• Volume 77, Issue 3

September 2011,   pages  405-597

• Preface

• On the Lie point symmetry analysis and solutions of the inviscid Burgers equation

Lie point symmetries of the ﬁrst-order inviscid Burgers equation in a general setting are studied. Some new and interesting solutions are presented.

• Isometric embeddings in cosmology and astrophysics

Recent interest in higher-dimensional cosmological models has prompted some signiﬁ-cant work on the mathematical technicalities of how one goes about embedding spacetimes into some higher-dimensional space. We survey results in the literature (existence theorems and simple explicit embeddings); brieﬂy outline our work on global embeddings as well as explicit results for more complex geometries; and provide some examples. These results are contextualized physically, so as to provide a foundation for a detailed commentary on several key issues in the ﬁeld such as: the meaning of `Ricci equivalent’ embeddings; the uniqueness of local (or global) embeddings; symmetry inheritance properties; and astrophysical constraints.

• Gravitational collapse with decaying vacuum energy

The effect of dark energy on the end state of spherical radiation collapse is considered within the context of the cosmic censorship hypothesis. It is found that it is possible to have both black holes as well as naked singularities.

• On the enigmatic 𝛬 – A true constant of spacetime

Had Einstein followed the Bianchi differential identity for the derivation of his equation of motion for gravitation, 𝛬 would have emerged as a true new constant of spacetime on the same footing as the velocity of light? It is then conceivable that he could have perhaps made the most profound prediction that the Universe may suffer accelerated expansion some time in the future! Further we argue that its identiﬁcation with the quantum vacuum energy is not valid as it should have to be accounted for like the gravitational ﬁeld energy by enlarging the basic framework of spacetime and not through a stress tensor. The acceleration of the expansion of the Universe may indeed be measuring its value for the ﬁrst time observationally.

• A note on the interplay between symmetries, reduction and conservation laws of Stokes’ ﬁrst problem for third-grade rotating ﬂuids

We investigate the invariance properties, nontrivial conservation laws and interplay between these notions that underly the equations governing Stokes’ ﬁrst problem for third-grade rotating ﬂuids. We show that a knowledge of this leads to a number of different reductions of the governing equations and, thus, a number of exact solutions can be obtained and a spectrum of further analyses may be pursued.

• Higher-order symmetries and conservation laws of multi-dimensional Gordon-type equations

In this paper a class of multi-dimensional Gordon-type equations are analysed using a multiplier and homotopy approach to construct conservation laws. The main focus is the analysis of the classical versions of the Gordon-type equations and obtaining higher-order variational symmetries and corresponding conserved quantities. The results are extended to the multi-dimensional Gordontype equations with the two-dimensional Klein–Gordon equation in particular yielding interesting results.

• Relativistic stellar models

We obtain a class of solutions to the Einstein–Maxwell equations describing charged static spheres. Upon specifying particular forms for one of the gravitational potentials and the electric ﬁeld intensity, the condition for pressure isotropy is transformed into a hypergeometric equation with two free parameters. For particular parameter values we recover uncharged solutions corresponding to speciﬁc neutron star models. We ﬁnd two charged solutions in terms of elementary functions for particular parameter values. The ﬁrst charged model is physically reasonable and the metric functions and thermodynamic variables are well behaved. The second charged model admits a negative energy density and violates the energy conditions.

• Temperature evolution during dissipative collapse

We investigate the gravitational collapse of a radiating sphere evolving into a ﬁnal static conﬁguration described by the interior Schwarzschild solution. The temperature proﬁles of this particular model are obtained within the framework of causal thermodynamics. The overall temperature evolution is enhanced by contributions from the temperature gradient induced by perturbations as well as relaxational effects within the stellar core.

• Charged ﬂuids with symmetries

We investigate the role of symmetries for charged perfect fluids by assuming that spacetime admits a conformal Killing vector. The existence of a conformal symmetry places restrictions on the model. It is possible to ﬁnd a general relationship for the Lie derivative of the electromagnetic ﬁeld along the integral curves of the conformal vector. The electromagnetic ﬁeld is mapped conformally under particular conditions. The Maxwell equations place restrictions on the form of the proper charge density.

• A note on the Lie symmetries of complex partial differential equations and their split real systems

Folklore suggests that the split Lie-like operators of a complex partial differential equation are symmetries of the split system of real partial differential equations. However, this is not the case generally. We illustrate this by using the complex heat equation, wave equation with dissipation, the nonlinear Burgers equation and nonlinear KdV equations. We split the Lie symmetries of a complex partial differential equation in the real domain and obtain real Lie-like operators. Further, the complex partial differential equation is split into two coupled or uncoupled real partial differential equations which constitute a system of two equations for two real functions of two real variables. The Lie symmetries of this system are constructed by the classical Lie approach. We compare these Lie symmetries with the split Lie-like operators of the given complex partial differential equation for the examples considered. We conclude that the split Lie-like operators of complex partial differential equations are not in general symmetries of the split system of real partial differential equations. We prove a proposition that gives the criteria when the Lie-like operators are symmetries of the split system.

• Effects of non-uniform interfacial tension in small Reynolds number ﬂow past a spherical liquid drop

A singular perturbation solution is given for small Reynolds number ﬂow past a spherical liquid drop. The interfacial tension required to maintain the drop in a spherical shape is calculated. When the interfacial tension gradient exceeds a critical value, a region of reversed ﬂow occurs on the interface at the rear and the interior ﬂow splits into two parts with reversed circulation at the rear. The magnitude of the interior ﬂuid velocity is small, of order the Reynolds number. A thin transition layer attached to the drop at the rear occurs in the exterior ﬂow. The effects could model the stagnant cap which forms as surfactant is added but the results apply however the variability in the interfacial tension might have been induced.

