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A new exact solution to the field equations in the Einstein-Gauss-Bonnet modified theory of gravity, for a six-dimensional spherically symmetric static distribution of a perfect fluid source is presented. The pressure isotropy equation is integrated after a form for the temporal potential proportional to the radius is postulated to close the system of equations. For a specific choice of the coupling parameter it is demonstrated that the matching of the interior and exterior spacetimes is explicitly achievable. The general model has been tested to be physically acceptable in this framework using criteria extrapolated from the standard four dimensional theory and after locating a suitable parameter space through fine-tuning. A vanishing pressure hypersurface signifying a boundary exists and the speed of sound is subluminal throughout the interior of the matter distribution. Furthermore, all energy conditions are satisfied. Finally, the Chandrasekhar adiabatic stability bound is satisfied.

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... For a study of the physical features in stellar models it is necessary to find exact solutions to the EGB field equations. Particular classes of exact solutions in static metrics have been found mainly in five and six spacetime dimensions [30][31][32][33][34][35] for neutral matter distributions with isotropic pressure. Other interesting models have been studied by [36][37][38][39]. ...

... Hence the dimension N has a dramatic effect on the gravitational behaviour of the model for N ≥ 6. Note that the condition of pressure isotropy for N = 5 was also presented by Hansraj et al. [31], and by Hansraj and Mkhize [33] when N = 6. Our result (11) holds for all dimensions N. Our intention is to find exact solutions for all N. ...

... This choice was also made by Hansraj and Mkhize [33] for the particular spacetime dimension N = 6. The integral in (16) can be evaluated and we obtain ...

We generate the Einstein–Gauss–Bonnet field equations in higher dimensions for a spherically symmetric static spacetime. The matter distribution is a neutral fluid with isotropic pressure. The condition of isotropic pressure, an Abel differential equation of the second kind, is transformed to a first order nonlinear canonical differential equation. This provides a mechanism to generate exact solutions systematically in higher dimensions. Our solution generating algorithm is a different approach from those considered earlier. We show that a specific choice of one potential leads to a new solution for the second potential for all spacetime dimensions. Several other families of exact solutions to the condition of pressure isotropy are found for all spacetime dimensions. Earlier results are regained from our treatments. The difference with general relativity is highlighted in our study.

... Tangphati et al. [25] have also found out that an anisotropic quark star in the context of EGB gravity leads to considerable change both in the structure of the star and the mass-radius relation. Hansraj and Mkhize [26] recently obtained exact solutions in EGB gravity in a six dimensional fluid sphere and used it to construct stellar models, using barotropic fluid in a higher dimensional Krori-Barua metric. However, for a given equation of state (EoS) the metric solutions can be obtained from the gravitational field equations. ...

We present a new class of relativistic anisotropic
stellar models with spherically symmetric matter distribution
in Einstein Gauss–Bonnet (EGB) gravity. A higher dimensional
Finch–Skea geometry in the theory is taken up here
to construct stellar models in hydrostatic equilibrium. The
Gauss–Bonnet term is playing an important role in accommodating
neutron stars. We study the physical features namely,
the energy density, the radial and tangential pressures and the
suitability of the models. It is found that the equation of state
of such stars are non-linear which is determined for a given
mass and radius of known stars. The stability of the stellar
models are also explored for a wide range of values of the
model parameters.

... A static charged anisotropic fluid sphere described by Krori-Barua metric is also studied [29] considering a coupled Einstein-Maxwell-Gauss-Bonnet field equations with a linear equation of state (EoS) different from MIT Bag model. Recently, Hansraj and Mkhize [30] obtained exact solutions for constructing stellar models in the Einstein-Gauss-Bonnet gravity in a six-dimensional spacetime. Considering Finch-Skea geometry, a new class of interior solutions of compact objects in five dimensional Einstein Gauss-Bonnet (EGB) gravity is obtained [31] with a linear equation of state (EoS) which permits compact objects. ...

