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We explore the recently introduced statefinder parameters. After reviewing their basic properties, we calculate the statefinder parameters for a variety of cosmological models, and investigate their usefulness as a means of theoretical classification of dark energy models. We then go on to consider their use in obtaining constraints on dark energy from present and future supernovae type Ia data sets. We find that it is non-trivial to extract the statefinders from the data in a model-independent way, and one of our results indicates that parametrizing the dark energy density as a polynomial of second order in the redshift is inadequate. Hence, while a useful theoretical and visual tool, applying the statefinders to observations is not straightforward.

Up to about four years ago the dynamical properties of the evolution of the universe were assumed to be well described by the Einstein-de Sitter universe model, which is a flat universe model dominated by cold matter. However, the discovery that the cosmic expansion is accelerating made it clear that, at the present time, the evolution of the universe is dominated by some sort of vacuum energy with repulsive gravitation. The most simple type of vacuum energy is the Lorentz invariant vacuum energy which has constant energy density during the expansion of the universe. This type of vacuum may be represented mathematically by including a cosmological constant in Einstein's field equations. Hence, the flat Friedmann-Lemaître model, which is a universe model with cold matter and vacuum energy, has replaced the Einstein-de Sitter model as the standard model of the universe. In this paper we give a pedagogical presentation of this model, including, among others, a new formula for the point of time when decelerated expansion changed into accelerated expansion.

We give a detailed description of the energy–momentum relationships of the Bondi–Gold field. The Schott field is separated into a near field and a radiation field. The physical significance of a plane acting as a barrier for transport of energy in the field of a charge with permanent hyperbolic motion, but which lets energy through in the case of hyperbolic motion for a finite time interval, is discussed. We deduce general expressions for energy and momentum of the near field and radiation field of charges with arbitrary motion. The field reaction on the particle producing the field is calculated. The formulae are applied to the Schott field and the Bondi–Gold field.

The question whether a hyperbolically moving charge emits radiation is discussed. In order to arrive at an unambiguous answer the question is considered from several points of view. The power spectrum of the electromagnetic field due to a hyperbolically moving charge is deduced. Also we analyse the field in order to clarify whether it contains a radiation zone. Our conclusion is that a charge with constant proper acceleration emits radiation in spite of the fact that the Poynting vector of the field vanishes everywhere in the instantaneous inertial rest frame of the charge.

The electromagnetic field of a charged particle with hyperbolic motion is described. The field of a particle with eternal hyperbolic motion is found by taking the limit of a situation with limited hyperbolic motion. The flow of energy in the electromagnetic field is discussed. The question whether the charged particle emits radiation is briefly touched upon in this first article in a series on the electrodynamics of radiating particles and their fields.

Two definitions of gravitational energy, Einstein's pseudo-tensor expression of energy in curved space-time and Lynden-Bell and Katz's (LBK) definition, are shown to give equivalent results, as applied to spherically-symetric static systems in the presence of arbitary sources. The conjectured expression for energy density of field plus matter for general static space times, given by Katz, Lynden-Bell, and Israel (KLBI), is applied to a system with internal and external Schwarzschild metric, and also to a vacuum dominated space-time with the De Sitter metric. The physical contents of the KLBI-expression is discussed by analyzing the Newtonian limit in a space filled with matter.

The research on viscous cosmological models is reviewed and carried further. Inflationary cosmological models of Bianchi type-I with shear, bulk, and nonlinear viscosity are studied. The inflation field energy is represented by a two-components cosmic fluid consisting of a vacuum fluid and a Zel'dovich fluid. It is shown that there exist models in which the viscous Zel'dovich fluid removes the initial singularity of the corresponding viscosity free models. The mean expansion of a Zel'dovich dominated unvierse is found to be independent of shear viscosity and anisotropy. Bulk viscosity and shear viscosity cause exponential decay of anisotropy, while nonlinear viscosity causes power-law decay of anisotropy.

In a recent discussion E. A. Desloge has performed an analysis of the kinematic nonequivalence of a uniformly accelerating reference frame (UAR) and a frame at rest in a uniform gravitational field (UGF). Desloge considered two-dimensional space-times. We here review the geometrical description of gravity in two, three, and four space-time dimensions in order to discuss thedynamical nonequivalence of UAR and UGF both in two and in four space-time dimensions. We also consider the motion of photons in UAR and UGF in order to illustrate some relativistic effects of a kinematic nature.

