# R. J. Torrence's research while affiliated with The University of Calgary and other places

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## Publications (45)

It is shown that a convenient identification gauge-invariant formalism for the study of cosmological perturbations has additional potential gauge freedom, which has not always been correctly identified, in solutions of the field equations.

Motions of finite Toda lattices are known to be associated with linear wave equations whose general solutions can be expressed in terms of progressing waves, and this association is known to generalize to finite non-Abelian Toda lattices of n x n matrices and systems of n coupled linear wave equations. We present a nontrivial family of non-Abelian...

The Bremmer series solution of the one-dimensional Helmholtz equation with variable velocity is generalized to obtain a similar series for the radial wave equation with a spherically symmetric velocity function. Since the leading term of Bremmer's series is the one-dimensional WKB approximation, we obtain an approximation for the radial wave equati...

A wave-splitting approach used elsewhere to solve a black-hole scattering problem is adapted to the formal solution of arbitrary self-adjoint wave equations in 1+1 dimensions and yields potentially useful results in this more general case. It is then shown that the self-adjoint wave equation being solved, and the non-self-adjoint linear wave equati...

A formalism suggested by Stewart for the study of perturbations of cosmological spacetimes is applied to the case of isentropic perfect fluid perturbations. Autonomous systems of differential equations for distinct types of perturbations, including gravitational waves, vorticity perturbations, and waves of density perturbations, arise in a natural...

The wave equation governing a scalar field of mass m on the de Sitter geometry is considered. It is shown that the multipole fields propagate without scattering whenever m, the constant alpha occurring in the metric, and the coupling constant xi , satisfy 2-12 xi -m2/ alpha 2=l'(l'+1) for some integer l'. In particular the minimally coupled massles...

A bijection is defined from the set of motions on the infinite Toda lattice of strings to a set of sequences of linear wave equations in 1+1 dimensions, the sequences being generated by a generalisation of the classical Darboux map. The bijection is applied to find probably all such wave equations for which characteristic initial data propagate wit...

Starting from a variational principle for perfect fluids, the authors develop a Hamiltonian formulation for perfect fluids coupled to gravity expressed in Ashtekar's spinorial variables. The constraint and evolution equations for the gravitational variables are at most quadratic in these variables, as in the vacuum case and in the coupling of gravi...

We discuss a family of variable-coefficient linear wave equations that are exactly solved by a generalization of progressing waves that spread out as they propagate. This can be viewed as a generalization to some variable-coefficient equations of the dispersion theory method that is standard for constant coefficient equations. The method is illustr...

Relates equivalence classes of coupled systems of N linear wave equations to motions of an N*N matrix dynamical systems, the two-dimensional non-Abelian Toda lattice. In particular, the correspondence is shown to relate those coupled systems of wave equations with progressing-wave general solutions to motions of the finite non-Abelian Toda lattice...

A structure-preserving bijection B:M to W from the set M of Toda lattice motions to a natural partition W of the set of linear wave equations in 1+1 dimensions is applied to obtain some simple and related results about both M and W. The Toda lattice motions derived emphasise the existence of qualitatively distinct Toda lattices depending on the ove...

The simplest wave equations are those whose general solutions comprise progressing waves. The author constructs a comparatively large, possibly exhaustive, family of self-adjoint acoustic equations in 1+1 dimensions that are simple in this sense.

We determine the large class of Robertson-Walker spacetimes whose first-order, linear, isentropic perturbations can be expressed in closed form, and the closed form perturbations are written down. It is shown that the class includes several well-known spacetimes including, for example, spatially flat dust and the radiation filled universe.

We consider a recently proposed theory of gravity, classically equivalent to Einstein's theory with the cosmological constant as an additional variable, in which spacetime volume plays the role of time. We develop a Hamiltonian formulation using Ashtekar's variables, set up the corresponding quantum theory, and show that the known loop state soluti...

A recent paper by Gottlieb [J. Math. Phys. 29, 2434 (1988)] provides examples of acoustic wave equations, in various dimensions, that have nontrivial families of solutions that are progressing waves of order 1, and relates this to whether or not these equations satisfy Huygens’ principle. A statement made in that paper related to Huygens’ principle...

Gerlach and Sengupta derived a master equation for arbitrary odd-parity gravitational perturbations of spherically symmetrical space-times. In this paper we show that for radiative purely gravitational perturbations of Robertson-Walker space-times the same equation governs the evenparity perturbations as well.

In a series of papers Janis and coworkers found a family of Robertson-Walker space-times whose most general radiative purely gravitational perturbations could be expressed as progressing waves. In this paper we significantly generalize, simplify, and extend, their results.

We consider conformally invariant massless spin-s field equations on a spherically symmetrical space-time. Precisely when these equations are consistent appropriately defined field components are shown to satisfy wave equations related by a generalization of the classical Darboux map.

It is known that a Kacs-van Moebeke lattice motion can be viewed as two interpolated Toda lattice motions, and that any Toda lattice motion corresponds to an equivalence class of linear wave equations. Together these results determine a correspondence between equivalence classes of linear wave equations, which preserves some properties of interest...