We obtain a class of new anisotropic relativistic solution in Einstein Gauss-Bonnet (EGB) gravity with Finch-Skea metric in hydrostatic equilibrium. The relativistic solutions are employed to construct anisotropic stellar models for strange star with MIT Bag equation of state $ p_{r}= \frac{1}{3} \left( \rho - 4 B_{g}\right)$, where $B_{g}$ is the Bag constants. Considering the mass and radius of a known star PSR J0348+0432 we construct stellar models in the framework of higher dimensions. We also predict the mass and radius of stars for different model parameters. The Gauss-Bonnet coupling term ($\alpha$) plays an important role in determining the density, pressure, anisotropy profiles and other features. The stability of the stellar models are probed analyzing the different energy conditions, variation of sound speed and adiabatic stability conditions inside the star. The central density and pressure of a star in EGB gravity are found to have higher values compared to that one obtains in Einstein gravity ($\alpha =0$). We also explore the effect of extra dimensions for the physical features of a compact object. For this we consider $D=5$ and $D=6$ to obtain a realistic stellar model and found that in the formal case both positive and negative values of $\alpha$ are allowed. But in the later case, only $\alpha <0$ permits compact object in the Finch-Skea metric. We determine the best fit values of the model parameters for a number of observed stars for acceptable stellar models.

... A static charged anisotropic fluid sphere described by Krori-Barua metric is also studied [28] considering a coupled Einstein-Maxwell-Gauss-Bonnet field equations with a linear equation of state (EoS) different from MIT Bag model. Recently, Hansraj and Mkhize [29] obtained exact solutions for constructing stellar models in the Einstein-Gauss-Bonnet gravity in a six-dimensional spacetime. Considering Finch-Skea geometry, a new class of interior solutions of compact objects in five dimensional Einstein Gauss-Bonnet (EGB) gravity is obtained [30] with a linear equation of state (EoS) which permits compact objects. ...

We obtain a class of new anisotropic relativistic solution in Einstein Gauss-Bonnet (EGB) gravity with Finch-Skea metric in hydrostatic equilibrium. The relativistic solutions are employed to construct anisotropic stellar models for strange star with MIT Bag equation of state p r = 1 3 (ρ − 4B g), where B g is the Bag constants. Considering the mass and radius of a known star PSR J0348+0432 we construct stellar models in the framework of higher dimensions. We also predict the mass and radius of stars for different model parameters. The Gauss-Bonnet coupling term (α) plays an important role in determining the density, pressure, anisotropy profiles and other features. The stability of the stellar models are probed analyzing the different energy conditions, variation of sound speed and adiabatic stability conditions inside the star. The central density and pressure of a star in EGB gravity are found to have higher values compared to that one obtains in Einstein gravity (α = 0). We also explore the effect of extra dimensions for the physical features of a compact object. For this we consider D = 5 and D = 6 to obtain a realistic stellar model and found that in the formal case both positive and negative values of α are allowed. But in the later case, only α < 0 permits compact object in the Finch-Skea metric. We determine the best fit values of the model parameters for a number of observed stars for acceptable stellar models.

... It is seriously difficult to locate exact solutions for perfect fluid matter in EGB because the extra curvature terms make the governing differential equations intractable. An additional solution for constant potentials in six dimensional EGB spacetimes was found in [27] and recently Hansraj and Mkhize generated a physically viable six dimensional model with variable potentials and density [28]. A greater number of the extra curvature terms survive in 6D as opposed to 5D making the differential equations even more difficult to work with. ...

It is known that the standard Schwarzschild interior metric is conformally flat and generates a constant density sphere in any spacetime dimension in Einstein and Einstein-Gauss-Bonnet (EGB) gravity. This motivates the questions: in EGB does the conformal flatness criterion yield the Schwarzschild metric? Does the assumption of constant density generate the Schwarzschild interior spacetime? The answer to both questions turn out in the negative in general. In the case of the constant density sphere, a generalised Schwarzschild metric emerges. When we invoke the conformal flatness condition the Schwarschild interior solution is obtained as one solution and another metric which does not yield a constant density hypersphere in EGB theory is found. For the latter solution one of the gravitational metrics is obtained explicitly while the other is determined up to quadratures in 5 and 6 dimensions. The physical properties of these new solutions are studied with the use of numerical methods and a parameter space is located for which both models display pleasing physical behaviour.