Abstract: Using the Weyl-type canonical coordinates, an integration of Einstein's field equations in the cylindrosymmetric case considered by Kurscedilunogbrevelu is reexamined. It is made clear that the resulting metric is not describing the spacetime in a rotating frame, but in astatic cylindrical elastic medium. The conclusion of Kurscedilunogbrevelu that ldquofor an observer on a rotating disk there is no way of escape from a curved spacetimerdquo is therefore not valid. The metric in an empty rotating frame is found as a solution of Einstein's field equations, and is not orthogonal. It is shown that the corresponding orthogonal solution represents spacetime in an inertial frame expressed in cylindrical coordinates. Introducing a noncoordinate basis, the metric in a rotating frame is given the static form of Kurscedilunogbrevelu's solution. The essential role played by the nonvanishing structure coefficients in this case is made clear.

Abstract: It is shown how a consistent kinematic resolution of Ehrenfest's paradox may be given in accordance with the special theory of relativity. Some statements by T. E. Phipps, Jr., connected with these matters, are commented upon. Problems connected with the relation between stress and strain are solved by a manifestly covariant formulation of Hooke's law.

Abstract: It is shown that the equation deduced by Marinov for the gravitational frequency shift does not follow from his assumptions. The correct equation is deduced. It is pointed out that the result of Marinov's absolute spacetime theory concerning the gravitational frequency shift is contained in general relativity as an approximate description. The need for experiments testing the validity of Marinov's measurements is emphasized.

Abstract: In a recent article in this journal, Kingsley has tried to show that the postulates of special relativity contradict each other. Here we show that the arguments of Kingsley are invalid because of an erroneous appeal to symmetry in a non-symmetric situation. The consistency of the postulates of special relativity and the relativistic kinematics deduced from them is restated.

Abstract: Einstein's gravitational field equations in empty space outside a massive plane with infinite extension give a class of solutions describing a field with flat spacetime giving neutral, freely moving particles an acceleration. This points to the necessity of defining the concept ldquogravitational fieldrdquo not simply by the nonvanishing of the Riemann curvature tensor, but by the nonvanishing of certain elements of the Christoffel symbols, called the physical elements, or the nonvanishing of the Riemann curvature tensor. The tidal component of a gravitational field is associated with a nonvanishing Riemann tensor, while the nontidal components are associated with nonvanishing physical elements of the Christoffel symbols. Spacetime in a nontidal gravitational field is flat. Such a field may be separated into a homogeneous and a rotational component. In order to exhibit the physical significance of these components in relation to their transformation properties, coordinate transformations inside a given reference frame are discussed. The mentioned solutions of Einstein's field equations lead to a metric identical to that obtained as a result of a transformation from an inertial frame to a uniformly accelerated frame. The validity of the strong principle of equivalence in extended regions for nontidal gravitational fields is made clear. An exact calculation of the weight of an extended body in a uniform gravitational field, from a global point of view, gives the result that its weight is independent of the position of the scale on the body.

Abstract: A rotating disk with angular acceleration is given a relativistic description as observed from the rotating rest frameR of the disk. It is shown how a non-Euclidean intrinsic spatial geometry develops inR, as the disk gets an angular velocity. The explanation of this as given by anR-observer is discussed. A recent description of the geometry inR presented by Grünbaum and Janis is criticized. The motion of light as described by use of coordinate clocks inR is discussed in connection with some recent work by R. C. Jennison and coworkers. The significance of the difference between directed signals and circular waves when clocks inR are to be synchronized by means of light emitted from the axis inR is made clear.

