Transport equation approach to calculations of Hadamard Green functions and non-coincident DeWitt coefficients

Abstract

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, σ (x, x ), the square root of the Van Vleck determinant, ∆ 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 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 [a7(x, x)] and V (x, x ) to order (σ 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. * adrian.ottewill@ucd.ie † barry.wardell@aei.mpg.de 1 The Hadamard and DeWitt coefficients also appear in the literature under several other guises. They may be called DeWitt, Gilkey, Minakshisundaram, Schwinger or Seeley coefficients, or any combination thereof (yielding acronyms such as DWSC, DWSG and HDMS).In the coincidence limit, it has been proposed that they be called Hadamard-Minakshisundaram-DeWitt (HaMiDeW) [8] coefficients. For the remainder of this paper, we will refer to them as either DeWitt (for the coefficients a k A B ) or Hadamard (for the coefficients Vr A B ) coefficients.arXiv:0906.0005v3 [gr-qc]

Diffusion in curved spacetimes

“…The higher-order terms involve higher derivatives of the squared distance function, which can also be expressed in terms of local curvature invariants[3,16].New Journal of Physics 14 (2012) 023019 (http://www.njp.org/)…”

Pseudodifferential Weyl Calculus on (Pseudo-)Riemannian Manifolds

“…A particularly efficient approach to calculating coefficients of covariant expansions of many important bitensors is Avramidi's method [2,3,9], especially the semi-recursive variant presented in [26]. Avramidi's method relies on deriving recursion relations for the coefficients from certain transport equations (such as (2.11)) for the bitensor.…”

“…Avramidi's method relies on deriving recursion relations for the coefficients from certain transport equations (such as (2.11)) for the bitensor. We refer to [26] for a full explanation and several examples.…”

“…For the remainder of this section, let u, v P T z M . Using Avramidi's method [2,3,9,26] applied to the transport equation (4.14), it is easy to calculate the covariant expansion of ζpz, z`uq to high orders. Up to fourth order, it is given by ζpz, z`uq " 1 12…”

Constitutive law of nonlocal gravity

“…For a general bitensor B ... with a given index structure, we have the following general expansion, up to the third order (in powers of σ y ): With the help of (45) we are able to iteratively expand any bitensor to any order, provided the coincidence limits entering the expansion coefficients can be calculated. We note in passing, that this expansion technique has also been applied extensively in the context of the equations of motion of extended test bodies [45][46][47][48][49][50][51] and in the gravitational self-force problem [44,52]. The expansion for bitensors with mixed index structure can be obtained from transporting the indices in (45) by means of the parallel propagator.…”