In this work, we are guided by the gravitational wave events GW 170817 and GW 190814 together with observations of neutron stars PSR J1614-2230, PSR J1903+6620 and LMC X-4 to model compact objects within the framework of Einstein-Gauss-Bonnet (EGB) gravity. In addition, we employ the complete gravitational decoupling method to explore the impact of anisotropy by varying the decoupling parameter. We model strange quark stars in which the interior stellar fluid obeys the MIT Bag equation of state which represents a degenerated Fermi gas comprising of up, down, and strange quarks. In order to close the system of field equations describing the seed solution, we employ the Buchdahl ansatz for one of the metric functions. The θ sector is solved under the bifurcation: $\epsilon =\theta ^0_0$ and $P_r=\theta ^1_1$ leading to two new families of solutions. In order to test the physical viability of the models, we vary the EGB parameter (α) or the decoupling constant (β) to achieve the observed masses and radii of compact objects. Our models are able to account for low-mass stars for a range of β values while α is fixed. Our models mimic the secondary component of the GW 190814 with a mass range of 2.5 to 2.67 M⊙ and radii typically of the order of 11.76$^{+0.14}_{-0.19}$ km for large values of the EGB parameter and the decoupling constant. The energy exchange between fluids inside the stellar object is sensitive to model parameters which lead to stable configurations.

Exploiting the use of curvature coordinates (also called Schwarzschild coordinates), new classes of exact solutions are discovered. By prescribing the timelike potential two new solutions are found one of which displays all the necessary qualitative features demanded of closed compact astrophysical objects. Since the analysis reduces to a single first order nonlinear differential equation, the single integration constant is obtained in terms of the mass and radius through matching of the interior and exterior metrics. Invoking the second matching condition results in constraining the value of the Gauss–Bonnet coupling constant in terms of the mass and radius of the star. Stability, causality and energy conditions are satisfied for a suitable choice of parameter space. In attempting to specify the radially oriented potential it was only possible to find a defective spacetime and to regain the interior Schwarzschild metric for an incompressible fluid sphere.

In this article, we investigate an anisotropic solution for compact static spherically symmetric objects as an alternative to neutron stars, which is a class of compact stars dubbed as strange stars in the context of five-dimensional Einstein-Gauss-Bonnet (EGB) theory by exploiting Tolman's metric. With current sensitivities, we also consider a nonlinear equation of state along with an anisotropic source of matter which formed the basis for generating the bounded compact stars. The unknown constants are determined via the boundary conditions along with the Boulware-Deser geometry to describe as an exterior space-time. Observational mass data of the recently discovered millisecond pulsars viz., PRS J1614-2230, PRS J1903+327, and LMC X-4, were used to predict radii via $M-R$ curve which are lying in the range of $R\le 10-12~\text{km}$. Furthermore, we discuss the stability of the model via adiabatic index and mass vs central mass density ($M-\rho_c$) profile. We find that a fine-tuning of the parameters coming from the theory, eventually, influence the inner geometry of compact stars and, therefore, influence important physical properties of realizable stellar structures. Moreover, the effects of EoS parameters on the mass and Bag constant $\mathcal{B}$ are shown by the $equi-plane$ diagrams. Conclusively, the results have shown that our stellar model is stable, physically acceptable, as well as it provides circumstantial evidence in favor of super-massive pulsars such as ultra-dense hypothetical strange stars in the background of five-dimensional EGB gravity.

We analyze the configuration of charged dust in the context of the higher dimensional and higher curvature Einstein–Gauss–Bonnet–Maxwell theory. With the prescription of dust, there remains one more prescription to be made in order to close the system of equations of motion. The choice of one of the metric potentials appears to be the only viable way to proceed. Before establishing exact solutions, we examine conditions for the existence of physically reasonable charged dust fluids. It turns out that the branches of the Boulware–Deser metric representing the exterior gravitational field of a neutral spherically symmetric Einstein–Gauss–Bonnet distribution, serve as upper and lower bounds for the spatial potentials of physically reasonable charged dust in Einstein–Gauss–Bonnet–Maxwell gravity. Some exact solutions for 5 and 6 dimensional charged dust hyperspheres are exhibited in closed form. In particular the Einstein ansatz of a constant temporal potential while defective in 5 dimensions actually generates a model of a closed compact astrophysical object in 6 dimensions. A physically viable 5 dimensional charged dust model is also contrasted with its general relativity counterpart graphically.