It is shown that the no interaction theorem, which has been an obstacle in formulating relativistic particle dynamics, does not exist in the asynchronous formulation. A manifestly covariant asynchronous description of two interacting particles is given. Differences between the synchronous and the asynchronous descriptions are exhibited by considering an illustrating example. Transformation formulae for the components of torque, as defined asynchronously, are deduced. It is pointed out that there is no polarization of inertia in the asynchronous formulation. Si mostra che il teorema della non interazione, che è stato un ostacolo alla formulazione della dinamica relativistica delle particelle, non esiste nella formulazione asincrona. Si dà una descrizione asincrona manifestamente covariante di due particelle interagenti. Si mostrano le differenze fra le descrizioni sincrone ed asincrone considerando un esempio illustrativo. Si deducono le formule di trasformazione per le componenti della coppia, definiti in modo asincrono. Si fa rilevare che non c’è nessuna polarizzazione d’inerzia nella formulazione asincrona. Показывается, что теорема об отсутствии взаимодействия, которая является помехой при формулировке динамики релятивистских частиц, не существует в асинхронной формулировке. Предлагается явно ковариантное асинхронное описание двух взаимодействуюших частиц. На примере показываются различия между синхронным и асинхронным описаниями. Выводятся формулы преобразований для компонент момента в соответствии с асинхронным описанием. Отмечается, что не существует поляризации инерции в асинхронной формулировке.

The synchronous and the asynchronous descriptions of a rod with rigid hyperbolic motion are compared. It is shown that the covariant description, emploging the energy-momentum four-vector, has an asynchronous character. The conclusion is that, in describing extended systems moving not too far from rigidly, the asynchronous formulation is more adequate than the synchronous formulation. Si confrontano le descrizioni sincrona ed asincrona di una barra in moto rigido iperbolico. Si dimostra che la descrizione covariante, che fa uso del quadrivettore energia-impulso, ha un carattere asincrono. La conclusione è che, nel descrivere sistemi estesi in cui il moto non è molto diverso da quello rigido, la formulazione asincrona è più adeguata di quella sincrona. Проводится сравнение синхронного и асинхронного описаний стержня в случае недеформируемого, гиперболического движения. Показывается, что ковариантное описание имеет асинхронный характер. Утверждается, что при описании протяженных систем, движущихся почти без деформаций, асинхронная формулировка является более адекватной, чем синхронная формулировка.

The vacuum field equations of the classical five-dimensional Kaluza-Klein theory are solved for a static, spherically symmetric system. It is shown that the projection of the solution into four-dimensional space-time represents the Reissner-Nordstrøm solution and that the five-dimensional geodesic equation of a properly chosen free particle represents the equation of motion of an electron in the field of a proton, as given by the Einstein-Maxwell equations. The electrical field of a charged particle is interpreted as the rotational inertial-dragging field outside a rotating particle in five-dimensional space-time.

The steady flow of a supercold gas, an ultrahot gas and a photon gas into a Schwarzschild black hole is investigated. The difference between an ultrahot, gas and a photon gas is made clear and their behaviour during steady flow into a black hole is compared.

The relativistic transformation of momentum and energy for open systems considered as continua of matter is studied. Generally the momentum and the energy of such systems do not transform as components of a four-vector, in the synchronous formulation. The calculations also show that, due to the relativity of simultaneity, the inertia of an open system with internal motion is direction dependent.

The charge distribution and electromagnetic field in a rotating conductor with a net electric charge under stationary conditions are described by Schiffs equations of electrodynamics in a rotating reference frame. The existence of a spatial charge distribution in a conductor at rest in a rotating reference frame is demonstrated.

The positions of the points on a rolling ring are calculated at retarded points of time, giving the “optical appearance” of a rolling ring.

The Einstein-Maxwell equations are solved with an energy-momentum tensor corresponding to the field of a standing electromagnetic wave outside a massive plane, in the limit where the test particle is much larger than the wavelength of the electromagnetic field. Physical properties of the solution are discussed.

The covariant definition of gravitational field energy given by Lynden-Bell and Katz is expressed in terms of Israel's theory of surface layers in general relativity. In this way an expression, valid for arbitrary radial coordinates, of the gravitational field energy in a static, spherically symmetric space-time, is deduced. This expression is applied to the Schwarschild and Reissner-Nordström space-times, and leads here to the same results as those given by Einstein's pseudotensor expression in isotropic coordinates.

The new class of interior Schwarzschild solutions found by Florides is generalized to the charged case. A particular solution within this class is found, which represents an electromagnetic mass-model of a neutral spherically symmetric system. The pressure is isotropic, decreasing monotonously with increasing radius and vanishes at the surface of the matter distribution. The solution is regular everywhere inside a radiusR, and is joined continuously to the exterior Schwarzschild solution at this radius.

