A complete set of integrals and solutions to the Szekeres system

The Szekeres system is an ODE system arising from the Szekeres cosmological model. The paper presents the way to solve it using its four global independent first integrals. It also contains the complete set of its solutions in the particular cases, related to the set where the integrals are indefinite.     

Szekeres spacetimes with heat flow

A new class of inhomogeneous exact solutions of Einstein field equations is given. They generalize the dust-filled models found by Szekeres. In the course of time a subclass of models present a friedmannian phase.     

Volume averaging in the quasispherical Szekeres model

This paper considers volume averaging in the quasispherical Szekeres model. The volume averaging became of considerable interest after it was shown that the volume acceleration calculated within the averaging framework can be positive even when the local expansion rate decelerates. This issue was intensively studied within spherically symmetric models. However, since our Universe is not spherically symmetric similar analysis is needed in non-symmetrical models. This papers presents the averaging analysis within the quasispherical Szekeres model which is a non-symmetrical generalisation of the spherically symmetric Lemaître–Tolman family of models. In the quasispherical Szekeres model the distribution of mass over a surface of constant t and r has the form of a mass-dipole superposed on a monopole. This paper shows that when calculating the volume acceleration, ä, within the Szekeres model, the dipole does not contribute to the final result, hence ä only depends on a monopole configuration. Thus, the volume averaging within the Szekeres model leads to literally the same solutions as those obtained within the Lemaître–Tolman model.     

On the dynamics of the Szekeres system

The gravitational Szekeres differential system is completely integrable with two rational first integrals and an additional analytical first integral. We describe the dynamics of the Szekeres system when one of these two rational first integrals is negative, showing that all the orbits come from the infinity of ${\mathbb{R}}^4$ and go to infinity.     

A relation between the Szekeres quasispherical gravitational collapse solution and the Robinson-Trautman metrics

A form of the Robinson-Trautman metric for pure radiation is obtained by applying a limiting procedure to the quasispherical gravitational collapse solution of Szekeres. The procedure can be interpreted as that of allowing the particles making up the dust in the Szekeres solution to become massless.Research supported in part by a scholarship awarded to the author by the Consejo Nacional de Investigaciones Cintíficas y Técnicas of Argentina.     

Complex null geodesics in the extended Schwarzschild universe

The generic null geodesic of the Schwarzschild–Kruskal–Szekeres geometry has a natural complexification, an elliptic curve with a cusp at the singularity. To realize that complexification as a Riemann surface without a cusp, and also to ensure conservation of energy at the singularity, requires a branched cover of the space-time over the singularity, with the geodesic being doubled as well to obtain a genus two hyperelliptic curve with an extra involution. Furthermore, the resulting space-time obtained from this branch cover has a Hamiltonian that is null geodesically complete. The full complex null geodesic can be realized in a natural complexification of the Kruskal–Szekeres metric.     

Szekeres's space-times have no Killing vectors

Szekeres obtained some exact solutions of Einstein's equations for dust which have applications in cosmology, in the theory of collapse, and in the study of gravitational radiation. It is shown that in their most general form these space-times have no Killing vectors.Dedicated to Achille Papapetrou on the occasion of his retirement.On leave from the Department of Mathematics, University of Malaya, Kuala Lumpur, Malaysia.Supported by the Public Services Department, Government of Malaysia.On leave from Centro Brasileiro de Pesquisas Fisicas.     

场方法的改进及其在积分Riemann-Cartan空间运动方程中的应用

和Hamilton-Jacobi方法类似,Vujanovi?场方法把求解常微分方程组特解的问题转化为寻找一个一阶拟线性偏微分方程(基本偏微分方程)完全解的问题,但Vujanovi?场方法依赖于求出基本偏微分方程的完全解,而这通常是困难的,这就极大地限制了场方法的应用.本文将求解常微分方程组特解的Vujanovi?场方法改进为寻找动力学系统运动方程第一积分的场方法,并将这种方法应用于一阶线性非完整约束系统Riemann-Cartan位形空间运动方程的积分问题中.改进后的场方法指出,只要找到基本偏微分方程的包含m(m≤ n,n为基本偏微分方程中自变量的数目)个任意常数的解,就可以由此找到系统m个第一积分.特殊情况下,如果能够求出基本偏微分方程的完全解(完全解是m=n时的特例),那么就可以由此找到≤系统全部第一积分,从而完全确定系统的运动.Vujanovi?场方法等价于这种特殊情况.     