The recent theoretical advance known as the minimal geometric deformation (MGD) method has initiated renewed interest in investigating higher-curvature gravitational effects in relativistic astrophysics. In this work, we model a strange star within the context of Einstein–Gauss–Bonnet gravity with the help of the MGD technique. Starting off with the Tolman metric ansatz, together with the MIT bag model equation of state applicable to hadronic matter, anisotropy is introduced via the superposition of the seed source and the decoupled energy-momentum tensor. The solution of the governing systems of equations bifurcates into two distinct models, namely, the mimicking of the θ sector to the seed radial pressure and energy density and a regular fluid model. Each of these models can be interpreted as self-gravitating static, compact objects with the exterior described by the vacuum Boulware–Deser solution. Utilizing observational data for three stellar candidates, namely PSR J1614–2230, PSR J1903+317, and LMC X-4, we subject our solutions to rigorous viability tests based on regularity and stability. We find that the Einstein–Gauss–Bonnet parameter and the decoupling constant compete against each other for ensuring physically realizable stellar structures. The novel feature of the work is the demonstration of stable compact objects with stellar masses in excess of M = 2 M ⊙ without appealing to exotic matter. The analysis contributes new insights and physical consequences concerning the development of ultracompact astrophysical entities.

The static isotropic gravitational field equation, governing the geometry and dynamics of stellar structure, is considered in Einstein–Gauss–Bonnet (EGB) gravity. This is a nonlinear Abelian differential equation which generalizes the simpler general relativistic pressure isotropy condition. A gravitational potential decomposition is postulated in order to generate new exact solutions from known solutions. The conditions for a successful integration are examined. Remarkably we generate a new exact solution to the Abelian equation from the well known Schwarzschild interior seed metric. The metric potentials are given in terms of elementary functions. A physical analysis of the model is performed in five and six spacetime dimensions. It is shown that the six-dimensional case is physically more reasonable and is consistent with the conditions restricting the physics of realistic stars.

In this endeavour, we model spherically, symmetric compact stellar configurations obeying a polytropic equation of state of the form pr=κρ2+βρ-γ within the framework of Einstein–Gauss–Bonnet gravity. We employ the Finch and Skea ansatz to complete the gravitational behaviour of the stellar fluid. The solution is smoothly matched to the exterior Boulware–Deser metric. The effect of the EGB coupling constant is highlighted by studying the thermodynamical properties and stability of the stellar model.

In this paper we present two new classes of solutions describing compact objects within the framework of five‐dimensional Einstein‐Gauss‐Bonnet (EGB) gravity. We employ the Complete Geometric Deformation (CGD) formalism which extends the Minimal Geometric Deformation (MGD) technique adopted in earlier investigations to generate anisotropic models from known isotropic solutions. The two solutions presented arise from mimicking the constraint for the pressure and density respectively which generate independent deformation functions. Rigorous physical tests show that contributions from CDG suppress the effective pressure but enhances the effective density and mass of the compact object, with the suppression/enhancement being modified by the EGB coupling constant. One of the highlights in our findings is that the deformation function along the radial component in CDG is nonzero at the boundary when we mimic both the pressure and density while in MGD we observe a vanishing of this deformation function at the boundary of the fluid configuration only for the pressure constraint. The difference in behavior of the deformation function at the surface predicts different stellar characteristics such as mass‐to‐radius and surface redshifts.

G\"{o}del universe is a homogeneous rotating dust with negative $\Lambda$ which is a direct product of three dimensional pure rotation metric with a line. We would generalize it to higher dimensions for Einstein and pure Lovelock gravity with only one $N$th order term. For higher dimensional generalization, we have to include more rotations in the metric, and hence we shall begin with the corresponding pure rotation odd $(d=2n+1)$-dimensional metric involving $n$ rotations, which eventually can be extended by a direct product with a line or a space of constant curvature for yielding higher dimensional G\"{o}del universe. The considerations of $n$ rotations and also of constant curvature spaces is a new line of generalization and is being considered for the first time.

It is possible to define an analogue of the Riemann tensor for $N$th order
Lovelock gravity, its characterizing property being that the trace of its
Bianchi derivative yields the corresponding analogue of the Einstein tensor.
Interestingly there exist two parallel but distinct such analogues and the main
purpose of this note is to reconcile both these formulations. In addition we
will show that any pure Lovelock vacuum in odd $d = 2N + 1$ dimensions is
Lovelock flat, i.e. any vacuum solution of the theory has vanishing
Lovelock-Riemann tensor. Further, in presence of cosmological constant it is
the Lovelock-Weyl tensor that vanishes.

