowskian coordinate systems used in flat space. This is accomplished by studying the intersection of the future light cone of interior space-time points with futu...

We formulate the vacuum Einstein equations as differential equations for two functions, one complex and one real on a six-dimensional manifold, M×S2, with M eventually becoming the space–time and the S2 becoming the sphere of null directions over M. At the start there is no other further structure available: the structure arising from the two funct...

Recently there has been developed a reformulation of General Relativity - referred to as {\it the null surface version of GR} - where instead of the metric field as the basic variable of the theory, families of three-surfaces in a four-manifold become basic. From these surfaces themselves, a conformal metric, conformal to an Einstein metric, can be...

A reduction of the self‐dual Yang–Mills (SDYM) equations is studied by imposing two space–time symmetries and by requiring that the connection one‐form belongs to a Lie algebra of formal matrix‐valued differential operators in an auxiliary variable. In this article, the scalar case and the canonical cases for 2×2 matrices are examined. In the scala...

We study the vacuum Maxwell theory by expressing the electric field in terms of its Faraday lines of force. This representation allows us to capture the two physical degrees of freedom of the electric field by means of two scalar fields. The corresponding classical canonical theory is constructed in terms of four scalar fields, is fully gauge invar...

A simple derivation is presented of the equations for the variation of the parallel propagator and the holonomy operators of Yang–Mills (YM) connections caused by variations of both the connection and the path. The derivation does not make any direct use of functional derivatives and is based on the solution of the varied parallel transport equatio...

The parallel propagator (associated with a Yang-Mills connection) taken along all null geodesics from a field point χ to null infinity is introduced as a basic variable in Yang-Mills theory. It is shown that the Yang-Mills connection can be reconstructed from this parallel propagator.
The Yang-Mills equations are expressed as an equation for the pa...

The field equations for two non-local variables, equivalent to the Einstein vacuum equations, are presented. These variables are the holonomy operator associated with special paths and the light cone cut function.Starting from these equations, one shows via a perturbation argument that a single, fourth-order equation for the cut function can be der...

The question is investigated as to what equations arise from the reduction of the anti-self-dual Yang–Mills equations by the imposition of three (space-time) translational symmetries and by the choice of the connection coefficients having values in the infinite-dimensional Lie algebras associated with one- and two-dimensional diffeomorphism groups...

Some new results are presented on the theory of Hamiltonian systems with first‐class constraints. In these systems it is possible to separate the physical part from the gauge part by transforming to canonical coordinates in which the constraints are a subset of the new momenta; this construction is accomplished by algebraic methods and the use of a...

An unusual and attractive system is studied that arises from the anti-self-dual (ASD) Yang–Mills equations with maximal translational symmetry and with gauge group the volume preserving diffeomorphisms of an auxiliary four-manifold M. The resulting equations lead to a system consisting of a volume form together with four independent vector fields o...