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Choptuik's critical phenomena in general relativity is revisited in the affine-null metric formulation of Einstein's equations for a massless scalar field in spherical symmetry. Numerical solutions are obtained by evolution of initial data using pseudo-spectral methods. The underlying system consists of differential equations along the outgoing null rays which can be solved in sequential form. A new two-parameter family of initial data is presented for which these equations can be integrated analytically. Specific choices of the initial data parameters correspond to either an asymptotically flat null cone, a black hole event horizon or the singular interior of a black hole. Our main focus is on the interior features of a black hole, for which the affine-null system is especially well adapted. We present both analytic and numerical results describing the geometric properties of the apparent horizon and final singularity. Using a re-gridding technique for the affine parameter, numerical evolution of initially asymptotically flat supercritical data can be continued inside the event horizon and track the apparent horizon up to the formation of the final singularity., Comment: 14 pages, 13 figures

The Minkowski background intrinsic to the Kerr-Schild version of the Kerr metric provides a definition of a boosted spinning black hole. There are two Kerr-Schild versions corresponding to ingoing or outgoing principal null directions. The two corresponding Minkowski backgrounds and their associated boosts differ drastically. This has an important implication for the gravitational memory effect. A prior analysis of the transition of a non-spinning Schwarzschild black hole to a boosted state showed that the memory effect in the nonlinear regime agrees with the linearised result based upon the retarded Green function only if the final velocity corresponds to a boost symmetry of the ingoing Minkowski background. A boost with respect to the outgoing Minkowski background is inconsistent with the absence of ingoing radiation from past null infinity. We show that this results extends to the transition of a Kerr black hole to a boosted state and apply it to set upper and lower bounds for the boost memory effect resulting from the collision of two spinning black holes., Comment: 17 pages, revised discussion section

We revisit Winicour's affine-null metric initial value formulation of General Relativity, where the characteristic initial value formulation is set up with a null metric having two affine parameters. In comparison to past work, where the application of the formulation was aimed for the timelike-null initial value problem, we consider here a boundary surface that is a null hypersurface. All of the initial data are either metric functions or first derivatives of the metric. Given such a set of initial data, Einstein equations can be integrated in a hierarchical manner, where first a set of equations is solved hierarchically on the null hypersurface serving as a boundary. Second, with the obtained boundary values, a set of differential equations, similar to the equations of the Bondi-Sachs formalism, comprising of hypersurface and evolution equations is solved hierarchically to find the entire space-time metric. An example is shown how the double null Israel black hole solution arises after specification to spherical symmetry and vacuum. This black hole solution is then discussed to with respect to Penrose conformal compactification of spacetime., Comment: 28 pages, matches (up to reference listing) published version

The Kerr-Schild version of the Schwarzschild metric contains a Minkowski background which provides a definition of a boosted black hole. There are two Kerr-Schild versions corresponding to ingoing or outgoing principle null directions. We show that the two corresponding Minkowski backgrounds and their associated boosts have an unexpected difference. We analyze this difference and discuss the implications in the nonlinear regime for the gravitational memory effect resulting from the ejection of massive particles from an isolated system. We show that the nonlinear effect agrees with the linearized result based upon the retarded Green function only if the velocity of the ejected particle corresponds to a boost symmetry of the ingoing Minkowski background. A boost with respect to the outgoing Minkowski background is inconsistent with the absence of ingoing radiation from past null infinity., Comment: 13 pages, matches published version

The Bondi-Sachs formalism of General Relativity is a metric-based treatment of the Einstein equations in which the coordinates are adapted to the null geodesics of the spacetime. It provided the first convincing evidence that gravitational radiation is a nonlinear effect of general relativity and that the emission of gravitational waves from an isolated system is accompanied by a mass loss from the system. The asymptotic behaviour of the Bondi-Sachs metric revealed the existence of the symmetry group at null infinity, the Bondi-Metzner-Sachs group, which turned out to be larger than the Poincare group., Comment: Scholarpedia review article, an up-to-date version can be found under http://scholarpedia.org/article/Bondi-Sachs_Formalism, includes general normalisation of unit sphere dyad, corrected typos, additional references

The gravitational memory effect leads to a net displacement in the relative positions of test particles. This memory is related to the change in the strain of the gravitational radiation field between infinite past and infinite future retarded times. There are three known sources of the memory effect: (i) the loss of energy to future null infinity by massless fields or particles, (ii) the ejection of massive particles to infinity from a bound system and (iii) homogeneous, source-free gravitational waves. In the context of linearized theory, we show that asymptotic conditions controlling these known sources of the gravitational memory effect rule out any other possible sources with physically reasonable stress-energy tensors. Except for the source-free gravitational waves, the two other known sources produce gravitational memory with E-mode radiation strain, characterized by a certain curl-free sky pattern of their polarization. Thus our results show that the only known source of B-mode gravitational memory is of primordial origin, corresponding in the linearized theory to a homogeneous wave entering from past null infinity., Comment: typos corrected, matches published version