We study the impact of dimension on the physical properties of the Finch-Skea astrophysical
model. It is shown that a positive definite, monotonically decreasing pressure and density are evident.
A decrease in stellar radius emerges as the order of the dimension increases. This is accompanied by a
corresponding increase in energy density. The model continues to display the necessary qualitative features
inherent in the 4-dimensional Finch-Skea star and the conformity to the Waleck a theory is preserved under
dimensional increase. The causality condition is always satisfied for all dimensions considered resulting in
the proposed models demonstrating a subluminal sound speed throughout the interior of the distribution.
Moreover, the pressure and density decrease monotonically outwards from the centre and a pressure-free
hypersurface exists demarcating the boundary of the perfect-fluid sphere. Since the study of the physical
conditions is performed graphically, it is necessary to specify certain constants in the model. Reasonable
values for such constants are arrived at on examining the behaviour of the model at the centre and
demanding the satisfaction of all elementary conditions for physical plausibility. Finally two constants of
integration are settled on matching of our solutions with the appropriate Schwarzschild-Tangherlini exterior
metrics. Furthermore, the solution admits a barotropic equation of state despite the higher dimension. The
compactification parameter as well as the density variation parameter are also computed. The models
satisfy the weak, strong and dominant energy conditions in the interior of the stellar configuration.

We obtain a new exact solution to the field equations in the EGB modified
theory of gravity for a 5-dimensional spherically symmetric static
distribution. By using a transformation, the study is reduced to the analysis
of a single second order nonlinear differential equation. In general the
condition of pressure isotropy produces a first order differential equation
which is an Abel equation of the second kind. An exact solution is found. The
solution is examined for physical admissability. In particular a set of
constants is found which ensures that a pressure-free hypersurface exists which
defines the boundary of the distribution. Additionally the isotropic pressure
and the energy density are shown to be positive within the radius of the
sphere. The adiabatic sound speed criterion is also satisfied within the fluid
ensuring a subluminal sound speed. Furthermore, the weak, strong and dominant
conditions hold throughout the distribution. On setting the Gauss-Bonnet
coupling to zero, an exact solution for 5-dimensional perfect fluids in the
standard Einstein theory is obtained. Plots of the dynamical quantities for the
Gauss-Bonnet and the Einstein case reveal that the pressure is unaffected while
the the energy density increases under the influence of the Gauss-Bonnet term.

The Einstein equations with quantum one-loop contributions of conformally covariant matter fields are shown to admit a class of nonsingular isotropic homogeneous solutions that correspond to a picture of the Universe being initially in the most symmetric (de Sitter) state.

The observable universe could be a 1+3-surface (the “brane”) embedded in a 1+3+d-dimensional spacetime (the “bulk”), with Standard Model particles and fields trapped on the brane while gravity is free to access the bulk. At least one of the d extra spatial dimensions could be very large relative to the Planck scale, which lowers the fundamental gravity scale, possibly even down to the electroweak (∼ TeV) level. This revolutionary picture arises in the framework of recent developments in M theory. The 1+10-dimensional M theory encompasses the known 1+9-dimensional superstring theories, and is widely considered to be a promising potential route to quantum gravity. At low energies, gravity is localized at the brane and general relativity is recovered, but at high energies gravity “leaks” into the bulk, behaving in a truly higher-dimensional way. This introduces significant changes to gravitational dynamics and perturbations, with interesting and potentially testable implications for high-energy astrophysics, black holes, and cosmology. Brane-world models offer a phenomenological way to test some of the novel predictions and corrections to general relativity that are implied by M theory. This review analyzes the geometry, dynamics and perturbations of simple brane-world models for cosmology and astrophysics, mainly focusing on warped 5-dimensional brane-worlds based on the Randall–Sundrum models. We also cover the simplest brane-world models in which 4-dimensional gravity on the brane is modified at low energies – the 5-dimensional Dvali–Gabadadze–Porrati models. Then we discuss co-dimension two branes in 6-dimensional models.

In Newtonian theory, gravity inside a constant density static sphere is independent of spacetime dimension. Interestingly this general result is also carried over to Einsteinian as well as higher order Einstein-Gauss-Bonnet (Lovelock) gravity notwithstanding their nonlinearity. We prove that the necessary and sufficient condition for universality of Schwarzschild interior solution describing a uniform density sphere for all $n\geq4$ is that its density is constant. Comment: 5 pages, title and abstract modified, one author added, considerable improvement in clarity and precision, one reference added, result unchanged. version to agree with the published paper in Phys. Rev. D

Spherically symmetric solutions of d-dimensional Einstein-Maxwell theory with a Gauss-Bonnet term are classified. All spherically symmetric solutions of d-dimensional Einstein gravity coupled to the Gauss-Bonnet and Born-Infeld terms are derived, classified, and compared with the previous solutions. Thermodynamic properties of the black holes are discussed and the black-hole temperatures derived. Unlike the solutions of Einstein-Maxwell theory the solutions with a Born-Infeld term do not appear to have a stable end point with regard to thermal evaporation.

