GRAVITATIONAL AND GEOMETRICAL SIGNIFICANCE OF SOME PLANE AND SPHERICALLY SYMMETRIC METRICS IN GENERAL RELATIVITY

Abstract

In Chapter I, which is introductory in nature, the basic mathematical and physical facts regarding Einstein's theory of gravitation are given* The conventional and notations used in the thesis are stated and explained.The chapter closes with a summary of main results obtained by the author in the present thesis* The main theme of chapter II is to study the properties of Taub's plane symmetric space-time from the point of view of its embedding in higher dimensional flat spaces* The isometric embedding of a Riemannian fourfold into higher dimensional pseudo-Euclidean space is of interest from various points of views* The idea has been very attractively shown to be in some relation to internal degrees of freedom of elementary particleejjie *eraan,196S • The concept of embedding of solutions of Einstein field equations have also found applications) in the discussion of singularities(Geroch,1968JHajicek,1970). The main results of this chapter can be stated as follows; (i) It is show?: that the plane-symmetric space-time can be imbedded in a flat space of seven dimensions. (ii) A sufficient condition for the plane symmetric space time to be of class two is obtained, (lii) An analogue of Karmarkar,e(1948) condition is obtained which is a sufficient condition for the plane symmetric space-time to be of class one* (iv) Explicit imbeddings of plane symmetric spaoe-tlme ie obtained in three special oases* The oase where Karmarkar*s condition breakdown is also discussed. In so far as the author is aware an exhaustive study of plane symm etric metric has not been done earlier from the point of view of The reference sighted in this preface may be seen at the end of chapters concerned. w imbedding* In chapter III we have considered the plane symmetric metric of chapter II from the point of view of possible perfect fluid dietrlbutio: •« It is shown that the sufficient condition for class one imbedding leads to two distinct types of perfect fluid distributions in general. One type is oonformal to a flat space-time and the other is not. These two types of perfect fluid distributions of class one have been obtained where the metric potentials are functions of time only and the geometri cal feature s of these models from the point of viev of their embedding in a five-dimensional flat space Is given.The cases of pressure free dust, disordered radiation (P=3p) and zeldovich fluid (P*p) are given in each case. Atime-dependent perfect fluid solution of class two has been obtained. Acosmo logies! model filled with pressure-free dust, which is of class two and not oonformal to a flat space-time, has been given.The embedding equations of two distinct types are given for class two models. The pressure free class two cosmologies! model reduces to an empty gravitational field when density vanishes. With reference to the metric (2.2.1) of chapter II, it is shown that the density and pressure are negative in case S«S(x) and the space-time Is of class one. As a consequence of this the static perfect fluid modrl of class one with plane symmetry turn out to be of negative density and pressure and hence physically untanable. The chapter is closed with the discussion of a plane symmetric null electromagnetic field of class two. The u\ canonical forms of the second fundamental tensor aj* and h^. are given in this case* In chapter IV we have discussed the plane symmetric space-time satisfying the Einstein field equation R^^O* This type of field has been discussed by Taub (1951) from the point of view of group of motion. He has also proved that a gravitational field of the plane symmetric type can always be reduced to the static form by a suitable coordinate transformation* In chapter II, section (2.5), we have shown that the most general plane-symmetric line-element (2.2.1) can be reduced to three distinct forms. We have proved in this chapter that the empty field equation Rj, « 0 lead to nontrivial solutions in only two oases. The metrics in these two cases are given explicitly. The non-static field is of cosmologlcal significance. We have discussed the static plane symmetric field in greater details. The main conclusions are summarised below? (i) The geodesic equations reveal that the gravitational field is repulsive in nature and a test particle at rest is expelled to infinity. The velocity of the test particle first Increases, attains a maximum value and then the velocity goes to zero at infinity. The acceleration of the particle attain its maximum velocity and after this stage it becomes negative ultimately reducing to aero at infinity. (ii) The plane symmetric metric can be looked upon as a limiting case of the spherioally symmetric gravitational field. w It is shown that three types of gravitational fields ars possible according as the velocity of the gravitating body is less than, equal to, or greater than the velocity of light which is equal to unity in the relativistlo units* The plane symmetric gravitational field corresponds to the case when the velocity f the gravitating particle is equal to that of light. In chapter V we have considered the spherically symmetric line—element with a cross term and a new gravitational metric has been obtained by solving the field equations Rj, * - 0. Some of the well known metrics, which are nonorthogonal, are derived as special cases of this more general form. The trans formation equations leading to the Schwaraschild standard metric,are also given. The asymmetrical behaviour of the field with respect to past and future is also demonstrated. A general solution of the equation R m0 has been obtained. This solution gives rise to a non-static generalisation of the Nordstrom metric of a changed mass. The generalised field describes a spherically symmetric relativisitic model of a changed mass moving with the fundamental velocity and ejecting null fluid. The content of this chapter has been published in Progress of Mathematics,Vol.8, No.l(197l),pp.17-22. In ohapter VI we have considered a spherioally symmetric static metric and we have constructed three static perfect fluid models of class two. The first of these models which develops a singularity at the centre is joined smoothly over the well£ nown Sohwaraschild interior model in such a manner that the metric potentials and their first derivatives are continuous at the interface. Physically this would imply that the press ure and density are continuous over the interface. This class two perfect fluid model is sandwiched between Gchwaraschild int erior and exterior solutions.The other class two model is wellbehaved at the centre and it has been joined with the exterior Schwarssschild model in such a manner that the metric tensor and its first derivatives are continuous across the interface. Consequently the pressure and density both vanish at MM interface. The third class two model is also joined with Schwarssschild exterior metric in a similar manner, but with a special feature of this model is that not only its density, but the first derivatives of the density also vanishes at the interface. An attempt has been made to construct a relativist!o model of a radiating star by a non-static generalisation of one of the class two statie fluid models. A radiating star model has been constructed which has the following interesting properties? (i) The interior of the radiating star is filled with a mixture of matter and radiation described by the energy tensor 2i4 « (p-f )v1v,.-pg1j4cwiw^, when v^ is the tine-like fluid flow vector and fe is the light-like radiation flow vector P, p and o are material density, hydrostatic pressure v/i and radiation density respectively* (ii) The metric tensor and its first partial derivatives are continuous across the Interface with Vaidya's pure radiation metric* (ill) The components of the energy tensor T^ for the model coincide with the components T| for Vaidya's radiation metric at the interface* (iv) The values of the radiation density or on two aides of the interface coincide at the interface. (v) The material density p and hydrostatic pressure p of the fluid, both vanish at the interface. (vi) The interface is a null surface across which some of the second derivatives of the metric tensor are discontinuous. A summary of this chapter has been published in curr ent Science, Vol.42, No.19, pp.674(1973).