Conformal symmetries in static spherically symmetric spacetimes

We obtain the conformal symmetry vector in static, spherically symmetric spacetimes, in terms of functions subject to a number of integrability conditions that also place restrictions on the metric. Some conformal symmetries found previously are regained as special cases.     

Parametrical embeddings of static spherically symmetric spacetimes

An analysis for a direct calculation of the embeddings in flat spacetimes of static spherically symmetric manifolds with Lorentz metric is worked out. For each manifold with non-constant curvature we arrive at a parametrical embedding which represents an infinite geometrical multiplicity of the embedded surface. The embeddings of manifolds with constant curvature are not parametrical and can be determined univocally. Examples concerning Schwarzschild, Reissner-Weyl, de Sitter and anti-de Sitter spacetimes are considered.     

Null charts and naked singularities in spherically symmetric,homothetic spacetimes

We give a unified derivation of a null chart for all spherically symmetric, homothetic spacetimes. These spacetimes contain an interesting class of naked singularities which we are also able to elucidate. Much use is made of graphical representation; in particular a chart of the spacetimes based on their homothetic group motions is introduced. Dust spacetimes, and two homogeneous examples with non-zero pressure (flat Robertson-Walker and a Kantowski-Sachs example) are studied in detail. We show the horizon structure in the null atlas, in comoving coordinates, in terms of the areal radius and comoving time, and in the homothetic diagrams. The critical delay between comoving observers for the onset of nakedness is interpreted in terms of a decreasing mass concentration in the spirit of Thorne's hoop conjecture. We also give a simple criterion for the existence of apparent horizons isolating the various singularities, and study in detail how this criterion is circumvented in the naked examples. We conclude that this type of naked singularity is a consequence of the imposed homothetic symmetry, by showing it to be generally present and timelike in the homothetic group chart even when it is not visible at comoving infinity (before the onset of criticality). It is the delayed final collapse of initially distant observers in inhomogeneous spacetimes that causes the initial singularity to become visible at comoving infinity. We conclude that these examples do not present an obstacle to the Event Horizon Conjecture as summarized by W. Israel (1984). That is, one can formulate criteria for the formation of apparent horizons that do not imply that all singularities are necessarily so enclosed. It is still possible that all singularities stronger than homothetic are isolated by an apparent horizon, in the spirit of Tipler's conjecture.On leave from Department of Physics, Queen's University, Kingston, Ontario, Canada     

Complete affine connection in the causal boundary: static,spherically symmetric spacetimes

The boundary at \(\mathcal {I}^+\), future null infinity, for a standard static, spherically symmetric spactime is examined for possible linear connections. Two independent methods are employed, one for treating \(\mathcal {I}^+\) as the future causal boundary, and one for treating it as a conformal boundary (the latter is subsumed in the former, which is of greater generality). Both methods provide the same result: a constellation of various possible connections, depending on an arbitrary choice of a certain function, a sort of gauge freedom in obtaining a natural connection on \(\mathcal {I}^+\); choosing that function to be constant (for instance) results in a complete connection. Treating \(\mathcal {I}^+\) as part of the future causal boundary, the method is to impute affine connections on null hypersurfaces going out to \(\mathcal {I}^+\), in terms of a transverse vector field on each null hypersurface (there is much gauge freedom on choice of the transverse vector fields). Treating \(\mathcal {I}^+\) as part of a conformal boundary, the method is to make a choice of conformal factor that makes the boundary totally geodesic in the enveloping manifold (there is much gauge freedom in choice of that conformal factor). Similar examination is made of other boundaries, such as timelike infinity and timelike and spacelike singularities. These are much simpler, as they admit a unique connection from a similar limiting process (i.e., no gauge freedom); and that connection is complete.     

Geometric inequalities in spherically symmetric spacetimes

In geometric inequalities ADM mass plays more fundamental role than the concept of quasi-local mass. This paper is to demonstrate that using the quasi-local mass some new insights can be acquired. In spherically symmetric spacetimes the Misner–Sharp mass and the concept of the Kodama vector field provides an ideal setting to the investigations of geometric inequalities. We applying the proposed new techniques to investigate the spacetimes containing black hole or cosmological horizons but we shall also apply them in context of normal bodies. Most of the previous investigations applied only the quasi-local charges and the area. Our main point is to include the quasi-local mass in the corresponding geometrical inequalities. This way we recover some known relations but new inequalities are also derived.     

Existence and stability of circular orbits in general static and spherically symmetric spacetimes

The existence and stability of circular orbits (CO) in static and spherically symmetric (SSS) spacetime are important because of their practical and potential usefulness. In this paper, using the fixed point method, we first prove a necessary and sufficient condition on the metric function for the existence of timelike COs in SSS spacetimes. After analyzing the asymptotic behavior of the metric, we then show that asymptotic flat SSS spacetime that corresponds to a negative Newtonian potential at large r will always allow the existence of CO. The stability of the CO in a general SSS spacetime is then studied using the Lyapunov exponent method. Two sufficient conditions on the (in)stability of the COs are obtained. For null geodesics, a sufficient condition on the metric function for the (in)stability of null CO is also obtained. We then illustrate one powerful application of these results by showing that three SSS spacetimes whose metric function is not completely known will allow the existence of timelike and/or null COs. We also used our results to assert the existence and (in)stabilities of a number of known SSS metrics.     

Universal time delay in static spherically symmetric spacetimes for null and timelike signals

A perturbative method of computing the total travel time of both null and lightlike rays in arbitrary static spherically symmetric spacetimes in the weak field limit is proposed.The resultant total time takes a quasi-series form of the impact parameter.The coefficient of this series at a certain order n is shown to be determined by the asymptotic expansion of the metric functions to the order n+1.For the leading order(s),the time delay,as well as the difference between the time delays of two types of relativistic signals,is shown to take a universal form for all SSS spacetimes.This universal form depends on the mass M and a post-Newtonian parameter y of the spacetime.The analytical result is numerically verified using the central black hole of galaxy M87 as the gravitational lensing center.     

The regularity of static spherically cylindrically and plane symmetric spacetimes at the origin

We find the necessary and sufficient conditions for the regularity of all scalar invariants polynomial in the Riemann tensor at the origin of spherically, cylindrically and plane symmetric static spacetimes under the assumption that the metric functions are sufficiently smooth there. These conditions turn out to be simple enough to allow a check for regularity by inspection.     

Quasi-local energy for spherically symmetric spacetimes

We present two complementary approaches for determining the reference for the covariant Hamiltonian boundary term quasi-local energy and test them on spherically symmetric spacetimes. On the one hand, we isometrically match the 2-surface and extremize the energy. This can be done in two ways, which we call programs I (without constraint) and II (with additional constraints). On the other hand, we match the orthonormal 4-frames of the dynamic and the reference spacetimes. Then, if we further specify the observer by requiring the reference displacement to be the timelike Killing vector of the reference, the result is the same as program I, and the energy can be positive, zero, or even negative. If, instead, we require that the Lie derivatives of the two-area along the displacement vector in both the dynamic and reference spacetimes to be the same, the result is the same as program II, and it satisfies the usual criteria: the energies are non-negative and vanish only for Minkowski (or anti-de Sitter) spacetime.