For static fluid interiors of compact objects in pure Lovelock gravity (involving ony one $N$th order term in the equation) we establish similarity in solutions for the critical odd and even $d=2N+1, 2N+2$ dimensions. It turns out that in critical odd $d=2N+1$ dimensions, there can exist no bound distribution with a finite radius, while in critical even $d=2N+2$ dimensions, all solutions have similar behavior. For exhibition of similarity we would compare star solutions for $N =1, 2$ in $d=4$ Einstein and $d=6$ in Gauss-Bonnet theory respectively. We also obtain the pure Lovelock analogue of the Finch-Skea model.

A method is developed for treating Einstein's field equations, applied to static spheres of fluid, in such a manner as to provide explicit solutions in terms of known analytic functions. A number of new solutions are thus obtained, and the properties of three of the new solutions are examined in detail. It is hoped that the investigation may be of some help in connection with studies of stellar structure. (See the accompanying article by Professor Oppenheimer and Mr. Volkoff.)

We show universality of isothermal fluid spheres in pure Lovelock gravity
where the equation of motion has only one $N$th order term coming from the
corresponding Lovelock polynomial action of degree $N$. Isothermality is
characterized by the equation of state, $p = \alpha \rho$ and the property,
$\rho \sim 1/r^{2N}$. Then the solution describing isothermal spheres, which
exist only for the pure Lovelock equation, is of the same form for the general
Lovelock degree $N$ in all dimenions $d \geq 2N+2$. We further prove that the
necessary and sufficient condition for the isothermal sphere is that its metric
is conformal to the massless global monopole or the solid angle deficit metric,
and this feature is also universal.

We obtain new exact solutions to the field equations in the Einstein-Gauss-Bonnet (EGB) modified theory of gravity for a five-dimensional spherically symmetric static matter distribution. By using a coordinate transformation, the study is reduced to the analysis of a single first-order nonlinear differential equation which is an Abel equation of the second kind. Three classes of exact models are generated. The first solution has a constant density and a nonlinear equation-of-state; it contains the familiar Einstein static universe as a special case. The second solution has variable energy density and is expressible in terms of elementary functions. The third solution has vanishing Gauss-Bonnet coupling constant and is a five-dimensional generalization of the Durgapal-Bannerji model. The solution is briefly examined for physical admissibility. In particular, a set of constants is found which ensures that a pressure-free hypersurface exists which in turn defines the boundary of the distribution. The matter distribution is well behaved and the adiabatic sound speed criterion is also satisfied within the fluid ensuring a subluminal sound speed. Furthermore, the weak, strong and dominant conditions hold throughout the distribution.

New exact solutions to the field equations in the Einstein-Gauss-Bonnet modified theory of gravity for a five-dimensional spherically symmetric static distribution of a perfect fluid are obtained. The Frobenius method is used to obtain this solution in terms of an infinite series. Exact solutions are generated in terms of polynomials from the infinite series. The five-dimensional Einstein solution is also found by setting the coupling constant to zero. All models admit a barotropic equation of state. Linear equations of state are admitted in particular models with the energy density profile of isothermal distributions. We examine the physicality of the solution by studying graphically the isotropic pressure and the energy density. The model is well behaved in the interior, and the weak, strong, and dominant energy conditions are satisfied.

