SCHWARZSCHILD SOLUTION OF THE MODIFIED EINSTEIN FIELD EQUATIONS

In general theory of relativity, Einstein’s field equations relate the geometry of space-time with the distribution of matter within it. These equations were first published by Einstein in the form of a tensor equation which related the local space-time curvature with the local energy and momentum within this space-time. In this article, Einstein’s geometrical field equations interior and exterior to astrophysically real or hypothetical distribution of mass within a spherical geometry were constructed and solved for field whose gravitational potential varies with time, radial distance and polar angle. The exterior solution was obtained using power series. The metric tensors and the solution of the Einstein’s exterior field equations used in this work has only one arbitrary function f(t,r,θ) , and thus put the Einstein’s geometrical theory of gravitation on the same bases with the Newton’s dynamical theory of gravitation. The gravitational scalar potential f(t,r,θ) obtained in this research work to the order of co, c-2 , contains Newton dynamical gravitational scalar potential and post Newtonian additional terms much importance as it can be applied to the study of rotating bodies such as stars. The interior solution was obtained using weak field and slow-motion approximation. The obtained result converges to Newton’s dynamical scalar potential with additional time factor not found in the well-known Newton’s dynamical theory of gravitation which is a profound discovery with the dependency on three arbitrary functions. Our result obeyed the equivalence principle of Physics.

Approximate Kerr- like interior and exterior solutions for a very slowly rotating star of perfect fluid is presented here which is valid for a very slowly rotating and hence very slightly oblate star of perfect fluid rotating with a very small constant angular velocity. Corresponding approximate exterior metric represents approximate Kerr exterior metric for a perfect fluid. The aim here is to obtain Kerr-like approximate exterior and interior solutions for a very slowly rotating star of perfect fluid simply following standard procedure used to obtain Schwarzschild solutions. Key words: Slowly rotating oblate star, approximate Kerr- like metric, exterior interior solution.

We obtain an approximate global stationary and axisymmetric solution of Einstein's equations which can be thought as a simple star model: a self-gravitating perfect fluid ball with a differential rotation motion pattern. Using the post--Minkowskian formalism (weak field approximation) and considering rotation as a perturbation (slow rotation approximation), we find approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic coordinates. Interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to get a global solution. The resulting metric depends on four arbitrary constants: mass density, rotational velocity at $r=0$, a parameter which takes into account of the change in the rotational velocity through the star and the star radius in the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined in terms of these four parameters.

In Chaps. 14 and 15 , we applied Eshelby’s interior solutions and the equivalent inclusion approach. In this chapter, we investigate the development of flanking structures as an example to use Eshelby’s exterior solutions. Flanking structures [Fig. 16.1; Passchier (J Struct Geol 23(6–7):951–962, 2001)] are deflections of linear and planar fabric elements around a cross-cutting element such as a fault, vein, dyke, or any other heterogeneity. Many authors have investigated the development of flanking structures by a combination of field, numerical modeling, analogue modeling, and analytical approaches [Exner and Dabrowski (J Struct Geol 32(12):2009–2021, 2010); Exner et al. (J Struct Geol 26(12):2191–2201, 2004); Grasemann et al. (J Struct Geol 25(1):19–34, 2003), (J Struct Geol 33(11):1650–1661, 2011); Grasemann and Stuwe (J Struct Geol 23(4):715–724, 2001); Kocher and Mancktelow (J Struct Geol 27(8):1346–1354, 2005), (J Struct Geol 28(7):1139–1145, 2006); Wiesmayr and Brasemann (J Struct Geol 27(2):249–264, 2005)]. These studies have greatly improved the understanding of flanking structures. The main conclusions are that the final geometrical features of flanking structures depend on the initial conditions of the cutting element (shape, orientation, relative rheology) and the characteristics of the flow field. One must therefore be cautious when interpreting the kinematics of flanking structures (e.g., Grasemann et al. (J Struct Geol 25(1):19–34, 2003); Wiesmayr and Brasemann (J Struct Geol 27(2):249–264, 2005)]. Because flanking structures result from the progressive deformation around a cutting element heterogeneity, we can regard the cutting element as an Eshelby ellipsoid and apply the exterior solutions (Chap. 11 ) to the development of flanking structures. Compared to previous investigations, the application of Eshelby’s exterior solutions has the following advantages. First, there is no limit to the flow type and the initial geometrical conditions of the cutting element (any ellipsoidal shape, arbitrary orientation with respect to the host element and the bulk flow field). Second, there is no limit to the amount of strain that can be reached because the approach does not rely on meshing. Third, the approach can handle anisotropic viscosity readily, although at a more expensive computational cost (Jiang and Bhandari 2018; Bhandari 2021). Furthermore, although the approach is based on linear viscous materials so far (Chaps. 10 and 11 ), it can be extended to power-law viscous materials by a linearization approach [Jiang (J Struct Geol 68:247–272, 2014), (Tectonophysics 693:116–142, 2016); Chap. 17 ]. This chapter first summarizes the principal equations related to the exterior mechanical fields in the vicinity of a heterogeneous ellipsoid. An outline of the algorithm for the numerical modeling of flanking structures is then presented. The algorithm is implemented in MATLAB. The MATLAB programs are provided in the online resource of the book. The program is applicable to anisotropic viscous materials under any flow field. As examples of using the numerical approach and the MATLAB program, we present numerical results for the development of flanking structures in plane-strain general shearing flows. The cutting element and the host medium are assumed to be isotropic and linear viscous materials. Furthermore, for simplicity, we limit ourselves to cutting elements with one principal axis parallel to the flow vorticity so that the whole deforming system has monoclinic symmetry.