The Reissner-Nordström repulsion is shown to be a consequence of Poincaré stresses in a static, charged object. The strong energy condition implies that the Reissner-Nordström repulsions do not stop a neutral test particle, falling freely from rest at infinity in the fields of a charged body, before it hits the surface of the body. However, if the particle falls from rest at a sufficiently small height above the surface of the body, it will not reach the surface due to the Reissner-Nordström repulsion.

The electrical field inside a uniformly charged, slowly accelerated spherical shell is calculated. The result is used to find the inertial translational dragging field inside a slowly accelerated spherical shell of dust particles, according to the linearized gravitational field equations. The relevance of this effect in connection with Mach's principle and the principle of relativity is discussed.

Solutions of the Einstein-Maxwell equations for static charged dust are discussed. Contrary to what has been asserted earlier it is found that cylindrically symmetric and plane symmetric solutions with the following properties, exist: (a) They are regular in the interior region; (b) the mass density is positive and vanishes at the boundary; (c) the metric, its first derivate, and the electrical field strength are continuous at the boundary; (d) the solutions are mirror symmetric in the plane symmetric case.

An exposition of the covariant asynchronous formulationAof relativistic statics and thermodynamics is given, and this description is compared with the traditional synchronous oneS. The spin equation is given a covariant formulation. Also the equation of motion for extended bodies is given inA. Further the interpretation inSand inAof the elements of the energy-momentum tensor is stated, and the result is employed on an ideal gas. The relativistic concept of heat is discussed. The temperature is introduced as a scalar. Some examples from the statics, the right-angled lever, Trouton and Noble’s experiment and Sears’ relativistic paradox, are discussed. A few topics from the relativistic thermodynamics, the transformation of energy and momentum of an ideal gas and the transformation of thermodynamical work between inertial systems, are treated in some detail. Also the exchange of heat between systems with a relative motion is described and discussed. Si riporta un’esposizione della formulazione asincrona covarianteAdella statica e della termodinamica relativistiche, e si confronta questa descrizione con quella sincrona tradizionaleS.Si dà all’equazione dello spin una formulazione covariante. Si dà anche inAl’equazione del moto per i corpi estesi. Inoltre si stabilisee inSed inAl’interpretazione degli elementi del tensore energia-impulso, e si impiega il ristdtato per un gas ideale. Si discute il concetto relativistico di calore. Si introduce la temperatura corne uno scalare. Si discutono alcuni esempi di statica, la leva ad angolo retto, l’esperimento di Trouton e Noble ed il paradosso relativistico di Sears. Si trattano alquanto dettagliatamente alcuni argomenti di termodinamica relativistica, la trasformazione dell’energia e dell’impulso di un gas ideale e la trasformazione di lavoro termodinamico tra sistemi inerziali. Si descrive anche e si discute lo scambio di calore fra sistemi in moto relativo. Приводится описание ковариантной асинхронной формули ровки. A, релятивистско й статики и термодина мики. Это описание сра внивается с о термодинамики. Это оп исание сравнивается с общепринятой синхр онной формулировкой, S. Приводится ковариан тная формулировка сп инового уравне общепринятой синхро нной формулировкой, S. П риводится ковариант ная формулировка спи нового уравнения. Так же в А записывается ур авнение движения для протяженных тел. Кром е того, предлагается и нтер ковариантная формул ировка спинового ура внения. Также в А запис ывается уравнение дв ижения для протяженн ых тел. Кроме того, пред лагается интерпрета ция в S и в А элементов т ензора энергии-импул ьса. Полученный резул ьтат применяется к ид еальному газу. Об в А записывается урав нение движения для пр отяженных тел. Кроме т ого, предлагается инт ерпретация в S и в А эле ментов тензора энерг ии-импульса. Полученн ый результат применя ется к идеальному газ у. Обсуждается реляти вистская концепция т еплоты. Температура в водится как скаляр. Об суждаются некоторые приме тел. Кроме того, предла гается интерпретаци я в S и в А элементов тен зора энергии-импульс а. Полученный результ ат применяется