Preface; List of tables; Notation; 1. Introduction; Part I. General
Methods: 2. Differential geometry without a metric; 3. Some topics in
Riemannian geometry; 4. The Petrov classification; 5. Classification of
the Ricci tensor and the energy-movement tensor; 6. Vector fields; 7.
The Newman-Penrose and related formalisms; 8. Continuous groups of
transformations; isometry and homothety groups; 9. Invariants and the
characterization of geometrics; 10. Generation techniques; Part II.
Solutions with Groups of Motions: 11. Classification of solutions with
isometries or homotheties; 12. Homogeneous space-times; 13.
Hypersurface-homogeneous space-times; 14. Spatially-homogeneous perfect
fluid cosmologies; 15. Groups G3 on non-null orbits V2. Spherical and
plane symmetry; 16. Spherically-symmetric perfect fluid solutions; 17.
Groups G2 and G1 on non-null orbits; 18. Stationary gravitational
fields; 19. Stationary axisymmetric fields: basic concepts and field
equations; 20. Stationary axisymmetiric vacuum solutions; 21. Non-empty
stationary axisymmetric solutions; 22. Groups G2I on spacelike orbits:
cylindrical symmetry; 23. Inhomogeneous perfect fluid solutions with
symmetry; 24. Groups on null orbits. Plane waves; 25. Collision of plane
waves; Part III. Algebraically Special Solutions: 26. The various
classes of algebraically special solutions. Some algebraically general
solutions; 27. The line element for metrics with
κ=σ=0=R11=R14=R44, Θ+iω≠0; 28.
Robinson-Trautman solutions; 29. Twisting vacuum solutions; 30. Twisting
Einstein-Maxwell and pure radiation fields; 31. Non-diverging solutions
(Kundt's class); 32. Kerr-Schild metrics; 33. Algebraically special
perfect fluid solutions; Part IV. Special Methods: 34. Applications of
generation techniques to general relativity; 35. Special vector and
tensor fields; 36. Solutions with special subspaces; 37. Local isometric
embedding of four-dimensional Riemannian manifolds; Part V. Tables: 38.
The interconnections between the main classification schemes;
References; Index.

A new analytical solution has been obtained for stellar models by solving Einstein's field equations for the spherically symmetric and static case. The density variation is found to be smooth and positive under all conditions. The authors show that if they change the variable r to x=Cr2, where C is a constant, the field equations are reduced to a form which is easier to solve. Two specific cases, namely P<or=1/ mod 3 rho and dP/d rho <or=1, are considered. The solution will find application in the case of a neutron star where one can reasonably assume a density rho <or=2*1014 g cm-3. For (dP/d rho )<or=1, the maximum mass of a neutron star model is found to be 4.15 M& and the surface red shift is found to be 0.845.

The Einstein tensor Gij is symmetric, divergence free, and a concomitant of the metric tensor gab together with its first two derivatives. In this paper all tensors of valency two with these properties are displayed explicitly. The number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors.

A new analytical solution has been obtained for stellar models by solving Einstein's field equation for the spherically symmetric and static case. The variation of density is smooth and gradual. The density remains positive under all conditions. For all finite pressures the configurations are stable under radial perturbations. For dP/dρ≤1, the maximum mass of a neutron-star model is 4.56M⊙, and the surface and the central red-shifts are 0.787 and 2.673, respectively. For an infinite central pressure the surface red-shift is 1.575 which is greater than that for any other analytical solution with varying density.

All tensors of contravariant valency two, which are divergence free on one index and which are concomitants of the metric tensor, together with its first two derivatives, are constructed in the four‐dimensional case. The Einstein and metric tensors are the only possibilities.

The objective of the study is to investigate the range of gravitational
redshifts which can be expected in the observation of emission lines
originating near the surface of neutron stars. The approach used here to
derive bounds on the gravitational redshifts from neutron stars is
analogous to the methods developed by Hartle (1978) to derive bounds on
the masses and moments of inertia for neutron stars. It is assumed that
general relativity correctly describes the gravitational interaction at
neutron star densities and that the stars are nonrotating, spherical,
and composed of fluid matter (i.e., matter having isotropic stresses).
For 1.4-solar-mass stars, the limits on the gravitational redshift are
estimated at 0.854-0.184. These limits are compared with redshifts
deduced from gamma-ray burst observations.

It is shown that the analytical stellar model of Duorah and Ray (1987) does not satisfy Einstein's field equations. All possible solutions derivable from the generalised density distribution are exhibited; these include a solution due to Durgapal and Bannerji. The correct solution following from the ansatz of Duorah and Ray is given. The equation of state for the model is obtained in terms of elementary functions, and the solution is shown to be both regular and physically realistic for a range of masses and radii. A comparison between the model and numerical integrations of neutron stars described by Walecka's relativistic mean-field theory description of neutron matter shows good overall agreement.

We suggest a mechanism by which four-dimensional Newtonian gravity emerges on a 3-brane in 5D Minkowski space with an infinite size extra dimension. The worldvolume theory gives rise to the correct 4D potential at short distances whereas at large distances the potential is that of a 5D theory. We discuss some phenomenological issues in this framework.