A family of spherically symmetrical spacetimes that are transparent to minimally coupled scalar multipole waves is investigated. The concept of transparency is reformulated in terms of the established concept of progressing waves. The set of spherically symmetric spacetimes and the family of wave equations for which the transparency condition can b...

It is shown that a portion of de Sitter space can be expressed in the formds
2=dt
2–A
2
(t)dr
2–B
2(t)(d
2+sin
2
d
2). It follows that it is a Kantowski-Sachs spacetime, according to the usual definition. This disproves the statement sometimes seen in the literature that all Kantowski-Sachs spacetimes are anisotropic.

We obtain a family of wave equations with progressive wave solutions of finite order that combines and generalizes a subfamily involving the nonreflecting Pöschl-Teller potentials and several subfamilies reported by Chang and Janis. The elementary equivalence of some of the known families is derived.

Within the class of second‐order linear self‐adjoint wave equations in 1+1 dimensions, an explicit construction is given of probably all those with the characteristic propagation property, that is, those whose solutions are without tails.

Using nonscattering potentials of Chang and Janis, a large class of spherically symmetric space-times is constructed on which all multipole solutions to the minimally coupled scalar wave equation are expressible in terms of characteristic data functions in essentially as simple a fashion as for flat space-time. The space-times are transparent to mu...

It is shown that only de Sitter space and Minkowski space can be expressed in more than one of the standard forms for Robertson-Walker space-times.

It has been asserted in the literature that a particular solution of the Liouville equation, ∂xtsigma = esigma, can be interpreted as an N-soliton. It is shown here that this solution id in fact familiar one-soliton written in a misleading form.

There are series solutions for characteristic boundary value problems for fields on black hole backgrounds that converge when the data are given on =– +, or on =– +, but may not converge when the data are given on – –, or on + +. We specialize to oscillatory data of frequency and calculate approximate reflection and transmission coefficientsR() and...

The extreme Reissner-Nordström geometry is shown to be conformally invariant under a spatial inversion. Generalization of this result to other geometries is briefly discussed.

Using the formalism of Cohen and Kegeles equations are obtained for arbitrary spin radiation field Debye potentials in Reissner-Nordström geometries, and a formal series solution is presented. For the case of integer spin the series is modified to what appears to be a more natural form. In the particular case of vanishing spin it is shown that in s...

The characteristic initial-value problem for the scalar wave equation in some spherically symmetrical geometries, including Reissner-Nordstrm exterior geometry, is considered. A modified Picard iteration making explicit use of the (known) static solutions yields a formula for the fields. The formulas extend in to the outer horizon, and from them ar...

A special class of N-soliton solutions of the Korteweg–deVries equation is examined. It is shown that the nonscattering wave equation based on the corresponding special reflectionless potentials can be obtained by a coordinate transformation of the ordinary wave equation and that these reflectionless potentials and others occur in the scalar wave e...

The Reissner-Nordstrm geometry is approximated by a certain sequence of conformally flat shells. As an example of the use of this approximation, spectral curves of transmission coefficientT(w) vs. frequencyw are calculated for spherical waves of monochromatic scalar radiation imploding on the charged black hole by solving the scalar wave equation w...

The Schwarzschild geometry is approximated by a certain set of conformally flat shells. As an example of the use of this approximation, the transmission coefficientT for spherical waves of monochromatic scalar radiation imploding on the black hole is calculated by solving the scalar wave equation within each conformally flat space and matching acro...

The inhomogeneous wave equation for a class of driving terms that arise in certain physical problems is analyzed by introducing a kind of ’’inner product’’ g with respect to which the 2l-pole solutions, &psgr;l, of the homogeneous wave equation are an orthogonal basis. This allows the condition that δ, the Lth multipole part of the driving term, wi...

A simple condition that is necessary and sufficient for the solution of the inhomogeneous wave equation to be a nonspreading wave is derived for a class of driving terms that arise in certain physical problems. The condition is applied to the analysis of the self−scattering of gravitational multipole radiation at second perturbative order. It is pr...

Newman and Penrose have given conditions on the asymptotic form of the Weyl tensor in empty space‐time that are sufficient to insure that the space‐time is asymptotically flat at null infinity and has the peeling property. We give considerably weaker conditions and show them to be sufficient for asymptotic flatness. Under the weaker conditions the...

A nonsingular, sourceless, first‐order pulse of gravitational radiation imploding from infinity to a focus and then exploding back out to infinity is examined to second order. It is found that, contrary to what might be expected, the nonsingular second‐order field contains no radiation. In space‐time regions outside of the pulse, the second‐order f...

We investigate the non-linear corrections to a non-singular retarded-minus-advanced pulse of linearized quadrupole gravitational radiation. The second order field is non-singular if and only if there is no scattering of the radiation. Radiation scattering must appear in third order.

A generalization of Born‐Infeld nonlinear electrodynamics, due to Plebanski, is reformulated in the context of general relativity theory. A class of nonsingular, static, spherically symmetric solutions of the modified Einstein—Maxwell equations are given, corresponding to a point‐charge source. The metric tensors of these solutions are shown to app...

