We present a general solution of the Einstein gravitational field equations for the static spherically symmetric gravitational interior spacetime of an isotropic fluid sphere. The solution is obtained by transforming the pressure isotropy condition,
a second order ordinary differential equation, into a Riccati type first order differential equation, and using a general integrability condition for the Riccati equation. This allows us to obtain an exact non-singular solution of the interior field equations for a fluid sphere, expressed in the form of infinite power series. The physical features of the solution are studied in detail numerically by cutting the infinite series expansions, and restricting our numerical analysis by taking into account only $n=21$ terms in the power series representations of the relevant astrophysical parameters. In the present model all physical quantities (density, pressure, speed of sound etc.) are finite at the center of the sphere. The physical behavior of the solution essentially depends on the equation of state of the dense matter at the center of the star. The stability properties of the model are also analyzed in detail for a number of central equations of state, and it is shown that it is stable with respect to the radial adiabatic perturbations. The astrophysical analysis indicates that this solution can be used as a realistic model for static general relativistic high density objects, like neutron stars.
New further integrability conditions of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ are presented. The first case corresponds to fixed functional forms of the coefficients $a(x)$ and $c(x)$ of the Riccati equation, and of the function $F(x)=a(x
)+[f(x)-b^{2}(x)]/4c(x)$, where $f(x)$ is an arbitrary function. The second integrability case is obtained for the reduced Riccati equation with $b(x)equiv 0$. If the coefficients $a(x)$ and $c(x)$ satisfy the condition $pm dsqrt{f(x)/c(x)}/dx=a(x)+f(x)$, where $f(x)$ is an arbitrary function, then the general solution of the reduced Riccati equation can be obtained by quadratures. The applications of the integrability condition of the reduced Riccati equation for the integration of the Schrodinger and Navier-Stokes equations are briefly discussed.
We present an extension of general relativity in which an $f(R)$ term `{a} la Palatini is added to the usual metric Einstein-Hilbert Lagrangian. Expressing the theory in a dynamically equivalent scalar-tensor form, we show that it can pass the Solar
System observational tests even if the scalar field is very light or massless. Applications to cosmology and astrophysics, and some exact solutions are discussed.
Electron accelerations of the order of $10^{21} g$ obtained by laser fields open up the possibility of experimentally testing one of the cornerstones of general relativity, the weak equivalence principle, which states that the local effects of a grav
itational field are indistinguishable from those sensed by a properly accelerated observer in flat space-time. We illustrate how this can be done by solving the Einstein equations in vacuum and integrating the geodesic equations of motion for a uniformly accelerated particle.
The higher dimensional Weyl curvature induces on the brane a new source of gravity. This Weyl fluid of geometrical origin (reducing in the spherically symmetric, static configuration to a dark radiation and dark pressure) modifies space-time geometry
around galaxies and has been shown to explain the flatness of galactic rotation curves. Independent observations for discerning between the Weyl fluid and other dark matter models are necessary. Gravitational lensing could provide such a test. Therefore we study null geodesics and weak gravitational lensing in the dark radiation dominated region of galaxies in a class of spherically symmetric brane-world metrics. We find that the lensing profile in the brane-world scenario is distinguishable from dark matter lensing, despite both the brane-world scenario and dark matter models fitting the rotation curve data. In particular, in the asymptotic regions light deflection is 18% enhanced as compared to dark matter halo predictions. For a linear equation of state of the Weyl fluid we further find a critical radius, below which brane-world effects reduce, while above it they amplify light deflection. This is in contrast to any dark matter model, the addition of which always increases the deflection angle.
We study the dynamical evolution of a phase-transition-induced collapse neutron star to a hybrid star, which consists of a mixture of hadronic matter and strange quark matter. The collapse is triggered by a sudden change of equation of state, which r
esult in a large amplitude stellar oscillation. The evolution of the system is simulated by using a 3D Newtonian hydrodynamic code with a high resolution shock capture scheme. We find that both the temperature and the density at the neutrinosphere are oscillating with acoustic frequency. However, they are nearly 180$^{circ}$ out of phase. Consequently, extremely intense, pulsating neutrino/antineutrino fluxes will be emitted periodically. Since the energy and density of neutrinos at the peaks of the pulsating fluxes are much higher than the non-oscillating case, the electron/positron pair creation rate can be enhanced dramatically. Some mass layers on the stellar surface can be ejected by absorbing energy of neutrinos and pairs. These mass ejecta can be further accelerated to relativistic speeds by absorbing electron/positron pairs, created by the neutrino and antineutrino annihilation outside the stellar surface. The possible connection between this process and the cosmological Gamma-ray Bursts is discussed.