Topological theories of gravity are constructed in odd-dimensional space-times of dimensions 2n + 1, using the Chern-Simons (2n + 1)-forms and with the gauge groups ISO(1, 2n) or SO(1, 2n + 1) or SO(2, 2n). In even dimensions the presence of a scalar field in the fundamental representation of the gauge group is needed, besides the gauge field. Supersymmetrization of the de Sitter groups can be performed up to a maximal dimension of seven, but there is no limit on the super-Poincaré groups. The different phases of the topological theory are investigated. It is argued that these theories are finite. It is shown that the graviton propagates in a perturbative sense around a non-trivial background.

On the occasion of the 25th anniversary of Asymptotic Freedom I describe the discovery of Asymptotic Freedom and the emergence of QCD.

A topological gauge theory of gravity in five dimensions is presented. This is based on the Chern-Simons five-form and the SO(1, 5) gauge group. The action contains a Gauss-Bonnet term, an Einstein term and a cosmological constant. Quantization and renormalizability of the theory are discussed. Indications of how to generalize to arbitrary odd dimensions are given.

We explore static spherically symmetric stars in the Gauss-Bonnet gravity
without cosmological constant, and present an exact internal solution which
attaches to the exterior vacuum solution outside stars. It turns out that the
presence of the Gauss-Bonnet term with a positive coupling constant completely
changes thermal and gravitational energies, and the upper bound of red shift of
spectral lines from the surface of stars. Unlike in general relativity, the
upper bound of red shift is dependent on the density of stars in our case.
Moreover, we have proven that two theorems for judging the stability of
equilibrium of stars in general relativity can be hold in Gauss-Bonnet gravity.

Expansion of supersymmetric string theory suggests that the leading quadratic curvature correction to the Einstein action is the Gauss-Bonnet invariant. It is shown that this model has both flat and anti-de Sitter space as solutions, but that the cosmological branch is unstable, because the graviton becomes a ghost there: the theory solves its own cosmological problem. The general static spherically symmetric solution is exhibited; it is asymptotically Schwarzschild. The sign of the Gauss-Bonnet coefficient determines whether there is a normal event horizon (for the string-generated sign) or a naked singularity. The effects of higher-curvature corrections, and an explicit cosmological term on stability, are discussed.

Conventional wisdom states that Newton's force law implies only four non-compact dimensions. We demonstrate that this is not necessarily true in the presence of a non-factorizable background geometry. The specific example we study is a single 3-brane embedded in five dimensions. We show that even without a gap in the Kaluza-Klein spectrum, four-dimensional Newtonian and general relativistic gravity is reproduced to more than adequate precision.

Models with a scalar field coupled to the Gauss-Bonnet Lagrangian appear
naturally from Kaluza-Klein compactifications of pure higher-dimensional
gravity. We study linear, cosmological perturbations in the limits of weak
coupling and slow-roll, and derive simple expressions for the main observable
sub-horizon quantities: the anisotropic stress factor, the time-dependent
gravitational constant, and the matter perturbation growth factor. Using
present observational data, and assuming slow-roll for the dark energy field,
we find that the fraction of energy density associated with the coupled
Gauss-Bonnet term cannot exceed 15%. The bound should be treated with caution,
as there are significant uncertainies in the data used to obtain it. Even so,
it indicates that the future prospects for constraining the coupled
Gauss-Bonnet term with cosmological observations are encouraging.

In spacetimes of dimension greater than four it is natural to consider higher order (in R) corrections to the Einstein equations. In this letter generalized Israel junction conditions for a membrane in such a theory are derived. This is achieved by generalising the Gibbons-Hawking boundary term. The junction conditions are applied to simple brane world models, and are compared to the many contradictory results in the literature. Comment: 4 pages

We ask the following question: Of the exact solutions to Einstein's equations extant in the literature, how many could represent the field associated with an isolated static spherically symmetric perfect fluid source? The candidate solutions were subjected to the following elementary tests: i) isotropy of the pressure, ii) regularity at the origin, iii) positive definiteness of the energy density and pressure at the origin, iv) vanishing of the pressure at some finite radius, v) monotonic decrease of the energy density and pressure with increasing radius, and vi) subluminal sound speed. A total of 127 candidate solutions were found. Only 16 of these passed all the tests. Of these 16, only 9 have a sound speed which monotonically decreases with radius. The analysis was facilitated by use of the computer algebra system GRTensorII.