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A connection between the symmetries of manifolds and differential equations is sought through the geodesic equations of maximally symmetric spaces, which have zero, constant positive or constant negative curvature. It is proved that for a space admitting so(n + 1) or so(n, 1) as the maximal isometry algebra, the symmetry of the geodesic equations of the space is given by so(n + 1) ⊕ d 2 or so(n, 1) ⊕ d 2 (where d 2 is the two-dimensional dilation algebra), while for those admitting so(n) ⊕ s R n (where ⊕ s represents semidirect sum) the algebra is sl(n + 2). A corresponding result holds on replacing so(n) by so( p, q) with p + q = n. It is conjectured that if the isometry algebra of any underlying space of non-zero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by h ⊕ d 2 , provided that there is no cross-section of zero curvature at the point under consideration. If there is a flat subspace of dimension m, then the symmetry group becomes h ⊕ sl(m + 2).

In the classical relativistic regime, the accretion of phantom-like dark energy onto a stationary black hole reduces the mass of black hole. Here we have investigated the accretion of phantom energy onto a stationary charged black hole and have determined the condition under which this accretion is possible. This condition restricts the mass to charge ratio in a narrow limit. This condition also challenges the validity of the cosmic censorship conjecture since a naked singularity is eventually produced as magnitude of charge increases compared to mass of black hole.

Ehlers and Kundt have provided an approximate procedure to demonstrate that gravitational waves impart momentum to test particles. This was extended to cylindrical gravitational waves by Weber and Wheeler. Here a general, exact, formula for the momentum imparted to test particles in arbitrary spacetimes is presented.

Using approximate symmetry methods for differential equations we have investigated the exact and approximate symmetries of a Lagrangian for the geodesic equations in the Kerr spacetime. Taking Minkowski spacetime as the exact case, it is shown that the symmetry algebra of the Lagrangian is 17 dimensional. This algebra is related to the 15 dimensional Lie algebra of conformal isometries of Minkowski spacetime. First introducing spin angular momentum per unit mass as a small parameter we consider first-order approximate symmetries of the Kerr metric as a first perturbation of the Schwarzschild metric. We then consider the second-order approximate symmetries of the Kerr metric as a second perturbation of the Minkowski metric. The approximate symmetries are recovered for these spacetimes and there are no nontrivial approximate symmetries. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the charged-Kerr metric has to be rescaled and the rescaling factor is r -dependent. This re-scaling factor is compared with that for the Reissner-Nordström metric.

Conditions are derived for the linearizability via invertible maps of a system of n secondorder quadratically semi-linear differential equations that have no lower degree lower order terms in them, i.e., for the symmetry Lie algebra of the system to be sl(n + 2, R). These conditions are stated in terms of the coefficients of the equations and hence provide simple invariant criteria for such systems to admit the maximal symmetry algebra. We provide the explicit procedure for the construction of the linearizing transformation. In the simplest case of a system of two second-order quadratically semi-linear equations without the linear terms in the derivatives, we also provide the construction of the linearizing point transformation using complex variables. Examples are given to illustrate our approach for two-and three-dimensional systems.

Approximate symmetries have been defined in the context of differential equations and systems of differential equations. They give approximately, conserved quantities for Lagrangian systems. In this paper, the exact and the approximate symmetries of the system of geodesic equations for the Schwarzschild metric, and in particular for the radial equation of motion, are studied. It is noted that there is an ambiguity in the formulation of approximate symmetries that needs to be clarified by consideration of the Lagrangian for the system of equations. The significance of approximate symmetries in this context is discussed.