Reverse Shock Emission in Gamma-ray Bursts Revisited

A generic synchrotron external shock model is the widely preferred paradigm used to interpret the broad-band afterglow data of gamma-ray bursts (GRBs), including predicted observable signatures from a reverse shock which have been confirmed by observations. Investigations of the nature of the reverse shock emission can provide valuable insights into the intrinsic properties of the GRB ejecta. Here we briefly review the standard and the extended models of the reverse shock emission, discussing the connection between the theory and observations, including the implications of the latest observational advances.Comment: Invited review, to be published in special issue on "GRB in Swift and Fermi Era" in Journal of Advances in Astronom

On the classification of warped product Einstein metrics

In this paper we take the perspective introduced by Case-Shu-Wei of studying warped product Einstein metrics through the equation for the Ricci curvature of the base space. They call this equation on the base the $m$-Quasi Einstein equation, but we will also call it the $(\lambda,n+m)$-Einstein equation. In this paper we extend the work of Case-Shu-Wei and some earlier work of Kim-Kim to allow the base to have non-empty boundary. This is a natural case to consider since a manifold without boundary often occurs as a warped product over a manifold with boundary, and in this case we get some interesting new canonical examples. We also derive some new formulas involving curvatures which are analogous to those for the gradient Ricci solitons. As an application, we characterize warped product Einstein metrics when the base is locally conformally flat.Comment: 29 pages. Minor changes and references updated. Submitted versio

Warped product Einstein metrics over spaces with constant scalar curvature

In this paper we study warped product Einstein metrics over spaces with constant scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is isometric to a product of Einstein manifolds. When the base is three dimensional and the dimension of the fiber is greater than one we show that the space is always rigid. We also exhibit examples of solvable four dimensional Lie groups that can be used as the base space of non-rigid warped product Einstein metrics showing that the result is not true in dimension greater than three. We also give some further natural curvature conditions that characterize the rigid examples in higher dimensions.Comment: 38 pages, 1 appendi

Warped product Einstein metrics on homogeneous spaces and homogeneous Ricci solitons

In this paper we consider connections between Ricci solitons and Einstein metrics on homogeneous spaces. We show that a semi-algebraic Ricci soliton admits an Einstein one-dimensional extension if the soliton derivation can be chosen to be normal. Using our previous work on warped product Einstein metrics, we show that every normal semi-algebraic Ricci soliton also admits a $k$-dimensional Einstein extension for any $k\geq 2$. We also prove converse theorems for these constructions and some geometric and topological structure results for homogeneous warped product Einstein metrics. In the appendix we give an alternative approach to semi-algebraic Ricci solitons which naturally leads to a definition of semi-algebraic Ricci solitons in the non-homogeneous setting.Comment: 28 pages. supersedes the earlier version of arXiv:1110.245