The Einstein-Cartan field equations are solved for a string source with spin-polarisation along the axis of symmetry. The interior solution is matched to an exterior vacuum space-time using Arkuszewski-Kopczynski-Ponomariev junction conditions. The exterior solution is a four dimensional extension of the space-time outside a spinning point particle in three dimensional Einstein theory. It reduces to the geometry outside a conventional straight cosmic string in the case of vanishing spin.

The contents of this paper refers to following main subjects: numerical adaptation of our cosmological parameters recently used within the framework of the Projective Unified Field Theory [1] to the new measured values of the Wilkinson Microwave Anisotropy Probe (WMAP) published by Ch.L. Bennett et al. [2]; study of the resulting consequences for our cosmological model of a closed homogeneous isotropic cosmos; approximate treatment of a quasi-static spherically symmetric perfect fluid sphere (interior and exterior solution) being under the adiabatic influence of the expanding cosmos; application of these results to sun-like and earth-like cosmic bodies; time dependence of the “empirical gravitational constant”; treatment of the Einstein effects, particularly of the periastron (perihelion) motion of an orbiting body (mercury) about a central body (sun) with a mass quadrupole moment and calculation of its numerical value for the sun. Further cosmological and astrophysical applications. An appendix points to a different set of parameters.

A semiinfinite crack extends nonuniformly against a body force that is antisymmetric about the fault plane but independent of the distance from the latter. Conversion of the inhomogeneous wave equation for SH motion into a radiation equation incorporates fault plane distributed mechanisms. An integral equation is then formulated for fault plane observations preceding the crack. Both the body force and the stress drop are assumed to be bounded over finite histories. Multiple inversion yields a formal exterior solution which behaves singularly near the crack edge. An analogy with a Barenblatt postulate cancels the singularity through a nonlinear integral equation governing the edge coordinate. The resultant non-singular displacement gradient turns out to be continuous at the crack edge; here, the force-induced contribution vanishes. The interior problem is also solved for the displacement along either crack face. An application is illustrated for an impulsive body force; this leads to an interesting corollary with totally uniform results. Finally, if the body force exists within a half-space away from which extension occurs, it contributes nothing to the exterior solution; furthermore, the interior solution can be evaluated in terms of only the stress drop and the edge locus, i.e. without explicit prescription of such a body force.

In veiw of a possible relevance of the concept of the C-field introduced by Hoyle and Narlikar to the problem of quasi-stellar objects or collapsing stars, an attempt is made to study permissible solutions of their field equations in an empty region with spherical symmetry. Two special, i.e. static and pseudo-staticl, solutions are obtained and their characteristic properties are examined in the light of Hoyle-Narlikar's exterior but non-empty solution. As a result, it is revealed that the problem of boundary conditions has a crucial importance in the decision of superiority of either one among the three exterior solutions.

Usually it is demanded that the metric and its 1st derivatives have to match at the boundary of two adjacent regions which are solutions to Einstein’s field equation. We propose a new linking condition concerning gravitational models based on surfaces which could be embedded into a higher dimensional flat space. We probe this condition for the Schwarzschild interior and exterior solution.