A first‐order quadrupole sandwich wave of gravitational radiation exploding from a first‐order Schwarzschild mass is examined to second order. If the second‐order field preceding the sandwich wave vanishes, it is shown that the region of space‐time following the sandwich wave contains a second‐order, imploding quadrupole wave. The rest of the secon...

A method is presented for studying asymptotically flat spaces possessing both incoming and outgoing gravitational radiation at infinity. The method uses multipole expansions and the invariance of general relativity under time reversal; calculations are facilitated by a small‐parameter perturbation approach. Some calculations are carried out to seco...

If one of the Ruse vectors of a field is assumed to be a geodesic having nonvanishing divergence θ, curl ω, and complex shear σ, the only vacuum metrics that exist are found to be of the cylindrical type, where the geodesic rays obey θ2 + ω2 = σσ.

A new solution of the Einstein-Maxwell equations is presented. This solution has certain characteristics that correspond to a rotating ring of mass and charge.

## Citations

... As we shall see, the emission of radiation is always accompanied by the appearance of vorticity of world lines of observers. Furthermore, the calculations suggest that once the radiation process has stopped, there is still a remaining vorticity associated to the tail of the wave, which allows in principle to prove (or disprove) the violation of the Huyghens principle in a Riemannian spacetime (see [43][44][45][46][47][48][49] and references therein for a discussion on this issue), by means of observations. ...

Reference: Deconstructing Frame Dragging

... [20, rel. (21) [8] studied: Blau, Guendelman, Guth (1987) [32] def. ...

... An early ansatz before was that of Gottlieb [17] with nonconstant coefficients, but for a circular symmetric equation. It was generalized by Bombelli et al. [18]. ...

... In [5] we have shown that D3-branes and their bound states in lower dimensions admit a symmetry under conformal inversions that generalize the Couch-Torrence (CT) transformations known to leave the metric of extremal Reissner-Nordström BHs invariant up to a Weyl rescaling [6]. CT transformations are known to exchange the event horizon with null infinity [6][7][8][9][10][11][12][13]. ...

... Clearly, if R has compact support [u 1 , u 2 ], the wave propagates permanently sandwiched between those two values of u. We have here no metric with which to verify whether this wave has tails in the sense defined above, but it is nevertheless obvious that " there are no tails with respect to u. " A slightly more general situation is that of relatively undistorted or simple progressing waves, those of the form φ(x) = f (x) R(u(x)), where the amplitude f is a function on M [5,6,16171819. An example in Minkowski spacetime is the spherical wave φ(t, x) = exp i(k|x| − ωt)/|x|, which progresses with speed ω/k in the radial direction, and is undistorted except for the decrease in the amplitude 1/|x|. ...

... At the time, anyone toiling at the abstract analysis of the Cauchy problem for the non-linear Einstein equations must have registered with surprise that the complicated global problem of characterizing the asymptotic behaviour of solutions should allow, in any generality, an answer in so simple and clean geometric terms as used in Definition 1.1. In fact, it has been argued from early on that the assumptions on the asymptotic behaviour underlying the work referred to above might be too stringent and that consistent formal expansions at null infinity can also obtained under more general assumptions [25], [33]. ...

Reference: Geometric Asymptotics and Beyond

... NED-GR solutions with electric or magnetic fields are currently widely discussed, probably beginning with finding a general form of an electric solution by Pellicer and Torrence [60]. Later on, a no-go theorem was proved [?, 23], showing that if NED is specified by a Lagrangian function L( f ) having a Maxwell weak-field limit (L ∼ f as f → 0), a static, spherically symmetric solution of GR with an electric field cannot have a regular center. ...

... The gravitational wave now has been detected, the first three observations GW150914 [1][2][3], GW151226 [4], GW170104 [5] by LIGO, the observation by the advanced Virgo detector and the two advanced LIGO detectors [6], and the gravitational wave produced by a binary neutron star merging [7,8]. The gravitational wave and the gravitational wave scattering have been studied for many years, e.g., historically, gravitational radiations at infinity in a asymptotically flat space [9], classical cross sections [10], inelastic cross sections of nonrotating and rotating black holes [11], scattering off a Kerr black hole [12], the weak-field gravitational scattering [13], gravitational wave scattering on a Schwarzschild black hole in the low-frequency limit [14], scattering of small wave amplitudes and weak gravitational fields [15], and differential cross sections of plane gravitational waves scattering from the gravitational field of sources in the weak-field approximation [16]. In observation, there is also an indirect evidence for the existence of gravitational wave was observed [17]. ...

... On the one hand, we want to explore the combined effect of rotation and electric charge parameters on the topological number of black holes, on the other hand, for the general consideration of the solution of four-dimensional black holes in the pure Einstein-Maxwell gravity theory, in this section, we turn to investigate the topological number of four-dimensional Kerr-Newman black hole [26,27], whose metric and Abelian gauge potential are ...

Reference: Topological classes of rotating black holes

... being the most common [87,166,172,173]. It is motivated by the similarity with the Coulomb gauge of electromagnetism (∇ · A = 0), since its gauge conditions are equivalent to B i ,i = 0 and C ij T ,i = 0, with C ij T being the traceless part of the perturbations of the spatial metric, C ij . ...