Studies of Singularities, Black Holes and Gravitational Radiation

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This project will study two aspects of gravity: (1) gravitational collapse and (2) gravitational radiation. (1) When an object's gravity becomes strong enough to trap light, the object becomes a black hole. Inside the black hole the object continues to become smaller and smaller under the influence of its own gravity until it becomes a point with infinite density and infinite gravitational field called a singularity. I will work out the properties of the singularities formed in gravitational collapse, and in particular the forces felt by any observer who approaches the singularity. (2) Just as the electric currents in a radio transmitter make radio waves, so moving masses make gravity waves. And just as radio receivers react to the presence of radio waves, so there is an ongoing effort to detect gravitational waves. One interesting aspect of gravitational radiation is something called gravitational wave memory: even after the wave has passed, there is a permanent change in the gravitational wave detector. I will study the causes and properties of this gravitational wave memory.It is conjectured that the singularity inside a black hole consists of two parts: a spacelike singularity at the center of the black hole and a null singularity that takes the place of the black hole's inner horizon. I will study both types of singularities using two different numerical methods. Previously, I have performed simulations of spacelike singularities; however these simulations revealed the presence of structure with a very small spatial scale (called "spikes") that was not resolved by the simulations. I will revisit these simulations using the technique of adaptive mesh refinement, which should serve to resolve the spikes. I will compare the results of these simulations to those of an analytic approximation to spike behavior using what are expected to be the leading terms of the field equations near the singularity. I will study null singularities using a "double null" formulation of the Einstein field equations. Here instead of the usual foliation of spacetime by spacelike surfaces of constant time t, one has a foliation by pairs of null surfaces of constant null coordinates u and v. The Einstein field equations and Bianchi identities then become equations for the expansion and shear of the null surfaces and the Weyl tensor. I will perform numerical simulations of these equations to see whether the Weyl tensor blows up on (what used to be) the inner horizon of the black hole. I will study gravitational wave memory from a variety of sources using a variety of methods. Memory from neutrinos will be treated using the full nonlinear Einstein-Vlasov equations. Memory in an expanding universe will be treated using cosmological perturbation theory. Memory for the vacuum case will be treated in second order perturbation theory. I will also study critical gravitational collapse using a new vacuum axixymmetric code. In addition, I will continue, using numerical methods, my studies of Einstein-Aether theory, gravitational collapse in anti de-Sitter spacetime, and charged black holes.