Transport Equation Approach to Calculations of Hadamard Green functions and non-coincident DeWitt coefficients

Building on an insight due to Avramidi, we provide a system of transport equations for determining key fundamental bi-tensors, including derivatives of the world-function, \sigma(x,x'), the square root of the Van Vleck determinant, \Delta^{1/2}(x,x'), and the tail-term, V(x,x'), appearing in the Hadamard form of the Green function. These bi-tensors are central to a broad range of problems from radiation reaction to quantum field theory in curved spacetime and quantum gravity. Their transport equations may be used either in a semi-recursive approach to determining their covariant Taylor series expansions, or as the basis of numerical calculations. To illustrate the power of the semi-recursive approach, we present an implementation in \textsl{Mathematica} which computes very high order covariant series expansions of these objects. Using this code, a moderate laptop can, for example, calculate the coincidence limit a_7(x,x) and V(x,x') to order (\sigma^a)^{20} in a matter of minutes. Results may be output in either a compact notation or in xTensor form. In a second application of the approach, we present a scheme for numerically integrating the transport equations as a system of coupled ordinary differential equations. As an example application of the scheme, we integrate along null geodesics to solve for V(x,x') in Nariai and Schwarzschild spacetimes.Comment: 32 pages, 5 figures. Final published version with correction to Eq. (3.24

Green Functions and Radiation Reaction From a Spacetime Perspective

The basis of this work is the first full application of the Poisson-Wiseman-Anderson method of `matched expansions' to compute the self-force acting on a point particle moving in a curved spacetime. The method employs two expansions for the Green function which are respectively valid in the `quasilocal' and `distant past' regimes, and which are matched together within the normal neighborhood. Building on a fundamental insight due to Avramidi, we provide a system of transport equations for determining key fundamental bi-tensors, including the tail-term, V(x,x'), appearing in the Hadamard form of the Green function. These bitensors are central to a broad range of problems from radiation reaction and the self-force to quantum field theory in curved spacetime and quantum gravity. Using their transport equations, we show how the quasilocal Green function may be computed throughout the normal neighborhood both numerically and as a covariant Taylor series expansion. These calculations are carried out for several black hole spacetimes. Finally, we present a complete application of the method of matched expansions. The calculation is performed in a static region of the spherically symmetric Nariai spacetime (dS_2 X S^2), where the matched expansion method is applied to compute the scalar self-force acting on a static particle. We find that the matched expansion method provides insight into the non-local properties of the self-force. The Green function in Schwarzschild spacetime is expected to share certain key features with Nariai. In this way, the Nariai spacetime provides a fertile testing ground for developing insight into the non-local part of the self-force on black hole spacetimes.Comment: Ph.D. thesis. University College Dublin, 2009. Advisor: Adrian C. Ottewil