The Szekeres Swiss Cheese model and the CMB observations

This paper presents the application of the Szekeres Swiss Cheese model to the analysis of observations of the cosmic microwave background (CMB) radiation. The impact of inhomogeneous matter distribution on the CMB observations is in most cases studied within the linear perturbations of the Friedmann model. However, since the density contrast and the Weyl curvature within the cosmic structures are large, this issue is worth studying using another approach. The Szekeres model is an inhomogeneous, non-symmetrical and exact solution of the Einstein equations. In this model, light propagation and matter evolution can be exactly calculated, without such approximations as small amplitude of the density contrast. This allows to examine in more realistic manner the contribution of the light propagation effect to the measured CMB temperature fluctuations. The results of such analysis show that small-scale, non-linear inhomogeneities induce, via Rees-Sciama effect, temperature fluctuations of amplitude 10⁻⁷–10⁻⁵ on angular scale ϑ < 0.24° (ℓ > 750). This is still much smaller than the measured temperature fluctuations on this angular scale. However, local and uncompensated inhomogeneities can induce temperature fluctuations of amplitude as large as 10⁻³, and thus can be responsible the low multipoles anomalies observed in the angular CMB power spectrum.     

On the Global Visibility of the Singularity in Quasi-Spherical Collapse

We analyze here the issue of local versus global visibility of a singularity that forms in gravitational collapse of a dust cloud, which has important implications for the weak and strong versions of the cosmic censorship hypothesis. We find conditions for when a singularity will be only locally naked, rather than being globally visible, thus preserving the weak censorship hypothesis. The conditions for the formation of a black hole or a naked singularity in the Szekeres quasi-spherical collapse models are worked out. The causal behaviour of the singularity curve is studied by examining the outgoing radial null geodesics, and the final outcome of collapse is related to the nature of the regular initial data specified on an initial hypersurface from which the collapse evolves. An interesting feature that emerges is that the singularity in Szekeres spacetimes can be directionally naked.     

Lie Algebras and Integrable Systems

A 3? 3 matrix Lie algebra is first introduced, its subalgebras and the generated Lie algebras are obtained, respectively. Applications of a few Lie subalgebras give rise to two integrable nonlinear hierarchies of evolution equations from their reductions we obtain the nonlinear Schrödinger equations, the mKdV equations, the Broer-Kaup (BK) equation and its generalized equation, etc. The linear and nonlinear integrable couplings of one integrable hierarchy presented in the paper are worked out by casting a 3? 3 Lie subalgebra into a 2? 2 matrix Lie algebra. Finally, we discuss the elliptic variable solutions of a generalized BK equation.     

Non-radiative solutions of Einstein's equations for dust

The quasi-spherical collapsing space-time of Szekeres is investigated. The arbitrary functions can be chosen so that it has positive density, and no Killing vectors; yet a ballr<r ₀ of it can joined to the Schwarzschild metric, and is therefore non-radiative.     

Covariant energy–momentum and an uncertainty principle for general relativity

We introduce a naturally-defined totally invariant spacetime energy expression for general relativity incorporating the contribution from gravity. The extension links seamlessly to the action integral for the gravitational field. The demand that the general expression for arbitrary systems reduces to the Tolman integral in the case of stationary bounded distributions, leads to the matter-localized Ricci integral for energy–momentum in support of the energy localization hypothesis. The role of the observer is addressed and as an extension of the special relativistic case, the field of observers comoving with the matter is seen to compute the intrinsic global energy of a system. The new localized energy supports the Bonnor claim that the Szekeres collapsing dust solutions are energy-conserving. It is suggested that in the extreme of strong gravity, the Heisenberg Uncertainty Principle be generalized in terms of spacetime energy–momentum.     

非完整转动相对论系统的Lindel?f方程

由转动相对论系统的Hamilton原理分别建立在广义坐标和准坐标下的Lindel?f方程及其改进形式,并从改进的Lindel?f方程导出新Chaplygin方程.最后说明由转动系统的相对论分析力学向普通分析力学过渡的方法. 关键词： 非完整约束 转动系统 相对论 Lindel?f方程 Chaplygin方程     

Extracting dynamical equations from experimental data is NP hard

The behavior of any physical system is governed by its underlying dynamical equations. Much of physics is concerned with discovering these dynamical equations and understanding their consequences. In this Letter, we show that, remarkably, identifying the underlying dynamical equation from any amount of experimental data, however precise, is a provably computationally hard problem (it is NP hard), both for classical and quantum mechanical systems. As a by-product of this work, we give complexity-theoretic answers to both the quantum and classical embedding problems, two long-standing open problems in mathematics (the classical problem, in particular, dating back over 70?years).     

Collision of plane electromagnetic-gravitational waves

An electromagnetic generalization of the Szekeres solution of the problem of interacting plane gravitational waves is used to obtain a general-relativistic solution of the collision of plane electromagnetic-gravitational waves. An analysis is made of this solution.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 91–96, 1979.The author wishes to thank Yu. G. Sbytov and Prof. D. D. Ivanenko for their interest in this work.