[Paper Review] Global Hadamard form for the Green Function in Schwarzschild space-time

Marc Casals, Brien C. Nolan|arXiv (Cornell University)|Jun 9, 2016

Astrophysical Phenomena and Observations4 citations

TL;DR

This paper derives a global Hadamard form for the retarded Green function in Schwarzschild spacetime that explicitly captures its complete singularity structure beyond the normal neighborhood, including contributions from multiple caustic crossings. Using Bessel function expansions and a link to Plebański-Hacyan spacetime, the authors represent the Green function as a sum of Hadamard forms indexed by the number of caustics traversed, providing a non-smooth, globally valid structure that enables improved self-force and field propagation calculations.

ABSTRACT

The retarded Green function of a wave equation on a 4-dimensional curved background spacetime is a (generalized) function of two spacetime points and diverges when these are connected by a null geodesic. The Hadamard form makes explicit the form of this divergence but only when one of the points is in a normal neighbourhood of the other point. In this paper we derive a representation for the retarded Green function for a scalar field in Schwarzschild spacetime which makes explicit its {\it complete} singularity structure beyond the normal neighbourhood. We interpret this representation as a sum of Hadamard forms, the summation being taken over the number of times the null wavefront has passed through a caustic point: the sum of Hadamard forms applies to the non-smooth contribution to the full Green function, not only the singular contribution. (The term non-smooth applies modulo the causality-generating step functions that must appear in the retarded Green function.) The singularity structure is determined using two independent approaches, one based on a Bessel function expansion of the Green function, and another that exploits a link between the Green functions of Schwarzschild spacetime and Pleba{ń}ski-Hacyan spacetime (the latter approach also yields another representation for the {\it full} Schwarzschild Green function, not just for its non-smooth part). Our representation is not valid in a neighbourhood of caustic points. We deal with these points by providing a separate representation for the Green function in Schwarzschild spacetime which makes explicit its (different) singularity structure at caustics of this spacetime.

Motivation & Objective

• To extend the local Hadamard form of the retarded Green function to a global representation valid beyond normal neighborhoods in Schwarzschild spacetime.
• To explicitly characterize the singularity structure of the Green function when null geodesics pass through caustic points.
• To provide a globally valid, non-smooth representation of the Green function that accounts for multiple caustic crossings via a sum of Hadamard forms.
• To enable more accurate self-force and field propagation calculations by resolving the full non-smooth contribution to the Green function.
• To derive a separate representation for the Green function at caustic points, where the global form breaks down.

Proposed method

• Derives a global representation of the retarded Green function using Bessel function expansions of the mode-sum decomposition in Schwarzschild spacetime.
• Establishes a link between the Green functions of Schwarzschild and Plebański-Hacyan spacetimes to derive an alternative global form.
• Interprets the Green function as a sum of Hadamard forms, each corresponding to a different number of caustic crossings (mod 4), capturing the four-fold singularity structure.
• Uses the causality-generating step function θ₊(x,x′) and the Synge world function σ₄d to define the causal structure and singular support.
• Applies the Hadamard series coefficients Uₖ and Vₖ to represent the non-smooth part of the Green function, with Uₖ derived from the first-order Hadamard coefficient U₀.
• Constructs the non-direct Green function Gₙd by subtracting the direct part G_d from the full GF, enabling analytic evaluation of the self-force tail integral.

Experimental results

Research questions

• RQ1 How can the singularity structure of the retarded Green function in Schwarzschild spacetime be fully characterized beyond the normal neighborhood?
• RQ2 What is the mathematical form of the Green function when null geodesics from x to x′ have passed through multiple caustics?
• RQ3 Can the global Green function be expressed as a sum of Hadamard forms, each corresponding to a different caustic crossing count modulo 4?
• RQ4 How does the link between Schwarzschild and Plebański-Hacyan spacetimes facilitate a global representation of the Green function?
• RQ5 What is the analytic structure of the Green function at caustic points, where the standard global form fails?

Key findings

• The global Green function is represented as a sum of Hadamard forms indexed by the number of caustics traversed, with the singularity type cycling through δ(σ₄d), PV(1/σ₄d), -δ(σ₄d), and -PV(1/σ₄d) for 4n, 4n+1, 4n+2, and 4n+3 caustics, respectively.
• The non-smooth contribution to the Green function is captured by summing over these Hadamard forms, providing a complete description of the leading and sub-leading singularities.
• Two independent methods—Bessel function expansion and Plebański-Hacyan spacetime correspondence—yield consistent global representations, validating the result.
• A separate analytic representation is derived for the Green function at caustic points, where the global sum form breaks down due to the concentration of multiple null geodesics.
• The non-direct Green function Gₙd is expressed analytically as a sum over k≥1 of terms involving Uₖ(2η)^k and Vₖ(η), enabling exact evaluation of the self-force tail integral.
• In the massless limit (M=0), the k≥1 sum vanishes term-by-term, confirming the absence of a tail in flat spacetime, as expected.