к идеа льному газу. Обсуждае тся релятивистская к онцепция теплоты. Тем пература вводится ка к скаляр. Обсуждаются некоторые примеры из статики: прямоугольн ый рычаг, эксперимент Траутона и Нобла, реля тивистский парадокс Сирса. Более п элементов тензора эн ергии-импульса. Получ енный результат прим еняется к идеальному газу. Обсуждается рел ятивистская концепц ия теплоты. Температу ра вводится как скаля р. Обсуждаются некото рые примеры из статик и: прямоугольный рыча г, эксперимент Трауто на и Нобла, релятивист ский парадокс Сирса. Б олее подробно рассма триваются некоторые вопросы релятивистс кой термодинамики: пр еобразование энерги и результат применяет ся к идеальному газу. О бсуждается релятиви стская концепция теп лоты. Температура вво дится как скаляр. Обсу ждаются некоторые пр имеры из статики: прям оугольный рычаг, эксп еримент Траутона и Но бла, релятивистский п арадокс Сирса. Более п одробно рассматрива ются некоторые вопро сы релятивистской те рмодинамики: преобра зование энергии и имп ульса идеального газ а и преобразование те рмодинамической раб оты между инерциальн ыми системами релятивистская конц епция теплоты. Темпер атура вводится как ск аляр. Обсуждаются нек оторые примеры из ста тики: прямоугольный р ычаг, эксперимент Тра утона и Нобла, релятив истский парадокс Сир са. Более подробно рас сматриваются некото рые вопросы релятиви стской термодинамик и: преобразование эне ргии и импульса идеал ьного газа и преобраз ование термодинамич еской работы между ин ерциальными система ми. Также исследуется обмен теплом между си стемами в случае отно сительного движения. вводится как скаляр. О бсуждаются некоторы е примеры из статики: п рямоугольный рычаг, э ксперимент Траутона и Нобла, релятивистск ий парадокс Сирса. Бол ее подробно рассматр иваются некоторые во просы релятивистско й термодинамики: прео бразование энергии и импульса идеального газа и преобразовани е термодинамической работы между инерциа льными системами. Так же исследуется обмен теплом между система ми в случае относител ьного движения. статики: прямоугольн ый рычаг, эксперимент Траутона и Нобла, реля тивистский парадокс Сирса. Более подробно рассматриваются нек оторые вопросы релят ивистской термодина мики: преобразование энергии и импульса ид еального газа и преоб разование термодина мической работы межд у инерциальными сист емами. Также исследуе тся обмен теплом межд у системами в случае о тносительного движе ния. Нобла, релятивистски й парадокс Сирса. Боле е подробно рассматри ваются некоторые воп росы релятивистской термодинамики: преоб разование энергии и и мпульса идеального г аза и преобразование термодинамической р аботы между инерциал ьными системами. Такж е исследуется обмен т еплом между системам и в случае относитель ного движения. рассматриваются нек оторые вопросы релят ивистской термодина мики: преобразование энергии и импульса ид еального газа и преоб разование термодина мической работы межд у инерциальными сист емами. Также исследуе тся обмен теплом межд у системами в случае о тносительного движе ния. термодинамики: преоб разование энергии и и мпульса идеального г аза и преобразование термодинамической р аботы между инерциал ьными системами. Такж е исследуется обмен т еплом между системам и в случае относитель ного движения. идеального газа и пре образование термоди намической работы ме жду инерциальными си стемами. Также исслед уется обмен теплом ме жду системами в случа е относительного дви жения. работы между инерциа льными системами. Так же исследуется обмен теплом между система ми в случае относител ьного движения. исследуется обмен те плом между системами в случае относительн ого движения. относительного движ ения.

The gravitational field equations in Dunn's scalar-tensor theory of gravitation are generalized by including a cosmological constant. The resulting equations are solved for a Robertson-Walker line-element with flat three-space. The solution represents a cosmological model that develops into an inflationary era.

Israel’s formulation of the theory of singular layers in general relativity is applied to González-Diaz’ solution of Einstein’s field equations, which was interpreted as describing space-time inside a black hole. It is shown that the solution demands the existence of a surface layer with infinite tangential stress. Thus the solution of González-Diaz is not dynamically stable. It is suggested that space-time inside a black hole must be described by avacuumsolution of Einstein’s field equations, in accordance with the standard picture.