We prove here a long standing conjecture in general relativity, that if barotropic perfect fluid is moving in a shear free way, then it must be either expansion free or rotation free.
We show that a general but shear-free perturbation of homogeneous and isotropic universes are necessarily silent, without any gravitational waves. We prove this in two steps. First we establish that a shear free perturbation of these universes are ac
celeration-free and the fluid flow geodesics of the background universe maps onto themselves in the perturbed universe. This effect then decouples the evolution equations of the electric and magnetic parts of the Weyl tensor in the perturbed spacetimes and the magnetic part no longer contains any tensor modes. Although the electric part, that drives the tidal forces, do have tensor modes sourced by the anisotropic stress, these modes have homogeneous oscillations at every point on a time slice without any wave propagation. We also show the presence of vorticity vector waves that are sourced by the curl of heat flux. This analysis shows the critical role of the shear tensor in generating cosmological gravitational waves in an expanding universe.
In this paper we consider conformally flat perturbations on the Friedmann Lemaitre Robertson Walker (FLRW) spacetime containing a general matter field. Working with the linearised field equations, we unearth some important geometrical properties of m
atter shear and vorticity and how they interact with the thermodynamical quantities in the absence of any free gravity powered by the Weyl curvature. As there are hardly any physically realistic rotating exact conformally flat solutions in general relativity, these covariant and gauge invariant results bring out transparently the role of vorticity in the linearised regime. Most interestingly, we demonstrate that the matter shear obeys a transverse traceless tensor wave equation, and the vorticity obeys a vector wave equation in this regime. These shear and vorticity waves replace the gravitational waves in the sense that they causally carry the information about local change in the curvature of these spacetimes.
We recast the well known Israel-Darmois matching conditions for Locally Rotationally Symmetric (LRS-II) spacetimes using the semitetrad 1+1+2 covariant formalism. This demonstrates how the geometrical quantities including the volume expansion, spacet
ime shear, acceleration and Weyl curvature of two different spacetimes are related at a general matching surface inheriting the symmetry, which can be timelike or spacelike. The approach is purely geometrical and depends on matching the Gaussian curvature of 2-dimensional sheets at the matching hypersurface. This also provides the constraints on the thermodynamic quantities on each spacetime so that they can be matched smoothly across the surface. As an example we regain the Santos boundary conditions and model of a radiating star matched to a Vaidya exterior in general relativity.
In this paper we show in a covariant and gauge invariant way that in general relativity, tidal forces are actually a hidden form of gravitational waves. This must be so because gravitational effects cannot occur faster than the speed of light. Any tw
o body gravitating system, where the bodies are orbiting around each other, may generate negligible gravitational waves, but it is via these waves that non-negligible tidal forces (causing shape distortions) act on these bodies. Although the tidal forces are caused by the electric part of the Weyl tensor, we transparently show that some small time varying magnetic part of the Weyl tensor with non zero curl must be present in the system that mediates the tidal forces via gravitational wave type effects. The outcome is a new test of whether gravitational effects propagate at the speed of light.
In this paper we excavate, for the first time, the most general class of conformal Killing vectors, that lies in the two dimensional subspace described by the null and radial co-ordinates, that are admitted by the generalised Vaidya geometry. Subsequ
ently we find the most general class of generalised Vaidya mass functions that give rise to such conformal symmetry. From our analysis it is clear that why some well known subclasses of generalised Vaidya geometry, like pure Vaidya or charged Vaidya solutions, admit only homothetic Killing vectors but no proper conformal Killing vectors with non constant conformal factors. We also study the gravitational collapse of generalised Vaidya spacetimes that posses proper conformal symmetry to show that if the central singularity is naked then in the vicinity of the central singularity the spacetime becomes almost self similar. This study definitely sheds new light on the geometrical properties of generalised Vaidya spacetimes.
In this paper we consider the novel scenario where a spherically symmetric perfect fluid star is undergoing continual gravitational collapse while continuously radiating energy in an exterior radiating spacetime. There are no trapped surfaces and the
collapse ends to a flat spacetime. Also the collapsing matter obeys the weak and dominant energy conditions at all epoch. Our analysis transparently brings out the role of the equation of state as well as the bounds on the thermodynamic potentials to realise such a scenario. We argue that, since the system of Einstein field equations allows for such a scenario for an open set of initial data as well as the equation of state function in their respective functional spaces, these models are generic and devoid of the problems and paradoxes related to horizons and singularities. The recent high resolution radio telescopes should in principle detect the presence of these compact objects in the sky.
In this paper, we perform a detailed investigation on the various geometrical properties of trapped surfaces and the boundaries of trapped region in general relativity. This treatment extends earlier work on LRS II spacetimes to a general 4 dimension
al spacetime manifold. Using a semi-tetrad covariant formalism, that provides a set of geometrical and matter variables, we transparently demonstrate the evolution of the trapped region and also extend Hawkings topology theorem to a wider class of spacetimes. In addition, we perform a stability analysis for the apparent horizons in this formalism, encompassing earlier works on this subject. As examples, we consider the stability of MOTS of the Schwarzschild geometry and Oppenheimer-Snyder collapse.
In this paper we investigate conformal symmetries in Locally Rotationally Symmetric (LRS) spacetimes using a semitetrad covariant formalism. We demonstrate that a general LRS spacetime which rotates and spatially twists simultaneously has an inherent
homothetic symmetry in the plane spanned by the fluid flow lines and the preferred spatial direction. We discuss the nature and consequence of this homothetic symmetry showing that a null Killing horizon arises when the heat flux has an extremal value. We also consider the special case of a perfect fluid and the restriction on the conformal geometry.
A problem in general relativity is, how the gravitational field can transfer energy and momentum between different distant places. The issue is that matter stress tensor is locally conserved, with no explicit interaction with the free gravitational f
ield, which is represented by the Weyl tensor. In this paper we explicitly construct an interaction tensor for free gravity and matter, that depicts the interplay between the energy momentum tensor of free gravity, which is taken to be the symmetric two index square root of Bell-Robinson tensor, and matter. This is examined both in the case of Coulomb-like Petrov type D spacetimes and radiation like Petrov type N spacetimes, where a unique square root exists. The first case generalises the Tweedledum and Tweedledee thought experiment regarding gravitational induction in Newtonian gravity to general relativistic scenarios, and the second gives a proposal for how gravitational radiation can transfer energy and momentum between distant objects separated by a vacuum.