• Chandrasekhar: The all rounder

This paper provides a very brief introduction to three of Chandrasekhar’s famous books on Stellar Structure, Hydrodynamics and Black Holes. In particular we summarize Chandra’s treatment of the “Thermal Instability” which plays such a crucial role in the understanding of convection zones in stellar atmospheres. We also outline three important ideas in ﬂuid dynamics which are inexplicably omitted from Chandrasekhar’s Hydrodynamic and Hydromagnetic Stability; the ﬁrst is the Brunt–Väisȧlä frequency which appears in internal gravity waves and is closely related to Schwarzschild’s stability criterion; the second is the baroclinic instability which is important in atmospheric dynamics and meteorology, and the third is the conservation of potential vorticity which is central to the understanding of the planetary scale – Rossby waves.

• Transient heat transfer in longitudinal ﬁns of various proﬁles with temperature-dependent thermal conductivity and heat transfer coefﬁcient

Transient heat transfer through a longitudinal ﬁn of various proﬁles is studied. The thermal conductivity and heat transfer coefficients are assumed to be temperature dependent. The resulting partial differential equation is highly nonlinear. Classical Lie point symmetry methods are employed and some reductions are performed. Since the governing boundary value problem is not invariant under any Lie point symmetry, we solve the original partial differential equation numerically. The effects of realistic ﬁn parameters such as the thermogeometric ﬁn parameter and the exponent of the heat transfer coefﬁcient on the temperature distribution are studied.

• An investigation of embeddings for spherically symmetric spacetimes into Einstein manifolds

Embeddings into higher dimensions are very important in the study of higherdimensional theories of our Universe and in high-energy physics. Theorems which have been developed recently guarantee the existence of embeddings of pseudo-Riemannian manifolds into Einstein spaces and more general pseudo-Riemannian spaces. These results provide a technique that can be used to determine solutions for such embeddings. Here we consider local isometric embeddings of four-dimensional spherically symmetric spacetimes into ﬁve-dimensional Einstein manifolds. Difﬁculties in solving the ﬁve-dimensional equations for given four-dimensional spaces motivate us to investigate embedded spaces that admit bulks of a speciﬁc type. We show that the general Schwarzschild–de Sitter spacetime and Einstein Universe are the only spherically symmetric spacetimes that can be embedded into an Einstein space of a particular form, and we discuss their ﬁve-dimensional solutions.

• Exact solutions of the generalized Lane–Emden equations of the ﬁrst and second kind

In this paper we discuss the integrability of the generalized Lane–Emden equations of the ﬁrst and second kinds. We carry out their Noether symmetry classiﬁcation. Various cases for the arbitrary functions in the equations are obtained for which the equations have Noether point symmetries. First integrals of such cases are obtained and also reduction to quadrature of the corresponding Lane–Emden equations are presented. New cases are found.

• Invariance analysis and conservation laws of the wave equation on Vaidya manifolds

In this paper we discuss symmetries of classes of wave equations that arise as a consequence of some Vaidya metrics. We show how the wave equation is altered by the underlying geometry. In particular, a range of consequences on the form of the wave equation, the symmetries and number of conservation laws, inter alia, are altered by the manifold on which the model wave rests. We ﬁnd Lie and Noether point symmetries of the corresponding wave equations and give some reductions. Some interesting physical conclusions relating to conservation laws such as energy, linear and angular momenta are also determined. We also present some interesting comparisons with the standard wave equations on a ﬂat geometry. Finally, we pursue the existence of higher-order variational symmetries of equations on nonﬂat manifolds.

• Estimate of stellar masses from their QPO frequencies

Kilohertz quasiperiodic oscillations (kHz QPOs) are observed in binary stellar systems. For such a system, the stellar radius is very close to the marginally stable orbit $R_{\text{ms}}$ as predicted by Einstein’s general relativity. Many models have been proposed to explain the origin of the kHz QPO features in the binaries. Here we start from the work of Li et al (Phys. Rev. Lett. 83, 3776 (1999)) who in 1999, from the unique millisecond X-ray pulsations, suggested SAX J1808.4−3658 to be a strange star, from an accurate determination of its rotation period. It showed kHz QPOs eight years ago and so far it is the only set that has been observed. We suggest that the mass of four compact stars SAX J1808.4−3658, KS 1731−260, SAX J1750.8−2900 and IGR J17191−2821 can be determined from the difference in the observed kHz QPOs of these stars. It is exciting to be able to give an estimate of the mass of the star and three other compact stars in low-mass X-ray binaries using their observed kHz QPOs.

• Linearization of systems of four second-order ordinary differential equations

In this paper we provide invariant linearizability criteria for a class of systems of four second-order ordinary differential equations in terms of a set of 30 constraint equations on the coefﬁcients of all derivative terms. The linearization criteria are derived by the analytic continuation of the geometric approach of projection of two-dimensional systems of cubically semi-linear secondorder differential equations. Furthermore, the canonical form of such systems is also established. Numerous examples are presented that show how to linearize nonlinear systems to the free particle Newtonian systems with a maximally symmetric Lie algebra relative to $sl$(6, $\mathfrak{R}$) of dimension 35.

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