In one of our previous papers, we obtained a general class of potentials inside the nucleus, such that the singular solution of the Dirac equation for the S-states of hydrogen atoms outside the nucleus can be matched with the corresponding regular solution inside the nucleus (the proton) at the boundary. The experimental charge density distribution inside the proton generates a particular case of such potentials inside the proton. In this way, the existence of the second kind of hydrogen atom was predicted: atoms having only S-states. This theoretical prediction was then evidenced by several different types of atomic experiments and by astrophysical observations. In the present paper we provide the following new results. First, we show that the solution of the Dirac equation inside the proton can be (and is) found within the class of functions that are non-analytic at r = 0—in distinction to the traditional practice of limiting the search for the solution by the class of analytic functions. We demonstrate that this non-analytic solution inside the proton can be matched at the proton boundary R with the corresponding singular solution outside the proton regardless of the particular value of R. Second, we show that the interior and exterior solutions are scale-invariant with respect to the change of the boundary R between these two regions. Such invariance is the manifestation of a new symmetry—in addition to the previously discussed symmetries of the Dirac equation in the literature. Third, based on the new, more accurate results for the wave function inside and outside the proton, we revisit the resolution of the neutron lifetime puzzle initially outlined in our previous papers. On the basis of the more accurate calculations, we reconfirm that (A) the 2-body decay of neutrons produces overwhelmingly the SFHA (rather than the usual hydrogen atoms) and (B) the strengthened-in-this-way branching ratio for the 2-body decay of neutrons (compared to the 3-body decay) is in excellent agreement with the branching ratio required for reconciling the neutron lifetime values measured in the trap and beam experiments, so that the neutron lifetime puzzle seems to be indeed resolved in this way.

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Energy levels: 1s 1/2, 2p 1/2, 2p 3/2, 3d 3/2, and 3d 5/2 of the muon in the spherical nuclei:120Sn,197Au and208Pb have been calculated under the assumption of harmonic oscillator potential. The levels are corrected for vacuum polarisation. The agreement with experimental values is better than 0·5%. An accurate method of solving the Dirac equation to obtain the energy eigenvalues is outlined. The importance of choosing the ‘classical turning point’ as the radius for matching the interior and exterior solutions is discussed.

We present a parametric study of the spacetime structures obtainable in Weyl conformal gravity's dyonic Reissner-Nordstr\"{o}m solution. We derive expressions for photon sphere radii and horizons for this metric in terms of the conformal gravity parameters, from which we then determine analytic formulae for extremal limits and Hawking temperatures. Due to the surprising lack of the inverse quadratic $1/r^2$ term in this fourth-order metric, there is no guarantee for the innermost horizon of a black hole spacetime to be a Cauchy horizon, which is in direct contrast to the corresponding metric in general relativity. For example, for certain parameter values, a ``nested black hole'' is seen to exist; in such a spacetime, we find a Cauchy horizon trapped between two event horizons, which is not a structure known to be obtainable in standard general relativity. In addition to such exotic spacetimes, we also find a critical value for the electric and magnetic charges, at which the stable and unstable photon spheres of the metric merge, and we obtain extremal limits where three horizons collide.

We study equilibrium shapes, stability and possible bifurcation diagrams of fluids in higher dimensions, held together by either surface tension or self-gravity. We consider the equilibrium shape and stability problem of self-gravitating spheroids, establishing the formalism to generalize the MacLaurin sequence to higher dimensions. We show that such simple models, of interest on their own, also provide accurate descriptions of their general relativistic relatives with event horizons. The examples worked out here hint at some model-independent dynamics, and thus at some universality: smooth objects seem always to be well described by both ‘replicas’ (either self-gravity or surface tension). As an example, we exhibit an instability afflicting self-gravitating (Newtonian) fluid cylinders. This instability is the exact analogue, within Newtonian gravity, of the Gregory–Laflamme instability in general relativity. Another example considered is a self-gravitating Newtonian torus made of a homogeneous incompressible fluid. We recover the features of the black ring in general relativity.

We propose a quantum mechanical theory of quantum spaces described by large N noncommutative geometry as a model for quantum gravity. The model admits a fuzzy sphere as a static solution. Over the fuzzy geometry, the quantum mechanics of the fermions is given by a sum of oscillators with equal frequency. The energy state where exactly half of the Fermi sea is filled contains the maximal amount of degeneracy. This state of the fuzzy sphere obeys the mass-radius relation of a Schwarzschild black hole if the fuzzy sphere is identified with the black hole horizon. Moreover, the set of states in the Fermi sea gives precisely the Bekenstein-Hawking entropy. We thus propose that quantum black holes are described by fuzzy spheres with a half-filled Fermi sea in our model. We also consider a system of two fuzzy spheres by embedding them as blocks in the matrix quantum mechanics. When the distance r between the two fuzzy spheres is small, the total energy of the system can be computed using perturbation theory. We show that, in the leading order of large N limit, the interaction energy depends on −GM1M2 exactly in the manner as in Newton gravity. To reproduce the correct r dependence in the long range, we expect the inclusion of large N corrections and quantum effects will be needed.